delMas & Liu

8/7/2019 delMas & Liu

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Students Understanding of Factors that Affect the Standard Deviation

Robert C. delMas, University of Minnesota, [email protected]

Yan Liu, Vanderbilt University, [email protected]

ABSTRACT

This study investigates students understanding of the concept of the standard deviation, inparticular, their understanding of the factors that affect, and how they affect, the size of the

standard deviation. Twelve students enrolled in an introductory statistics course participated in

the study. The interview activities engaged students in arranging a number of bars of different

frequencies on a number line to produce the largest and smallest standard deviation, and tocompare the size of the standard deviation of two distributions. Initial analysis identified

rules/strategies that students used to construct their arrangements and make comparisons. The

paper discusses these distinctive rules and how they are affected by task characteristics and types

of interviewer questions.

Keywords: conceptual understanding, strategies, standard deviation, variability, interviews

1. INTRODUCTION

Garfield and Ahlgren (1988) argued that little research has been done on how students cometo understand statistical concepts. Mathews and Clark (1997) interviewed eight college students

enrolled in an introductory statistics course and found that while all were highly successful in thecourse, none of the students demonstrated a conceptual understanding of the introductory level

material. One area of statistical instruction that has received very little attention is studentsunderstanding of variability (Reading & Shaughnessy, in press). This is the case despite the

central role the concept plays in statistics and an apparent conceptual gap in studentsunderstanding of variability (Shaughnessy, 1997). Investigations into students understanding of

sampling variability (Reading & Shaughnessy, in press) and instructional approaches that affectthis understanding (Meletiou-Mavrotheris & Lee, 2002) have been conducted. Reading and

Shaughnessy (in press) present evidence of different levels of sophistication in elementary andsecondary students reasoning about sample variation. Meletiou-Mavrotheris and Lee (2002)

found that an instructional design that emphasized statistical variation and statistical processproduced a better understanding of the standard deviation, among other concepts, in a group of

undergraduates. Students in the study saw the standard deviation as a measure of spread thatrepresented a type of average deviation from the mean. They were also better at taking both

center and spread into account when reasoning about sampling variation in comparison to

findings from earlier studies (e.g., Shaughnessy, Watson, Moritz, & Reading, 1999). However,little is known about students understanding of measures of variation, how this understandingdevelops, or how students might apply their understanding to make comparisons of variation

between two or more distributions. The latter ability represents an important aspect of statisticalliteracy that is needed both for interpreting research results and for everyday decision making.

The main purpose of the current study is to gain a better picture of the different ways thatstudents look at the standard deviation as this concept develops at the beginning of an

introductory statistics course.


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2. A CONCEPTUAL ANALYSIS OF STANDARD DEVIATION

In order to have a working understanding of standard deviation, a student needs to coordinate

several underlying statistical concepts from which the concept of standard deviation isconstructed. Distribution is one of these fundamental concepts. The students in the current

study worked with distributions of discrete variables, so the concept of distribution in this paperwill be described in those terms. Essentially, an understanding of distribution requires a

conception of a variable and the accumulating frequency of its possible values. Therefore, avisual or graphical understanding of distribution involves the coordination of values and density.

Such coordination allows the student to consider a question such as What proportion of thedistribution lies inclusively between the values of 4 and 7?

A second fundamental concept is that of mean, defined as the arithmetic average. A

conceptual understanding of the standard deviation requires more than the knowledge of aprocedure for calculating the mean, either in procedural or symbolic form (e.g., Sx/n). Imagery

that metaphorically considers the mean to behave like a self-adjusting fulcrum on a balancecomes close to the necessary conception. Such imagery supports the development of the third

foundational concept, deviation from the mean. It is through the coordination of distribution (asrepresented by the coordination of value and frequency) and deviation (as distance from the

mean) that a dynamic conception of the standard deviation is derived as the relative density ofvalues about the mean. A student who possesses this understanding can anticipate how the

possible values of a variable and their respective frequencies will, independently and jointly,affect the standard deviation. An illustration from the computer program used in the presentstudy may help to clarify this conception.

Figure 1a presents a distribution of a variable along the number line. Each bar is composed ofa certain number of rectangles each of which represents one datum. The location of a bar

indicates the value of all the data represented by the bar (e.g., all the data in the red [tallest] barhave a value of 4). The point location of the mean is indicated by a blue arrow along the number

line, and the standard deviation is reported and represented by a black horizontal bar as adistance below and above the mean. In addition, the deviation from the mean of each value is

printed within each rectangle. Therefore, the program provides the potential to see how a changein value (i.e., bar location) simultaneously affects the mean, deviations from the mean, and thestandard deviation.

For example, a student can be asked to anticipate how the mean, deviations, and standarddeviation are affected by changing the value of the green (shortest) bar from 2 to 1. A student

with a fully coordinated conception of standard deviation can anticipate that moving the greenbar (the leftmost bar) to a value of 1 shifts the mean to a slightly lower value. In addition, all

deviations from the mean will change simultaneously. The student would anticipate thedeviations of the red and blue bars (representing frequencies of 8 and 6, respectively) from the

mean to increase, the deviations of the yellow bar to decrease, and possibly the deviations of thegreen bar to increase. Since the student is able to coordinate density (or frequency) with

deviation, she would realize that the larger frequencies of the red and blue bar coupled withincreases in deviation are likely to outweigh the few values in the yellow bar that had a slight

decrease in deviation. This would result in a larger density of values away from the mean and,therefore, an increase in the standard deviation. Figure 1b illustrates the results of actuallymoving the green bar from a value of 2 to a value of 1.


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a. b.

Figure 1. A graphic representation of standard deviation and its related concepts.

A student who understands such dynamic relationships would be able to reliably compare the

standard deviation of two distributions. For example, consider the pairs of graphs presented inTable 2. Knowing only the location of the mean in both distributions, a student might reason that

the graph on the left in pair 2 has a larger standard deviation because there is a lower density ofvalues around the mean. The same conception would lead to the conclusion that the graph on the

right in pair 4 has the larger standard deviation.

3. RESEARCH DESIGN

3.1 SETTING

The subject pool was comprised of thirteen students registered in an introductory statistics

course at a large Midwest research university during the spring 2003 term. At the time of theinterview the course had introduced students to distributions, methods for constructing graphs

(stem-and-leaf plots, dot plots, histograms, and box plots), measures of center (mode, median,and mean), and measures of variability (range, interquartile range, and the standard deviation).With respect to the standard deviation, students had participated in an activity exploring factors

that affect the size of the standard deviation. Students compared nine pairs of graphs todetermine which one had a larger standard deviation. Students worked together, identified

characteristics of the graphs thought to affect the size of the standard deviation, recorded theirpredictions, and received feedback. The goal of the activity was to help students see that the size

of the standard deviation is related to how values are spread out and away from the mean, theonly factor associated with the correct choice across all nine pairs.

3.2 NATURE OF THE INTERVIEW

Each student participated in a one-hour interview. During the interview, a student interactedwith a computer program. A digital video camera was used to capture a students utterances and

actions. The computer program wrote data to a separate file to capture students actions andchoices. A separate record in the file represented each movement or choice with each recordtime stamped so that it could be coordinated with the digital video recording.

The interview had two parts. Part 1 was designed to help students learn about factors thataffect the standard deviation. Each student was presented with five different sets of bars (see


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Table 1), one set at a time, and asked to arrange the bars in a graphing area to accomplish aspecified task. The number of bars ranged from two in the first set to five in the last. The first

task asked a student to find an arrangement that produced the largest possible value for thestandard deviation, followed by the task of finding a second arrangement that also produced this

same value (see Figure 2). The third through fifth tasks required three different arrangements

that each produced the smallest value for the standard deviation. Each set of bars along with theaccompanying tasks was referred to as a game. Each student played five games during Part 1.

A button labeled CHECK was provided to determine if an arrangement met the statedcriterion for a task. Before checking, students were asked to describe what they were thinking,

doing, or attending to. In addition, the investigator (the first author) asked students to state whythey thought an arrangement did or did not meet a criterion once feedback was received. The

investigator also posed questions or created additional bar arrangements when a statementappeared to reveal a misunderstanding or to explore the stability of students reasoning.

The interviewer attempted to promote two instructional goals during Part 1. The first goal

was to help students develop a mean-centered understanding of how deviation from the meanand frequency combined to determine the value of the standard deviation. The second goal was

to promote an understanding of how the shape of a distribution is related to the size of thestandard deviation (e.g., given the same set of bars, a bell-shaped distribution tends to have a

smaller standard deviation than a skewed distribution). When students seemed to have difficultyfinding an arrangement to meet a task criterion, or when they appeared to be moving in the

direction of one of the goals, the interviewer used several approaches to support the developmentof one of the understandings. A mean-centered conception of the standard deviation was

promoted by drawing attention to the values of the deviations, by asking students howhypothetical movements of the bars would affect the mean, and by the interviewer modeling

reasoning of how the distribution of deviation densities affected the values of the mean andstandard deviation. The second goal was supported by asking students to predict how the mean

and standard deviation would be affected by bell-shaped and skewed arrangements of the same

bar sets or by asking students to judge the extent to which a distribution was bell-shaped,whether the bars could be arranged to produce a more bell-shaped (or symmetric) distribution,and drawing student attention to the change in the standard deviation.

Part 2, the test phase, was designed to assess students understanding of factors that influence

the standard deviation. Students were asked to answer 10 test questions where each test

presented a pair of histograms (see Table 2). For each pair of histograms, the average and

standard deviation were displayed for the graph on the left, but only the average was displayedfor the graph on the right (see Figure 3). The students were asked to make judgments on whether

the standard deviation for the graph on the right was smaller than, larger than, or the same as the

graph on the left. Once a student made a judgment, the investigator asked the student to explain

his/her justification and reasoning. The student then clicked the CHECK button to check the

answer, at which time the program displayed the standard deviation for the graph on the rightand stated whether or not the answer was correct.


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Table 1

Bar sets for the five games with possible solutions

Game Bars Largest SD Smallest SD1

2

3

4

5


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Table 2

Test items presented in the second part of the interview

1 6

2 7

3 8

4 9

5 10


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Figure 2. Example of Information Displayed by the Variability Game During Game 4 of Part 1.


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Figure 3. Example of Information Displayed by the Variability Game During Test 2 of Part 2.


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4. FINDINGS

Twelve interviews have been transcribed and analyzed. We have identified several strategies

and rules that students use to accomplish the tasks for the five sets of bars. While some studentsappear to start the interview with a fairly sophisticated understanding of factors that affect the

standard deviation and how these factors work together, most students have a very simple, ruleoriented approach. Brief descriptions of the various approaches are presented. Video and

transcript excerpts that illustrate each approach can be found in delMas and Liu (2003).

4.1 GENERAL STRATEGIES AND RULES

Students used several general approaches to make arrangements that produced either the

largest or the smallest standard deviation.

Mirror Image

The student would often create a mirror image of a distribution to produce a distribution withthe same standard deviation. Some students expressed this approach as swapping or

switching the position of the bars, while others stated that the arrangement was flip-flopped.The student would state that since the bars were still as close together or as far apart as before

(either relative to each other or relative to the mean), the standard deviation should be the same.

Location

In this strategy, the student moved all the bars an equal distance, maintaining the same

relative arrangement and standard deviation. The student would note that the standard deviationshould be the same because the relative distances between the bars or from the mean were

maintained.

Mean in the Middle

The Mean in the Middle approach was indicated by a statement that the bars were arrangedso that the mean appeared to be somewhat in the middle of the distribution. This is an interesting

statement since the mean is technically always at the center of the density of the distribution.However, this made more sense when coupled with strategies such as Far Away -Mean and

Balance that are described later. In these approaches a somewhat symmetric distribution wasproduced, thereby placing the mean in the middle. This rule suggests a concept of deviation

where the student realizes that near equal amounts of values need to be positioned below andabove the mean to minimize or maximize the standard deviation.

Balance

Arranging the bars so that near equal amounts of values are placed above and below the

mean was taken as an indication of a Balance strategy. Typically, the bars were arranged in arecognizable order; descending inward from the opposite extremes of the number scale to

produce the largest standard deviation, or descending outward from the mean to produce thesmallest standard deviation.


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4.2 LARGEST STANDARD DEVIATION: STRATEGIES AND RULES

For the first game, most students placed one bar at each of the extreme positions of the

number line to produce the largest standard deviation. The subsequent games with more than twobars challenged students, revealing more details of the their thinking and strategies. In general,

the arrangements represented placing the bars far apart or having them spread out. The typicalorder was to place the tallest bars at the extremes with subsequent bars placed so that heights

decreased toward the center of the distribution. Three general strategies were identified based onstudents justifications for their arrangements.

Equally Spread Out

In the Equally Spread Out strategy, the student believed that the largest standard deviation

occurred when the bars were spread out across the entire range of the number line with equalspacing between the bars. The Equally Spread Out strategy may have resulted from the in-class

activity described earlier. Students may have translated spread out away from the mean to

equally spread out. For example, with three bars, the student places one bar above a value of 0,one above 9, and then places the third bar above 4 or 5 (see Figure 4). Some students using thisapproach realized that the third bar could not be placed with equal distance between the lowest

and highest positions, so they would move the third bar back and forth between values of 4 and 5to see which placement produced the larger standard deviation.

a. b.

Figure 4. Equally Spread Out arrangements for the largest standard deviation in Game 2.

Far Away Values

Some students stated that an arrangement should produce the largest possible standard

deviation if the values are placed as far away from each other as possible. No mention was madeof distance (or deviation) from the mean in these statements. Examples from several students arepresented in Figure . (Did you mean to insert a figure here? If so, the figure number in the

remainder of the paper should change from x to x+2, otherwise, x+1. For now Ill leave thefigure number unchanged.)


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Far Away Mean

This rule is similar to Far Away-Values, except the student stated that the largest standarddeviation is obtained by placing the values as far away from the mean as possible. For example,

when using a Balance approach to produce the largest standard deviation, Alice stated that way

they'd both be as far away from the mean as possible. Carl provided a clear expression of thisrule in the first game, although he was not sure that it was entirely correct.

Wellumwhat I know about standard deviation is, um, given a mean, the morenumbers away, the more numbers as far away as possible from the mean, is that

increases the standard deviation. So, I tried to make the mean, um, in between both ofthem, and then have them as far away from the mean as possible. But I dont, yeah, I

dont know if thats right or not, but, yeah.

Later, in Game 2, Carl tried to arrange three bars to produce the largest possible standarddeviation. He first used an Equally Spread Out approach (see Figure 5a) and then appeared

frustrated that it did not produce the largest standard deviation. The interviewer intervened,trying to help him extend the Far Away Mean rule he had used earlier to the new situation. Carl

then produced the graph in Figure 5b. When asked why this arrangement produced the largeststandard deviation, Carl stated basically because I have the most amount of numbers possible

away from the mean. Um, because if I weresay I have two bars that are bigger than the onesmaller bar. If I were to switch these two around, the blue [frequency of 4] and the yellow

[frequency of 2], the standard deviation would be smaller then because theres more of thenumbers closer to the mean. Carl did not automatically apply the Far Away Mean rule to the

three bar situation, but he readily accommodated the new situation into the scheme once it wassuggested that the rule was applicable.

a. b.

Figure 5. Carls arrangements for the largest standard deviation in Game 2.

4.3 SMALLEST STANDARD DEVIATION: STRATEGIES AND RULES

Nearly all of the students placed the bars next to each other in some arrangement to produce

the smallest possible standard deviation. There were several distinct rules that seemed to beapplied to these arrangements.


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a. b.

Figure 7. Monas arrangements for the smallest standard deviation in Game 5.

The Bell-Shaped rule was also used to justify choices during the test phase. This rule did not

lead to a correct choice when the two graphs represented different numbers of values and bars,

such as the pair of distributions in Test 8.

More Bars in the Middle

Some students stated that one of the reasons the standard deviation would be the smallest was

because more values or the tallest bars were placed in the middle of the distribution or close tothe mean. This approach is similar to the Bell-Shaped approach, except that symmetry or bell-

shape is not identified as a guiding factor. Mona provided a More Bars in the Middle statementin her Contiguous arrangement for the smallest deviation in Game 4 when she stated, a lot of

them have to be closer to the mean.Nancy also used a More Bars in The Middle justification for her first smallest standard

deviation arrangement in Game 4. She started off with an arrangement that was near optimal (seeFigure 8a) and then decided to move the green and yellow bars to produce the graph in Figure

8b. Although the result was fairly symmetric and bell-shaped, she talked only about placing lotsof values close to the mean.

a. b.

Figure 8. Nancys graphs for the smallest standard deviation in Game 4.


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Tallest Bar in the Middle

A statement that the arrangement should produce the smallest standard deviation because thetallest bar was in the middle indicated a Tallest Bar in the Middle approach. This was usually

associated with the second game that has three bars and may be a special case of More Bars in

the Middle. In Game 4, Troy believed he had an arrangement with the smallest possible value forthe standard deviation because he saw it as symmetrical and because the mean was below thetallest bar (see Figure 9a). He learned that the distribution did not meet the criterion and, with

some trial and error, came up with a new distribution that did (see Figure 9b). When asked tojustify the latter arrangement, Troy said, Uh, I mean if I tried this at first I thought it would be a,

not the lowest number I guess, so, um, but since the mean is more centered here

a. b.

Figure 9. Troys first and second attempts for the smallest standard deviation in Game 4.

Troy saw the mean as being more centered under the tallest (red) bar in Figure 9b, whenactually it appeared to be more centered in the first graph. Many of the students saw what they

were looking for or may have been describing a discernable difference between prior and currentdistributions when trying to justify why a new arrangement met the criterion. Just having the

tallest bar in the middle or centered over the mean is not a sufficient condition for producing thesmallest standard deviation, and does not indicate a coordination of how deviation and frequency

affect the size of the standard deviation.

4.4 TEST PHASE STRATEGIES AND RULES

There were several strategies and rules that were stated as justifications primarily during thetest phase of the interview.

Range

The student stated that a distribution with a larger range would have a larger standard

deviation. This type of statement was seen most often during the test phase of the interview.Alex provided an example when reasoning about Test 8. Alex first based his choice on the range

of the bars. He then considered other factors, but finally went back to the range as the decidingfactor, stating Im going to go with larger since theres more of a spread between the


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twotheres more of, theres a larger range. This one has four bars for the range and this one hasseven bars.

More Values

More Values thinking was represented by a statement that a distribution that hadmore values relative to a second distribution would have a larger standard deviation. ForTest 10, Alice stated that theres more values, so its spreading it out more when

describing the variation in the left-hand graph. As such, More Values may be part of theRange rule. This was also suggested by Troys thinking during Test 10 where he

mentioned that the graph on the right had less bars, but seemed to focus more on therelative spread of the bars: Yeah, I guess, that one, yeah maybe it is smaller because it's

tighter. It's smaller numbers, and these are spread out, spread out more.

4.5 AN IDIOSYNCRATIC RULE

Big Mean

Troy was the only student who initially believed that a distribution with a higher mean would

have a larger standard deviation. The first statement of this belief came in Game 1 after he foundan arrangement that produced the largest standard deviation: I was thinking that if the mean

was, I knew the mean was going to be smaller. I thought that would make the standard deviationsmaller.

Troys reasoning seemed to be primarily one of association: the mean gets lower, so thestandard deviation will get lower. While he appeared to have an understanding of how value and

frequency affect the mean, he had not coordinated these changes with changes in the standarddeviation. Even though Troy witnessed that the standard deviation did not change when he

produced a mirror image of the distribution, his belief that the size of the mean affects thestandard deviation persisted. After Troy found a distribution with the smallest standard deviation

in Game 1, he was not sure that moving the same arrangement to a new location would conservethe value of the standard deviation. He did move the bars to higher values, then seemed surprised

to find that the standard deviation stayed the same: Again I guess it's thinking that if the meangoes up that the standard deviation goes up. But obviously that's not true a case. Troy seemed

to have learned that the location of an arrangement did not affect the standard deviation, althoughhe did not appear to understand why at this point.

4.6 BASIC CONCEPTS

There are four basic concepts that students needed to coordinate in order to have a fullunderstanding of factors that affect the size of the standard deviation.

Frequency (F)

Students with an understanding of frequency indicated that the height of a bar represents a

frequency or the number of values at a particular position on the number line. Alexdemonstrated this understanding when reasoning in Game 2 about a distribution he created for


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the largest standard deviation. Alex stated, I kind of figured that the two bars that had the highercounts would have to be on the ends because theres going to be, theres more of, theres more,

more, because they have more people, or more, of the, I mean theres more counts on each end,so theres going to be more of, more leaning to that direction.

Value (V)

Statements that the position of a bar represents a value along the number line were taken as

an indication that students distinguished the representation of value from frequency in thedistributions. An example comes from Alex in Game 1 after he created a mirror image of a

distribution that resulted in the largest standard deviation (see Figure 19). When asked why thearrangement would produce the same value for the standard deviation, Alex replied, Um, well

when I had the 9, when I had the 8 of the nines, it was actually on the other side. It was eight ofthe zeros, and the mean was, the mean basically has changed, like, we cut it down the middle,and its split, and it was on the other side, and it was closer to the count of eight.

Mean (M)

A full conception of the mean requires the coordination of frequency and value such that the

student can predict the effects of adding values to a distribution, or assigning different values todata. Students did not tend to spontaneously demonstrate this understanding. Statements about a

students conception of the mean usually followed a prompt by the interviewer. After Alexcreated a graph for the largest standard deviation in Game 4 (see Figure 10), the interviewer

asked what would happen to the mean if the positions of the yellow (frequency of 5) and green(frequency of 2) bars were switched. Alex responded, The mean will move more towards the

higher numbers.

Figure 10. Alexs first arrangement for the largest standard deviation in Game 4.

Deviation (D)

Students made a variety of statements indicating some conception of the relative distance

between a value and the mean. Phrases such as away from the mean or close to the meanwere taken as evidence of this understanding. Statements of this concept did not usually appearin isolation and tended to accompany explanations of why an arrangement produced the largest


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or smallest standard deviation. To justify why an arrangement produced the smallest standarddeviation, Carl stated Because, putting all the numbers as close together as possible, you get,

um, you get the mean and the standard deviation, and since all the numbers are as close as theycan be to the mean, youll get the smallest standard deviation. Here, we take as close as they

can be to the mean as evidence that Carl was considering the distance between the mean and the

values.

4.7 COORDINATED CONCEPTIONS

Transcripts were scanned for evidence that students were coordinating basic conceptions in

order to predict or justify why an arrangement would produce the largest or smallest value for thestandard deviation (Part 1), or why the standard deviation for one distribution would be smaller,

larger, or the same when compared to a second distribution (Part 2). Coordination refers to astudents ability to treat each concept as a dynamic entity and to consider the mutual effects of

changes in one entity on changes in another. Each code presented below starts with an upper caseC to indicate coordination of concepts followed by letters that identify the concepts that are

coordinated. The lower case letters sd are used to indicate that the effect of the coordinationon the size of the standard deviation is included as part of the coordinated conception.

C-MV

C-MV is indicated by knowing that the location of the mean is dependent on the value (or

location) represented by a bar, but not necessarily taking the frequency of the values intoconsideration. This coordination represents a conceptual understanding of deviation (D). Nancy

demonstrated this understanding when she justified why a mirror image would produce the samevalue for the standard deviation by stating the mean is different, but theyre still the same

amount away from the mean, because its just switching them.

C-MVF

When a bar is moved to a new location, a student with a C-MVF conception is able tocoordinate how both the new value and the frequency represented by the bar height affect the

mean. Alice provided an example of reasoning with this concept. She made the arrangement inFigure 11 to obtain the largest standard deviation with the fourth set of bars in Part 1. The

interviewer asked Alice to consider what would happen if she changed the position of the gold(frequency of 5) and green (frequency of 2) bars. She indicated that the mean would get pulled

towards the gold and the red bar because theres more frequency over there than on the othertwo. Because they aren't pulling the mean that much more over.

C-VFsd

Knowing that once the relative distances among bars are decided, that the absolute location

of the bars doesnt affect the size of the standard deviation indicates a C-VFsd coordination.This is essentially the concept behind the Location rule. Carl gave an example of this

coordination when justifying why an arrangement of the first set of bars results in the smalleststandard deviation, stating that It doesnt matter, really, um, it doesnt really seem to matter


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what the values of the numbers are, as long as theres, um, the same amount of the numbers assquished together as possible, then youre still going to get the standard deviation.

Figure 11. Alices arrangement for the largest standard deviation in Game 4.

C-Dsd

This coordination of how the relative sizes of the deviations affect the standard deviation was

seen primarily during the test phase in Part 2, and only with the fifth and seventh test items.Based on a students statements, we only know that he was considering the overall size of the

deviations from the mean when judging the relative sizes of the respective standard deviations.In these items, frequency could be judged irrelevant given that the same bars are used in both

distributions and the same relative order is maintained. Typically the student refers to gaps,more gaps, or larger gaps. For example, when justifying his choice for the relative size of

the graph on the right in Test 5, Carl stated Im definitely going to have to say smaller. Thegaps really throw off standard deviation. Because, then you have zero, then you have, like, kind

of like outliers, or gaps now. So, Im definitely going to have to say that Oh, well Im sorry,this ones smaller.

Jeff provided an example of a student who did not have this coordination for the same testproblem: Uh, only, because the only reason I think they're going to be the same is they are all

arranged in the same position and, uh, they are the same distance, or not the same distance apart,but the, you know, the space in between them is both equal. So I would think it would be the

same.

C-DFsd

This concept represents the coordination of the four basic concepts with the size of thestandard deviation. A student would state how the standard deviation was affected by

considering the combined effects of changes in the deviation of values and the frequencies of thedeviations.

Carl demonstrated C-DFsd reasoning while answering some questions about the arrangement

he made for the second task in Game 4 (see Figure 12). The interviewer asked Carl if there wasany other arrangement that would produce the largest standard deviation. Carl responded, I


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dont really think so but suggested switching the places of the yellow and green bars, althoughhe thought it would still to be a little too small. Carl made the switch, found that the standard

deviation was smaller, and switched the bars back. The interviewer asked why Carl thought thearrangement produced a smaller standard deviation, and Carl responded that it was because the

mean moved toward bars with higher frequencies.

Figure 12. Carls second arrangement for the largest standard deviation in Game 4.

Alice provided another example of C-DFsd. Alice made the arrangement in Figure 13 to

produce the largest standard deviation in Game 2.

Figure 13. Alices arrangement for the largest standard deviation in Game 2.

The interviewer observed that Alice had moved the blue bar (frequency of 4) next to the redbar (frequency of 6) at one point and asked what she had noticed. She stated that the standard

deviation became smaller. When asked why she thought this happened, Alice answered,

Because, um, it's going to move, I think, um, the mean this way [towards the lower end of thenumber scale]. Yeah. Towards, you know, where more of the values are. So then the deviationthere, for, it's going to be a deviation for more of the numbers are going to be smaller and then

there's just a few over here that are going to be further away.Alice knew that the location of the mean moved toward the larger mass of data, that this

caused the deviations in both the red bar and the blue bar to get smaller, which resulted in alarger number of smaller deviations that were not counterbalanced by the two large deviations ofthe yellow bar, with an overall outcome of a smaller deviation.


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4.8 TASK CHARACTERISTICS AND STUDENTS STRATEGIES

Table 3 presents counts of the different approaches that students presented evidence of using

or considering when arranging the bars to meet the criterion of Task 1 (largest standarddeviation) or the criterion in Task 3 (smallest standard deviation) in each game of Part 1. The

two bar set of the first game required only that students place the bars as far apart as possible toproduce the largest standard deviation, or that they place them next to each other to produce the

smallest standard deviation. As would be expected, most of the students either stated the FarAway approach or the Contiguous approach for the respective criteria. Four of the twelve

students appeared to understand that the values needed to be placed so they were as far awayfrom the mean as possible, but most did not note this.

The three-bar set of the second game produced more revealing information regardingstudents initial understanding of standard deviation. For the first task, slightly more than half of

the students produced an initial arrangement that qualified as the Equally Spread Out approachwhere an attempt was made to evenly space the bars across the range of the number scale. Once

these students found that the Equally Spread Out arrangement did not produce the largeststandard deviation, they typically engaged in moving the bars around and observing the standard

deviation value, what we are referring to as a guess and check approach. When studentsfinally found a qualifying arrangement for the largest standard deviation, they tended to present

justifications along the lines of the Far Away or Balance rules.A contiguous arrangement was most often the first arrangement attempted by a student to

produce the smallest standard deviation, but, surprisingly, three-fourths of the students first madean arrangement where the bars were in ascending or descending order instead of placing the

tallest bar in the middle. In justifying the ordered arrangements, some students thought that theorder did not matter (and may have been using a preferred organizational approach), some stated

that the order was necessary to produce the smallest standard deviation, and others claimed theyfirst placed them in order just to have them together and then they were going to test out different

arrangements. After completing the first two tasks for the largest standard deviation, the barswere not in either ascending or descending order so that students using the latter justification for

an ordered contiguous arrangement did not come about the arrangement by simply sliding thebars over to each other. When students did finally find a qualifying arrangement, it appeared to

happen primarily a guess and check process. Some students offered the Tallest Bar in theMiddle and More Bars in the Middle rules as explanations for why the arrangements produced

the smallest standard deviation, with about half of these explanations referencing the mean, butmost students did not demonstrate a reasonable understanding of how the arrangement met the

criterion.The third game presented a four-bar set where all the bars were of the same frequency. This

was intended to reduce the complexity of the task in anticipation of the fourth game whichpresented a four-bar set with bars of different frequencies. Most students did not have difficulty

with this task, producing a balanced distribution with two bars at opposite ends of the scale forthe largest standard deviation and a contiguous, uniform distribution for the smallest standard

deviation. A Far Away strategy was cited by all the students, with slightly over half noting thebalance of the arrangement, and only a few stating that the distribution placed the value as far as

possible from the mean. All of the students noted the contiguity of the arrangement thatproduced the smallest standard deviation with two noting that the arrangement placed the mean

in the middle of the four bars. Very little guess and check behavior was observed during thisgame.


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The fourth game proved to be a little more challenging. Six students displayed guess andcheck behavior for the first task, although they tended to first move pairs of bars to opposite ends

of the scales, then move the bars and note the changes in the standard deviation. Three fourths ofthe students justified the arrangements that met the largest standard deviation criterion by noting

the balance in the number of value at each end of the scale (e.g., 10 at the low end and 11 at the

high end; see Table 1). Many of these students also stated that the arrangement placed the valuesfar away from each other, with a few noting that the values were as far away from the mean aspossible. Again, a few observed that an arrangement placed the mean at the perceived center of

the scale (Mean in the Middle).For the task of producing the smallest standard deviation, students tended to produce

contiguous distributions with the tallest bar in the perceived center, with only one studentcreating an initial arrangement that was ordered. Most students appeared to have learned that an

ascending or descending order was not optimal. The Tall in the Middle and More in the Middlerules were offered by three-fourths of the students as justifications, with six students referencing

the position of the mean relative to the bar or bars with the highest frequencies. Most of theinitial arrangements did not meet the criterion, which resulted in guess and check behavior to

fine tune the arrangement. The interviewer typically took advantage of this behavior by askingthe student to describe the shape of an arrangement that was being tested, most of which were

somewhat symmetric and bell-shaped. The interviewer asked if the bars could be arranged toproduce an even more bell-shaped or symmetric distribution. This usually resulted in the

students finding an optimal arrangement of the bars.The fifth game presented another challenge with five bars all representing different

frequencies. Most of the students used a Balance strategy, but an arrangement that resulted inthe largest standard deviation was not obvious since there were two different ways to place near

equal amounts at the opposite ends of the number scale. This resulted in some guess and checkbehavior for some before finding an arrangement that met the criterion. The Bell-Shaped

strategy was a little more prevalent for the smaller standard deviation task than was seen withother games. This was probably a result of the interviewers questions and modeling of the bell-

shaped strategy during game 4. The odd number of bars made it possible to have a clear peakwith tallest bar that anchored the center of the distribution. The other bars could be alternated to

the left and right of the tallest according to height, producing a nearly symmetric distribution andan optimal arrangement for the smallest standard deviation. However, only two students

operated with an alternating strategy. Others positioned the tallest bar first, then placed two barsto the left and two bars to the right of the tallest bar, but not in an optimal arrangement. This was

followed by guess and check behavior to produce an arrangement that met the criterion.All students developed the strategy of swapping the positions of the bars (Mirror Image) to

produce a second distribution that met the larger standard deviation criterion. Most students alsogeneralized this strategy to producing a second or third arrangement that met the smaller

standard deviation criterion. Only one student did not spontaneously use this strategy after thesecond game, suggesting that she did not see Mirror Image as a general strategy that would

always conserve the size of the standard deviation. A few students independently developed anunderstanding that once an arrangement was found that produced the smallest standard deviation,

moving the arrangement to a new location preserved the value of the standard deviation. If bythe fourth task of the second game the student had not displayed the Location rule, the

interviewer intervened on the fifth task by moving two of the bars to a new location whilemaintaining the relative position and order of the bars, and asked the student if the third bar


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could be moved to produce the smallest standard deviation. All students were successful at thistask.

4.9 TEST CHARACTERISTICS AND STUDENTS STRATEGIES

Table 4 presents the rules and strategies that were cited as justifications for studentsresponses to the 10 tests in Part 2. Tests 8 and 10 provided the most difficulty to the students,and Test 4 proved difficult for two of the students. The remainders of the tests were solved

correctly by all the students, except for Test 5 for which Adam gave an incorrect response.Students justifications for their responses tended to be along the lines of the rules identified

during Part 1, and tended to reflect relevant characteristics of the graphs. For example, allstudents indicated that the two graphs had the same standard deviation for the first test, noting

that the arrangements were the same but in different locations. Similarly for Test 6, studentsrecognized that one graph was the mirror image of the other and that the standard deviations

were the same. Tests 2 and 3 contrasted bell-shaped distributions with U-shaped distributions.Most of the students justifications reflected the Tallest in the Middle and More in the Middle

rules, although Test 3 also prompted Far Away types of justifications, probably because thegraph on the right was U-shaped.

Students found Test 4 to be a little more difficult than the first three tests. Most noted thatthe only difference between the two graphs was the placement of the black (shortest bar). Only

about half of the students provided a justification for their responses, which tended to be eitherthat the graph on the left was more bell-shaped, or that it had more values in the middle or

around the mean compared to the graph on the right. A few students made reference to the left-hand graph have more of a skew. Two students (Adam and Lora) judged incorrectly that the

graph on the left would have a smaller standard deviation. Adam noted the different location ofthe black bar in both graphs, but did not offer any other justification for his response. Loras

justification was actually a description of each graphs characteristics and did not compare thetwo graphs on features that would determine how the standard deviations differed. When asked

to explain why she thought the graph on the right would have a smaller standard deviation, shesaid, Because they are just in order, basically like steps, except that one. This could be

reflecting a Contiguity with Descending Order rule, but appears to be only a description of thegraph. Similarly, when asked how the graph on the right compared to the graph on the left, Lora

responded, That, well that's kind of the same thing as this one, but this one is, um, the smallestone is on the end instead, and along with the other one. No other justifications were offered

before checking. The difficulty with Test 4 may have resulted from the nature of the Part 1 tasks.The tasks in Part 1 required solutions that had some degree of symmetry and did not require

students to produce skewed distributions, although skewed distributions were often generatedand checked before an optimal distribution was found for the third task. The arrangements

produced for the largest standard deviation were U-shaped, although with a large gap through themiddle, while the optimal arrangements for the smallest distribution were bell- or mound-shaped.

This experience in Part 1 probably facilitated the comparison in tests 2 and 3, but did notnecessarily provide experience directly related to the comparison needed to solve Test 4.

Nonetheless, the majority of the students correctly identified the graph on the right as having alarger standard deviation.

Almost all of the students justifications for Test 5 involved statements of contiguity. A fewstudents noted that the range was wider in the left-hand graph so that the right-hand graph should


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have a smaller standard deviation given that the bars were the same and in the same order in bothgraphs. Two students, Adam and Jeff, appeared to ignore the gaps between bars in the left-hand

graph and responded that the two graphs would have the same standard deviation, suggestingthat shape was the main feature they considered in their decisions and an insensitivity to spread

or deviation from the mean. The first four tests involved graphs with contiguous bar placement,

so that Test 5 provided the first test of sensitivity to gaps between the bars. Test 7 also testedstudents understanding of how gaps affected the standard deviation, although in a more subtleway. All students answered Test 7 correctly, using arguments of contiguity and more values in

the middle for the left-hand distribution as the reason why the right-hand distribution would havea larger standard deviation. Some students also noted that the right-hand distribution had

relatively more values away from the mean. Test 9 also presented gaps in the two distributionsand challenged the misconception that having the bars evenly or equally spaced produces a large

standard deviation. All students answered Test 9 correctly and offered justifications similar tothose used in Test 7, indicating that this initial misconception had been overcome by those who

first displayed it.Tests 8 and 10 were created in response to students answers in a pilot study. Students

statements during the pilot study indicated that many developed a rule that bell-shapeddistributions always had smaller standard deviations compared to other distribution shapes.

Tests 8 and 10 were designed to challenge this rule by presenting situations where a perfectlysymmetric, bell-shaped distribution had the larger standard deviation between the pair. In Test 8,

the left-hand graph was U-shaped, but had fewer values and covered a smaller range than thesymmetric, bell-shaped distribution on the right. All but one of the students incorrectly indicated

that the graph on the right would have a smaller standard deviation for Test 8. Linda answeredcorrectly that the symmetric, bell-shaped graph would have a larger standard deviation. Her

reasoning was that there are more bars. There, um, even though like I think they are clumpedpretty well, theres still, um, a higher frequency and so, theres three more, so because of like the

extra three theres going to be more like room for deviation just because theres more addedvalues.

Other students noted that there were more bars or values in the graph on the right, or that ithad a larger range, but still selected the smaller option as an answer. Students tended to note the

bell-shape or the perception that more values were in the middle of the right-hand distributioncompared to the one on the left as reasons for their responses. After checking the answer and

finding out it was incorrect, the interviewer attempted to guide the students attention to theambiguity that stemmed from a comparison of characteristics between the two graphs. Students

attention was drawn to the differences in the number of values and the range, pointing out thatwhile the bell shape suggested the right-hand graph would have a smaller standard deviation, the

larger range made it possible for it to have a larger standard deviation, and the different numberof values (or bars) presented a different situation than was presented either during Part 1 or in the

previous tests. The interviewer suggested that in ambiguous situations it might be necessary tocalculate the standard deviations, and demonstrated how the information provided by the

program could be used to do so.When students came to Test 10, they were more likely to note the difference in the number of

bars or values between the two graphs and resort to calculating the standard deviation for thegraph on the right. Nine of the students came to a correct decision predominantly through

calculation. Two of the students (Troy and Jane) initially predicted the graph on the right to havea larger standard deviation, but then noted the discrepancy in range or number of values,


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calculated the standard deviation, and changed their choice. Three students (Jeff, Lora, andLinda) did not perform calculations and incorrectly responded that the right-hand graph would

have a larger standard deviation. Because Linda correctly answered Test 8, she did not receivethe same guidance from interviewer received by the other students, which may account for her

incorrect response on Test 10.

4.10 COORDINATED CONCEPTIONS AND REASONING

Table 5 lists the games in Part 1 and the tests in Part 2 where students presented evidence ofbasic concepts and coordination of basic concepts. The information in Table 5 was used to

tabulate the number of coordinated concepts a student displayed during Part 1 (a measure ofconceptual richness), and the number of games on which a student presented evidence of the C-

DFsd coordination (a measure of the extent of C-DFsd reasoning). These measures were used torank order the students by sorting first on the number of games with C-DFsd coordination, and

then by the total number of coordinated concepts (see Table 6). Table 6 also indicates -whetheror not the students gave correct responses for each test, and tallies the number of tests on which a

students initial response was incorrect, regardless of whether or not the response was changed tothe correct choice before checking. In general, correct responses were not dependent on the

richness of concepts displayed by the students. This suggests that the rules students developedduring Part 1 were adequate for comparing variability during Part 2. Students who demonstrated

the C-DFsd coordination were more likely to use calculation on Test 10 to answer correctly. Thismay suggest that students with rich conceptual understanding were able to consider the

ambiguity of the information regarding the relative size of the standard deviations in Test 10,whereas this was less likely to occur with other students, possibly because their conceptual

frameworks could not represent more than one or two aspects of the distributions at a time.

5. DISCUSSION

This ensemble of rules, strategies, and concepts found in this study indicate that students inan introductory statistics course form a variety of ideas as they are first learning about the

standard deviation. Some of these ideas, such as the Contiguous, Range, Mean in the Middle,and Far Away-Values rules, capture some relevant aspects of variation and the standard

deviation, but may represent a cursory and fragmented level of understanding. Others such as theFar Away-Mean, Balance, More Values in the Middle, and Bell-Shaped rules, represent much

closer approximations to an integrated understanding. There are still other ideas, notably theprevalent Equally Spread Out rule and the idiosyncratic Big Mean rule, that are inconsistent with

a coherent conception of the standard deviation. Most students did not persist with thesemisconceptions such as the Equally Spread Out rule and developed a subset of the richer rules as

they progressed to more complex situations in Part 1. Three-fourths of the studentsdemonstrated an ability to coordinate several concepts simultaneously, and half of them appeared

to coordinate the effects of several operations on the value of the standard deviation, anindication of a more integrated conception. In general, the students moved from a one -

dimensional representation towards a multidimensional representation of the standard deviation.Most of the students, however, used a rule-based approach to comparing the variability

across distributions instead of reasoning from a conceptual representation of the standarddeviation. Even among the students with apparently richer representations, their explanations


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were usually based on finding a single distinguishing characteristic between the two distributionsrather than reasoning about the size of the standard deviation through a conception that reflected

how density was distributed around the mean in each distribution. This may have been the resultof them having only a single hour of exploration and suggests that students first take a rule-

based, pattern recognition approach when trying to perform the tasks presented in Part 1 of the

interview. A pattern recognition, rule-based approach is consistent with a goal of finding theright answer by noting characteristics that differentiate one state from another (i.e., onearrangement from another) and noting how this relates to the size of the standard deviation. This

approach was promoted by the software in that the value of the standard deviation was alwaysavailable and there were no penalties for an arrangement that did not meet a criterion.

The interviewer also attempted to extend students conceptual understanding by trying todraw student attention to relevant aspects of the distributions and changes in the distributions,

and by modeling the desired conceptual understanding. The software was designed to presentstudents with feedback that could potentially reveal students misunderstandings, and, to a

certain extent, it appeared that it did so effectively. The software was not designed with modelbuilding or the promotion of model building in mind. However, several changes to the program

could be introduced to support model building. A simple change would be to remove the displayof the standard deviation value while bars were being arranged and only providing feedback on

the value when the students checked if an arrangement met the task criterion. This might resultin less guess and check behavior, although the students could still click the CHECK button after

every change. However, the interviewer could -prompt the students to give an explanation beforechecking, and explore their understanding by asking them to consider alternative arrangements

without the benefit of knowledge of the standard deviation value. This may help to focusstudents attention on relational characteristics of the distribution and to conjecture how they

might affect the standard deviation. Similarly, information displayed on the right side of thewindow regarding the sum of the squared deviations may need to be removed so that a reliance

on the calculated values is not promoted over a conceptual coordination of concepts.Several other features could be added to the software to further support model building.

Students often could not recall a previous arrangement of value of the standard deviation, whichmade it difficult to compare the results of one arrangement with another. A review window

equipped with forward and backward buttons would allow students to revisit prior arrangementsthat were present when the CHECK button was clicked, or a separate snapshot button could be

presented so that students could capture an arrangement before moving trying something else.This would allow students to consider not only the arrangements that met the task criteria, but to

also test out hypotheses and conjectures more reliably.The software currently promotes focusing attention on a single bar rather than visually being

able to see how characteristics change simultaneously. Although the deviation values in all barschange simultaneously as a single bar is moved, this is not a visually salient change. An

alternative might be to create a second display above the graphing area that presents horizontalbars that extend to the left and right of the mean point as a representation of deviation from the

mean. There would be one bar per value in the distribution, with the bars colored to match thecorresponding bar colors. As a single bar is moved to a new location, the lengths of all the bars

would change simultaneously, which might provide a more noticeable representation of thedynamics. Such a visual display would also emphasize the density of the various deviations.

Many short bars with relatively few long bars would indicate a smaller standard deviation,whereas more long bars relative to the short bars would be associated with a larger standard


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deviation. The review window would need to display this representation as well as the histogramrepresentation of the distribution in order to be effective. Controls for displaying only one of

both representations could be added to see if students preferred one over the other, or found onemore informative than the other.

Finally, the interview protocol needs to be modified to include model eliciting prompts and

probes. This can be done through eliciting conjectures from students about how changes toarrangements will affect the value of the mean and deviations, and how these changessubsequently affect the standard deviation, promoting a coordination of the concepts and a

relational structure that models their mutual effects. This contrasts with the current protocolwhere the interviewer modeled the thinking and reasoning for the student rather than supporting

students to produce their own conjectures and test their implications. A model eliciting approachis more likely to produce the system-as-a-whole thinking (Lesh & Carmona, 2003) that is

needed for the C-DFsd coordination, and to allow students to develop a more integratedrepresentational system - (Lehrer & Schauble, 2003) rather than - a collection of separate and

potentially conflicting rules.


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Table 3.

Number of Students Who Presented Evidence of Each Approach for Tasks 1 and 3 for Each Game in Part 1

Task 1: 2 Bars Freq = 8,4 Task 2: 3 Bars Freq = 6,4,2 Task 3: 4 Bars Freq = 5,5,5,5 Task 4: 4 Bars Freq = 8,6,5,2 Task 5: 5 Bars Freq = 8,6,5

Largest SD

Far Away 10

Mean 4

Value 2

Range 1

Big Mean 1

Bell 1

Mean Middle 1

Contiguous 1

Guess & Check 4

Largest SD

Far Away 7

Value 2

Mean 1

Equal 7

Order 3

Balance 5

Mean in Middle 1

Guess & Check 8

Largest SD

Far Away 12

Mean 4

Balance 7

Mean in Middle 3

Equal 1

Guess & Check 2

Largest SD

Balance 9

Far Away 8

Mean 4

Mean Middle 3

Guess & Check 6

Largest SD

Balance 9

Far Away 5

Mean

Value

Guess & Check 5

Smallest SD

Contiguous 12

Location 2

Mean Middle 1

Guess & Check 1

Smallest SD

Contiguous 11

Order 9

Tall in Middle 5

Mean 4

Balance 2

Location 1

Mean in Middle 1

Bell 1

Guess & Check 7

Smallest SD

Contiguous 12

Mean in Middle 2

Location 1

Balance 1

More in Middle 1

Mean 1

Guess & Check 1

Smallest SD

Contiguous 7

Order 1

Tall in Middle 4

Mean 2

More in Middle 5

Mean 4

Symmetric or Bell 2

Balance 1

Guess & Check 7

Smallest SD

Symmetric or Bell 5

Alternating 2

Tall Mid 2

Mean

More in Middle 1

Mean

Guess & Check 4


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Table 4.

Number of Students Who Referenced Each Approach to Justify Choices for Tests in Part 2

TEST 1 TEST 2 TEST 3 TEST 4 TEST 5 TEST 6 TEST 7 TEST 8 TEST 9 TEST 10

Location 12 More Mid 6

TallMid 3

Bell 3Range 1

Contiguous 1

TallMid 3

More Mid 3

FarAway 4Mean 2

Value 1

Bell 2

Contiguous 1

Bell 3

More Mid 2

Tall Mid 1

Balance 1

Contiguous 7

Mean Mid 1

More Mid 1

Spread Out 1Range 1

Swap 11

Mean Mid 1

Contiguous 7

More Mid 5

FarAway

Mean 5Mean Mid 1

Range 1

Calculation 7

Mean Mid 4

Bell 4More Values 4

Range 2

More Mid 3

Tall Mid 1

Contiguous 2

More Mid 5

Mean Mid 3

FarAway

Mean 3Calculation 2

Contiguous 2

Tall Mid 1

Calculation 9

Range 4

More Mid 2

More Values 2M&M 1

Bell 4

Frequency 1


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Table 5

Games* and Tests** Where Concepts and Coordination of Concepts Were Exhibited

BASIC CONCEPT COORDINATION OF CONCEPTS

STUDENT VALUE FREQ MEAN DEV C-MV C-MVF C-DF C-Vsd C-VFsd C-Dsd C-DFsd

JEFF 9

2

9 9 9

TROY7

1

7 7 7

JANE2 3

2 5 8

3

2

2 3

2 5 8

2 3

2 5 8 5 2

ADAM2 4 2 4 2 3 4 5

2 3 7 9

2 4 4

LORA1 2 3 4

3

2 3 4

7

2 3 4

LINDA1 2 4

3

2 4

3

1 2 3 4 5

3 7 9

2 4

3

1 4

3

4

3

MONA1 3 4

2 4 5 7 9

3

4 7

1 3 4

2 4 5 7 9

3 4

2 4 5 7 9

1

5 9

3

4 7

NANCY1 2 3 4

2 3 4 9

2 4

2 3 4 0

1 2 3 4

2 3 4 9

1 2 4

2 3 4 9

2 3 2 4

2 3 4

MARY2 3 4

1 3 4 6 9

2 4

1 3 4 6 9

1 2 3 4

1 3 4 6 7 9

2 3 4

1 3 4 6 9 4

2 3 4 2 4

1 3 4 6 9

ALEX1 2 4

2 3 7 8 9 0

1 2 4

2 3 8 0

1 2 3 4

2 3 7 8 9 0

1 2 4

2 3 7 8 9 0

1 1 2 2 4

2 3 8 0

ALICE1 2 3 4

2 4 7 9

2 3 4 5

2 4 9

1 2 3 4 5

2 4 7 9

1 2 3 4 5

2 4 7 9

4

7

2 3 4 5

2 4

CARL 1 2 3 4 52 3 4 5 7 8 9

1 2 3 4 52 3 4 8

1 2 3 4 52 3 4 5 7 8 9

1 2 3 42 3 4 5 7 8 9

4 5 2 4 1 15

3 42 3 4 8

*Numbers on the first line of each cell indicate the games during which the concept was exhibited.

**Numbers on the second line of each cell indicate the tests on which the concept was exhibited. A 0 (zero) represents the 10 th test.

Value: Concept implied by codings of Deviation, Far Away (V), or any coordination involving Value or Deviation

Mean: Concept implied by codings of Deviation, Mean in the Middle, Far Away (M), C-MV or any coordination involving Mean or Deviation


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Table 6

Relationship Between Test Responses and Coordination of Concepts

PART 1 CORRECT RESPONSE ON TEST

STUDENT

Number of

Coordinated

Concepts

Number of

Games with

C-DFsd

1 2 3 4 5 6 7 8 9 10

Number of

Initially Wrong

Responses

JEFF 0 0 YES YES YES YES NO YES YES NO YES NO 3

TROY 0 0 YES YES YES YES YES YES YES NO YES NO - YES 2

JANE 0 0 YES YES YES YES YES YES YES NO YES NO - YES 2

ADAM 1 0 YES YES YES NO NO YES YES NO YES YES 3

LORA 1 0 YES YES YES NO YES YES YES NO YES NO 3

LINDA 3 0 YES YES YES YES YES YES YES YES YES NO 1

MONA 2 1 YES YES YES YES YES YES YES NO YES YES 1

NANCY 3 1 YES YES YES YES YES YES YES NO YES YES 1

MARY 2 2 YES YES YES YES YES YES YES NO YES YES 1

ALEX 3 2 YES YES YES YES YES YES YES NO YES YES 1

CARL 6 2 YES YES YES YES YES YES YES NO YES YES 1

ALICE 2 4 YES YES YES YES YES YES YES NO YES YES 1


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