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Presentations on Day Two
Capturing Patterns and Functions: Variables and Joint Variation (G. Lappan)
Functions and Relations: A Unifying Theme for School Algebra in Gracles 9-12 (C. Hirsch)
Micic3le School Algebra from a Modeling Perspective (G. Kleiman)
Why Mocleling Matters by. GoclboIcl)
Modeling: Changing the Mathematics Experience in Postseconciary Classrooms (R. Dance)
Algebraic Structure in the Mathematics of Elementary-Schoo} Children (C. Tierney)
Structure in School Algebra (Micic3le School) (M. van Recuwijk)
The Role of Algebraic Structure in the Mathematics Curriculum of Gracles ~1-14 (G. Foley)
Language and Representation in Algebra: A View from the Micic3le (R. BilIstein)
Teaching Algebra: Lessons Learned by a Curriculum Developer (D. Resek)
The Nature and Role of Algebra: Language and Representation (D. Hughes Hallett)
ss
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Cap Aiding Patterns and Functions:
Va~ialoles and Joint Variation
Glenda Lappan
Michigan State University
East Lansing, Michigan
In ordinary English context helps distinguish among possible meanings of common words. As a representation
of ideas, a word stated free of some meaningful context does not communicate very well. For example, the
definition of a word in the dictionary usually includes several possible meanings. The word used in context helps
the listener or reader to differentiate among the various meanings and understand what the speaker or writer intends.
In mathematics, we face the same dilemmas. Many mathematical words have different or at least different shades
of meaning. In learning to understand both how to communicate in and how to decipher the language of
mathematics, students have to determine meaning from contexts of use. Two of the key concepts in developing a
deep understanding of functions are variables and joint variation. Variable is one of those mathematical words that
has many meanings that must be determined from context of use.
In the new curricula that have been developed as a response to the NCTM's Curriculum and Evaluation
Standards for School Mathematics and Professional Teaching Standards for School Mathematics, students learn
mathematics through engagement with problems embedded in interesting contexts. This means that students have
to learn to interpret different kinds of contexts. Students have to negotiate the "story" of the context whether the
"story" is from the real world, the world of whimsy, or the world of mathematics to find ways to mathematicize
the situation, presented in the story to manipulate the representations to find solutions, to interpret the solution in the
original context, and to look for ways to generalize the solution to a whole class of problems. In addition, students
have to interpret the context of the mathematization of the situation. Is the meaning of variable that of a place holder
for an unknown? Or is the meaning of variable that of a domain of possible values for one of the changing
phenomena in the context? Or is variable used in yet another way in the mathematization?
Joint variation of variables is the heart of understanding patterns and functions. As students grow in their
ability to derive meaning for variables in contexts, they encounter variables that are changing in relation to each
other. This pattern of change, or joint variation, becomes the object of study as certain kinds of change produce
recognizable tables, graphs, and symbolic expressions. At the middle-school level, these important families of
predictable joint variation are linear, quadratic, and exponential functions. In order to see how students can grow in
their understanding of variable and joint variation, let us turn to a series of examples of significant stages of
development in one of the reform middle-school curricula, the Connected Mathematics Project.
In the first situation given on the next page, students are challenged to determine variables of interest and to find
ways to represent how these variables change in relation to each other over the day. The story line is that six college
students are in the process of planning a five-day bicycle tour from Philadelphia to Williamsburg in the summer as
a money-making business. They are exploring the route of the tour while gathering data on each day's trip. Day
five encompasses a trip from Chincoteague Island to Norfolk, Virginia. The data are presented as word notes on the
trip.
57
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58
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
Malcolm and Sarah's notes:
We started at 8:30 A.M. and rode into a strong wind until our midmorning break.
About midmorning, the wind shifted to our backs.
· We stopped for lunch at a barbecue stand and rested for about an hour. By this time, we had traveled about halfway
to Norfolk.
· Around 2:00 P.M., we stopped for a brief swim in the ocean.
· At around 3:30 P.M., we had reached the north end of the Chesapeake Bay Bridge and Tunnel. We stopped for a few
minutes to watch the ships passing by. Since bikes are prohibited on the bridge, we put our bikes in the van and drove
across the bridge.
· We took 7 1/2 hours to complete the day' s 80-mile trip.
In this problem, the variables of time and distance traveled change in relation to each other but not in a
mathematically predictable way. The elements, terrain, creature comforts, and rules of bridge use change the rates
at which the distance is changing as time passes. The graph may be linear in parts, curved in parts, and constant in
parts.
In the second situation, below, students are challenged to solve a problem that is stated in an open-ended way.
To solve the problem, students have to find ways to represent and think about what the variables are and about two
pairs of variables that are changing at different rates relative to each other.
Bonne challenged his older brother, Amel, to a walking race. Amel and Bonne had figured their walking rates. Bonne
walks at 1 meter per second, and Amel walks at 2.5 meters per second. Amel gives his brother a 45 meter head start. Amel
knows his brother would enjoy winning the race, but he does not want to make the race so short that it is obvious his
brother will win. What would be a good distance for the race if Amel wants his brother to win but wants it to appear to
be a close race?
Here students have to decide how to represent the progress of each brother so that they can determine what a
good length would be for the race. Since the rate of change between time and distance for each brother is constant,
each brother's time versus distance relationship is linear. The variables of time and distance in this situation change
in a predictable way. One can predict the distance traveled for any number of seconds, which is a very different
pattern of joint variation from the first situation.
Students can graph both functions on the same axes and see that the two lines cross. They have to figure out the
significance of the point at which the lines cross and how this relates to determining a good distance for the race. Of
course, students may reason from tables of data for each of the brothers. But, in any case, they have to identify the
variables and make sense of which variable depends on the other and how the two pairs of variables relate to each
other. Here the expression 2.5t can represent Amel's progress over t seconds. The variable stands for a whole
domain of possible times. Students could write the functions = 2.5t or d = 2.5t to show the independent variable,
time, and the dependent variable, distance. They might also write 2.5t = 45 + It to show the equation that must be
solved to find the point of intersection. Here the variable is standing for an unknown value, the value of t that makes
the equation true.
In the third situation, below, students again have to deal with variables and the pattern of change between
related variables, but the nature of the change or variation is different from either of the first two situations.
U.S. Malls Incorporated wants to build a new shopping center. The mall developer has bought all of the land on the
proposed site except for one square lot that measures 125 meters on each side. The family that owns the land is reluctant
to sell the lot. In exchange for the lot, the developer has offered to give the family a rectangular lot of land that is 100
meters longer on one side and 100 meters shorter on another side than the square lot. Is this a fair trade?
Here students may talk about the problem at a general level or solve the particular case by comparing areas.
However, the question of whether the results are true in general for a beginning square of any size remains. Here the
problem can be restated with smaller numbers as follows:
What happens if you own a square piece of land that is n meters by n meters and you are offered a piece of land that is
2 meters shorter on one side and 2 meters longer on the other? How does the area of the new lot compare to the original?
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PRESENTATIONS ONDAY TWO
Increase:
59
Here is a table that many groups of students make to record what is happening as the original square sides
Original Square
Length
3 m
4m
5 m
n
New Rectangle
Area
9 m2
16 m2
25 m2
.
n2
Length
5 m
6 m
7 m
.
n+2
Width
1 m
2m
3 m
Change in Area
Area
5 m2
12m2
21 m2
.
(n+2)(n-2) n2- (n+2)(n-2) = 4
4m2
4m2
4m2
The students notice that the change in areas from the square to the rectangle seems to be constant, 4 m. Since
the students also develop a symbolic expression that tells what they do each time to find the change in area between
the square and the rectangle, n2- (n + 2~(n -2), the question of equivalence of expressions naturally arises. Why
does n2- (n + 2~(n - 2) = 4? Students have various ways of looking at this equivalence. They can graph the data in
different ways to observe the behavior of the graphs. There are different graphs that can be made from the data and
that show different kinds of change. The functions, A = n2, A = (n + 2~(n - 2), where A is area and n is the length of
the side of the square, show quadratic growth, and D = n2- (n + 2~(n - 2), where D is the difference between the
areas, gives a constant value and hence a horizontal graph. But the students also are motivated to examine different
ways to transform the symbolic expression. What is another way to express (n + 2~(n -2~? By looking at a rectangle
that is n + 2 on one edge and n -2 on the other, students can use the distributive property to see that this expression
is equivalent to n2 _ 4.
n 2
n-2
n2 _ 2n
2n-4
Therefore, the whole expression n2 _ (n2 _ 4) is always equal to 4.
The three problems I have presented here would be appropriate for different stages of a student' s development
of algebraic skill, but, nonetheless, all three serve to illustrate the centrality of variable and joint variation in
understanding and using function to make sense of situations.
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Functions and Relations:
A Unifying Theme for School Al,gelora in Grades 9-12
Christian Hirsch
Western Michigan University
Kalamazoo, Michigan
One of the most important transitions from middle- to high-school mathematics is the emergence of algebraic
concepts and methods for studying general numerical patterns, quantitative variables and relationships among those
variables, and important patterns of change in those relationships. The mathematical ideas that are central to that
kind of quantitative reasoning are variables, functions, and (to a somewhat lesser extent) relations and their
representations in numerical, graphic, symbolic, and verbal forms. Organizing school algebra around the study of
the major families of elementary functions (linear, exponential, quadratic and polynomial, rational, and periodic)
offers the opportunity to bring greater coherence to the study of algebra. Situating that study in explorations of
contextual settings can provide more meaning to algebra and can provide a broader population greater access to
algebraic thinking.
GOALS AND APPROACHES
From a functions and relations perspective, the continued study of algebra at the high-school level should
enable all students to develop the ability to examine data or quantitative conditions; to choose appropriate algebraic
models that fit patterns in the data or conditions; to write equations, inequalities, and other calculations to match
important questions in the given situations; and to use a variety of strategies to answer the questions. Achievement
of these goals would suggest that the study of algebra be rooted in the modeling of interesting data and phenomena
in the physical, biological, and social sciences, in economics, and in students' daily lives. Through investigations of
rich problem situations in which quantitative relations are modeled well by the type of function under study,
students can develop important ideas of recognizing underlying mathematical features of problems in data patterns
and expressing those relations in suitable algebraic forms.
Answering questions about the situations being modeled leads to questions such as the following, some of
which are at the heart of a traditional algebra program. For a given function modeling rule fix), find
· fix) for x = a;
· x so that f~x)=a;
· x so that maximum or minimum values offer) occur;
· the rate of change inf near x = a;
· the average value off over the interval (a,b).
Early work by Fey and his colleagues at the University of Maryland (cf. Fey and Good, 1985; Fey, Held, et al.,
1995) using computer utilities demonstrated the promise of such a modeling and function-based approach. The
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62
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
emergence of hand-held graphics calculators puts such an approach in reach of all students and teachers of high-
school mathematics.
Graphics calculator technology provides powerful new visual, numenc, and even symbolic approaches to
answering questions such as those given on the previous page. The technology also facilitates exploration of more
general properties of each family of functions and of all functions collectively; these can then be formally organized
and verified at a later point in the curriculum.
A UNIFYING CONCEPT
Functions are a central and unifying concept of school algebra and, more generally, of school mathematics (c.f.
Coxford et al., 1996~. For example, symbolic expressions for function rules provide compact representations for
patterns revealed by data analysis. Fundamental concepts of statistics, such as transformations of un~vanate or
bivanate data, and of probability, such as probability distnbutions, are expressed and understood through the idea of
function. The function concept can be generalized naturally to mappings such as (x, y) ~ (3x, 3y) or the following,
x O -!
Y 1 0
'lye
which describe transformations of the plane. In discrete mathematics, early experiences with recursive descriptions
of linear change (NEXT = NOW + b) and exponential change (NEXT = NOW x b) lead naturally to more general
modeling with difference equations. Again, matrices are linked with transformations, and matrix methods are
dependent on syntax and inference rules of algebraic symbolism. Finally, the mathematics of continuous change or
calculus is fundamentally the study of the behavior of functions, including rates of change and accumulation.
SUMMARY
Organizing school algebra around functions and their use in mathematical modeling can provide a meaningful
and broadly useful path to algebra for all students. Algebra as a language and means of representation is a natural
by-product of this approach. Patterns that emerge through modeling with functions and studying families of
functions can motivate at a later stage a study of the structure of algebra through deductive methods. Finally, the
theme of functions and relations offers a way to provide a more unified approach to the high-school mathematics
curriculum.
REFERENCES
Coxford, Arthur F., James T. Fey, Christian R. Hirsch, Harold L. Schoen, Gail Burrill, Eric W. Hart, Ann E. Watkins, Mary Jo
Messenger, and Beth Ritsema. (1996.) Contemporary Mathematics in Context: A Unified Approach. Chicago: Everyday
Learning Corporation.
Fey, James T., and Richard A. Good. (1985.) "Rethinking the Sequence and Priorities of High School Mathematics Curricula."
In The Secondary School Mathematics Curriculum, 1985 Yearbook of the National Council of Teachers of Mathematics,
edited by Christian R. Hirsch and Marilyn J. Zweng, pp. 43-52. Reston, VA: NCTM.
Fey, James T., and M. Kathleen Held, with Richard A. Good, Charlene Sheets, Glendon W. Blume, and Rose Mary Zbiek.
(1995.) Concepts in algebra: A TechnologicalApproach. Dedham, MA: Janson Publications.
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Middle School Algebra from a Modeling Perspective
Glenn M. Kleiman
Education Development Center, Inc.
Newton, Massachusetts
To begin, we place algebra within the very general framework shown in Diagram 1 below.
( Situation ')
Extracting and
Representing
-
-
~/
~ Mathematical
<: Representations
Interpreting
and Applying
- ` Mathematical ,'
J
Diagram 1
Mathematical ~`
Findings
This framework emphasizes that mathematics is more than working with mathematical symbols and tools. It
also includes (a) extracting information from a situation and representing that information mathematically the
process of "mathematizing"; and (b) interpreting and applying mathematical findings to have meaning within
specific situations.
This same framework can be applied to any area of mathematics. Algebra is defined by the representations,
tools, techniques it provides, and the types of problem situations it enables one to address. Some specifics, focusing
on algebra grades 6-8, are given in this paper. These are expansions of each of the three corners of Diagram 1 above.
The organizing theme of this framework is modeling. The other themes are incorporated within modeling. The
language and representation theme is reflected in the processes of extracting and representing bringing the
original situation into a mathematical form and interpreting and applying translating back from a mathematical
63
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64
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
form to the situation. As we will see below in Diagram 3, the Unctions and relations and the structure themes are
reflected in the mathematical analyses and mathematicalfindings components of the diagram.
First, let us expand upon possible types of situations in Diagram 2 below.
Pictorial or
_ Physical
Physical Arrangement Mathematical
Experiment . Prohl~m
1 ' ~
I Real World I Ad\ \
| Data | ~\\
\\
/~
Diagram 2
~ .. a.... 1
id I Game or |
/ ,~| Puzzle |
it_ ~~
The middle-school curriculum should include a wide variety of types of situations. An appropriate situation
has the following characteristics: it is engaging for many middle school students; it can lead to significant
mathematical explorations at an appropriate level of complexity; and it is manageable within the classroom.
Next, let' s expand upon the mathematical representations for the middle-school algebra curriculum. The link
across the representations in Diagram 3 below is a reminder that understanding the relationships among these
representations is also important.
~ \
/ My Mathematical ~~\
RepresentationsI F: J
Let
Pictures Tab
Diagrams |
~ _
;{ inequalities
1
Scatter Plot | | Linear
1 1 1 1 1 1
Line ~ ~Quadratic
Diagram 3
- ~
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PRESENTATIONS ONDAY TWO
65
Next, in Diagram 4 below are some of the types of mathematical findings we emphasize within the middle-
school algebra strand
Mathematical
Representations
Mathematical >
Analyses
-
Mathematical
Findings
Identify I Solve for I I Test If/Then
I Patterns I ~ Unknowns ~Conjectures
Diagram 4
Identify Functional
Relationships
Students should be able to use these findings to do such things as (1) use patterns to predict new cases in the
situation; (2) interpret solutions of unknowns in terms of filling in missing information about the situation; (3) make
if-then statements about the situation; and (4) use knowledge of functions to predict what will happen when one
thing changes in the situation.
To get to these four types of findings, students need a repertoire of mathematical tools and understandings.
Diagram 5 on the next page shows some categories of patterns students should understand; knowledge and
techniques students will need to solve equations and inequalities; some tools for testing conjectures; and types of
functions that students should become able to recognize and apply to understanding situations. All of these can be
introduced in the middle-school curriculum, in many cases at an informal, context-based level, that forms a
conceptual base for the more formal and abstract understandings that will develop in later grades.
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THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
The fact that t3/t2 equals t is algebra, that t3 for large t is much larger than t2 is calculus, and that 1000/100 = 10
is arithmetic.
When talking about school algebra, I mean something other than mathematical algebra. Algebra in the context
of school algebra is a coherent integration of elements from the three domains: arithmetic, algebra, and calculus.
ALGEBRA AT THE MIDDLE GRADES
Over the past five years, the Freudenthal Institute has been involved in a curriculum development project in
which a complete, new mathematics curriculum named MiC has been developed for American students between 10
and 14 years-of-age. One of the content strands in this curriculum is "algebra," and it contains a collection of topics
from different mathematical domains. In the algebraic structure session, I will use some examples to illustrate the
philosophy and approach in this curriculum towards algebra. In this paper, I have restricted myself to outlining the
philosophy and approach in general terms.
ALGEBRA IN MATHEMATICS IN CONTEXT
The algebra strand in MiC emphasizes the study of relationships between variables, the study of joint variation.
Students learn how to describe these relationships with a variety of representations and how to connect these
representations. The goal is not for students merely to learn the structure and symbols of algebra but for them to use
algebra as a tool to solve problems that arise in the real world. For students to use algebra effectively, they must be
able to make reasonable choices about what algebraic representation, if any, to use in solving a problem.
MiC APPROACH TO ALGEBRA
The MiC curriculum especially the algebra strand is characterized by progressive formalization. In other
words, students rely heavily, first, on their intuitive understanding of a concept, then they work with the concept
more abstractly. The realistic problem contexts support this progression from informal, intuitive understanding to a
more formal, abstract understanding. Students can move back and forth from informal to formal depending on the
concepts and the problem contexts. Their ability to understand and to use algebra formally develops gradually over
the four-year curriculum. By the end of the four-year curriculum, students have developed an understanding of
algebraic concepts and are able to work quite formally with algebraic symbols and expressions. Algebra in MiC
lays a solid groundwork for mathematics at the high-school level.
EXAMPLE
Even and odd numbers can be visualized by dot patterns. Dot patterns also can be used to visualize and to
investigate more complex (number) patterns. Symbols, expressions, and formulas (recursive and direct) can be
used to describe the patterns. The formulas themselves can then become an object of study that lead to re-inventing
such mathematical properties as distributivity and factorization. For example, when students investigate the
structure of rectangular and triangular numbers, they can use visual representations to support finding appropriate
algebraic expressions. In the algebraic structure session, I will illustrate this example with problems from the
curriculum materials.
NO ALGEBRAIC STRUCTURES BUT STRUCTURE IN SCHOOL ALGEBRA
Algebra at the middle-grade levels builds on students' intuitive and informal knowledge of arithmetic, of
symbols, patterns, regularities, processes, change, and so on as developed in the early grades. Algebra in the
middle grades does not need to lead to a complete and formal understanding of (the parts of) algebra. It is not the end
of students' education. High school follows, and that is the place to formalize the concepts.
Algebraic structure as described by Greg Foley, for example is "number theory." We should be careful
about making topics from number theory the focus of the mathematics curriculum, especially at the middle grades.
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PRESENTATIONS ONDAY TWO
85
Factoring, divisibility rules, prime factorization, manipulating symbols and expressions, and other such topics have
been the focus of the algebra curriculum, and students have not then had an opportunity to develop a meaningful
understanding of the underlying concepts. The focus should be a long-term learning strand in which students can
re-invent the algebra themselves, with the result being a mathematical system that is meaningful to students. The
MiC algebra strand serves as an example of how this goal can be achieved.
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The Role of Al,~ebraic St~nctnre in the
Mathematics Cliche of Grades I 1-14
Gregory D. Foley
Sam Houston State University
Huntsville, Texas
MATHEMATICAL THEORY VERSUS RELEVANCE
The so-called "new math" movement of the 1960s brought such logic-based organizing themes as set theory
and algebraic structure to the fore of school mathematics in the United States. Mathematical reasoning, axiomatic
structure, and within-mathematics connections were driving forces of a reform movement motivated by American-
Soviet competition and led by research mathematicians. The goal of preparing a cadre of highly capable engineers
and scientists caused us to focus on the most able students. By contrast, the National Council of Teachers of
Mathematics (NCTM) Standards-inspired school mathematics reform of the l990s has been driven by calls for
relevance realistic applications, modeling, genuine data, and mathematics in context and by a powerful collec-
tion of emerging instructional technologies. Mathematical communication, problem solving, and cross-disciplinary
connections drive the current reform. The need for a generally well-educated population to remain competitive in a
global economy has led us to conclude that "everybody counts" and that we need algebra for everyone. In the
1960s, we sought to motivate the mathematics; in the l990s, we seek to motivate the students.
This, of course, is an oversimplification. A careful reader of the NCTM Curriculum and Evaluation Standards
(1989) will notice an overarching theme of "Mathematics as Reasoning" and will see that the document says high-
school students should learn about matrices, abstraction and symbolism, finite graphs, sequences, recurrence
relations, algorithms, and mathematical systems and their structural characteristics, and that, in addition, college-
intending students should gain facility with formal proof, algebraic transformations, operations on functions, linear
programming, difference equations, the complex number system, elementary theorems of groups and fields, and the
nature and purpose of axiomatic systems. The American Mathematical Association of Two-Year Colleges'
(AMATYC) Crossroads in Mathematics (1995) contains similar calls for the content of introductory college
mathematics. Ideally, there should be a balance between solid mathematics and relevance to the student and
societal needs. The Standards documents for Grades 11-14 recognize this.
TECHNOLOGY AS CURRICULAR CATALYST
The influence of technology should not be downplayed. Technology is affecting the mathematics curriculum
in several ways. Compared to the past, current technology gives students access to relatively advanced mathemat
87
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88
THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
ical concepts and allows them to explore, descnbe, and display data with relative ease. Modern hand-held
computers, such as the TI-92, have powerful features that allow students
· to operate with integers, rational numbers, real numbers, or complex numbers;
· to define, algebraically manipulate, graph, and tabulate functions of one vanahle. narametnc equations
sequences, polar equations, and functions of two vanables;
· to solve equations, find zeros of functions, and factor and expand expressions;
· to define, algebraically manipulate, graph, and tabulate sequences, polar equations, and real-valued
functions of two vanables;
· to operate on lists, vectors, and matrices whose entries are integers, rational numbers, real numbers, or
complex numbers;
· to organize, display, process, and analyze data;
· to wnte, store, edit, and execute programs; and
· to construct and explore geometric objects dynamically and interactively.
, ~
,
In addition, modern technology and the related emergence of computer science make the knowledge of discrete
mathematical structures more important. Technology has indirectly increased the use of statistics throughout
society. It is no wonder, then, that the University of Chicago School Mathematics Project has a two-year sequence
of Functions, Statistics, and Trigonometry (Rubenstein et al., 1992) followed by Precalculus and Discrete
Mathematics (Peressini et al., 1992~. There is simply more appropriate content after second-year algebra in
preparation for postsecondary work in statistics, discrete mathematics, calculus, and linear algebra than in past
decades. Technology makes this both possible and desirable.
WHAT IS THE ROLE OF ALGEBRAIC STRUCTURE?
Teachers of mathematics in Grades 11-14 must understand algebraic groups, nngs, fields, and the associated
theory. They need, for example, to recognize the importance of the complex numbers being a field and the
significance of the fact that matrix multiplication is noncommutative and that matrices have zero divisors. They
should see a loganthm~c function as an isomorphism between groups and recognize geometric transformations as
forming a group under composition. Furthermore, in keeping with the NCTM curriculum standards for college-
intending students, high-school teachers need to be able to convey such understanding to their upper level students.
This should be reinforced, amplified, and extended in lower division postsecondary mathematics courses, especial-
ly those in discrete mathematical structures and linear algebra.
We must, however, be careful not to make algebraic structure the overriding focus of mathematics in Grades
11-13, except, possibly, for the most gifted and talented students. On the other hand, it is essential that, in Grades
14-16, students acquire a clear vision of the "big picture" provided by a structural understanding of algebra. While
we help students acquire this vision, we continually should call their attention to the numerous specific examples of
groups, nngs, and so on, as they learn the common structure and associated theory. There are abstract algebra
textbooks, such as Fraleigh's (1989), that do a good job of this.
Abstract algebraic structures can serve as important organizing tools for the mathematics curnculum, but we
should not fall into the trap of creating a new "new math."
REFERENCES
American Mathematical Association of Two-Year Colleges. (1995.) Crossroads in Mathematics: Standards for Introductory
College Mathematics Before Calculus. Memphis, TN: Author.
Fraleigh, J. B. (1989.) A First Course in Abstract Algebra, 4th Ed. Reading, MA: Addison-Wesley.
National Council of Teachers of Mathematics. ( 1989.) Curriculum and Evaluation Standards for School Mathematics. Reston,
VA: Author.
Peressini, A. L., et al. (1992.) Precalculus and Discrete Mathematics (University of Chicago School Mathematics Project).
Glenview, IL: Scott-Foresman.
Rubenstein, R. N., et al. (1992.) Functions, Statistics, and Trigonometry (University of Chicago School Mathematics Project).
Glenview, IL: Scott-Foresman.
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Language and Representation in Algebra:
A View from the Middle
Rick Billstein
Director, STEM Project
University of Montana
Missoula, Montana
Algebra can be thought of as a language, and students learn language best in their early years. We should begin
teaching informal algebraic concepts in the elementary grades and continue to develop the concepts throughout the
middle-school years. Algebra too often is taught as rules and tricks without an understanding of the concepts. Then
the jump to the formal level is often made too quickly for the concepts to be mastered. Topics should be developed
slowly and informally without symbol manipulation as the primary goal. The jump into symbol manipulation
should come only after students recognize the need for it.
Algebra has been described as a way of thinking about and representing many situations. Unfortunately, many
textbooks confine algebra to solving equations and manipulating symbols. Other representations, such as graphs,
tables, patterns, diagrams, and other visual displays, should be used as appropriate. Visual representations are
powerful because they help abstract mathematical ideas to become concrete. Since different representations may
provide new or fresh insights about a problem, each representation is important and plays a role in the learning of
algebra. There should be many opportunities for students to make transitions between the various representations.
As students mature mathematically, they learn which types of representations are most useful for which kinds of
problems. Students need to describe various representations in their own words. After a representation has been
used, it is important to discuss it in terms of the original context. Interaction between teachers and students is
important in development of language and representation skills.
At the middle-school level, students might be asked to translate between words, tables, graphs, and equations.
Given any one of these representations, they could be asked to determine any of the other three. Having students
work in groups and share representations makes them aware that different representations can be equivalent yet look
quite different. This is a powerful experience in middle school and will pay huge benefits at the high-school level.
Instead of always translating from words to representations, as in the traditional curriculum, we now ask students to
translate representations to words.
With the use of technology on the rise, new understandings of symbol manipulation are needed to model
situations that can be entered into a computer. For example, spreadsheets can be used to analyze complex numerical
data from a problem situation. Algebra becomes important because it is the language used to communicate with the
technology. Spreadsheet formulas are but one example of a form of algebra. Technology allows students to
experiment, to investigate patterns, and to make and test conjectures. Technology allows us to go where we could
not go before because the mathematics became too "messy." Students need experiences making representations
with and without technology.
Students should be involved in "doing" mathematics at the middle-school level. It is important that they
investigate problems and be involved in hands-on activities. "Doing" mathematics provides students with
opportunities to communicate about algebra. There is little communication in a typical algebra textbook. The
language that students use will develop as they become more mathematically mature. Curriculum materials not
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THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
only must contain good problems, but also the problems must be structured in such a way that solving them will help
students achieve the desired learning outcomes.
New materials are now becoming available in which algebraic ideas are taught throughout the curriculum.
Teachers must be made aware of these materials and support must be provided to train teachers to use these
materials. Teachers must understand that when we say we want to include algebra in the middle school, we are not
talking about the algebra that they had when they were in school.
Hugh Burkhardt of the Shell Center for Mathematics Education in England describes algebra as "inherently
slippery" and has said that "having separate algebra courses is one of the United States' great self-inflicted wounds"
(1997~. Most National Science Foundation (NSF)-funded middle-school curriculum developers have struggled
with this and are constantly asked about the role of algebra in the materials and how their curricula fit with an
Algebra I course. The "Six Through Eight Mathematics" (STEM) project response has always been that algebra
should not be a separate course taught at a particular grade level but, rather, that it should be a strand taught within
the mathematics curriculum at every grade level. The teaching of algebra should be integrated with the teaching of
other mathematical strands, such as statistics or geometry.
The traditional Algebra I course should not be a required eighth-grade course as it is in many schools because
this means the sixth, seventh, and eighth-grade curriculum must be covered in only two years. Two years is not
enough time to develop adequately topics in probability, statistics, measurement, discrete math, number theory, and
geometry. Many of the negative feelings that develop towards mathematics as a result of an "algebra course" might
be eliminated if the algebra were integrated into the curriculum as a strand. Students would no longer remember
algebra as a course in manipulating expressions and solving symbolic equations. Student experiences in middle-
school mathematics courses might then actually prepare them for and encourage them to take additional
mathematics courses, especially if those courses were taught in the reform mode of the new NSF high-school
projects.
STEM has found that one way to teach algebra effectively is to make it useful to students. Real contexts that are
meaningful to students play a major role in algebraic learning. Real contexts do not mean that all problems have to
come from students' everyday lives but, rather, that problems must make sense to students. Algebraic abstraction is
motivated by the need to represent the patterns found in the context. Algebraic thinking is more important than
algebraic manipulating. To develop algebraic thinking, we need to include informal work with algebraic concepts
in the middle school and not move too quickly to the abstract level. For example, being able to set up graphs or
tables in various problem settings brings mathematical power and understanding to students. Experiences with
graphs should include a detailed plotting of points to determine a graph as well as experiences with the overall
shapes of graphs based on the information in the problem. If students are given a graph, they should be able to write
a story about it. Having students communicate about mathematics is a worthwhile goal in the new middle-school
curricula.
Don Chambers wrote, "Algebra for all is the right goal at the right time. We just need to find the right algebra"
, ~
(1994~. The NSF-funded middle school projects are taking us closer to finding the right algebra.
REFERENCES
Burkhardt, H. (1997.) Personal Conversation at the National Council of Teachers of Mathematics National Meeting in
Minneapolis, MN.
Chambers, D. (1994.) "The Right Algebra for All." Educational Leadership, 51, 85-86.
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Teaching Algebra:
Lessons [earned by a C?~ric?~?~m Developer
Diane Resek
San Francisco State University
San Francisco, California
In this symposium, I am approaching the issue of teaching algebra from the perspective of a curriculum
developer. I am one of the developers of the National Science Foundation's (NSF's) secondary curriculum,
"Interactive Mathematics Program" (IMP). The original design for the program was based on the experiences of the
developers in past curriculum projects and in their own teaching. In this new project, we have had experience
teaching algebra to students in elementary school, high school, and college. The original IMP curriculum has been
rewritten three times shaped by the authors' observations of the curriculum as it was taught in different
classrooms, by the comments and suggestions of teachers, and by student work.
What follows are some statements about what I now believe about the teaching of algebra.
INTUITION SHOULD COME FIRST
People often ask for evidence that the new curriculum projects "work." There is some evidence of this, but
there are mountains and mountains of evidence that the traditional methods don't "work." Now, exactly why the
traditional curriculum does not work is open to question. My personal belief is that the chief culprit is the teaching
of manipulative skills in a way that does not allow adequate intuition to come into play about what the symbols
mean and why the manipulation is valid and useful.
It is not that we do not know how to teach manipulation in meaningful ways. Many projects have shown us how
to do this for years. One way to develop intuition about manipulating equations symbolically is by tapping into both
the students' familiarity with the fact that equations are statements about functions and the students' comfort in
associating the symbolic form of functions with other representations of functions, graphical or numerical. Student
familiarity and comfort must be developed over time.
It is my hope that much of the work in elementary-school and middle-school algebra will be on developing
student comfort in moving between representations of functions. In general, we can decide what we need to teach
at various levels by looking at what is difficult to teach later on. Traditionally, we have looked at what skills we
thought were needed for success at one level and then taught those skills at the lower level. I am suggesting that we
look at what skills or understandings are difficult for students at one level and try to develop intuition at lower levels
that might serve as a basis for those skills or understandings.
UNDERSTANDING DOES NOT COME IN DISCRETE PACKAGES
One difficulty with building students' use of intuition is that this requires time often several years. Teaching
one concept over time conflicts with the traditional idea of organizing teaching around subject matter. Traditional-
ly, one studies a chapter on linear equations at one time and that subject is then checked off. However, if we want
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THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
students to understand linear functions in different representations, this must be worked on over several years.
Striving for long-term exposure to subject matter creates a difficult bookkeeping problem. It becomes difficult to
check off the skills that students have. As the public pushes us for accountability, they push for a neat and tidy
assessment system. Unfortunately, that kind of assessment system does not match the way students learn.
At any level first grade, high school, or college students do not study a deep topic and suddenly "get it."
Understanding comes gradually. It develops over time. Anyone who has asked students to write about a topic
knows this. Reading what they have written, we can see that there are things the student seems to understand and
other things he or she has not yet come to terms with. Rarely do we get a picture of perfect understanding. Our
curriculum must be structured so that students can work on tasks at different levels and so that everyone in a class
will grow.
I am not saying that teachers should not be accountable for students' learning or that students should not
eventually master key ideas. I am saying that we need to take into account how students learn and that this is
gradual. We must not let the difficulty of assessing understanding sway us from trying to teach effectively.
WE CAN LEAD STUDENTS TO WATER, BUT ...
Once we have decided what we want to teach and how we want to teach it, we have to wrap it in the right paper.
This is not because students are lazy or do not have good taste. It is because their minds cannot actively work with
material if they have no way to relate to it. Contexts and relationships to other subject matter can provide students
with a door to approach new mathematics.
This is not to say we should never teach mathematics without a "real life" context. I am saying that we have to
introduce the mathematics in a "real life" situation that students can relate to or in the context of other mathematics
that they are working with. Once they have gone into the mathematical ideas, they can and will go on to work on the
"bare" mathematics.
A few students do think well symbolically and do not require much of a context. Most of us here were that kind
of student. In the past, success in algebra was reserved for us and others like us. Algebraic knowledge is too
important to reserve for so few. It also is not clear that people who think in this way have the most to contribute even
to pure mathematics. We have to open the doors to others. It is not that hard to do.
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The Nature and Role of Algebra.
I~an~na~e and Representation
Deborah Hughes Hallet
Harvard University
Cambridge, Massachusetts
Students arriving at college should be familiar with verbal, symbolic, graphical, and numerical representations.
They need to be able to manipulate each one and be able to convert one to another. Manipulating each of the
representations requires some degree of technical skill supported by conceptual understanding. This understanding
must comprise both an understanding of how the representation works in general and of each particular object being
represented.
To work with graphs, students need to understand how graphs are constructed. For example, they need to
understand that values of inputs to a function are measured horizontally, whereas values of outputs from the
function are measured vertically. This will enable students to interpret intercepts and to estimate values of limits
and asymptotes. Technology is changing how much technical skill students need to have in drawing graphs by
hand, but it has not changed the fact that students need to understand how the zeros and symmetries of a function
appear on a graph, where to expect asymptotes, and what sort of scale will show all the features of the graph.
Numerical data, usually in a table, is for many students the least familiar representation. Students need to
understand how the data were generated (from an experiment, by using a formula, for example). They should be
able to work with numerical data, such as rounding, interpolating, and extrapolating (where these make sense). The
ability to find patterns in data, such as where values are increasing or where there are constant differences, is a
useful skill.
The manipulation of symbols traditionally has formed the largest part of an algebra course. It is still central.
Students must be able to solve equations, collect terms, simplify, and factor. The degree of skill and the speed
required may be altered by changes in technology. For example, methods of factoring higher degree polynomials
are probably not as important as they used to be. However, what it means to factor a polynomial (for example, that
it is not usually useful to write x2 - 2x = xtx) - 2x) is as important as ever. Experience and observation will suggest
the most effective balance between paper-and-pencil work and technology. Currently, there is a wide range of
opinion about the best way to develop manipulative skill, ranging from not allowing technology to allowing it to be
used heavily. As we try to figure out how to teach this skill, we should be mindful of the fact that we were not very
successful at teaching symbol manipulation before technology complicated the situation. It is tempting to gild the
past, but weaknesses that we currently observe are probably not the result of technology. Our charge is to figure out
how to fix them.
Besides acquiring skill in manipulating graphs, numerical data, and symbols, students need to be able to move
easily between these representations. For example, given a straight line graph, students should be able to figure out
its (approximate) equation. Given a table of data from an exponentially growing population, students should be able
to figure out a formula for the function. Given a data set, students should be able to make a mental sketch of the data
or match data with the correct sketch.
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THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM
Mastery of the language of algebra requires a two-pronged approach:
1. What does it mean?
2. How do wedoit?
Equal emphasis on both of these leads to students who know both what algebra means and how to use it
correctly.
The future is likely to change the balance between these two because the skills required to do algebra are likely
to change. However, we always will need to make sure students can use graphs, tables, symbols, and verbal
descriptions fluently.
Representative terms from entire chapter:
school algebra
