Management Information Systems Assignment | Get Paper Help

looking for 0 similarity work . the homework will be uploaded to the Turnitin WebsiteBringing Out the Algebraic Character of Arithmetic: From Children’s Ideas to
Classroom Practice
Article · January 2013
DOI: 10.4324/9780203827192
CITATIONS
69
READS
684
3 authors, including:
Some of the authors of this publication are also working on these related projects:
The Poincare Institute for Mathematic Education View project
Everyday Mathematics View project
Analucia Schliemann
Tufts University
62 PUBLICATIONS 2,657 CITATIONS
SEE PROFILE
David Carraher
TERC
61 PUBLICATIONS 2,670 CITATIONS
SEE PROFILE
All content following this page was uploaded by David Carraher on 23 August 2015.
The user has requested enhancement of the downloaded file.
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
2
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
3
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
4
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
6
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)?
7
• 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
8
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
9
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,
10
(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
12
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.

 

Don't use plagiarized sources. Get Your Custom Essay on
Management Information Systems Assignment | Get Paper Help
For $10/Page 0nly
Order Essay
Calculator

Calculate the price of your paper

Total price:$26

Need a better grade?
We've got you covered.

Order your paper