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Classroom Practice

Article · January 2013

DOI: 10.4324/9780203827192

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1

Bringing Out the Algebraic Character of Arithmetic:

From Children’s Ideas to Classroom Practice

Analúcia D. Schliemann

David W. Carraher

Bárbara M. Brizuela

Studies in Mathematical Thinking and Learning Series.

Lawrence Erlbaum Associates

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Table of Contents

Bringing Out the Algebraic Character of Arithmetic: ……………………………………………… 1

From Children’s Ideas to Classroom Practice ………………………………………………………… 1

Analúcia D. Schliemann…………………………………………………………………………………….. 1

David W. Carraher …………………………………………………………………………………………… 1

Bárbara M. Brizuela………………………………………………………………………………………….. 1

Studies in Mathematical Thinking and Learning Series. Lawrence Erlbaum Associates . 1

Table of Contents……………………………………………………………………………………………… 2

Preface: Rethinking Early Mathematics Education …………………………………………………. 6

Rethinking the Relationships Between Arithmetic and Algebra ………………….. 7

Rethinking Our Views About Young Students……………………………………….. 10

The Structure of the Book…………………………………………………………………… 12

Acknowledgements …………………………………………………………………………… 13

Chapter 1: Prior Research and Our Approach………………………………………………………. 15

(a) Students’ difficulties with algebra …………………………………………………… 15

(b) The transitional approach ………………………………………………………………. 18

(c) Another view: Algebra in Elementary School …………………………………… 25

(c) Young children doing Algebra………………………………………………………… 27

Our Approach…………………………………………………………………………………… 31

Introduction to Part I: Interview Studies…………………………………………………………….. 35

Chapter 2: Young Children’s Understanding of Equivalences ……………………………….. 37

Contexts for understanding equivalence………………………………………………… 39

Method……………………………………………………………………………………………. 44

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Results…………………………………………………………………………………………….. 48

Children’s answers and justifications:………………………………….. 48

Age and Context Effects on Children’s Justifications……………… 51

Effects due to numerical information…………………………………… 54

Discussion ……………………………………………………………………………………….. 54

Chapter 3: Can Young Students Solve Equations?………………………………………………… 57

Study 1: Recognizing Invariance Despite Change: Strategies and

Representations…………………………………………………………………………………………… 58

Method…………………………………………………………………………… 58

Results …………………………………………………………………………… 59

Study 2: “But How Much, How Many?”: The Paradox Inherent In

Representing Unknown Quantities …………………………………………………………………. 65

Method…………………………………………………………………………… 66

Results …………………………………………………………………………… 67

General Conclusions………………………………………………………………………….. 77

Chapter 4: Addition operations as functions ……………………………………………………….. 80

A Bare Bones Example………………………………………………………………………. 83

Additive Functions and Word Problems………………………………………………… 88

Instantiation…………………………………………………………………….. 90

Students’ use of algebraic notation ……………………………………… 93

How do the above examples bear upon early algebra? …………………………….. 95

Lessons for Early Algebra ………………………………………………………………….. 98

Structure versus Context…………………………………………………………………… 100

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Chapter 5: From Quantities to Ratio, Functions, and Algebraic Notation………………… 103

Functions and Rate ………………………………………………………………………….. 105

Function and Ratio in Non-School Settings………………………………………….. 107

The intervention and its results ………………………………………………………….. 110

First Week: Children’s Initial Strategies ……………………………………………… 112

Task 1: Filling out function tables ……………………………………. 112

Task 2: Different ways to go from one number to another…….. 114

Second Week: Developing New Strategies and Algebraic Notation …………. 114

Task 1: Focusing on any number (n)…………………………………. 114

Task 2: Breaking the columns’ pattern………………………………. 115

Task 3: Developing a notation for the function …………………… 118

Task 4: Finding the rule from two pairs of numbers…………….. 118

Discussion ……………………………………………………………………………………… 122

Chapter 6: On Children’s Written Notation to Solve Problems ……………………………… 124

Using Notations to Solve Problems…………………………………………………….. 127

Using Notations to Show a Solution …………………………………………………… 131

Sara Explains her Use of Notations to Solve Problems…………………………… 133

Discussion ……………………………………………………………………………………… 135

Chapter 7: Discussion …………………………………………………………………………………… 138

Development versus Learning……………………………………………………………. 140

Contexts ………………………………………………………………………………………… 141

Theme 3: The Roles of Representational Systems in Mathematical Learning143

Future developments………………………………………………………………………… 144

5

References …………………………………………………………………………………………………… 147

Figure Titles…………………………………………………………………………………………………. 218

Footnotes …………………………………………………………………………………………………….. 221

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Preface: Rethinking Early Mathematics Education

People generally believe that arithmetic should precede algebra in the curriculum.

And they can find ample evidence to support their view: Arithmetic is easy; algebra is

difficult. Arithmetic is about operations involving particular numbers; algebra is about

generalized numbers. Arithmetic appears in all cultures; algebra appears only in some,

and, even in those, it made its appearance only recently.

But what if there are good reasons for thinking otherwise? What if there were

compelling research showing that young students can learn algebra? What if there were

good mathematical justifications for teaching algebra early? What if history is not always

a trustworthy guide for ordering topics in the mathematics curriculum?

Recently, following proposals by researchers in mathematics education, the

National Council of Teachers of Mathematics (NCTM, 1997, 2000) has endorsed early

algebra and recommended that algebraic activities start at the very first years of schooling

and that algebraic notation be introduced between grades 3 and 5.

Many people react to these ideas with puzzlement and outright skepticism. They

wonder about issues of:

• Possibility: Can young students learn algebra? Can teachers teach algebra to

young students?

• Desirability: Is it important or useful for students to learn algebra early (or at all)?

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• Implementation: How can the recommendations for early algebra be put into

effect?

This book is part of an ongoing effort to establish a research basis for the

introduction of algebraic concepts and notation in elementary school, an area of studies

that came to be known as Early Algebra. Our research helps clarify the question of

possibility, specifically, the issue of whether young students can reason algebraically.

Generally, the results are encouraging. But we need more than a simple yes or no

answer. For Early Algebra is not the Algebra I syllabus taught to young students. And

algebraic reasoning is not synonymous with methods for using algebraic notation and for

solving equations. As Kaput (1995) suggests, algebra encompasses pattern generalization

and formalization, generalized arithmetic and quantitative reasoning, syntactically-guided

manipulation of formalisms, the study of structures and systems abstracted from

computations and relations, the study of functions, relations, and joint variation, and the

modeling and phenomena-controlling languages.

Below we will describe our rationale for rethinking the roles of arithmetic and

algebra in the mathematics curriculum and for Early Algebra. The chapters that follow

provide evidence bearing directly on the issue of children’s capability to reason

algebraically and to learn algebra.

Rethinking the Relationships Between Arithmetic and Algebra

Early Algebra is not so much about when as about what, why, and how. It rather

concerns diverse views about arithmetic, algebra, and how they are related.

For many years the prevailing view in the United States has been that arithmetic

and algebra are largely distinct subject matters standing in a particular order. As Figure

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1.1 illustrates, arithmetic and algebra are conceived as mostly separate. Certain ideas,

techniques, and representations are common to both; these correspond to the region of

intersection (with hatched lines). Advocates of this view tend to speak about the need for

“bridging arithmetic and algebra”. And they tend to think of this bridging as occurring

somewhere between the “end” of arithmetic and the “beginning” of algebra.

INSERT FIGURE 0.1 ABOUT HERE

“Pre-algebra” courses, generally offered one or two years prior to Algebra I,

embrace this view of arithmetic and algebra. In these courses, special attention is given

to the expanded uses and meanings of symbols from arithmetic. But anything but a

simple transition occurs. Consider, for example, the equals sign. In most arithmetic

instruction the equals sign corresponds to the notion of “yields.” Students show that they

subscribe to this notion when they accept, for instance, “3+5 = 8”, but reject “8=3+5”

and “3+5=7+1”. They may even gladly accept expressions such as “3+5 = 8 + 4 = 12.”

Early mathematics instruction offers little for students to rectify the issue.

After years of unnecessarily limited arithmetic instruction, students are expected

to treat the equals sign as a comparison operator expressing an equivalence relation. But

as the above examples suggest, of the three properties of an equivalence

relation—reflexivity, symmetry, and transitivity—(the latter) two do not apply to the

equals sign.

This makes the difficult task of introducing algebraic equations all the more

daunting. Consider, for example, an equation with variables on each side (e.g., 3x = 5x –

14). Students have to contend with the idea that the expressions on each side of the equals

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sign can be viewed as a function having variation. The equals sign effectively constrains

the values each function can have to the same, solution set. x is free to vary; but the

equation is only true under the solution set. The reader can imagine the difficulty

students will have in comprehending these weighty ideas if they are still working with a

constricted understanding of the equals sign. The situation is even more challenging

when one considers the variety of purposes equations serve (Usiskin, 1988): equations as

formulas, sentences to solve, identities, properties, functions, etc.

There is an alternative to having students relearn mathematics when they take

Algebra I. That alternative rests on a strikingly different view about what arithmetic and

elementary mathematics are about. The key idea behind this new view is that arithmetic

is a part of algebra (see Figure 1.2), namely, that part that deals with number systems, the

number line, numerical functions and so on. This does not mean that every idea, concept

and technique from arithmetic is manifestly algebraic; however, it does imply that each

one is potentially algebraic1. It also means that concepts such as equivalence can and

should be treated early on in ways consistent with their usage in more advanced

mathematics.

INSERT FIGURE 0.2 ABOUT HERE

That arithmetic is a part of algebra is neither obvious nor trivial, especially for

those of us who followed the condescending “arithmetic first, algebra much later” route

through the curriculum. It didn’t have to be this way. In mathematics isolated examples

can always be treated as instances of something more general. The number, 327, stands

for (3 x 100)+(2 x 10)+(7 x 1), which is just an instance of the more general expression,

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(a x 100)+(b x 10)+(c x 1), which can be expressed even more generally for an arbitrary

radix or base. Opportunities for generalizing, for thinking about functions and variables,

for using algebraic notation, abound in elementary mathematics.

To consider arithmetic as a part of algebra encourages us to view isolated

examples and topics as instances of more abstract ideas and concepts. Addition, for

example, is not only a computational routine. It is also a function with certain general

properties. Likewise multiplication by two is not simply a table of facts (1×2 =2; 2×2=4;

3×2=6; 4×2=8) but a function that maps a set of input values to unique output values.

This idea can be expressed algebraically, for instance, through the mapping notation, x‡

2x, or the equation y=2x, or the graph on a Cartesian plane of such a relation between x

and y.

It helps to bear these points in mind while reading the research chapters. When

you first look at the problems we gave children to solve you may think that they are

arithmetical. Upon looking more closely you might conclude that they are algebraic. The

categories, arithmetical and algebraic, are not mutually exclusive when one adopts the

view that arithmetic is part of algebra.

Rethinking Our Views About Young Students

Pupils and mathematicians from Aristotle’s era would likely show immense

difficulty in understanding certain topics that a high school student of today could easily

solve. While ancient Greeks represented problems essentially through natural language

and geometry, today’s high school students have access to modern algebraic notation that

emerged only in the last handful of centuries (Harper, 1987). Even such “simple”

representations such as place value notation and column multiplication were uncommon

11

before 1500. Psychologists and educators may see a “developmental pattern” in the

emergence of concepts such as multiplication, ratio, and proportion—overlooking the fact

that students in today’s schools are given explicit instruction about these topics from

around nine years of age. The same may hold for algebraic concepts: given the proper

circumstances and learning experiences, school children may be able to make

mathematical generalizations and to deal with algebraic concepts and algebra notation

much earlier than is presently thought.

The mathematics education community sorely needs careful analyses of children’s

understanding of mathematical rules, of their own ways of approaching and representing

algebra problems in different contexts, and of instructional models for initiating algebra

instruction. We need to conduct such research under special circumstances (in or out of

classrooms) because typical classrooms make restrictive assumptions about what students

are capable of learning.

We have been charting children’s initial understandings and how their algebraic

reasoning evolves as they participate in classroom activities designed to bring to the

forefront the potentially algebraic nature of arithmetic and thinking about quantities. Our

approach to the study of early algebraic reasoning includes analyses of children’s

reasoning processes and of classroom activities, with a deep concern with mathematics

content, mathematical notation, and with the relationship between arithmetic and algebra.

It deals with the pressing issues faced by teachers and faculty in teacher preparation

programs regarding how to implement the NCTM goals of introducing algebra in the

early grades. Through the video-taped clips we will show that this is possible. We will

discuss some of the conditions under which this may be achieved. As such the book will

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constitute a rich source for researchers in mathematics education, for researchers in

cognitive development, for pre-service and in-service teachers, and for anyone concerned

with mathematics education and curriculum development and implementation.

We hope this book will convince some skeptical readers that algebra has an

important role to play in early mathematics education. For those who already accept this

view, we hope it will provide glimpses into children’s thinking that will help them to

move forward in this enterprise.

The Structure of the Book

Chapter 1 reviews research on algebraic reasoning and describes our approach to

early algebra. Chapter 2 looks at how young Brazilian and American children make

sense of algebraic ideas in different contexts. Chapter 3 looks at the notations children

produce in solving algebraic problems. Chapter 4 focuses on how students reason about

addition and subtraction as functions. Chapter 5 focuses on how students understand

multiplication when it is presented as a function. Chapter 6 examines how children use

notations in algebraic problems involving fractions. The accompanying CD-ROM

includes a videopaper for chapters 4 through 6 that acts as a sidebar for the respective

chapter. Each videopaper is a self-standing research paper, with embedded video footage,

in which we look closely at students reasoning in classroom situations and, occasionally,

in interviews outside of class.

The final chapter (Chapter 7) summarizes the findings and highlights the

implications for early mathematics education.

In writing this book we had a fairly diverse audience in mind: researchers,

practitioners, curriculum developers, and policy makers.

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