EXPANDING THE NUMERICAL CENTRAL CONCEPTUAL …hk239ky7693/negative-numbers [for online...expanding the numerical central conceptual structure: first graders’ understanding of integers

EXPANDING THE NUMERICALCENTRAL CONCEPTUAL STRUCTURE:

FIRST GRADERS’ UNDERSTANDINGOF INTEGERS

A DISSERTATIONSUBMITTED TO THE SCHOOL OF

EDUCATION AND THE COMMITTEE ONGRADUATE STUDIES

OF STANFORD UNIVERSITYIN PARTIAL FULFILLMENT OF THE

REQUIREMENTSFOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Laura Christine Bo�erdingMay 2011

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/hk239ky7693

© 2011 by Laura Christine Bofferding. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

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http://purl.stanford.edu/hk239ky7693

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Aki Murata, Primary Adviser


Shelley Goldman


Daniel Schwartz


Yukari Okamoto

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

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A B S T R A C T

In working with integers, students have di�culties that may extend intomiddle school and even adulthood. However, even young children can dis-play insights into negative numbers well before receiving formal instruction.Using a pre-test, instruction, post-test design, this study explores how 61 �rstgraders reason about negative number properties and operations and howtheir understanding changes depending on the instruction they receive. Re-sults of the study indicate that children build on their existing whole numberunderstanding to develop a central conceptual structure for integers. Further-more, the process by which they extend their numerical central conceptualstructure di�ers among students; their initial schemas, together with the formof the integer instruction, in�uence how they reason about and solve integeraddition and subtraction problems. �ese results highlight the need to revisitthe placement, duration, and content of integer instruction in curricula.

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A C K N O W L E D G M E N T S

As the humble Piglet once said, “It’s so much more friendly with two.”Likewise, my dissertation work has been much more friendly with the helpand support of numerous people, and my work re�ects their in�uence.

Before I begin, I would like to acknowledge my funding. �is dissertationwork is supported by a SUSE dissertation support grant. �anks to the grant,I was able to buy the materials necessary for the study and give back to thosewho helped me without expecting anything in return.

First I would like to acknowledge the members of my committee: AkiMurata (my advisor); Dan Schwartz; Shelley Goldman; Yukari Okamoto;and Michael Saunders (my chair). Dan Schwartz discussed any and all top-ics related to integers with me. He was also instrumental in pushing metoward using experimental methods, which have greatly strengthened mywork. Shelley Goldman is responsible for welcoming me to Stanford in the�rst place, and for that I am extremely grateful. She helped me stay groundedin the implementation of the study and focused on the children. YukariOkamoto’s enthusiastic willingness to help me understand the central concep-tual structures framework and to provide detailed comments on my �rst dra�is amazing. I cannot thank her enough for the time and help she provided.Michael Saunders, deserves special mention as he was also my husband’sPh.D. advisor. Not only has Michael been a dear friend to us, but he con-tinues to best my grammar skills! It was a pleasure having him chair mycommittee.

Aki Murata is both my advisor and friend, and as such, she has helped megrow as a researcher and teacher. Her openness and trust in others pervadesher teaching and research, and I attribute my growth as a teacher and a morere�ective researcher to her. Our discussions about children’s understandingof numbers has enriched my research experiences, and I look forward tomany more such conversations in the future!

Aside my from my committee members, I received enormous help andsupport from my colleagues.

April Alexander was a life-saver. She helped me interview students sothat I would have enough time to complete my study before the end of theschool year, and she spent numerous hours with me discussing children’sunderstanding of numbers.

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Ben Hedrick and Melissa Kemmerle have endless amounts of patience.�ey helped me hone my coding protocol and painstakingly code students’responses. I am deeply grateful to them for setting aside their precious time(usually dedicated to family and school work) to help me.

Lambrina Mileva translated my consent forms into Spanish in record-breaking time, so that I was able to get my study approved by the deadline.�e math education group has been a huge support over the past years,

and I would like to thank all members who listened to my ideas in theirbeginning stages, asked just the right questions, and provided important sug-gestions. In particular, Julie Cohen, Sarah Kate Selling, Erin Baldinger, andApril Alexander edited earlier versions of some key chapters and listened tomy presentations to the point where they had them memorized. All the othermembers – Megan Westwood Taylor, Bindu Pothen, Ben Hedrick, JessicaTsang, Nicole Hallinen, Melissa Kemmerle, Hilda Borko, Jo Boaler, KathySun, Cathy Humphreys, Charmaine Mangram, and Dan Meyer – providedthoughtful feedback over the years as well.

No doctoral experience is complete without two music-loving, keepin’-it-real o�cemates, and I was lucky to have two of the best. Matt Kloser andShayna Sullivan listened to and commented on a multitude of ideas over theyears and also provided needed silliness when work became too engul�ng.

I am deeply indebted to my friends for their encouragement and for help-ing me keep my life balanced.

Debbie and Jason Azicri are some of my biggest cheerleaders. �ey arealways interested in hearing about my work and are excited at each step of myjourney. �ey helped me relax – no one can beat their entertaining stories –and gain perspective on my work.�e Fletchers – Hugh, Jane, Les, and Lindsey – have been an amazing

support. �eir door is always open for a delicious meal and great company,and they have become my California family.�e original ICME clan not only provided me with great friendship but

also free statistics support! Chris Maes, Paul Constantine, Nicole Taheri,Andrew Bradley, Nick West, Mike Atkinson, Mike Lesnick, Esteban Arcaute,and Nick Henderson all inspired me with their excitement about math andlearning.

�roughout my life, my parents, Mark and ReNee Bo�erding, have alwaysencouraged my passion for education and teaching. When I was a child,both of my parents went back to school to pursue additional degrees andencouraged me to do the same. I thank them for being great role-models.

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�is acknowledgment section would not be complete without mentioningDavid Gleich (my husband). His passion for knowledge is a true inspiration,and I am thankful for his continual encouragement. He is the ultimate sound-ing board, continually pushes my thinking, and due to his LaTeX help, mydissertation looks almost as beautiful as his.

David’s parents were also very supportive. It’s clear where he got his thirstfor knowledge.

Finally, I want to thank the groups of people involved in my study (somein more direct ways than others). I worked with the STEP students of 2008-2009 as their TA for their math methods class. During one of these classes– based on their request – we talked about how to teach negative numbers.As I watched the teacher candidates wrestle with understanding the repre-sentations used to teach integers, I �rst began to wonder how children cometo understand negative numbers. �eir simple request to learn more turnedinto a three-year journey for me.

My pilot studies and dissertation work would not have been possible with-out the cooperation of two amazing principals, several teachers, and countlessfamilies. �eir belief in the power of research is humbling and encouraging.

Last, but never least, I must thank the dozens of children who let me intotheir minds and hearts. �eir insights never fail to surprise me, and theycontinue to motivate my work.

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C O N T E N T S

Abstract v

Acknowledgments vii

1 Introduction 1

2 Background L iterature 7

3 Theoretical Framework 19

4 Instructional Philosophy 29

5 Pilot Studies 37

6 Methods 41

7 Analysis 59

8 Results – Problem Accuracy 77

9 Results – Integer Schema Changes 89

10 Results – Schema Integration 107

11 Overall Discussion 123

12 Conclusion 131

a Lesson Plans 139

b Pilot I Questions 189

c Pilot II Questions 193

d Pilot II Codes 195

e Pilot II Schema Categories 199

f Integer Problem Types 203

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g Integer Problems 205

h Pre- and Post-Test protocols 207

i Schema Diagrams 211

j Final Codes and Protocol 223

k Normality Tests 245

l Homogeneity of Variance 251

m Total Ordered Value Schemas 253

References 255

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L I S T O F TA B L E S

Table 1.1 Cancellation Model 3Table 1.2 Negative Number Reasoning 4

Table 2.1 Counting Principles 8Table 2.2 �ree Meanings of the Minus Sign 11

Table 6.1 Participants 42Table 6.2 Question Categories 45Table 6.3 Ordering Task 46Table 6.4 Value Tasks 47Table 6.5 Symbol Identi�cation Tasks 47Table 6.6 Integer Operations Property Questions 47Table 6.7 Integer Arithmetic Question Types 49Table 6.8 Post-test Only Directed Magnitude Questions 50Table 6.9 Pre-test Coding and Ranking for Strati�cation 52Table 6.10 Student Demographics Between Instructional Groups 53Table 6.11 Schedule of Lessons 55Table 6.12 Instruction Schedule for the �ree Instructional Groups 57

Table 7.1 Notation for Students’ Schema Diagrams 60Table 7.2 Examples of Original Codes 61Table 7.3 Revised Codes 64

Table 8.1 Total Test ANOVA Results 77Table 8.2 Sche�é Tests for Instructional Groups 78Table 8.3 Mean Total Scores for Instructional Groups 78Table 8.4 Sche�é Tests for Performance Groups 79Table 8.5 Mean Total Scores for Performance Groups 79Table 8.6 Mean Total Scores for Performance by Instructional Groups 81Table 8.7 “Easiest” Pre-test Questions 81Table 8.8 “Easiest” Post-test Questions 82Table 8.9 Integer Property Questions ANOVA Results 83Table 8.10 Integer Property Items - Sche�é Tests for Instructional Groups 83Table 8.11 Integer Property Scores for Instructional Groups 84

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Table 8.12 Integer Property Items - Sche�é Tests for Performance Level Groups 85Table 8.13 Integer Property Scores for Performance Level Groups 85Table 8.14 Integer Property Scores for Performance by Instructional Groups 87

Table 9.1 Schema Categories for Identifying Negative Numbers 90Table 9.2 Schema Changes for Identifying Negative Numbers in Isolation 91Table 9.3 Schema Changes for Identifying Negative Numbers in Context 91Table 9.4 Schema Changes for Identifying Negative Numbers Overall 92Table 9.5 Schema Categories for Counting Backward 94Table 9.6 Schema Changes for Counting Backward 95Table 9.7 Order Schema Categories for Completing a Number Line 96Table 9.8 Order Schema Changes for Completing Number Lines 97Table 9.9 Order Schema Categories for Ordering Integers 98Table 9.10 Order Schema Changes for Ordering Integer Cards 99Table 9.11 Value Schema Categories for Integer Values 100Table 9.12 Value Schema Changes for Greatest/Least Items 101Table 9.13 Value Schema Changes for Comparing Two Integers 102Table 9.14 Directed Magnitude Schema Categories 103Table 9.15 Schema Changes for Directed Magnitude 104Table 9.16 Subtraction Property Questions 104

Table 10.1 Student 117’s Schema Levels 107Table 10.2 Student 118’s Schema Levels 110Table 10.3 Student 111’s and 119’s Schema Levels 112Table 10.4 Student 108’s and 204’s Schema Levels 115Table 10.5 Student 102’s, 217’s, and 419’s Schema Levels 117

Table 11.1 Performance of Students by Instructional Group in Shi�ing Schemas 129

Table 12.1 Pedagogical Content Knowledge for Integers 135

Table L.1 Levene’s Test on Total Test Di�erence Scores 251Table L.2 Levene’s Test on Integer Properties Di�erence Scores 251

Table M.1 Value Schema Changes for Both Value Items 253

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L I S T O F F I G U R E S

Figure 3.1 Central Conceptual Structure for Whole Numbers 21Figure 3.2 Whole Number Schemas 24Figure 3.3 Central Conceptual Structure for Integers 25

Figure 6.1 Pre-test, Post-test Study Design 43

Figure 8.1 Mean Di�erence Scores for Instructional Groups by Performance Level 80Figure 8.2 Mean Percentage Di�erence Scores for Instructional Groups by Performance Levels

on Integer Property Items 86

Figure 10.1 Student 102’s Number Line 119

Figure K.1 Total Test Di�erence Score Histograms 246Figure K.2 Total Test Di�erence Score QQ-plots 247Figure K.3 Properties Percentage Di�erence Score Histograms 249Figure K.4 Properties Percentage Di�erence Score QQ-plots 250

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Learning mathematics involves accumulatingideas and building successively deeper and

more re�ned understanding.

—National Council of Teachers of Mathematics

1 I N T R O D U C T I O N

In the quest to promote an optimal learning progression for mathematicsstudents, researchers, educators, and policy makers in the United States (andthe rest of the world) spend enormous e�ort determining what studentsshould learn and when they should learn it. �e result, which teachers useto guide their teaching and support students’ learning, is a set of standards.According to the National Council of Teachers of Mathematics (2000), “Ina coherent curriculum, mathematical ideas are linked to and build on oneanother so that students’ understanding and knowledge deepens and theirability to apply mathematics expands” (p. 14).

Sequencing certain topics is fairly straightforward and based on researchof children’s learning trajectories (see Sarama & Clements, 2009); for example,counting and understanding how many objects are in a set is placed beforeunderstanding the place-value structure of the number system. However, as isthe case with the recent Common core standards for mathematics, when learn-ing pathway research is not available, the sequence of standards is based on“the basis of state and international comparisons and the collective experienceand collective professional judgment of educators, researchers and mathe-maticians” (Council of Chief State School O�cers & National Governor’sAssociation, 2010, p. 5).

If not research, what is our professional judgment based on? To some ex-tent committees chose which topics to highlight in curricula based on culturalpriorities (National Research Council, 2001). Due to these research and cul-tural priorities, number and operations form the bulk of elementary learningstandards in the US. According to the common core standards, kindergartenstudents should learn to count, order, compare, and add and subtract withnumbers to 10. As they move to �rst grade, children should add and sub-tract within 20 and learn that the digits in a two-digit number represent tensand ones. Between third and ��h grades, students should add and subtractlarger numbers and also learn multiplication and division. Additionally, theyshould apply these operations to fractions.

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2 1 ⋅ introduction

�e order of some standards, however, is based on their perceived com-plexity – which is tied to how and when the concepts arose historically; suchis the case for negative numbers, which were hard for mathematicians toaccept historically, and therefore (many claim) must be di�cult for childrenas well (�omaidis, 1993). �e common core standards indicate that studentsshould �rst learn about negative numbers in the sixth grade and use themin operations in the seventh grade (Council of Chief State School O�cers &National Governor’s Association, 2010); �e National Council of Teachersof Mathematics (2000) suggests that students in grades 3 – 5 learn of theexistence of negative numbers but wait until grades 6 – 8 to compare andoperate with them. Some people might argue that negative numbers are tooabstract for younger students to understand any earlier. Yet as mathematicianDevlin (1999) illustrates, all number concepts are abstract:

Explain what the number three is without using the word three orthreeness. You cannot do it, yet none of us worries about that. �esame is true of all the other mathematical abstractions: althoughthey may have their roots in the real world, the abstractionsthemselves are only concepts, having no existence outside ourminds (p. 39).

Since they are not the �rst abstract concepts students must decipher, whatmakes negative numbers so di�cult? Negative numbers are one of the �rstconcepts children encounter that contradicts their previous learning at astructural level. Before this point, subtracting a smaller number from a largernumber (e.g., 3−9) is not allowed because it does not result in a whole numberanswer. With the introduction of negative numbers, students must learn thatnegative numbers are less than zero and are ordered in such a way that thefurther from zero they are, the less their value. Furthermore, students needa new way to think about addition and subtraction since adding a negativenumber to a positive one no longer results in a larger number.

Curricula help students learn about negative numbers in three main ways:a cancellation model, a number line model, and number rules. �e cancella-tion model is meant to help students think of negative numbers as quantitiesthat are the opposite of the positive numbers. Using a set of positive and neg-ative counters (o�en colored red and black or yellow and red), students learnthat one positive counter will cancel out one negative counter (+1 + -1 = 0),that taking away a negative counter will result in a greater total, and thatadding a negative and a positive counter will not change the overall total (seeTable 1.1 for example problems).

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Table 1.1 – Cancellation Model. Example solutions to addition and subtraction problems withintegers using the cancellation model.

-3 + 5

Start with threenegative counters.

Add �ve positive counters. Cancel out pairs of positive andnegative counters.

Answer: 2

3 − 5

Start with threepositive counters.

Add pairs of positive andnegative counters (zero pairs)until there are �ve positivecounters.

Subtract �ve positive counters. Answer: -2

�e number line model focuses on integers as directed magnitudes ormovements. In the enVisionMATH curriculum, students learn a series ofrules for adding and subtracting on the number line. Addition means facethe positive direction; subtraction means turn around. For -3 + 5, studentsare taught to picture themselves standing on -3 but facing toward the positivenumbers because it is an addition problem. Since the next number is positive,they move forward �ve spaces, ending at 2. If the number were negative (-5),students would move backward �ve spaces, ending at -8. Now picture thesame problem with subtraction: -3 − 5. Students once again start at -3 butmust “turn around” so they are facing the negative numbers. Since the nextnumber is positive �ve, they move forward �ve spaces, ending up at -8. If the�ve had been negative, students would walk backwards �ve spaces, endingup at 2 (Pearson Education, 2009).

Finally, using the experiences with the cancellation and number line mod-els as a reference (or sometimes as a separate supplement), curricula andteachers help students generate a list of rules for adding and subtracting inte-gers. For example, to add a positive and a negative number, they might tellstudents to subtract the smaller number from the larger one and then use thesign of the larger number (e.g., for 3 + -5, do 5 − 3 = 2 and since 5 is largerthan 3, use the negative sign to get -2).

Regardless of when students �rst encounter negatives, the introduction ofnegative numbers causes them di�culty. Children’s con�icts with negativescontinue to resurface each time they appear in a new context. �e following


Table 1.2 – Negative Number Reasoning. Examples of students’ reasoning about negativenumber problems across grades.

Topic Problem Explanation

Recognizingnegatives

-5 “I’ve seen it on thermometers; it’s less than zero” (Murray, 1985,p. 148) – 10 year old

Subtractionwith wholenumbers

3 − 5 “�ree minus �ve is zero because you have 3 and you cant [sic]take away 5 so take away the 3 and it leaves you with zero (Callie,kindergarten)” (Bishop et al., 2010, p. 698)

Adding withnegatives

-7 + -1 “It’s confusing cause there’s a plus and a minus...I would do a sixor an eight.” – First grader (408)

Negatives asSubtraction

-9 + 5 “It says zero minus nine, nine minus zero, and so it would benine...and then I did nine plus �ve. (Student solved 0 − 9 as 9 − 0,then did 9 + 5” (Bo�erding, 2010, p. 706) – Second grader

Multiplyingnegatives

7 × -5 “�en there will be positive thirty �ve...because the plus [the posi-tive] is bigger” (Sfard, 2001, p. 6) – Seventh grader

Simplifyingequations

20 + 8 − 7n − 5n “You put the ’n’ with the ’n’ and the numbers alone with the numbersalone, so Imake 20+8, that gives 28 minus...and 7n−5n; so, 28−2n”(Vlassis, 2004, p. 476) –Eighth grader

examples illuminate how students attempt to make sense of negatives as theyappear throughout the curriculum (see Table 1.2).

Children learning mathematics in the US have the possibility of encoun-tering negatives in their environments: on thermometers, game shows, com-puter games, etc. �is means they have opportunities to start thinking aboutnegative quantities much earlier than they are taught in school. �e examplesabove (and many more) provide further evidence that younger students arestarting to think about negative numbers long before they are presented informal instruction. Given that our current sequencing of negative numbers solate in the curriculum is not rooted in a research body on learning trajectoryresearch, it is worthwhile to explore how children develop an understandingof negative numbers in more depth. �is dissertation research presents awindow into the learning process for students at a much younger age, �rstgrade.

Investigating the process of children’s learning is like reading a detectivestory: it is engaging, full of unexpected turns, and leaves one with new in-sights. �is story is no di�erent. To set the stage, I explore how students learnabout positive numbers and what we know about students’ understandingof negative numbers. Like any detective would, I then present a theory ofhow students learn about negative numbers: they build on their knowledge

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(or schemas) of positive numbers (such as their order, value, and symbolicrepresentations) and number relations. By providing the �rst grade studentswith instruction in negative numbers, I explore not only how they try to makesense of negative numbers initially but also a�er having di�erent types ofinstruction.

What is the mystery I am trying to solve? What are children’s schemas ofnegative numbers, how do they change, and what role do they play in solvinginteger addition and subtraction problems?

Come! �e game is afoot!

—Sherlock Holmes


2 B A C KG R O U N D L I T E R AT U R E

Developing number sense of whole numbers is an important skill on whichchildren’s future understanding builds (Resnick, 1989) and should be thefocus of learning in preschool through early elementary grades (NationalResearch Council, 2005; National Council of Teachers of Mathematics, 2006).Number sense encapsulates a wide range of understanding about numbers:their order, value, symbolic representation, and the relations among theseconcepts. Case and colleagues describe these elements and their relations asthe central conceptual structure for whole number (Case, 1996; Okamoto &Case, 1996).

Whole Number Order

Children’s understanding of whole number quantity and order form thefoundation of the conceptual structure; however, these concepts initiallydevelop separately. As young as 2 years-old, children begin to string togethera series of number names; by the time they are �ve, many can count to 20 or30 (Sarama & Clements, 2009). Counting is a fundamental process (Fuson,1992; Sarama & Clements, 2009; Sophian, 1987; Ste�e, 1992), and knowingthe correct number sequence is an important �rst step (Fuson, 1992), whichhelps students gain an initial knowledge of order.�rough experience, children transition from reciting a list of number

words to learning a series of counting principles that they use to quantify sets(see Table 2.1 for brief description of each): one-to-one correspondence, thestable-order principle, the cardinal principle, the abstraction principle, andthe order-irrelevance principle (Gelman & Gallistel, 1978). Some suggest thatcardinality develops as children compare amounts they subitize with the �nalnumeral assigned to those sets; regardless, over time young children learnthat they can use counting to determine the quantity of a set (Baroody, 1992;Sarama & Clements, 2009; Wynn, 1992).

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8 2 ⋅ background literature

Table 2.1 – Counting Principles.

Counting Principles Description

One-to-one correspondence (alsocalled “one-one principle”)

Students match one number nameto each object being counted.

Stable order principle Students use a consistent numbersequence each time they count.

Cardinal principle Students know that the last num-ber in a count refers to the quan-tity of the set.

Abstraction principle Students know that they can counta group of unlike items or non-visible items, like sounds.

Order-irrelevance principle Students know that they can countthe items in any order and thiswill not impact the result of thecount.

Whole Number Values

Aside from counting, preschool children also make judgments about rela-tive sizes and amounts of objects without measuring or counting; this is alsoknown as protoquantitative knowledge (Resnick, 1989) or, when speci�c torecognizing and naming the size of small sets, subitizing (Gelman & Gallistel,1978). Although children may know how to count, they may fail to countand continue to rely on their protoquantitative knowledge when comparingtwo quantities. For example, when shown a collection of jars and lids andasked if there were enough lids for each jar to get one, the majority of 3 and3.5 year-olds did not count to verify the answer (Sophian, 1987). Furthermore,they can visually determine that adding to a set leads to more in the set, whiletaking away leads to fewer or less (Fuson, 1992).

Over time, young children integrate their counting knowledge with theirprotoquantitative knowledge and can use the counting sequence to determinewhich number is greater (Baroody, 1992; Resnick, 1989). Indeed, early ele-mentary students from middle-class families have high accuracy on tasks ofnumerical magnitude comparison (Laski & Siegler, 2007; Siegler & Robinson,1982). Additionally, Ramani and Siegler (2008) found that preschool childrenwho counted while playing a linear board game with numbers made numeri-cal magnitude comparisons with increased accuracy (73% to 83%) comparedto preschoolers who played a similar game with colors. �ese results also

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held for children who came from low-income families and had less numer-ical experience to begin with. Based on these �ndings, Ramani and Sieglersuggested that “conventional board games [like Chutes and Ladders] providea physical realization of the mental number line, hypothesized by Case andOkamoto(1996) and Case and Gri�n (1990), to be the central conceptualstructure underlying early numerical understanding” (p. 377).

Negative Integers

While children’s development of counting and quantity principles is well–researched for positive numbers, the same is not true for negative numbers.Some believe that negative number concepts “require a long training periodand external storage of information to overcome cognitive limitations” (DeCruz, 2006, p. 319)(Fosnot & Dolk, 2001); yet, a handful of researchers andeducators argue that negative numbers are the most natural extension of thewhole numbers and should be taught earlier (preferably before fractions)(Davidson, 1987; Freudenthal, 1973; Werner, 1973). De Cruz (2006) explains,“Negative numbers are structured by analogy with the positives: they arerepresented on a mental number line that runs backward” (p. 318). However,not only is counting backwards harder for students (Baroody, 1992; Fuson,1992), but children’s counting does not usually result in negative numbers(Gelman & Gallistel, 1978). Furthermore, Gaer (1969) found that 3 to 6 yearolds have di�culty even understanding the language of negation. Severaleducators have described or proposed the use of linear games or instructionto help students extend numbers into the negatives (Aze, 1989; Chilvers,1985; Dabell, 2007; Mauthe, 1969; McAuley, 1990; Swanson, 2010), but theire�ectiveness has not been explored su�ciently.

Negative Integer Order

How students transition from whole number to integer understandingremains under-researched, but there are a few studies that provide a pictureof students’ conceptual variance when talking about the order and valueof negative numbers. Peled (1991) suggested that students’ understandingof the order of negative numbers is the �rst level of integer understanding.As with positive numbers, knowledge of integer order involves the conceptthat numbers further to the right on the number line are larger. In theirinterviews, Peled, Mukhopadhyay, and Resnick (1989) found that 1st gradersand some 3rd graders ordered negatives next to their positive counterpartson the number line (e.g., 0, -1, 1, 2, -2, -3, 3 – essentially treating them as


positive numbers) or treated the negatives as zeros. However, those whoacknowledged negatives said negatives fall on the other side of zero, and evenmost third graders could put them in the correct order. Additionally, whendescribing the meaning of negatives, some fourth graders spontaneously drewnegatives to the le� of zero (Hativa & Cohen, 1995).

Negative Integer Values

Students who order negatives next to their positive counterparts or zeromay think that the negative numbers do not exist or are worth zero (Hativa& Cohen, 1995). Even students who are aware of negative numbers and theirorder may interpret them as zeroes when solving problems; for example, theysolve 7 + -2 as 7 + 0 (Schwarz, Kohn, & Resnick, 1993). In contrast, whenasked the meaning of -3, fourth graders’ responses ranged from “a numberbelow zero; smaller than zero; smaller than all numbers [positive]” to “to thele� of zero; zero minus something; and subtracting a large number from asmaller number” (Hativa & Cohen, 1995, p. 425). Older students (4th–7thgraders) provided more speci�c de�nitions for -5; for instance, -5 is �ve belowzero or you need �ve to get to zero (Murray, 1985). Based on their reasoningof integers’ quantities, almost all 5th graders, as well as 7th and 9th graders,knew that -4 > -6; however, only one 1st grader could express this relationship(Peled et al., 1989).

Symbols for Representing Whole Numbers and Operations

Although children’s early experiences with natural numbers involve phys-ical items, which they can count and add, children also need to learn thesymbolic representations of numbers because “�ere is evidence that thedevelopment of mathematics concepts on one hand and notational systemson the other interact, each supporting the other (Brizuela, 2004)” (cited inSarama & Clements, 2009, p. 70). When young children are asked to createrepresentations or symbols for amounts that they can see, they do so at oneof three levels.

At level 1, they draw a series of marks that do not match the exact quantity,at level 2 they draw the same number of objects with symbols or numerals inone-to-one correspondence, and at level 3, they write the cardinal value ofthe set using traditional numerals or words (Kato, Kamii, Ozaki, & Nagahiro,2002); Hughes (1986) found similar results but classi�ed responses as idiosyn-cratic, as pictographic or iconic, and as symbolic, respectively. Children movetoward writing the cardinal value as they get older, and those at the higher

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Table 2.2 – Three Meanings of the Minus Sign.

Meaning of theMinus Sign

Explanation Example

Binary Function Subtraction 9 − 3 (operation)

Unary Function Negative Number -7 (numeral + sign)

Symmetric Function Taking the Opposite -(4 + 2) = -(6) = -6

levels also tend to write the correct numerals when given number names.Ramani and Siegler (2008) further found that experience with numericalboard games is associated with correctly identifying more numerals.

While children see numerals in their environment, they typically onlysee operational symbols, such as the plus, minus, and equal sign, in school(Hughes, 1986; Tolchinsky, 2003). Additionally, children o�en describe theplus as meaning and and the minus as meaning take away (Hughes, 1986).Regarding single-digit numerals, children have the most di�culty with zero.When asked to record the amount in a box with no objects, many childrenwill not write anything (Hughes, 1986; Tolchinsky, 2003).

Symbols for Representing Integers

Unlike with positive numbers, negative numbers are “�ctional numerosi-ties” (Gelman & Gallistel, 1978) and physical analogs for negative numbersare contrived (De Cruz, 2006); they exist as abstract quantities representedby a numeral and a sign. While children can extrapolate from whole numbersto understand negatives as numbers, when introduced to negative integers,children must reinterpret the meaning of the minus sign. Instead of one signrepresenting an operation, the minus sign now has three meanings: the bi-nary function, the unary function, and the symmetric function (see Table 2.2)(Gallardo & Rojano, 1994; Vlassis, 2008, 2004).

Children who do not understand the unary nature of the minus sign mayignore negatives in situations where it does not look like subtraction. Forinstance, in a group of 6 to 10.5 year-olds, some students represented both 5and -5 with �ve blocks, making no distinction between them (Hughes, 1986).However, interpreting the minus as subtraction is so entrenched in early op-erations learning that others will interpret the minus sign as subtraction nomatter how it is presented; Hughes found some of the students in the samegroup interpreted “-5” as 5− 5 = 0. To avoid the confusion between the nega-tive sign and minus sign, researchers who explore students’ understanding of


negatives sometimes let students make up their own notations for negativenumbers, such as writing them in red or circling them (Liebeck, 1990), orwrite out the names of the operations (Williams & Linchevski, 1997).

In their interviews of 8th graders, Fagnant, Vlassis, and Crahay (2005)highlight other rules that students use to manipulate negatives:

...placed at the beginning of the expression, the minus is consid-ered as attached to the number. Placed between two like terms,students explain that the minus is used for subtracting, and be-tween two unlike terms, that it is used for splitting, operating,or making the following term negative (p. 89).

Similarly, although even second graders acknowledge the multiple meaningsof the minus sign in their solutions to problems, they do not always use themcorrectly (Bo�erding, 2010).

Whole Number Addition and Subtraction

Once students have integrated their schemas for whole number order andvalue, they can use this knowledge together with their understanding of addi-tion and subtraction symbols to solve single-digit arithmetic problems. Whilesubtraction is harder for students to master than addition (Kamii, Lewis, &Kirkland, 2001), students use similar strategies for solving both. Childrenprogress through three conception of quantities levels (and sometimes a tran-sition level) as they advance from using counting methods to solve additionand subtraction problems to manipulating chunks (Fuson, 2003; Fuson et al.,1997; Murata, 2004). More speci�cally, children move from counting all, tocounting on for addition or counting up or back for subtraction, to breakingapart numbers and operating with friendlier numbers, such as ten. �roughthis process, students learn about the commutative property of addition–youcan add numbers in any order–and develop stronger connections betweenaddition as increasing and subtraction as decreasing (Gelman & Gallistel,1978).

Transitioning to Integer Addition and Subtraction

Children’s struggles with negative numbers and operations with themare unsurprising considering they spend grades K–3 learning the meaningsof addition, subtraction, and the minus sign in terms of positive numbersand then must rework their “organization of schemata“ or the forms of theirthoughts about all three concepts to properly understand negatives (Gri�n,2004a, p.271):

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Researcher – Why do you think negatives are so tricky to thinkabout?

4th Grader – Because you don’t learn them until like fourth gradeand then you’re so used to regular numbers (Interview, July 7,2009).

When unfamiliar with the unary meaning of the minus sign, studentswill subtract negatives as if they are no di�erent from positive numbers. Forexample, students of all ages ignore negative signs and solve problems as ifthey are positive (Peled et al., 1989), such as when students answer 2 for 7− -5(Murray, 1985). Problems with two signs next to each other cause particulardi�culty (Vlassis, 2008). Students may ignore one of the signs or use bothoperations, adding and then subtracting 2 in the case of 8+ -2 or subtracting2 twice in the case of 8 − -2 (Murray, 1985); others simply do not respond tothe questions when they do not know the answers (Vlassis, 2004).

As students transition to learning about negative numbers, they mustreinterpret the meaning of addition and subtraction, the binary meaning ofthe minus sign. According to Bruno and Martinon (1999), at the beginningof this process, students continue to associate adding with increasing andsubtracting with decreasing so will confuse the meaning of minus and negativesigns in their explanations. �ey may say, “I lost -6” instead of “I lost 6.”Mukhopadhyay, Resnick, and Schauble (1990) found that students could adda positive number to a negative (e.g., -5 + 2) when it did not cross the zeroboundary or change the meaning of addition, but adding to get a greaternegative balance in the context of money (e.g., -5 + -2) was confusing forsome.

Similar to whole number operations, students have more di�culty solv-ing integer subtraction problems than addition problems (Galbraith, 1975;Janvier, 1985; Liebeck, 1990; Mukhopadhyay et al., 1990). Students especiallystruggle with problems where subtraction does not mean getting smaller, suchas when the problems involve both a negative and a positive number (e.g.,5 − -6) (Mukhopadhyay et al., 1990; Murray, 1985) or when subtracting anegative from a negative requires students to cross the zero boundary in theircalculations (e.g., -5 − -8) (Linchevski & Williams, 1999; Mukhopadhyay etal., 1990; Murray, 1985).

Smaller Number Minus Larger Number

Crossing the zero boundary is hard, especially for preschool through �rst-grade students, even when the problems are presented verbally in a game-like


context (Davidson, 1987). �is may be because young students’ arithmeticoperations are generally governed by the numbers (and quantities) that theycan count (Gelman & Gallistel, 1978); getting a new type of number as an an-swer may be confusing for them. While negatives are not typically introduceduntil fourth grade, students’ di�culties with them originate even earlier dur-ing initial subtraction instruction. During this time, children learn eitherimplicitly or explicitly that you cannot subtract a larger from a smaller num-ber; therefore, crossing the zero boundary is not allowed (e.g., 4 − 9) (Ball &Wilson, 1990; Hughes, 1986). �is incomplete conception most frequentlyresults in the “smaller-from-larger bug” where students reverse the numbers(Ball, 1993; Murray, 1985; Peled et al., 1989; Vlassis, 2004; VanLehn, 1982):

2nd Grader – <solving 4−9> You have to do it backwards...so it’dbe nine minus four (Interview, December 1, 2009).

Following these rules contributes to students’ struggles with related con-cepts. For example, many students incorrectly solve two-digit subtractionproblems that require regrouping by subtracting the smaller number fromthe larger number in each column, regardless of their placement (e.g. solving62 − 48 = 26) (Fuson, 2003). On the other hand, some children–especiallythose who know that the commutative property does not hold for subtraction–claim that there are not enough to take away and answer zero (Peled et al.,1989):

2nd Grader – <solving 4 − 9>�ere’s no nine in four, so it wouldbe zero (Interview, November 17, 2009).

Without knowing about the negative numbers, their internal number linesstop at zero. Students might also circumvent the issue by incorrectly chang-ing the minus to a plus (Tatsuoka, 1983) or by borrowing from an imaginarynumber, changing a problem like 3 − 7 into 30 − 7 = 23 (Peled, 1991). How-ever, they can still solve smaller whole number minus larger whole numberproblems, provided they are within a story context about going into debt(Mukhopadhyay et al., 1990). Even if they know about negative numbers,students might not operate through the zero point, possibly because theyoperate with a divided number line (Peled et al., 1989). �ese students alsotalk about subtracting on the “negative” side. Students who are able to movebeyond the zero point may have trouble overcoming the previous mantrathat it is not possible to subtract a larger number from a smaller one. �eymay continue to use this mantra even whilst solving the problems correctly:

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3rd Grader – <solving 5−7> Because �ve minus seven, you reallycan’t do that, so you have to go into the negatives (Interview, July7, 2009).

Once they focus on ordinal relations, students solve problems by movingup and down a number line (Murray, 1985), counting, talking about removingor taking away parts (Schwarz et al., 1993), or focusing on the zero boundaryby calculating to the zero point and then beyond. For example, to solve -5+8,the student solves (-5 + 5) + 3 = 0 + 3 (Hativa & Cohen, 1995; Schwarz et al.,1993).

Based on students’ arithmetic solutions to these and other negative numberproblems, Peled (1991) posited four levels of integer understanding. At level1, students know the order of negative numbers and that numbers furtherto the right on the number line are larger. At level 2, students can cross thezero boundary when subtracting a larger from a smaller number (e.g., 5 − 7)or when adding a positive number to a negative number (-4 + 9). At level 3,students realize that adding means getting larger in positivity or negativity,and subtraction means getting smaller in positivity or negativity. Due to thisdeveloped understanding, students can add negatives to other negatives (e.g.,

-2+ -3) but can only take away negatives from other negatives as long as thereare enough of them to take away (e.g., -5 − -3). By level 4, on the other hand,students can solve all combinations and know that the operation and sign ofthe second number are the key elements.

Bruno and Martinon (1999) believe levels 3 and 4 are just one level, indi-cating that students can distinguish between minus and negative signs andcan describe situations where addition results in decreasing (e.g. 5 + -3) andsubtraction results in increasing (e.g. 5 − -3). However, results from Murray(1985) contradict both models. He found that even ninth grade students couldsolve some level 3 questions, such as -2 + -5, but not level 2 questions, suchas 5 − 7. �ese results suggest that other factors contribute to the relativedi�culties of the problems. In the latter case, Murray suggests that studentsover rely on the commutative property, so they solve 5 − 7 incorrectly even ifthey can solve other problems involving negatives.

In other instances, students manipulate the negative signs and obtainnegative answers by taking the opposite of positive ones. For example, theymay use the symmetrization strategy (Schwarz et al., 1993), solving problemsas if they are positive and then adding a negative back in at the end (Murray,1985; Peled et al., 1989). Additionally, in cases such as -7−5, some students usebrackets to show that they deal with the negative last (e.g., -(7 − 5)) (Vlassis,


2004); whereas, others change the position of the negative, transforming7 − -5 into -7 − 5 (Peled, 1991) to get negative answers.

Sometimes students’ symmetric strategies arise out of algebraic rules suchas, “two minuses makes a plus” that they misapply (e.g., -9 + -4 = +13) orsometimes use correctly (Vlassis, 2004). Other rule-based strategies createdby students, such as “-5 − -2 = -3 because 5 − 2 = 3 and these are minuses”(Murray, 1985, p. 149), illustrate their inclination to add negatives to answers,but their rules do not provide reasoning for why this is the case. As opposedto rules, due to experience working with positive numbers, students maydraw analogies with positive numbers and reason that “-5 − -8 = 3 because if5 − 8 = -3, -5 − -8 must be +3” (Murray, 1985, p. 150). However, without thisdepth of logic, students may guess answers based on the number combina-tions possible or may choose computations based on contemplating how thequantities will change (Schwarz et al., 1993).

To complicate matters further, students can use any of these sign strategiesin combination with the number strategies to form hybrid strategies (Schwarzet al., 1993), like using the inverse strategy by switching the numbers, solving,and also changing the sign of the answer (Schwarz et al., 1993; Tatsuoka, 1983).

Identifying Students’ Strategies

Rather than relying on students’ reports to determine levels of understand-ing, a group of researchers in the measurement �eld investigated ways toidentify the rules students use to solve integer addition and subtraction prob-lems from their response patterns to a set of questions (Tatsuoka, 1981, 1983;Tatsuoka, Birenbaum, & Arnold, 1989). Based on the combination of absolutevalue strategies and sign strategies, Tatsuoka and Baillie found 89 di�erentrules that students use to solve negative addition and subtraction problems(e.g., subtract the smaller absolute value from the larger and add the sign ofthe �rst number) (Tatsuoka, 1983). However, their method for determiningstudents’ rules requires that students use the same rule across di�erent prob-lems; in one case this meant they only identi�ed fewer than half of students’subtraction rules (Tatsuoka et al., 1989).

Investigating Integer Understanding

�e lack of understanding how children approach integer problems leadssome researchers to classify certain problems (such as subtracting a negative)as di�cult (Murray, 1985) while others claim problems like -7− -7 are amongthe easiest (Human & Murray, 1987). Deeper investigation reveals a more

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nuanced situation: students can solve -7−-7 correctly because they can ignorethe negatives, subtract, and get the same answer; they can obtain the correctanswer using an incorrect procedure (Bo�erding, 2010; Tatsuoka, 1983), butthey may not understand why it works. Aside from disagreement on thedi�culty of some problem types, what is less clear is what makes problemseasier or more di�cult, what contributes to students’ understanding of oneproblem type over another, and to what extent children’s changing beliefsabout over-reliance on the commutative property and the changing meaningof addition, subtraction, and the minus sign in�uence their solution strategiesand answers.


3 T H E O R E T I C A L F R A M E WO R K

Conceptual Change

�e question of how students’ thinking develops and shi�s and what in�u-ences this process is at the center of research on conceptual change. Concep-tual change involves a progression whereby children transition from havingan incomplete understanding of a concept to having the formal understand-ing of that concept (based on what the culture considers correct). Studentsgenerally experience conceptual change when learning a new concept inter-feres with their previous understanding of a topic (Vosniadou & Verscha�el,2004). For example, children’s initial schemas or initial mental structures(Gri�n, 2004a) of numbers include the presupposition or assumption thatnumbers are discrete (Gallistel & Gelman, 1992; Vamvakoussi & Vosniadou,2004). �ese initial schemas re�ect information gathered from their physi-cal and observable experiences in the world around them (Vosniadou, 1994,2007). Based on their experiences with objects and number order and theirpresupposition that numbers are discrete, when children �rst encounter deci-mals, they have an initial, naïve schema that there are no numbers in-between0.5 and 0.6, just as they originally thought there were no numbers between 5and 6 (Vamvakoussi & Vosniadou, 2004).

In the conceptual change model, children expand or restructure theirschemas as they encounter information or situations that are not part of theircurrent structure of ideas. Reasoning about similarities and di�erences be-tween their current knowledge and new information can also prompt changesin children’s schemas. If the new information is consistent with their currentschemas, students can easily add it on to their initial schemas, expandingtheir schemas; for example, learning that 0.6 > 0.5 does not con�ict withtheir initial schemas of decimals as discrete or with their schemas of six beinggreater than �ve. However, when novel information con�icts with students’existing schemas, the process of conceptual change is slow and occurs over alonger period of time (Vosniadou, 1994). For instance, when children learnabout the continuous nature of decimals (not only does .5 exist, but so do

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20 3 ⋅ theoretical framework

.51, .52, etc.), they take this formal knowledge and incorporate it into theirexisting initial schemas, creating a synthetic schema.

Synthetic schemas, also commonly referred to as misconceptions, re�ecta particular restructuring of children’s schemas and represent intermediaryknowledge, which does not re�ect formal knowledge of the concept. �eseschemas are relatively unstable and can change depending on the child’s con-text (Vosniadou, 2007). �erefore, children can advance from having aninitial schema to having one of several synthetic schemas. In the case ofdecimals, children might interpret .51 as greater than .6 since 51 is greaterthan 6 (Resnick, 1989); this schema preserves their understanding of wholenumbers while allowing for .51 to exist (Vamvakoussi & Vosniadou, 2004).Other children might develop a rule that shorter decimals are always greater(Steinle & Stacey, 2003). With additional support, which may include ex-ploration and interactions with peers, students reorganize their schemas tore�ect the formal understanding of decimals.�e process through which children move from initial schemas to syn-

thetic schemas and �nally formal schemas is aligned with Piagetian andneo-Piagetian notions of di�erentiation and integration (Case, 1996; Gri�n,2004a; Siegler & Chen, 2008). In the previous example, children develop amore di�erentiated understanding of number when they expand their wholenumber schemas to include decimals. However, they o�en integrate theirknowledge of either decimal order with decimal values or whole number val-ues with the decimal values in nonstandard ways before restructuring theirschemas to re�ect the formal schema for decimal understanding.

Central Conceptual Structure for Whole Numbers

Since there is no physical analog for negative numbers (De Cruz, 2006),children generally learn about them a�er learning about positive numbers.In the terms of conceptual change theory, children develop presuppositionsabout numbers through their experiences with whole numbers and then drawon this number framework as they build their initial schemas of negative num-bers. As described in chapter 2, children spend several years expanding andintegrating their schemas for number order, value, and symbolic representa-tions. While children may understand elements of this structure separately at�rst, they eventually integrate some and then all of them together. �e integra-tion of these concepts results in the central conceptual numerical structure(Case, 1996), referred to here as the central conceptual structure for naturalnumbers to emphasize that this original framework did not include zero; this

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Figure 3.1 – Central Conceptual Structure for Whole Numbers. �e central conceptual structurefor whole numbers, modi�ed from Case (1996), indicates the integration (shown by thevertical lines) of written symbols, number word order, a tagging procedure for pointing whilecounting, and knowledge of set values.

structure is also referred to as a “mental number line [italics added] on whichaddition and subtraction depend” (Gri�n, Case, & Capodilupo, 1995).

Although the original central conceptual structure did not include zero,children in preschool and kindergarten can solve problems involving zero(Sarama & Clements, 2009). �e addition of zero to the framework resultsin the central conceptual structure for whole numbers. While the entire struc-ture represents the schema of number understanding needed for additionand subtraction, each row represents a particular schema involved in wholenumber understanding (e.g. a schema of number order). Figure 3.1 showsthe modi�ed central conceptual structure for whole numbers (rather thanjust natural numbers) and the integration of the individual schemas as theyare mapped onto each other.

Children who have developed the central conceptual structure for wholenumbers know their culture’s list of number words and can recite them inorder. �is understanding is re�ected in the row labeled “number wordorder” in Figure 3.1. �e arrows from one word to the next indicate thatchildren know that saying the next number in the sequence is counting up andsaying the previous word in the sequence is counting down. �e row labeled“tagging routine” in Figure 3.1 illustrates that along with the counting words,children have a method for tagging items in one-to-one correspondence asthey count. �e arrows indicate the movement from one object to the next(or one �nger to the next), a process that children eventually do mentally(Case, 1996). Children might only use this tagging procedure for zero whencounting backward, so this dot is gray instead of black. �e row labeled“values” represents children’s recognition of quantities. �e arrows betweenquantities show children’s understanding that each tag leads to the next or


previous quantity and that adding one more to a collection increases thequantity by one and taking away one reduces the quantity by one.

Furthermore, the dotted lines connecting the “number word order”, “tag-ging routine”, and “value” rows indicate that these three schemas are inte-grated and mapped onto each other. As discussed previously, when �rstlearning positive number order and value, students may have only partiallyintegrated connections between these concepts. For example, children maybe able to correctly state the numbers in order and assign these tags to acollection of items as they count, but they might not be able to use this in-formation to determine the quantity of the set. With additional experiences,this integration becomes complete.

At the top of the diagram, the row labeled “written symbols“ is slightly setapart as in the original diagram and corresponds to children’s understandingof the symbolic form of the numerals as well as the addition and subtractionsymbols. In the original conceptual structure diagram, the value row was la-beled with “+1 and −1”. In the modi�ed diagram shown here, the operationalsymbols appear in the written symbols row to emphasize that these notationsare part of our formal notation system. �e arrows between numerals inthe written symbol row along with the “+1” and “−1” symbols represent thearithmetic relationship between successive whole numbers: one-unit incre-ments in one direction and decrements in the reverse direction. �is symbolicunderstanding of addition and subtraction parallels students’ quantity under-standing.�e integration line, shown by vertically stacked dots, that connects the

written symbols to the three schemas below looks di�erent from the othersbecause some cultures do not have symbol systems, and the symbols are notrequired for computation (Gri�n et al., 1995; Sarama & Clements, 2009).However, children in the U.S. must “learn to map addition and subtractionoperations onto their numerical representations” (Gri�n et al., 1995, p. 126),which is emphasized in the value row. �e brackets at either end of thediagram emphasize that movement to the right with the mental number linemeans getting greater in quantity, while moving to the le� means getting lessin quantity.

Once students develop the fully-integrated central conceptual structure forwhole numbers, they can use this structure to solve increasingly more di�cultproblems. According to the central conceptual structure theory, integrationof the previously discussed concepts and their subsequent development rely

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on maturation to a signi�cant extent. �at is, increases in students’ work-ing memory and neuronal connections allow them to coordinate thinkingabout multiple concepts or dimensions of thought at once. Students whocan use their conceptual structure for whole numbers to determine whichof two single-digit quantities is greater or to solve single-digit addition andsubtraction problems are at the unidimensional level of thought; they canoperate with one mental number line to solve problems involving knowledgeof ones. At the bidimensional level of thought, young students (approximately6–8 years of age) become more familiar with using this mental number lineand start to use a second mental number line in order to di�erentiate be-tween multiple dimensions of number, for example, tens and ones. �en, atthe integrated bidimensional level of thought, children integrate these twomental number line schemas and can use them to solve problems involvingregrouping (Case, 1996).

Central Conceptual Structure for Integers

If the preceding are the individual schemas that comprise the central struc-ture for whole numbers, then from what do negative numbers arise? Sinceorder, value, and notational concepts involved with negative numbers con-�ict with children’s whole number schemas (Fagnant et al., 2005; Vlassis,2004), the inclusion of negative numbers in this number system representsan elaboration of and reintegration with the central conceptual structurefor whole numbers and not a separate structure. In other words, childrendo not start over from scratch when learning about negative numbers, nordo they immediately intuit negative numbers from the existence of wholenumbers. Children’s whole number schemas serve as the set of presuppo-sitions from which children begin to reason about negative numbers. Forexample, when �rst encountering negative numbers, children treat negativesigns as subtraction signs or negative numbers as positive numbers due totheir similarities.

Figure 3.2 shows a more general view of the number and sign conceptsstudents must di�erentiate and integrate in the transition from whole numberto integer understanding. As described in the conceptual structure diagramfor whole numbers and shown here in the le� box, students use their knowl-edge of the order and value of numbers and the operation signs togetherwith their understanding of more and less in relation to addition and sub-traction. Students develop addition and subtraction strategies based on theirwhole number understanding; these strategies as well as their schemas change


Figure 3.2 –Whole Number Schemas. Schemas involved in whole number and negative numberunderstanding, with the minus sign taking on three di�erent meanings.

as students take the sign of numbers into consideration. �e horizontal ar-rows represent the in�uence of instruction or other experiences on children’sschemas and strategies.

As children learn about negative numbers, they must di�erentiate betweenthe three meanings of the minus sign as discussed in chapter 2. �e binaryminus sign refers to the subtraction operation and the symmetric minus signrefers to taking the opposite of a number. �e unary meaning of the minussign, represented in the overlap of signs and numbers in Figure 3.2, is uniqueand signals that when attached to a number, the minus sign designates anew type of number: negative numbers, which have a di�erent order andvalue than the positive numbers. �us the unary meaning of the minus signspans both the sign and number categories. When solving addition andsubtraction problems with negative numbers, students may still use theirfamiliar operations strategies, but they must also determine when and howto incorporate negative signs in their solution processes.

Figure 3.3 shows how the acceptance of the unary minus sign (negativenumbers) and the schemas involved in this process �t into the proposedextension of the central conceptual structure of whole numbers to the cen-tral conceptual structure of integers. Since the extension requires a shi� inthinking about the written symbols and an additional value schema, thesechanges are also marked in the whole number structure. Unlike the originalcentral conceptual structure of numbers described by Case (1996), the pro-posed elaboration of the structure to incorporate negatives relies primarily oninstruction to advance children’s thinking rather than on further maturation.

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Figure 3.3 – Central Conceptual Structure for Integers.�e central conceptual structure forintegers, extended from the framework presented by Case (1996), indicates the integration(shown by the vertical lines) of written symbols, number word order, a tagging procedurefor pointing while counting, knowledge of ordered values, and knowledge of directed magni-tudes.

As indicated in row labeled number word order in Figure 3.3, when chil-dren learn about negative numbers, they need to extend their number wordsequence before zero, saying the word “negative” before each number in theirtraditional string. Moving from one number to the next still involves movingup or down to the next number in the sequence. While the pointing schemeinvolved with whole numbers and represented in “tagging routine” row ofFigure 3.3 could easily extend into the negatives, there are no physical in-stantiations of the negative numbers (De Cruz, 2006). �erefore, childreneither need to use physical objects or a written number line to represent theabstract entities so they can tag each one, or they can use their mental pro-cess of tagging to interpret each number in their count as an item that theysimultaneously tag. Extending their understanding of number values into thenegatives (see the row labeled “value”) also requires some abstraction becausestudents cannot see -3 things. Children can use their whole number schemafor value to determine that moving to the right on the mental number linemeans getting larger and moving to the le� means getting less. From thisknowledge they can judge that two is greater than negative four and negativefour is greater than negative six–here, and in the rest of the paper, value refersto ordered value, and these terms may be used interchangeably.

However, in order to use the central conceptual structure of integers to sup-port addition and subtraction, another value schema is necessary: directedmagnitude, which is shown in the bottom row of Figure 3.3. �is schemarelates to the understanding that numbers can be more or less in two ways:relative to the positive numbers or relative to the negative numbers. Adding apositive value to a negative value (e.g., -5+ 3) still results in a greater ordered


value than the initial number (shown by the arrows pointing more positive),and removing a positive value from a negative value (e.g., -5 − 3) results inan ordered value less than the initial number (shown by the arrows pointingless positive).

Adding and subtracting negative quantities are also possible, so childrenneed to di�erentiate their schemas for more and less to apply to these separateconditions. Now, instead of more indicating a movement only to the right onthe mental number line, it represents a movement in the direction of the typeof quantity added (more positive or more negative). Similarly, subtraction,or getting less, indicates a movement away from the type of quantity beingsubtracted (less positive or less negative). �e directed magnitude schema,then, represents the schema that most directly supports addition and subtrac-tion with integers–of course the other schemas support this process; whereas,in the central conceptual structure for whole numbers, the ordered valueschema serves this purpose.

Negative numbers involve both numerals and negative signs, such as thosein the row labeled written symbols in Figure 3.3. With the introduction ofnegative numbers, children need to discriminate between subtraction andnegative signs, and integrate the negative signs with the numerals they al-ready know to create a new set of signed numerals; in doing so, childrenindicate knowledge of the unary meaning of the minus sign. When workingwith integers, they must discriminate between the signed numerals and theunsigned numerals.1 1 In some cases, people write pos-

itive numbers with a plus sign infront of them. When this is thecase, students must discriminatebetween the two signs when de-termining if a number is positiveor negative. Here, descriptions ofpositive numbers refer to the morecommon unsigned numerals, andsigned numerals refer to negativenumbers. Children do not necessar-ily pay more attention to negativenumbers when they are boxed toset them apart (Bo�erding, 2010),so parentheses were not used.

Finally, just as students must add to their schemas of more and less todistinguish between ordered value and directed magnitude, they also mustintegrate this newly formed directed magnitude schema with the operationsigns to formalize the schema for adding and subtracting positive or negativenumbers. �e written symbols row indicates this new information with theaddition of “+-1” and “−-1” to the �gure.

While it is possible that children could encounter negative numbers (e.g.,on game shows or the weather channel) (Clemson & Clemson, 1994) beforethey fully integrate and develop their central conceptual structure for wholenumbers, the research presented here assumes that the subjects do have anintegrated whole number schema. Given this assumption and the proposedcentral conceptual structure for integers, students can expand their wholenumber schemas to include integers by �rst learning the signed numerals,ordering a list of negative numbers, tagging an unknown quantity below zero,or reasoning about values less than zero. According to the conceptual change

27

view, learning in any one of these areas can lead to a variety of synthetic orintermediary schemas. Additional instruction could then encourage childrento revise their synthetic schemas into more advanced synthetic schemas orto reintegrate their schemas into a formal schema, re�ective of the centralconceptual structure for integers.

As with the conceptual structure for whole numbers, the vertical lines inthe conceptual structure for integers (see Figure 3.3), represent the integrationof the underlying schemas. Before children develop a formal schema forintegers and integer operations, they may have one of many connected orpartially connected integration lines. �e following examples illustrate how�ndings based on the integer research discussed in chapter 2 might �t intothe initial and synthetic schema framework.

Children’s initial integer schemas are based on their prior number expe-riences, which involve the central conceptual structure for whole numbers.Using this structure, students might ignore the negative signs and treat neg-ative numbers as they would positive numbers in numerical tasks (Murray,1985; Peled et al., 1989). Other students might draw on their schemas for thesubtraction sign and solve 5 + -2 by �rst adding two and then subtractingtwo (Murray, 1985). A�er exposure to negatives as separate from positivenumbers, students might determine that positive numbers are larger thannegative numbers but still treat negative numbers as positive when addingand subtracting, creating a synthetic schema.

Students’ responses to integer addition and subtraction problems suggestthat there are many di�erent paths (along with the one described) that stu-dents follow when transitioning from initial to formal schemas for negativenumbers. Since students’ initial schemas change when they encounter in-formation that challenges their schemas (Vosniadou & Verscha�el, 2004),instruction can play a role in how their schemas change.�is research seeks to clarify how young students’ schemas change de-

pending on whether they receive instruction on integer properties (writtensymbols, including the unary meaning of the minus sign; negative numberword order; and the ordered values of negatives), integer operations (the dualmeaning of addition and subtraction as operations on directed magnitudesas well as the non-commutativity of subtraction), or both. In particular, thisresearch focuses on how students build on their whole number schemas todevelop schemas of negative integer identi�cation, order, value, and directed


magnitudes; additionally it explores the various ways in which students in-tegrate these initial, synthetic or intermediary, and formal schemas and usethem to reason about addition and subtraction problems with integers.

4 I N S T RU C T I O NA L P H I L O S O P H Y

Providing students with instruction is one way to encourage the concep-tual change process (Vosniadou, 2007; Vosniadou & Verscha�el, 2004). Sincethis study explores students’ schemas before and a�er instruction, it is worth-while to explore integer instruction (both historically and in this study) fur-ther before talking about the methods of the study more speci�cally.

Previous Negative Number Instructional Research

Previous studies have attempted to determine the best instruction forteaching children negative number operations (Bruno & Martinon, 1999;Janvier, 1985; Liebeck, 1990). However, they were o�en concerned with whichteaching situation (using a number line representation versus a cancellationrepresentation) produced better accuracy. �e cancellation model focuses onteaching negative numbers (the unary meaning of the minus sign) along withdirected magnitudes; the number line model focuses on the unary meaningof the minus sign along with integer order. Unbeknownst to the researchers,they were comparing not only two di�erent representations and methodsof teaching negatives but also two di�erent underlying schemas of negativenumber understanding.

Additionally, students in the di�erent conditions did not always havesimilar experiences. In one case, third and fourth graders who learned how touse the number line to solve integer problems had trouble solving problemswith three numbers (e.g. 1 + 3 + -2); their classmates, who were comparablein age and ability, learned cancellation with chips and correctly solved almostall of them (Liebeck, 1990). �e students who used the chips, though, alsopracticed on problems where they added more than two numbers, so thecomparison was not equal. Additionally, the children who used the chipsmade up their own notation for negatives while the children who used thenumber line had to make sense of the multiple meanings of the minus sign(as a subtraction and negative symbol).

29

30 4 ⋅ instructional philosophy

Another set of instructional studies focused on determining if studentscould learn about negative numbers with a particular form of instruction(Schwarz et al., 1993; �ompson & Dreyfus, 1988; Linchevski & Williams,1999) or through guided practice (Hativa & Cohen, 1995). Unlike the compar-ison studies, these focused on either number line or cancellation instruction,and all authors found that students increased their understanding of opera-tions with negatives. Additionally, their results indicated that some studentshad better understanding than others at the end of the studies.

A large gap in these previous studies is that researchers did not investigatewhat impact the instruction had on students’ integer schemas except in termsof accuracy on the addition or subtraction problems. It is unclear to whatextent students developed formal schemas of integer identi�cation, order,value, and directed magnitudes.

Current Integer Instruction

An underlying philosophy of the lessons provided in this study is thate�ective instruction should build o� of students’ current knowledge and un-derstanding. It should help them make connections among new conceptsand what they already know to enable them to better understand the struc-ture of the subject (Bruner, 1960/2003). Additionally, the instruction shouldtarget areas that traditionally serve as stumbling blocks for students initiallylearning the concepts. With this in mind, I developed the curriculum for thisstudy based on information gained about students’ integer understandingfrom previous research and two pilot studies. Further details of the di�erentinstructional conditions are provided in chapter 6 on methods.

Students across all age ranges have di�culty in three main topic areasrelated to negative numbers. �ey have trouble distinguishing between themultiple meanings of the minus sign (Bo�erding, 2010; Vlassis, 2008), inter-preting the new meanings of addition and subtraction when negatives areinvolved (Bruno & Martinon, 1999), and solving positive subtraction prob-lems where the minuend is smaller than the subtrahend (e.g., 3− 9) (Murray,1985). �e curriculum for this study focuses on underlying principles of thesetopics (see Appendix A for complete lesson plans), such as symbolic consis-tency and inconsistency. Because students who have just recently integratedthe central conceptual structure for whole numbers are typically at the unidi-mensional level of thought (Case, 1996), all questions and instruction involvesingle-digit problems.

31

Aside from the content, instructional activities need to engage studentsand provide them with the extended opportunities to think about the con-tent (i.e., ordered value of negatives, moving less negative, etc.). �e recent(and successful) use of linear games designed to promote early mathematicallearning in research (Ramani & Siegler, 2008) and curricula (Gri�n, 2004b)inspired the use of games in this curriculum. �e games for learning aboutnegative numbers include the card games Memory and War, ordering games,and using a number line to help determine which integers are larger; thegames for addition and subtraction include moving on both horizontal andvertical number lines.

While number lines are o�en criticized as being di�cult for children to usebecause they confuse the meaning of the operations and numbers involved(Fuson, 1984; Sarama & Clements, 2009), they serve an important purposein relation to the central conceptual structure of whole numbers. Gri�n(2004b) asserts:

Perhaps the most important transition that children must make,as they move from the world of small countable objects to theworld of abstract numbers and numerical operations, is to treatthe physical addition or subtraction of objects as equivalent tomovements forward or backward along a line. All children even-tually make this correspondence; however, until they do, they areunable to move from physical operations to mental operationswith any insight (p. 332).

Studies comparing number line instruction to cancellation instruction foradding and subtracting negative numbers revealed that students have lesssuccess with number line instruction (Janvier, 1985; Liebeck, 1990) or thatthere is no natural way of thinking about subtracting a negative using thenumber line (Ball, 1993). However, these viewpoints refer to a speci�c useof the number line, which places addition and subtraction into unhelpfulcontexts. For example, both researchers and textbooks teach adding andsubtracting negatives on number lines in the following way: when adding,face in the positive direction and when subtracting face in the negative direc-tion. Start at the initial number and if the second number is positive, moveforward; if the second number is negative, move backward (Liebeck, 1990;McGraw-Hill, 2004). �is instruction forces students to think about additionand subtraction in contrived ways that are not consistent with the meaning ofaddition and subtraction or that do not address the value of negative versuspositive numbers.


Unlike the number line instruction for addition and subtraction in previ-ous studies, the instruction for this curriculum builds on students’ centralconceptual structure for whole numbers. Students naturally associate addi-tion with getting more and subtraction with getting less (Fuson, 1992); thecurriculum takes advantage of this pairing but pushes students to think aboutthe directed magnitudes of the numbers and about what they are getting moreor less of.

Focusing on the language is important because it supports students’ math-ematics learning (Sarama & Clements, 2009). Sfard (2002) suggests that sincewe o�en refer to negative quantities with labels rather than numbers (e.g., “Iowe $5”), it is hard for negative numbers to become part of everyday discourse.By presenting negative numbers as quantities separate from positive numberson the number line and distinguishing between operating with positive andnegative numbers, the curriculum pushes students to make the distinctionbetween positive and negative quantities. Van Oers (2002) proposes thatchildren’s mathematical development progresses in terms of symbol devel-opment, meaning development or both as children use their own intuitivelanguage, model the language of others, and receive reinforcement from peersand adults. To encourage students to use the language modeled by the author,the lesson structures invite students to work in pairs and participate in dis-cussions about the activities. Finally, planned and speci�c discussions aboutthe symbols students use in the games serve to facilitate students’ e�orts tomake connections between the content and symbols.

Integer Properties Group Instruction

�e purpose behind the Integer Properties group’s instruction is to helpstudents understand the unary meaning of the minus sign. �erefore, instruc-tion targets the written symbols, number word order, and ordered value rowsof the central conceptual structure for integers (see Figure 3.3). Because stu-dents in this condition do not receive instruction on how to add and subtractnegative numbers, the condition provides information on what students areable to apply from integer property instruction to arithmetic problems.

Rather than telling the students that the minus sign is di�erent from thenegative sign, the lessons �rst expose students to the idea that one symbolcan be used in multiple ways or can have multiple meanings. For example,the lessons on negatives begin with an investigation of how signs can lookthe same but mean something di�erent (e.g. the number eleven versus twotally marks). From there, students explore the similarities and di�erences

33

between how the minus and negative signs look and are used. Since thelessons continue to focus on the di�erence between positive and negativenumbers in the contexts of order and value, only the �rst couple of lessonsfocus on written symbols exclusively.

�e other lessons focus on ordering negative numbers relative to the pos-itive numbers and judging their values but continue to encourage studentsto notice the di�erences between the numbers. For example, in one of theordering games, students need to draw cards until they have three sequentialnumbers (e.g., -2, -1, 0 or 2, 3, 4). To sequence the numbers correctly, stu-dents need to distinguish between 3 and -3 as symbols and know that onemore than 0 is 1 and one less than zero is -1. In the pilot studies, some stu-dents reversed the order of the negatives, so the lessons help students focuson seeing zero as a key point with the positive and negative numbers goingin opposite directions from that point.

Value and order concepts are closely aligned, but the lessons have studentsplay value games with isolated number cards as well as within a number linecontext to ensure that they have practice thinking about the numbers withoutthe extra ordering sca�old.

Integer Operations Group Instruction

�e purpose of the Integer Operations group instruction is to help studentsunderstand the binary meaning of the minus sign. �erefore, instruction tar-gets the directed magnitude row of the conceptual structure for integers (seeFigure 3.3) as well as the understanding that subtraction is a non-commutativeoperation. Because students in this condition do not receive instruction onwhat negative numbers are, the condition provides information on what stu-dents are able to glean from directed magnitude instruction concerning thenature of negative numbers. It also provides information regarding students’methods for solving integer arithmetic problems when they do not have deepunderstanding of negatives.

Similar to the integer operations instruction, the lessons on addition andsubtraction begin with a more abstract lesson where students explore whenorder does and does not matter in terms of an end state. For example, eatinga banana and then an orange leads to the same end result (all food is gone)as eating an orange and then a banana; whereas, standing up and then sittingdown leads to a di�erent end state than sitting down and then standing up.Students continue exploring this reasoning to determine if order matters inaddition or subtraction by moving an elevator above and below ground level.


It is important for students to see both pictorially and numerically that theelevator ends on di�erent �oors for 3 − 1 versus 1 − 3, so the �oor levels arelabeled with the correct integer. However, the numerals are not named in theinstruction, so students may read “-1” as “one” or “minus one.” Regardlessof how the students name them, the instructor only writes the numbers butdoes not verbally label them.

As mentioned previously, the number line instruction for addition andsubtraction takes advantage of students’ inclination to think of addition asgetting more and subtraction as getting less (Fuson, 1992). Before pairingthe more and less language with negative and positive, students explore theopposite nature of more and less with colors and feelings (where getting morehappy is equivalent to getting less sad). Later, adding a negative is taught asmoving more negative, while subtracting a negative is taught as moving in theless negative direction. Similarly, adding a positive number relates to movingin a more positive direction (or getting more positive), and subtracting apositive number means moving in a less positive direction. Students practicethese actions with the addition and subtraction board games that look likenumber lines without the numerals. It is important for this group to seemostly blank number lines to keep them separate from the group that hasinstruction on integer order. Because this directed magnitude language isconfusing for beginning students, the bulk of the operations instruction ison these concepts.

Full Instruction Group Instruction

�e purpose of the Full Instruction group’s instruction is to help studentsunderstand both the unary and binary meanings of the minus sign. �erefore,instruction targets all rows of the central conceptual structure for integers (seeFigure 3.3) as well as the understanding that subtraction is a non-commutativeoperation. �is condition provides information on how students solve inte-ger addition and subtraction problems when they have more understandingof negative number order and value. �is group compared to the integeroperations group illustrates di�erences due to having some versus no integerproperties instruction and when compared to the integer properties groupillustrates di�erences due to having or not having integer operations instruc-tion.

Although the Full Instruction group’s instruction covers the material fromthe integer properties and integer operations groups, the instruction canstill only last eight days. �erefore, only select lessons from the other two

35

groups comprise the instruction for the full instruction group. However,these lessons are the primary lessons, so the group only misses the extrapractice sessions. Additionally, unlike students in the Integer Operationsgroup, students in the Full Instruction group always see the number lineslabeled with the integers because they learn about negative numbers beforethe directed magnitude games. Further details of the di�erent instructionalconditions are provided in chapter 6 on methods.


5 P I L O T S T U D I E S

�e majority of the interview questions, some of the coding schemes, andthe negative number instruction ideas for this dissertation study arose out ofpilot work, so before exploring the methods of the current study, it will behelpful to review the two previous studies.

Pilot Study I

In July of 2009, I interviewed 2nd (N=8), 3rd (N=10), and 4th (N=4)graders on several integer tasks: integer value comparisons and sorting, or-dering integers, and solving integer addition and subtraction problems (seeAppendix B). Students’ performance on the arithmetic problems varied bothwithin and across grade levels. Percentage correct scores for second, third,and fourth graders ranged from 0–80%, 0–80%, and 70–100% respectively.Only three of the students correctly solved 2−? = 4 and two of them weresecond graders who used logical reasoning to solve the problem.

Similar to the �ndings by Human and Murray (1987), many students usedstrategies that resulted in incorrect answers on one problem but which hap-pened to give them correct answers on other problems. Some problems whereone number was a multiple of the other were harder for students than similarproblems with di�erent numbers. A possible hypothesis for this was that stu-dents are overly primed to solve positive variants of these problems. Studentsalso had di�culty merging the language they typically use to talk about posi-tive number addition and subtraction with the quantities in negative integerproblems. For example, when solving -2 − -5, one girl said, “If I’m plussingthen I would be going down because then I’m just plussing and that would begoing up.” �is student had trouble reconciling the idea of adding as goingup versus adding as going towards numbers with larger absolute value. Manystudents also ignored the negatives in their explanations or solutions.

While students did not always report what strategies they used to solvethe problems, based on their answers to the integer value and integer orderquestions, students fell within one of three categories: positive-only thinkers,

37

38 5 ⋅ pilot studies

transition thinkers, and negative-knowledgeable thinkers. �e three positive-only thinkers did not fully accept the unary meaning of the minus sign. Oneignored all negative signs and interpreted negatives as positive. �e other twointerpreted negatives as worth zero and ignored negative signs when solvingproblems; their median percentage correct score on the arithmetic problemswas 0%.

Eight students were transition thinkers who sometimes said negativeswere worth zero and sometimes were less than zero, although they orderedthem correctly. While they o�en ignored the negative signs when solving theproblems, they also added negatives to their answers; their median percentagecorrect score was 45%.

Finally, eleven students were negative-knowledgeable and knew the correctorder and quantity of negatives. �ese students generally used countingmethods to solve the arithmetic problems but had particular trouble solvingthose that involved understanding the changing meaning of subtraction withnegatives (that getting less negative is di�erent than getting less positive);their median percentage correct score was 70%. �ese results suggest thatstudents understand the unary function before rethinking the binary functionin terms of negatives and that understanding of the unary function can helpstudents solve some integer arithmetic problems.

Pilot Study II

In November and December of 2009, I conducted additional interviewswith twenty-two 2nd graders. To explore 1) question types not included inthe original pilot, 2) di�erent presentation formats or number combinations,and 3) to what extent students use the three interpretations of the minussign, I gave one group (N=13) items where boxes were inserted around thenumbers on half of the items (to see if this resulted in them paying moreattention to the negative signs) and another group (N=9) items where halfof the items used numbers that were multiples of each other (e.g. 4 + -2as opposed to 5 + -3) (see Appendix C for the list of questions). Studentswere asked how they solved the problem a�er each question. �ere were nodi�erences between percent correct for the boxes versus un-boxed problemsor the multiples versus not multiples problems.

Students’ solutions were coded for types of strategies they used to set upthe problem (e.g., ignored negatives then reversed the numbers, used anarithmetic rule: subtracting a negative is adding a positive, treated negativesas zero) and to obtain their answer (e.g., recalled answer, counted with �ngers,

39

used a related fact) (see Appendix D for complete list). �eir verbal reportswere also coded according to their descriptions of integer quantities, the orderof integers, and the meaning of addition, subtraction, and negative signs.�rough developing and working with these integer and operation codes, Iidenti�ed general codes to represent possible schemas (re�ecting incompleteto full understanding of negatives) that students might have within each ofthose categories (e.g., the value of negatives are zero or negatives are orderedlike positive numbers from smallest to largest magnitude) (see Appendix E).

Overall, students’ scores ranged from 0%–53% correct (average 20%).Among the group of 22 second graders and 17 problems, students acknowl-edged all three roles of the minus sign. Together, they used–although notalways correctly–the binary function 53% of the time, the symmetric function20% of the time, and the unary function 10% of the time. Within problemtypes solved on the test, 15 of the 22 students were almost completely consis-tent in the strategies they used to solve the problems while the other sevenchanged their strategies more o�en. Over half of the students (13/22) ignoredall of the negative signs and solved the main arithmetic problems as if theywere positive. Additionally, 14 of the 22 students reversed numerals in sub-traction problems to avoid subtracting a larger number from a smaller one.�e only problems which no students answered correctly were ones that re-quired them to subtract a negative (e.g. 3 − -5 or 9 − -1) and understand thechanging meaning of subtraction. Students’ language from these questionssuggested that they have trouble bridging both the new signs and the changein meaning of addition and subtraction.

40 5 ⋅ pilot studies

6 M E T H O D S

As a step toward clarifying the development of the central conceptualstructure for integers as an extension of whole number understanding, thisstudy investigates the following research questions:

research questions

1. What are �rst grade students’ schemas1 of negative integers (in terms of 1 Schemas are children’s mentalmodels or how they think aboutand interpret the content or opera-tions (Gri�n, 2004a).

the elements underlying the central conceptual structure for integers:written symbols, order, value, and directed magnitude)?

2. How does the relation among students’ integer schemas manifest itselfin students’ approaches to integer addition and subtraction problems2? 2 �ere are 32 problem types deter-

mined by the signs of the numbers,the operation, and whether thelarger number is �rst or second inthe problem (see Appendix F forthe complete list).

3. How does instruction in various elements of the central conceptualstructure for integers in�uence students’ understanding of negativesas measured through their changing schemas, language, arithmeticaccuracy, and approaches to integer arithmetic problems? To whatextent is students’ math performance level a factor in this change?

participants and setting

Students from the four �rst grade classrooms at Numerica ElementarySchool (pseudonym) in northern California participated in this study dur-ing the spring of 2010. Numerica provided an ideal location for the studybecause of its student diversity, size, and commitment to supporting prospec-tive teachers and contributing to research. During the year of the study, therewere three predominant ethnic groups attending Numerica: Asian (34.5%),Hispanic or Latino (28.1%), and White not Hispanic (24.8%). Additionally,47.2% of students were English Learners and 29.6% quali�ed for free andreduced lunches (California Department of Education, 2010). Because theschool had four �rst grade classrooms, it was large enough for recruiting the

41

42 6 ⋅ methods

Table 6.1 – Participants. Class participation percentages and breakdown.

Class 1 Class 2 Class 3 Class 4 All FirstGraders

Total Students 20 20 20 19 79

Returned Consent Forms 19(95%) 19(95%) 13(65%) 18(95%) 69(87%)Agreed to Participate 18(90%) 16(80%) 11(55%) 18(95%) 63(63%)Gender 10 male

8 female8 male

8 female4 male

7 female8 male

10 female30 male

33 female

needed number of participants. Furthermore, the school’s frequent mentor-ing of prospective teachers meant that the students were accustomed to beingvideotaped and having unfamiliar people work with them. �e teachers wereusing the math curriculum enVisionMATH for the �rst time, and althoughthe curriculum does not acknowledge negative numbers in the �rst grade,all four teachers indicated that at some point during the academic year theexistence of negative numbers arose brie�y in their classes before the studybut was not a topic they talked about otherwise.

All 79 �rst graders took home permission slips. Table 6.1 shows the numberof students in each class who returned the parental consent forms and thenumber who agreed to participate3, along with corresponding percentages 3 Two of the female students from

class 4 who agreed to participatewere excluded from the studybecause they could not completethe initial interviews in English.

of the class.All students who received parental consent and could complete the inter-

views in English were included in the study. Consequently, two non-Englishspeakers were dropped from the study, leaving 61 participants (77% of the�rst graders), 30 male and 31 female. Nine students (15%) came from familieswith Spanish as the primary home language. Students who missed instructionduring the study were still included in analysis to better re�ect the realitiesof regular instruction. Upon their �rst day of interview, participants rangedin age from 74.6 to 92.7 (82.9 X̄, 4.3 SD) months old. Students’ ages werecalculated using http://easycalculation.com/date-day/age-calculator.php andwere rounded to the nearest tenth of a month (based on a 31-day month).

An additional 16 ��h-graders (9 male; 7 female) from the same school,and representing a range of math performance levels, participated in just thepost-test to serve as a comparison for students’ thinking, but these data arenot discussed here.

study design 43

Figure 6.1 – Pre-test, Post-test Study Design.

study design

�is study used a pre-test, post-test design with all students also complet-ing one of three instructional interventions. Figure 6.1 presents an overallpicture of the design, and each phase (pre-test, instruction, and post-test) isdescribed in more detail in the following sections.

pre-test interviews

�e �rst phase of the study began two weeks prior to the instructionalintervention and consisted of a pre-test designed to gather information aboutstudents’ initial integer schemas. I (Interviewer 1) designed an interviewprotocol based on pilot work and trained a graduate student in math edu-cation (Interviewer 2) to use the protocol by talking through the document.Interviewer 2 then watched Interviewer 1 use the protocol with a practiceparticipant. Following a discussion of the interview in regards to the protocol,Interviewer 2 administered the protocol on a second practice participant andreceived feedback during and a�er the session. Interviewer 1 continued toprovide feedback to the second interviewer throughout the study to ensurethe protocol administrations remained similar.

�e interviews followed a format modeled a�er methods used by (Siegler,1996). Students saw one question at a time, and a�er each question theinterviewer asked, “How did you solve it?” Students did not receive feedbackon whether their answers were correct or incorrect to reduce the interviewers’in�uence on students’ thought processes. All students solved the questionsin the same order, except for the �nal set of arithmetic questions, which werecounterbalanced using a balanced Latin Square design (Shuttleworth, 2009)to reduce the chances that question order in�uences the group’s behavior.

Using a randomized list of the 61 participants, the two interviewers re-moved individual students from their classrooms and brought them to a

44 6 ⋅ methods

separate multi-purpose room. �e randomization removed any biases theinterviewers would have had in selecting certain students �rst or last and alsoensured that some students from each class were interviewed sooner or later.�e interviewers took students in order from the list unless a student wasabsent or unavailable. In these cases, the next student was chosen and themissed student was interviewed when available.

Students sat at a table to the right of the interviewer and facing the wall.A�er receiving assent, the interviewer turned on a voice recorder and camera.�e camera, positioned slightly in front of and to the right of the student,captured each student’s facial expressions and use of �ngers. In the case whenboth interviewers were working with students at the same time, the studentswere placed on opposite sides of the room facing away from each other. Ina few instances, students had to leave (for lunch, recess, etc.) during themiddle of an interview. When this happened, the student returned as soonas possible a�erwards to complete the interview.

�e pre-test questions were based on items from pilot studies and extrap-olations of the schemas that comprise the conceptual central structure fornatural numbers (Case, 1996); the questions probed students’ understand-ing of a) names for integers, b) integer order, c) integer values, d) directedmagnitude (concepts of more and less in terms of movement), and e) in-teger addition and subtraction. A few additional whole number questionsprovided a check that the �rst graders had some prerequisite understandingof numbers and operations. See Table 6.2 for a brief look at the questioncategories listed in the order in which students solved them. �roughout thestudy negative signs were presented as smaller than minus signs and slightlyraised.

Operations

�e operations questions provide an opportunity to determine how stu-dents de�ne addition and subtraction operations before they solve the integerarithmetic problems. �ese questions occur �rst to encourage students totalk less formally before seeing more traditional math problems.

Counting

�is question aims to determine the extent to which students know thedecreasing sequence of number names, a foundational part of the numericalconceptual structure. Students start counting backwards from �ve to get themin the rhythm of counting backwards before reaching the negative integers.

pre-test interviews 45

Table 6.2 – Question Categories. Question categories listed in the order students solved them.

QuestionCategory

Items Example/Description

SchemasTargeted

Operations 2 What is addition? What doesit mean to add? What is sub-traction? What does it mean tosubtract?

Whole num-ber additiveoperations

Counting 1 Start at �ve and count back-wards as far as you can. Is thereanything before <last numberstated>?

Integer Order

NumberLine

1 Fill in the missing numbers onthe number line.

1

Integer Orderand Identi�ca-tion

Ordering 2 Put these number cards in orderfrom least to greatest.2 −3 0 −9 3 8 −5

Integer Orderand IntegerValues

GreaterOrderedValue

7 What are these two numbers?3 −9

Circle the one that is greater.

Integer Iden-ti�cation andValues

Symbols 4 Circle all the numbers in thisequation: 6 + -2 − 7 = -3 Whichnumbers did you circle?

Integer Identi-�cation

OperationsProperties

6 Do you think these two prob-lems will give you the same ordi�erent answers? 3− 1 and 1− 3

Properties ofAddition andSubtraction

2-digit Sub-traction

2 31-27

BidimensionalOperations

DirectedMagnitude

4If the cat moves four steps morelow, draw an X where she willbe.

Directed Mag-nitude, Moreand Less

Integer Ad-dition &Subtrac-tion

26 Solve -4 − -7. Integer Ad-dition andSubtraction

46 6 ⋅ methods

Table 6.3 – Ordering Task. Sets of integers for each ordering task.

Integers (as laid out for students to order)

Set A 2 -3 0 -9 3 8 -5

Set B 6 -6 4 -7 3 -1

Since counting backwards before zero is not a typical routine in �rst grade,interviewers prompt students if they stop before the negatives by asking if theycan count backwards any further. �e counting backwards question appearsbefore students see negative numbers printed as in subsequent questions.

Number Line

Similar to the counting task, the goal of the number line task is to deter-mine students’ order schemas for integers, only this time as applied to thespeci�c instructional representation. Additionally, this task aims to deter-mine which symbols students use to represent their verbal integer sequence.Because students must produce their own symbols, this task appears beforethe other questions that could expose them to negative numbers. �e num-ber line contains the number 1 already written so that students must decidewhether anything comes before one.

Ordering

As with the number line task, the purpose of the card ordering tasks is todetermine students’ integer order schemas in relation to their value schemas.�e inclusion of zero and opposites (e.g., 6 and -6) helps determine if studentsorder negatives before or a�er zero and how they handle numbers of equalabsolute value (see Table 6.3 for integer sets). Having students identify thegreatest and least values provides insight into whether they use the order todetermine the answers.

(Ordered) Value

�is set of questions provides another measure of what ordered valuesstudents attribute to the symbols (both numerals and signs) and evaluatesstudents’ ability to identify integers. �e positive integer comparison ensuresthat students can identify positive numbers before comparing a mixed pair(one positive and one negative integer) or two negative integers. For themixed case, the negative integer always has a larger absolute value. �e goal


Table 6.4 – Value Tasks. Symbolic and situational integer value tasks.

What are these numbers? Circle the one that’s greater.

(Symbolic) 8 vs. 6 3 vs. -9 -2 vs -7 -5 vs. 3 -8 vs. -2Two children are playing a game and trying to get the highest score. Circle who is winning.

(Situational Context) Abigail: 4 vs. Joseph: -7 Crystal: -7 vs. Leon: -3

Table 6.5 – Symbol Identification Tasks.

Circle the numbers. Circle the signs which tell you whether toadd or subtract.

-5 + 3 − -3 = 1 -4 − 3 + -1 = -8

6 + -2 − 7 = -3 5 − -3 + 2 = 10

of the �nal two comparison questions is to determine if students can cor-rectly interpret the same task within a situational context (see Table 6.4 forquestions).

Symbols

�e aim of the symbols task–adapted from a task used by (Fagnant et al.,2005)–is to determine to what extent students identify the unary meaning ofthe minus sign and to what extent they confuse negative signs with minussigns (see Table 6.5 for items).

Operations Properties

Students’ explanations on the properties problems provide informationabout their understanding of addition and subtraction properties. Morespeci�cally they highlight whether students believe they can switch the or-der of the numbers in subtraction problems and whether they recognizeimportant numerical distinctions between problems (see Table 6.6 for items).

Table 6.6 – Integer Operations Property Questions.

Will these two problems give you the same answer?

4 + 5 3 − 1 6 − 4 5 − 8 3 + 2 9 − 6

vs. vs. vs. vs. vs. vs.

5 + 4 1 − 3 7 − 4 8 − 5 3 + 3 6 − 9

48 6 ⋅ methods

2-digit Subtraction

As mentioned previously, students o�en reverse numbers in subtraction.�e 2-digit subtraction problems probe whether students also reverse numer-als when the ones digit of a two-digit minuend is smaller than the ones digitof a two-digit subtrahend (i.e., 23 − 15 and 31 − 27). �e two problems werewritten vertically to make the place values more salient.

Integer Addition and Subtraction Problems

�e arithmetic questions consist of a subset of problems from the 32 prob-lem types for negative number addition and subtraction (see Appendix F)which, based on the pilot studies, appear to discriminate among students’di�erent schemas of integer quantities, order of integers, meanings of theminus sign, and de�nitions of addition and subtraction.

Other studies investigating students’ addition and subtraction strategiesused 25 - 28 questions (Siegler & Shrager, 1984); this study includes 26 arith-metic problems. �ey vary according to characteristics commonly exploredin relation to the di�culty of problems: size of addend or subtrahend, sumof the two numbers, di�erence of the two numbers, and whether the largeror smaller number is �rst (Siegler, 1987) and vary according to other hy-pothesized characteristics such as presence of one versus two negative signs,addition versus subtraction, whether the answer is positive or negative, andwhether the negative number is �rst or second (see Appendix G). Addends,minuends, and subtrahends in the problems are numbers from -9 through 9.�e number of questions per problem type is mainly based o� of the reliabil-ity of the questions as determined from the pilot studies. �ere is a balanceof questions in which the values of integers should help students solve theproblem and in which the directed magnitudes–thinking of problems as get-ting more or less negative or positive–should help students solve the problem.As shown in Table 6.7, the questions consist of the following problem typeswhere L > S > 0, and “L” stands for larger number, “S” stands for smallernumber, and “X” stands for any number.

In some cases, problems involve the same numerals but have a di�erentoperation or a di�erent combination of negative signs (e.g., -6+ -4 and -4+6)to help tease apart strategies students use to solve the problems.


Table 6.7 – Integer Arithmetic Question Types. On the pre-test and post-test, students answered26 integer arithmetic questions, of which there were 11 di�erent types.

Type (Example) # ofQ’s

Reasons for Using the Questions ValueSchema

S − L (3 − 9) 3 Helps determine if students understand that the commu-tative property doesn’t work for subtraction and if theyknow of negative numbers.

OrderedValue

-S − L (-5 − 9) 2 Helps determine students’ de�nition of subtractionwithin the negatives.

OrderedValue

-S + L (-1 + 8) 3 Helps determine if students ignore the negative, use it assubtraction, count in the negatives, or make the answernegative.

OrderedValue

-L + S (-9 + 2) 2 Helps determine if students use the negative as a minusand whether they switch it with the operator.

OrderedValue

-X − -X (-8 − -8) 2 While students can correctly answer these by ignoringthe negatives, this question helps determine if studentsbelieve in -0.

OperationsProperty(X−X = 0)

-L + -S (-7 + -1) 2 Helps determine if students switch the negative with theplus.

DirectedMagnitude

-L − -S (-8 − -5) 2 Helps determine if students ignore the negatives. DirectedMagnitude

-S − -L (-4 − -7) 3 Helps determine if students make the answer negative. DirectedMagnitude

L − -S (5 − -3) 3 Helps determine if they change the problem to addition,add the negative to the answer, or subtract twice.

DirectedMagnitude

S − -L (6 − -7) 2 Helps determine if they reverse numbers, change theproblem to addition, add the negative to the answer, orsubtract twice.

DirectedMagnitude

L + -S (5 + -2) 2 Presents a comparison with -S + L to see if order changestheir strategy.

OrderedValue/DirectedMagnitude

50 6 ⋅ methods

Table 6.8 – Post-test Only Directed Magnitude Questions. On the post-test, students answeredtwo new directed magnitude questions related to the language of more or less positive andnegative.

More Positive, More Negative

0

Put your �nger at zero. Move your �nger more positive 2 lines and drawa triangle.

Put your �nger at zero. Move your �nger more negative 4 lines and drawa square.

Less Positive, Less Negative

0

Put your �nger at zero. Move your �nger less positive 4 lines and draw atriangle.

Put your �nger at zero. Move your �nger less negative 2 lines and draw acircle.

post-test interviews

�e �nal stage of the study involved re-interviewing the students on thesame categories from the pre-test. Only the integer addition and subtractionproblems were identical to those on the pre-test, but each student receivedthese questions in a di�erent order than they did on the pre-test. �e othercategories contained the same questions as on the pre-test but with di�erentnumbers. In the case of �lling in the empty number line, students saw thesame horizontal number line with “1” labeled, and they also received a newvertical number line with “0” labeled. Finally, on the post-test, students com-pleted one additional category of questions: Positive, Negative (see Table 6.8for explanation). �is category was not part of the pre-test because the con-tent of the questions was only taught to two groups. While learning from thetest is always a risk in pre-/post-test designs, due to the quantity and vari-ety of questions, it is unlikely that students remembered the exact questionsbetween administrations. Additionally, since participants were never told ifthey correctly answered any of the pre-test problems, it is more likely thanany performance changes on the post-test were due to the instruction theyreceived between tests.

instructional treatment 51

Post-test administration followed the same protocol as the pre-test de-scribed previously (see Appendix H for both interview protocols), began theweek a�er the last day of instruction, and ended within two weeks of the lastlesson. Both interviewers took students individually from their classroomsbased on a new randomized order. Students within each instructional groupwere randomly assigned to be in the �rst third, second third, and �nal thirdof the post-test order to eliminate any time e�ects from treatment betweengroups.

instructional treatment

Creating Instructional Groups

During the instructional phase of the study, the students participated ineight lessons within one of three instructional groups. As a step toward creat-ing the groups, the researcher had teachers of the four �rst grade classroomsat Numerica Elementary School rate their students’ general mathematics per-formance according to “low = 1”, “medium = 3”, and “high = 5”. Studentsplaced in-between two categories formed two extra groups: “medium-low= 2” and “medium-high = 4”. Correlations between teachers’ indirect judg-ments of students’ mathematics performance and students’ performance onstandardized tests historically are moderate to strong (Hoge & Coladarci,1989); and, student performance on the pre-test interviews provided addi-tional information on students’ initial performance levels. Table 6.9 explainsthe rating criteria used to determine students’ pre-test levels. Refer back tothe Pre-test Interviews section for explanations of the questions.

Students were sorted �rst by teacher’s performance level placement andthen by pre-test performance level to create four main categories: 1) Ratedhigh by teacher, mid to high pre-test level; 2) Rated high by teacher, lowpre-test level; 3) Rated medium by teacher, mid to low pre-test level; and4) Rated low by teacher, low pre-test level. Within each category, studentswere randomly assigned to one of three instructional groups (to be describedlater); stratifying students before randomly assigning them ensured that eachgroup would have an equal mix of performance levels. Although students’performance on the integer arithmetic problems was not used to place stu-dents into levels, the resulting instructional groups had similar median andaverage correct scores. A�er random assignment, two students from thesame category switched groups to create a better balance of male and female

52 6 ⋅ methods

Table 6.9 – Pre-test Coding and Ranking for Stratification. Students were rated on three cate-gories. �eir ratings on the categories were averaged to determine students’ pre-test levels,which were used for strati�cation.

Order: Ordering number cards

Low = 1 Several numbers out of place (regardless of whether they are ordered as positive)

Medium Low = 2 Numbers ordered as positive

Medium = 3 One pair of neighboring numbers transposed in each question; Positive or nega-tive numbers ordered backwards; or one group of numbers ordered correctly andone ordered at level 1 or 2.

Medium High = 4 One group of numbers ordered correctly and one ordered at level 3

High = 5 Both groups ordered correctlyOrder: Fill in number line

Low = 1 Numbers are out of order or wrap around the number line

Medium Low = 2 Only uses positive numbers; all numbers are positive on both sides of zero

Medium = 3 Positive or negative numbers are written backwards; number line is missing zero

Medium High = 4 2 numbers interchanged

High = 5 Correct order with negatives and positivesOrdered Value: Compare integer values

Low = 1 No comparisons are correct or only the positive comparison is correct.

Medium Low = 2 One additional comparison correct in addition to positive comparison; 2 mixedcomparisons correct or 2 negative-only comparisons correct.

Medium = 3 All mixed comparisons are correct or all comparisons within or without the situa-tional context are correct.

Medium High = 4 All mixed comparisons are correct. At least one negative-only comparison is cor-rect.

High = 5 All comparisons are correct.Pre-Test Level Average each student’s values for the order questions and round down to the near-

est integer. Student’s overall pre-test performance level is determined by takingthe average of the order and value scores.


Table 6.10 – Student Demographics Between Instructional Groups. A�er randomization, theinstructional groups looked similar in terms of student characteristics.

Full Instruction Integer Op-erations

Integer Properties

Number of Males 10 10 10

Number of Females 10 11 10

% from Spanish-speaking Households 20% (4/20) 14% (3/21) 10% (2/20)

Students from class 1 7 6 5




(Teacher) High, (Pre) High 3 3 3

(Teacher) High, (Pre) Mid-Low 4 6 5

(Teacher) Medium, (Pre) Mid-Low 9 7 8

(Teacher) Low, (Pre) Low 4 5 4

Average Integer Arithmetic Score 4.25 3.90 4.15

(Median Score) (3.5) (3.0) (3.0)

students in the groups. Table 6.10 presents the breakdown of students in eachgroup.

Description of Instructional Groups and Lessons

An underlying assumption of this research is that students need knowl-edge of both integer properties and integer directed magnitudes to e�ectivelyreason about addition and subtraction problems with negatives. Addressinginteger properties involves helping students develop schemas for the order,identi�cation, and values of negative integers in comparison to positive in-tegers. Likewise, to understand directed magnitudes in relation to negativeintegers, students must develop a schema for subtracting a negative versussubtracting a positive–students also need to distinguish between adding anegative versus adding a positive.

In order to determine how these elements play a role in students’ devel-oping schemas, students were randomized into one of three instructionalgroups. �e students in Group 1, the Full Instruction Group, learned aboutboth integer properties and operations in an e�ort to develop all schemasinvolved in the central conceptual structure for integers. Students in Group 2,the Integer Operations Group, only learned about directed magnitudes and

54 6 ⋅ methods

the directed nature of adding and subtracting integers. �eir core lessonswere the same as those included in Group 1’s last �ve days of instruction,but they also reviewed and practiced the skills in more depth so that theyhad the same amount of instruction time as Group 1. Finally, Group 3, theInteger Properties Group, only learned about identifying negative numbers,their order, and their values. �eir lessons were the same as those included inGroup 1’s �rst three lessons but also included extra games and time to practiceso that they also had the same amount of instruction time as the other twogroups. Table 6.11 outlines the schedule of lessons among groups.


Table6.11

–ScheduleofLessons.

Sche

dule

ofle

sson

topi

csfo

rthe

thre

eins

truc

tiona

lgro

ups.�

eFul

lIns

truc

tion

grou

pha

ssom

eles

sons

that

arei

dent

ical

tole

sson

suse

dw

ithth

eInt

eger

Ope

ratio

nsgr

oup

and

som

etha

tare

iden

tical

tole

sson

suse

dw

ithth

eInt

eger

Prop

ertie

sgro

up.

Day

1D

ay2

Day

3D

ay4

Day

5D

ay6

Day

7D

ay8

Full

Instr

uctio

n(N=

20)

Disc

ussa

ndex

plor

esym

bols

Expl

orem

inus

sign

vers

usne

gativ

esig

n

Neg

ativ

eson

ave

rtic

alnu

mbe

rlin

e;Gam

eW

hich

isgr

eate

r?

Expl

orec

omm

utat

ivep

rope

rty

and

lack

ofco

mm

utat

ivity

fors

ubtr

actio

nco

mpa

red

toad

ditio

n

Mor

e(m

ove

right

)vs.

Less

(mov

ele�

)

Mor

epos

itive

=la

rger

,mor

ene

gativ

e=sm

alle

r

Less

posit

ive=

smal

ler,

less

nega

tive=

larg

er

Inte

ger

Ope

ratio

ns(N=

21)

Expl

orec

omm

utat

ivep

rope

rty

and

lack

ofco

mm

utat

ivity

fors

ubtr

actio

nco

mpa

red

toad

ditio

n,no

spec

i�cm

entio

nof

nega

tives

Mor

e(go

right

onN

L)ve

rsus

Less

(go

le�

onN

L)

Mor

epos

itive

=la

rger

,mor

ene

gativ

e=sm

alle

r

Less

posit

ive=

smal

ler,

less

nega

tive=

larg

er

Mor

epos

itive

,les

spos

itive

,mor

ene

gativ

e,le

ssne

gativ

e

Inte

ger

Prop

ertie

s(N=

20)

Disc

ussa

ndex

plor

esym

bols

Expl

orem

inus

sign

vers

usne

gativ

esig

n

Mat

chin

gne

gativ

enu

mbe

rsvs

.po

sitiv

enu

mbe

rs

Neg

ativ

eson

aver

tical

num

berl

ine;

Gam

eWhi

chis

grea

ter?

War

Whi

chin

tege

ris

grea

ter?

Gam

eGet

thre

eorf

ourc

onse

cutiv

ein

tege

rsin

arow

56 6 ⋅ methods

Description of Lesson Logistics

Rather than training the individual �rst grade teachers, I taught all instruc-tional lessons for all groups. To remain as impartial to the groups as possibleand maintain integrity between lessons, I followed strict lesson plans. If atopic or question arose in one group, I covered the same topic or questionwith the other groups as appropriate to their treatments. Additionally, alllessons were videotaped. At the beginning of a group’s lesson time, I took thestudents from their respective classes and brought them to the multi-purposeroom for instruction. Each instructional period lasted 45 minutes, with thelast 5–10 minutes reserved for a short worksheet to monitor their learning.Results from the worksheets are not presented here as they are beyond thescope of the dissertation. At the end of the lesson, I returned the students totheir respective classes and took the next group of students as indicated onthe schedule shown in Table 6.12. Overall, each group received eight lessons.�e groups met for instruction at di�erent times over the week and in

di�erent orders to control for e�ects of time of day on learning. On Mondaysand Tuesdays, Group 1 was the only group available during the �rst timeslot because four children from the other groups had a mandatory tutoringsession during those times. �ese restrictions made truly randomizing themeeting times impossible, but the groups received instruction �rst, second,and last two times each.


Table 6.12 – Instruction Schedule for the Three Instructional Groups. Group 1 = Full Instructiongroup, Group 2 = Integer Operations group, and Group 3 = Integer Properties group.

Monday,May 3rd

Tuesday,May 4th

�ursday,May 6th

Friday,May 7th

A) 8:40 – 9:25 Group 1 Group 1 Group 3

B) 9:35 – 10:20 Group 2 Group 3

C) 10:50 – 11:35 Group 2 Group 3 Group 1

D) 12:50 – 1:35 Group 3 Group 2 Group 1 Group 2Monday,May 10th

Tuesday,May 11th

�ursday,May 13th

Friday,May 14th

A) 8:40 – 9:25 Group 1 Group 1 Group 2

B) 9:35 – 10:20 Group 1 Group 2

C) 11:15 – 12:00 Group 3 Group 2 Group 3

D) 12:50 – 1:35 Group 2 Group 3 Group 3 Group 1

58 6 ⋅ methods

7 A NA LY S I S

research question 1

What are �rst grade students’ schemas of negative integers (in terms of the ele-ments underlying the central conceptual structure for integers: written symbols,order, value, and directed magnitude)?

A major goal of this study is to explore students’ schemas for integeridenti�cation, order, values, and directed magnitudes, and a �rst step inexploring students’ thinking was to transcribe their responses to the interviewquestions. Audio data were transcribed �rst, since the audio-recorder wasplaced close to the students during the interviews. Long pauses are indicatedby “...”, interviewer interjections and encouraging remarks such as “um hmm”or “yeah?” are placed inside brackets. I also compared the transcriptions fromthe audio to the videos to catch additional missed words, and I added notesregarding students’ pointing, use of �ngers, and other gestures or writing.

Based on the answers students provided on the pre-test, as well as informa-tion from previous research and pilot studies, I developed a set of codes foreach of the schema categories, which I organized in relation to the proposedinteger central conceptual structure. �e integer central conceptual structurediagram (refer to Figure 3.3) served as a framework for creating diagrams ofstudents’ possible schemas for the di�erent question types in the interviews.Each code in a schema category is represented by a series of connected cir-cles, where each circle represents part of a row (e.g., identifying numerals,knowing the order of numerals, etc.) in the conceptual structure diagram.Table 7.1 presents the notation.

59

60 7 ⋅ analysis

Table 7.1 – Notation for Students’ Schema Diagrams.�e following notations were used tomake schema diagrams, which characterize students’ solutions to the interview questionspictorially and are then ordered according to complexity.

Ip Positive integeridenti�cation

I0 Identi�cation of zero In Negative integeridenti�cation

Op Positive integer order O0 Order of zero beforepositive numbers

On Negative integer order

Vp Positive integer value V0 Value of zero Vn Negative integer value

Dp Directed Magnitudes:More positive, Lessnegative

Dn Directed Magnitudes:More negative, Lesspositive

Solid circle: Formalschema

Dotted-circle:Initial-syntheticschema

Solid line: Formalconnection

Dotted line:Developingconnection

Dotted arrow:Possible in�uence onschema

For example, a student who gives positive integer names to all integers(positive or not) is considered to be operating with a positive-only identi�ca-tion schema. A student who correctly names all positive and negative integersillustrates that negative and positive numbers are di�erent and demonstratesformal understanding of their names. �e two schemas are represented usingthe notation above as follows:

Ip IpIn

positive-only negative and positive identi�cation schemaidenti�cation schema (clear distinction between positive and negative);

zero not mentioned

Similar schema diagrams were created to capture the types of responsesstudents gave on the pretest for integer order, values, and directed values (seeAppendix I for complete list). Next, I ordered the schema diagrams basedon whether they represented initial, synthetic, or formal schemas and ontheir connection complexity. For example, two solid circles connected bya solid line is considered more complex than two dotted circles connectedby a solid line. Rather than drawing the corresponding diagram for eachstudent’s response, I labeled each diagram with a letter and used those lettersto code students’ responses in each schema category. A student who, on thepre-test, named all integers as positive would get an “A” for their pre-test

research question 2 61

Table 7.2 – Examples of Original Codes. Original coding categories with sample codes from thesecond pilot study.

Problem Set Up Codes Description

2Ignore Student ignores both negatives in the prob-lem.

NegOp Student uses the negative to replace the oper-ator when it follows the plus sign. (E.g., 4+-2,does 4 − 2).

Problem Strategy Codes Description

CAFZ Student counts away from zero.

Related Facts Student relates the problem to an “easier”one and compensates for the change ifneeded (E.g., 9 + 5 = 10 + 5 − 1).

Answer Strategy Codes Description

AN Student adds a negative or says that the an-swer needs to be negative.

integer naming schema. Finally, I calculated the frequencies of the codes inthe schema categories to illustrate the variety of schemas these �rst gradestudents have about negative numbers.

research question 2

How does the relation among students’ integer schemas manifest itself in stu-dents’ approaches to integer addition and subtraction problems?

�e set of codes used for analyzing students’ solutions to the arithmeticproblems originated from codes used in the second pilot study and wentthrough a couple rounds of revisions. �e original codes targeted three mainparts of students’ solutions: how they set up the problem or manipulated thesign, how they solved the problem, and how they modi�ed their answers.Table 7.2 shows a few examples of codes in each category (see Appendix Dfor full original list).

While the codes from the pilot study were varied, given the more diversedata set from this study, it was necessary to create more categories to dealwith situations where students used more than one code in a category. Forexample, when solving a problem some students both counted away fromzero and used a related fact, indicating a need for the counting strategies tobe separate from other fact strategies. Similarly, when setting up the problem,

62 7 ⋅ analysis

students could ignore one or more negatives, use a di�erent operation, ora combination of the two. Consequently, these codes were separated intothree categories. Additionally, because one of the goals of this study was toexplore how students’ language changes, new codes were added to addresstheir use of directional language (e.g., whether they say they are going “up”,“down”, “more positive”, etc.), operational language (use of “plus” or “minus”),identi�cation of negatives, and talk about negative numbers in general.

Coding for Reliability: Round 1

Students’ responses were coded on each of these categories for each of the26 arithmetic questions. If students changed their minds a�er providing ananswer, their original solution and subsequent solution were coded separately.�ree mathematics education Ph.D. students were trained to help with thecoding. During the initial training, I explained each code, a�er which thecoders coded the �rst question for student 1. A�er each question, we discussedtheir responses and resolved inconsistencies; as the head researcher, I madethe �nal call on any discrepancies. A�er this initial training, the coders codedthe answers for a second student on their own and sent their spreadsheetsto me for comparison. Additionally, the coders provided feedback on thewording of the codes; this feedback was used to clarify the wording and makeit more speci�c. Armed with the wording changes, I continued double codinga sample of the students with just one of the coders. Due to time constraints,the other two coders did not continue at this point.

For the �rst round of double coding, we independently coded the �rststudent’s answers, discussed the codes, and reconciled di�erences a�er eachquestion. Following this practice, we coded three additional students’ an-swers to all arithmetic problems. �is time we compared codes a�er eachstudent but even a�er discussing problematic coding categories, we were notable to get high enough reliability in our coding. Because students o�en usedsimilar reasoning across problems, if there was a coding discrepancy withina category, it was prevalent; whereas, some categories had few to no disagree-ments. Based on analysis of the areas where we were not reliable, I revisedthe codes again, making them more speci�c and less open to interpretation.

One area of particular di�culty involved interpreting what students weredoing with the negative signs and operations when solving a problem. Forexample, given the problem 5 + -2, a student who answers “3”, could beinterpreting the problem as “5 − 2” by ignoring the negative sign and usingthe opposite operation or by using the negative sign as a minus and ignoring


the plus sign. Additionally, the students could know that adding a negative isequivalent to subtracting.

While this granularity was desired, students’ responses were o�en vagueenough that two people could interpret their responses in more than one way.�erefore, the sign and operation codes were reworked to focus on the typeof answers students reached. A couple of categories focused on how studentstalk about negatives were infrequently used. �ese codes were removed and aspace was provided for coders to write notes about statements students madeabout negatives. Table 7.3 presents the �nal list of coding categories withexamples (see Appendix J for complete list of codes for each coding categoryand �nal coding protocol) as compared to the initial categories and codes.Categories or codes that stayed the same span all columns. In some cases, acategory or code stayed the same but its description became more speci�c.

64 7 ⋅ analysis

Table7.3

–C

odes

whi

chw

erer

evise

dfro

mth

eorig

inal

listf

orcla

ssify

ing

addi

tion

and

subt

ract

ion

solu

tions

.

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Orig

inal

How

dostu

dent

srea

d/ta

lkab

outt

heor

igin

alpr

oble

m?

Nam

efor

nega

tives

and

nega

tives

igns

How

dostu

dent

sref

erto

nega

tives

and

nega

tive

signs

?

Min

usSt

uden

tsta

tes/

read

sthe

prob

lem

and

says

the

nega

tives

igns

as“m

inus

”.M

inus

�es

tude

ntre

fers

tone

gativ

esig

nsas

“min

usor

take

away

sign.”

Ope

ratio

nLa

ngua

geW

hato

pera

tion(

s)do

stude

ntst

alk

abou

tusin

g?

Min

us/S

ubtr

act

�es

tude

ntta

lksa

bout

subt

ract

ing

orm

inus

ing

orre

ads/d

escr

ibes

thep

robl

emw

ith“m

inus

”,“s

ubtr

act”,

or“ta

keaw

ay”.

Ifstu

dent

start

sto

say

min

us(e

.g.,

“min

”)it

coun

ts.If

stude

ntss

ayth

eyar

e“ta

king

”anu

mbe

rbut

does

n’tsa

y“ta

keaw

ay”,

itco

unts

ifth

eans

wer

isco

nsist

entw

ithan

oper

atio

nan

dno

tmea

ning

“I’m

goin

gto

start

with

”or“

I’mgo

ing

tom

ovei

t”,et

c.

Dire

ctio

nW

hatd

irect

iona

llan

guag

edo

stude

ntsu

se?

Forw

ard

Stud

ents

say

they

arec

ount

ing

forw

ard

orth

eyar

egoi

ngfo

rwar

d.Fo

rwar

dSt

uden

tssa

yth

eyar

ecou

ntin

gfo

rwar

dor

they

areg

oing

forw

ard

orin

front

.

Con

tinue

don

next

page

...


Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Sign

SetU

pW

hatd

ostu

dent

sdo

with

nega

tives

igns

befo

reso

lvin

gpr

oble

ms?

Solu

tions

Inw

hich

solu

tion

cate

gory

dostu

dent

s’an

swer

sfa

ll?

Igno

re1s

tFo

rpro

blem

swith

two

nega

tives

,stu

dent

igno

res

the�

rst.

PosE

qui

Stud

entg

etsa

posit

ive/

zero

answ

erth

atw

ould

beex

pect

edif

thes

tude

ntig

nore

dal

loft

hene

gativ

esig

nsor

,in

thec

aseo

fpro

blem

slik

e4–

7,re

fuse

sto

oper

atei

nto

then

egat

ives

.

Ope

ratio

nSe

tUp

Wha

tdo

stude

ntsd

ow

ithth

eope

ratio

nsig

nsin

itial

ly?

Opp

PosE

qui

Stud

entg

etsa

posit

ive/

zero

answ

erth

atw

ould

beex

pect

edif

thes

tude

ntig

nore

dal

loft

hene

gativ

esig

ns(o

rkep

tnum

bers

posit

ive)

and

used

the

oppo

siteo

pera

tion.

Opp

Op

Stud

ents

tate

s/re

adsa

prob

lem

with

theo

ppos

iteop

erat

ion.

Opp

Neg

Mod

Stud

entg

etsa

nega

tivea

nsw

erth

atw

ould

beex

pect

edif

thes

tude

ntig

nore

dal

loft

hene

gativ

esig

ns,s

olve

dth

epro

blem

with

theo

ppos

iteop

erat

ion,

and

then

mad

ethe

answ

erne

gativ

e(or

mad

eall

num

bers

nega

tive�

rst).

(For

smal

ler

min

usla

rger

prob

lem

,stu

dent

solv

esas

ifad

ding

then

umbe

rsan

dm

akin

gth

emne

gativ

e.)

Con

tinue

don

next

page

...

66 7 ⋅ analysis

Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Sign

Op

SetU

pW

hats

ign

and

oper

atio

nco

mbi

natio

nstr

ateg

ies

dostu

dent

suse

?

AN

SSt

uden

tsta

test

hata

ddin

gan

egat

ivei

ssu

btra

ctin

gap

ositi

ve.

AN

MN

(AN

S)St

uden

tsta

test

hata

ddin

gan

egat

ivei

sgoi

ngm

oren

egat

ive(

toth

ele�

onth

enum

berl

ine,

tow

ard

nega

tiven

ettle

s)or

subt

ract

ing

(apo

sitiv

e).

Num

eral

SetU

pW

hatd

ostu

dent

sdo

tonu

mbe

rsbe

fore

solv

ing

prob

lem

s?N

umbe

rOrd

erW

hato

rder

ofnu

mbe

rsdo

stude

ntsu

se?

Reve

rse

Stud

enti

ndic

ates

that

they

switc

hed

theo

rder

ofth

enum

eral

s.�

eym

ayex

plic

itly

state

itor

show

itth

roug

hth

eirs

trat

egie

s.

Reve

rse(

impl

ied)

�ea

nsw

erex

plan

atio

nor

answ

erm

akes

sens

eif

thes

tude

ntth

inks

thep

robl

emis

equi

vale

ntto

onew

here

then

umbe

rsar

erev

erse

d(b

utw

ear

en’t

100%

sure

that

’sw

hatt

hestu

dent

did.

Adju

stN

umbe

rsW

hatd

ostu

dent

sdo

with

then

umbe

rsw

hen

solv

ing

prob

lem

s?Ad

just

Num

bers

How

dostu

dent

scha

nget

henu

mer

alsi

nth

epr

oble

m?

Con

tinue

don

next

page

...


Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Add1

0St

uden

tadd

sten

toan

initi

alnu

mbe

rto

avoi

dne

gativ

es(it

’slik

ethe

yar

ereg

roup

ing

toge

tmor

eon

es,e

xcep

tthe

yha

veno

thin

gto

regr

oup

from

.)E.

g.,f

or3

–7,

does

13–

7=

6.

Med

ium

Wha

tdo

stude

ntsu

seto

solv

ethe

prob

lem

?

Fing

ers

Stud

entu

ses�

nger

sto

solv

ethe

prob

lem

.Fi

nger

sSt

uden

tuse

s�ng

erst

oso

lvet

hepr

oble

man

dm

ayor

may

nota

lsoco

unto

utlo

ud.S

tude

ntdo

esno

tne

edto

usea

ll�n

gers

nece

ssar

yto

mod

elth

epr

oble

mor

may

also

just

look

atth

em.

Num

berL

ine

Wha

tare

stude

nts’

inte

rnal

num

bers

chem

as?

Num

bers

Wha

tnum

bers

dostu

dent

suse

whe

nso

lvin

gth

epr

oble

ms?

Opp

Dir

Stud

entu

sesa

num

berl

ines

chem

awith

nega

tives

orde

red

back

war

ds.

Neg

Opp

Dir

Stud

entc

ount

sint

oth

eneg

ativ

esbu

tthe

nega

tives

arei

nba

ckw

ards

orde

rort

heon

lyw

ayth

eyco

uld

have

gotte

nth

eneg

ativ

eans

wer

isif

they

thou

ghtt

hene

gativ

esw

erei

nba

ckw

ards

orde

r.E.

g.,(

3–

9),g

ets-

4.

Con

tinue

don

next

page

...

68 7 ⋅ analysis

Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Cou

ntin

gH

owdo

stude

ntsc

ount

?

CAFZ

Stud

entc

ount

saw

ayfro

mze

roin

eith

erpo

sitiv

eor

nega

tived

irect

ion.

Whe

nC

ount

isPr

esen

tW

hen

dostu

dent

scou

nt?

Befo

reYo

uca

nte

llba

sed

onstu

dent

s’us

eof�

nger

s,m

ovin

gon

anum

berl

ine,

draw

ing

circ

les,

verb

aliz

ing

,etc.

that

thes

tude

ntis

coun

ting

befo

rew

ritin

gth

eira

nsw

erO

Rbe

fore

the

inte

rvie

wer

asks

him

how

hego

tthe

answ

er.

Cou

ntEr

ror

Wha

terr

orsd

ostu

dent

smak

ewhi

leco

untin

g?

CIN

Stud

ents

tart

scou

ntw

ithin

itial

num

ber(

oran

swer

isco

nsist

entw

ithth

is)(e

.g.5

+3

does

“5,

6,7.”

)

Con

tinue

don

next

page

...


Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Fina

lOpe

ratio

nA

restu

dent

sadd

ing

orsu

btra

ctin

gap

ositi

veor

nega

tiven

umbe

r?Fi

nalA

nsw

erH

owdo

esth

eval

ueof

thea

nsw

erco

mpa

reto

the

initi

alnu

mbe

rasd

eter

min

edby

stude

nt?

AddN

egSt

uden

tadd

sane

gativ

enum

ber.

Gre

ater

Stud

ents’

answ

eris

grea

tert

han

thea

ctua

lval

ueof

then

umbe

rthe

stude

ntsta

rted

with

.

Add/

Sub

Sche

ma

Wha

tare

stude

nts’

addi

tion

orsu

btra

ctio

nsc

hem

as?

Larg

erAV

Stud

entg

etsa

larg

erab

solu

teva

lue.

Sour

ceH

owdo

stude

ntsr

easo

nab

outg

ettin

gth

eir

answ

ers?

Sour

ceA

sidef

rom

coun

ting

and

fact

strat

egie

s,w

here

are

stude

nts’

answ

erco

min

gfro

m?

Less

onSt

uden

tref

erst

oon

eoft

hele

sson

s.Le

sson

Stud

entr

efer

sto

oneo

fthe

expe

rimen

tall

esso

ns.

Ans

wer

Man

ip.

How

dostu

dent

sman

ipul

atet

heir

answ

ers?

Mism

atch

PNSt

uden

tsay

sthe

answ

eras

posit

iveb

utw

rites

itas

nega

tive.

Con

tinue

don

next

page

...

70 7 ⋅ analysis

Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

Fact

/Rec

all

Stra

tegy

Wha

tmat

hfa

ctso

rnum

berr

elat

ions

dostu

dent

sus

e?Fa

ct/R

ecal

lStr

ateg

yW

hats

trat

egie

sdo

stude

ntsu

seto

rem

embe

rthe

answ

er?

Ana

lyze

Stud

enta

naly

zest

hepr

oble

min

term

sof

simila

ritie

sto

anot

herp

robl

em,l

ikeo

new

ithou

tne

gativ

es,(

E.g.

,-4

+5

is-9

beca

use4

+5

=9)

.

Sim

ile1)

Stud

enta

naly

zest

hepr

oble

min

term

sof

simila

ritie

sto

thes

amep

robl

emw

ithou

tne

gativ

es,(

E.g.

,-4

+5

=-9

beca

use4

+5

=9.�

ere

vers

eisa

lsopo

ssib

le:S

tude

ntac

know

ledg

esne

gativ

esin

theo

rigin

alpr

oble

mbu

tsay

sit’s

just

liket

hepo

sitiv

epro

blem

.“N

egat

ivef

ourp

lus�

veis

like4

+5

=9”

.

2)St

uden

tind

icat

esth

atit

will

bene

gativ

ebe

caus

ethe

num

bers

aren

egat

ive(

impl

ying

that

they

�rst

solv

edth

epro

blem

and

then

adde

da

nega

tive)

.Stu

dent

smig

htta

lkab

outt

hepr

oble

mas

posit

iveb

utth

engi

vean

egat

ivea

nsw

er.

3)St

uden

tana

lyze

sthe

prob

lem

inte

rmso

fone

whi

chin

volv

esas

imila

rstr

ateg

y(7

–1

islik

e5–

1,yo

uju

stpu

tdow

non

e�ng

er).

Con

tinue

don

next

page

...


Tabl

e7.3

–C

ontin

ued

Initi

alCa

tego

riesa

ndC

odes

:Des

crip

tions

and

Exam

ples

Revi

sed

Cate

gorie

sand

Cod

es:D

escr

iptio

nsan

dEx

ampl

e

*Com

parin

gto

apro

blem

with

thes

amen

umbe

rs,

sam

eord

er,a

ndop

posit

eope

ratio

nco

unts

ifstu

dent

gets

theo

ppos

itean

swer

.(E.

g.,4

+5

=9

so4

–5

=-9

)

Neg

Sche

ma

Wha

tsta

tem

ents

dostu

dent

smak

eabo

utne

gativ

enum

bers

?

Neg

Zero

Says

nega

tives

arew

orth

zero

.

OpS

chem

aW

hata

restu

dent

s’sc

hem

asof

oper

atio

ns?

SML

Stud

ents

aysy

ouca

n’tso

lves

mal

lerm

inus

larg

erpr

oble

ms.

72 7 ⋅ analysis

Coding for Reliability: Round 2

Using the new coding scheme, a second coder and I double-coded a newsample of questions. Since the �rst round of coding revealed that coding all26 questions for one student led to a lot of redundancy in student responses,we chose to sample from more students for this round of coding. Ten students’tests (5 pre-tests and 5 post-tests) were randomly selected to be coded. Foreach test, a sequential group of �ve questions was randomly selected. In somecases, students provided two answers for a question, in which case an extraquestion was coded for that student, resulting in a total of 53 questions. Ratherthan coding one question using all categories before moving on to the nextquestion, we started with one coding category and used it to code all questionsin the sample before moving on to the next category. �is meant that we couldconcentrate on one aspect of the student’s response without getting distractedby the other coding categories. Prior to coding each category, we discussedwhat the codes meant. �en we coded the entire sample of 53 questions,comparing a�er each group of questions per student.

A�er coding for each category, we had over 96% agreement in our codesfor 11 out of the 15 categories and 90% agreement in one additional category.For the other three categories, number order, facts/recall, and source, agree-ment was still high (88.7%, 88.7%, and 86.5% respectively). For the numberorder category, the coder did not realize that a student could be scored asimplicitly reversing the numbers just based on the answer. For example, astudent who solves 6 − 8 = 2 and says, “I did six minus eight” could not havesolved the problem in this order and still get 2, so it is coded as reverse (im-plicit). �is clari�cation was emphasized in the coding document to serve asa reminder. A�er discussing the other two categories (facts/recall and source),the researcher adjusted the wording to the fact/recall codes and chose to theremove codes from the source category related to whether students “knew”the answer. We considered that these codes were too subjective as writtenand were not capturing important information.

Using a random sample of four more students, we coded for these threecategories once again. We had 100% agreement on number order and sourcecategories. However, we only had 80% agreement on the fact/recall category.�rough discussion of what the codes should be capturing, we realized thatthe wording was still not clearly encapsulating these ideas. I once againadjusted the wording of the codes and provided a list of examples for theproblematic codes in the fact/recall category to guide coding decisions. Whenusing these new descriptions for the �rst time, we improved our reliability


slightly (84% agreement on thirteen more items) and were reliable by thesecond time (100% agreement on an additional twelve items).

At this point, I coded the remaining questions for all students, markingany questionable codes. Coding each student for all 15 categories meant I readeach student’s response over 10 times, which provided several opportunitiesfor me to catch any coding mistakes. A�er completing the coding, I met withthe second coder once again to go over the questionable items. �e codersuggested how she would code each item and, in all cases, her codes matchedmine.

Students’ solutions to arithmetic problems rely on schemas of numberorder and value. When factoring in the use of negative signs to the equation,the possible ways students solve the problems increases tremendously. Dueto the wide variety of schemas present in the interviews, it was not feasibleto describe all the possible arithmetic schemas; rather, I chose to highlightstudents who presented interesting cases and comparisons. For each student,I calculated frequencies of codes (such as identifying negative numbers)–for areas related to the students’ integer schemas of particular interest–toillustrate the students’ general patterns of solving the arithmetic problems.

research question 3

How does instruction in various elements of the central conceptual structurefor integers in�uence students’ understanding of negatives as measured throughtheir changing schemas, language, arithmetic accuracy, and approaches to inte-ger arithmetic problems? To what extent is students’ math performance level afactor in this change?

Quantitative Analysis

Since students’ accuracy on the interview questions provides a more gen-eral picture of how formalized students’ integer schemas are, accuracy isdiscussed �rst. For both pre-test and post-test, each student received a set offour scores: total number correct (and total percent correct), percent correcton the integer property items, percent correct on the operation properties anddirected magnitude items, and percent correct on the arithmetic items. Sincethe integer properties items consisted of integer identi�cation, integer order,and integer value questions, students’ percentages correct for each categorywere calculated, and their overall percent correct was taken as the average

74 7 ⋅ analysis

of those subsets. Di�erence scores were calculated by subtracting students’scores on the pre-test from their scores on the post-test.

�e following hypotheses concern the four types of scores: a) total test, b)arithmetic items, c) operations properties and direct magnitude items, andd) integer properties items.Hypothesis 1H0 – �ere is no di�erence in mean gain scores among the three instruc-

tional groups.H1 – �ere is a di�erence in mean gain scores among the three instructional

groups.Hypothesis 2H0 – �ere is no di�erence in mean gain scores among the three perfor-

mance levels.H1 – �ere is a di�erence in mean gain scores among the three performance

levels.Hypothesis 3H0 – �ere is no di�erence in mean gain scores among the three perfor-

mance levels within instructional groups.H1 – �ere is a di�erence in mean gain scores among the three performance

levels within instructional groups.To determine if there were signi�cant e�ects of instruction, performance

level, or a combination of both on the di�erence scores, I ran a 3 x 3 (instruc-tional group by performance level indicated by teacher) Factorial ANOVA oneach set of di�erence scores using SPSS 19. Post-hoc (She�é) tests providedpairwise comparisons of the mean gain scores of the instructional groups orperformance levels. Complex comparisons were done by hand, also usingShe�é tests. In cases where the interaction between instruction and perfor-mance level was signi�cant, I ran a one-way ANOVA on the interaction termand conducted comparisons by hand to compare the mean gain scores forperformance groups within and/or between instructional groups. Sche�é’swas chosen because it can handle uneven cell-sizes, multiple comparisons,and is conservative (Shavelson, 1996).�e ANOVA relies on three assumptions: 1) independence, 2) normality,

and 3) homogeneity of variance.

Independence

Each student was randomly assigned to one instructional group. Studentsin the other groups were in their own classrooms and did not see what the


third group was doing when they had their instruction. While students couldgo back to their classrooms and talk with their classmates, teacher reportsindicate that students would remark that they had fun or played games. It isunlikely that students provided details of what they did in a manner su�cientto in�uence other students.

Normality

Values for the Kolmogorov-Smirnov (K-S) test above 0.05 (or .2 for a morestringent test) indicate that pre- to post-test di�erence values are normallydistributed within a group. In this study, all but a few of the K-S valuesindicate that the di�erence scores are normally distributed. Some of theexceptions include instances where the sample sizes of groups are too low tocalculate the statistic or for di�erence scores on the operation properties anddirected magnitude items in instructional groups, but ANOVA is robust tothis assumption (see Appendix K for all K-S values as well as histograms andqq-plots).

Homogeneity of Variance

Levene’s test of equality of error variance was not signi�cant for any of theANOVA tests (values greater than 0.05). In each case we fail to the reject thenull hypothesis and conclude that there is homogeneity of variances, so thisassumption is met (see Appendix L for values).

Qualitative

As was done for the pre-tests, I coded students’ responses in each schemacategory on the post-tests using the letters assigned to the schema categorydiagrams. �en, students’ pre- and post-test schema codes were alignedto highlight transitions. For example, a student who named all integers aspositive on the pre-test was assigned an “A” for the pre-test integer namingschema. If the student named all integers correctly on the post-test, he orshe would be assigned an “F” for the post-test integer naming schema. �eirtransition is shown by the two letters AF.

Within group totals for each transition were calculated to highlight pat-terns in how students’ thinking shi�ed from pre- to post-test among thegroups. �ese were calculated for the integer identi�cation, integer order,and integer value tasks. Finally, I compiled tables for each group and eachtask showing the percentage of students in each group who started (at pre-test) with a particular schema, the percentage of students in each group who

76 7 ⋅ analysis

ended (at post-test) with a particular schema, and the percentage gains foreach schema level within groups.

Because students’ integer schemas were so varied, rather than reportingon all possible arithmetic solutions and approaches within and between in-structional groups, I selected two students from the same instructional group.A�er aligning their arithmetic codes from pre- and post-tests, I compared thefrequency of codes in each area to look for di�erences in their responses. Us-ing these codes and excerpts from the transcripts, I describe how the students’schemas and arithmetic responses compare before and a�er instruction.

To illustrate how students’ schemas interacted when they solved the prob-lems, I chose students (from each instructional group) who showed a speci�cstarting point and/or transition in their thinking from pre- to post-test. �etransitions are described through descriptions of their general solution pat-terns and excerpts from the interviews. By comparing and contrasting thefrequency of select codes from the arithmetic problems, I highlight somesuggestive di�erences between the groups as well as within the groups.

8 R E S U LT S – P R O B L E M A C C U R A C Y

As suggested previously, the accuracy scores provide a general overviewof how students’ schemas changed (or did not change) from pre- to post-test,so these results are presented before speci�cs on students’ schema shi�s.

total scores : gains from pre- to post-test

Results of the factorial ANOVA on di�erence scores indicate signi�cante�ects (at the .05 level) for the main e�ect of instruction (F=8.82, signi�cance=.001), the main e�ect of performance level (F=5.03, signi�cance = .010), andthe interaction e�ect of instruction by performance level (F=2.74, signi�cance= .038) (see Table 8.1). Based on the omega-squared values for the strengthof association, the proportion of variance explained by performance levelsand the interaction between instruction and performance level is medium(8.7% and 7.5% respectively), and the proportion of variance explained bythe type of instruction is high (16.8%) (Cohen, 1988; Dimitrov, 2008); it isworthwhile to look at these results further to better understand the e�ects.

Table 8.1 – Total Test ANOVA Results. Results of 3 x 3 (Instruction X Performance level) fac-torial ANOVA run on total test di�erence scores (Post-test − Pre-test). R2

= .434 (AdjustedR2

= .347)

Tests of Between-Subjects E�ects

Source Type IIISum ofSquares

df MeanSquare

F Sig.

Instruction 960.65 2 480.32 8.82 .001

PerformLevel 547.88 2 273.94 5.03 .010

Instruction × PerformLevel 596.15 4 149.04 2.74 .038

Error (Within) 2831.03 52 54.44

Corrected Total 5001.77 60

77

78 8 ⋅ results – problem accuracy

Table 8.2 – Scheffé Tests for Instructional Groups. Sche�é tests for multiple comparisons fortotal score mean di�erences between pre- and post-tests for the instructional groups. Basedon observed means. �e error term is Mean Square(Error) = 54.443. �e mean di�erence issigni�cant at the .05 level

95% Conf. Int.

(I) Instruction (J) Instruction MeanDi�. I-J

Std.Error

Sig. LowerBound

UpperBound

Full Instruction Operations 6.45∗ 2.31 .026 .64 12.26

Integer Properties −3.25 2.33 .386 −9.13 2.63

Operations Full Instruction −6.45∗ 2.31 .026 −12.26 −.64

Integer Properties −9.70∗ 2.31 .000 −15.51 3.89

Integer Properties Full Instruction 3.25 2.33 .386 −2.63 9.13

Operations 9.70∗ 2.31 .000 3.89 15.51

Table 8.3 –Mean Total Scores for Instructional Groups.Mean scores and standard deviations forthe total pre-test, post-test, and di�erence scores in each instructional group.

Group Pre-Test,Mean (SD)

Post-Test,Mean (SD)

Di�erence,Mean (SD)

Full Instruction (n = 20) 21.70 (9.59) 37.20 (11.81) 15.50 (9.98)

Integer Properties (n = 20) 21.75 (8.55) 40.50 (6.59) 18.75 (6.70)

Integer Operations (n = 21) 20.86 (10.65) 29.90 (14.71) 9.05 (7.92)

Total Di�erence Score, E�ect of Instruction

Overall, the instructional groups averaged a 14.4 point gain from pre- topost-test. According to pair-wise Sche�é’s tests (see Table 8.2), on average, stu-dents in the Integer Properties and Full Instruction groups had signi�cantlyhigher gains on the interview items than students in the Integer Operationsgroup. Students who had the opportunity to learn about integer properties foreither part or all of their instructional time made greater gains overall than stu-dents who did not receive this instruction. Table 8.3 illustrates each group’smeans and standard deviations for the pre-test, post-test, and di�erencescores. Furthermore, on average, students in the Integer Properties group,who had extended experiences with these concepts, had signi�cantly highergains than the average gains of the other two groups, who received some tono exposure to integer values and order (tobserved = 3.22 > tcritical = 2.52).

Total Di�erence Score, E�ect of Performance Level

Performance groups on average gained 13.4 points on the post-test. Re-sults of the Sche�é test for multiple comparisons (see Table 8.4) indicate

total scores : gains from pre- to post-test 79

Table 8.4 – Scheffé Tests for Performance Groups. Sche�é tests for multiple comparisons fortotal score mean di�erences between pre- and post-tests for the performance groups. Basedon observed means. �e error term is Mean Square(Error) = 54.443. �e mean di�erence issigni�cant at the .05 level.

95% Conf. Int.

(I) PerformLevel (J) PerformLevel MeanDi�. I-J

Std.Error

Sig. LowerBound

UpperBound

Low Medium −8.60∗ 2.54 .006 −15.01 −2.20

High −6.94∗ 2.54 .031 −13.34 −.53

Medium Low 8.60∗ 2.54 .006 2.20 15.01

High 1.67 2.13 .738 −3.70 7.03

High Low 6.94∗ 2.54 .031 .53 13.4

Medium −1.67 2.13 .738 −7.03 3.70

Table 8.5 –Mean Total Scores for Performance Groups.Mean scores and standard deviations forthe total pre-test, post-test, and di�erence scores for each performance group.


Post-Test,Mean (SD)


High Perform. (n = 24) 28.25 (11.35) 43.42 (9.90) 15.17 (7.85)

Medium Perform. (n = 24) 17.58 (4.31) 34.42 (10.94) 16.83 (9.79)

Low Perform. (n = 13) 15.92 (3.66) 24.15 (8.17) 8.23 (7.79)

that on average high and medium performance group students had signif-icantly higher gains than students whose teachers considered their mathperformance to be low. Although high math performers started out withhigher average scores on the pre-test (X̄high = 28.25 versus X̄medium = 17.58),the medium math performers made similar gains as the high performersdid on average. On the contrary, while the medium and low math perform-ers started out with similar average pre-test scores (X̄medium = 17.58 versusX̄low = 15.92), the medium performance group’s average gains were twicethat of the low performance group’s average gains. Table 8.5 illustrates themean di�erence scores for each group.

Total Di�erence Score, E�ect of Instruction by Performance Level

As previously stated, there were statistically signi�cant di�erences in gainsbetween instructional groups and between performance levels. However, thedi�erence in gains for students at all performance levels within the instruc-tional groups was not signi�cantly di�erent. In other words, low performingstudents in the Integer Properties group showed similar gains to the medium


and high performing students in the same instructional group. �e same isnot true for performance levels between instructional groups.

High performing students performed well across all instructional groups,but medium performance students in the Integer Properties group signi�-cantly improved compared to the medium performance students in the Op-erations group (tobserved = 4.21 > tcritical = 4.12). Although the averagegains for medium performers in the Full Instruction group were not signif-icantly di�erent from those in the Operations group, medium performerswho learned about integer properties to any extent signi�cantly improvedcompared to medium performance students who did not get this instruction(tobserved = 4.45 > tcritical = 4.12).

Figure 8.1 –Mean Difference Scoresfor Instructional Groups by Perfor-mance Level.

�e plot of performance level group average gains within each instruc-tional group (see Figure 8.1) suggests that students classi�ed as low perform-ing in the Integer Properties group improved compared to the low studentsin the other two groups. However, this di�erence was not signi�cant. Sincestudents classi�ed as low performers and medium performers had similarscores on the pre-test, the average gains of these combined groups were ex-plored. Non-high performers in the Integer Properties group improved sig-ni�cantly compared to the non-high performers in the Operations group(tobserved = 4.37 > tcritical = 4.12) (see Table 8.6 for means and standarddeviations for each group).

arithmetic scores : gains from pre- to post-test

Results of the factorial ANOVA on gains scores for just the arithmeticportion of the test were not signi�cant. On average, students in all threeinstructional groups had no di�erence in gains (average gain=8.3%), andstudents at all three performance levels had no di�erence in gains (averagegain=7.5%).

Even though the between group di�erences were not signi�cant for thearithmetic questions, students’ responses overall show an interesting patternfrom pre- to post-test. On the pre-test, students had the highest performanceon the questions they could get correct even if they ignored the negative signs.For example, students answered -4 − -7 correctly by ignoring the negativesigns and solving 7 − 4 = 3. Table 8.7 lists these questions and the percentageof all students who answered them correctly.

Students continued to do well on -5 − -5 and -8 − -8 on the post-test,but performance on the other questions dropped. Instead, on the post-test,

arithmetic scores : gains from pre- to post-test 81

Table 8.6 –Mean Total Scores for Performance by Instructional Groups.Mean scores and stan-dard deviations for the total pre-test, post-test, and di�erence scores for each performancelevel in each instructional group.

Group Pre-Test, Mean(SD)

Post-Test, Mean(SD)

Di�erence, Mean(SD)

Full Instruction

High Perform. (n = 7) 29.71 (12.53) 46.0 (4.90) 16.29 (9.83)

Medium Perform. (n = 9) 18.33 (3.08) 38.22 (9.22) 19.89 (8.81)

Low Perform. (n = 4) 15.25 (1.89) 19.50 (2.65) 4.25 (1.71)

Integer Properties

High Perform. (n = 8) 29.13 (8.56) 45.38 (5.7) 16.25 (6.23)

Medium Perform. (n = 8) 16.00 (2.45) 38.50 (5.2) 22.50 (4.90)

Low Perform. (n = 4) 18.50 (5.57) 34.75 (4.1) 16.25 (8.66)

Integer Operations

High Perform. (n = 9) 26.33 (13.47) 39.67 (14.48) 13.33 (8.05)

Medium Perform. (n = 7) 18.43 (6.83) 24.86 (12.77) 6.43 (7.28)

Low Perform. (n = 5) 14.40 (2.07) 19.40 (4.51) 5.00 (5.52)

Table 8.7 – “Easiest” Pre-test Questions. Questions with highest percentages correct on thepre-test.

Pre-Test Question -5 − -5 -8 − -8 -2 − -6 -4 − -7 -6 − -9

% of students whoanswered correctly(N = 61)

72.1% 68.9% 34.4% 29.5% 23.0%


Table 8.8 – “Easiest” Post-test Questions. Questions with highest percentages correct and high-est gains on the post-test.

Post-Test Question -7 + -1 -6 + -4 -4 − -3 -8 − -5

% of students whoanswered correctly(n = 61)

45.9% 41.0% 41.0% 32.8%

% of students whogained (n = 61)

32.8% 27.9% 32.8% 24.6%

students did better on questions that they could get correct if they solvedthe problem as positive and made those answers negative; these questionsalso showed the largest gains in terms of number of students who answeredthem correctly. Table 8.8 lists these questions, the percentage of students whoanswered them correctly on the post-test, and the percentage of students whogained on the questions from pre-test to post-test.

operation property scores : gains from pre- to post-test

Results of the factorial ANOVA on gain scores for items that measureunderstanding of operations properties and directed magnitude were notsigni�cant. On average, students in all three instruction groups had no dif-ference in gains (average gain=11.1%), and students at all three performancelevels had no di�erence in gains (average gain = 9.8%).

integer properties scores : gains from pre- to post-test

Results of the factorial ANOVA on gains scores for the integer propertyitems indicate signi�cant e�ects (at the .05 level) for the main e�ect of in-struction (F=12.76, signi�cance =.000), the main e�ect of performance level(F=5.00, signi�cance = .010), and the interaction e�ect of instruction byperformance level (F=2.61, signi�cance = .046) (see Table 8.9). Based onthe omega-squared values for the strength of association, the proportion ofvariance explained by performance levels and the interaction between in-struction and performance levels is medium (7.7% and 6.2% respectively),and the proportion of variance explained by the type of instruction is high(22.8%); it is worthwhile to look at these results further to better understandthe e�ects.

integer properties scores : gains from pre- to post-test 83

Table 8.9 – Integer Property Questions ANOVA Results. Results of 3 x 3 (Instruction X Perfor-mance level) factorial ANOVA run on Integer Property di�erence scores (Post-test − Pre-test).R2

= .491 (Adjusted R2= .412)

Tests of Between-Subjects E�ects

Source Type IIISum ofSquares

df MeanSquare

F Sig.

Instruction 11662.15 2 5831.03 12.56 .000

PerformLevel 4567.05 2 2283.54 5.00 .010

Instruction × PerformLevel 4769.90 4 1192.44 2.61 .046

Error (Within) 23770.75 52 457.10

Corrected Total 46681.67 60

Table 8.10 – Integer Property Items - Scheffé Tests for Instructional Groups. Sche�é tests formultiple comparisons for total score mean di�erences between pre- and post-tests for theinstructional groups. Based on observed means. �e error term is Mean Square(Error) =457.130. �e mean di�erence is signi�cant at the .05 level

95% Conf. Int.


Std.Error

Sig. LowerBound

UpperBound

Full Instruction Operations 26.16∗ 6.68 .001 9.32 42.99

Integer Properties −8.73 6.76 .440 −25.77 8.30

Operations Full Instruction −26.16∗ 6.68 .001 −42.99 −9.32

Integer Properties −34.89∗ 6.68 .000 −51.72 −18.06

Integer Properties Full Instruction 8.73 6.76 .440 −8.30 25.77

Operations 34.89∗ 6.68 .000 18.06 51.72

Integer Properties Di�erence Scores, E�ect of Instruction

On average, the instructional group improved by 35% from pre- to post-test. According to pair-wise Sche�é tests, students in the Integer Propertiesand Full Instruction groups had signi�cantly higher percentage gains on theinteger property items than students in the Operations group (see Table 8.10).Students who had the opportunity to learn about integer properties eitherfor part or all of their instructional time made greater gains on these typesof items. �e e�ect was not just due to the passage of time or taking the testagain because students in the Operations instructional group did not showsimilar gains.

Table 8.11 illustrates the mean di�erence scores for each group. Further-more, on average, students in the Integer Properties group, who had extended


Table 8.11 – Integer Property Scores for Instructional Groups.Mean percent correct scores andstandard deviations for the total pre-test, post-test, and di�erence scores in each instructionalgroup.


Post-Test,Mean (SD)


Full Instruction (n = 20) 25.80 (22.52) 66.91 (28.16) 41.10 (26.35)

Integer Properties (n = 20) 28.93 (27.26) 79.76 (17.05) 49.84 (24.61)

Integer Operations (n = 21) 27.54 (28.14) 42.49 (33.24) 14.95 (20.49)

experiences with these concepts, had signi�cantly higher gains than the av-erage gains of the other two groups, who received some to no exposure tointeger properties (tobserved = 3.74 > tcritical = 2.52). Extended experienceswith identifying and determining the order and value of negative numbersbene�ted these �rst graders more than having little to no instruction. Stu-dents in the Integer Properties instructional group averaged 79% correct onthe integer property items on the post-test.

Integer Properties Di�erence Score, E�ect of Performance Level

Across performance levels, students’ percentage correct scores improved32.4% on average. Results of the Sche�é test for multiple comparisons indi-cate that on average medium performance group students had signi�cantlyhigher percentage gains than students whose teachers considered their mathperformance to be low (see Table 8.12). While the medium and low mathperformers started out with similar average percent correct scores on the pre-test (X̄med ium = 16.17% versus X̄ l ow = 14.50%), the medium performancegroup’s gains were just over twice that of the low performance group gains.Table 8.13 illustrates the mean percentage di�erence scores for each group.Although the high performance group’s percentage gains were not signi�cantcompared to the low performance group, the high performers averaged 46%correct on the pre-tests and had a higher average percentage correct on thepost-test than the medium performers.

Integer Properties Di�erence Scores, E�ect of Instruction by Performance Level

Similar to the whole test results, there were no statistically signi�cantgains for performance levels within instructional groups. However, therewere signi�cant gains for performance groups between instructional groups.Once again, medium performance students in the Integer Properties groupsigni�cantly improved compared to the medium performance students in theOperations group (tobserved = 4.99 > tcritical = 4.12). Although the average

integer properties scores : gains from pre- to post-test 85

Table 8.12 – Integer Property Items - Scheffé Tests for Performance Level Groups. Sche�é testsfor multiple comparisons for mean percent correct di�erences between pre- and post-tests forthe performance level groups on integer property items. Based on observed means. �e errorterm is Mean Square(Error) = 457.130. �e mean di�erence is signi�cant at the .05 level.

95% Conf. Int.


Std.Error

Sig. LowerBound

UpperBound

Low Medium −25.81∗ 7.36 .004 −44.37 −7.26

High −15.76 7.36 .111 −34.31 2.80

Medium Low 25.81∗ 7.36 .004 7.26 44.37

High 10.06 6.17 .274 −5.49 25.61

High Low 15.76 7.36 .111 −2.80 34.31

Medium −10.06 6.17 .274 −25.61 5.49

Table 8.13 – Integer Property Scores for Performance Level Groups.Mean percent correct scoresand standard deviations for the integer properties items for pre-test, post-test, and di�erencescores for each performance level group.


Post-Test,Mean (SD)


High Perform. (n = 24) 45.69 (32.72) 80.05 (25.05) 34.36 (28.72)

Medium Perform (n = 24) 16.17 (8.75) 60.59 (29.16) 44.42 (27.75)

Low Perform (n = 13) 14.50 (4.12) 33.12 (18.43) 18.61 (19.08)


gains for medium performers in the Full Instruction group were not signif-icantly di�erent than those in the Operations group, medium performerswho learned about integer properties to any extent signi�cantly improvedcompared to medium performance students who did not get this instruction(tobserved = 4.77 > tcritical = 4.12).�e plot of performance level group average gains within each instruc-

tional group (see Figure 8.2) suggests that students classi�ed as low perform-ing in the Integer Properties group improved compared to the low studentsin the other two groups. However, this di�erence was not signi�cant. Sincestudents classi�ed as low performers and medium performers had similarscores on the pretest, the average gains of these combined groups were ex-plored. Non-high performers in the Integer Properties group improved sig-ni�cantly compared to the non-high performers in the Operations group(tobserved = 4.82 > tcritical = 4.12) (see Table 8.14 for means and standarddeviations for each group).

Figure 8.2 –Mean Percentage Differ-ence Scores for Instructional Groupsby Performance Levels on IntegerProperty Items.

discussion of quantitative results

�e quantitative results suggest that the groups who learned about negativesigns, integer order, and values had a shi� in their schemas for these topics.Additionally, learning about integer properties for eight days was more bene�-cial on average than having some to no instruction on these topics. �is latterdi�erence is largely due to the gain of students considered low-performers inthe Integer Properties instructional group. Since the lower performing stu-dents had lower number sense in general, they probably bene�ted from theextended practice. Because there were no statistically signi�cant di�erencesin percentage gains for the other items, it is unclear to what extent instruc-tion might have in�uenced their changing schemas in these areas. However,for the groups in general, students provided more negative answers to thearithmetic problems a�er the instruction, suggesting that students were moreopen to the existence and use of negative numbers. Qualitative analysis willprovide more insight for all of these areas.

discussion of quantitative results 87

Table 8.14 – Integer Property Scores for Performance by Instructional Groups.Mean percentagecorrect scores and standard deviations for the integer property pre-test, post-test, and di�er-ence scores for each performance level in each instructional group.

Group Pre-Test, Mean(SD)

Post-Test, Mean(SD)

Di�erence, Mean(SD)

Full Instruction

High Perform. (n = 7) 44.02 (30.63) 92.28 (7.93) 48.26 (31.51)

Medium Perform. (n = 9) 16.77 (7.12) 65.51 (19.29) 48.75 (17.35)

Low Perform. (n = 4) 14.26 (1.61) 25.65 (11.34) 11.39 (11.65)

Integer Properties

High Perform. (n = 8) 51.48 (31.98) 88.61 (10.54) 37.13 (25.70)

Medium Perform. (n = 8) 12.96 (3.43) 80.60 (12.62) 67.64 (14.27)

Low Perform. (n = 4) 15.74 (7.64) 55.38 (14.63) 39.64 (20.15)

Integer Operations

High Perform. (n = 9) 41.83 (37.75) 62.92 (33.56) 21.09 (26.96)

Medium Perform. (n = 7) 19.05 (13.72) 31.38 (31.82) 12.33 (19.10)

Low Perform. (n = 5) 13.70 (1.66) 21.26 (5.90) 7.56 (6.18)


9 R E S U LT S – I N T E G E R S C H E M A C HA N G E S

Although the previous analyses address how students’ accuracy di�eredbetween groups, the following results focus on students’ schemas at the begin-ning and end of the study and address the question of how students’ integerschemas change depending on their instructional groups. Further, they high-light the changes (or lack thereof) for students considered low performing ineach group. In each case presented, the possible schemas are listed accordingto complexity, from least to most formal.

written symbols : identifying negative numbers

All three groups demonstrated some improvement in identifying negativenumbers from the pre-test to the post-test, although students considered low-performing in math bene�ted most from the integer properties instruction.Students identi�ed negative numbers speci�cally during two of the interviewtasks. In one case, students completed six questions where they saw twointegers in isolation and had to name them; in the other case, students sawtwo equations, circled the numbers they saw, and named the numbers. �eiractual and potential responses to how they identi�ed the integers fell into sixcategories as shown in Table 9.1.

When identifying negative numbers in isolation, students’ responses fellalmost exclusively in categories A, C, and F. �ey either interpreted all inte-gers as positive, inconsistently identi�ed negatives (or called some positivenumbers negative) or named all integers correctly. Students’ ability to nameintegers was interpreted rather liberally. Even adults sometimes interchangethe terms “negative” and “minus”, so the term “minus” as well as “penalty”were accepted as correct for these items. As shown in Table 9.2 at least 1/3 ofstudents in each instructional group transitioned from an initial or syntheticschema at pre-test to formal understanding on post-test. Close to half of thestudents in the Operations instructional group continued to have an initialschema for integer identi�cation at post-test. �is is unsurprising because

89

90 9 ⋅ results – integer schema changes

Table 9.1 – Schema Categories for Identifying Negative Numbers.

Schema Label Description Example

Initial A Names all integers positive. Calls -3, ”three.”

Synthetic B Names some positive numbers as neg-ative AND names some negative num-bers as positive.

Calls -3, 4, -5, and6, ”three, four, nega-tive �ve, and negative six” respectively.

Synthetic C Names some positive numbers as nega-tive OR names some negative numbersas positive.

Calls -3, 4, -5, 6, ”three, four, negative�ve, six” respectively.

Synthetic D Misnames some or all positive ANDnegative numbers but distinguishesbetween them.

Calls 6 and 4, ”nine and three” respec-tively and calls -3, ”three and a half.”

Synthetic E Misnames some positive numerals ORmisnames some negative numerals.

Calls 7, 2, -3, and -1, ”six, two, threeand a half, and negative one” respec-tively.

Formal F Names integers correctly. Calls -3, 4, -5, and 6, ”negative three,four, negative �ve, and six” respectively.

they did not receive instruction on what negative numbers are. While thesestudents heard the term negative, they would need to pair this term with theconcept of numbers occurring before zero to be successful on this task. Allstudents in the Integer Properties group transitioned from an initial schema(A) to a synthetic (B–E) or formal schema (F).

Although at least half of the students in the Full Instruction and IntegerProperties groups still identi�ed negatives within the context of an equation(see Table 9.3), both groups had more students interpret the numbers aspositive in equations than they did when working with negatives in isolation(refer back to Table 9.2). �e Operations group had particular di�culty withidentifying negatives in equations (see Table 9.3), and 67% of them did notidentify any of the integers as negative in this task. Students in this group didnot have practice distinguishing negatives from minus signs or learning thatthe negative sign is an important feature to pay attention to when namingnumbers.

To determine students’ overall integer identi�cation schemas, I combinedtheir performance on both identi�cation tasks and treated their answers asone task. Based on the overall classi�cations, the Integer Properties groupwas the only one who had students classi�ed as low performers develop aformal schema for integer identi�cation. �ree out of the four low performers

written symbols : identifying negative numbers 91

Table 9.2 – Schema Changes for Identifying Negative Numbers in Isolation. Percentage of totalstudents in each instructional group who started and ended at each schema level along withtheir gains. nL = number of low performing students.

Identifying Negatives in IsolationFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)

Schema Pre-Test Post-Test

Gain Pre-Test Post-Test


Gain

A 70% 15%3L−55% 60% 0% −60% 71% 43%3L

−28%B 0% 0% 0% 5% 0% −5% 0% 0% 0%C 20% 35%1L

+15% 10% 35% +25% 5% 5% 0%D 0% 0% 0% 0% 0% 0% 0% 0% 0%E 0% 0% 0% 0% 5% +5% 5% 0% −5%F 10% 50% +40% 25% 60%4L

+35% 19% 52%2L+33%

Table 9.3 – Schema Changes for Identifying Negative Numbers in Context. Percentage of totalstudents in each instructional group who started and ended at each schema level along withtheir gains. nL = number of low performing students.

Identifying Negatives in ContextFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 85% 45%4L−40% 90% 25% −65% 81% 71%5L

−10%B 0% 0% 0% 0% 0% 0% 0% 0% 0%C 15% 0% −15% 5% 15%1L

+10% 0% 5% +5%D 0% 0% 0% 0% 0% 0% 0% 0% 0%E 0% 0% 0% 0% 5% +5% 0% 0% 0%F 0% 55% +55% 5% 55%3L

+50% 19% 24% +5%


Table 9.4 – Schema Changes for Identifying Negative Numbers Overall. Percentage of total stu-dents in each instructional group who started and ended at each schema level along with theirgains. nL = number of low performing students.

Identifying Negatives OverallFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 70% 15%3L−55% 60% 0% −60% 71% 43%3L

−28%B 0% 0% 0% 5% 0% −5% 0% 0% 0%C 30% 40%1L

+10% 30% 55%1L+25% 10% 33%2L

+23%D 0% 0% 0% 0% 0% 0% 0% 0% 0%E 0% 0% 0% 0% 5% +5% 0% 0% 0%F 0% 45% +45% 5% 40%3L

+35% 19% 24% +5%

moved from identifying all negatives as positive on the pre-test to correctlyidentifying them as negative on the post-test. Furthermore, all of the studentsin the Integer Properties group developed at least a synthetic schema foridentifying integers. While the students in the Full Instruction group hadsimilar performance to the Integer Properties group, three of the �ve lowperformers in this group persisted in identifying all integers as positive.

Additionally, students need help connecting the term negative to the sym-bols. Almost half of the students in the Operations group were not able toconnect these concepts and called all integers positive. Only two studentsfrom this group transitioned to calling them negative consistently. Table 9.4shows the percentages of students with each type of schema for the identi�-cation tasks combined.

Discussion – Identifying Negatives

Consistent with the quantitative results, the integer identi�cation schemasfor each instructional group suggest that participating in speci�c activities,which require students to talk about negative numbers, helped the Full In-struction and Integer Properties groups identify negative numbers. Not tak-ing performance levels into account, both the Full Instruction and IntegerProperties groups had almost half of their students name negative numbersconsistently overall. Additionally these groups had few or no students onlyidentify them as positive. �e Operations group, however, had almost half ofits students continue to identify negatives as positive.

number word order – counting backward 93

For all groups, identifying negatives in isolation was easier; more studentsin each group identi�ed negatives as positive when they were within equa-tions. For the equations questions, students �rst had to circle the numbersand then name them. It is possible that some students who could name thenegatives in isolation did not consider the negative signs as part of numbers;thus, when they were speci�cally asked to �nd the numbers, they ignored thenegative signs and identi�ed them as whole numbers.

Although the Integer Properties group only had one more lesson than theFull Instruction group on identifying negatives speci�cally, their more inten-sive focus on this as well as integer values and integer order proved importantfor students who are considered lower performers in math to develop the cat-egorization structure of positive and negative. �e low performing studentsin the Full Instruction group might have been overwhelmed by the extra in-struction on operations, and the low performing students in the Operationsgroup did not have practice identifying negatives.

number word order – counting backward

When counting backward on the pre-test, many students stopped at oneor zero, even if they knew about negative numbers. �is is unsurprising ascounting tasks in �rst grade involve only positive numbers. However, onceasked if anything was before zero or if they could count back any further, somestudents demonstrated synthetic or formal schemas of integer word order,with twice as many students in the Full Instruction and Integer Propertiesgroups developing formal schemas than in the Operations group. Table 9.5describes the counting schemas.�ree to four students in each instructional group counted back into the

negatives on both the pre-test and on the post-test. Aside from these stu-dents, both the Full Instruction and Integer Properties groups had 45% ofits students transition from only counting to zero on the pre-test to count-ing correctly into the negatives on the post-test. Only one low-performingstudent from the Integer Properties group developed this formal schema forinteger word order.

Over half of the students in the Operations group stopped counting at zeroor one on both the pre- and post-tests. �is is twice as many students thanthose in the Integer Properties group. �e students in the Operations groupmoved through zero on board games, but they did not map this movementonto the number word sequence. Students in the Full Instruction group had


Table 9.5 – Schema Categories for Counting Backward.


Initial A Counts back to “one”. ”Five, four, three, two, one.”

Initial B Counts back to “zero”. ”Five, four, three, two, one, zero.” re-spectively.

Synthetic C Counts back to repeating zero. ”Five, four, three, two, one, zero, zero,zero.”

Synthetic D Counts back to “negative zero”; “zero”excluded.

”Five, four, three, two, one, negativezero.”

Synthetic E Counts back to “negative zero”; “zero”included.

“Five, four, three, two, one, zero, nega-tive zero.”

Synthetic F Counts back to negatives; skips zero. ”Five, four, three, two, one, negativeone.”

Synthetic G Counts back into the negatives; nega-tives ordered backward.

”Five, four, three, two, one, zero, nega-tive nine, negative eight...”

Formal H Counts back into the negatives. ”Five, four, three, two, one, zero, nega-tive one, negative two...”

some experience counting and had fewer students with initial schemas atpost-test. �is pattern strengthened for students in the Integer Propertiesgroup who had the most experience counting into the negatives (see Table 9.6for percentages).

Discussion – Counting Backward

Since students in the Full Instruction and Integer Properties groups prac-ticed counting from positive to negative numbers and vice versa, it is surpris-ing that at least one-fourth of students in each of these groups did not countpast 0 on the post-test. A possible reason for this result is that the practiceof counting backward to zero is more familiar and usually what a �rst gradeteacher expects in response to the request of a student to count backward.Students may have stopped at zero due to this expectation. However, they didnot continue even when prompted. Additionally, only one student consideredlow performing in math transitioned from counting to zero to counting intothe negatives.

One student in the Operations group counted back and repeated zero.Another student counted back to negative zero. Two students in the IntegerProperties group skipped zero and counted into the negatives. Based on these

number order – completing a number line 95

Table 9.6 – Schema Changes for Counting Backward. Percentage of total students in each instruc-tional group who started and ended at each schema level along with their gains. nL = numberof low performing students.

Order: Counting BackwardFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 15% 0% −15% 0% 0% 0% 10% 10%1L 0%B 70% 40%4L

−30% 80% 25%3L−55% 71% 43%4L

−28%C 0% 0% 0% 0% 0% 0% 0% 5% +5%D 0% 0% 0% 0% 0% 0% 0% 0% 0%E 0% 0% 0% 0% 0% 0% 0% 5% +5%F 0% 0% 0% 0% 10% +10% 0% 0% 0%G 0% 0% 0% 0% 0% 0% 5% 0% −5%H 15% 60% +45% 20% 65%1L

+45% 14% 38% +24%

suggestive results, it is possible that some students stopped counting at zerobecause they consider negatives to be equivalent to zero. On the post-test,each group had two students who treated negatives as worth zero on thearithmetic problems and who also stopped counting at zero. Without furtherprobes, though, it is not clear how related these two observations are.

number order – completing a number line

A�er counting backward, students were asked to �ll in a horizontal num-ber line with only “1” marked (pre-test and post-test) and a vertical numberline with “0” marked (post-test). Once again, the Full Instruction and IntegerProperties group excelled. �is task investigated students’ schemas of numberword order and their association of symbols (numerals and negative signs)to the words, and all students in the Full Instruction group transitioned tosynthetic or formal schemas. �is task had the largest variety of potential re-sponses, which fell into one of four categories: wrapped, repeated, one-sided,or symmetric (see Table 9.7).

Overall, on the pre-test, about 41% of all students only �lled in the wholenumbers on the number line (initial schema E). �e most popular initialschema on the post-test was E (13% of all students), and the most popularsynthetic schema on the post-test was the symmetric version of H (13% of allstudents). Only 15% of students in the Operations group moved to formalunderstanding of the integer number line by the post-test. However, 45%of the students in the Full Instruction group and 55% of the students in the


Table 9.7 – Order Schema Categories for Completing a Number Line.


Initial A Positive numbers only, some or all inwrong direction.

Wrapped: 9 8 7 6 1 2 3 4 5Repeated: 5 4 3 2 1 4 3 2 1One-sided: 5 4 3 2 1 _ _ _ _Symmetric: 5 4 3 2 1 2 3 4 5

Initial B Positive numbers only, correct direc-tion.

One-sided: _ _ _ _ 1 2 3 4 5

Initial C Whole numbers, some or all in wrongdirection, zero not always next to 1.

Wrapped: 8 7 6 5 0 1 2 3 4Repeated: 4 3 2 1 0 4 3 2 1Symmetric: 5 4 3 2 0 2 3 4 5

Initial D Whole numbers, some or all in wrongdirection, zero next to “1”.

One-sided: 5 4 3 2 1 0 _ _ _Symmetric: 3 2 1 0 1 2 3 4 5

Initial E Whole numbers, correct. One-sided: _ _ _ 0 1 2 3 4 5

Synthetic F Integers, no zero, wrong direction. Wrapped: 5 4 3 2 1 -6 -7 -8 -9Symmetric: 5 4 3 2 1 -2 -3 -4 -5

Synthetic G No zero, written as positive but callednegative and/or negatives not con-nected to positives correctly.

Wrapped: 9 8 7 6 1 2 3 4 5 (9 = -9)Repeated: -1 -2 -3 -4 1 2 3 4 5

Synthetic H No zero, negatives and positive each goin the correct direction.

Wrapped: -9 -8 -7 -6 1 2 3 4 5Symmetric: -4 -3 -2 -1 1 2 3 4 5or -5 -4 -3 -2 1 2 3 4 5

Synthetic I Includes zero; wrong direction. Wrapped: 5 4 3 2 1 0 -6 -7 -8Repeated: 5 4 3 2 1 0 -3 -2 -1One-sided: 5 4 3 2 1 0 0 0 0

Synthetic J Positive or negative not ordered cor-rectly and/or not connected to positivescorrectly.

Wrapped: -8 -7 -6 0 1 2 3 4 5Repeated: -1 -2 -3 -4 0 1 2 3 4One-Sided: 0 0 0 0 1 2 3 4 5Symmetric: 5 4 3 2 1 0 -1 -2 -3

Synthetic K Alternative notation for negatives. Symmetric: N3 N2 N1 0 1 2 3 4 5

Formal L Integers correctly ordered. Symmetric: -3 -2 -1 0 1 2 3 4 5

number order – completing a number line 97

Table 9.8 – Order Schema Changes for Completing Number Lines. Percentage of total studentsin each instructional group who started and ended at each schema level along with their gains.nL = number of low performing students.

Order: Number LineFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

Aw 5% 0% −5% 5% 0% −5% 14% 10%1L−4%

Ar 0% 0% 0% 0% 0% 0% 0% 5%1L+5%

As 10% 0% −10% 10% 15%1L+5% 5% 5% 0%

B 10% 0% −10% 5% 0% −5% 10% 5% −5%Do 0% 0% 0% 0% 0% 0% 10% 0% −10%Ds 10% 0% −10% 0% 0% 0% 0% 0% 0%E 35% 0% −35% 50% 10%2L

−40% 38% 29%2L−9%

Fs 0% 5%1L+5% 0% 0% 0% 0% 0% 0%

Gw 0% 5% +5% 0% 0% 0% 0% 0% 0%Gr 0% 0% 0% 0% 0% 0% 0% 5%1L

+5%Hs 5% 25%2L

+20% 0% 5% +5% 0% 10% +10%Jo 10% 10%1L 0% 10% 0% −10% 5% 5% 0%Js 5% 5% 0% 0% 0% 0% 0% 0% 0%K 5% 0% −5% 5% 0% −5% 5% 0% −5%L 5% 50% +45% 15% 70%1L

+55% 14% 29% +15%

Integer Properties group transitioned from having an initial or syntheticschema to having a formal schema of the integer number line (see Table 9.8for the schema percentage charts for each group).

Discussion – Completing the Number Line

Although students in the Operations group saw the negative integers la-beled on a number line in one set of their lessons, this exposure did nothelp them on the task. Students in the Full Instruction and Integer Proper-ties groups bene�ted from labeling the number lines while playing gamesand connecting the number words and symbols with their positions on thenumber line. As with the counting backward task, students’ use of, or ratherexclusion of, zero in their number lines is interesting. Students may believethat negatives come before zero and thus do not list zero. It is possible, asproposed previously, that students consider negatives to be worth zero, andso exclude zero. Either way, this is an area that should be investigated further.Additionally, students’ primary use of synthetic schemas involving symmetryis interesting and suggests that this characteristic is highly salient for them,even if they do not always use it correctly.


Table 9.9 – Order Schema Categories for Ordering Integers.


Initial A Random order. 1 -4 5 -2 -8 7

Initial B Orders as positive, two numbers transposed. 1 -2 5 -4 7 -8

Initial C Orders as positive. 1 -2 -4 5 7 -8

Synthetic D Orders negative and positive separately, negative backward. 1 5 7-2 -4 -8

Synthetic E Orders negatives backward a�er positive. 1 5 7 -2 -4 -8

Synthetic F Orders negatives backward. -2 -4 -8 1 5 7

Synthetic G Orders correctly with two numbers transposed. -2 -8 -4 1 5 7 or-2 -4 -8 1 7 5

Synthetic H Negative and positive switched. 0 3 8 -9 -7 -5

Synthetic I Sometimes random order, sometimes ordered correctly. 1 5 -8 -4 7 -2 and-9 -7 -5 0 3 8

Synthetic J Orders negative and positive separately but in correct order. 1 5 7-8 -4 -2

Formal K Orders integers correctly. -8 -4 -2 1 5 7

number order – ordering integers

Students’ schemas for ordering the integers were similar to their schemasfor �lling in the number lines, without the numerous subcategories (seeTable 9.9). Although all groups contained students who remained at initialschema levels a�er the instruction, students in the Integer Properties andFull Instruction groups had several more students develop formal schemasas compared to students in the Integer Operations group.

Both the Full Instruction and Integer Properties groups had similar per-centages of students who developed a formal schema for integer order (40%and 50% respectively). In stark contrast, only 10% of the Operations groupdeveloped a formal schema. Rather, 48% of them continued to treat thenegative numbers as positive.

Although most of the students classi�ed as low-performers in the IntegerProperties group did well on identifying negative numbers, recognizing thenegatives was not enough to help them complete this task successfully. �istask seemed to be especially di�cult for two of them. Two of the studentsorganized them randomly on both the pre- and post-tests, but when asked

number order – ordering integers 99

Table 9.10 – Order Schema Changes for Ordering Integer Cards. Percentage of total students ineach instructional group who started and ended at each schema level along with their gains.nL = number of low performing students.

Order: Number LineFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 20% 5%1L−15% 10% 20%4L

+10% 10% 10%1L 0%B 0% 10%1L

+10% 10% 0% −10% 5% 5% 0%C 60% 10%1L

−50% 50% 0% −50% 67% 48%4L−19%

D 0% 5% +5% 5% 0% −5% 0% 0% 0%E 0% 5%1L

+5% 5% 5% 0% 0% 0% 0%F 5% 10% +5% 0% 10% +10% 5% 14% +9%G 5% 10% +5% 5% 5% 0% 0% 0% 0%H 5% 0% −5% 0% 0% 0% 0% 0% 0%I 0% 0% 0% 5% 0% −5% 0% 0% 0%J 0% 0% 0% 0% 0% 0% 0% 0% 0%K 5% 45% +40% 10% 60% +50% 14% 24% +10%

to identify the greatest and least among the numbers, did not choose thenumbers at the beginning and end of their sequence. It is possible that theydid not fully understand the task.

Not including students who demonstrated formal understanding of inte-ger order on the post-test, 29% of the remaining students across all groups(10/35) had schemas D, E, or F, which involved organizing the negative num-bers backward (see Table 9.10 for percentages of students at each schema inthe three groups).

Discussion – Ordering Integers

Overall, these results �t in well with the conceptual change model. Ifstudents who used to treat all negatives as positive learned that they aredi�erent, a logical next step would be to place them into separate groups butstill order them as they do positive numbers. Organizing integers in terms ofvalue is more abstract than merely ordering them without factoring in values,which could account for some of the students’ di�culty on this task–even a�erexploring the values of integers–compared to counting backward or �lling ina number line. Furthermore, it appears that numerical value information isharder to discern without some instruction because not all students in theOperations group who �lled in numbers correctly on a number line wereable to order the numbers in terms of value; this information was not alwaysintuitive from using the number line representation.


Table 9.11 – Value Schema Categories for Integer Values. Value schema categories for judginggreatest/least and comparing integers


Initial A Random. 2, 5, 3: says 3 is greatest.

Initial B Absolute value, no zero. 0, -3, 3, -5, -9, 8, 2: says 2 is the least.

Initial C Absolute value. 0, -3, 3, -5, -9, 8, 2: says -9 is the great-est.

Synthetic D Ignores negatives. -1, -6, -7, 3, 4, 6: says 3 is smallest

Synthetic E Negatives equal zero. -3 vs. -5: says neither is greater

Synthetic F More negative = larger; zero is least. 0, -3, 3, -5, -9, 8, 2: 0 is smallest; -3 vs.-5: says -5 is greater

Synthetic G More negative = larger; negatives lessthan zero.

-3 vs. -5: says negative �ve is greater

Synthetic H Sometimes more negative = larger;sometimes correct.

-3 vs. -5: says negative �ve is greater; -2vs. -4: says negative two is greater

Synthetic I Sometimes treats as positive; some-times correct.

-3 vs. -5: says �ve is greater than three-2 vs. -4: says -2 is greater

Formal J Positive > negative; less negative >more negative.

-3 vs. -5: says -3 is greater

integer values

Students judged the value of integers on two tasks. A�er ordering a setof integers, students had to identify which was the least and which was thegreatest. Correctly ordering the numbers should help students identify theleast and greatest, but this was not always the case. Students o�en answeredthe questions as if they were completely separate tasks. On the second task,students chose which of two integers was greater. �e schema categories forthese tasks take into account students’ answers on both types of questions(see Table 9.11). In cases like initial schema A, students were more likely tobe at this level when choosing the number from a set than when comparingjust two integers. However, for both types of questions the schema results inthe groups followed a similar pattern as with the previous questions; studentsin the Full Instruction and Integer Properties groups had at least four timesas many more students reach formal schemas than students in the IntegerOperations group.

integer values 101

Table 9.12 – Value Schema Changes for Greatest/Least Items. Percentage of total students ineach instructional group who started and ended at each schema level along with their gains.nL = number of low performing students.

Ordered Value: Greatest/LeastFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 15% 5%1L−10% 0% 15%3L

+15% 0% 5% +5%B 0% 10%1L

+10% 15% 0% −15% 5% 5%1L 0%C 65% 20%1L

−45% 60% 0% −60% 76% 52%4L−24%

D 0% 0% 0% 5% 0% −5% 0% 0% 0%E 0% 0% 0% 0% 0% 0% 0% 0% 0%F 10% 0% −10% 0% 0% 0% 0% 0% 0%G 0% 0% 0% 0% 15%1L

+15% 5% 14% +9%H 0% 10% +10% 0% 0% 0% 0% 0% 0%I 0% 15%1L

+15% 5% 0% −5% 0% 0% 0%J 10% 40% +30% 15% 70% +55% 14% 24% +10%

Compared to the previously described items, the task of identifying thegreatest or least integer from a set distinguished between the groups mostdistinctly. Over half of the students in the Integer Properties group (55%)transitioned from having an absolute value schema of the integer values tounderstanding the formal values. About a third of all students in the FullInstruction group (30%) and only 10% of students in the Operations grouptransitioned in this way. It is important to note that as with the ordering task,no students classi�ed as low-performers in any group developed the formalschema, although one low student from both the Full Instruction and IntegerProperties groups did develop a synthetic schema (see Table 9.12 for moredetails).

In contrast to comparing a set of values, when comparing just two integervalues, similar percentages of students in the Full Instruction and IntegerProperties groups transitioned from an initial or synthetic schema to the for-mal schema for integer values (55% and 60% respectively). As seen previously,over half of the students in the Operations group continued to demonstrate aninitial schema for integer value (see Table 9.13). All low performing studentsin the Integer Properties group transitioned to having at least a syntheticschema; however low performing students in the other two groups tended toassign positive values to the negative numbers.


Table 9.13 – Value Schema Changes for Comparing Two Integers. Percentage of total studentsin each instructional group who started and ended at each schema level along with their gains.nL = number of low performing students.

Ordered Value: Comparing Two IntegersFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 0% 0% 0% 0% 0% 0% 0% 0% 0%B 0% 0% 0% 0% 0% 0% 0% 0% 0%C 80% 25%3L

−55% 75% 0% −75% 81% 57%4L−24%

D 0% 0% 0% 0% 0% 0% 0% 0% 0%E 0% 0% 0% 0% 5% +5% 0% 0% 0%F 5% 0% −5% 0% 0% 0% 0% 0% 0%G 0% 5% +5% 5% 10%2L

+5% 5% 14% +9%H 0% 0% 0% 0% 5% +5% 0% 0% 0%I 0% 0% 0% 10% 10%1L 0% 0% 5%1L

+5%J 15% 70%1L

+55% 10% 70%1L+60% 14% 24% +10%

Discussion – Integer Values

It is interesting that the Full Instruction group did better when comparingtwo integers than when they had to judge values of several integers at once.Perhaps the extra experiences playing with a variety of integers helped theInteger Property group manage several values at once; whereas students inthe Full Instruction group had fewer experiences with sets of integers beforefocusing on operations. �eir schemas might not have been as stable, since aquarter of them answered correctly only occasionally.�e results of the value task also provide some justi�cation for the hy-

pothesis posed in the section on counting backward. Recall that althoughstudents in the Full Instruction and Integer Properties groups had practicecounting through zero, several of them did not do so on the post-test. Basedon students’ overall value schemas in these groups, 25% of students in theFull Instruction group treated negatives as having the same value as theirpositive counterparts or sometimes treated more negative values as greaterthan less negative values. Similarly, 20% of students in the Integer Propertiesgroup treated more negative values as larger than less negative values someor all of the time (see Appendix M for overall value schema percentages).�is behavior suggests that students thought about negatives sometimes asseparate but also as parallel to positive numbers.

directed magnitude 103

Table 9.14 – Directed Magnitude Schema Categories. Directed magnitude schema categories fordetermining more/less high/low


Initial A All high or all low. Student always moves the cat higher OR always movesthe cat lower.

Initial B More low or more high. Student always moves the cat lower except when askedto move the cat less high OR always moves the cathigher except when asked to move the cat less low.

Synthetic C More and less. Student moves the cat higher when “more” is used andlower when “less” is used.

Synthetic D High and low. Student moves the cat higher when “high” is used andlower when “low” is used.

Synthetic E Less high. Student moves correctly, except goes higher for lesshigh.

Synthetic F Less low. Student moves correctly, except goes lower for less low

Formal G More/Less High/Low. Student correctly interprets the pairs of words.

directed magnitude

Students explored the movements of more high, less high, more low, andless low in the context of a cat moving on some stairs. Some students focusedon one of the two words in a pair, which led to a set of synthetic schemas(see Table 9.14). Unlike with the previous tasks, the Integer Properties groupshowed the least growth in directed magnitude schema development.�e majority of students (48%) had synthetic schema F on the pre-test.

While students were able to interpret the opposite nature of moving “less high”as really moving lower, they continued to interpret “less low” as moving lower.Unlike in previous tasks, the fewest number of students (5%) to develop theformal schema came from the Integer Properties group. On the other hand,40% of students in the Full Instruction group and 38% of students in theOperations groups developed a formal understanding of “less low” meaning“go higher” (see Table 9.15).

Discussion – Directed Magnitude

Since the Full Instruction and Operations groups had experience movingmore positive, less positive, more negative, and less negative, these resultssuggest that experiences interpreting the opposite language can help studentsunderstand what it means – even when exploring the language in three lessons


Table 9.15 – Schema Changes for Directed Magnitude. Percentage of total students in eachinstructional group who started and ended at each schema level along with their gains. nL =number of low performing students.

Directed MagnitudeFull Instruction Integer Properties Integer Operations

(n = 20) (n = 20) (n = 21)




Gain

A 20% 15%1L−5% 15% 5%1L

−10% 10% 0%1L−10%

B 10% 5%1L−5% 5% 5%1L 0% 5% 5% 0%

C 0% 0% 0% 0% 0% 0% 6% 0% 0%D 0% 0% 0% 25% 10% −15% 29% 29%2L 0%E 5% 0% −5% 0% 10% +10% 14% 5%1L

−9%F 60% 35% −25% 45% 55%2L

+10% 38% 24%1L−14%

G 5% 45%1L+40% 10% 15% +5% 5% 38%1L

+33%

Table 9.16 – Subtraction Property Questions. Average number of subtraction problems (out of 3)and their reversals that students said would not be equal.

Group Pre-Test Post-Test Gain

Full Instruction 1.10 1.05 −0.05

Integer Operations 0.33 1.09 +0.76

Integer Properties 0.45 1.45 +1.00

as students in the Full Instruction group did. �ese results further suggestthat for �rst graders, this language, especially “less low” (or “less negative”),was not intuitive to them without some explanation.

subtraction as non-commutative

On both the pre- and post-test, students were asked to look at a pairof arithmetic problems and judge if the answers would be the same. Oneproblem involved commuting addition and two were addition or subtractionproblems with one number changed in one of the sets (e.g., 7 − 3 vs. 7 − 4).�e last three problems involved reversed subtraction problems (e.g., 1 − 4vs. 4 − 1). Since most students consistently answered the �rst three problemscorrect, only the results for the reversed subtraction problems are shown.Table 9.16 lists the average number of problems students answered correctlyon the pre- and post-tests.

subtraction as non-commutative 105

Based on the average gain scores, students in the Full Instruction groupdid not bene�t from the lesson where they learned that subtraction is notcommutative. However, students in the Operations group who received twolessons on this concept did improve slightly. Strangely, the Integer Propertiesgroup improved the most, although not signi�cantly so. �e results suggestthat students’ understanding of subtraction needs to be explored further.

Discussion – Operation Properties and Directed Magnitude Items

Recall that the factorial ANOVA on operation properties and directedmagnitude test items–the questions probing students’ understanding of moreor less high and more or less low plus the operations questions–was notsigni�cant. Although students in the Full Instruction group and Operationsgroup were more likely to develop a formal schema for the directed magnitudeitems, this was not captured in the grouped measure. �e grouped measurelooked at the di�erence in number of questions students answered correctly.�erefore, students moving from an initial schema to a synthetic schemacan have the same score as students moving from a synthetic schema to theformal schema. While the Full Instruction group gained more points on thedirected magnitude questions, they also did not gain on the subtraction asnon-commutative questions, whereas the other two groups did.


10 R E S U LT S – S C H E M A I N T E G R AT I O N

integration of formal and non-formal schemas

Much like children �rst learning whole number symbols, values, and order,some of the students in this sample demonstrated formal understanding ofone aspect of negative numbers but initial or synthetic understanding ofanother. �e following two cases serve as examples that children can developformal understanding of any one of the integer schemas before the others andas examples of how the integration of formal and non-formal schemas of thevarious integer concepts can play a role in how students reason about integerarithmetic. In total, 16 of the 61 students had formal schemas in one or moreareas and initial or synthetic schemas in the others–the rest of the studentsstarted with no formal schemas. �e integration of formal and non-formalinteger schemas leads to a synthetic schema of the integer central conceptualstructure as a whole.

student 117

High performance in mathInteger Properties Instructional GroupPre-test score = 3/26Post-test score = 10/26

Table 10.1 – Student 117’s Schema Levels. Student 117’s integer schema levels on the pre-testand post-test.

IdentifyIntegers

CountBackward

Number Line Integer Order Integer Value DirectedMagnitude

SubtractionProperties

Pre Synthetic Formal Formal Synthetic Synthetic Initial 2/3B H L I I D

Post Formal Formal Formal Formal Formal Synthetic 3/3F H L K J F

107

108 10 ⋅ results – schema integration

Student 117, Pre-Test

On the pre-test, Student 117 counted backward saying, “Five, four, three,two, one, zero.” When asked if anything comes before zero, he responded,“Minus one, minus two.” He was also able to �ll in both positive and negativenumbers correctly on the number line. His performance on the other tasks,however, was less consistent. When asked to identify integers, he sometimescalled negative numbers positive and sometimes called positive numbers neg-ative. He did not have a stable categorization for the symbols. Furthermore,he used the term “minus” to label negative numbers.

Likewise, when asked to order a set of integers, he treated one set as posi-tive and ordered them incorrectly, but correctly ordered a second set. �ispattern carried over into identifying the greatest integer in a set or the greaterof two integers. Some of the time he choose the integer with the greatestabsolute value (-7 > 4), and sometimes he answered correctly (-2 > -7). Onthe arithmetic problems, Student 117 solved 16 out of the 23 problems con-taining negatives (70%) as if they contained only positive numbers. For theother problems, he provided negative answers but either guessed or said he“knew it” even if the answer was incorrect. In some cases he talked about theproblems as if they were positive but wrote negative answers. For example,he solved -4 − -7 = -6 saying, “Four minus seven, uh, (writes -6)...I alreadyknow.”

Student 117, Post-test

A�er receiving instruction on integer properties for 8 lessons, Student 117demonstrated a formal understanding of the integer concepts on the post-testTable 10.1. He correctly identi�ed all integers (and used the term “negative”),ordered them correctly, and could say which was greater (and less). �is timeon the arithmetic problems, he solved the problems as positive on only 7 outof the 23 problems with negatives (30%), a decrease of 40%.

When acknowledging the negatives in the problems, he used a couple ofdi�erent strategies. On 50% of the problems involving subtracting a negative–excluding problems where a negative was subtracted from itself (e.g., -5− -5)–Student 117 treated the second negative as a minus and subtracted twice. Forexample, on 9 − -2, he said, “Nine minus two equals um seven and sevenminus two equals �ve.” Interestingly, he was even able to use this incorrectstrategy correctly when the �rst number was negative. On -2 − -6, he got ananswer of -14. Furthermore, on the 10 problems involving adding a positive

student 118 109

number or subtracting a positive number, he correctly solved 7 of them, evenwhen the �rst number was negative (e.g., -5 − 9 = -14).

Student 117, Summary

Overall, Student 117 expanded his conceptual structure to include nega-tive numbers and integrated this order with the value of the integers wherenumbers to the right on the number line equal more than the numbers to thele�. Using his ordered value schema he could solve many problems involvingadding and subtracting positive numbers. However, he did not develop aformal understanding of directed magnitudes, and was unable to correctlysolve problems asking him to subtract a negative.

student 118

Medium performance in mathInteger Operations Instructional GroupPre-test score = 5/26Post-test score = 11/26

Student 118, Pre-Test

Student 118 presents the case of someone who knew about negatives butwas unfamiliar with their formal notation, which led to an interesting patternof behavior on the arithmetic problems. When counting backward, Student118, said, “Four, three, two, one, zero, minus, negative one, negative two,negative three, negative four, negative �ve.” He counted into the negativeswithout prompting. As he �lled out the number line, he wondered, “Howcould I write negative one?” Ultimately, he decided to represent it by writingan “N” before the 1. When identifying integers, he used the term “minus”to label the negative numbers just as the previous student had. However, heordered all of the integers as though they were positive and chose the greatest(or least) number as if they were all positive.

His pattern of responses to the arithmetic problems suggests that minusnumbers and negative numbers were two separate concepts for him. On everyquestion that had a negative sign, Student 118 included a negative sign in hisanswer but called it “minus”. �is included answering “minus zero” to thequestion -8− -8. To him a minus number was used as an adjective describingthe numbers much in the way a number could be colored green. �is isfurther illustrated by two examples. When solving subtraction problems


Table 10.2 – Student 118’s Schema Levels. Student 118’s integer schema levels on the pre-testand post-test.

IdentifyIntegers

CountBackward



Pre Formal Formal Synthetic Initial Initial Formal 1/3F H K C C G

Post Formal Formal Formal Formal Formal Synthetic 2/3F H L K J D

such as 1 − 4, he counted, “Zero, negative one, negative two” and wrote “N3”.�en, for -5− 9 he counted, “Minus four, minus three, minus two, minus one,minus zero, minus negative one, minus negative two...minus negative four!”and wrote “-N4.”

Because his schema of minus numbers was not aligned with his schemas ofnegative numbers (as they were for other students), Student 118 was only ableto correctly answer the problems like 1− 4 or problems where both numeralswhere negative; to him, -8 − -5 was the same thing as 8 − 5 but “minus.”

Student 118, Post-test

A�er receiving the operations instruction, Student 118 demonstrated for-mal understanding of the integer concepts but had di�culty interpreting lesslow and less high. It may seem surprising that he reached a formal understand-ing of integer value, except that he already demonstrated this understandingwhen he solved 3−9 = N6. He just needed to realize that the negative sign wasequivalent to the “N” that he used on the pre-test. Although he successfullyinterpreted negative numbers as values below zero on the post-test, he stillfrequently applied the negative sign to his answers as he did on the pre-test.For example, he still answered -5 − -5 with -0, and solved 5 − -3 = -2 byreasoning, “Five minus three equals two and �ve minus negative three equalsnegative two.” Once again, he correctly answered problems which requirednegative answers; however he also answered -2 − -6 and -4 − -7 by starting atthe �rst number and counting through zero. Interestingly, both times whencounting he called zero “negative zero.”

Student 118, Summary

Student 118’s responses underscore the importance of understanding howstudents solve integer problems rather than relying solely on answer correctscores; students can answer correctly without having conceptual understand-ing of why the answer is correct or can answer incorrectly while having

integrating integer schemas – within group cases 111

conceptual understanding. Overall, he developed a formal understanding ofnegatives but had di�culty with directed magnitudes, which is re�ected inhis problem solutions. He was able to reason that -7+ -1 would be -8 becauseit is “more less than zero”; yet, he was o�en unable to explain his thinking andwas inconsistent in whether he solved the problem as positive and added anegative or whether he counted forwards or backwards when solving similarproblems.

Discussion – Two Cases of Schema Integration

Although both boys originally referred to negative signs as “minuses,” theythought about them di�erently. A�er learning about negatives, Student 117used the negative signs as a second minus when they appeared next to otherminus signs; whereas, Student 118 used them to designate that the answercame from a problem with negative signs. Student 117 was also able to addand subtract positive numbers from negative numbers, demonstrating thathe was able to apply his current understanding of addition and subtractionto negatives. Student 118 was less consistent in his approach to addition andsubtraction. It is possible that the addition and subtraction instruction wasnot as helpful to him without the explicit mapping of the integers onto thegame board. Overall, these two cases illustrate that students can developformal schemas of some integer concepts before others, and this process doesnot look the same for all students.

integrating integer schemas – within group cases

�e previous two cases provide general examples of how students’ schemasplay a role in the integer arithmetic solutions, but the students started outwith dissimilar integer schemas and had di�erent instruction. �e followingtwo pairs of case studies highlight di�erences for students who had the sameinstruction.

case 1 – students 111 and 119

Case 1 presents two students who are from the same instructional groupand who have formal integer schemas on the post-test, even though theirschemas on the pre-test were slightly di�erent. �e case illustrates that stu-dents can still show di�erences in arithmetic performance on the post-testeven if they demonstrate the same schemas a�er instruction.


Table 10.3 – Student 111’s and 119’s Schema Levels. Student 111’s and 119’s integer schemalevels on the pre-test and post-test.

Student 111, Male Student 119, Female

High performing in math High performing in mathOperations instructional group Operations instructional group

Pre-Test = 4/26 Post-Test = 4/26 Pre-Test = 13/26 Post-Test = 13/26IdentifyIntegers

CountBackward



111 Formal Formal Formal Formal Formal Synthetic 0/3Pre F H L K J D

111 Formal Formal Formal Formal Formal Formal 3/3Post F H L K J G

119 Formal Formal Formal Formal Formal Synthetic 3/3Pre F H L K J F

119 Formal Formal Formal Formal Formal Formal 2/3Post F H L K J G

�e two students presented here (see Table 10.3 for details) both startedwith formal schemas on all integer identi�cation, order, and value tasks. Also,both students only had di�culty interpreting the language of less low onthe directed magnitude task but gained formal understanding of directedlanguage by the post-test. A main di�erence between these two studentson the pre-test was that Student 111 thought subtraction was commutative;whereas, Student 119 stated that subtraction problems and their reversalswould result in di�erent answers.

Students 111 and 119, Pre-Test

Students 111 and 119 appeared to have a similar understanding of the integerconcepts, yet they applied them to the arithmetic problems in distinct ways.On the pre-test, Student 111 reversed the order of the numerals on 7 out of15 subtraction problems (not counting -5 − -5 or -8 − -8). Student 119 onlyreversed numerals on 2 of these problems. Student 111 also guessed on overhalf of the questions (15/26). When not guessing, he treated the numbersas if they were positive, and only provided negative answers on three of thequestions. Student 119, on the other hand, started at the initial number inthe problems and counted on 15 of the 26 questions. Alternatively, she solvedthe problem as positive and sometimes added a negative to the answers. Forexample, on -4− -3, she said, “�ree plus one is four, and there was a negative,so I put negative one.” Overall, student 111 answered 15% of the problemscorrectly and student 119 answered 50% correctly.

case 1 – students 111 and 119 113

Students 111 and 119, Post-Test

A�er receiving instruction on the operations, the students’ strategies onthe arithmetic problems changed in subtle ways. Student 111 still reversedthe numerals on 5 out of the 15 of the subtracting problems even though hehad said that subtraction was not commutative, and he said he guessed onover half of the problems (17/26). However, his guesses were much morerealistic, and were all negative (or zero) instead of positive. On the pre-test,he guessed 100 for the problem -7 + -1; on the post-test, he answered -6. Healso treated negatives as worth zero on 6 of the 26 problems, ignoring thenegative number but keeping the negative sign in his answers. For example,on 5 − -3, he wrote “5” and then wrote a negative before it. He explained, “Ithas a negative number, and there’s a �ve.”

On the post-test, Student 119 counted on 17 of the 26 problems. However,rather than talking about counting backward as she did on the pre-test, sheused the language “less positive”, “less negative”, and so forth, as discussed inthe lessons. �is strategy only worked for her, though, when both numeralswere negative. In the lessons, students learned that adding a negative meansmoving in the more negative direction. When looking at a problem, such as7 + -3, the plus signs tells them that they are moving more and the numberfollowing the plus sign tells them whether they are getting more negative ormore positive. In the example, the -3 indicates that they are getting morenegative. Rather than using the second number for this information, Student119 used the �rst number. In the example, she would see that the 7 is positiveand move more positive three spaces and incorrectly answer 10. However,on -8 − -5, she correctly counted less negative and wrote -3.

Even though the students’ strategies changed slightly from pre- to post-test,Student 111 still only answered 15% of the questions correctly, and Student 119still answered 50% of the questions correctly. In some cases they answeredthe same questions correctly but not always.

Discussion – Case 1

Although both Student 111 and Student 119 ended with formal integerschemas, their di�erent approaches to the problems on the pre-test carriedover into the post-test. Student 119 seemed more comfortable using a strategy(counting) or adding a negative to a possible number combination to try andsolve the pre-test problems. Student 111, however, was less willing to try tosolve the problem and instead resorted to guessing, even though he appearedto understand the relation between negative numbers and positive numbers.


He also avoided the need for negative numbers by reversing the numerals inproblems like 3 − 9 even though he had previously said that would give hima di�erent answer.

Why did Student 111 rarely provide negative answers? �e results do notprovide a clear answer; however, he was much more willing to provide nega-tive answers on the post-test a�er hearing the operations language and mov-ing his game piece onto the negative side of the board, so it is possible thatthese experiences made him feel more comfortable about negative answersbeing possible. Perhaps without further instruction on how to use negatives,he could not transfer the knowledge on his own.

Since Student 119 was already willing to accept negative answers and try touse her knowledge of negatives, this may explain why she was more receptiveto the more negative, less negative style of language presented in the lessons.At the same time, her misapplication of the language suggests that for thisinstruction to be successful, students might bene�t from more experienceslinking the language to the quantities they manipulate.

case 2 – student 108 and student 204

Case 2 presents two students who come from the same instructional groupand have the same integer schemas on the pre-test. �is case illustrates howstudents who start out with the same integer schemas do not always end withthe same schemas, even a�er having the same instruction.�e two students presented here (see Table 10.4 for details) both started

with initial schemas on all integer identi�cation, order, and value tasks. Bothalso had trouble interpreting the language of less low on the directed magni-tude (cat) task. Additionally, both students initially believed that subtractionwas commutative.

Students 108 and 204, Pre-Test

On the pre-test, both students provided only positive answers or answersof zero. Student 108 answered zero 4 times, and three of these times it was dueto miscounting and subtracting too much. She identi�ed the negative signs as“minus” on 8 of the problems, but even a�er identifying them, she frequentlysolved the problems as if they were positive. On half of the problems, though,Student 108 treated the negative sign as a minus sign but did so in a strangeway. Regardless of whether the problem involved addition or subtraction,she counted out the absolute value of both numbers on her �ngers and then

case 2 – student 108 and student 204 115

Table 10.4 – Student 108’s and 204’s Schema Levels. Student 108’s and 204’s integer schemalevels on the pre-test and post-test.

Student 108, Female Student 204, Male

Low performing in math High performing in mathOperations instructional group Operations instructional group

Pre-Test = 0/26 Post-Test = 3/26 Pre-Test = 2/26 Post-Test = 9/26IdentifyIntegers

CountBackward



108 Initial Initial Initial Initial Initial Synthetic 0/3Pre A B Do C C F

108 Synthetic Initial Synthetic Initial Initial Synthetic 1/3Post C B Aw C C F


204 Initial Formal Formal Initial Initial Formal 2/3Post A H L C C G

took away the second number. For 4 − -5, she said, “Oh, there’s those twominuses...four and �ve (puts up four �ngers on le� hand and �ve on righthand). Take away �ve more. One, two, three, four, �ve. Makes four.”

On the other hand, Student 204 answered zero 11 times. He answeredzero to all but one problem where he had to subtract a number with largerabsolute value from a number with smaller absolute value. For example, whilehe answered 4 − -5 = 0, -3 − 5 = 0, and 3 − 9 = 0, he solved 1 − 4 = 3. As withthese problems, he solved all the others as if the numbers were positive.

Students 108 and 204, Post-Test

On the post-test, a�er having eight lessons on operations, Student 108continued to provide positive answers or answers of zero. While she identi�edtwo numbers as negative, she continued to solve the problems as though theywere positive. �is is unsurprising because she showed little change in herinteger schemas on the other questions. Even though she only added and tookaway the same number twice this time, she also gave many more unrealisticanswers without providing su�cient justi�cation of them. For example, on

-5 − -5, she said, “Five and �ve...ah, yes (writes 4) four.” When asked how shegot her answer, she merely said, “I think in my brain.” On occasion, Student108 got an answer correct, but it is unclear based on her descriptions whethershe knew why.

Unlike Student 108, Student 204 developed formal schemas for the numberline, counting, and directed magnitude tasks, the tasks most supported by


the instruction he received. �e ordering task should have been easy if hethought about it in relation to a number line; however, the question is framedin terms of value, which could have made it more di�cult for him. Onthe post-test, he provided 16 negative answers on the arithmetic questions.Without prompting, he pointed out negatives on 17 of the problems and usedthe speci�c language “more positive”–as used in the lessons–correctly. For

-4+6, he counted from -4 through zero to correctly answer 2. Student 204 wasonly able to correctly answer problems where he was adding or subtractinga positive number. However, he could do this regardless of whether the �rstnumber was positive or negative.

Discussion – Case 2

Even though Students 108 and 204 demonstrated the same schemas on thepre-test, their performance on the arithmetic problems and their subsequentlearning was varied. Student 108 was considered to be low-performing inmath in general. Her attempts to solve the arithmetic problems match thisassessment. Not only did she have di�cultly determining whether to addor subtract, but she frequently miscounted or incorrectly used her �ngers tohelp her. Although she was able to identify a couple negative numbers on thepost-test, she was not always accurate in applying positive number conceptsto solve the arithmetic problems, so it is likely this made it di�cult for her togain, let alone apply, any new integer number sense from the lessons.

In contrast, Student 204 was comfortable with positive number arithmeticand considered high-performing in math. Although negative numbers werenot speci�cally discussed in his lessons, he did pick up on the order of negativenumbers, which were shown on one of his game boards. He was not able toapply the instruction on adding or subtracting negatives, perhaps becausehe did not understand the values of the negatives. Using a mental integernumber line, he was able to start from a negative number and add and subtractpositive amounts correctly on the post-test, demonstrating he could applyhis current operations knowledge to the new concepts. Overall, the results ofthese two students support the hypothesis that having an integrated centralconceptual structure for whole numbers is an important factor in learningnew number concepts, like integer addition and subtraction.

integrating integers schemas – between groups case 117

Table 10.5 – Student 102’s, 217’s, and 419’s Schema Levels. Student 102’s, 217’s, and 419’sinteger schema levels on the pre-test and post-test.

Student 102 Student 217 Student 419

Gender Male Female FemaleMath Performance Medium Medium High

Group Full Instruction Integer Properties Integer OperationsPre-Test 3/26 2/26 2/26Post-Test 11/26 7/26 4/26

IdentifyIntegers

CountBackward




102 Synthetic Initial Synthetic Initial Initial Synthetic 1/3Post C B Aw C C F





integrating integers schemas – between groups case

�is �nal case explores one student from each of the three instructionalgroups. �ese students started with the same integer schemas according tothe pre-test, and a�er their instruction, their schemas had changed in speci�cways.

As shown in Table 10.5, Student 102, Student 217, and Student 419 all inter-preted negatives as positive numbers on the pre-test. Additionally, all threestudents thought that subtraction was commutative on at least two of thesubtraction properties questions.

Students 102, 217, and 419, Pre-Test

�e three students’ performance on the pre-test was remarkably similar.All three students provided positive answers or answers of zero. On 15 ofthe 26 problems, all three students answered identically. While Student 102skipped an additional 6 questions, Students 217 and 419 provided the sameanswer on 8 of the remaining problems. Each of these students answered zerofor problems where they had to subtract a number with larger absolute valuefrom a number with smaller absolute value (e.g.,3 − 9, -5 − 9, and-2 − -6),


which is slightly surprising since they thought subtraction problems and theirreversals would give them the same answer. For the other problems on whichthey agreed, students solved the problems as if the numbers were positive.A�er participating in their respective instruction, these three students nolonger solved the problems identically.

Student 419, Post-Test

Student 419 received the Operations instruction and did not appear todevelop a more sophisticated integer schema. In fact, on the counting back-ward and number line tasks, she le� out zero and stopped at 1. However, shedid identify negatives within the arithmetic problems, which suggests thather integer naming schema was tied to the problems similar to those shehad been using in the lessons and was not applicable to the isolated numberquestions.

On the post-test she continued to solve problems like 4− -5 by treating thenumbers as positive and answering “0” for �ve of the problems. However, infour additional cases, she answered “-0”. While she answered a few problemscorrectly, the only problem type she consistently answered correctly was

-L + -S, where L > S > 0, such as -6 + -4. �is is one of the problems shecould get correct without having a formal value schema; she only needed toadd a negative to the positive answer to 6 + 4. On other addition problemsshe appeared confused about whether she should have a positive or negativeanswer, although she always got an answer with a larger absolute value. Forexample, when solving -4 + 6, she treated both numbers as positive, countedon her �ngers and answered “10”. However, on -1+8, she wrote “-9” and stated,“It’s more.” She did not have a stable schema for when she was supposed touse the negative, and her belief that -9 is “more” than -1 or 8 is consistentwith her treating negatives as positive on the integer comparison tasks.


Student 217 received instruction in identifying negatives and negative signs,as well as the value and order of negative numbers. By the post-test, she haddeveloped a formal schema for integer values and numerical order but didnot always identify negatives in equations. She also continued to misinterpretthe meaning of less low.

On the arithmetic questions, she still solved -5 − -5 and -8 − -8 correctly,answered zero for two problems, and skipped one problem. Unlike eitherof the other students, on the rest of the arithmetic problems (21 out of 26)


she only supplied negative answers. For 18 out of the 26 problems, she pro-vided answers consistent with solving the problems with positive numbersand adding a negative sign; sometimes she also reversed numerals to avoidsubtracting a larger number from a smaller one even though she had indi-cated that subtraction is not commutative. A�er solving -5 − 9 = -4, Student217 explained, “I know nine minus �ve equals four, and I saw a negative on it(points to the negative).” Student 217’s strong schema for negatives, togetherwith a lack of instruction on negative number operations, may have in�u-enced her application of negative signs to the majority of the problems. Fromthe instruction, she knew negatives were important; therefore, she made mostof her answers negative.


Figure 10.1 – Student 102’s NumberLine.

Student 102 received abbreviated forms of the other two students’ instruc-tion; he learned about negative number order, value, identi�cation, opera-tions properties, and directed magnitudes. It initially appears that he did notdevelop formal schemas of the integer concepts, especially order; however,these results might partially be due to the structure of the interview. On thehorizontal number line, Student 102 correctly �lled in the missing positiveand negative numbers, but he le� out zero. �e number line on the next pagelooked identical but with the paper turned to make it vertical. Student 102�lled it out with the negatives on top (which is where you would put the neg-atives if the number line was horizontal). �is switch in orientation causedsome di�culty for other students and may have been troublesome for him aswell because, as seen in Figure 10.1, he later produced his own number line,which was correct.

Furthermore, when ordering the integer cards, Student 102 originally hadthe �rst set ordered correctly -8, -4, -2, 1, 5, 7. He had said that seven was thelargest and one was the least. When I pushed him on this, he rearranged thenumbers so that 1 was �rst but continued to talk about negative two as beingsmall. Although he had di�culty determining which number was greatestor least, he correctly determined the greater of two integers on all trials ofthe comparison task. �ese results suggest that while he has some formalunderstanding of integer value, it is still weak and in�uenced by context.

On the post-test, Student 102 provided 12 positive answers and 11 negativeanswers. Rather than solving the problems as positive and sometimes oralways answering with negatives like the other two students, Student 102 useda vertical number line to solve 24 of the 26 problems. While he did not use


the less positive, less negative language discussed in the lessons, he seemed togeneralize this language to “up” and “down.” �is is slightly surprising sincethe games for these concepts were on horizontal number lines. When solving

-8 − -5, he explained, “If it does a minus [and there’s a negative sign], countup. If it does a plus and there’s a negative sign, go down.”

Although he correctly solved 11 of the 26 problems, he sometimes countedin both directions for the same type of problem. For example, for -3 + 1, heanswered “-4”; whereas, for -9 + 2, he answered, “-7.” He also miscountedeight times by counting the initial number as the �rst number subtracted insubtraction problems. For 1 − 4, he started counting at 1 for his �rst count,then moved down to zero [“2”], negative one [“3”], and reaching his fourthcount answered “-2.” If he had counted correctly, Student 102 would haveanswered 5 additional questions correctly.

Discussion – Students 102, 217, and 419

Much like what was revealed through the quantitative results and theschema changes discussed previously, Student 217 from the Integer Propertiesgroup developed the most formal schema of integer identi�cation, value, andorder, Student 102 from the Full Instruction group developed an intermediaryschema that was close to formal, and Student 419 from the Operations groupmaintained an initial schema. However, without instruction on operations,Student 217 was unable to change her notions of addition and subtraction toincorporate negative numbers. Rather, she and Student 419 to some extent,added or subtracted numbers as if they were positive and then made themnegative; they maintained an initial understanding of addition and subtrac-tion. Student 102, on the other hand, was still solidifying his integer schemasbut was able to use the vertical number line to begin to use negative numbersas part of the operations.�e performance of these three students highlights two of important

points. One, as demonstrated by Student 217, students may demonstratea solid understanding of integer concepts in isolation but may have troubletransferring this knowledge to arithmetic problems, especially if they havenot had any exploration of integer addition and subtraction. Perhaps negativenumbers only take on true meaning once students have to manipulate themor use them in relation to positive numbers. �is also seems to be the casewith reversing subtraction problems. Students’ decision whether to reversesubtraction problems was not closely related to whether they previously saidthis would result in the same answer or not.


Two, as demonstrated by Student 419, without the underlying conceptsto support them, students have di�culty using and attaching meaning torepresentations like the number line. Furthermore, without a schema for in-teger order and values, Student 419 was unable to build a schema for directedmagnitude that she could relate to the number line. Rather she continued tosolve problems as positive and added negatives haphazardly to some of heranswers.


11 O V E R A L L D I S C U S S I O N

�e quantitative and qualitative results of this study provide some initialanswers to the research questions posed previously.

What are �rst grade students’ schemas of negative integers (in terms of the ele-ments underlying the central conceptual structure for integers: written symbols,order, value, and directed magnitude)?

Before formal instruction, about half of the �rst graders correctly in-terpreted the directed magnitude language, except for less low. Similarly,students had the most di�culty interpreting the meaning of less negative.Previous studies on children’s understanding of the terms more and lessfound that while young children can distinguish between these terms, theyhave a more di�cult time interpreting the meaning of not more and not less(Wannemacher & Ryan, 1978). �e results of this study suggest a similarpattern for less positive and less negative, with the latter proving the mostchallenging of all the pairs. Children’s di�culty interpreting less negativemay partly explain why previous studies, and the present one, have foundthat children have a harder time solving problems involving subtracting anegative.�e di�culty that young students have with the language of negatives

suggests that while learning plays a large role in helping students extendtheir numerical central conceptual structure into the negatives, maturationplays a role as well. As mentioned previously, students at a bidimensionallevel of numerical reasoning are able to reason about two dimensions, suchas tens and ones simultaneously (Case, 1996). Similarly, to interpret themeaning of less negative, students need to coordinate their reasoning aboutthe meaning of less with the meaning of negative. In general, children donot reach bidimensional thought until ages 7 – 10, so the directed magnituderesults might be a re�ection of this type of maturation.

Furthermore, it is possible that directed magnitude understanding is nota separate row of the central conceptual structure for integers but a re�ection

123

124 11 ⋅ overall discussion

of students’ understanding when they employ logic in combination with bi-dimensional reasoning. �at is, once students are able to think about positiveand negative quantities or parts of the number line simultaneously, they canutilize logic to explore the relationship between these dimensions. �is studyexplored the logic involved in relating negatives and opposites to the ideasof more and less; however, applying logic to the symmetrical structure of theintegers could also lead students to uncover the additive inverse property(e.g., -5+ 5 = 0). Even if the directed magnitude row is part of bidimensionalinteger reasoning, it does not mean that logic is only part of the bidimensionallevel. Students solving integer problems involving one dimension also usedlogic by attributing negatives with the properties of positive numbers. Forexample, some students solved -8 − -5 by reasoning that if they had negativeeight things and took negative �ve of them away, they would have negativethree le�.

Aside from directed magnitudes, only two �rst graders had a formal un-derstanding of all integer properties before instruction. Nine students hadformal understanding of one or more integer concepts but initial or syntheticschemas of the others, and an additional three students demonstrated formalschemas for a concept when asked in one context (e.g., identifying integersin isolation) but then had di�culty when the same concept was presented ina di�erent context (e.g., identifying integers within equations). �ese resultsare consistent with previous studies in which authors have found childrenacross grade levels who demonstrated some understanding of negative num-bers before formal instruction (Murray, 1985). Framing students’ schemasin terms of the elements that compose the central conceptual structure forintegers, however, helps illuminate the variety of synthetic schemas studentsmight have.

In the current study, the majority of students who already knew aboutnegative numbers also mentioned that they heard or learned about themfrom their family or friends. It is likely that other young students who showsome understanding of negative numbers have had speci�c encounters withthem, rather than generating understanding of them on their own. Eventhough some students may demonstrate integer understanding, their pre-instructional understanding does not look the same across children. Somestudents only had formal understanding of integer values while others couldonly identify negatives and count into the negatives. Furthermore, just be-cause a student could count into the negatives did not mean he or she couldorder them when given integer cards.

125

Such inconsistent results among children �t within the proposed frame-work for the central conceptual structure of integers. �e framework suggeststhat students not only can but will develop formal schemas for the variousinteger properties at di�erent rates, as was possible when they learned wholenumbers. Additionally, children’s primary responses to the integer questionsbefore instruction were consistent with treating negative numbers as posi-tive numbers or treating negative signs as subtraction signs. Such behaviorprovides support for the assertion that students do not develop a separateconceptual structure for integers but that negative number knowledge buildson whole number knowledge.

How does the relation among students’ integer schemas manifest itself in stu-dents’ approaches to integer addition and subtraction problems?

Children’s integer schemas in�uenced their arithmetic solutions in twoprimary ways (and many more subtle ways). On the pre-test, the majorityof students had initial integer property schemas. Similarly, students solvedproblems as if the numbers were positive, essentially treating the arithmeticproblems as whole number problems that result in positive and zero answers.Peled et al. (1989) also reported on �rst- and third-graders who solved arith-metic problems this way. By the post-test, many more students developedformal schemas for integer properties, and likewise students who solved prob-lems as if the numbers were positive now added negatives to their answers.Students who recognized negatives, especially those in the Integer Propertiesgroup, were about 10% more likely to supply negative answers than studentsin the other groups at post-test. While Schwarz et al. (1993) identi�ed thisstrategy in their sample of ��h-graders, the results here provide a pictureof one way in which students transition from only using positive numbersto getting negative answers a�er exposure to and instruction in negativenumbers.

Aside from solving problems as positive and adding negative signs, stu-dents who demonstrated formal understanding of integer order also countedor used number lines to solve the problems. In one case, a student reasonedabout operations with negative numbers without having a formal notationfor them. As with the whole number central conceptual structure, it appearsthat formal notation of negative numbers is not necessary before one canadd and subtract them. Rather, this case suggests that students could extendtheir conceptual structure from whole numbers to integers and then learn theformal addition and subtraction notation rather than learning the notation


while developing the whole number structure and then applying the notationas they learn about negative numbers.

On the contrary, proper development of whole number order and valuedoes seem necessary prior to understanding negative number order and value.Students who randomly ordered the integer cards on the pre-test had lowschemas in general on the post-test. Additionally, most of them only wrotepositive answers on the post-test. Beyond their integer property schemas,students’ integer operations schemas in�uenced their arithmetic solutions.As mentioned in the quantitative results, one of the “easiest” problem types forstudents on the pre-test was subtracting a larger negative from a smaller one(e.g., -4−-7) because they could ignore the negatives and solve 7−4 = 3. Otherresearchers (Bruno & Martinon, 1999; Peled, 1991) claim that these types ofproblems are di�cult for students because solving them involves crossing thezero point and interpreting the meaning of subtracting a negative. However,these results show that theoretical di�culty and practical di�culty are muchdi�erent and both should be addressed in theories on problem di�culty.

Students’ desire to start with the larger absolute value when solving a prob-lem led some to solve 1 − 3 = 2, even a�er asserting that 1 − 3 would have adi�erent answer than 3 − 1. Aside from their “start with the larger number”rule, students also had di�culty shi�ing from thinking about addition andsubtraction in terms of changes in ordered value to changes in directed mag-nitudes. In terms of the central conceptual structure for whole numbers, thisresult makes sense, since the directed magnitude reasoning line is not partof that original structure and students use their schemas of ordered value toadd (get more) or subtract (get less). Developing the directed magnitude un-derstanding (whether it is part of the underlying structure or bidimensionalreasoning) represents a conceptual–and to some degree maturational–shi�.�e rules students develop from working with only positive numbers did notgeneralize well to negative number operations and were resistant to change.

How does instruction in various elements of the central conceptual structurefor integers in�uence students’ understanding of negatives as measured throughtheir changing schemas, language, arithmetic accuracy, and approaches to inte-ger arithmetic problems?

In general, students who received integer property instruction developedmore sophisticated schemas in those areas; students in the Full Instructionand Integer Properties groups signi�cantly improved on the integer property

127

items compared to the students in the Integer Operations group. Further-more, although middle- and high-performing students had signi�cant gainscompared to the students classi�ed as low-math performers, non-high per-formers had signi�cant gains in the Integer Properties group, leading to arejection of the null hypotheses that their mean gain scores were equal inthese cases. Students’ performance on the integer problems, then, was notjust a result of which type of instruction they received but also which schemasthey started with before instruction. Just as students who are missing partsof the whole number conceptual structure have di�culty learning how tosolve addition and subtraction problems (Gri�n et al., 1995), students wholacked integer properties understanding had di�culty using this knowledgebecause of the foundation they lacked.

For example, students who received instruction on properties of opera-tions and directed magnitudes developed more advanced schemas in thoseareas, but they were less likely to apply this knowledge to the problems, par-ticularly in the case of solving 3 − 9 and other similar problems. It appearsthat students need an understanding of negative number properties for thedirected magnitude concepts to hold meaning. �erefore, in regards to thequestions most reliant on directed magnitude understanding and the three hy-potheses presented previously, there were no signi�cant di�erences in meangain scores among the three instructional groups, the three performancelevels, or the interaction of the two for the arithmetic, operations properties,and directed magnitude items.

Based on these results, certain aspects of the instruction worked betterthan others for di�erent students. In general, the integer operations instruc-tion on its own was not successful. While students in the Integer Operationsgroup were able to move more and less negative or positive on the numberline board games, they had di�culty connecting these motions to the numer-ical problems. Furthermore, they had less success in using the language ofdirected magnitude to help them order or judge the values of negative integers.Without the understanding of what negatives are, the directed magnitudeinstruction was not meaningful.

However, a few students in the Integer Operations group already knewabout negatives before the instruction. �ese students had a more formalcontext for understanding the directed magnitude language, and they startedpaying more attention to both dimensions of the arithmetic problems. For oneof these students, this meant he was able to solve all but one problem correctly.While other students started to use the directed magnitude language, they


did so incorrectly; instead of interpreting the problem 5 + -2 as “Five, morenegative two,” they determined whether to get more negative or positive bylooking at the sign of the �rst number and interpreted the problem as “Five,more positive two.” �erefore, the results suggest that while students canbegin to use directed magnitudes once they have some understanding ofnegatives, the directed magnitudes need to be continually mapped onto theinteger values to ensure they develop meaningful connections between theconcepts.

Contrary to the integer operations instruction, the integer properties in-struction helped the majority of students to progress in understanding neg-ative numbers. Students in the Full Instruction group had three days ofinstruction focused on integer properties, and a�er this brief introduction,most of them transitioned to having synthetic or formal schemas for theconcepts. Additionally, because they spent almost every lesson using num-ber lines, all of these students moved away from having initial number lineschemas.�e extended focus on negatives that students in the Integer Properties

group received, however, bene�ted non-high-performing students more thanin the Full Instruction group. Students considered lower-performing in mathtend to be those students who, compared to their peers, have less sophisti-cated number sense and may need more time to process new concepts. �eextra practice time gave them the opportunity to think about the conceptsmore thoroughly, and many of them advanced to more sophisticated schemasthan similar peers in the Full Instruction group. Additionally, students whostruggled with whole number concepts on the pre-test advanced in theirwhole number understanding–although to a lesser extent in their integerunderstanding–through participating in the lessons focus on number valueand order. Table 11.1 summarizes which types of children bene�ted from thedi�erent instruction.

Finally, students in the Full Instruction group who had the combinationof instructional lessons were more likely to use counting to solve the integerproblems. While their counting did not always align with the correct directedmagnitude, it did help about half of them count through zero to get positiveor negative answers. Even if maturation plays a strong role in students’ under-standing of directed magnitude, using directed magnitude as a frameworkfor counting forward and backward on the number line encouraged many ofthe students who had developed formal integer schemas to use their internalnumber line to help them solve the problems.

129

Table 11.1 – Performance of Students by Instructional Group in Shifting Schemas. Students (ac-cording to general math performance level) who bene�ted from each form of instruction.Codes for the table are as follows: students in this group developed partial or formal schemasfor the majority of items (Majority), for at least half of the items (≥ 1/2), or for less than halfof the items (< 1/2).

Integer Operations Instruction Integer Properties Instruction Full Instruction

Low-Performing < 1/2 ≥ 1/2 < 1/2

Medium-Performing < 1/2 Majority ≥ 1/2

High-Performing Majority (but only if alreadyhad some knowledge of nega-tives)

Majority Majority


12 C O N C LU S I O N

implications

�is research investigated �rst grade students’ schemas of negative num-bers and explored how these schemas changed and helped or hindered stu-dents as they solved integer addition and subtraction problems. A main goalof the conceptual structures research has been to demonstrate that studentsuse an integrated set of central concepts to reason about problems; with ad-ditional maturation and experience, students can navigate problems withmore than one dimension (Case, 1996). �erefore students who use theirinternal number line structure to solve single-digit addition can move towardusing two number lines to solve problems that require manipulation of twodimensions, such as tens and ones.

�e results of this study indicate that aside from integrating multiple wholenumber lines together to solve multidimensional problems, students canbuild upon the conceptual structure of whole numbers and expand it, cre-ating a conceptual structure of integers. �is study, like other studies of thecentral conceptual structure for number discussed by (Case, 1996), providesfurther evidence that students can understand di�erent types of abstractionmuch earlier. �is point is important because it suggests that the di�culty ofmathematics concepts–and negative numbers especially–is to a large extentdependent on what prerequisite knowledge students have and less on howabstract a concept is. However, while the extension of the central conceptualstructure to include negative numbers occurred due to instruction ratherthan maturation, the directed magnitude concepts do seem more reliant onmaturation. Directed magnitude reasoning, supported by logical reasoning,might be another instance of bidimensional thinking rather than another rowadded onto the central conceptual structure.

In the same way that students can learn about the value and order ofpositive numbers separately before integrating this knowledge, students’ un-derstanding of negative number value does not guarantee understanding of

131

132 12 ⋅ conclusion

negative number order and vice versa. �is similarity to whole number devel-opment suggests that students need more time than we currently provide ininstruction to work with negative number concepts. For example, currently,in the enVisionMATH (Pearson Education, 2009) curriculum for ��h grade,students receive a total of three lessons on the order and value of negativenumbers. Following this short introduction students have two lessons onaddition and subtraction with integers.

Although curriculum developers, researchers, and policy makers continueto place negative numbers late in the curriculum, this study demonstrates thatstudents are quite capable of learning about integers much earlier than ��hgrade. Starting integer instruction earlier could allow students more time towrestle and become familiar with these concepts before using them in alge-braic contexts. Providing students with extra time to wrestle with negativenumber concepts would particularly bene�t students who are considered lowperforming in math. As discussed previously, the low-performing studentswho consistently improved in moving towards formal understanding of inte-gers were those who spent eight lessons working with the integer properties.Cutting this time of focused exposure down to three days resulted in the lowperforming students maintaining their initial understanding. Researchersand educators need to reexamine curricula and standards and evaluate waysnegatives could be introduced for a longer period of time (if not many gradessooner).

One barrier to introducing negative numbers earlier is time. Teachersalready cover an enormous amount of material in the early elementary yearsand making room for negatives would likely mean that some other conceptis removed. A possible way to incorporate exposure to negative numbersearlier is, for example, through the use of games–much like those used in theinstruction in this study. Many teachers make games available to studentsduring free-time or during a certain time of the week, and they could addnegative number games into the mix. �ese games could also be sent homefor students to play with their caregivers or siblings. Another alternative isto include discussions of negative numbers when whole number order andvalue concepts are taught throughout the school year; short but cumulativeexperiences, such as these, could also provide students with longer exposureto negative numbers earlier. �us, there are multiple ways to address the timeissue in introducing this concept in earlier grades.

Another barrier to early integer instruction is testing. If curricula requireteachers to teach students about negative numbers earlier, it is likely that

implications 133

state tests will include negative number questions earlier. While the purposeof introducing negatives earlier is to allow students more time to becomefamiliar with the concepts, current approaches to testing do not align withthis philosophy. In an ideal world, students would have a few years to grapplewith a concept before being tested on them on high-stakes tests, and teachersacross several grades would be responsible for helping students reach formalunderstanding.

Currently, multiple teachers do in�uence students’ learning of conceptsover time, but only students’ current teachers are held responsible for stu-dents’ performance on their grade-level tests. Given the current testing andaccountability climate, if negative number concepts were not included onthe state tests, it is likely that teachers would focus on teaching other topicswhich are on the tests in lieu of the negative number instruction. Although itwould be unfortunate if negative number instruction could only be pushedto earlier grades if they were also tested at those grades, the longer term ben-e�t of providing students extra time to develop understanding of negativesmight outweigh the near-term testing pressures. Moreover, the students inthe Integer Properties group in this study demonstrated that even studentsconsidered low-performing can bene�t from an extended focus on the topic,so worries about testing the concepts earlier might be overstated.

Although interpreting the meaning of more negative, less negative, morepositive, and less positive was di�cult for the students, learning the directednature of addition and subtraction without formal understanding of negativeswas not helpful. Addressing the directed magnitude de�nition of addition andsubtraction, then, would not be helpful for students without having a contextfor why this information is useful. However, recognition of negative numbersand knowledge of their order and value is not su�cient to ensure that studentsknow how to apply it towards solving addition and subtraction problems.Again, students need more time to work through the changing meanings ofaddition and subtraction; their inconsistency in deciding which way to countsuggests that they are trying to make sense of the new information.

An important part of this research is the breadth of knowledge it suppliesregarding students’ synthetic schemas for the central conceptual structureof integers. Students’ intermediary thinking can provide pre-service andin-service teachers with valuable information about how students’ thinkingprogresses, along with typical areas of di�culty students have with negativenumbers. �is knowledge will help teachers plan instruction to target theseareas of di�culty.


Furthermore, the central conceptual structure for integers frameworktogether with the conceptual change framework serve as helpful tools foridentifying concepts which might confuse and challenge students. For in-stance, in the integer conceptual structure, zero stands out as a unique point,which does not have an opposite and serves as an endpoint for the wholenumbers. Its placement in the structure suggests that students will not havea complete understanding of negatives without a complete understanding ofzero. �e conceptual change framework would suggest that children wouldhave trouble accepting zero as worth nothing and as a quantity that has mean-ing and greater value compared to negative numbers. In practice, studentsdid struggle to distinguish between zero and negative numbers and this isanother area in which future instruction should focus.

As discussed in the previous chapter, students will bene�t from di�erenttypes of instruction depending on their current conceptions. For example,students considered high-performing in the Integer Operations group whoalready knew that negative numbers existed and had some ideas about thevalue of negative numbers bene�ted from instruction on how to use the valueof negatives together with the meaning of addition and subtraction. However,students who ignored negative signs did not receive instruction tailored totheir needs and made less progress in their integer understanding. Table 12.1illustrates aspects of integer learning identi�ed in this study that are importantfor teachers to understand as they develop pedagogical content knowledgeof integers. Using this information and further research as a guide, teacherscan tailor their integer instruction to address the needs of their students asthey support students’ developing integer understanding.

limitations

As with all research, this study has limitations that reduce its power orrestrict the types of conclusions researchers and educators can draw fromit. �e period of instruction for this study was quite short. Even thoughthere were still di�erences between instructional groups, ideally students inthe full instruction group would have had the complete integer instructionplus the complete addition and subtraction instruction. However, this wouldhave meant students in the other groups would have had a lot of practicein their concepts. Because all students had to have the same amount ofinstructional time, this problem was unavoidable; consequently, it is unclearwhy the Full Instruction group did not surpass the Integer Properties group

limitations 135

Table 12.1 – Pedagogical Content Knowledge for Integers. Possible student conceptions ofintegers, types of strategies aligned with integer conceptions, and instruction that targetsthese conceptions. As illustrated previously in Table 11.1, students will bene�t from havingextended experiences exploring the meaning, order, and value of negative numbers beforeworking with the directed magnitude understanding that is important for addition and sub-traction.

Students’ Integer Conceptions Integer Addition and Subtrac-tion Strategies

Possible Targets for Instruction

Negative signs have no meaning(no such thing as negatives).

Ignore negatives, read problemsas positive, solve 1 − 3 by stop-ping at zero or reversing thenumbers to avoid the possibilityof negatives.

Instruction showing the needfor negative numbers: temper-ature, distances under water,�oors under the ground, penaltypoints

Negatives mean subtract. Use the negative as a subtrac-tion sign, so to solve 9 − -3would subtract twice, treats thenumber as already subtractedand worth zero

Instruction focusing on the dif-ferences between negative signsand minus signs

Negatives designate a quality ofthe problem.

Students solve the problem aspositive and add a negative totheir answers no matter if oneor both numbers in the problemare negative.

Instruction focusing on nega-tives as speci�c points, practicecounting starting at negatives toadd and subtraction

Negatives are worth zero. May only consider the negativeworth zero if you are adding orsubtracting it. May solve -9 + -4as -9 + 0 or as 0 + 0.

Instruction focusing on thevalue di�erences of the num-bers. Focus on changes in dis-tance from points.

Negatives are less than zero,but their values mirror positivenumbers (-5 > -3).

May confuse in which directionto go to add or subtract. For

-9 + 5, may get -14 because it isgetting larger in negatives.

Instruction focusing on thevalue di�erences of the num-bers; e.g. more negative meansbeing further underground.

Zero can be negative. Student may include negativezero in their counting sequenceor solve a problem like -8 − -8 aspositive and make it negative.

Focus on zero as the referencepoint from which numbers getmore or less positive and moreor less negative.

Negatives are less than zero; neg-atives larger in absolute valueare smaller.

If students hold onto the be-lief that you cannot subtract asmaller number from a largerone, they will claim that youcannot answer problems like7 − -5.

Focus on helping students rea-son about adding and subtract-ing as more or less positive ornegative.


or why the lower performing students did not do as well in this group as theircounterparts did in the Integer Properties group. It could be because theyhad abbreviated versions of both types of instruction or because the additionand subtraction instruction created too much cognitive dissonance on top oftheir new learning.

Also related to the timing of the study, students completed the post-test in-terviews within two weeks of the instruction. Consequently, students’ perfor-mance on the post-test might re�ect more short-term and unstable learningrather than a more permanent elaboration or restructuring of the central con-ceptual structure of integers. Further exploration of students’ understandingover time would help clarify the stability of their conceptions.

Unlike classroom instruction where students’ performance one day candrive instruction for the next day, the lessons for this study were carefullyplanned so that all common lessons were consistent among groups. �isrestriction (and the rapid pace of the instructional unit) limited my abilityto respond to needs of the students. �erefore, I had to keep on track withthe lesson plans even when I felt like students might bene�t from an extradiscussion or further day of instruction. Although it was appropriate for theintegrity of the study, the e�ect of the instruction might have been strongerwith changes to and/or more freedom in the instruction.

During the lessons, students were allowed to pick their own partners towork with. �is was done in part for management issues and students wereencouraged to work with new partners; the social dynamics of the groupsmay have in�uenced why some students saw more gains than others did.If the students classi�ed as high performing tended to work with otherssimilar to them, this might help explain why the low students made fewergains. On the other hand, if high performing students chose to work withlow-performing students in the just the Integer Properties group, this couldintroduce another possible explanation for why they improved relative tolow-performing students in the other groups.

Finally, due to the variety of students’ schemas on the pre-test, it is hardto generalize how students’ thinking changed because so few students startedat the same level, especially within each instructional group. Additionally,aside from the arithmetic questions, there were few questions that requiredstudents to apply their understanding of integer properties (especially orderand value) in less familiar contexts. Because of this, it is hard to determineif they had trouble applying the concepts in general or speci�cally to thearithmetic problems, which interfered with their understanding of positive

future research 137

number operations. Also, due to the length of the interviews and students’variable tolerance for being asked how they solved each problem, we didnot always ask students’ for their reasoning if they appeared to use the samemethod on subsequent problems. �is, however, limited how much we wereable to say con�dently about certain students’ reasoning in retrospect.

future research

Based on the results and limitations, there are several future directions forthis research. First, I would like to explore the use of the integer addition andsubtraction instruction involving more positive, more negative, less positive,and less negative with older students. I would like to see if this instructionhelps them develop a deeper understanding of operations with negatives thancurrent number line instruction, which focuses on a set of strange rules. Ifthey have di�culty with this instruction it would provide further insight intowhy only a handful of the students in the present study picked up on thislanguage. If the instruction is helpful, I would like to explore teaching a longernegative number unit to �rst graders to see if combining the Operations andInteger Properties instruction in their entirety would help more studentse�ectively navigate the arithmetic questions.

Along with this, I would also like to explore to what extent understandingof addition and subtraction of negatives is related to understanding of doublenegatives. Students had a hard time determining the meaning of less lowand less negative. Additionally, problems like 5 − -3 were some of the mostdi�cult for them to solve. If language plays a role in students’ understandingof negatives, we would need to address this in future instruction.

To tease apart the development of the central conceptual structure for in-tegers, it would be helpful to explore how instruction in just negative numberorder or value in�uences students’ integer schemas (especially on tasks thatdraw on both concepts) as well as their approaches to the addition and sub-traction problems. One way to do this is to teach separate groups of studentsonly value or integer order concepts.

One of the greatest di�culties with this study was capturing all aspectsof students’ schemas. In particular, questions involving zero o�en providedadditional insight into students’ thinking about negative numbers than ques-tions that did not involve zero. For future interviews, it will be importantto develop a set of questions that not only target students’ understanding ofzero on its own but also in relation to negative numbers.


�rough systematic exploration of the concepts involved in negative num-ber understanding, we can slowly work towards building an understandingof the integer learning progression. From here, we can explore the stability ofthis foundational integer understanding as students move into more complexapplications of negatives: multiplication, division, and algebra.

A L E S S O N P L A N S

properties lesson – symbol consistency vs . inconsistency

Full Instruction Group – Lesson 1Integer Properties Group – Lesson 1

Enduring Understandings1. Symbols can have more than one meaning depending on their context

and how they are arranged.2. Integers are symbols that represent speci�c quantities, which can be

ordered on a number line.

Materials – 2 sticks per pair, sticky notes, circle, rectangle, tape, camera

Objective1. Students will identify situations where symbols keep their same mean-

ing and change their meaning.

Introduction

(�e �rst part of the activity is meant to acknowledge that sometimeschanging the orientation of a symbol doesn’t change its meaning.)

T – Show a cutout of a rectangle lying on its long side. “What does thispicture or symbol mean? What do we call it?”

S – May not understand the word “symbol.” (A picture or sign that standsfor something like women sign on bathroom.)

S – May say “box”, “rectangle”, “shape” (Prompt: what kind of shape?)T – Turn the rectangle so the long side is standing up. What do we call

this one?S – Will likely give the same answers or say it’s the same thing. Or they

may say it looks like a “door” now while it didn’t before.T – Okay, so we can turn it and we have the same thing. It still means

rectangle or box. Have students use thumbs-up or thumbs-down to agree ordisagree.

139

140 A ⋅ lesson plans

S – Some may think it’s not the same when turned.(�is next part helps children see that sometimes adding another symbol

doesn’t change the meaning of the original symbol.)T – Show a cutout of a circle to the le� of the rectangle (not touching).

“What do these pictures or symbols mean or what would we call them?”S – May not know what you mean. (Ask students what shape the circle is.

Tell them when we see that symbol, we think “circle” because that’s what itmeans.)

T – Keep track of the words students use.S – May call the circle “circle” or “round”S – May call the rectangle “rectangle” or “box”S – May claim that the two shapes look like a picture (refer to this when

talking about how symbols change meaning)T – Move the circle to the right side of the rectangle and ask the students

what these picture or symbols mean or what would you call them?S – May give the same answers as before.S – May say that it �ipped. (Ask students if that changes what the symbols

are or mean) Students might claim that they look like a di�erent picture now(refer back to this)

S – Might be confused because the shapes didn’t change; they just moved.T – “Oh, so it doesn’t matter how they are arranged? �ey still mean

the same thing? Shapes might look di�erent because they are in di�erentpositions, but they still remain the same shape.”

(�is part helps students see that sometimes when we add a symbol, de-pending on where it is added, the meaning does change. In the pilot studies,students o�en ignore negative signs. �is instruction will help students getin the habit of looking at how they can change the meaning of symbols toprepare them to pay attention to the negative signs.)

T – Move the circle just above the rectangle, so it looks like the letter “i”.“What about now? What do these symbols mean?”

S – Might give the same answers as before. (prompt them to think howthe way they are arranged could mean anything else).

S – “It’s the letter i.”T – “Wait! So now it does matter how they are arranged! Sometimes we

get the same things if we move symbols around and sometimes they giveus something with a new meaning.” Show an arrow facing upward. “�inkabout what this symbol means. Share your ideas with a partner.”

S – Might say: “an arrow” (prompt what does this arrow tell us to do?)

properties lesson – symbol consistency vs . inconsistency 141

S – “up”, “back” or “behind” or “forward”T – “When the arrow faces upwards it means something speci�c to us, like

‘go up’. Now talk with your partner about what else the arrow could mean ifyou moved it in some way.”

S – Might think moving it means sliding it in one direction and may thinkit means the same thing. (Prompt, can you do anything else to move it besidessliding it?)

S – Might picture turning the arrow to get: “Down”, “Le�”, “Right”, “Over”,“�at way (points)”

T – “So the meaning of the arrow changes depending on which way itis pointing. Cool. Sometimes the same symbol can have more than onemeaning.”

Exploration

T – “Now I’m going to challenge you. I’m going have you work witha partner and together you are going to try and �gure out all the di�erentsymbols you can make with these two lines. Can someone think of an exampleto get us started?”

S – Might be confused about what they are supposed to do (put the twolines next to each other in parallel and ask what it means)

T – Have students think of an alphabet chart for help.S – Possible answers include: perpendicular sign (they probably don’t

know what this means); T, L, V, II, =, >, <, _ _, -|, +, XT – Give students some sticky notes so they can record each symbol they

make on a di�erent sticky note. Challenge students to �nd at least 8 possiblesymbols. As pairs work, circulate and provide encouragement.

S – Might get hung up on putting the lines together in a certain way.(Encourage them to connect the lines in a new way, suggest that the linesmight not touch).

S – Might put the lines together in many ways but not consider turningthe symbols. (Encourage students to turn their symbols to see if they stay thesame.)

T – If students �nish early, encourage them to �nd more or challenge themto use three lines and see what they can make.

Sharing

T – Gather students together and have them share the symbols on theirpost-its one at a time. Ask students what each symbol means.


S – Some students might try and bend a line (remind them to keep eachpart of the symbol as it is but to put them together in di�erent ways).

T – Show a thumb-up if your group had the same symbol. “Do you agreeor disagree with the meaning of the symbol? What do you think the symbolmeans? Who came up with a di�erent symbol?”

S – May show perpendicular sign (they probably don’t know what thismeans, may think it is a stand)

S – May show T, L, V, X : lettersS – May show II : 2 or eleven or parallelS – May show = : equals, roadS – May show >, < : greater than and less than (chomping mouth, arrows)S – May show _ _ two blanks; two dashesS – May show +: plus or “and”S – May show -1: minus one, take away one, negative oneT – “What were di�erent ways you made the symbols? What strategies

did you use to make a new one? Did you have to move the lines?”T – “Are there any other ways we can change the meaning of a symbol

besides just moving them around?”(�is is to review that you can add something new to a symbol to make it

change its meaning.)S – “�ey change when they look di�erent”S – We recognize it as something new. We have a di�erent name for it.S – Make up a new meaning. Add something to it. (If a group used three

lines, prompt them to share what new meanings they got).T – Close by telling students that they learned that symbols can have a

di�erent meaning depending on how they are put together and that nexttime they will look at how certain symbols, numbers, take on new meaningdepending on how they are put together.


properties lesson – symbol consistency vs . inconsistency

Full Instruction Group – Lesson 2Integer Properties Group – Lesson 2

Enduring Understandings

1. Symbols can have more than one meaning depending on their contextand how they are arranged.

2. Integers are symbols which represent speci�c quantities which can beordered on a number line.

Materials – sorting cards, whiteboard marker

Objective1. Students will identify negative and positive numbers.

Introduction

T – Remind students that last time they learned that when symbols (likethe lines) are put together in di�erent ways, they have di�erent meanings.

T – “Now, we are going to explore some di�erent groups of symbols. Ihave a bunch of number sentences, which I’ve put into groups based on secretrules. Look at the groups and then turn and talk to a partner about why I putthe problems into these groups.”

B C4 + 3 4 − 37 + 2 7 − 26 + 1 6 − 1

S – “�ey all have two numbers and a sign” (prompt: what is di�erentbetween the two groups?)

S – “�ey have di�erent answers” (Prompt: how come?)S – “B is addition. C is subtraction.”T – What tells you it is an addition problem? What tells you it is a subtrac-

tion problem?S – “B has plusses. C has minuses.”T – (Cover up column C and show A)

A B-4 + -3 4 + 3-7 + 2 7 + 26 + -1 6 + 1


S – About (A): �ere are more signs. (Prompt: what else can you tell meabout the signs?)

S – About (A): Addition with subtraction or minuses (prompt: what doyou mean by subtraction? How do these signs compare to the subtractionsigns in C? – show C again)

S – About (A): Has more lines (Prompt: what can you tell me about thelines?)

S – About (B): All are addition problems.S – “�ey have plusses.” (Prompt: how are the addition problems in A

di�erent than those in B?)(�e point of the next two parts is to explicitly address the unary form of

the minus sign (negative sign) and help students see it as di�erent from theminus sign (binary meaning).)

T – Show the next comparison.

A C-4 + -3 4 − 3-7 + 2 7 − 26 + -1 6 − 1

S – About (A): All are addition problems.S – About (C): All are subtraction problems.S – “�ey have minuses.”S – “Both have lines” (Prompt: how are they the same or di�erent?)S – “�ey look like minuses.”S – About (A): �ere are lines in front of the numbers.S – About (C): �e lines are between numbers.T – Show the next comparison.

C D4 − 3 -4 − 37 − 2 7 − -26 − 1 −6 − -1

S – About (D): Two subtraction signs (Prompt tell me more about the twosigns - do the signs look the same?)

S – About (D): �ere are extra minuses (Prompt - tell me more about theminuses. Do they all look the same?)

T – “How are 4 − 3 and -4 − 3 the same or di�erent (show on chart)?”S – “Both have a four, a three, and a minus sign.”S – “�e second problem has another minus sign (negative sign, line,

mark)”


T – “Do the lines all look the same?”S – “Yes. �ey are all lines” (prompt them to look at how long they are and

where they are written)S – “No. Some are shorter/longer. Some are higher/lower. Some are closer

to the numbers.”T – “A lot of times when we see this symbol (write “-”) we think of sub-

traction.” Write 5 − 3. “�e minus sign is between two numbers.”T – “Sometimes, though, this minus sign is attached to a number like -1

(the symbol we made last time) -2 or in -2 + -3. (It may even look shorter -when we use it, it will look shorter–or it might be slightly raised.) When thishappens, the sign and the number together are called negative numbers. Let’ssay that together.”

S – “Negative numbers”T – “�e sign that makes a number negative is called the negative sign.

Let’s say that together.”S – “Negative sign.”T – “Have you ever seen negative numbers (numbers with the negative

sign) before? Where?”S – Some may say no.S – Some may say it reminds them of subtraction (refer back to the previous

explanation of the di�erence between subtraction and negatives)S – “Temperature, like when it’s really cold.”S – “When you lose lots of points in a video game.” (Penalties)S – “Below sea level.”T – Show -2. “What do you think this negative number means?”S – “Minus 2, or minused 2”S – “Take away two”S – “Negative 2”T – “Negatives are numbers below zero or less than zero, so -2 would be

two less than or under zero, -2 degrees is two degrees below or colder thanzero degrees. -2 points would be two points less than 0 points.” (Show onnumber line and thermometer)

Exploration

T – Let’s sort some math problems into two groups. One group will haveproblems with negative numbers and the other will only have regular, positivenumbers. Give your partner a thumbs-up if you think the card has a negative


number and a thumbs-down if it has a regular, positive number. Put theminto di�erent piles.

S – Sort 4 − 1; -5 + -2; 7 − -5; 6 + 2 − 1; -3 − 1; 8 − 3 − 2; 0 + -9S – Pairs might sort them by addition and subtraction (Prompt: Look

for problems with negative numbers. How do you know if a problem has anegative number?)

T – If students �nish early, they can write their own problems to go in thecategories.

Sharing

T – “Does this problem have a negative number?”S – “Yes.”S – “No. It’s just subtraction.”S – “No. �e sign is between two numbers.”T – Have students agree or disagree and explain why.S – “Disagree - it’s just minus. It looks like the other problem.”T – Ask if the next problems have negatives, one at a time: -5 + -2; 7 − -5;

6 + 2 − 1; -3 − 1; 8 − 3 − 2; 0 + -9; (others students came up with)S – Might think they are all subtraction because they see minus signs

(Prompt them to look at the example problems at the top of each category)T – “Great! You are starting to �gure out which problems have negative

numbers. Next time, we will learn more about what negative numbers are.”

properties lesson – numerical symbols 147

properties lesson – numerical symbols

Integer Properties Group – Lesson 3


and how they are arranged.2. Integers are symbols which represent speci�c quantities which can be


Materials – pictures of objects with numbers; notes of each group’s sort

Objective1. Students will identify negative and positive numbers.

Introduction

T – Ask for a volunteer to write a negative number on the board.S – May write a positive number (Ask students to agree or disagree, refer

them to the chart from the previous day)S – May write a negative number.T – “How do we know if a number is negative?”S – “It looks di�erent” (Prompt: What is di�erent?)S – “It has a minus sign” (Show 5−3 and -5−3 and have students describe

the di�erences.)(�e point of this part is to give children extra practice identifying negative

numbers in di�erent contexts.)T – “Last time we sorted math problems into groups: those with negative

numbers and those no negative numbers. Today, we are going to sort a fewmore pictures and problems as a class.”

Exploration

T – Give each student a card with a picture or problem on it. Tell studentsthey need to �rst �nd their partner and then decide together if it does or doesnot have negative numbers.

T – Each pair will decide on a problem or pictures with numbers in them.For example,


Positive Negative�ermometer ClockGame Points Book

7 − 2 -19 + 1124 − 10 − 2 14 + -1 − -2

S – Some students might continue to ignore negative numbers. (Havethese students look for the short line that looks a little di�erent than theminus sign)

S – Might confuse the negatives and minuses.

Sharing

T – Ask students to suggest one of the problems or pictures to put into acategory. Ask the rest of the class to show a thumbs-up or thumbs-down toshow if they agree or disagree. Have them check their piles if they are unsure.

S – Might interpret all “-” as minuses. (Prompt: Ask students how to tellif a number is negative)

T – “Why do you agree? Disagree?”T – “How did you decide to put this in the negative/not negative group?”S – Might ask what negative points means. (Ask others to share - less than

0 points.)

Exploration

T – Have students play memory, matching the negative numbers with eachother and positive ones with each other (e.g., -5 and -5 vs. 5 and 5)

S – Might match -5 and 5. (Remind them to look at the numbers carefully;refer them back to previous examples and encourage their partners to helpthem check their answers.)

T – To dismiss students to line up, ask them to name one thing that is thesame or di�erent about these problems: 7 + 5 vs. 7 + -5 vs. 7 − 5 or 6 − -1 vs.6 − 1

S – “All have 7, 5.”S – “Two have +”S – “Two have -” (prompt if the “-” looks the same. How are they di�erent?)S – “One has a minus; one has a negative sign.”S – May answer one of the problems (e.g., “12” “5”) (Tell them it’s great

that they are thinking about the numbers but to tell what they notice that isdi�erent or the same about how the problems look.)

properties lesson – order and quantity 149

properties lesson – order and quantity

Full Instruction Group – Lesson 3 Integer Properties Group – Lesson 4




Materials – Mountain and water number lines, people tokens, positive andnegative number cards

Objectives1. Students will determine which integer represents a greater quantity.2. Students will order integers.

Introduction

T – Remind students of the new numbers they learned about last time.If no one remembers what they are called, tell them and have all studentsrepeat, “Negative numbers.”

S – “Negative numbers”T – Write -9 and ask students, “How much is -9?”S – Students may say it is worth 9, minus 9, 9 − 9 which equals 0, I don’t

know, or nine below zero.(�is instruction is meant to prime students to think about how the quan-

tity numbers can represent can change depending on how the numbers arecombined with other numbers or symbols.)

T – Show a vertical number line next to a picture of a cli� with a waterline next to it. Tell the students that we will put zero at the level of the waterand show a person �oating in the water. (Maybe show a topographic map)

T – Ask, “If we climbed two feet out of the water, who can show me wherethe person would be on the number line?”

S – May miscount by starting with zero. (Ask another student to checkand then have them talk about who is right.)

S – May put the 2 below zero. (Ask who has a di�erent answer. When both2’s are up, ask students what the di�erence is between them.)

T – Reinforce that 2 is 2 more than zero.S – May point to the 2 in the correct spot above 0.


T – Ask, “If we dove two feet under the water, who can show me wherethe person would be on the number line?”

S – May miscount (have other students agree or disagree until there isconsensus)

S – May put the 2 in the correct spot below 0.T – Tell students that negatives seem a lot like positive numbers but that

they are numbers below zero, so -2 is two less than zero or two below zero(just like we showed in the picture). Make a point of showing the negative signand tell them that mathematicians use the negative sign to remind themselvesthat the number is less than zero.

S – May need to be reminded that the positive numbers mean they areabove 0 and negative numbers mean they are below zero.

T – Ask students what they notice about the numbers above the zero andbelow the zero.

S – “�ey are the same.” (Prompt: what’s di�erent?)S – “�ey are opposite. On top the go up, on bottom they go down.”T – “Because the numbers go in the opposite direction, �ve is a bigger

than four but negative four is bigger than negative �ve.”T – Show the swimmer at positive 5 and tell the students he is going to

dive in and wants you help him count where he is at. Show him divingand help count 5, 4, 3...as he passes the landmarks. When he gets to -5, tellthe students he wants to swim up to take a breath. Ask them to help count

-5, -4, -3, -2, -1, 0.S – Might echo the counting.S – Might race ahead in their counting.S – Might forget to say “negative” before the negative numbers. (Emphasize

the word when counting)

Exploration

T – Model playing a number line game with the students: Tell studentsthat each person starts their person at the zero level in the water and theyplay a game like the card game “War”. (�e point here is to stress the di�erentmeaning of the negative sign.)

T – “Each person gets the same number of cards and students pick the topone o� their pile. Players count up or down to the number on the card, movetheir people, and write in the number on the board. Whoever is highest (outof the water or closest to out of water if both are below the water line) gets totake the other player’s card.”


T – “Each player puts their people back at zero and draws the next topcard. �e person with the most cards at the end of the time wins.”

S – May forget to write the negative sign. Remind them to use it to showthat it is less than zero.

S – May confuse magnitude with which person is at the greater or higherposition.

Sharing

T – Based on their game boards, have students help you �ll in a numberline with all the numbers.

S – Might forget to write the negative signs.T – If students forget the negative signs ask them how the number is

di�erent than the positive counterpart.S – Might miscount or forget a number.(�is part of the activity will help students focus on which quantities are

more.)T – Pick two numbers, such as -4 and 3 and ask which is greater. Have

students discuss with partners before calling on someone: -4 and -2; 5 and 6;-3 and -5

S – Might say that -4 is greater because 4 > 3. (Remind students to thinkabout the swimmer. Which number will get us higher out of the water, havethem refer to the number line, ask other students to remind us what negativenumbers mean–less than zero)

(In the pilot studies, some students thought the quantity of negatives wasthe absolute value. �is explanation helps students think about the orderednature of the numbers as a key to �guring out their quantities.)

T – Show students a horizontal number line with 0, 1, 2, 3 �lled in. Ask ifstudents have seen the number line before and how it is similar to the verticalnumber line from the game.

S – May have seen them in their room or textbooks (or on a ruler).T – Ask for a volunteer to show where the number -2 would be and explain

why.S – “�ey both have numbers going in order.”S – May try and put it where “2” is. (Prompt: remember what the sign

attached to the number means)T – Reinforce that we can draw the number line either way, but negative

always means less than zero.


properties lesson – order and quantity





Materials – Place value worksheets, number cards for place value game, Moun-tain and water number lines, people tokens, positive and negative numbercards

Objectives1. Students will determine which integer represents a greater quantity.2. Students will order integers.

Introduction

Introduction T – Practice counting up from -10.(�e following is a review of the same game played in the last session.)T – Ask which is greater: 4 or 6 and how they know.S – “6 because it’s larger” (how do we know it’s larger).S – “6, it’s more things.”S – “6, it comes a�er four when we count”T – “If we count up, we start with the smaller numbers and count up to

the biggest number.”T – Have students help you remember the rules for the cli� and water

game and review which number is greater: 7 or 5; -5 or 4; -3 or -8S – Might still think the absolute value number is greater. (Have students

show on the cli� game which is greater or higher)T – “If we count up from the negative numbers, we count up to the larger

number, so -4 is greater than -5.”

Exploration

T – Have students play the game with the same rules as previously. Halfwaythrough, have students combine groups with another group that has a di�er-ent color of cards (one group may need to split up).


Sharing

T – Have students share if they used any strategies. Review which numberis greater: 7 or 5; -5 or 4; -3 or -8

S – Might continue to confuse absolute value with quantity. (Have studentsexplain their thinking–refer them to the game and which is highest. Remindthem that mathematicians decided that negative numbers are small becausethey are less than 0.)

T – Tell students they will get to play a new game next time to practicemore and less.


properties lesson – order and quantity (war)





Materials – Positive and negative number cards

Objective1. Students will determine which integer represents a greater quantity.

Introduction

T – Show children two cards: -4 and 1 and ask which is greater.S – Might say 4 if they are thinking of absolute value (four is more than 1).T – “How do you know?”T – “Do you agree or disagree and why?”S – “1 because four is less than zero.”T – Show children two cards: -6 and -9 and ask which is least or smallest.T – “How do you know?”S – Might say -6 if they are thinking of absolute value (remind them to

think of the number line and which number is furthest to the le� or downT – Have students help you demonstrate how to play “War” with positive

and negative number cards.

Exploration

T – Have students play war in groups of 2 or 3 only have them play to getthe smallest number (if they seem ready for this).

S – Might forget about the negative signs when playing. (Prompt: remindstudents to think about negative numbers as below zero, so they are small;give students a number line if they need extra help.)

T – Keep track of any confusion during the game, especially any compar-isons that caused confusion. Allow students to use the diving number line ifthey are having trouble.

properties lesson – order and quantity (war) 155

Sharing

T – “Were any pairs di�cult to �gure out?” S – Might have been particu-larly confused by comparisons such as -5 and 5. T – Discuss any comparisonsthat were tricky for groups. “Why were they tricky?” S – “I forgot they werenegative” (Prompt: how do we know if a number is negative?) S – “�eylooked the same.” (Prompt: what is di�erent?) T – “Next time, we’ll play anew game that helps us think about the order of negative numbers.”


properties lesson – order and quantity (in a row)






Objective1. Students will order integers.

Introduction

T – Show the cli� and water number line. Have students help you countas the diver dives from 10 down to -10. Have students help you count as thediver swims up to the surface from -10 to 0.

S – Some students might count more enthusiastically than others (encour-age all students to help count, start over as necessary)

(�is game is to help students practice focusing on the order of numbers.)T – Show students how to play the “In a Row” game. Draw 3 cards. �e

goal is to get your cards so that the numbers are three numbers in a row (e.g.,-3, -2, -1 or -1, 0, 1). Have students help you pick which card to discard. Drawa new one. Have students help you decide if you can make three in a row.

S – Might suggest playing a low card or one not close to the other two.T – “What numbers could I get on the next turn to win?”S – Might suggest a number related to the absolute value or which is not

reasonable. (Prompt: ask for their reasoning. Point to the number line andshow the numbers you have.)

T – Show students that they have to put their cards down in order on thegame board when they have 3 in a row and cover the three numbers withcolored cellophane.

T – Tell students they will play with a partner and take turns. �e personwho gets three in a row wins.

properties lesson – order and quantity (in a row) 157

Exploration

T – Have students play the game with a partner. If students get really goodat getting 3 in a row, have them play with four cards and try to get four in arow.

S – Students might get rid of a card which is consecutive to another card(e.g. get rid of -2 in 6, -3, -2) (Prompt: ask students to circle their numberson a number line to see which numbers to keep)

S – Might interpret a card as positive instead of negative.

Sharing

T – Have students share if they used any strategies.S – “Kept any numbers which were close to each other.”S – “Tried to always get positive numbers.”S – “I used the number line.”T – Show two card hands: -4, -5, 1 and -8, -2, 3. “Who could win on the

next turn? How do you know?”S – �e �rst person. She could get -3 or -6 because these two numbers are

together.S – Might see -2 as 2: �e second person they could get 1 or 4 (Prompt:

Can you show us on the number line which numbers the student has?)T – Tell students they will get to play the game some more the next time.


properties lesson – order and quantity (in a row)

Integer Properties Group – Lesson 8 Enduring Understandings

1. Symbols can have more than one meaning depending on their contextand how they are arranged.

2. Integers are symbols which represent speci�c quantities which can beordered on a number line.


Objective1. Students will order integers.

Introduction

T – Have students help you order the numbers -9 to 9.S – Might want to order the numbers with -9 next to 0. (Ask students how

they can tell which number is greater).(�is game is to help students continue practicing the order of numbers.)T – Have students help you remember the rules for the 3 in a row game.

Emphasize having them think about the order of the numbers.T – Tell students they can play the game with 4 cards and try to get four

in a row.

Exploration

T – Have students play the game with the same rules as previously onlywith four cards.

(If students are getting really good, allow them to play with �ve cards.Walk around and take notes on strategies students are using and di�cultiesthey have.)

Sharing

T – Demonstrate any hands that were tricky for students and ask for sug-gestions on what to do.

T – “Are these cards in order?” -5, 4, -3S – “Yes” (Prompt; have student show it on the number line; have students

agree or disagree).S – “No. Two are negative and 4 is positive. �ey aren’t next to each other.”T – “Great! You guys learned a lot about what negative numbers are, which

is greater, and how to put them in order.”

operations lesson – order with operations 159

operations lesson – order with operations

Full Instruction Group – Lesson 4Integer Operations Group – Lesson 1

Enduring Understandings1. Order sometimes matters in operations but you can still get an answer.2. When dealing with opposites, getting more of one thing means you

have less of the other and vice versa. (opposites lie on a continuum,where more of one is less of the other)

Materials – Cake cutouts (some with cutting lines); Recording sheets.

Objective1. Students will identify situations in which the order of two actions leads

to similar or di�erent end results.

Introduction

T – “Have you ever helped someone make a recipe? Lots of recipes tellyou what to do in order. Like �rst mix the �our and sugar. Next add theeggs. Sometimes if you don’t follow the order, the recipe doesn’t work. I waswondering if following order mattered in other areas, and I hope you canhelp me.”

(�e purpose of the activity is to help students see that the order in whichwe do actions sometimes doesn’t matter for the end result (even if the processlooks di�erent).)

T – Ask students what it means to turn around in a circle.S – “You turn all the way around.” (prompt them to show the class).T – “Since we are all going to be turning in a circle, we will say that you

need to turn all the way around so you are facing in the same direction.”T – Have half of the class turn around once in a circle and then stand up;

have the other half stand up and then turn around once in a circle.S – May have trouble turning on the �oor, but encourage them to scoot as

best as they can.T – Record the two orders of actions.T – Ask someone from the �rst group to share the result of these two

actions; ask someone from the second group to share the result of the actions.S – “We are standing up”S – “We are facing front”T – “Did it matter what order we did the actions? Why or why not?”


S – “No. We all ended up standing.”S – “Yes. It was harder to turn while sitting” (Who agrees or disagrees?

Help students focus on the end result)T – “�at’s true; it might be harder while we are doing it. How about at

the end. Do we have the same result at the end?”S – “Yes. We are all standing.”(�is part is to help students understand that changing the order of actions

can sometimes lead to di�erent outcomes.)T – Ask if anyone can think of two actions that give di�erent results if you

do them in di�erent orders.S – “Stand up and then sit versus sit then stand.” (Encourage students to

say more.)S – “Getting dressed.”S – May not be able to come up with an example.T – Have half of the class sit for �ve seconds and then stand for �ve seconds;

have the other half stand for �ve seconds and then sit for �ve seconds.T – “What happened?”S – “We did the same things.”S – “Some people are standing and some are sitting.”S – “We look di�erent.”T – “Sometimes order does matter because we get something di�erent

at the end. Here we had some people standing at the end and some peoplesitting.”

Exploration

(�is part of the activity is meant to help students see that actions withnumbers can end with di�erent results depending on which number you startwith. �e motivation for addressing this is that students in the pilots o�enreversed the numerals in subtraction problems. While these problems dealwith division, the point is the same.)

T – “Now we are going to see if order in which we use numbers mattersin some math problems. Have students work in partners. Your groups willhave pretend cakes which you want to share with some people.”

T – Show an example recording sheet. At the top of the sheet is the divisionproblem (e.g. 4÷3 “four shared with three”) with labels or pictures underneathit (cakes shared with people). Have children help you �gure out how to sharefour cakes with three people and model how to cut and glue the cake pieces


onto the paper. Point out that the light lines on the cakes can help us cut soeveryone gets an equal share.

S – Children may say to give each person 1 cake and forget about thele�over (Prompt: how can we break apart the last cake? How many piecesshould we cut it into so each person gets the same amount of cake from thislast one?)

S – Children may suggest cutting each cake into three pieces and sharingfrom each cake.

T – Give each group a set of problem sheets: (4 ÷ 2 & 2 ÷ 4 with 2 ÷ 2 asextra; 8 ÷ 2 & 2 ÷ 8 with 8 ÷ 8 as extra; or 8 ÷ 4 & 4 ÷ 8 with 4/4 as extra)to complete in their pairs. (If students �nish early, they can try the extraproblem and then a di�erent combination on their own.)

S – May start out with the wrong number of cakes. (Remind them to lookat the number of cakes they should use as listed on their papers.)

S – May forget to share all of the cake (Encourage them to split up theremainder so that all of the cake is given away.)

S – May split up cakes unevenly. (Encourage them to use the lines asguides and emphasize that each person should get pieces that are the same,so it is fair)

S – Might share out the whole cakes �rst and then divided up the le�overcake.

S – Might divide up each cake and share a piece with each person.

Sharing

(�e point of this segment is to help students see the visual di�erence inthe results of switching the order of the numbers in the division problems.)

T – Gather students back together with their recording sheets. Ask fora volunteer to share one of their cake splitting sheets. Record the groups’answers in di�erent rows to highlight the pattern going across (that the samenumbers lead to di�erent answers when they are in the opposite order).

T – “Let’s look at our results. What do you notice about the cakes in eachrow?”

S – Might notice the number of pieces of cake, regardless of their size.(Prompt students to look at the size of the cakes)

S – “Some cakes are small and some are large.”S – “People get more cake in this one.”T – “Why are they di�erent sizes?”S – “We had to share with more people.”


S – “�ere were fewer cakes.”S – “�ere were fewer cakes and more people.”T – “We have di�erent pairs.” Read them o� (e.g. Share 2 cakes with 8

people or 8 cakes with 2 people – “we used the same numbers but in di�erentorder; 4 cakes with 8 people or 8 people with 4 cakes, etc”.)

T – “What happens when we change the order of the numbers?”S – “For 2 and 2 it’s the same thing either way.”S – “Whenever the numbers are the same it’s 1.”S – “For 2 people and 8 cakes, you get less cake than if it’s 8 cakes for 2

people.”S – “If the larger number is �rst, you get more cake.”S – “You get di�erent amounts of cake.”T – “What if we shared 6 cakes with 3 people? Would it be the same as

sharing 3 cakes with 6 people?” (Have students show a thumbs up for yes anda thumbs down for no.)

S – Might be unsure (demonstrate the �rst half of the problem and havestudents predict if the people will get larger or smaller cakes for the secondpart)

S – “Yes. You are using the same numbers” (Try it out with the paper –take predictions on which will have larger cakes.)

S – “No. If you have six cakes, each person can get at least one, but if thereare three cakes and more people, then you won’t even get one.”

T – “Interesting. Sometimes in math it matters in what order we use ournumbers, like when we are dividing or sharing. (Point to the di�erent ordersof the numbers on the papers and the di�erent cake results). We’ll explorethis some more next time.”

operations lesson – order with operations (elevator) 163

operations lesson – order with operations (elevator)

Operations Group – Lesson 2



Materials – Elevator sheets



Introduction

T – Ask for someone to share what we learned with sharing cakes lasttime.

S – “Sharing 2 cakes with 8 people was di�erent than sharing 8 cakes with2 people.”

S – “Sometimes we get di�erent answers; sometimes they are the same.”T – “Sometimes when we are dividing numbers in math (like when we

shared the cakes for people), the order of the numbers mattered. Today, youare going to see if order matters for addition and subtraction.”

(�e purpose of the activity is to extend the exploration of order intoaddition and subtraction.)

T – Tell groups they are going to play a game with their partners. Eachgroup will get a hotel elevator sheet. Explain how the hotel numbers its �oors.

T – “When you walk in, you are on the ground �oor, which we call the 0�oor. Why might they call it the ground �oor?”

S – “It sits on the ground?”S – “It’s not up or down. It sits on the ground.”T – “Sometimes apartment buildings say that the �oor on the ground is

the 1st �oor. Why might we call it the 0 �oor instead?”S – “You don’t have to do anything to get to it.”(By using an elevator context, there is a purpose for using negative num-

bers.)


T – “�e next �oor up is labeled 1 because it is one �oor above the ground�oor. �e next �oor is 2 because it is two �oors above the ground �oor. Whatwould the next one be and why?”

S – “3” (who agrees or disagrees and why? – prompt for reason) “It comesa�er 2.”

S – “3. It is 3 �oors above the ground.”T – “Now this hotel also has �oors below the ground.” (Point to �rst �oor

underground.)T – “I’ll just label the �oors this way” (Label -1, -2, etc. don’t explain what

they are called or that you are putting them in order and don’t explain whatnegatives are.)

T – “At the start of each game, the elevator starts at the ground �oor. Onyour turn, draw a card (draw an example of an up and a down card). It saysFloor, this stands for the number of the �oor you are on. What number �oorare we on right now?”

S – “Zero!”S – “Ground �oor” (and what number is the ground �oor?)T – “A�er the �oor is the action. It has a minus and says less (#). If we are

at �oor zero, and go less (#), who can show where we will be?”S – May miscount (have others agree or disagree)S – May count in wrong direction.S – Moves correct number down.T – “Yes, going less (#), is the same as going down (#) (and we have a hint

on the card).”T – “�is one has a plus and says more (#). If we are at �oor zero, and go

more (#), who can show where we will be?”S – May miscount (have others agree or disagree) or go in wrong direction.

(prompt them to look at the symbols on the card)S – Moves correct # up T – “Yes, going more (#), is the same as going up

(#) (see hint on card)”(Recording the problems will help students see if they use the same two

numbers in di�erent orders and whether this leads to di�erent outcomes.)T – Show students the recording chart. Tell them that their goal is to �ll

out the whole chart to win, but they can only record an answer when theymove the elevator using those numbers. Remind students that “Floor” meansthe �rst number in the problem. “If we are on �oor zero, we are trying to �ndthe answers to problems which start with zero.”


S – Help �ll out the starting �oor, whether we add (up) or subtract (down),the number we move, and the ending �oor.

T – “Turn and talk to a partner and tell them which problems start withzero. Record where you ended up (model this).” Ask for a volunteer to pointout the problems which start with zero.

S – May need help �nding a partner. Students may point out all problemswith a zero. (remind them it has to start with a zero).

T – Draw a card (+1) and have students help you �nd if a problem matchesit. Prompt students to say the whole problem out loud, starting with theircurrent �oor before moving the elevator.

S – “Zero more one.”S – Might select the problem with the wrong operation. Remind them

they have to match the problem exactly.T – Have a volunteer move the elevator and have students turn and tell a

partner which �oor it ended up on. Model how to �ll in the answer on therecording sheet.

T – Ask for a volunteer to �nd the problems which start with one. Draw acard (+3) and ask if there is a problem which matches it.

S – Might be confused and try and pick a problem that is close. Some willsay that there isn’t a match (have students agree or disagree).

T – If there isn’t a match, you just move the elevator and try again. Re-member, the game is over when you �ll up the sheet.

T – “Let’s say you now draw a +3 card. Can we move three more? Why orwhy not?”

S – “Yes, because you can go down.”S – “Yes, you can go part of the way.”S – “No, you would go out of the building.”T – “If you cannot move all of the spaces, you stay where you are and try

again. You don’t move part of the way, the elevator will only move if it canmove all of the spaces.”

T – “As you play, look for times when you use the same numbers but indi�erent orders (e.g. starting at �oor 2 and going up 1 versus starting a 1 andgoing up 2), and we’ll talk about it at the end.”

Exploration

T – As students play the game, circulate around and ask them if they haveany of the same numbers in the opposite order. If so, ask what they notice.

S – May need prompting to use their recording sheets correctly.


S – May think I mean 3 − 2 = 1 and 1 + 2 = 3 are opposite order.S – Might start counting from the initial �oor. (Encourage their partners

to check each other’s answers.)S – Might be hesitant to solve problems where they subtract a larger from

smaller. Have them think about the game and move their elevator to see ifthey can do it.

Sharing

T – Allow students to keep playing until students have used a few numberpairs in both orders (either within the same group or across groups if time isrunning out, make note of which they are).

T – Gather students back together. If certain groups used numbers in bothorders, ask groups to share their results if they used numbers in both orders.Record them up front, making a separate column for ones that involve adding(going up) versus ones that involve subtracting (going down).

S – Might read their problems with the terms “up” and “down” insteadof “plus” and “minus”. (If they do, reinforce both ways of talking about theproblems).

T – If no group used a pair of numbers in both orders, ask questions like,“Who had a problem where they started at the second �oor?” (e.g. 2 + 3),then pick students who did a problem with numbers that another group used,so that there will be a pair if you then ask, “Who had a problem where theystarted at the third �oor?” (e.g. 3 + 2).

S – Might repeat the same problems. (Put a checkmark next to repeats).S – Might look at the wrong columns on their paper (remind them to

look at the problem and not the answer; have a student demonstrate whereto look).

T – (If no one did an addition combination or subtraction combinationwith numbers reversed, make one up and add it to the chart.) “What do younotice about the problems that have the same numbers in di�erent orders?”

S – Might focus on which column has more: “�ere are more problemswith addition.”

S – “�e numbers are �ipped.”S – “�e addition (up) problems have the same answer. �e order doesn’t

matter.”S – “�e subtraction (down) problems have di�erent answers. (unless the

problem uses the same number like start at �oor 2 and go down 2)”


T – “So what did this activity tell us about the order of numbers in additionand subtraction problems?”

S – “It doesn’t matter.”T – “Who wants to add something to that?”S – “Order doesn’t matter for addition. You get the same answer. Order

matters for subtraction. You get di�erent answers.”T – “Let’s say I started at �oor 1 (write it down) and drew a card to go up

4 (write it down). It’s easier for me to count up 1 than to count up four. Is itokay for me to count 4, 5?”

S – “No, because then it would be like you started on �oor 6.”T – “Who agrees/disagrees and why? Will I get the same answer (end up

on the same �oor)?”S – “Yes. You get the same answer.”S – “Yes. It looks di�erent but you end up in the same place.”T – “What about if I start at �oor 3 (write it down) and drew a card to go

down 4 (write it down). I think it’s easier to do 4 − -3. Can I do that?”S – “Yes. We switched them for the other problem and got the same

answer.”T – “Who agrees or disagrees and why? Will I get the same answer?”S – “No. You will end up in a di�erent spot.”S – “No. �at’s a new problem. It wouldn’t look the same.”T – “So when we say that order doesn’t matter or order does matter, what

do we mean?”S – “If it doesn’t matter, we can use the numbers either way.”S – “We can add one way or the other way.”S – “We can do 4 up 3 or 3 up 4. �e order doesn’t matter because we end

up at the same place.”S – “If order does matter, we can’t switch the numbers around. 4 down 3

is di�erent from 3 down 4. We get di�erent answers.”T – “Great, so if we will get the same answer, we can switch the numbers

around when trying to solve the problem. But if we will get a di�erent answer(like when we go down), we need to keep the numbers in the same orderwhen we solve the problem.”


operations lesson – order with operations (elevator ii)

Full Instruction Group – Lesson 5 Integer Operations Group – Lesson 3



Materials – elevator sheets



Introduction

(�e �rst part is for the full instruction group only.)T – Ask for someone to share what we learned with sharing cakes last

time.S – “Sharing 2 cakes with 8 people was di�erent than sharing 8 cakes with

2 people.”S – “Sometimes we get di�erent answers, sometimes they are the same.”T – “Sometimes when we are dividing numbers in math (like when we

shared the cakes for people), the order of the numbers mattered. Today, youare going to see if order matters for addition and subtraction.”

(�e purpose of the activity is to extend the exploration of order intoaddition and subtraction.)

T – Tell groups they are going to play a game with their partners. Eachgroup will get a hotel elevator sheet. Explain how the hotel numbers its �oors.

T – “When you walk in, you are on the ground �oor, which we call the 0�oor. Why might they call it the ground �oor?”

S – “It sits on the ground?”S – “It’s not up or down. It sits on the ground.”T – “Sometimes apartment buildings say that the �oor on the ground is

the 1st �oor. Why might we call it the 0 �oor instead?”S – “You don’t have to do anything to get to it.”T – “�e next �oor up is labeled 1 because it is one �oor above the ground

�oor. �e next �oor is 2 because it is two �oors above the ground �oor. Whatwould the next one be and why?”

operations lesson – order with operations (elevator ii) 169

S – “3” (who agrees or disagrees and why? – prompt for reason) “It comesa�er 2.”

S – “3. It is 3 �oors above the ground.”(By using an elevator context, there is a purpose for using negative num-

bers.)T – “Now this hotel also has �oors below the ground. (Point to �rst �oor

underground.) What should we call this �oor?” (Point to �rst �oor under-ground).

S – “One.”T – “How can we write it di�erently so that we know it is one under the

ground and not the same as this other �oor?”S – “Circle it.”S – “Write it in a di�erent color.”S – “Write basement 1.”S – “Put a line above it.”S – “Negative one.”T – “Just like with the water, the �rst �oor under the ground is labeled

with -1 (negative one) because it is one less than the 0 �oor.”T – “What would this next �oor be?”S – “2” (who agrees or disagrees and why? Prompt for how it is di�erent

than other 2 and remind them of what they decided to call the previous �oor).S – Might say, “Minus two. Negative two. Below zero two or Two in the

basement” (is there another name for this?)T – Follow similar procedure as for previous �oor, then �ll in the other

numbersT – Have students count as you move the elevator from -5 to 3.T – “Which is larger or greater? 2 or -3?”S – Might think -3 is larger if they think of absolute value. (ask for others

to agree or disagree).T – Remind them that as we go up, the numbers get larger and that num-

bers below zero are smaller than numbers above zero.(�e rest of the plan applies to the Full Instruction and Operations Groups.

�e purpose of the activity is to practice the exploration of order or operationswith addition and subtraction.)

T – Show the students two game sequences:Start: 0 + 5 =Start: 5 + 0 =


T – Have students show you with a thumbs-up (same) or thumbs-down(di�erent) if the answer will be the same. “Why do you think they will be thesame?”

S – “�umbs-up: �ere are the same numbers. Both are addition.”S – “�umbs-down: �ey start at di�erent places and go up by di�erent

amounts.”T – Have a student help show where each elevator would end up and model

how to �ll out the �rst problem.S – Might be confused about what it means to move the elevator zero.

�ey may think they should move the elevator to 0.T – Have students look at the two answers in the chart. “Do they have the

same answer? Did we end up on the same �oor?”S – “No.” (Have them double check).S – “Yes.” (Tell them, they would circle yes.)T – Tell students they a�er they use the elevator to help them solve the

problems and determine if they are the same no matter what the order is,they will get to use the playing cards to play the elevator game.

T – “At the start of each game, the elevator starts at the ground �oor. Onyour turn, draw a card (draw an example of an up and a down card). It saysFloor, this stands for the number of the �oor you are on. What number �oorare we on right now?”

S – “Zero!”S – “Ground �oor” (and what number is the ground �oor?)T – “A�er the �oor is the action. It has a minus and says less (#). If we are

at �oor zero, and go less (#), who can show where we will be?” (Count outloud)

S – May miscount (have others agree or disagree)S – May count in wrong direction.S – Moves correct # down.T – “Yes, going less (#), is the same as going down (#) (and we have a hint

on the card). �is one has a plus and says more (#). If we are at �oor zero,and go more (#), who can show where we will be?”

S – Student may miscount (have others agree or disagree) or go in wrongdirection. (prompt them to look at the symbols on the card)

S – Moves correct # upT – “Yes, going more (#), is the same as going up (#) (see hint on card)”

operations lesson – order with operations (elevator ii) 171

T – “If you cannot move all of the spaces, you stay where you are and tryagain. You don’t move part of the way, the elevator will only move if it canmove all of the spaces. If you land on 0, the ground �oor, you get a point.”

T – “As you play, look for times when you use the same numbers but indi�erent orders (e.g. starting at �oor 2 and going up 1 versus starting a 1 andgoing up 2), and we’ll talk about it at the end.”

Exploration

T – Have students �ll out the problem sheets in pairs. (Watch to see howwell they are recording and moving their elevators.) When students �nish,ask them what patterns they notice on their recording sheet. Ask them iforder matters for addition and subtraction.

S – “�e addition problems are at the top, the subtraction are at the bot-tom.”

S – “�ey all have the same numbers” (are they all positive or negative orboth?–are they exactly the same?).

S – “�e answers for addition are the same but are di�erent for subtrac-tion.”

T – If students got an incorrect answer, have them model how they �guredit out using their elevator.

T – When they are ready, give them the playing cards and remind themthat they get a point every time they move to the ground �oor, or zero.

Sharing

T – “What did you �nd?”S – “You get the same answer when you add in either order, but di�erent

with subtraction.”S – “With 1 − 1 it doesn’t matter what order because the numbers are the

same.”T – Wrap up with another example subtraction problem: 2 − 4 vs 4 − 2T – “Great, next time we will talk about adding and subtracting some more

with a fun exploration.”


operations lesson – order with operations

Full Instruction Group – Lesson 6Integer Operations – Lesson 4

Enduring Understanding1. Order sometimes matters in operations but you can still get an answer.2. When dealing with opposites, getting more of one thing means you


Materials – Emotion faces, water, cups, food coloring, q-tips

Objective1. Students will identify the degree of an opposite in terms of both oppo-

sites.

Introduction

T – “Last time we looked at how an elevator moved up and down. Up isthe opposite of down.”

T – “What is the opposite of happy?”S – Possible responses include, “Not happy. Unhappy. Sad.”(�e purpose of this activity is to get students comfortable talking about

opposites along with more and less.)T – Write down sad and happy on opposite ends of a continuum. “What

would be in the middle. What if you are not happy but you aren’t really sadeither. What word could we use for that?”

S – Possible responses include, “Okay. Not happy or sad. Blah. Nothing.”T – Label the middle with one of the suggestions. Show a picture of a very

unhappy face. “If I am really unhappy, where could we put the face on thisline?” (Have a volunteer put it up.)

S – Might put it at the end of the line or somewhere between the end andthe middle.

T – Have students give a thumbs-up if they agree or thumbs-down if theythink it should be somewhere else. (Try and get consensus or put it in a spotthat is a compromise depending on where students want it.) Tell the studentsthat the person got a little bit happier (show “+ smile”) and show the facewith a smaller frown. “Where should we put this person?”

(�ese manipulations parallel the type of thinking needed to reason aboutadding a little positive number to a larger negative number.)


S – May want to put him on the happy side because he got a little happier–they focus on the word happier. (Prompt students to look at his face.)

T – “Can anyone think of a time when they were sad and someone triedto cheer you up but you were still a little bit sad?”

S – May put him close to the other frowny face or closer to the middle.T – Ask for a volunteer to place the face that is a little less sad (“− sad”),

the super happy face, the face that is a little less happy (“−smile”), and theface that is a little more sad (“+ sad”).

T – “Here we have a really sad person at one end, a really happy person onthe other end, and if we put the two people together we would have someonehow is so-so or okay. What would happen if we had two colors on the end?Let’s say blue is on this side and yellow on the other. If we mixed them together,what would we get?”

S – Possible responses include, “A new color, green, or I don’t know.”T – “Today we are going to work with colors to see what happens if we

add a little bit more of blue or yellow to these colors. We have to be reallycareful with the materials because we don’t want any of the color to get onus.” Tell students that they will have to follow along with the directions.

(�is activity will give students hands-on time to explore what happenswhen adding a little or a lot of “an opposite.”)

Exploration

T – Give students letters (A, B, C, D) and have them line up behind a setof clear, plastic cups �lled with water.

T – “Scientist D. Find your blank card and draw a blank number line on itlike this: one dash in the middle and one on each end.”

T – (BLUE) “What do you think will happen if we add 1 blue drop to cupA?”

S – “Nothing.”S – “It will turn blue.”S – “It will be light blue.”T – “Scientist A, gently squeeze ONE drop of the blue dye into the water.

You have to be really careful or you will get too much. Scientist B, take thestick and gently mix up the water.”

S – May add too much of one color. �ey can empty out the cup and startover in extreme cases.

T – “What happened?”S – “It swirled.”


S – “It is blue.”S – “It looks dark.”T – “Scientist C, take a cotton swab and hold it into the water. Be careful

to only get the top in the water. Count to 5 and take it out. Put it close to themiddle on your number line. What color does it look like?”

S – “It looks light.”S – “Barely blue. A little blue.”S – “Blue and white. Blue (light or dark blue?).”T – “What do you think will happen if we add more blue to cup A?”S – “Nothing.”S – “It will stay blue.”S – “It will get darker. It will be more blue or darker blue.”T – “Scientist B, gently squeeze SIX drops of the blue dye into the water

(count together). You have to be really careful or you will get too much.Scientist C, take the stick and gently mix up the water.”

S – Students who are color blind may have more di�culty distinguishingbetween some of the shades. �eir group members can help as well.

T – “What happened?”S – “It swirled again.”S – “It got darker.”S – “It’s really blue.” (Did it get MORE or LESS blue?)T – “So if we add MORE blue, it goes this way (show on number line), in

the direction of more blue.”T – “Scientist D, take a cotton swab and hold it into the water. Be careful

to only get the top in the water. Count to 5 and take it out. Put it at the le�end of number line. What color does it look like?”

S – “Same blue.” (Have students hold it up next to the other blue to moreclosely compare the colors.)

S – Possible responses might include, “Bright blue. Darker blue. Moreblue.”

T – (YELLOW) “What do you think will happen if we add 1 yellow dropto cup C?”

S – “Nothing.”S – “It will turn yellow.”S – “It will be light yellow.”T – “Scientist C, gently squeeze ONE drop of the yellow dye into the water.

You have to be really careful or you will get too much. Scientist D, take thestick and gently mix up the water.”


S – Students may add too much of one color. �ey can empty out the cupand start over in extreme cases.

T – “What happened?”S – “It swirled.”S – “It is yellow.”S – “It looks dark.”T – “Scientist A, take a cotton swab and hold it into the water. Be careful

to only get the top in the water. Count to 5 and take it out. Put it close to themiddle on your number line on the other side of the blue. What color does itlook like?”

S – “It looks light.”S – “Barely yellow. A little yellow.”S – “Yellow and white. Yellow” (light or dark yellow?).T – “What do you think will happen if we add more yellow to cup C?”S – “Nothing.”S – “It will stay yellow.”S – “It will get darker. It will be more yellow or darker yellow.”T – “Scientist D, gently squeeze TEN drops of the yellow dye into the

water (count together). You have to be really careful or you will get too much.Scientist A, take the stick and gently mix up the water.”

S – Students who are color blind may have more di�culty distinguishingbetween some of the shades. �eir group members can help as well.

T – “What happened?”S – “It swirled again.”S – “It got darker.”S – “It’s really yellow.” (Did it get MORE or LESS blue?)T – “So if we add MORE yellow, it goes this way (show on number line),

in the direction of more yellow.”T – “Scientist A, take a cotton swab and hold it into the water. Be careful

to only get the top in the water. Count to 5 and take it out. Put it at the le�end of number line. What color does it look like?”

S – “Same yellow.” (Have students hold it up next to the other blue to moreclosely compare the colors.)

S – “Bright yellow. Darker yellow. More yellow.”T – (GREEN) “What do you think will happen if we add 1 yellow and 1

blue drop to cup B?”S – “Nothing.”S – “It will turn green.”


S – “It will turn black or dark. It will look yellow. It will look green.”T – “Scientist A, gently squeeze TWO drops of the yellow dye and TWO

drop of the blue dye into the water. You have to be really careful or you willget too much. Scientist B, take the stick and gently mix up the water.”

S – May add too much of one color. Have them add an equal amount ofthe other dye.

T – “What happened?”S – “It swirled.”S – “It is yellow and blue.”S – “It looks dark.”S – “It’s green.”T – “Scientist C, take a cotton swab and hold it into the water. Be careful

to only get the top in the water. Count to 5 and take it out. Put it in the middleon your number line (between the light blue and yellow). What color does itlook like?”

S – “It looks light.”S – “It’s yellow and blue (and what new color does that make?) It’s light or

dark green.”T – “What would happen if we took out some of the yellow drops from

cup C?”S – “It would still be yellow.” (Prompt: Would it be the same yellow? Or

would it get lighter or darker?–have students check what the swab with lessyellow looked like).

S – “Less or lighter yellow.”T – “What would happen if we took out some of the blue drops from cup

A?”S – “It would still be blue.” (Prompt: Would it be the same blue? Or would

it get lighter or darker?–have students check what the swap with less bluelooked like).

S – “Less or lighter blue.”T – “What do you think will happen if we add blue to the cup C?”S – Possible responses include, “It will turn blue. It will turn green. Noth-

ing.”T – “Scientist B, gently squeeze ONE drop of the blue dye into the water.

You have to be really careful or you will get too much. Scientist C, take thestick and gently mix up the water.”

T – “What happened?”S – “It looks dark again. It looks a bit green.”


T – “Scientist D, take a cotton swab and hold it into the water. Be carefulto only get the top in the water. Count to 5 and take it out. What color doesit look like?”

S – “It’s green” (does it look the same as the other green? What other colordo you see in there? “Yellow”)

T – “Is it more yellow or less yellow than before?”S – “More yellow” (have them compare)S – “Less yellow”T – “Put it on the yellow side but moving toward blue.”T – “What do you think will happen if we add yellow to the cup A?”S – Possible answers include, “It will turn yellow. It will turn green. Noth-

ing.”T – “Scientist C, gently squeeze TWO drops of the yellow dye into the

water. You have to be really careful or you will get too much. Scientist D, takethe stick and gently mix up the water.”

T – “What happened?”S – “It looks dark again. It looks a bit green.”T – “Scientist A, take a cotton swab and hold it into the water. Be careful

to only get the top in the water. Count to 5 and take it out. What color doesit look like?”

S – “It’s green” (does it look the same as the other green? What other colordo you see in there?) “Blue.”

T – “Is it more blue or less blue than before?”S – “More blue” (have them compare)S – “Less blue.”T – “Put it on the blue side but moving toward blue.”T – Have students clean up. (D puts cotton swabs, sticks, and paper towel

in baggie; A empties A cup. B empties B cup. C empties C cup and stacks itnext to sink.)

Sharing

T – “What happens if you have blue and add more blue?”S – “It stays blue (did the blue change?) Gets more blue.”T – Write: Blue + Blue = More BlueT – “What if we had blue and took some of it away?”S – “Less blue.”T – Write: Very Blue - Blue = Less BlueT – “What happened when you added a little yellow to the blue?”


S – Possible responses include, “Nothing. It is still blue. It looks a littlemore blue.”

T – Write: Very Blue + Yellow = Less BlueT – “If we get less blue, what do we move towards?”S – “Green” (and then what?) “Yellow.” (show on number line)T – “What do you notice with taking away blue and with adding yellow?”S – “�ey do the same thing.”T – “If we added lots and lots of yellow, what do you think would happen?”S – “It would look more yellow. It would be less blue even more.”(�e point here is for students to notice di�erences in the jars and realize

that adding a little yellow can make the jar look both a little more yellow anda little less blue.)

T – Have students predict what will happen when you add color to newcups: “When I add lots of blue to yellow water, will it get more yellow or lessyellow?”

S – May be confused by the language and ignore “blue” and think addingmore will give you more yellow. (Refer them to the chart and their cups)

T – Have students show a thumb-up or down to agree or disagree.S – “More blue.”S – “Less yellow.”S – “You aren’t adding more yellow, so it won’t look as yellow.”S – May think it will get less yellow because it will still look a little blue.S – “It will get more yellow because you are adding yellow.”T – Tell students that next time they will explore more and less in a game.

operations lessons – more positive , more negative 179

operations lessons – more positive , more negative

Full Instruction Group – Lesson 7 Integer Operations Group – Lesson 5



Materials – game boards, tokens, cards

Objective1. Students will identify more positive or more negative in terms of adding

positive and negative numbers.

Introduction

T – Remind students that last time they played with color and found outthat adding yellow to blue, made the color less blue or more yellow and addingblue to yellow made the color less yellow and more blue (point this out on acolored number line).

T – Explain that the same thing happens in negative and positive land.Label zero (explain it is not negative or positive), the positive side, and thenegative side with pictures (zoo, park, nettle patch).

S – Might say they like going to the park. (Yeah, the park is fun. We havepositive or good experiences there usually.)

S – Might wonder what the nettle patch is. (Stinging nettles are plants thatcan prick you and make you itchy. �at is a negative or bad experience.)

T – Show students that they will start at zero zoo. “If I get more negative(toward negative nettle patch), which direction will I move in?”

S – Might point to the le� or say to the le�. If students say to the right,have them point to make sure they aren’t misusing the directional word.

T – “Right, just like if I get more blue, I will move in the blue direction, if Iget more negative, I will move in the negative direction. If I get more positivewhich direction will I move in?”

S – Might point right or say right.T – (Show a card) “�e plus sign helps us know that we will want more,

and the number tells us if we want to go more positive (point to park at 10)or (Show another card) more negative (point to nettle patch at -10).”


T – Show the game board to the students. Students will see a zoo at 0with a sidewalk on the le� side, becoming progressively blue and ending at a“nettle patch” and a sidewalk on the right side, becoming progressively yellowand ending at a park.

S – Might suggest that you write the numbers on the board (includingnegatives) (If they suggest it write it in.)

(�e purpose of this activity is to help students distinguish between addingas getting more positive or more negative but using the idea that addition ismore, which students express.)

T – Demonstrate the game with students’ help. Both players start at thezoo and are trying to get to the park. Player one chooses a card from the�rst deck. If he gets a plus, he says “more” (practice �ipping cards and sayingmore when the plus appears) and then draws a number card (#) from thesecond deck. If it is positive (has the positive park on it), he says “positive #”,and if it is negative (has the negative nettle on it), he says “negative #” (havestudents practice reading some number cards). He then moves accordinglyeither more negative or more positive the correct number of spaces. �enext player goes. (Introduce fun voices and hand motions to get studentsexcited about using the verbal frames–hands showing more, �ngers can shownumber, happy voice versus scared or sad voice for positive versus negative.)

S – Might forget to say positive or negative. (Refer them to the colors ofthe words and remind them we need to know which direction to move. Askwhat about the number helps us know if it is negative or not.)

T – If a player gets a WILD card, player follows the directions on the card(e.g., go to zero zoo, switch places, closest to the park = one point, closest tohouse = two points, closest to nettle patch = 1).

S – Might get caught up in saying more and might say it accidentally whena WILD card comes up.

T – When someone reaches the park, they get a point and go back to thezoo.

S – Might forget what these cards mean or have trouble reading them.Encourage them to ask their peers for help (they will have pictures to helpthem.)

S – Might forget to say “more” “positive or negative” “#” on each of theirturns. (Have partners make sure their partners follow directions. May haveto give students a point every time they say it to encourage it. Remind themto use their interesting voices to tell their partner what move they will make.)

operations lessons – more positive , more negative 181

Exploration

(�is will give students an opportunity to work with the idea of adding toget more positive or more negative.)

T – Have students play with their partners. Walk around to see if they arecorrectly identifying the numbers and moving correctly. Watch to see if theyanswer the WILD cards correctly.

S – May need help remembering what it means to move more in thenegative direction versus the positive direction (especially if they are at 7 andhave to move more negative, 3 and won’t actually get to the negative side).

Sharing

T – Gather students together to talk about the game. “If I am here (-3) onthe board, and I draw more (+) positive 4, where will I end up?”

S – May just add absolute value: 7 or move in wrong direction -7 (En-courage child to show on the number line; prompt: if you are moving morepositive, which direction do you need to go in?)

T – “How did you think about it? Who agrees or disagrees?”S – “1–move more positive (right) 4 spaces”T – “Do we always end up on the positive side if we move more positive?

(Do we always get a positive answer if we move more positive?)”S – “Yes.” (Have students agree or disagree. Have them start at -4 and

move 1 more)S – “No. We might move that way but not get there.”T – “If I am here (-3) on the board, and I draw more (+) negative -4, where

will I end up?”S – Some may think it is the same question: “1” (Prompt them to think

about what it means to get more negative.)S – Point at -7.S – Some may say -7.T – “Do we always end up on the negative side if we move more negative?

(Do we always get a negative answer if we move more negative?)”(�is is slightly more challenging because they need to think about which

card they would need to draw.)S – “Yes.” (Have students agree or disagree. Have them start at 6 and go -2

more)S – “No. You might move closer but not get there.” (Can you show an

example?)


T – “If I am here (2) on the board, what do I need to get to reach the zoo?(Do I need to move more positive or more negative? By how many?) In whichdirection? To reach the park?”

S – Student may miscount.S – “2, move le�.”T – “To reach the park?”S – “More negative 8.”T – Tell students that tomorrow they will get to play the same game, only

with subtraction.

operations lesson – less positive , less negative 183

operations lesson – less positive , less negative

Full Instruction Group – Lesson 8 Integer Operations – Lesson 6 Enduring

Understanding1. Order sometimes matters in operations but you can still get an answer.2. When dealing with opposites, getting more of one thing means you


Materials – board game, negative number cards

Objectives1. Students will identify less positive or less negative in terms of subtract-

ing positive and negative numbers.2. Students will identify more or less positive or more or less negative

depending on the numbers and symbols involved in the problems.

Introduction

(�is introductory sort is to help students review what they did previ-ously with “more positive” and “more negative” and check how well studentsunderstand it.)

T – Show number line from previous day. Pass out addition numbersentences and have children sort the problems according to whether theywould move more positive or more negative like in the game the previousday (e.g. 4 + -3 = more negative; -4 + 3 = more positive). (Positive numbershave a picture of the park next to them and negative numbers have a pictureof the nettles next to them.)

S – Students might sort problems like -4+3 as more negative because theystart out with negative numbers. (Have someone model the problem on thenumber line.)

T – Have students put their problems into categories on the board and askstudents if they agree or disagree with where they are placed.

S – May put 4+-3 in the more positive group because the answer is positive.(Prompt them to think about which direction they will move–in the positivedirection or the negative direction)

T – Model any problems that cause disagreement.T – Have a student demonstrate what solving 3− 2 looks like on the game

board from the previous day (put marker at three and draw “−2”).S – Starts at three and moves 2 more towards the house.


T – Reinforce the actions by saying when we subtract positive numbers,we get less positive (just like if we took away yellow paint, our color wouldget less yellow). We are moving away from the positive side.

S – Should remember this from the review and the previous day. Havethem use the number line to help them.

T – Show a marker at zero zoo. “If I get more negative (toward negativenettle patch), which direction will I move in?” (move marker into negativewhile saying you are moving more in the negative direction)

S – “To the le�” (may point).(�e purpose of this part of the activity is to help students understand the

changing meaning of subtraction with negatives.)T – “Which direction will I move in if I get LESS negative?”S – Might focus on the term less and point to the le� because negatives

get smaller (Prompt them to think about the colors: taking away some blue,made the color less blue and look more yellow)

T – (Show operation cards as appropriate: - means less; + means more;and sign cards: - means negative; these may be color-coded depending onthe di�culty students had with them in previous lessons.)

T – “�e minus sign helps us know that we will want less, and the numbertells us if we want to go less positive (point to park at 10) or less negative(point to nettle patch at -10).”

T – “Today, you will play the game like last time, only now, when you drawa minus sign, you say ‘less’. Let’s try it together. Let’s say I am here on theboard (put marker at -3). What do we say if I draw a minus sign?”

S – “less”T – (Show a negative two), “and what do we say when we turn over the

number?”S – “Negative two” (students may forget the negative–draw their attention

to the sign)T – “Less negative two.” Okay. If we are at here (-3), and want to get less

negative, 2 spaces, where should we move?”S – “It’s the same thing as 3 − 2 because we take away two.” (Ask if taking

away blue and taking away yellow would have the same e�ect. If taking awayyellow, makes something less yellow, taking away blue makes something lessblue, and taking away a positive number makes a number less positive, thentaking away a negative should make the number less...)

S – “It will go less negative or more positive.”

operations lesson – less positive , less negative 185

T – Tell students that when they subtract or take away a negative num-ber, they are getting less negative. Have a volunteer show how to move lessnegative.

S – (Students may think that getting less negative means the answer has tobe positive. Ask students what they think about this–refer back to the colorsas well.)

Exploration

T - Have students play the game again, only this time they always subtractand the numbers are either positive or negative.

Sharing

Sharing T – Have students work with their pairs or table groups to sortproblems into ones which will get less positive and ones which will get lessnegative.

S – Might ignore the negative signs and classify numbers based on theiranswers.

T – When all have sorted, talk about where they go as a class. Have studentsexplain whether they agree or disagree and show what they are thinking onthe game board.

S – Might think problems can only be less positive if their answers arenegative (refer back to the number line).

T – “If I am here (-5) on the board, and I draw “less (-) negative -4, wherewill I end up?”

S – May just subtract absolute value: 1 or move in wrong direction -9(Encourage child to show on the number line; prompt: if you are moving lessnegative, which direction do you need to go in?)

T – “How did you think about it? Who agrees or disagrees?”S – “-1: move less negative (right) 4 spaces; move more positive”T – “Do we always get a negative answer if we move less negative?”S – “Yes.” (Have students agree or disagree. Have them start at -2 and

move 4 more)S – “No. We might move that way but cross zero.”T – “If I am here (6) on the board, and I draw less (-) positive 4, where

will I end up?”S – “6 − 4” (what’s the answer?)S – “10: Some might add” (Prompt them to think about what it means to

get less positive.)


S – “2.”T – “Do we always get a positive answer if we move less positive?”S – “Yes.” (Have students agree or disagree. Have them start at 6 and go

less positive by 4)S – “No.” You might go past zero. (Can you show an example?)T – “If I am here (2) on the board, what do I need to get to reach the zoo?

(Do I need to move less positive or less negative? By how many?) In whichdirection?”

S – Student may miscount.S – “2, move le�, less positive”T – “To reach the park?”S – “Less negative 8.”

operations lesson – more or less , positive or negative 187

operations lesson – more or less , positive or negative

Integer Operations Group – Lesson 7







Introduction

T – Review the game with the students. Ask them to show how to move:move positive, more negative, less positive, and less negative. Tell them thattoday, they will get to use all cards together.

S – Might have forgotten the movements. Remind them of the colorsexercise.

T – Show some examples and have volunteers show how to move.

Exploration

T – Have students play the game. Walk around and make note of anytroubles students have.

Sharing

T – Bring up any tricky movements that students had trouble with.T – Place a person at 7 on the board. Ask students what the person would

need to get to make it to the park.S – “More positive 3.”S – “Add 3.”S – Might miscount.S – Might not realize that less negative 3 also works.


operations lesson – more or less , positive or negative

Integer Operations Group – Lesson 8







Introduction

T – Place a person at -7 on the board. Ask students what the person wouldneed to draw to get to make it to the nettle patch.

S – “More negative 3.”S – “Add negative 3.”S – Might miscount.S – Might not realize that less positive 3 also works.

Exploration

Exploration T – Have children play the game again. Students can addin the rule that if they land on the same space as their partner, it sends thepartner to the zoo. Students can also choose to make the goal of the gamereaching the negative nettle instead of the positive park (or both).

B P I L O T I Q U E S T I O N S

1. What is the smallest number you can think of?If child’s smallest number is positive: What if you had (child’s smallest

number) dollars, and you owed me (child’s smallest number + two) dollars?What would happen? (If child suggests it is a smaller number or they wouldowe money: How could you write it?)2. How do you know it is the smallest?3. How would you write the number?4. How could you draw it?5. What are these numbers? Which is more? How do you know?

A. 8 3B. 2 4 Draw a picture of these two numbers.C. 7 9D. 9 5 (Give if child struggles with A-C.)E. 3 6 (Give if child struggles with A-C.)F. -5 0 Draw a picture of these two numbers.

G. 1 -9H. 2 -4 Draw a picture of these two numbers.I. -7 3 (Optional)J. -9 2 (Optional)

K. -5 -2L. -8 -2

M. -1 -9N. -3 -1 (Optional)O. -4 4

6. Which is closer to 10?A. 5 3B. -4 2C. 6 -9D. -8 -4E. -2 -6

189

190 B ⋅ pilot i questions

7. Which is closer to 0?A. 3 5B. 2 -4C. -9 6D. -6 -2

8. Put or sort these numbers into di�erent groups.1 2 -7 0 -4 5 7 -1 8 3 -9 -6 -3a. Tell me about your groups.b. Can you sort the numbers another way?c. Tell me about your new groups.d. If child does not sort groups by small, medium, and large: Put these

numbers into groups by whether they are small, medium, or large.e. Tell me about your groups.

9. Put these number cards in order.-4 8 0 4 -2 3 9 -5

a. Tell me how you put the cards in order.b. Which one is smallest?c. Which one is largest?

10. Alternate the order of the following pairs of questions:a. What would you call this number: 6.How would you draw a picture of it?b. What would you call this number: -6.How would you draw a picture of it?Show numbers on number cards

11. Two children are playing a game. �e boy has -3 points and the girl has 2points.

a. Who is winning?b. How many more points does the boy/girl have?c. How do you know?

12. What if the boy had -2 points and the girl had -6 points?a. Who is winning?b. How many more points does the boy/girl have?c. How do you know?

191

13. Question from TIMSS

14. Solve the following problems:A. 8 − 5 = ◻ B. ◻ + 5 = 8 C. 9 − 7 = ◻D. ◻ + 6 = 9 E. 5 − 7 = ◻ F. -2 + 7 = ◻G. 2 − ◻ = 4 H. 4 + -2 = ◻ I. -2 − -5 = ◻J. -3 + 4 = ◻ K. 3 + -6 = ◻ L. -8 + 4 = ◻

M. -5 − -3 = ◻ N. 6 + -1 = ◻

192 B ⋅ pilot i questions

C P I L O T I I Q U E S T I O N S

Group 1: Boxed andNon-Boxed

Group 2: Doubles andNon-Doubles

What is addition? What does it mean to add?

What is subtraction What does it mean tosubtract?

6 − 6 6 − 6

4 − 9 4 − 9

9 − -1 2 − -4

6 − -4 1 − -2

7 − -2 3 − -5

5 − -3 4 − -6

-4 − -4 -3 + -7

-7 − -7 -2 + -5

-3 − -3 -4 + -8

-8 − -8 -3 + -6

-2 − 5 -3 − 2

-3 − 8 -7 − 4

-1 − 6 -2 − 1

-2 − 7 -6 − 3

-8 + 4 -8 + 4

-9 + 5 -9 + 5

-3 + 1 -3 + 1

193

194 C ⋅ pilot ii questions

Group 1: Boxed andNon-Boxed

Group 2: Doubles andNon-Doubles

-4 + 2 -4 + 2

-5 − ◻ = 0 What answer do you thinkthe student got?

-6 + ◻ = 0 -5 − -3 = ◻ “I started atnegative �ve and countedback.”

-2 + ◻ = -2 -2 − -5 = ◻ “I took away bygoing up the numbers.”

D P I L O T I I C O D E S

Problem Set Up Description

AsIs Student does not change the problem

1Ignore Student ignores the only negative in the problem

2Ignore Student ignores both negatives in the problem

Ignore1st Student ignores 1st negative only if there are more than one (for -3 + -5does 3 + -5)

Ignore2nd Student ignores 2nd negative only if there are more than one (for -3 + -5does -3 + 5)

Compares Quantities Student compares the quantities of the two numbersNotEnough ...and determines there won’t be enough to subtractSame ...and notices they are the same, so subtraction rules in zeroBelowZero ...and identi�es the subtrahend as larger than the minuend, resulting in

a number below zeroReverse Student switches the order of the numbers

OppOp Student says he or she is using the opposite operation or reads the prob-lem with opposite operation

NegOp Student uses the negative to replace the operator when it follows theplus sign (4 + -2, does 4 − 2)

ReverseNegOp Student reverses the negative in front of a number with the operation(-8 + 4 = 8 − 4)

NegSubSelf Student uses negatives as a minuses and subtracts the same number (e.g.-9 + 5 = 9 − 9 + 5)

NegSubAgain Student uses negatives as minuses and subtracts twice (e.g. 9 − -1 =9 − 1 − 1)

195

196 D ⋅ pilot ii codes

Problem Set Up Description

NZ Student replaces negative numbers with zeroes

AP Student adds a plus sign (E) – erases it

ANN Student adds a negative to one of the numbers

OppDir Student uses an incorrect ordering of the negatives (-8, -9, 0, 1, 2); num-ber which goes to the le� of zero varies

Equivalent Student uses an algebraic rule to change the operation (e.g., says -4+ 2 isthe same as taking away two of the negative fours or solving -4 − -2)

(* ) *strategy code corresponding to the hypothesized strategy a student usebased on his or her answer and explanation

Guessed Student indicates that he or she guessed

Other/Unknown It is not clear if the child recalled the answer, especially if it took a whileto answer and the child might have been counting in his/her head

Problem Solution Description

Written Student writes out numbers when counting

Fingers Student counts with �ngers

Counts Students claims to have counted but no details given.

CtowardZ Student counts toward zero

CthroughZ Student counts through the zero boundary

CAFZ Student counts away from zero

Cdist Student counts the distance between two numbers

CP Student counts numbers as positive instead of negative, doesn’t say an-swer as negative, but may add a negative to the answer

CPN Student counts numbers as positive instead of negative but ver-bally states the answer as negative “Negative eight, nine, ten, eleven,twelve...negative twelve”

FactFamily Student relates the problem to one using similar numbers (e.g., for 2 − 4,talks about 4 − 2; for 3 − 5, talks about 2 + 3 = 5)

197

Problem Solution Description

RelatedFacts Student changes the problem to an easier one and compensates forchange if needed (9 + 5 = 10 + 5 − 1)

BA Break-apart the numbersBAMZ Break-apart-to-make-zero; student solves 6 − 8 by stating that 6 − 6 is 0

and two more would be two below zero.BAMT Break-apart-to-make-ten; students solves -5 − 9 by stating that -5 − 5 is

-10 and then minus four more.

Previous Student refers to a previous problem in answering (it’s like 6 − 6)

Analyze Student analyzes the problem in terms of similarities with another prob-lem (e.g. nine plus one equals ten, so minus nine plus minus one wouldequal minus ten)

Rule Refers to ruleSNL Subtracting a negative is getting largerANS Adding a negative is getting smaller< 0 Negative is less than 0

Operation Justi�es an answer by making a general comment about the operation.

Add=Higher Added by going “higher”

Sub=Lower Subtracted by going “lower”

Recall-Fingers Recall by looking at �ngers

Recall Recall answer

Algorithm Talks about the standard algorithm procedure

Answer Strategy Description

None Student does not manipulate the sign of the answer

Mismatch Num Written and verbal answers don’t match in numeral

Mismatch Sign Written and verbal answers don’t match in sign

Repeat Justi�es answer by repeating problem

Opposite It has to be opposite (some indication of knowledge about opposite op-erations)

198 D ⋅ pilot ii codes

Answer Strategy Description

Negative Student makes the answer negative without talking about why

NN Student adds a negative because one or both of the numbers is negative(“they’re both signs; that’s a negative four”)

RN Removes negative

AN Student says he or she is adding a negative or indicates that the answerneeds to be negative

E P I L O T I I S C H E M A C AT E G O R I E S

negative sign

Schema Categories Description

Ignore Students do not acknowledge negatives as important elements of arith-metic problems. �ey read negative numbers as positive and ignore thenegative symbols completely.

Replace Students replace the operator with the negative, typically one at thebeginning of the problem (e.g. for -8 + 4, students change the negative toa minus and solve 8 − 4).

Subtract Students interpret negatives as minus signs. If the negative is �rst, theymight subtract the number from itself (e.g., -8 + 4 would be 8 − 8 − 4). Ifthe negative follows a minus sign, they might subtract the number twice(e.g., 7 − -2 would be 7 − 2 − 2).

Answer Students ignore the negative signs when operating with the numbers butadd a negative sign to the answer.

Number Students acknowledge negative signs as part of the number both in theirlanguage and in their manipulations of the numbers.

199

200 E ⋅ pilot ii schema categories

negative number quantities


Absolute Value Students think that -9 = 9 or that -7 > 5.

Zero Students think negatives are worth zero because they have been sub-tracted from themselves (-9 stands for 9 − 9 = 0).

Taken Away Students think that negatives are less than positive numbers becausethey taken away, but -9 > -8 because taking away nine is more thantaking away eight.

Less �an Zero Students think that negatives are less than positive and that -1 > -2.

negative number order


Positive Students only use the positive number line. 0, 1, 2, 3, 4...

Two-way Positive Positive numbers go in both directions. ...3, 2, 1, 0, 1, 2, 3...

Negatives Interspersed Negatives are next to their positive counterparts. 0, -1, 1, -2, 2, -3, 3...

At Zero Negatives are stacked or next to zero. 0, -1, -2, -3...1, 2, 3...

Negatives Reversed Negatives are to the le� of zero but their order is reversed. �e numberto the le� of 0 ranges from -9 to -in�nity. ...-7, -8, -9, 0, 1, 2, 3...

Negative Positive Correct ordering of the negative numbers in relation to the positivenumbers. ...-3, -2, -1, 0, 1, 2, 3...

negative addition and subtraction 201

negative addition and subtraction


Absolute Value Students believe adding means you’ll get a number with larger absolutevalue and subtracting leads to a smaller absolute value.

Order Students believe adding means going right on the number line andsubtracting means going le� on the number line.

Opposite and AbsoluteValue

Students believe adding results in a larger absolute value and subtrac-tion results in a smaller absolute value but also thinks that adding isreally subtracting when there are negatives and vice versa.

Opposite Order Students believe adding means moving right and subtraction meansmoving le� on the number line but also thinks that adding is reallysubtracting when there are negatives and vice versa.

Quali�ed Amount Students know that adding a positive means getting more positive,adding a negative means getting more negative, subtracting a positivemeans getting less positive, and subtracting a negative means gettingless negative.

Quali�ed Order Students know that adding a positive means moving in the positive di-rection, adding a negative means moving in the negative direction, sub-tracting a positive means moving away from the positive direction, andsubtracting a negative means moving away from the negative direction.

202 E ⋅ pilot ii schema categories

F I N T E G E R P R O B L E M T Y P E S

For each problem type: L, S, or the answer may be unknown. L stands forlarger positive number, S stands for smaller positive number, and X refersto a speci�c positive number

Integer Problem Typeswhere L > S > 0, X > 0

Doesn’tCross Zero

CrossesZero

EqualsZero

One Sign(+ or −)

Two Signs(+- or−-)

1 (L) + (S) = + X X

2 (L) − (-S) = + X X

3 (S) + (L) = + X X

4 (S) − (-L) = + X X

5 (X) + (X) = 2X X X

6 (X) − (-X) = 2X X X

7 (L) + (-S) = + X X

8 (L) − (S) = + X X

9 (-S) + (L) = + X X

10 (-S) − (-L) = + X X

11 (S) + (-L) = − X X

12 (S) − (L) = − X X

13 (-L) + (S) = − X X

14 (-L) − (-S) = − X X

15 (-X) + (X) = 0 X X

16 (-X) − (-X) = 0 X X

17 (X) + (-X) = 0 X X

18 (X) − (X) = 0 X X

Continued on next page ...

203

204 F ⋅ integer problem types

For each problem type: L, S, or the answer may be unknown. L stands forlarger positive number, S stands for smaller positive number, and X refersto a speci�c positive number

Integer Problem Typeswhere L > S > 0, X > 0

Doesn’tCross Zero

CrossesZero

EqualsZero

One Sign(+ or −)

Two Signs(+- or−-)

19 (-L) + (-S) = − X X

20 (-L) − (S) = − X X

21 (-S) + (-L) = − X X

22 (-S) − (L) = − X X

23 (-X) + (-X) = −2x X X

24 (-X) − (X) = −2x X X

25 (X) + 0 = X X X

26 (X) − 0 = X X X

27 0 + (X) = X X X

28 0 − (-X) = X X X

29 (-X) + 0 = −X X X

30 (-X) − 0 = −X X X

31 0 + (-X) = −X X X

32 0 − (X) = −X X X

G I N T E G E R P R O B L E M S

P = positive answer; N = negative answer; 1st = negative comes �rst, 2nd= negative comes second, both = both numbers are negative

Problemx , y

Add orSub

∣x∣ + ∣y∣ ∣x∣ − ∣y∣ AnsP, N , 0

∣x∣ > ∣y∣ x > y # of neg.Neg. 1st,2nd, B

9 − -2 subtract 11 7 P Yes Yes 1 2nd8 − -7 subtract 15 1 P Yes Yes 1 2nd5 − -3 subtract 8 2 P Yes Yes 1 2nd6 − -7 subtract 13 -1 P No Yes 1 2nd4 − -5 subtract 9 -1 P No Yes 1 2nd-7 + -1 add 8 6 N Yes No 2 Both-6 + -4 add 10 2 N Yes No 2 Both6 − 8 subtract 14 -2 N No No 0 None3 − 9 subtract 12 -6 N No No 0 None1 − 4 subtract 5 -3 N No No 0 None-5 − 9 subtract 14 -4 N No No 1 1st-3 − 5 subtract 8 -2 N No No 1 1st-4 + 6 add 10 -2 P No No 1 1st-2 + 7 add 9 -5 P No No 1 1st-1 + 8 add 9 -7 P No No 1 1st-9 + 2 add 11 7 N Yes No 1 1st-3 + 1 add 4 2 N Yes No 1 1st-6 − -9 subtract 15 -3 P No Yes 2 Both-4 − -7 subtract 11 -3 P No Yes 2 Both-2 − -6 subtract 8 -4 P No Yes 2 Both-8 − -5 subtract 13 3 N Yes No 2 Both-4 − -3 subtract 7 1 N Yes No 2 Both-8 − -8 subtract 16 0 0 na na 2 Both-5 − -5 subtract 10 0 0 na na 2 Both7 + -3 add 10 4 P Yes Yes 1 2nd5 + -2 add 7 3 P Yes Yes 1 2nd

205

206 G ⋅ integer problems

H P R E - A N D P O S T-T E S T P R O T O C O L S

Pre-test Post-test

What is addition?If child is stuck or unsure:

What does it mean to add? (What happens when you add?)What is subtraction?

If child is stuck or unsure:What does it mean to subtract? (What happens when you subtract?)

“Start at 5 and count backwards as far as you can.”If child stops at 1: “Keep counting backwards as far as you can.”If child stops at 0: “Keep counting backwards as far as you can.”If child will not go further: “What comes before zero? How do you know?”If child stops somewhere in the negatives: “Can you go any further?”If child indicates the numbers keep going, ask, “What would the last number be?”

“Fill in the rest of the numbers on the number line shown here.” (if child labels negatives inreverse order, e.g., -7, -8, -9, 0, 1, ask what would come before -7 and so on to see if he/shegets to -0 and what he/she thinks comes before that. If child has trouble understanding whatto do, relate the question to the previous one on counting). (Post-Test also contains a verticalnumber line)

1

Lay down cards in front of child. “Put thefollowing number cards in order from leastto greatest.” When they �nish ask, “Whichis greatest? How did you think about it?”2, -3, 0, -9, 3, 8, -56, -6, 4, -7, 3, -1

Lay down cards in front of child. “Put thefollowing number cards in order from leastto greatest.” When they �nish ask, “Whichis greatest? How did you think about it?”-8, 2, -3, 5, 0, -2, 9-3, 7, -4, 4, -6, 1

207

208 H ⋅ pre- and post-test protocols

Pre-test Post-test

What are these numbers? Which is greater?How do you know?Show 8 and 6If students get 1.1 wrong, give next two:Show 1 and 7Show 4 and 2Show 3 and -9Show -2 and -7Show -5 and 3Show -8 and -2

What are these numbers? Which is greater?How do you know?Show 9 and 7If students get 1.1 wrong, give next two:Show 2 and 8Show 5 and 3Show -4 and -9Show 2 and -8Show -7 and -1Show -6 and 4

Two children are playing a game and tryingto get the highest score. Who is winning?How do you know?

Abigail: 4Joseph: -7

Crystal: -7Leon: -3How many points does <name of losing per-son according to student> need to get tohave the same number of points as <nameof winning person according to student>?

Two children are playing a game and tryingto get the highest score. Who is winning?How do you know?

Abigail: -8Joseph: 5

Crystal: -6Leon: -2How many points does <name of losing per-son according to student> need to get tohave the same number of points as <nameof winning person according to student>?

Circle all of the numbers in the equation.How did you decide what to circle?-5 + 3 − -3 = 16 + -2 − 7 = -3

Circle all of the plus signs and minus signswhich tell you whether to add or subtract.How did you decide what to circle?-4 − 3 + -1 = -85 − -3 + 2 = 10

Circle all of the numbers in the equation.How did you decide what to circle?-8 + 4 − -5 = 13 + -6 − 4 = -7

Circle all of the plus signs and minus signswhich tell you whether to add or subtract.How did you decide what to circle?-2 − 9 + -2 = -128 − -1 + 4 = 13

209

Pre-test Post-test

Look at the problems. Will these problemsgive you the same answer? How do youknow?4 + 5 5 + 43 − 1 1 − 36 − 4 7 − 45 − 8 8 − 53 + 2 3 + 39 − 6 6 − 9

Solve:23 − 531 − 7

Look at the problems. Will these problemsgive you the same answer? How do youknow?4 + 3 3 + 45 − 6 6 − 58 − 4 7 − 49 − 2 2 − 97 + 2 3 + 74 − 1 1 − 4

Solve:24 − 732 − 9

(Show cat on stairs) Put an X on the stairwhere the cat will be if she moves 2 stairsless high?How did you solve it? Will the cat be morehigh or more low?

(Show cat on stairs) Put an X on the stairwhere the cat will be if she moves 1 stairmore low?How did you solve it? Will the cat be morehigh or more low?

(Show cat on stairs) Put an X on the stairwhere the cat will be if she moves 3 stairsless low?How did you solve it? Will the cat be morehigh or more low?

(Show cat on stairs) Put an X on the stairwhere the cat will be if she moves 7 stairsmore high?How did you solve it? Will the cat be morehigh or more low?

(Show cat on stairs) Is the cat more high ormore low? How do you know? Put an X onthe stair where the cat will be if she moves 1stair more low?How did you solve it?

(Show cat on stairs) Is the cat more high ormore low? How do you know? Put an X onthe stair where the cat will be if she moves 4stairs more high?How did you solve it?

(Show cat on stairs) Is the cat less high orless low? How do you know?Put an X on the stair where the cat will be ifshe moves 2 stairs less high?How did you solve it?

(Show cat on stairs) Is the cat less high orless low? How do you know?Put an X on the stair where the cat will be ifshe moves 3 stairs less low?How did you solve it?

210 H ⋅ pre- and post-test protocols

Pre-test Post-test

-4 − -3 -9 + 2-1 + 8 -5 − 9-5 − -5 -2 − -6-3 + 1 3 − 96 − -7 -4 + 6-2 + 7 4 − -5-6 + -4 9 − -21 − 4 5 − -3-3 − 5 -7 + -1-4 − -7 6 − 8-8 − -5 -8 − -85 + -2 7 + -38 − -7 -6 − -9

-6 + -4 -1 + 85 + -2 7 + -33 − 9 -2 + 7-9 + 2 -7 + -1-4 − -3 8 − -71 − 4 -6 − -96 − 8 6 − -7-5 − -5 -2 − -6-8 − -8 -3 + 1-5 − 9 -4 − -7-3 − 5 4 − -55 − -3 9 − -2-4 + 6 -8 − -5

(Show a number line with zero marked)Number line 1:Put your �nger at zero. Move your �ngermore positive 2 lines and draw a triangle.Put your �nger at zero. Move your �ngermore negative 4 lines and draw a square.Number line 2:Put your �nger at zero. Move your �ngerless positive 4 lines and draw a triangle. Putyour �nger at zero. Move your �nger lessnegative 2 lines and draw a circle.

I S C H E M A D IA G R A M S

written symbols – integer identification

A) Names all integers as positive. Ip

B) Name some positive numbers as negative ANDnames some negative numbers as positive.

IpIn

C) Names some positive number as negative ORnames some negative numbers as positive.

IpIn

IpIn

D) a. Names all positive numbers as negative ANDnames all negative numbers as positive b. Misnamessome or all positive numerals AND misnames some orall negative numerals.

IpIn

E) Misnames some positive numerals OR misnamessome negative numerals. (e.g. -7 = seven and a half;6 = nine)

IpIn

IpIn

F) Names negative integers correctly as negative. IpIn

211

212 I ⋅ schema diagrams

integer order – counting backwards

A) Counts back to one. “Five, four, three, two one.” Op

B) Counts back to zero. “Five, four, three, two, one,zero.”

OpO0

C) Counts back to repeating zero. “Five, four, three,two, one, zero, zero, zero...”

OpO0

V0

On

D) Counts back to negative zero, skipping zero. “Five,four, three, two, one, negative zero.”

OpO0On

E) Counts back to negative zero. ”Five, four, three, two,one, zero, negative zero.”

OpO0On

F) Counts back into negatives, skipping zero. “Five,four, three, two, one, __, negative one, negative two.”

Op

V0

On

G) Counts back into the negatives, which are orderedbackwards. “Five, four, three, two, one, zero, negativenine, negative eight...”

OpO0

Vp

On

H) Counts back into the negatives correctly. “Five,four, three, two, one, zero, negative one, negative two...”

OpO0On

integer order – completing the number line

integer order – completing the number line 213

Aw) Writes positive numbers wrapped around. Ar)Writes positives as repeated in the same direction.Ao) Writes positive numbers on le� side. As) Writessymmetric positive numbers.

number line

Ip

Op

Bo) Writes positive numbers on right side

number line

Ip

Op

Cw) Writes whole numbers wrapped around. Cr)Writes whole numbers as repeated in same direction.Cs) Writes symmetric positive numbers with zero(missing numbers)

number line number line

Ip

Op

I0

O0

Do) Writes whole numbers on le� side. Ds) Writessymmetric whole numbers.


Ip

Op

I0

O0

E0) Writes whole numbers on right side.


Ip

Op

I0

O0

Fw) No zero; Writes negatives as wrapped positivecontinuation. Fs) No zero; Writes symmetric posi-tive/negative numbers.


Ip

Op

In

On


Gw) No zero; Writes positives wrapped around, butones on negative side are called negative.


Ip

Op

In

On

Gr) a. No zero; Writes negatives as repeated positives(in�uenced by the order or value of positive num-bers).


Ip

Op

Vp

In

On

b. No zero; Writes positive numbers backward


Ip

Op

In

On

Hw) No zero; Writes negatives as wrapped positivecontinuation. Hs) No zero; Writes symmetric posi-tive/negative numbers.


Ip

Op

In

On

Io) Writes whole numbers with repeated zero.


Ip

Op

I0

O0

V0

On

integer order – completing the number line 215

Iw) Writes negatives as wrapped whole continuation.

number linenumber line number line

Ip

Op

I0

O0

In

On

Ir) a. Positive and negative numbers �ipped. Writesnegatives as repeated whole numbers.


Ip

Op

I0

O0

In

On

b. Positive and negative numbers �ipped. Writes neg-atives as repeated whole numbers (in�uenced by theorder or value of positive numbers).


Ip

Op

Vp

I0

O0

In

On

Jo) Writes whole numbers with repeated zero.


Ip

Op

I0

O0

V0

On

Jr) a. Writes negatives as repeated whole numbers.


Ip

Op

I0

O0

In

On


b. Writes negatives as repeated whole numbers (in�u-enced by the order or value of positive numbers).


Ip

Op

Vp

I0

O0

In

On

Jw) Writes negatives as wrapped whole continuation.


Ip

Op

I0

O0

In

On

Js) Writes symmetric whole/negative numbers.


Ip

Op

I0

O0

In

On

K) Writes symmetric whole/negative numbers (ownnotation).


Ip

Op

I0

O0

In

On

L) Writes symmetric whole/negative numbers.


Ip

Op

I0

O0

In

On

integer order – ordering integer cards

integer order – ordering integer cards 217

A) Random: Lacks order (more than neighborsswitched) 1 -4 5 -2 -8 -7

Ip

Op

Vp

I0

O0

V0

In

On

Vn

B) Ordered as if all positive, mixed (two transposed) Ip

Op

Vp

I0

O0

V0

In

C) Orders as if whole numbers. -8 -7 5 -4 2 1 (or�ipped)

Ip

Op

Vp

I0

O0

V0

In

D) Orders as negative and whole numbers on separatelines with negative numbers ordered backwards:

1 5 7-2 -4 -8

Ip

Op

Vp

I0

O0

V0

In

On

Vn


E) Ordered as negative and whole numbers connected,but negatives are ordered backward and before posi-tives: 1 5 7 -2 -4 -8

Ip

Op

Vp

I0

O0

V0

In

On

Vn

F) Ordered as negative and whole numbers connectedbut negatives are ordered backward: -2 -4 -8 1 5 7

Ip

Op

Vp

I0

O0

V0

In

On

Vn

G) a. Negatives ordered before zero but mixed (twotransposed), positive correct b. Negatives orderedcorrectly, positive mixed (two transposed)

Ip

Op

Vp

I0

O0

V0

In

On

Vn

H) Sometimes ordered randomly; sometimes cor-rectly

Ip

Op

Vp

I0

O0

V0

In

On

Vn

integer ordered value – greatest (greater) and least 219

I) Ordered as negative numbers and whole numberson separate lines

1 5 7-8 -4 -2

Ip

Op

Vp

I0

O0

V0

In

On

Vn

J) Ordered as negative and whole numbers connected,but the position of negative and positive numbers isswitched: 0 3 8 -9 -7 -5

Ip

Op

Vp

I0

O0

V0

In

On

Vn

K) Orders numbers correctly -8 -4 -2 1 5 7 Ip

Op

Vp

I0

O0

V0

In

On

Vn

integer ordered value – greatest (greater) and least

A) Ordered Randomly.Out of several positive numbers, may choose wrong

one as greatest.

Ip

Vp

I0

V0

In


B) Absolute value, no zero. Decides greatest/least basedon absolute value, but zero is not considered.

Ip

Vp

In

C) Absolute value.Treats all numbers as positive, where largest absolute

value is largest.

Ip

Vp

I0

V0

In

D) Ignore negatives.Only considers positive numbers when judging

largest/smallest.

Ip

Vp

I0

V0

In

Vn

E) Negatives equal zero.Positives > negatives, but all negatives equal zero

Ip

Vp

I0

V0

In

Vn

integer ordered value – greatest (greater) and least 221

F) Backward negatives; zero least.Treats more negative as larger than less negative

values; negatives consider as greater than or equal tozero

Ip

Vp

I0

V0

In

Vn

G) Backward negatives.Treats more negative as larger than less negative

values.

Ip

Vp

I0

V0

In

Vn

H Backward vs. formal.Sometimes treats negatives as backward, sometimes

correct. (Mix of G and J)

Ip

Vp

I0

V0

In

Vn

I) Formal vs. absolute.Sometimes treats as positive, sometimes correctly

solves; (Mix of C and J)

Ip

Vp

I0

V0

In

Vn


J) Formal value.Positive > Negative; Less negative> more negative

Ip

Vp

I0

V0

In

Vn

directed magnitude – more/less high/low

A) All high or low.Always moves up or always moves down.

Dp

Dn

B) More low or more high.Always moves lower except for less high OR always

moves higher except for less low.Dp

Dn

C) More and less.Moves higher for “more” and lower for “less.”

DpDn

D) High and low.Moves higher for “high” and lower for “low.”

DpDn

E) Less high/More low.Moves correctly except goes higher for less high or

more low OR moves correctly except goes lower forless low or more high.

Dp

Dp

Dn

Dn

F) Less low.Moves correctly except goes lower for less low.

DpDn

G) More/Less, High/Low.Correctly interprets the pairs of words.

DpDn

J F I NA L C O D E S A N D P R O T O C O L

general directions

Code each category separately, going through all answers for the studentbefore moving on to the next category. (�e only exception is Counting,When Count, and Count Errors should be scored together.)

If students’ explanations are unclear, use up to three questions before anda�er the current one to help you make a judgment. If students write an answerand then change the numeral that they write (not if they do it immediatelybut a�er a long pause or additional dialog), code answers as two problems.If the student just adds or removes a negative from his/her response, theseanswers count as one problem. If student changes answer, code up until thestudent starts to question the �rst answer, then start coding the next sectionseparately.

name for negatives & negative signs

How do students refer to negatives and negative signs?

(Read the student’s response to the question. Determine what the studentcalls negative signs when talking about the problem, talking about how he/sheis solving the problem (such as counting), or talking about the answer. He orshe must use the language for it to be coded. If the interviewer says, “�at’sa negative?” and the student agrees without saying the word, then it doesnot count. If student repeats the word a�er the interviewer, it does countas long as the student has used that word without prompting in a questionbeforehand. Except for “minus” and “negative,” if students use more than oneterm in the same question, code for the higher level term (the one furtherdown on the list).)

223

224 J ⋅ final codes and protocol

CODE EXPLANATION

na �e problem and answer do not have any negatives or the student doesnot provide a name for the negative signs. �is may be because the studentdoes not talk about the problem or answer or is ignoring all of the negativesigns.

Vague �e student refers to negative signs as “dash”, “line”, “blank” or a similarlyvague word (such as “this”, “these” – if accompanied by a gesture to it). E.g.,(-1 + 8) “Blank one plus eight.”

Minus �e student refers to negative signs as “minus or take away sign.” If prob-lem involves adding a negative and student refers to both adding and sub-tracting but it obvious that this is because they are treating the negative asan additional minus sign, it would also count here. Student might refer tothem as “minus” numbers. E.g., (-5 − -5) “Minus �ve minus minus �ve.”E.g., “�e answer is minus three.”

Penalty �e student refers to negative signs as “penalties” or something “owed” or“taken away” or “missing” or “below zero.” E.g., (-1 + 8) “Penalty one pluseight” or “Below zero one plus eight.”

Negative �e student refers to negative signs as “negative”. Alternatively, the studentcould say a number is “not negative” if referring to a positive number. O�-side comments about negatives in general do NOT count. Comments needto be targeted on the actual numbers. E.g., (-5 − -5) “Negative �ve minusnegative �ve.” E.g., “It’s 3 + 5 and since there’s a negative it’s negative eight.”

Min-Neg Student refers to negative signs as both “negative” and “minus” in the samequestion.

operation language

What operation(s) do students talk about using?

(Read the student’s response to the question. Identify what operation (ifany) students talk about using during the problem. Like previous category,students must use the language, not just agree to interviewer’s suggestion.If interviewer asks how they knew to go up/down and student says becauseof plus/minus sign, it would be coded. Do not count use of these words inother contexts; for example “I added a negative sign.” Students might refer toprevious solution – if so, code applies to that previous problem, not currentone. (E.g., “I thought it was a plus.”))

CODE EXPLANATION

na Student does not talk about the operation. �e student may just say and/orwrite the answer or just count.

Continued on next page . . .

direction 225

. . . continued from previous page

CODE EXPLANATION

Vague �e student refers to negative signs as “dash”, “line”, “blank” or a similarlyvague word (such as “this”, “these” – if accompanied by a gesture to it). E.g.,(-1 + 8) “Blank one plus eight.”

Plus/Add �e student talks about adding or plussing or reads/describes the problemwith “plus/add.” If students say part of the word (e.g., pl) it counts if it isclear that student is adding (in terms of absolute value). Student may alsotalk about doing 4 + 5 = 9 to solve 9 − 4. If student says, “It’s plus, notminus,” it would only be coded as plus.

Minus/Subtract

�e student talks about subtracting or minusing or reads/describes theproblem with “minus”, “subtract”, or “take away”. If student starts to sayminus (e.g., “min”) it counts. If students say they are “taking” a numberbut doesn’t say “take away”, it counts if the answer is consistent with anoperation and not meaning “I’m going to start with” or “I’m going to moveit”, etc. Be careful that the “minus” is in terms of the operation and not thenegative sign. If student says, “It’s minusing, not plussing”, it would only becoded as minus.

Both �e student uses words associated with adding and subtracting when de-scribing what they are doing. �is may be because they believe they shouldadd and subtract when they see +- or because they mistakenly read it andare correcting themselves or because they use one operation to reasonabout what would happen with the other operation. Students may start tosay part of the word before changing their minds. E.g. (-5 − -5) “Five pl,minus �ve.” E.g. (-8 − -8) “Eight plus eight is sixteen but you minus, so it’s

-16.”

direction

What directional language do students use?

(�is category refers to language students use about which direction tomove, count, calculate, etc. It doesn’t count if students use these words inother contexts, such as talking about quantity di�erences between two num-bers (that one’s higher) or an amount le� over (one more le�). Students muststate this language on their own without interviewer prompting (unless inter-viewer gives two opposite choices to choose from: did you go up or down?).If the interviewer says it, it doesn’t count even if student then uses that wordfor the same question explanation. If the student moves le� on the numberline but doesn’t say they are moving le�, it is not coded. Codes further downthe list trump those at the top of the list.)


CODE EXPLANATION

na Student does not mention any directions.

More Students say they are counting more/bigger/larger or going more. If stu-dents use another directional word a�er more, code for the following word.E.g., One more up is coded as “up”, not “more”.

Less Students say they are counting less/smaller or they are going less orsmaller.

Up Students say they are counting up or they are going up.

Down Students say they are counting down or they are going down or “it’s down”.

Higher Students say they are counting/going higher or say “it’s high”.

Lower Students say they are counting/going lower or say “it’s low”.

Before Students say they counted before or are going before a number.

A�er Students say they counting/going a�er a number or to the “next”.

Backward Students say they are counting backward or they are going backward/back.

CountOn Students says they counted on.

Forward Students say they are counting/going forward or in front.

Right Students say they are counting right or they are going right.

Le� Students say they are counting le� or they are going le�.

Opposite Student indicates that the answer or problem is opposite/going the otherway.

MoreLess Students say they are moving more and less

UpDown Students say they are going up which is down (or vice versa)

RightLe� Students say they are going right and le�

HigherLower Students say they are going higher and lower

BeforeA�er Students say they are going both before and a�er.

ForwardBack Students say they are counting forwards which is really backwards (viceversa).

MorePositive Students say they are counting more positive or they are going more posi-tive.

MoreNegative Students say they are counting more negative or they are going more nega-tive.

LessPositive Students say they are counting less positive or they are going less positive.

LessNegative Students say they are counting less negative or they are going less negative.

Other Use if we need to create a new category.

number order 227

number order

What order of numbers do students use?

(Read student’s solution and determine if the student keeps the numbersin the original order or not. GIVE PREFERENCE TO STUDENT’S COUNTor USE OF FINGERS. If students say they didn’t reverse but they clearly didbased on their counting, it would be reverse. If they don’t use counting/�ngers,GIVE PREFERENCE TO WHAT THEY DO BEFORE ANSWERING. If for3 − 9 they say “nine minus three is six” but then later say they did 3 − 9 = 6,it would be reverse. If students explain AFTER answering and they say boththe original and reversed versions, (e.g. 6−8): Six minus eight is two becauseeight minus six is two (Reverse – keyword “because”); eight minus six is twoand six minus eight is two (Reverse – reversed one said �rst); six minus eightis two and eight minus six is two also (AsIs – original problem stated �rst).)

CODE EXPLANATION

AsIs Student keeps numerals as written in the problem. If it is a problem whereboth numbers are the same (e.g., -8 − -8), and student just gives an answer,this code also applies. If student skips the problem but reads it in the orderit is written �rst, code as “AsIs.”

Unknown Student just gives an answer and provides no explanation which helpsdetermine which number they started with. If it is a positive subtractionproblem, the answer should provide some evidence of which number thestudent started at; however, for many addition problems, the student couldhave started at either number (if the signs on the numbers are the sameor interpreted as the same) and for negative subtraction problems thestudent could have applied the negatives in strange ways. If student skipsthe problem and doesn’t read it, it is unknown.

Reverse Student indicates that they switched the order of the numerals in the prob-lem. �ey explicitly state it or rewrite the problem or it’s obvious from theway they are counting. For -3−5 says, “5 minus 2 is 3 so 3 minus 5 is 2,”thiswould be reverse even though the student read the problem AsIs. As longas student justi�es it using the reverse, count it as reverse.

Reverse(Implied)

�e answer explanation or answer makes sense if the student thinks theproblem is equivalent to one where the numbers are reversed (but we aren’t100% sure that’s what the student did – when subtracting a larger fromsmaller number students will o�en get an answer of zero. If student gets ananswer of zero but would have needed to reverse the numbers to subtract alarger from smaller (in absolute value or otherwise), this goes here).E.g., (-3 − -7), does 3 + 4 = 7 (fact family) and then adds a negative. �eonly way this would make sense using the fact family is if it were -7 − -3.




CODE EXPLANATION

E.g., (6 − -7),“I did 6 − 7 = 1 and then it’s negative one because there’sa negative there.” For 6 − 7 = 1, mathematically student would have toreverse.E.g., (5−-3), student gets an answer of zero (makes sense if he did 3−5 = 0)

adjust numbers

How do students change the numerals in the problem?

(Read through the student’s solution and decide what they are doing withthe numbers. Default to assuming that the student is using the numbers givenif there is no explanation, unless this is impossible given their answer.)

CODE EXPLANATION

na Students do not change which numerals they are working with or there isno explanation and the answer is reasonable (any number in the problem–negative or positive–or any combination of the numbers) given the num-bers. If student reads the problem correctly but gives an incorrect answer,it still counts as “na” – they may be recalling incorrectly. If students countand are one o� on the answer and there is no evidence that they purposelychanged the numbers they were working with, count as “na” and codefor a counting error. If students are one o� on the answer and there is noevidence that they purposely changed the numbers, code as “na”.

Unclear Student changed the numbers they are working with but it’s unclear what(there is no way they could have gotten the answer they did without doingsomething to the numbers or changing the problem). If students do notread the problem but count or show a di�erent number of �ngers thanwould be expected and get an answer inconsistent with the original prob-lem, it is unclear if they miscounted or changed the numbers. If studentreads the problem correctly but uses a di�erent amount of �ngers it is acounting error, not a change in numbers and should be marked “na”. How-ever if student doesn’t read the problem, says they know it, and gets thewrong answer, it would be unclear. For problems such as 6 − 8, 6 − -7,answers of 0 are considered reasonable and don’t fall under “unclear.”

Value Student reads problem correctly but chooses to ascribe a new value to oneor both numbers. E.g., “. . . take away negative seven, it’s like �ve, you takeaway �ve more.”

ANN When both numerals are positive, student adds a negative to one or both.

Add10 Student adds ten to an initial number to avoid negatives (it’s like they areregrouping to get more ones, except they have nothing to regroup from.)E.g., (3 − 7), does 13 − 7 = 6.


numbers 229


CODE EXPLANATION

R1w2 Student replaces the �rst number with the second one. E.g., (2 − 5), does5 − 5.

R2w1 Student replaces the second number with the �rst one. E.g., (2 − 5), does2 − 2.

Nothing1stHalf

Student ignores the �rst part of the problem, seems to not realize that othernumbers are there, or treats the �rst number as nothing or somethingthat is no longer there, and answers the 2nd number. If student keeps thenegative from the part they ignored, select NegMod and this. “1st half ” isconsidered the �rst half of the original problem.

Nothing2ndHalf

Student ignores the second part of the problem or seems to not realize thatother numbers are there, or treats the �rst number as nothing or some-thing that is no longer there, and answers the 1st number. If student keepsthe negative from the part they ignored, select NegMod and this. “2ndhalf ” is considered the second half of the original problem.

NothingBoth Student ignores both the �rst and second parts of the problem or treatsthem both as nothing.

PartofPart Student considers the smaller number as part of the second. When adding,the �rst number isn’t added on but counted as part of the other. (E.g., 2 + 4is one, two. �ree, four.)

numbers

What numbers do students use when solving the problems?

(Read the student’s solution. Determine what numbers students use/ac-knowledge to solve the problem. Students may mention a certain type ofnumber but not use them when working on the problems or use a negativenumber in their answer a�er working with whole numbers.)

CODE EXPLANATION

Unknown Can’t be determined what numbers students used to solve the problems.E.g., (-2 + 7), student writes 5. It is unclear if student counted up fromnegatives or just did 7 − 2. Students do not mention positive or negativenumbers (e.g., they do not read the problem). E.g., (-5−-5) student answers0. �ey could have done 5 − 5 or -5 − -5.

WholeNumbers

Student counts with whole numbers or names numbers as positive, andgets a positive answer. Negatives are not mentioned or given as answers(there is no evidence that negative numbers were used).




CODE EXPLANATION

TenEnd Student counts in the positive numbers but won’t go past ten even thoughthey need to in order to use up the numbers.E.g., (8 − -7), does 8 + 7 but stops counting on at 10.

NegOppDir Student counts into the negatives but the negatives are in backwards orderor the student gets a negative answer but the only way they could havegotten this answer is if they thought the negatives were in backwards orderE.g., (3 − 9), gets -4.

NegZero Student counts and says -0 or student solves a problem like -5 − -5 aspositive and then adds the negative at the end to get -0.

Integer Student talks about or uses positive and negative (or just negative) num-bers. Student might just give a negative answer or answer depends uponrecognizing the numbers as negatives (e.g., -2 + 7, says “I counted up sevenand got to �ve.”)

solutions

In which solution category do students’ answers fall?

Read the student’s response to the question. A�er taking into accountwhether students reversed the numbers in the problem or changed some ofthe numbers, determine in which category their answer falls. See answer tablefor help with this. If an answer falls in more than one category, use students’explanations to help decide where it should fall. If the explanation does nothelp or is not present, choose unknown. �e �nal acronym codes shouldonly be used if the student speci�cally verbalizes them. Use info prompted byinterviewer.

CODE EXPLANATION

na �e student skipped the problem.

Unknown Student gets an answer that �ts in more than one category and it isn’t ex-plained or the explanation doesn’t help clarify which category it goesin, or student gets an answer that is not a reasonable combination of thetwo numbers (possibly because they miscounted or guessed) and it is notpossible to tell what answer they would have gotten if they knew the factcorrectly. If students’ answers are 1 o� of something reasonable, interpretthem as the reasonable number (e.g., 7 − 3 = 3, treat as 7 − 3 = 4).


solutions 231


CODE EXPLANATION

PosEqui Student gets a positive/zero answer that would be expected if the studentignored all of the negative signs or, in the case of problems like 4−7, refusesto operate into the negatives. Some reasons for this could be 1) the studentignores negatives; 2) the student interprets negatives as similar to positives;3) the student believes adding a negative = subtraction or subtracting anegative = subtraction. (Overall patterns might elucidate this.)E.g., (-6 + -4) “Minus six plus minus four. 10.”E.g., (5 − -3), “Five minus three is two.”E.g., (1 − 4), “You can’t take more away. It’s zero.”

NegPosEqui Student gets a negative answer that would be expected if the student ig-nored all of the negative signs but the resulting problem could result in anegative answer or in the case of problems like 4 − 7 gets a negative answerwithout showing counting. A possible reason for this may be that studentsdon’t recognize negatives within the problem but can get a negative answer.E.g., (2 − -5), “Two minus �ve. . .negative three.”

OppPosEqui Student gets a positive/zero answer that would be expected if the studentignored all of the negative signs (or kept numbers positive) and used theopposite operation. Some reasons for this could be 1) the student knowsequivalency rules (e.g., 5 − -3 = 5 + 3); 2) the student ignores negativesigns and mistakenly uses the opposite operation. (Overall patterns mightelucidate this.)E.g., (-3 − 8) “�ree plus eight is eleven.” E.g., (4 − 6) “Four plus six is ten.”

CountLe� Student gets an answer that would be expected if the student started at oneof the original numbers on a number line (even if negative) and counted tothe le� (more negative) the absolute value of the other number of spaces.�ere must be physical/verbal evidence (use of �ngers–just showing theinitial amount is okay if answer is consistent with them then countingle�–, verbal counting, tapping, drawing, student says “I counted or movedthree spaces”). Just agreeing with the interviewer’s count does not applynor does just saying “I used the number line” if there is no evidence of howthey used the number line. If student counts, writes answer, and a�er moredialog adds a negative sign, then look at “Negmod” codes.

CountRight Student gets an answer that would be expected if the student started at oneof the original numbers on a number line (even if negative) and counted tothe right (more positive) the absolute value of the other number of spaces.�ere must be physical/verbal evidence (use of �ngers–just showing theinitial amount is okay if answer is consistent with them then counting right–, verbal counting, student says “I counted or moved three spaces”). Justagreeing with the interviewer’s count does not apply nor does just saying “Iused the number line” if there is no evidence of how they used the numberline. If student counts, writes answer, and a�er more dialog adds a negativesign, then look at “Negmod” codes.




CODE EXPLANATION

NegMod Student gets a negative* answer that would be expected if the student ig-nored all of the negative signs, solved the problem, and then made theanswer negative (or made all numbers negative �rst and then solved theproblem). (For smaller minus larger problems, student gets a correct nega-tive answer only if student reversed.) Students might 1) relate this problemto a positive one except “it’s lower/negative”; 2) they might think the an-swer needs to be negative because the problem has a negative symbol; 3)they might solve the problem by counting within the negatives in theirheads (so it’s not visible), getting a negative answer. *If student answers0 because they specify that there is no negative zero, it can go here. E.g.,(-5 − -5) “Five minus �ve = 0 and there’s no negative zero, so it’s 0.” **Ifstudent answers -N2 because they recognize the negative sign as some-thing else that modi�es the problem but they also can get negative answers,choose this category and write a comment.E.g., (4+-6) “Negative ten because four plus six is ten and add the negative.”

OppNegMod Student gets a negative* answer that would be expected if the student ig-nored all of the negative signs, solved the problem with the opposite oper-ation, and then made the answer negative (or made all numbers negative�rst). (For smaller minus larger problem, student solves as if adding thenumbers and making them negative.) Students might 1) relate this prob-lem to a positive one but mix up the operations; 2) they might think theanswer needs to be negative because the problem has a negative symboland also uses an equivalency rule; 3) they might solve the problem by say-ing they are using the correct operation but counts in the wrong direction.*If student answers 0 because they specify that there is no negative zero, itcan go here.E.g., (1 − 4) “Four plus one is 5 and you go down because it’s minusing, so

-5”E.g., (-5 − 3) “Negative �ve minus three. Negative four, three, negative two.”E.g., (-6+-4) “Negative two because six minus four is two and the negative.”

NegZeroAll Student gets an answer that would be expected if the student treated 1)all negatives as if they were zero, 2) treated subtracted numbers in positivenumber problems as if they were zero (E.g. for 6−9 thinks “-9” is 0 becauseyou took 9 away), or 3) treated both a negative and subtracted number aszero (e.g., -5− 3). �is applies if the student substitutes zero for the numberor if they ignore one half of the problem because it is worth zero. Studentsmight do this because 1) they think negatives are worth zero; 2) they thinknegatives are taken away, 3) they think adding a negative means you haveto add and then subtract the number. Student might treat the number aszero but still keep the negative sign. �ere must be some indication thatthis is what they are doing. Students might alternatively ignore negativesigns on the remaining numbers.E.g., (-3 − 5), does 3 − 3 − 5 or just ignores the �rst half and says, “Five”E.g., (5 − -2), does 5 − 0


solutions 233


CODE EXPLANATION

NegZeroPick Student gets an answer that would be expected if the student treated one(of two) of the negatives as if it was zero or subtracted numbers (but notthe negative) as if they were zero (E.g. for -6 − 9 thinks “-9” is 0 becauseyou took 9 away). �is applies if the student substitutes zero for the num-ber or if they ignore one half of the problem because it is worth zero. Stu-dents might do this because 1) they think adding or subtracting a negativedoesn’t change the initial number; 2) they don’t realize both numbers havenegatives, 3) they use an equivalency rule to change subtracting a negativeto adding or adding a negative to subtracting, so only one negative is le�or think they just need to add and subtract one of the numbers. Studentmight treat the number as zero but still keep the negative sign. �ere mustbe some indication that this is what they are doing. Students might alterna-tively ignore negative signs on the remaining numbers.E.g., (-7 + -1) “It’s -7 because -1 doesn’t do anything.”

PosZero Student gets an answer that would be expected if the student treated apositive number as if it was zero (in any position for addition problems butonly in the initial position for subtraction problems).E.g., (-1 + 3) “It’s -1. 3 does nothing” E.g., (3 − 6) “It’s -6.”

NegSub Student gets an answer that would be expected if the student used the neg-ative sign as subtraction and chose a number to put before the subtractionsign. Student must say what they are doing or this will be unknown.E.g., (-3 − 5), does 1 − 3 − 5.

DoubleSub Student gets an answer that would be expected if the student used thenegative sign as “subtract” again when it follows a minus sign.E.g., (7 − -2), does 7 − 2 − 2.

�e codes below trump those above if student explicitly states the rule in a recognizable form(they may use slightly di�erent language)

SNLN (SNA) Student states that subtracting a negative is going less negative/smaller(to the right on the number line, toward positive park) or adding (a posi-tive)/getting larger.OR Student states that subtracting the same number/same negative is zeroor justi�es an answer of zero because the numbers are the same.

SPLP (SPS) Student states that subtracting a positive is going less positive (to the le� onthe number line, toward negative nettles) or subtracting (a positive).

ANMN(ANS)

Student states that adding a negative is going more negative (to the le� onthe number line, toward negative nettles) or subtracting (a positive).

APMP (APA) Student states that adding a positive is going more positive (to the right onthe number line, toward positive park) or adding (a positive).

WrongRule Student states that adding a negative is like adding a positive (going righton the number line or more positive) or subtracting a negative is like sub-tracting (going le� on the number line or more negative)


PosE

qui

Neg

Pos

Equi

Opp

Pos

Equi

Neg

Mod

Opp

Neg

Mod

Neg

Zero

(all)

Neg

Zero

(pic

k)Po

sZer

oD

oubl

eSu

bC

ount

Le�

Cou

ntRi

ght

SNLN

SNA

SPLP

SPS

AN

MN

AN

SA

PMP

APA

5−

- 32(

0*)

Na

8-2

-85/

-5N

a±

30/

-12(

-8*)

8(2/

0*)

88−

- 71(

0*)

Na

15-1

-15

8/-8

Na

±7

0/-6

1(-1

5*)

15(1

/0*)

159−

- 27(

0*)

Na

11-7

-11

9/-9

Na

±2

57(

-11*

)11

(7/0

*)11

4−

- 50(

1*)

-19

-0/0

(-1*

)-9

4/-4

Na

±5

0/-6

0/-1

(-9*

)9(

-1*)

96−

- 70(

1*)

-113

-0/0

(-1*

)-1

36/

-6N

a±

70/

-80/

-1(-

13*)

13(-

1*)

13- 7+

- 18

Na

6(0*

)-8

-6(-

0*/0

*)0

±7/±

1N

aN

a-8

-6(6

/0*)

-8- 6+

- 410

Na

2(0*

)-1

0-2

(-0*

/0*)

0±

6/±

4N

aN

a-1

0-2

(2/0

*)-1

03−

90(

6*)

-612

-0(-

6*)

-12

3/-3

±3

±9

Na

0/-6

(6*)

12-6

6−

80(

2*)

-214

-0(-

2*)

-14

6/-6

±6

±8

Na

0/-2

(2*)

14-2

1−

40(

3*)

-35

-0(-

3*)

-51/

-1±

1±

4N

a0/

-3(3

*)5

-3- 5−

90(

4*)

-414

-0/0

(-4*

)-1

49/

-9±

5±

9N

a-1

4(4*

)0/

4(14

*)-1

4- 3−

50(

2*)

-28

-0/0

(-2*

)-8

5/-5

±3

±5

Na

-8(2

*)0/

2(8*

)-8

- 1+

89

Na

0(7*

)-9

-0/0

(-7*

)8/

-8N

a±

1N

a-9

(7*)

0/7(

9*)

7- 2+

79

Na

0(5*

)-9

-0/0

(-5*

)7/

-7N

a±

2N

a-9

(5*)

0/5(

9*)

5- 4+

610

Na

0(2*

)-1

0-0

/0(-

2*)

6/-6

Na

±4

Na

-10(

-2*)

0/2(

10*)

2- 3+

14

Na

2(0*

)-4

-2(-

0*/0

*)1/

-1N

a±

3N

a-4

(0/-

2*)

-2(4

*)-2

- 9+

211

Na

7(0*

)-1

1-7

(-0*

/0*)

2/-2

Na

±9

Na

-11(

0/-7

*)-7

(11*

)-7

- 4−

- 70(

3*)

-311

-0/0

(-3*

)-1

10

±4/±

7N

a-1

8(-1

5*)

-11

0/3(

-3*)

3- 2−

- 60(

4*)

-48

-0/0

(-4*

)-8

0±

2/±

6N

a-1

4(-1

0*)

-80/

4(-4

*)4

- 6−

- 90(

3*)

-315

-0/0

(-3*

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50

±6/±

9N

a-2

4(-2

1*)

-15

0/3(

-3*)

3- 4−

- 31(

0*)

Na

7-1

-70

±4/±

3N

a-1

0(-1

1*)

-7-1

(1/0

*)-1

- 8−

- 53(

0*)

Na

13-3

-13

0±

8/±

5N

a-1

8(-2

1*)

-13

-3(3

/0*)

-3- 8−

- 80

Na

160/

0-1

60

±8

Na

-24

-16

00

- 5−

- 50

Na

100/

0-1

00

±5

Na

-15

-10

00

7+

- 310

Na

4(0*

)-1

0-4

(-0*

/0*)

7/-7

Na

±3

Na

4(-1

0*)

10(4

*)4

5+

- 27

Na

3(0*

)-7

-3(-

0*/0

*)5/

-5N

a±

2N

a3(

-7*)

7(3*

)3

medium 235

medium

What do students use to solve the problem?

(Read the student’s solution and determine what the student uses to answerthe problem before he or she states or writes the answer – whichever comes�rst. Information that students provide a�er writing the answer can be usedto clarify the codes; for example, if student just writes answer (Head) butthen says they were picturing objects in their head, it would be HeadObjects.If students use multiple mediums (such as �ngers and then tallies), codefor the prominent one – although once students use something other thanhead/verbal, you cannot use this code for the question.)

CODE EXPLANATION

Head Student shows no outward physical/verbal signs of calculation or says sheused her head or just knew it. Mouthing words without us hearing themalso counts as Head. Students might say the problem or answer or theymay later say they were counting – this is still considered just “head.”

Verbal Student counts out loud or as a whisper without using anything to keeptrack of the count or student talks about the facts or relations they areusing while solving the problem without using anything else (no manip-ulatives or representations). If counting, students need to say at least onenumber in the counting series (apart from the initial number or answer. Ifstudent uses �ngers and counts aloud, code as Fingers. Phrases which doNOT count towards the verbal code include “It’s hard”, repeating part ofthe problem (including “plus”, “minus”, or answer), or “I don’t know”

Fingers Student uses �ngers to solve the problem and may or may not also countout loud. Student does not need to use all �ngers necessary to model theproblem or may also just look at them.

VNL Student uses a written vertical number line to solve the problem or mo-tions moving up/down as if picturing a number line.

HNL Student uses a written horizontal number line to solve the problem ormotions moving le�/right as if picturing a number line.

Circles Student uses written circles to solve the problem.

Objects Student uses objects to solve the problem.

Tallies Student uses written tallies, marks, or lines to solve the problem.

Numerals Student writes a list of numerals to solve the problem.

HeadFingers Students say they pictured �ngers in their heads.

HeadNL Students say they pictured a number line in their heads.

HeadCircles Students say they pictured circles in their heads.

HeadObjects Students say they pictured objects in their heads.Continued on next page . . .



CODE EXPLANATION

HeadTallies Students say they pictured tallies in their heads.

HeadNumerals Students say they pictured numerals in their heads.

counting

How do students count?

(�is category refers to how students visibly count; moving �ngers w/otalking is still counting. Code for last counting attempt before they give theiranswer. Determine the direction of counting based on which way studentsgo from the �rst number they use. If interviewer asks them to show how theycounted, this information can be used to inform a code but does not countas its own code. If student says they counted and you know which numberthey started at and/or how many they counted or if student just shows theinitial amount using �ngers, this information can be used to imply how thestudent counted, such as away from zero, toward zero, etc. (see “implied”categories). If students count as part of another problem (not the one given),it still counts, but if counts for actual problem as well, code just for givenproblem. If problem involves adding or subtracting one or “one more” andstudent states the answer originally, it is counted as counting. If they just writethe answer and then say they counted, it would be “Counts”.)

CODE EXPLANATION

notvisible No counting is apparent or mentioned. Students might say they used anumber line or went backwards/forwards, up/down but they do not showwhat that looked like.

Counts Students says they counted but it is not clear how they did so because theygave no outwards signs of counting. and it’s unclear how based on theiranswer (especially if their description doesn’t match their answer) or whatthey were doing made no sense. �is category does NOT apply if the stu-dent only says they used a number line or “went backwards, up, or down”since this language is o�en used to describe “addition” and “subtraction.”�ey need to mention counting or taking steps or moving spaces. If stu-dent counts out the �rst number and then gives that number as an answer,it is “counts.”


when count is present 237


CODE EXPLANATION

Cdist Student counts the distance between the two numbers (starting with the sec-ond number – e.g., 5 − 3 is “three..four, �ve. Two.” – student would show two�ngers in this instance; starting with �rst number – e.g., 5 − 3 is “�ve...four,three. Two.” – student would show two �ngers here as well.)

Cdist(implied)

Student says he counted the distance between the two numbers but showsno evidence. Just saying he counted and got 5 − 3 = 2 isn’t enough todetermine that he used distance. He may have counted down (this wouldjust be “counts”).

CAFZ Student counts away from zero in either positive or negative direction.

CAFZ(implied)

Student says they counted or moved up/down and gets an answer consis-tent with counting away from zero. (E.g., 2 + 7 = 9 must be CAFZ)

CtoZ Student counts to zero and stops.

CtoZ(Implied)

Student says they counted and gets an answer of zero.

CtowardZ Student counts toward zero but doesn’t get to zero.

CtowardZ(Implied)

Student says she counted or moved up/down and gets an answer consistentwith counting toward zero.

CthroughZ Student counts through zero from positive to negative or vice versa.

CthroughZ(Implied)

Student says he counted or moved up/down and gets an answer consistentwith counting through zero.

CAFZ-CtowardZ

Student �rst counts away from zero and then toward zero.

CtowardZ-CAFZ

Student �rst counts toward zero and then away from zero.

CAFZ-CthroughZ

Student �rst counts away from zero and then through zero.

CthroughZ-CAFZ

Student �rst counts through zero and then away from zero.

when count is present

When do students count?

(�is category refers to whether students count before writing an answer,a�er, or both.)


CODE EXPLANATION

na Student doesn’t count.

Unknown Student says “I counted” or thought of a number line but there is no evi-dence (�ngers, verbal number counting) to verify when/if they actually didit. (Interviewer may say, “So you counted backward” and child may agreebut we still have no other evidence that this is true, so it is unknown.)

Before You can tell based on students’ use of �ngers, moving on a number line,drawing circles, verbalizing, etc. that the student is counting before writingtheir answer OR before the interviewer asks him how he got the answer. Ifstudent counts, states an answer, recounts and gets a new answer, this is allconsidered “before” the �nal answer.

A�er A�er writing the answer or a�er the interviewer asks him how he got theanswer, student counts (like he/she is explaining or checking the answer).Unlike “Unknown” if student says he counted and then demonstrates how,it would be this category (we still don’t know if that’s really how the studentcounted beforehand). If the interviewer asks the student to show howhe/she counted, this category does NOT apply and the information canonly be use to clarify how the student counted. It is only if the studentvolunteers it.

Both You can tell based on students’ use of �ngers, moving on a number line,drawing circles, verbalizing, etc. that the student is counting before writingan answer or before the interviewer asks him how he got the answer ANDa�er writing the answer or a�er the interviewer asks him how he got theanswer, the student counts again.

count error

What errors do students make while counting?

(�is category refers to what errors students make while counting.)

CODE EXPLANATION

na Student doesn’t count or counts with correct order of numbers/process.

Unknown Student made some sort of counting error, but it is not clear what the erroris (or it is not listed below or student made several errors)

CIN Student starts count with initial number (or answer is consistent with this)E.g., (5 + 3), does 5, 6, 7 instead of 6, 7, 8.

CFS Student counts from second number E.g., (8 − 5), does 4, 3, 2, 1, 0 insteadof 7, 6, 5, 4, 3.

Extra Student counts extra numbers (using 1 to 1 correspondence) or reportsextra �ngers.


answer manipulation 239


CODE EXPLANATION

Fewer Student counts fewer numbers or reports too few �ngers (does not count ifstudent stops at zero because they don’t believe there is anything below it).

SkipNum Student skips a number while counting.

Lack1to1 Student puts down too many �ngers per number or says too many/fewnumbers for the correct number of �ngers.

WrongIN Student starts counting as if the initial number is something else.

Report Student counts correctly but then reports the wrong number (i.e. the num-ber counted).

Subdist Student subtracts the distance between the two numbers instead of thesubtrahend. For 8 − 6 knows the distance is 2 and does 8 − 2 = 6.

answer manipulation

How does the student manipulate their answers?

(�is category applies to di�erences in what students’ say versus what theywrite as well as what they do to the answer a�er writing it down.)

CODE EXPLANATION

na Students’ spoken and written answers match, and they do not change orstudent doesn’t speak an answer but also doesn’t change the written answer.

MismatchPN Student says the answer as positive when talking about the given problem(not the corresponding positive problem) but writes it as negative. Do notmark this if student says something like, “It’s like 6 + 8 = 14 but negative”and writes -14. It should be more obvious. “It’s 14.” Writes -14.

MismatchNP Student says the answer as negative but writes it as positive.

MismatchNum Student says one number but writes a di�erent one. �e emphasis here ison the numeral, not the sign.

ChangeNum Student changes the answer a�er writing it down (the focus here is on thenumeral, not the sign). Chose this if student erases/crosses out an answerand writes a new one (put in 2nd row).

AN Student adds a negative to the answer a�er writing it down and there hasbeen additional dialog or they were moving on to the next problem. E.g.,says/writes 5 and comes back to the problem and adds a negative later.

RN Student removes the negative from the answer a�er writing it down.


fact/recall strategy

What strategies do students use to remember the answer?

(�is category refers to how students use math facts or number relations.Counting by itself is NOT counted. IF student’s answer appears in morethan one category, choose the one further down the list. IF student refersto a previous question and says they did it the same way, you can use thesame code as long as student doesn’t appear to do anything di�erent and thestrategy is consistent with the answer.)

CODE EXPLANATION

notstated Student’s explanation is unclear, you cannot understand what the child issaying, or no fact or recall strategy (from list below) is stated. �is may bebecause:1) Students may answer quickly but not say what they were thinking about.2) Students might count and not describe any further strategies3) Students might just say “I knew it.”4) Students might restate the problem or just talk about the quantitiesinvolved but not how they are dealing with them (e.g., for 6 − 8, “Well, sixis less than eight. I’ll put -6.”

Skip Student doesn’t state/write a �nal answer or doesn’t know and won’t guess.

Guess �ere are some possible reasons to choose guess:1) Student says he guessed (even if the answer is correct/reasonable)2) Student says they don’t know but will choose a number.3) Student’s answer is far beyond anything considered reasonable, even ifthe student says he/she knew the answer.

If student does one of the three guessing options but then explainsfurther and it can be classi�ed as something below, choose the categorybelow.

UnrelatedProb Student volunteers a math problem as an explanation that is not related tothe current problem (e.g. when solving 8 − 8 says 8 minus 8 is 0 but thenadds: 8 plus 8 is 16) AND doesn’t use one of the other codes below this onein the table.

FactFamily Student relates the answer to a problem using similar numbers rearrangedin a di�erent order (the emphasis is NOT on how the negative changes theproblem).1) Student refers to a problem with the opposite operation to help themdetermine the missing number in the given problem. (e.g. 3 − 9 = 6because 3 + 6 = 9 or 9 − 3 = 6 or one more than seven is eight for 8 − 7 = 1)2) Student solves 1+ -3 = -2 because -3+ 1 = -2 and states this as what theyare doing (otherwise this is just reverse).3) �e numbers somehow help the student determine the answer: 4 + 4 = 8so 8 − 8 = 4.


fact/recall strategy 241


CODE EXPLANATION

RelatedFacts Student changes the problem to an easier one and compensates for thechange or uses simpler problems to help them build up to more di�cultones. E.g., (3 + 9), does 9 + 1 = 10, so 9 + 2 = 11 and 9 + 3 = 12 (if BA,would do 3 + 7 = 10 and 2 more is 12) E.g., (3 + 9), does 3 + 3 = 6 and then+6 more = 12. *If student solves a problem to get to 10 or 0, then chooseone of the “Break Apart (BA)” codes below.

Simile Students relate/talk about/use positive and negative forms of the problem.1) Student analyzes the problem in terms of similarities to the same prob-lem without negatives, (E.g., -4 plus 5 equals -9 because 4 + 5 = 9. �ereverse is also possible: Student acknowledges negatives in the originalproblem but says it is just like the positive problem. “Negative four plus �veis just like 4 + 5 = 9”.2) Student may just indicate that it will be negative because the numbersare negative (implying that they �rst solved the problem and then added anegative). Students might talk about the problem as positive but then give anegative answer.3) Student analyzes the problem in terms of one which involves a similarstrategy (7 − 1 is like 5 − 1, you just put down one �nger). *Comparing to aproblem with the same numbers, same order, and opposite operation onlycounts if student gets the opposite answer. (E.g., 4 + 5 = 9 so 4 − 5 = -9)




CODE EXPLANATION

QuantRules Student expresses and uses a quantity-related rule to solve the problem,such as:1) For subtraction, if two numbers are the same, the answer will be zero.Student cannot just repeat the answer or just point to the two numbers butneeds to indicate that the number is minusing or taking away itself/thesame thing or explicitly state that the numbers are the same and get ananswer of zero/negative zero.2) When subtracting a larger from a smaller, the answer will be zero/neg-ative. Student must state the rule or indicate that they can’t subtract all ofa number because the second number is larger or too many (or you don’thave enough to take that amount away). Student might show that one num-ber is bigger than the other on their �ngers.3) Adding/Subtracting 1 is moving up or down 1. When adding/subtract-ing one, student says the answer before explaining and explains that it isone more/one less or one before/a�er. When subtracting two sequentialnumbers, student knows it is one because they are next to each on a num-ber line (or are close/next to each other). Saying, you just “add one” doesNOT count.4) Student treats negatives as zeros. Student must indicate that they con-sider the value of negatives as zero, so the number doesn’t change. Onlysaying that the number is taken away or that it “was seven” is not su�cient;they must indicate that it is like or is worth zero or “is no more” or “isn’tthere” and provide an answer consistent with this.5) Student indicates that you cannot subtract a negative from a positive, anegative, or a larger/smaller number (likewise with adding).6) Students may adjust a previous rule when they encounter a new type ofproblem. (e.g., a student who treats negatives as worth zero may decide tokeep one of the numbers if both are negative. If they provide some expla-nation for this, count it.) Just talking about quantities is NOT su�cient forthis category. �e following would NOT be part of this category: E.g., “�ink-ing about the fours...pretend there’s six circles.” E.g., “Pretend you have thesecandies.”

SimileQuant Student does both “Simile” and “QuantRules.” E.g., Numbers greater thanten will not be negative, so I didn’t add a negative. Numbers under ten arenegative, so I made it negative.

BAD Student breaks apart a number but then discards part of it. It must be clearthat the student is breaking apart one of the numbers.

BAMT Student breaks apart a number to make a ten/negative ten and then oper-ates beyond the ten. Student might also count to ten and then know howmany a�er ten the answer is. E.g., “-5 + -8, I did -5 + -5 is -10 and threemore is -13.”


source 243


CODE EXPLANATION

BAMZ Student breaks apart a number to get to zero and then operates beyond it.Student might also count to zero and then know the answer without havingto �nish the count. (For -5 + 8: “-5, -4, -3, -2, -1, 0, (sees �ve �ngers le� onhand) oh, it will be �ve.”)

source

Aside from counting and fact strategies, where are students’ answers comingfrom?

(�is category refers to reasons students knew this problem. Responsesfurther down the list trump those at the top.)

CODE EXPLANATION

na Student doesn’t provide any explanation for their answer or solution.

Previous Students say the problem is like a previous one they solved or says theirreason for solving the problem is the “same” as a previous one, even if theydon’t re-explain what this explanation was.

WrongOp Students change answer because they used the wrong operation when they�rst tried to solve it OR because they think they went the wrong way forthe same operation. If student has already written an answer, this codeshould appear on the second row for the problem as a justi�cation forwhy they changed the answer. If student wrote an answer but changed itimmediately before doing anything else, this code is in the same row.

WrongNum Students change answer because they used the wrong number (not relatedto using the wrong operation; e.g., maybe they miscounted the �rst time).If student has already written an answer, this code should appear on thesecond row for the problem as a justi�cation for why they changed theanswer. If student wrote the answer and changed it right away before doinganything else, this code is in the same row.

Tutor Student mentions they know the problem because they do it at Kumon orwith a tutor.

Someone Student comments that someone helped them/taught them. Student isvague about who/where this might have been.

Family Student comments about a family member helping them.

Friend Student comments about a friend or “someone” helping or telling them.

Teacher Student comments about a teacher teaching them or doing problems likethis in class or in work.




CODE EXPLANATION

Lesson Student refers to one of the experimental lessons (E.g., diving, positivepark, negative nettles, war, elevator).

final answer

How does the value of the answer compare to the initial number as determinedby student?

(Determine which number in the problem the student started with (countreverse-implied as reverse). Based on that, is the answer greater than, lessthan, or equal to it? Use the number as given in the problem. E.g., if thestudent interprets -3 + -5 as 5 + 3, determine if their answer is >, <, or = to

-5.)

CODE EXPLANATION

na No answer given.

Unclear It is not clear which number the student started with and the answer isgreater than one of the numbers and less than the other number.

Greater Student’s answer is greater than the actual value of the number studentstarted, OR it’s unclear which number students started with but students’answer is greater than the actual values of both numbers in the problem.

Less Student’s answer is less than the actual value of the number student startedwith, OR it’s unclear which number students started with but student’sanswer is less than the actual values of both numbers in the problem.

Equal Student’s answer is equal to one of the numbers in the problem but it isunclear which number the student started with.

E1 Student’s answer is equal to the �rst number in the problem (as he/shesolved it).

E2 Student’s answer is equal to the second number in the problem (as he/shesolved it).

K N O R M A L I T Y T E S T S

total test items

Kolmogorov-Smirnov test with Lillefors signi�cance correction

K-S Test

Instruction Statistic df Sig.Di�erence Scores

Full Instruction .11 20 .200*

Operations .13 21 .200*

Integer Properties .11 20 .200*

Performance LevelDi�erence Scores

Low .18 13 .200*

Medium .09 24 .200*

High .13 24 .200*

Performance Level by InstructionDi�erence Scores

Full Instruction, Low .19 4 —Full Instruction, Medium .19 9 .200*

Full Instruction, High .23 7 .200*

Operations, Low .17 5 .200*

Operations, Medium .18 7 .200*

Operations, High .18 9 .200*

Integer Properties, Low .35 4 —Integer Properties, Medium .12 8 .200*

Integer Properties, High .15 8 .200*

* �is is a lower bound of the true signi�cance.

245

246 K ⋅ normality tests

Full instruction

Operations

Integer Properties

Low

Medium

High

Full instruction, Low

Full instruction, Medium

Full instruction, High

Operations, Low

Operations, Medium

Operations, High

Integer Properties, Low

Integer Properties, Medium

Integer Properties, High

Figure K.1 – Total Test Difference Score Histograms. For the most part, the total test di�erencescores appear normally distributed (as shown by the normal curves).

total test items 247

Full instruction

Operations

Integer Properties

Low

Medium

High




Operations, Low

Operations, Medium

Operations, High




Figure K.2 – Total Test Difference Score QQ-plots. Although there are outliers in a few groups,the majority of the points fall along the line.


integer properties test items

Kolmogorov-Smirnov test with Lillefors signi�cance correction

K-S Test

Instruction Statistic df Sig.% Di�erence Scores

Full Instruction .12 20 .200*

Operations .20 21 .031Integer Properties .15 20 .200*

Performance Level% Di�erence Scores

Low .22 13 .096Medium .10 24 .200*

High .13 24 .200*

Performance Level by Instruction% Di�erence Scores

Full Instruction, Low .23 4 —Full Instruction, Medium .23 9 .181Full Instruction, High .20 7 .200*

Operations, Low .32 5 .111Operations, Medium .25 7 .200*

Operations, High .27 9 .054Integer Properties, Low .34 4 —Integer Properties, Medium .17 8 .200*

Integer Properties, High .15 8 .200*

* �is is a lower bound of the true signi�cance.

integer properties test items 249

Full instruction

Operations

Integer Properties

Low

Medium

High




Operations, Low

Operations, Medium

Operations, High




Figure K.3 – Properties Percentage Difference Score Histograms. Except for a few groups, theproperties percentage di�erence scores appear normally distributed (see normal curves).


Full instruction

Operations

Integer Properties

Low

Medium

High




Operations, Low

Operations, Medium

Operations, High



Integer Properites, High

Figure K.4 – Properties Percentage Difference Score QQ-plots. Although there are outliers in afew groups (especially the ones with smaller samples), the majority of the points fall along theline.

L H O M O G E N E I T Y O F VA R IA N C E

Table L.1 – Levene’s Test on Total Test Difference Scores. Tests the null hypothesis that the errorvariance of the total di�erence scores is equal across groups. �e value is not signi�cant, sowe reject the null hypothesis and conclude that the error variances are equal across groups.Design: Intercept + PerformLevel + Instruction + PerformLevel*Instruction.

F Value degrees offreedom (1)

degrees offreedom (2)

Signi�cance

1.001 8 52 .447

Table L.2 – Levene’s Test on Integer Properties Difference Scores. Tests the null hypothesis thatthe error variance of the integer properties di�erence scores is equal across groups. �evalue is not signi�cant, so we reject the null hypothesis and conclude that the error vari-ances are equal across groups. Design: Intercept + PerformLevel + Instruction + Perform-Level*Instruction

F Value degrees offreedom (1)

degrees offreedom (2)

Signi�cance

1.602 8 52 .147

251

252 L ⋅ homogeneity of variance

M T O TA L O R D E R E D VA LU E S C H E M A S

Table M.1 – Value Schema Changes for Both Value Items. Percentage of total students in eachinstructional group who started and ended at each schema level along with their gains. nL =number of low performing students.

Total Ordered Value

Full Instruction Integer Properties Integer Operations

(n=20) (n=20) (n=21)




Gain

A 0% 0% 0% 0% 0% 0% 0% 0% 0%B 0% 0% 0% 15% 0% −15% 5% 0% −5%C 80% 20%2L

−60% 60% 0% −60 76% 57%4L−9%

D 0% 0% 0% 0% 0% 0% 0% 0% 0%E 0% 0% 0% 0% 0% 0% 0% 0% 0%F 5% 0% −5% 0% 0% 0% 0% 0% 0%G 0% 0% 0% 0% 10%2L

+10% 5% 14% +9%H 0% 10% +10% 5% 5% +10% 0% 0% 0%I 5% 30%2L

+25% 10% 15%2L+5% 0% 5%1L

+5%J 10% 40% +30% 10% 60% +50% 14% 24% +10%

253

254 M ⋅ total ordered value schemas

R E F E R E N C E S

Aze, I. (1989). Negatives for little ones? Mathematics in School, 18(2), 16-18. Cited on page 9.Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. �e Elementary

School Journal, 93(4), 373-397. Available from http://search.ebscohost.com/login.aspx?direct=true&db=a2h&AN=

9305125747&site=ehost-live Cited on pages 14 and 31.Ball, D. L., & Wilson, S. M. (1990). Knowing the subject and learning to teach it: Examining assumptions about becoming a mathematics

teacher (Tech. Rep. No. 90-7). East Lansing, Michigan: National Center for Research on Teacher Education, Michigan StateUniversity. Available from http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=

ED323207 Cited on page 14.Baroody, A. J. (1992). Remedying common counting di�culties. In J. Bideaud, C. Melijac, & J. Fischer (Eds.), Pathways to number:

Children’s developing numerical abilities (p. 307-323). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Cited on pages 7, 8,and 9.

Bishop, J. P., Lamb, L., Philipp, R., Schappelle, B., & Whitacre, I. (2010, October). A developing framework for children’s reasoningabout integers. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North AmericanChapter of the International Group for the Psychology of Mathematics Education (Vol. 4, p. 695-702). Columbus, OH: �e OhioState University. Cited on page 4.

Bo�erding, L. (2010, October). Addition and subtraction with negatives: Acknowledging the multiple meanings of the minus sign.In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter ofthe International Group for the Psychology of Mathematics Education (Vol. 4, p. 703-710). Columbus, OH: �e Ohio StateUniversity. Cited on pages 4, 12, 17, 26, and 30.

Bruner, J. S. (1960/2003). �e process of education: A landmark in educational theory. Cambridge, MA: Harvard University Press. Citedon page 30.

Bruno, A., & Martinon, A. (1999, November/December). �e teaching of numerical extensions: �e case of negative numbers.International Journal of Mathematical Education in Science and Technology, 30(6), 789-809. Cited on pages 13, 15, 29, 30,and 126.

California Department of Education. (2010). Dataquest. Available from http://data1.cde.ca.gov/dataquest/ (Retrieved1/28/2011) Cited on page 41.

Case, R. (1996). Introduction: Reconceptualizing the nature of children’s conceptual structures and their development in middlechildhood. Monographs of the Society for Research in Child Development, 61(1-2), 1-26. Cited on pages 7, 20, 21, 23, 24, 25, 30, 44,123, and 131.

Chilvers, P. (1985). A consistent model for operations on directed numbers. Mathematics in School, 14(1), 26-27. Cited on page 9.Clemson, D., & Clemson, W. (1994). Mathematics in the early years: Teaching and learning in the �rst three years of school. New York,

NY: Routledge. Cited on page 26.Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Cited on page

77.Council of Chief State School O�cers & National Governor’s Association. (2010, June). Common core state standards for mathematics.

Available from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf (Retrieved March 10, 2010) Citedon pages 1 and 2.

Dabell, J. (2007). Would you play to win one? Times Educational Supplement(4754), 46. Available from http://search.ebscohost

.com/login.aspx?direct=true&db=a2h&AN=27256145&site=ehost-live Cited on page 9.Davidson, P. (1987). How should non-positive integers be introduced in elementary mathematics. In J. Bergeron, N. Herscovics,

& C. Keeran (Eds.), Proceedings of the 11th International Conference for the Psychology of Mathematics Education (Vol. 2,p. 430-436). Montreal, Canada: University of Montreal. Cited on pages 9 and 14.

255

http://search.ebscohost.com/login.aspx?direct=true&db=a2h&AN=9305125747&site=ehost-live


http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED323207

http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED323207

http://data1.cde.ca.gov/dataquest/

http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf



256 References

De Cruz, H. (2006). Why are some numerical concepts more successful than others? an evolutionary perspective on the history ofnumber concepts. Evolution and Human Behavior, 27(4), 306-323. Cited on pages 9, 11, 20, and 25.

Devlin, K. (1999). Mathematics:�e new golden age. New York: Columbia University Press. Cited on page 2.Dimitrov, D. M. (2008). Quantitative research in education: Intermediate & advanced methods. Oceanside, NY: Whittier Publications,

Inc. Cited on page 77.Fagnant, A., Vlassis, J., & Crahay, M. (2005). Using mathematical symbols at the beginning of the arithmetical and algebraic learning.

In L. Verscha�el, E. De Corte, G. Kanselaar, & M. Valcke (Eds.), Powerful environments for promoting deep conceptual andstrategic learning (p. 81-95). Leuven, Belgium: Leuven University Press. Cited on pages 12, 23, and 47.

Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth,NH: Heinemann. Cited on page 9.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht-Holland: D. Reidel Publishing Company. Cited on page 9.Fuson, K. C. (1984). More complexities in subtraction. Journal for Research in Mathematics Education, 15(3), 214-225. Cited on page 31.Fuson, K. C. (1992). Relationships between counting and cardinality from age 2 to age 8. In J. Bideaud, C. Melijac, & J. Fischer (Eds.),

Pathways to number: Children’s developing numerical abilities (p. 127-149). Hillsdale, New Jersey: Lawrence Erlbaum Associates,Inc. Cited on pages 7, 8, 9, 32, and 34.

Fuson, K. C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, W. G. Martin, & D. Schi�er(Eds.), A research companion to principles and standards for school mathematics (p. 68-94). Reston, VA: �e National Councilof Teachers of Mathematics, Inc. Cited on pages 12 and 14.

Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., et al. (1997). Children’s conceptual structures formultidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2),130-162. Cited on page 12.

Gaer, E. P. (1969). Children’s understanding and production of sentences. Journal of Verbal Learning and Verbal Behavior, 8(2),289-294. Cited on page 9.

Galbraith, M. J. (1975). Smith and jones: Children’s thinking about some problems involving displacement along the number-line.International Journal of Mathematical Education in Science and Technology, 6(3), 287-302. Cited on page 13.

Gallardo, A., & Rojano, T. (1994). School algebra. syntactic di�culties in the operativity with negative numbers. In D. Kirshner (Ed.),Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology ofMathematics Education (p. 159-165). Baton Rouge, Louisiana: Louisiana State University. Cited on page 11.

Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44(1-2), 43-74. Cited on page 19.Gelman, R., & Gallistel, C. R. (1978). �e child’s understanding of number. Cambridge, MA: Harvard University Press. Cited on pages 7,

8, 9, 11, 12, and 14.Gri�n, S. (2004a). Contributions of central conceptual structure theory to education. In A. Demetriou & A. Ra�opoulos (Eds.),

Cognitive developmental change:�eories, models and measurement (p. 264-295). New York, NY: Cambridge University Press.Cited on pages 12, 19, 20, and 41.

Gri�n, S. (2004b). Number worlds: A research-based mathematics program for young children. In D. H. Clements, J. Sarama, &A. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (p. 325-340).Mahwah, NJ: Lawrence Erlbaum Associates. Cited on page 31.

Gri�n, S., Case, R., & Capodilupo, A. (1995). Teaching for understanding: �e importance of the central conceptual structures inthe elementary mathematics curriculum. In A. McKeough, J. L. Lupart, & A. Marini (Eds.), Teaching for transfer: Fosteringgeneralization in learning (p. 123-151). Hillsdale, NJ: Erlbaum. Cited on pages 21, 22, and 127.

Hativa, N., & Cohen, D. (1995, June). Self learning of negative number concepts by lower division elementary students through solvingcomputer-provided numerical problems. Educational Studies in Mathematics, 28(4), 401-431. Cited on pages 10, 15, and 30.

Hoge, R. D., & Coladarci, T. (1989). Teacher-based judgments of academic achievement: A review of literature. Review of EducationalResearch, 59(3), 297-313. Cited on page 51.

Hughes, M. (1986). Children and number: Di�culties in learning mathematics. New York, NY: Basil Blackwell, Inc. Cited on pages 10,11, and 14.

Human, P., & Murray, H. (1987). Non-concrete approaches to integer arithmetic. In J. Bergeron & C. Herscovics N. & Kieran (Eds.),Proceedings of the 11th International Conference for the Psychology of Mathematics Education (p. 437-443). Montreal, Canada:

References 257

University of Montreal. Cited on pages 16 and 37.Janvier, C. (1985). Comparison of models aimed at teaching signed integers. In L. Stree�and (Ed.), Proceedings of the Nineth Annual

Meeting of the International Group for the Psychology of Mathematics Education (p. 135-140). Utrecht, �e Netherlands. Citedon pages 13, 29, and 31.

Kamii, C., Lewis, B. A., & Kirkland, L. D. (2001). Fluency in subtraction compared with addition. �e Journal of MathematicalBehavior, 20(1), 33-42. Cited on page 12.

Kato, Y., Kamii, C., Ozaki, K., & Nagahiro, M. (2002). Young children’s representations of groups of objects: �e relationship betweenabstraction and representation. Journal for Research in Mathematics Education, 33(1), 30-45. Cited on page 10.

Laski, E. V., & Siegler, R. S. (2007). Is 27 a big number? correlational and causal connections among numerical categorization, numberline estimation, and numerical magnitude comparison. Child Development, 78(6), 1723-1743. Cited on page 8.

Liebeck, P. (1990). Scores and forfeits: An intuitive model for integer arithmetic. Educational Studies in Mathematics, 21(3), 221-239.Available from http://www.jstor.org/stable/3482594 Cited on pages 12, 13, 29, and 31.

Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in ‘�lling’ the gap in children’s extension of their numberconcept to include the negative numbers. Educational Studies in Mathematics, 39(1-3), 131-147. Available from http://

search.ebscohost.com/login.aspx?direct=true&db=a2h&AN=2997559&site=ehost-live Cited on pages 13 and 30.Mauthe, A. H. (1969). Climb the ladder. Arithmetic Teacher, 16, 354-356. Cited on page 9.McAuley, J. (1990). "please sir, i didn’t do nothin". Mathematics in School, 19(1), 45-47. Cited on page 9.McGraw-Hill. (2004). Everyday mathematics: Fourth grade teacher’s lesson guide. Chicago, IL: Author. Cited on page 31.Mukhopadhyay, S., Resnick, L. B., & Schauble, L. (1990). Social sense-making in mathematics: Children’s ideas of negative numbers. In

Proceedings of the 14th International Conference for the Psychology in Mathematics Education (p. 281-288). Oaxtepex, Mexico.Cited on pages 13 and 14.

Murata, A. (2004). Paths to learning ten-structured understandings of teen sums: Addition solution methods of japanese grade 1students. Cognition and Instruction, 22(2), 185-218. Cited on page 12.

Murray, J. C. (1985). Children’s informal conceptions of integer arithmetic. In L. Stree�and (Ed.), Proceedings of the Ninth AnnualInternational Conference for the Psychology of Mathematics Education (p. 147-153). Utrecht, �e Netherlands. Cited on pages 4,10, 13, 14, 15, 16, 27, 30, and 124.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Councilof Teachers of Mathematics. Cited on pages 1 and 2.

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: Aquest for coherence. Reston, VA: National Council of Teachers of Mathematics. Cited on page 7.

National Research Council. (2001). Adding it up: Helping children learn mathematics (J. Kilpatrick, J. Swa�ord, & B. Findell, Eds.).Washington, D.C.: National Academy Press. Cited on page 1.

National Research Council. (2005). How students learn: Mathematics in the classroom. Washington, D.C.: Division of Behavior andSocial Sciences and Education. �e National Academies Press. Cited on page 7.

Okamoto, Y., & Case, R. (1996). Exploring the microstructure of children’s central conceptual structure in the domain of number.Monographs of the Society for Research in Child Development, 61(1-2), 27-58. Cited on page 7.

Pearson Education. (2009). enVisionMATH California (grade 5). Glenview, IL: Author. Cited on pages 3 and 132.Peled, I. (1991). Levels of knowledge about signed numbers. In F. Furinghetti (Ed.), Proceedings of the Fi�eenth Annual Conference

of the International Group for the Psychology of Mathematics Education (p. 145-152). Assisi, Italy. Cited on pages 9, 14, 15, 16,and 126.

Peled, I., Mukhopadhyay, S., & Resnick, L. B. (1989). Formal and informal sources of mental models for negative numbers. InG. Vergnaud, J. Rogalski, & M. Artique (Eds.), Proceedings of the 13th Annual International Conference for the Psychology ofMathematics Education (Vol. 3, p. 106-110). Paris, France. Cited on pages 9, 10, 13, 14, 15, 27, and 125.

Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledgethrough playing number board games. Child Development, 79(2), 375-394. Cited on pages 8, 9, 11, and 31.

Resnick, L. B. (1989). Developing mathematical knowledge. American Psychologist, 44(2), 162-169. Cited on pages 7, 8, and 20.Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New

York, NY: Taylor & Francis. Cited on pages 1, 7, 10, 21, 22, 31, and 32.

http://www.jstor.org/stable/3482594



258 References

Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993). Positives about negatives: A case study of an intermediate model for signednumbers. �e Journal of the Learning Sciences, 3(1), 37-92. Cited on pages 10, 15, 16, 30, and 125.

Sfard, A. (2001). Learning mathematics as developing a discourse. In R. Spieser, C. Maher, & C. Walter (Eds.), Proceedings of the21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education(p. 23-44). Columbus, Ohio. Cited on page 4.

Shavelson, R. J. (1996). Statistical reasoning for the behavioral sciences. Needham Heights, MA: Allyn and Bacon. Cited on page 74.Shuttleworth, M. (2009). Counterbalanced measures design. Available from http://www.experiment-resources.com/

counterbalanced-measures-design.html (Retrieved 4/1/2010) Cited on page 43.Siegler, R. S. (1987). �e perils of averaging data over strategies: An example from children’s addition. Journal of Experimental

Psychology, 116(3), 250-264. Cited on page 48.Siegler, R. S. (1996). Emerging minds:�e process of change in children’s thinking. New York, NY: Oxford University Press, Inc. Cited on

page 43.Siegler, R. S., & Chen, Z. (2008). Di�erentiation and integration: Guiding principles for analyzing cognitive change. Developmental

Science, 11(4), 433-448. Cited on page 20.Siegler, R. S., & Robinson, M. (1982). �e development of numerical understandings. In H. W. Reese & L. P. Lisitt (Eds.), Advances in

child development and behavior (Vol. 16, p. 242-312). New York, NY: Academic Press. Cited on page 8.Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian

(Ed.), Origins of cognitive skills:�e eighteenth annual carnegis symposium on cognition (p. 229-293). Hillsdale, NJ: LawrenceErlbaum Associates. Cited on page 48.

Sophian, C. (1987). Early developments in children’s use of counting to solve quantitative problems. Cognition and Instruction, 4(2),61-90. Cited on pages 7 and 8.

Ste�e, L. P. (1992). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J. Fischer (Eds.), Pathwaysto number: Children’s developing numerical abilities (p. 83-98). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Cited on page7.

Steinle, V., & Stacey, K. (2003). Grade-related trends in the prevalence and persistence of decimal misconceptions. In Proceedings of the27th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, p. 259-266). Honolulu,Hawaii. Cited on page 20.

Swanson, P. E. (2010). �e intersection of language and mathematics. Mathematics Teaching in the Middle School, 15(9), 516-523. Citedon page 9.

Tatsuoka, K. K. (1981). An approach to assessing the seriousness of error types and the predictability of future performance. (Tech. Rep.No. 81-1). Computer-based Education Research Laboratory, University of Illinois. Cited on page 16.

Tatsuoka, K. K. (1983). Rule space: An approach for dealing with misconceptions based on item response theory. Journal of EducationalMeasurement, 20(4), 345-354. Cited on pages 14, 16, and 17.

Tatsuoka, K. K., Birenbaum, M., & Arnold, J. (1989). On the stability of students’ rules of operation for solving arithmetic problems.Journal of Educational Measurement, 26(4), 351-361. Cited on page 16.

�omaidis, Y. (1993). Aspects of negative numbers in the early 17th century. Science and Education, 2, 69-86. Cited on page 2.�ompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19, 115-133. Cited

on page 30.Tolchinsky, L. (2003). �e cradle of culture and what children know about writing and numbers before being taught. Mahwah, New

Jersey: Lawrence Erlbaum Associates. Cited on page 11.Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach.

Learning and Instruction, 14(5), 453-467. Cited on pages 19 and 20.Van Oers, B. (2002). �e mathematization of young children’s language. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verscha�el

(Eds.), Symbolizing, modeling and tool use in mathematics education (p. 29-58). Dordrecht, �e Netherlands: Kluwer AcademicPublishers. Cited on page 32.

VanLehn, K. (1982). Bugs are not enough: Empirical study of bugs, impasses and repairs in procedual skills. �e Journal of Mathemati-cal Behavior, 3(2), 3-71. Cited on page 14.

http://www.experiment-resources.com/counterbalanced-measures-design.html

http://www.experiment-resources.com/counterbalanced-measures-design.html

References 259

Vlassis, J. (2004). Making sense of the minus sign or becoming �exible in ’negativity’. Learning and Instruction. Special Issue:�eConceptual Change Approach to Mathematics Learning and Teaching, 14(5), 469-484. Cited on pages 4, 11, 13, 14, 15, 16, and 23.

Vlassis, J. (2008). �e role of mathematical symbols in the development of number conceptualization: �e case of the minus sign.Philosophical Psychology. Special Issue: Number as a test case for the role of language in cognition, 21(4), 555-570. Cited on pages11, 13, and 30.

Vosniadou, S. (1994). Capturing and modeling the process of conceptual change. Learning and instruction, 4(1), 45-69. Cited on page19.

Vosniadou, S. (2007). �e conceptual change approach and its re-framing. In S. Vosniadou, A. Baltos, & X. Vamvakoussi (Eds.),Reframing the conceptual change approach in learning and instruction (p. 1-15). Oxford, UK: Elsevier. Cited on pages 19, 20,and 29.

Vosniadou, S., & Verscha�el, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. Learningand Instruction, 14, 445-451. Cited on pages 19, 27, and 29.

Wannemacher, J. T., & Ryan, M. L. (1978). "less" is not "more": A study of children’s comprehension of "less" in various task contexts.Child Development, 49(3), 660-668. Cited on page 123.

Werner, S. M. (1973). �e case for a more universal number-line model of subtraction. Arithmetic Teacher, 20(1), 61-64. Cited on page9.

Williams, J. S., & Linchevski, L. (1997). Situated intuitions, concrete manipulations and the construction of the integers: Comparing twoexperiments. Paper presented at the Annual Conference of the American Educational Research Association. Cited on page 12.

Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24(2), 220-251. Cited onpage 7.

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