PGDE_algebra_v1

8/14/2019 PGDE_algebra_v1

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Algebra

what is

(school)

algebra?

algebra

learning

algebra vs algebraic thinking

arithmetic & algebra

transition

integration

context

technological age

algebra for all

approaches

generalization

express generality

multiple expressions

compare

manipulate

problem solving & modelling

concept

notion of equivalence

operational / relational thinking

concept of numbers and operations

meaning of algebraic symbols

function


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Thornton, S. J. New approaches to algebra: have we missed the point?.

MathematicsTeachingintheMiddleSchoolv. 6 no. 7 (March2001) p.388-92 [http://library.hku.hk/record=b2014009~S6]

388 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

CU R R I C U L U M M O V E M E N T S I N T H E

United States and Australia, characterized by

such documents as Curriculum and Evaluation

Standards for School Mathematics (NCTM1989) andA National Statement on Mathematics for

Australian Schools (AEC 1991), have challenged the

conventional view of algebra as formal structure, ar-

guing that algebra is fundamentally the study of pat-

terns and relationships. Increased emphasis hasbeen given to developing an understanding of vari-

ables, expressions, and equations and to presentinginformal methods of solving equations. The empha-

sis on symbol manipulation and on drill and prac-

tice in solving equations has decreased (NCTM1989).

Has the net effect of these changes been merely

to replace one kind of procedural knowledge with

another? This article looks at three approaches to

algebra: (1) a patterns approach, in which studentsare asked to generalize a relationship; (2) a sym-

bolic approach, in which students learn to manipu-

late algebraic expressions; and (3) a functions ap-

proach, which emphasizes generation andinterpretation of graphs. This article examines the

nature of thinking inherent in each approach and

asks whether any or all of these approaches are, in

themselves, sufficient to generate powerful alge-

braic reasoning.

The Patterns Approach,or Matchstick Algebra

THE PATTERNS APPROACH TO ALGEBRA IN THE

middle school is typified by the matchstick pattern

shown in figure 1. Faced with this problem, stu-dents almost invariably describe the rule as add 3.

Most students look at the table of values horizon-tally, observing that each time a square is added, the

number of matches needed increases by three.

Well-intentioned teachers often help students find ageneral rule from this observation, saying, for exam-

ple, that if one adds 3 each time, the rule is of the

STEVE THORNTON,[email protected], is direc-

tor of teacher development at the Australian Mathematics

Trust, University of Canberra, Australia 2601. His inter-

ests include mathematical rigor and enrichment for talent-

ed students.

New Approachesto Algebra:Have We Missed the Point?

S T E P H E N J. T H O R N T O N

Fig. 1 Matchstick pattern

Examine the following pattern, complete thetable, and find a rule that shows how the num-

ber of matches (m) depends on the number of

squares (s).

Rule: m = ________

s 1 2 3 4 5 100

m 4 7 10

Copyright 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

VOL. 6, NO. 7 . MARCH 2001 389

form m = 3s + k, and suggesting that students try a

few numbers to determine the value of the constant.

The students regard this approach as goodteaching because it helps them obtain the correct

answer. The teacher is similarly reinforced in the

belief that he or she is acting in the students best

interests, because the students are able to find the

rule for this pattern and, perhaps, even a generalrule for other linear cases. The ability to find these

rules is, arguably, a useful skill, but do the students

understand any more about the nature of algebra

than if the subject had been introduced in a formal,

symbolic way? Students who use this heuristic tofind the constant and thus the general rule have, in

reality, looked at the specific rather than the gen-

eral. They have not necessarily acquired any well-

developed notion of the general nature of the pat-

tern but have merely learned a procedure to

develop a correct symbolic expression. The alge-braic essence of the problem is absent.

The Matchstick Pattern Problem is not about

finding a general rule. The answer to the problem,

that is, the rule itself, is unimportant. The problem

is really about alternative representations. It is a vi-

sualization exercise in which different ways of look-ing at the pattern produce differ ent expressions. Vi-

sualizing the pattern in different ways and writing

corresponding algebraic relationships help stu-

dents understand the nature of a variable and be-

come familiar with the structure of algebraic ex-pressions. This particular pattern can be visualized

in at least four different ways (seefig. 2).

Writing down the number pattern in a table, an ac-

tivity commonly found in textbooks and on work-

sheets, does not help students visualize the generalityinherent in the matchstick constructions. A much

more constructive approach is to ask students to build

one element of the pattern physically and explain how

it is put together, not in terms of numbers but in terms

of its underlying physical structure. The different alge-

braic structures then have direct physical meanings.Numerous other visual approaches to algebra are

possible (Nelsen 1993). For example, students could

be asked to visualize the pattern shown in figure 3in different ways so as to generate a relationship be-tween the number of shaded squares (b ) and the

length of the side of the white square (n). Again, at

least four different representations are possible

(seefig. 4). The point of the exercise is not to ob-

tain the answerb = 4n + 4 or any of its variants butrather to understand how the pattern can be visual-

ized and how these different visualizations can be

described symbolically. If we are to foster powerful

algebraic thinking in our students, we must encour-

age a variety of well-justified generalizations of thepattern. Rather than be an end in itself, the purpose

of generating rules is to develop insight into pat-terns and relationships. As Gardner (1973, p. 114)

writes, There is no more effective aid in under-

standing certain algebraic identities than a good di-agram. One should, of course, know how to manip-

ulate algebraic symbols to obtain proofs, but in

many cases a dull proof can be supplemented by a

geometric analogue so simple and beautiful that the

truth of a theorem is almost seen at a glance.

The SymbolicApproach, or FruitSalad Algebra

THE FORMAL, SYMBOLIC

approach to algebra, in which

variables are defined as letters that stand for num-bers, has been criticized as lacking meaning

(Chalouh and Herscovics 1988) and has been iden-

tified as the source of many difficulties faced by be-

ginning algebra students (Booth 1988). OlivierFig. 2 Different ways to visualize the matchstick pattern

Pattern built of one match plus three for eachsquare, orm = 3s + 1

Pattern built of four matches for the first

square plus three for each subsequent square,

orm = 4 + 3(s 1)

Pattern built of two horizontal rows joined byvertical links, orm = 2s + (s + 1)

Pattern built of four matches for each square,

with the overlapping match removed from all

but one of the squares, orm = 4s (s 1)


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Baroudi, Z. (2006). Easing Students' Transition to Algebra.Australian Mathematics Teacher.

Vol.62, Issue 2, pp.28-33. [http://library.hku.hk/record=b2513964~S6]

amt 62 (2) 200628

Traditionally, students learn arithmetic

throughout their primary schooling, and this

is seen as the ideal preparation for thelearning of algebra in the junior secondary

school. The four operations are taught and

rehearsed in the early years and from this, it is

assumed, children will induce the funda-

mental structure of arithmetic (Warren &

Pierce, 2004, p. 294). Recent research has

shown that the emphasis on computation can

actually lead to many misconceptions in the

students minds, which in turn will make the

learning of algebra more difficult.

This article will focus on two categories of

student misconceptions, the first concerns

difficulties with the notion of equivalence and

the second concerns difficulties with the appli-cation of the four operations. The last section

of the article presents suggestions on easing

the transition to algebra through problem-

solving.

Student misconceptionsof equivalence

Falkner, Levi and Carpenter (1999) asked 145

American grade 6 students to solve the

following problem:

8 + 4 = + 5

All the students thought that either 12 or

17 should go into the box. Referring to the

same study, Blair (2005) reports: It became

clear through subsequent class discussions

that to these students, the equal sign meant

carry out the operation. They had not

learned that the equal sign expresses a rela-

tionship between the numbers on each side of

the equal sign. This is usually attributed tothe fact that in the students experience, the

equal sign always comes at the end of an

equation and only one number comes after it

(Falkner et al., 1999, p. 3). One expert has

suggested to me that another possible origin of

this misconception is the = button on many

calculators, which always returns an answer.

Figures 1 and 2 show two typical responses

from my Year 8 class to the question: Explain

the meaning of the = sign (Toth, Weedon &

Stephens, 2004). One third of the class (nine

out of 27 students) gave an operational defini-

tion despite the fact that we had previously

discussed the meaning of the equal sign inthat class.

ZIAD BAROUDI

Figure 1. Relational and operational understanding

of the equal sign by a Year 8 student.

Figure 2. Relational and operational understanding

of the equal sign by a Year 8 student.

amt 62 (2) 2006 29

While all of my Year 8 students are capable

of solving problems such as the one above,

those with an operational understanding of

the equal sign perform the sum on the left

hand side (8 + 4 = 12) and then resort to

different strategies to find the missing number

on the right. Having found that number, they

then perform the operation on the right hand

side in order to verify their answer. In contrast,

those with a relational understanding of the

equal sign recognise that the missing number

must be one less than 8, since it is being

added to a number that is one more than 4.

Figures 3 and 4 contrast two students justifi-

cations of the truth of the equation: 46 + 33 =

45 + 34. Clearly, those with an operational

understanding can establish the truth of the

statement, but their understanding proves tobe a hindrance when learning algebra.

equal sign and change their sign. I recently

asked my Year 11 students to explain to me

how they understood this method, and their

answers are best summed up by the following

statement from one of them: It gets zapped by

the equal sign! Clearly, while the justification

of that method may have been taught, the

practice of taking terms over to the other side

does nothing to address students misunder-

standings of equivalence.

Equivalence andteacher discourse

Booth (1986) suggests that teachers should

emphasise the equivalence of an equation in

the way they read number sentences. Forinstance, when working with the sentence

2 + 3 = 5, teachers should sometimes read

the left hand side as the number that is 3

more than 2, and avoid reading the equal sign

as makes as this reinforces the operational

meaning of the sign (p. 4). This use of

language is not lost on curriculum writers.

The following performance indicator comes

from level 1 (prep.) of the Curriculum

Standards Framework II, used in Victorian

schools at the time of writing: Use materials

and models to develop and verbalise

partwhole relationships (e.g., 6 is 5 and 1

more, two more than 4, one less than 7,double 3) (2000, p. 31).

Building generalisationsin arithmetic

Recent research is suggesting that students

need to be helped, from an early age, to

construct valid generalisations of the arith-

metic operations. Fuji and Stephens (2001)

introduce a concept built on elements of the

Japanese program which they call a quasi-

variable. They define this term as a number

sentence or group of number sentences thatindicate an underlying mathematical relation-

ship which remains true whatever the

numbers used (p. 260). For instance, before

students are able to understand an equation

expressed as ab + b = a, they can be intro-

duced to equations such as 78 49 + 49 = 78.

The truth of this relationship is independent of

Figure 3. Application of the relational and

operational understandings of the equal sign.

Figure 4. Application of the relational and

operational understandings of the equal sign.

When teaching students to solve equations,

we teach them the necessity of doing the same

thing to both sides. This is particularly impor-

tant as students encounter and learn to solve

algebraic equations with operations on both

sides of the symbol (e.g., 3x 5 = 2x + 1)(Knuth, Alibali, McNeil & Weinberg, 2005,

p. 69). Unless a student understands that this

rule exists to preserve the equality of both

sides, then that student will have little chance

of experiencing success. Teachers often pass

over these difficulties by teaching their

students to take terms to the other side of the


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arithmetic to

algebra, difficulty &

possibility

notion of

equivalence

intuitive strategies vs

standard algorithms

building

generalisations

in arithmetic

meaning of equal sign

relational or

operational

understanding

teachers' use of language

quasi-variable

problem

modelling

and solving

guess-check-

generalize

amt 62(2)200628

Traditionally, students learn arithmetic

throughout theirprimary schooling,and this

is seen as the ideal preparation for the

learningo f algebrain the jun iorsecondary

school.The four operations are taught and

rehearsedinthe earlyyears andfromthis,it is

assumed, children will induce the funda-

mental structure of arithmetic (Warren&

Pierce,2004, p.294). Recent researchhas

shownthat the emphasis oncomputationcan

actuallylead to manymisconceptions inthe

students minds,whichinturn will make the

learningofalgebra more difficult.

This article will focus ontwo categories of

student misconceptions,the first concerns

difficulties withthe notionofequivalence and

the secondconcerns difficulties withthe appli-

cationofthe fouroperations. The last section

ofthe article presents suggestions oneasing

the transitionto algebrathrough problem-

solving.

Student misconceptionsofequivalence

Falkner,Levi andCarpenter(1999) asked145

American grade 6 students to solve the

followingproblem:

8+ 4= + 5

All the students thought that either12 or

17should go into the box.Referring to the

same study,Blair(2005) reports:It became

clear throughsubsequent class discussions

that to these students,the equal signmeant

carry out the operation. They had not

learnedthat the equal signexpresses arela-

tionshipbetweenthe numbers oneachside of

the equal sign.This is usuallyattributed to

the fact that inthe students experience,the

equal signalways comes at the endof an

equationandonlyone numbercomes afterit

(Falkneret al., 1999,p.3). One expert has

suggestedto me that anotherpossible originof

this misconceptionis the =buttonon many

calculators,whichalways returns ananswer.

Figures 1and2 showtwo typical responses

frommy Year8class to the question:Explain

the meaningof the = sign(Toth, Weedon&

Stephens,2004).One third ofthe class (nine

out of27students) gave anoperational defini-

tiondespite the fact that we hadpreviously

discussedthe meaningof the equal signin

that class.

ZIADBAROUDI

Figure1.Relationalandoperationalunderstandingoftheequalsignbya Year 8student.

Figure2.Relationalandoperationalunderstandingoftheequalsignbya Year 8student.

Baroudi (2006)

Al

Cuoco

Fennell

Falkner


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Student Misconception of Equivalence

illustrated with findings ofFalkner, Levi & Carpenter (1999),response to a 'simple' open

number sentence

diference between operational and relationalunderstanding

relationalthinking


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the '=' sign is used in equations to show

that whatever is on the right of the '='sign is equal to whatever is on the leftof the '=' sign e.g. 49=7x7, 3+3=5+1

etc


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the '=' sign specifies that you are comingto the conclusion of a sum of somethinge.g. 3x2=6 '=' means you are going to

have a total


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46 + 33 = 45 +34

True, because if you ... subtract 1 from34 and add it onto 45 you get 33 and

46 ... just like the other side


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In the following expression,what number do you thinkbelongs in the box?

8+4 = ( )+5

How do you think students in

the early grades or middleschool typically answer this

question?


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Relational Thinking

How can students answer these?

7 6 4+

( )= +

534 533 176+ ( ) = +


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics Teaching Developmentally.Boston: Allyn & Bacon. p.262

Relational Thinking

When a student observes and uses numericrelationships between two sides of the equal

sign rather than actually computing theamounts, the thinking involved is referred toasrelational thinking.

Relational thinking goes beyond simplecomputation and instead focuses on how one

operation or series of operation is relatedanother.


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1

Algebra in Elementary Schools:

Its Not About xs and ys

2007 NCTM Conference

Atlanta, Georgia

Tad WatanabeKennesaw State University

[email protected]

http://science.kennesaw.edu/~twatanab

What should elementary school algebra look

like?

What are the fundamental ideas of algebra

that are appropriate for investigation by

elementary school students?

What ideas related to algebra are discussed

in the Japanese elementary school

curriculum?

http://science.kennesaw.edu/~twatanab/NCTM07.pdf

Homepage of Tad Watanabe:

http://science.kennesaw.edu/~twatanab/


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Watanabe, 2007


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Watanabe, 2007


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Watanabe, 2007


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Watanabe, 2007


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A correct understanding of the equalsign is the first form of symbolism that

is addressed.

Students from grade 1 through middle

school continue to think of= as a

symbol that separates problem from

answer.

Out of an exploration of open

sentences an understanding of variable

can and should develop.

Meaning Of Equal Sign


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics Teaching Developmentally.Boston: Allyn & Bacon. pp.260-261

Meaning of Equal Sign

First, it is important for students to see andunderstand the relationships in our number system.The equal sign is a principal method of representing

these relationships. For example,

6 7 5 7 7 = +When basic ideas, initially and informally developedthrough arithmetic, are generalized and expressed

symbolically, powerful relationships are availablefor working other numbers in a generalized manner.


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics Teaching Developmentally. Boston: Allyn & Bacon. pp.260-261

Meaning of Equal Sign

A second reason is that when students fail tounderstand the equal sign, they typically havedifficulty when it is encountered in algebraicexpressions. Even solving a simple equation such as

5 24 81x!

=

requires students to see both sides of the equal signas equivalent expressions. It is not possible to "do" theleft-hand side.


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics Teaching Developmentally. Boston: Allyn & Bacon. p.261

True or False

Introduce true/false sentences or equations withsimple examples to explain what is meant by a trueequation and a false equation. Then put several

simple equations on the board, some true and somefalse.

Students decide which are true and which are false.For each response they are toexplain theirreasoning.

5 2 7+ =

4 1 6+ =

4 4 8+ =

8 10 1= !


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True or False

Afterwards, have students explore equations that areless traditional in form:

Do not try to explore all variations in a single lesson.Listen to the types of reasonsand plan additional

equations accordingly for subsequent days.

4 5 8 1+ = +

4 5 4 5+ = +9 5 14 0+ = +

3 7 7 3+ = +

9 5 14+ =

6 3 7 4!

=!

8


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Open Sentences

Write several open sentences on the board. To beginwith, these can be similar to the true/false sentencesthat you have been exploring.

5 2+ = ( )

4 5 1+ =

( )!

4 6+ ( ) =

3 7 7+ = + ( )

The task is to decide what number can be put into thebox to make the sentence true. Of course, an

explanation is also required.

( ) + =4 8

( ) =!

10 1

6 7 4!

( )= !


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics Teaching Developmentally.Boston: Allyn & Bacon. p.262

Writing True/False Sentences

After students have had ampletime to discuss true/false andopen sentences, challenge them

to make up their own true/falsesentences that they can use to

challenge their classmates.


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics TeachingDevelopmentally. Boston: Allyn & Bacon.

http://wps.ablongman.com/ab_vandewalle_math_6/0,12312,3548294-,00.html

A recent focus of research in algebraic

reasoning is on structure or generalizationsbased on properties of our number system.

Generalization is again the main interest.For example, when students note that 4 x 7

= 7 x 4, what helps them to understand

that this is true for all numbers?

Making Structure in the Number System

Explicit


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van de Walle, J.A. (2007) Elementary and Middle School Mathematics Teaching Developmentally.

Boston: Allyn & Bacon.http://wps.ablongman.com/ab_vandewalle_math_6/0,12312,3548294-,00.html

Further, how do students go about

"proving" that properties such as the

commutative property or relationships on

odd and even numbers are always true?

By helping students make conjectures

about the truth of equations and open

sentences, they can then be challenged to

make decisions about the truth of these

conjectures for all students.

Making Structure in the Number System

Explicit


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Routes to, Roots of Algebra

Expressing Generality

Rearranging and Manipulatingexpressions as entities

building and stripping expressions

Possibilities and Constraints

getting a sense of X as a variable

Generalized Arithmetic

becoming explicitly aware of the rules thatoperations on numbers satisfy

seeing, saying and recording

Mason, Graham, Pimm & Gowar, (1985)


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What is school algebra about? What is it

for? When does it begin?

http://www.ncetm.org.uk/mathemapedia/AlgebraMathemapedia

School algebra is about developing amanipulable language for expressinggeneralities about numbers and their

properties, making it much easier toconjecture & convince that it would beusing only words and pictures.


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What is school algebra about? What is it

for? When does it begin?

http://www.ncetm.org.uk/mathemapedia/Algebra

Mathemapedia

The manipulation of algebraic symbols issometimes referred to as arithmetic withletters and as generalized arithmetic ......

Traditionally algebra has been introduced asmanipulating letters as if they werenumbers, which lacks motivation ......, and

efficacy and suggests that there are a lot ofrules to be learned in order to do algebra.


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What is school algebra about? What is it for? When

does it begin?

http://www.ncetm.org.uk/mathemapedia/Algebra

Mathemapedia

Another approach is to prompt learners toexpress generalities as early as possible, and

gradually to move to more and more succinctexpressions, often in situations in which thereare multiple expressions for the same thing.

Learners then develop the wish to work out howto transform one expression into another, andthey can draw upon their knowledge of

arithmetic in doing this. Thus algebra emergesas a language which can be used to expressand to manipulate generalisations.


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First Encounters of Expressing Generality

... algebra is most usefully seen as a language inwhich toexpress generalities, usually to do withnumbers.

Learners will only understand algebra as alanguage of expression if theyperceive andexpressgeneralities for themselves.

At first this takes time, but in this way learnersbecome effective users of algebra.

Experience with expressing generality in differentcontexts leads tomultiple expressionsfor the samething.

Mason et al 2005, p.23

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PGDE_algebra_v1