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Appendix E
A Framework for Constructing a Vision of Algebra:
A Discussion Document
Working Draft
This document has been adapted from the "Algebra in the K-12 Curriculum: Dilemmas and Possibilities,"
submitted in March 1995 by the Algebra Working Group to the National Council of Teachers of
Mathematics. This document reflects the comments and suggestions on the original document by the
National Council of Teachers of Mathematics Board of Directors and reviewers from the mathematics
community. Permission to photocopy materials from this document is granted to individuals and groups who
want to use it for discussion purposes.
May, 1997
145
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Copyright 1997 by
The National Council of Teachers of Mathematics, Inc.
1906 Association Drive, Reston, VA 22091-1593
All rights reserved.
Printed in the United States of America
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NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS
THE ALGEBRA WORKING GROUP
Gail F. Burrill
University of Wisconsin-Madison
Madison, Wisconsin
Jonathan Choate
Groton School
Groton, Massachusetts
Joan Ferrini-Mundy
University of New Hampshire
Durham, New Hampshire
Steven Monk
University of Washington
Seattle, Washington
Beatrice Moore-Harris
Fort Worth Public Schools
Fort Worth, Texas
Mary M. Lindquist, Board Liaison
National Council of Teachers of Mathematics
Reston, Virginia
147
Elizabeth Phillips
Michigan State University
East Lansing, Michigan
Merrie L. Schroeder
Price Laboratory School
Cedar Falls, Iowa
Jacqueline Stewart
Okemos Public Schools
Okemos, Michigan
Lee V. Stiff
North Carolina State University
Raleigh, North Carolina
Erna Yackel
Purdue University-Calumet
Hammond, Indiana
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CONTENTS
Preface
Introduction
A Promising Practice
Critical Issues
A Framework Building a Dynamic View of Algebra
Embedding Algebraic Reasoning in Contextual Settings
Bringing Coherence to the Algebra Curriculum Organizing Themes
Summary
Examples from Contextual Settings
Bringing Meaning to the Framework
Example 1: From the Contextual Setting of Growth and Change
Example 2: Contextual Settings Within Size and Shape
Example 3: Contextual Settings Within Number
Using the Framework
Bibliography
149
151
153
155
158
160
160
161
164
165
165
166
173
179
185
187
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PREFACE
In 1994, the National Council of Teachers of Mathematics Board of Directors created an Algebra Working
Group and charged it to
produce a document that: Expands the vision of algebra for all that begins with experiences in early
elementary school and extends through secondary school; elaborates this vision by including example,
practical ideas, and promising practices, and helps school systems raise questions about the process of
change.
The Working Group met in the summer and fall of 1994 and developed a draft document that was circulated
broadly within the mathematics, mathematics education, and school community for review and comment. This
document entitled A Framework for Constructing a Vision of Algebra was presented to the National Council of
Teachers of Mathematics Board of Directors in 1995 as a final report of the Working Group. It has been circulated
since that time upon request to many groups and individuals interested in questions about school algebra.
In anticipation of the May, 1997 National Council of Teachers of Mathematics (NCTM)/Mathematical
Sciences Education Board (MSEB) symposium on the nature and role of algebra in the K-14 curriculum, the
document has been revised and updated to serve as a discussion and background document for the symposium.
Gail Burrill
Joan Ferrini-Mundy
Algebra Working Group Members
May, 1997
151
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INTRODUCTION
The release of the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for
School Mathematics (NCTM, 1989) marked a new era in K-12 mathematics education. The Curriculum Standards
call for a rethinking of the mathematical goals and emphases of school mathematics. In particular, the document
outlines ways in which the subject matter of algebra can be organized as a strand occurring throughout the K-12
grade span, rather than confining algebra to the typical two courses in high school. The Patterns and Relationships
standard for grades K-4, for example, calls for students to
· recognize, describe, extend, and create a wide variety of patterns;
· represent and describe mathematical relationships; and
· explore the use of variable and open sentences to express relationships (NCTM, 1989, p. 60~.
The standards for grades 5-8 include a standard called Patterns and Functions and an Algebra standard. For
grades 9-12, there is an Algebra and Functions standard as well as a Mathematical Structure standard. Taken as a
whole, these five standards offer one sketch of a K-12 algebra strand.
Curriculum developers, textbook authors, and others have elaborated such conceptualizations of K-12 algebra
in their publications and materials. The NCTM through its publications (Algebra for Everyone; the Addenda Series
for grades K-6, Making Sense of Data and Patterns; for grades 5-8: Dealing with Data and Change and Patterns
and Functions; for grades 9-12: Algebra in a Technological World and Data Analysis and Statistics across the
Curriculum; and the February, 1997 special issues of Teaching Children Mathematics, Mathematics in the Middle
Grades, and the Mathematics Teacher) also has provided further elaboration and discussion of how a K-12 focus on
algebra and algebraic thinking might be formulated.
This document contributes further to the ongoing examination and work of shaping the school algebra
curriculum, largely through a proposed framework for organizing discussion about algebra in the K-12 curriculum.
We also offer extended examples of how algebraic reasoning might be developed and encouraged across the grades.
A number of current pressures contribute to the need for ongoing examination of the algebra dimension of the
school curriculum. Problems in the workplace, in industry, and in everyday life involve algebraic concepts.
Fundamental mathematical ideas in the areas of growth and change, patterns and regularity, quantity, size, shape,
and data are often expressed with the tools and symbols of algebra.
Increasingly sophisticated technology opens a wide range of possibilities of rethinking the emphases that have
been traditional in school algebra, and raises a set of serious questions. Fundamental issues about the type and
amount of symbol manipulation and procedural activity that is appropriate for students can now be examined and
debated within the context of heretofore unavailable technological tools. New conceptualizations of "symbol
153
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154
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
sense" (Arcavi, 1994) and "function sense" (Eisenberg, 1992) have emerged within the general discussion of
algebra teaching and learning. In addition to considering how the available technologies might be used to help
students understand the concepts of algebra and the procedures of algebra, there is now the dual question of how
these technologies might themselves necessitate changed emphases and new additions to the content of school
algebra.
Research and practice provide compelling evidence that children engage in significant mathematical reasoning
at early ages, and that algebraic thinking can be nurtured and encouraged early in the curriculum (Bastable &
Schifter, in preparation; Kaput, in preparation). How can the school algebra curriculum be formulated to develop
across the grades and to capitalize on these understandings? What might such early introduction of key algebraic
concepts and processes mean for the revision of the traditional secondary school algebra curriculum?
Currently, there is a strong trend toward algebra for all in the nation' s eighth grades and secondary schools.
Yet, various sources of evidence indicate that, for many students, their experiences with algebra in middle and
secondary schools are not leading to high levels of understanding or proficiency (Beaton et al., 1996; Reese et al.,
1997~. Compounding the situation, curriculum and instructional materials currently available provide a wide, and
sometimes confusing, array of distinct possibilities of how the algebra curriculum might be organized.
The context and climate around algebra as a K-12 element of the mathematics curriculum is ready for
discussion. The framework and examples that follow are intended as a contribution to the process of continuing
deep discussion about this important area of mathematics education.
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A PROMISING PRACTICE
Many people assume that "algebra" means working with symbols, but in recent years there has been a great
deal of discussion among the mathematics education research community on reasoning that promotes understand-
ing of important algebraic concepts at early levels (Bastable & Schifter, 1997; Confrey, 1995; Confrey & Smith,
1995; Harel & Confrey, 1994; Thompson, 1995~. The following discussion parallels an episode that occurred mid-
year in a heterogeneous sixth grade class.* The students were studying a unit on rational numbers, and the intent of
the problem was to develop understanding of and methods for comparing ratios. The problem illustrates how
students' reasoning can be used to develop understanding of algebraic ideas.
Description of the Problem: The school is hosting a lunch for the senior citizens in the area, and a class is asked
to test four different recipes for mixing punch made from sparkling water and cranberry juice. First, they have to
decide which of the four has the most cranberry juice, and then determine how many cups of juice and water are
needed to mix 120 cups of punch.
Recipe A: 2 cups cranberry juice
3 cups water
Recipe B: 4 cups cranberry juice
8 cups water
Recipe C: 3 cups cranberry juice
5 cups water
Recipe D: 1 cup cranberry juice
4 cups water
Discussion: The interaction that ensued was very lively. Students interpreted the question of which recipe has the
most cranberry juice in a variety of ways. Some looked at the absolute number of cups of juice in each recipe.
Others looked at the ratio of juice to water. Still others used the part-whole relationship of the number of cups of
juice to the total number of cups in the recipe. After comparing methods and discussing choices, the class split into
four groups to decide how to adapt each recipe to make enough punch for 120 cups. The groups reasoned as
follows.
*This vignetteisbasedon an episode from a sixth grade class taughtbyMaryBouck, who was piloting materials from the Connected Mathematics
Project (CMP), a National Science Foundation funded middle school mathematics curriculum project (Fey et al., 1995).
155
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180
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
The search for generalizations and methods of representation have many origins in the setting of number. As
students reason about calculations, they may observe numerical patterns or represent quantities pictorially or
geometrically; with advances in technology reasoning about number has broadened to include use of spreadsheets
~ ~ . ~., . ~ ~
and sets of Instructions for computers. Ultimately, reasoning about calculations produces a need for symbolic
notation, both for representing the process and for modeling relations. The focus of instruction should not, however,
be on the manipulation of symbols but rather on the conceptual understanding of the meaning of the symbolic
representation. Choosing number as a setting provides opportunities to illustrate how student understandings of any
of the themes can be developed singly or in conjunction with others.
The properties of the real number system allow us to rewrite expressions in an equivalent form, to represent a
situation, and then to work within that representation free from context. A focus on the distributive property as one
illustration of how number is incorporated into the framework highlights the structural aspect of algebra. Other
properties of number could serve the same purpose. Other organizing themes, however, are also present as students
develop and apply their understanding.
Understanding the distributive property allows students to
· think about different characteristics of the same situation:
· represent, generalize, and confirm conjectures;
· gain new information about a situation from an equivalent representation.
With the distributive property, students can think about the calculation embedded in a situation as a sum of
quantities, each of which has two or more factors, ab + ac, or they can think of the same situation as a product of
quantities, each of which may be a sum of two or more addends, am + c). Generalizing and gaining new information
from a representation are equally possible for many of the properties of number.
The following problem situations illustrate these different aspects of the distributive property. However,
because these outcomes are not disjoint, the problems are not separated according to specific goals. Instead, the
problems are grouped according to developmental level of students. They are intended to serve as a springboard for
thinking about the role of the distributive property in reasoning algebraically and for further discussion of the
experiences necessary for students as they construct a view of algebra from their work with number.
[saying the Foundation
Early childhood experiences that help children understand the setting of number include investigations into
patterns and regularities. Reasoning about the relationship between quantities and about efficient ways to link two
mathematical representations involves using properties such as the distributive property as a matter of course. Using
situations that have two equivalent interpretations will help build understanding of the distributive property.
Experiences with the distributive property can occur in many situations in the elementary grades: simplifying
computation 3 x lo + 3 x 9 = 3~10 + 9) or using an area model to show how to express multiplication of two-digit
numbers.
10 9
311111111111 L411 11113
10 + 9
C_ _ _ _ _ I _ _ _ _ _
3 _ _ _ _ _ l _ _ _ _ _ I I I r T l
_ _ _ _ _ I _ _ _ _ _ _ ~ T T I I I I
= 3 x 10 + 3 x 9
=3x (10+9)
A visual representation allows students to assimilate the equivalence of the two ways to think.
· Why is it important for students to have alternative ways to calculate?
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APPENDIX E
181
· What does this area model add to student understanding of the multiplication algorithm for whole,
fractional, and decimal numbers?
Building on the Foundation
As students in the middle grades become more familiar with expressing relationships symbolically, individual
students may create different but equivalent expressions, depending on how they reasoned about the problem. The
following example illustrates how the distributive property provides a link between different ways of thinking.
The Telephone Network Problem: There are many different houses in a particular
region. How many different telephone paths are necessary if each house is directly
connected to every other house?
Looking into the classroom: Ann and Juan decide to begin with some small examples
and see if they can find a pattern. Ann begins with 4 and then 5 houses and draws the
number of connections; Juan uses 6 and then 7. They make a table and discover they
can tell how many connections they need for 8 houses, then 9, then 10. They can see a
way to get to the NEXT entry in the table, but they cannot see how to get any general
rule.
Number Number
of Houses of Paths
4
5
6
8
6
10
15
21
NEXT = previous Number of Houses
previous Number of Paths
Another student writes, "If there are N houses, you won't connect a house to itself,
A ~ D
HE
ELF
1 1
A
TIC
ME
\G
- , etc.
so each will have to be connected to one less or n- 1 houses. This means there will be
ntn - 1 ) connections. For example, for 8 houses...
If house A is connected to house B. though, it is the same as if house B is connected to
house A, so my answer should be divided by 2. There will be ntn - 1~/2 connections."
Another pair of students makes a chart where each "1" represents a connection
between the houses and a "O" represents no connection.
A B C
o
o
D... n
A
B
C
1 ... 1
, ... .
O 1 1
n 1 1 1 1 ... 0
0 = no connection
~ = connection
The chart is a square, so if there are n houses, there will be n 2 connections in the chart.
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182
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
But the diagonal is not helpful because that shows a house is connected to itself. There
are n houses on the diagonal, so the number of connections is n 2 - n. The connections
on each half of the diagonal are the same (A to B is the same as B to A), so really there
are only in 2 - nils connections. Tina rewrote the rule as
1x nx (n-1~/2
She feels sure that this is equivalent because she recognizes she can apply the
distributive property to n 2_ n. She comments that the formula for the area of a triangle is
2 be and wonders if this relates to the matrix she and her partner made.
Questions to ask: Do all of the rules generate the same answers as Ann's NEXT rule?
Are the rules equivalent to each other? How do you know? One of the rules looks like the
formula for the area of a triangle, but where is the triangle? Would a graph facilitate
understanding of different reasoning processes?
Where Is the Algebra?
Searching for recursive patterns can provide a solution but is not efficient in many cases.
Relations are often difficult to describe in closed form. Those who try to find a more direct rule
may think very differently and use different symbolic representations. One link that shows these
different ways of thinking are mathematically equivalent is the distributive property. Tina's
contribution is somewhat different in nature. She has deliberately applied the distributive prop-
erty and linked the form of the result to something else she knows. Students can try to connect her
new representation to something concrete in the problem. Some representations of this problem
will have a triangular aspect because the numbers generated are the triangular numbers. Some
students may recognize these from Pascal's triangle. Students may reason from the sequence of
numbers or the ordered pairs, as follows,
Number of Number of Ordered
houses paths pairs
1 0 (1,0)
2 1 (2,1)
3 3 (3,3)
4 6 (4,6)
and think about what relation operating on the first would give the second. Others may reason
from a graph of the ordered pairs. They should see that the resulting graph is like the graph of
y = x2 but with a smaller rate of change and different vertex. This is an occasion to discuss which
of the two representations, y = x2 -x ory = Ox - 1), gives the most insight on estimating answers.
For very large numbers, the graphs of y = x2 and y = x2 - x are almost identical.
· How do the representations capture the reasoning process of the students? What is the
.
. .
advantage of having different representations?
How are the suggestions qualitatively different from each other? How can you systemati-
cally build on such thinking?
At the high school level, the problem can be extended to a counting problem where there are n
ways to make the first choice, n - 1 ways to make the second choice, and so ntn -1) ways to
choose two things. Dividing by 2 reduces the repetition. The triangular numbers that result, 1, 3,
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APPENDIX E
183
6, 10, ..., can be related to the binomial theorem, (x + yin as the coefficients of the terms in
expanded form, a powerful application of the distributive property. When students learned to
multiply two-digit numbers, a geometric model of the distributive property can provide them with
an understanding of the algonthm. For example, a student might write 3~27) = 3~20 + 7) = 60 + 21
= 81. When the computational process is extended to 27~42), the distributive property leads to
(20 + 71~40 + 2), or the product of two binomials. Initially, some students may say symbolically
that (x + y)2 = X2 + y2 but drawing an area model can convince them the square (x + y) has area x2
+ My + y2 and reinforces use of the distributive property.
x
x + Y
2 my
Y _ Vet
The geometric model reinforces the need for the cross products fly. The use of (x + y) as a factor
being distributed as an entity, (x + Ax + (x + ply, is an extension of the thinking established in
the early grades and helps explain symbolically why the sum of the two squares as an answer to
(X + y)2 iS insufficient. Students should be able to think and reason about the process and the
algorithm they use. Expanding a tnnom~al or cube enables students to think of reasonable ways to
extend their geometric model both in two and three dimensions and allows them to begin to
develop patterns that will later lead to the binomial expansion theorem
(X+y~n=xn+nxn-ly+ + yn
Physical models can help students understand a situation, but students should become more and
more comfortable with symbols as ways to represent situations, as well as ways to generalize
anthmetic. The follow-up questions deliberately focus on structural issues: When does this
property apply? What is the pattern in the terms? The comparative usefulness of the geometric
model or the symbolic representation highlights choices of representations.
· How much emphasis should be placed on structure at each level? What essential under-
standings do students need? How are these to be developed in contexts that motivate?
· How much exposure and over what length of time does it take for students to recognize and
be able to use the distributive property?
Within the setting of number and the organizing theme of structure, it becomes important to
reflect on certain patterns and behaviors in very general ways. Some situations seem to share
certain charactenstics. These common attributes allow you to generalize and treat in similar ways
systems that, on the surface, seem to have little in common. The distributive property is actually
a specific instance of a more general structural property. College- bound students might
investigate the following:
The Function Problem: Think about the functions you have studied. Which of these
functions have the property Pa + by = f(a) + f(b)?
Looking into the classroom: Students check the functions they recall. For example they
write "Is sinta + by = sin a + sin b?" Some check the validity of each conjecture by making
substitutions. Others discuss the nature of the function: "The value of sin x is never
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184
Summary
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
greater than 1, but sin a + sin b could be." The students continue to investigate other
functions.
Questions to ask: What do the functions you chose have in common? Would the
property, ha * by = f(`a) * f(`b), apply to operations other than adding for the functions you
identified?
Where Is the Algebra?
Students may conclude that Ma + b) =;ff~a) + fibs applies only to linear functions such as those
of the form y = kx. Some students find a geometric argument convincing:
f (a+b)
f (b) -
f (a)
7
I f(n)
f(~)
a b a+b
Those who have studied abstract algebra will recognize the general question: what operations
are preserved by a mapping from one algebraic structure to another? For the linear function,
y = kx with k fixed, each real number x is mapped onto the real number kx. Understanding what is
preserved under what conditions allows flexibility in analyzing situations; in some cases, it is
easier to analyze the range elements after the transformation, in others, to begin with the original
domain elements then make the transformation. This study of what is preserved under different
mappings leads to the concept of homomorphic functions, those for which Ma * b) = fiat * fibs.
The distributive property of multiplication over addition is a particular example of a homomor
. .
phi mapping.
What are the connections between the distributive property of multiplication over addition and
linearity?
While the setting for this example is number, the theme of structure comes to the forefront. The distributive
property also plays a role in operating on and understanding relationships in systems where symbols represent
objects other than number. Scalar multiplication distributes over addition of matrices; if the dimensions are aligned,
matrix multiplication distributes over matrix addition.
To use the structural properties of an algebraic system, students have to be introduced to these properties in
thoughtful and meaningful ways in the early grades, using problem settings that are familiar to both students and
teachers. The ability to generalize relationships is at the heart of algebra, and the structural nature of the system
allows relationships to be expressed in different forms, some more useful in particular situations than others. Each
different representation can add to the knowledge gained from one of the other representations. The use of symbols
and the properties governing the behavior of operations with those symbols allow students to represent a situation in
symbols, to manipulate those symbols temporarily free from the situational meaning to gain insights and informa-
tion about the situation, and then to return to the situation to make sense of the symbols.
· What is the interface between technology and structure?
· How much symbol manipulation is necessary to function effectively with symbolic representations?
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USING THE FRAMEWORK
Groups from the mathematics education community can use the ideas in this document to organize their
thinking and discussion of school algebra to develop a vision of algebra in a K-12 curriculum and as a means to
move toward that vision. Possible questions might include the following:
· What is the essential nature of each theme?
· How do themes help organize ideas?
· Are some themes more appropriate at different grade levels than at others?
· Is there a hierarchy to the themes?
· What contextual settings can be fruitful grounds for exploring algebraic concepts?
· How do the themes focus algebra in grades 9-12?
· What are examples of curricula that provide a coherent and balanced algebra curriculum in grades K-12?
· How can adequate articulation and continuity be built into a K-12 algebraic sequence?
· What should be done to help students develop depth in their algebraic understanding?
· How and when should algebraic understanding be assessed?
· What are some characteristics of algebraic reasoning at different developmental levels?
The demands of a fast-changing society and the presence of technology require a vision of school algebra
that is dynamic and fluid enough to keep pace with future needs of society, yet retains the essential aspects of
algebra that have made it so significant in the history of mathematics. The workforce needs citizens who can adapt
to new technologies, identify problems, reason about problems, and communicate their findings using symbols,
graphs, tables, pictures, and words. Studies from other countries demonstrate that students can learn to reason
algebraically much earlier than grade 9 and that all students can do so. Developing and implementing a coherent
and balanced algebra curriculum for grades K-12 requires a complete rethinking of the entire mathematics
curriculum, a task that is already underway by some involved in curriculum development.
The success of implementing any vision of school algebra ultimately lies in creating conditions, policies,
assessment, curriculum materials, and support that enable teachers to provide the kind of algebra experience that is
essential for all students. Discussions and policies on "who takes algebra when" must have the full participation of
teachers who will be responsible for enacting the changes. The algebra that is called for in this document is quite
different from the algebra that most teachers have been taught or have been teaching. A framework such as this one
can support teachers in a critical and reasoned review and adaptation of new curriculum materials purporting to
exemplify a rethinking of algebra. Long-term professional development activities and preservice programs must be
examined in light of a framework and the issues raised while equal efforts must be made to help "lay people"
185
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186
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
understand the reasons for and nature of a different view of algebra. Reconfiguring algebra as a K-12 endeavor will
take time, commitment, and deep thinking on the part of the entire mathematics education community to make a
successful algebraic experience a reality for all students.
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BIBLIOGRAPHY
This bibliography will be a useful resource for anyone interested in the nature and role of algebra in the K-12 curriculum.
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(Ed.), Employing Children's Natural Powers to Build Algebraic Reasoning in the Content of Elementary Mathematics.
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THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
Coxford, A.F., & Shulte, A.P. (Eds.) (1988~. Ideas of Algebra, K-12. Reston, VA: NCTM.
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Representative terms from entire chapter:
contextual settings
