DANIEL CHAZAN
ON TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT
EXPLORATION: A PERSONAL STORY ABOUT TEACHING A
TECHNOLOGICALLY SUPPORTED APPROACH TO SCHOOL
ALGEBRA
In the United States, examples of elementary school mathematics teaching
(Ball, 1993b; Lampert, 1990; Peterson, Fennema and Carpenter, 1991;
Wood, Cobb, Yackel and Dillon, 1993) have been celebrated as exemplars of broader trends to reform students’ experience of schooling and
(re?)focus their attention and efforts on understanding core ideas of traditional academic disciplines.1 For example, Magdalene Lampert’s (1990)
“When the problem is not the question and the solution is not the answer”
lays out one influential image of teaching aimed at student understanding.
If traditional mathematics instruction links mathematics problems and
their solutions to teacher-taught solution methods, Lampert severs that
connection. In the described episode, students are not solving the problem
by applying a method taught previously in class. Instead, they are posed
a problem for which they have not been taught an algorithm. However,
they have methods for checking the validity of proposed solutions. Using
conclusions developed from previous classroom work, they then seek both
to find the answer to the problem and to find a more general method of
solution to problems of this type.
In other contexts, teaching of this sort goes by other names: inquiry
teaching, a problem-solving orientation, child-centered instruction, mathematical investigations, or mathematical activity (Cuban, 1993; Jaworski,
1994; Lester, 1994; Love, 1988; Morgan, 1998). What does such teaching
require of the teacher? Is it simply a matter of withholding, of not teaching
solution methods? Is it simply the will to interact differently with students,
to listen to their mathematical ideas? Or, do attempts to create childcentered mathematics classrooms in which students explore and conjecture
require anything special of the teachers’ knowledge of mathematics?2 Are
there qualities of the teachers’ knowledge which are crucial to this sort of
teaching? If so, do advances in calculator and computer technology have
any role to play?
International Journal of Computers for Mathematical Learning 4: 121–149, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
122
DANIEL CHAZAN
To approach these questions, I will reflect on my own experiences
teaching beginning algebra and the sorts of mathematical understandings
I could call upon as a teacher. In particular, I will compare and contrast
two three-year-long experiences with different pedagogical approaches to
beginning algebra. I will compare teaching I did between 1982 and 1985
with Dolciani and Wooton’s Algebra One text (1970/73) with teaching of a
functions-based approach to school algebra between 1990 and 1993. Each
of these two approaches is a conceptualization of algebra designed for
use in teaching. Each seeks to provide teachers with a perspective on the
content which will aid in providing students access to a set of mathematical
ideas.3
In this paper, I explore ramifications of these two curricular approaches
for teaching. In particular, I focus on differences in the nature of the subject
matter understandings which each provided me as a teacher. I explore
whether these differences had an impact on my capacity to support student
classroom exploration of mathematical ideas. I argue that the functionsbased approach provided a type of knowledge of the subject matter which
supported involvement of students in the exploration of the subject and that
it provided opportunities for engagement with students’ questions about
the purposes of studying mathematics.
ON THE ROLE OF TECHNOLOGY IN ALGEBRA REFORM
Technology plays a pivotal, but often indirect, role in this story. Much of
what changes between 1982 and 1990 is related to technology available
for the teaching of high school mathematics.
For neo-Vygotskian activity theorists (like Tikhomirov, 1972/1981),
it is not surprising that technology would have such an impact. They
suggest that technology – by virtue of the way it restructures activity –
has the potential to play a critical role in the shaping of thought. Crucial
to the appreciation of such arguments is the recognition that such shaping
does not necessarily require the actual presence of the technology in the
context of particular interactions; the opening up of new technological
possibilities, even in the absence of the relevant technology in a particular
interaction, can have profound impact. This sort of analysis suggests that,
in mathematics education, technology – by virtue of the ways in which
it restructures activity and provides new opportunities for analysis and
solution of problems – can shape the thinking of teachers and curriculum
developers, in addition to supporting the learning of students.
For me, an eloquent example of this sort of impact of technology is
found in recent developments in the teaching and learning of x in school
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
123
algebra. In the past, school algebra has been dominated by a conceptualization of x as unknown; solving equations to find the “unknown” number
has been a central activity. Yet, in the past fifteen years (for example, since
the early eighties, van Barneveld and Krabbendam (1982), and since the
entrance of microcomputers and graphing calculators into schools), particularly in North America, school algebra curriculum and research projects
have begun to explore a more central role for the conceptualization of x as
a variable.4
Arguably (see critics of this development, for example, Pimm, 1995),
much of this change has been influenced by computer technology. Specifically, the capacity of graphing calculators and computers to carry out many
calculations rapidly supports the transition from examination of single
cases towards the examination of groups of cases at once. This possibility
is supported in technological environments by the possibilities of graphing
the output of an expression for “all” real values of x in an interval, of
creating a table of values of the output of an expression involving x, of
linking such graphical and tabular representations, or even the possibility
of having a spreadsheet recalculate a series of expressions as a particular
cell is varied. Thus, arguably, technology has helped it become easier to
conceptualize x as taking on a series of values.
To my view, these technological capabilities have had an impact even
on projects which claim to continue earlier traditions of examination of
x as unknown; even projects which claim to focus on traditional algebra
problem solving or algebra as the generalization of arithmetic illustrate
the influence of a variational approach. Thus, projects using spreadsheets
to have students solve for unknowns (e.g., Rojano, 1996) involve students
in guessing and checking answers by trying out different values for the
unknown number. Or, in the complete absence of technology, arithmetic
generalizations become statements about all possible numbers, rather than
statements about some particular, unidentified number (Mason, 1996).
In an effort to illustrate the impact of technologically-supported
curriculum innovation on teachers’ subject matter knowledge and to
provide fodder for discussion of the nature of teachers’ subject matter
knowledge, in this paper I will analyze my own teaching experiences. In
each case, I am interested in understanding the resources which each way
of understanding algebra provided to me as a teacher interested in having
students explore beginning algebra.
124
DANIEL CHAZAN
A PRELIMINARY STORY OF TWO TEACHING EXPERIENCES
In the 1980s, for three years, I taught Algebra One in a small, Jewish,
private, suburban school in the Northeastern United States. I used the
revised edition of Dolciani and Wooton’s Modern Algebra: Structure
and Method (Book One) (Dolciani and Wooten, 1970/73).5 For Dolciani
and Wooton expressions with x’s and y’s are expressions for particular
numbers. Based on this view, the book focuses on topics like: solving linear
equations (Chapters 4 and 5), factoring quadratic trinomials (Chapter 7),
and three methods for solving systems of linear equations (Chapter 11). In
retrospect, underlying the book’s focus on technique there is a consistent
story about generalizing arithmetical procedures to create algebraic ones.
This generalization is supported by the notion that in both arithmetic and
algebra one is dealing with the same sorts of numbers.
While the school community was quite happy with my teaching and I
enjoyed the students greatly, during these years, I was quite frustrated. I
was concerned that my teaching was focused on a long list of techniques,
that students depended on me and on the text to tell them right from wrong
and exercised little independent judgment, and that students did not understand what the course was all about. This last concern is in the vein of
John Dewey’s criticism of an overemphasis on curriculum at the expense
of the child. Dewey (1902/1990) argues that when educators forget that
a subject has two aspects “one for the scientist as a scientist; the other
for the teacher as a teacher” (p. 22) a lack of a “motive for the learning”
(p. 25) is the result. In Dewey’s words, “When material is directly supplied
in the form of a lesson to be learned as a lesson, the connecting links of
need and aim are conspicuous for their absence . . . there is no craving, no
need, no demand” (p. 203). I was concerned that my teaching exhibited
this absence.
While the textbook I was using was well structured and rigorous in its
development of Algebra and though my students were college intending
and intent on being successful in school, I felt that the course was focused
inwardly to an unhealthy degree. With each topic that I taught, I could find
no intrinsic justification; topics were always justified by their relationship
to further chapters in the book or to further course work. Though I was
teaching students a course focused on numbers, mathematical objects with
which I hoped they had much comfort, I felt that I could only justify the
value of learning the algorithms I was teaching to students by making
reference to problem types which they would encounter later in the year
or in high school or college. Though my students at the time did not often
question me in this way, I felt I was on quite shaky ground; I could only
refer to experience they did not have. I was teaching them methods to solve
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
125
problems which they had not yet encountered and which I thought they
could not understand. Furthermore, I myself could not see connections
between the algorithms I was teaching them and the activity of people in
the world around me. These algorithms were primarily useful in solving
problems in school or on academic tests.
In addition, since I taught the same students mathematics and Biblical
criticism, I was disturbed to note stark differences in the nature of
classroom discourse in the two subject. While in Bible class, students
would debate various interpretations, critique ones that I offered, and offer
their own, in mathematics class, students rarely exercised their own judgment. They always deferred to my mathematical authority. Ironically, it
was the mathematics class which felt like less of an intellectual experience.
When contrasting my teaching of Algebra with teaching of Geometry
that I did at the same school, I began to focus on my own understanding
of Algebra as the source of my frustration. I had mastered the techniques
taught in the Algebra course and could help students solve the problems
present in the Algebra text, but I felt that my own understanding of the
course was too much focused on technique. I could not give an overview
of the course without referring to sections of the text to illustrate the kinds
of problems which would be solved. I did not feel that I had the sort of
conceptual understanding of the material covered in the course necessary
to support the sort of teaching I wanted to do.
And, I did not think I was alone in this predicament. I did not see
different sorts of understandings of school Algebra in the texts I reviewed
and conference presentations I attended. It seemed to me that many people
sidestepped this issue in the teaching of school Algebra by saying that
school Algebra was not a field of mathematical study, but rather the
language in which much of mathematics is written (Lacampagne et al.,
1995; Lee, 1996; National Council of Teachers of Mathematics, 1994,
September; Usiskin, 1987). Others seemed to dismiss my frustration by
suggesting that mathematics is not supposed to have the sort of meaning I
was after.
Recently, in North America,6 in the context of pressures on school
Algebra, there has been much interest in alternatives to the standard
curriculum. In part because of availability of technology – like graphing
calculators – which allows for links between input-output tables, Cartesian
graphs, and algebraic symbols, one popular alternative is often called a
“functions-based” Algebra curriculum (Chazan, 1993; Computer-Intensive
Curricula in Elementary Algebra, 1991; Confrey and Smith, 1995; Kieran,
Boileau and Garancon, 1996; Lacampagne et al., 1995; National Council
of Teachers of Mathematics, 1994, September; Schwartz and Yerushalmy,
126
DANIEL CHAZAN
1992; van Barneveld and Krabbendam, 1982; Yerushalmy and Schwartz,
1993). Rather than organize school Algebra around the continued study
of numbers and focus on a long list of symbolic manipulations, this
approach organizes introductory Algebra experiences around functions,
their representations, and operations on functions.
In the 1990s, for three years, I team-taught a lower track Algebra One
course using an approach of this kind. I taught with Sandra Callis Bethell
of Holt High School, a high school with students in grades 10–12. Most
students in our classes were not college intending; in order to arrive in
this course, many of the students had failed a previous course; most had to
pass the course in order to graduate high school (Chazan, 1996). Though
the teaching at Holt was often quite difficult and my Holt students were not
shy about asking directly or indirectly why school Algebra was important,
I did not feel as frustrated when using this approach to the curriculum.
Taking a different approach to the course content, I felt better equipped to
help my students understand what the course was about, how the parts of
the course were connected, and how Algebra related to the world around
them.
Ironically, making functions and their standard representations – new
mathematical objects for students – central to the course, changed my
experience. This approach helped me express the problems I posed to
students in a way that allowed them to understand the desired goals. At the
same time, it gave them resources which they could use to solve the problems even before being taught standard methods. Standard methods could
then be introduced to students as ways of solving problems which they
already understood. I felt on less shaky ground with respect to a motive for
learning. At the same time, I felt better equipped to help students see the
mathematics we were studying in the activity of people they knew, across
a range of professions, vocations, and avocations.
In search of an understanding of how alternative views of a subject
matter differ in what they provide “the teacher as a teacher” (qualities of
a teacher’s subject matter knowledge), the remainder of this paper will
be devoted to a comparative analysis of the two ways in which I have
taught Algebra. In particular, I will focus on two issues in the introductory
teaching of Algebra: solving quadratic equations and helping students find
Algebra in the world around them. But, before proceeding to this analysis,
I will first outline the approach that Sandy and I took in our teaching of
Algebra One.7
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
127
OUR VERSION OF A “FUNCTIONS-BASED” APPROACH
TO SCHOOL ALGEBRA
Based on ideas about functions-based approaches to school Algebra (we
were particularly familiar with Schwartz and Yerushalmy, 1992; Yerushalmy and Schwartz, 1993), Sandy and I conceived of the course as the
study of a particular set of mathematical objects – relationships between
quantities which can be mathematized as functions.8 We used a range of
technological tools to support this approach: from programs like Interpreting Graphs (Dugdale and Kibbey, 1986) and The Function Supposer
(Schwartz, Yerushalmy and EDC, 1989) to numerical calculators. Our
approach was different than traditional approaches that we had both taught
before in terms of its treatment of the x’s and y’s of school Algebra. If x
sometimes can be seen as a variable which takes on many possible values,
or as an unknown, particular number, the approach we took makes the first,
variable view of x central and the second, particular unknown view of x
background, reversing the traditional relationships between these views.
This change makes various representations of functions an important
part of instruction right from the beginning of the course. It delays the
introduction of equations and the solving of equations.
Our Algebra One course can be thought of in three strands which are
staggered in time, but which overlap. Early in the year, students become
acquainted with relationships between quantities and methods for representing them. Based on our experiences in our early years, in the later
years, we began by working with students to understand what we meant by
quantities (in the sense of Thompson, 1993). We had students identify the
aspects of their experience which could be, at least theoretically, measured,
counted, or computed from other quantities.
Having helped students identify quantities in the world around them,
we then began to examine various representations of relationships between
quantities. Each year we tried a slightly different order of introduction
to these representations and would not suggest that one order is optimal
in any sense. For example, one year, we began to examine relationships between an independent and dependent quantity by having students
examine sketches of graphs on the Cartesian plane. As part of this work,
students learn to use the words increasing, decreasing, and constant to
describe both the behavior of a dependent quantity and its rate of change
(Schwartz and Yerushalmy, 1995)
Next, inspired by a comment of August Comte in which he suggests
that mathematics originated in attempts to measure or count quantities
inaccessible to direct measuring or counting (Wheeler, 1993), we had
student examine algorithms for computing quantities, algorithms found in
128
DANIEL CHAZAN
their hobbies or in the work lives of their parents or other local business
people (this project will be described in more detail below). Students then
examined and made up algorithms on numbers.
To examine and understand these algorithms, we introduced the class
to input-output tables, showed them how to plot points in the Cartesian
plane, and how such points could constitute a graph. By the end of this
section of the course, we expected our students to be comfortable with
aspects of what Alan Schoenfeld and others have called the Cartesian
Connection – relationships between the tabular, graphical, and “algebraic”
representations of a function (Moschkovich, Schoenfeld and Arcavi, 1993;
Schoenfeld, Smith and Arcavi, 1990); students were prepared to work with
tools like graphing calculators, or in our case, The Function Supposer.
A second strand of the work asked students to use technological tools to
deepen their understandings of representations of functions by examining
changes to those representations (many of these changes would be tedious
to carry out by hand). Inspired by a matrix in Yerushalmy and Schwartz’s
work and their emphasis on operations in tabular, graphical and symbolic
representations (Schwartz and Yerushalmy, 1992), in our work we distinguished between two sorts of operations on a representation of a single
function. There are operations on a representation which preserve the function: for example, using an identity like the distributive law to rewrite the
symbolic expression of a function in a new form, creating a new table for
the same function, or graphing the function at a different scale. At the same
time, there are operations which create a new function: incrementing a
single coefficient in an algebraic expression, translating a graph, or shifting
the alignment between a column of outputs and a column of inputs in a
table. In using technological tools to examine such operations, we asked
students to describe the families of functions which can be created with
particular operations. For example, what family of functions results from
stretching x 3 horizontally and vertically.
While in the first strand students examined a wide range of functions – polynomial functions, sketched graphs not expressible in algebraic
symbols – the second strand focused mainly on linear functions (without
explicitly addressing the assumption that their domain was the real
numbers). However, there was some work with quadratic, absolute value,
exponential, and step functions, as well as functions on a discrete
domain.
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
Objects
Processes
Maintaining the
same function
129
Creating new
functions
Linear functions
Quadratic functions
Miscellaneous functions:
Functions on integers; constant,
absolute value, exponential, step
functions
A depiction of the second strand: operations on a single function
Thus, this second strand encompassed work on simplifying, expanding,
and multiplying expressions; work found early in a traditional curriculum
focused on algebraic manipulation.
As captured in the matrix below, the third strand of the course asked
students to work with pairs of functions in two different ways.
Objects
Processes
Creating a new
function
Comparing two
functions
Pairs involving linear and
constant functions
Pairs of polynomial functions
up to quadratic
Pairs involving absolute value
and a linear or constant function
A depiction of the third strand: operations on pairs of functions
Creating a new function – by adding, subtracting, multiplying, dividing,
or composing two functions (as supported by The Function Supposer) –
was contrasted with comparing two functions. The comparisons of two
functions included comparisons of the values of the outputs of two functions for values in a shared domain. This sort of examination lead to
questions like: for which values of the shared domain (potential members
of the solution set) will these two functions produce the same output?
Answering such questions lead to the traditional solving of equations and
inequalities. We introduced these manipulations as operations on pairs
of functions which preserved solution sets, though students sometimes
created their own symbolic algorithms for accomplishing such tasks or
reduced such tasks to previously solved problems (see below).
While this cursory overview describes the introductory Algebra course
Sandy and I taught, it also lays out a broader perspective on continued
130
DANIEL CHAZAN
study of algebra and calculus (see for example Yerushalmy and Schwartz,
1997). With this perspective, much of the traditional curriculum can be
understood as a further deepening of students’ understandings of the
representations of functions and of operations on functions and pairs of
functions, as they are introduced to new classes of functions and new operations. Thus, in further study of Algebra, students are introduced to classes
of transcendental functions and to new operations, like creating functions
which will invert a given function. In calculus, students are introduced
to other operations which start with a given function and produce a new
function: integrating, differentiating, and approximating.
MANIPULATION OF SYMBOLS:
SOLVING QUADRATIC EQUATIONS
In what way did a change to this sort of technologically-supported
approach change my understanding of school Algebra? How did my 1980s
understanding of Algebra shape instruction differently than my 1990s
understanding? I will begin to address this question by examining a particular set of symbolic manipulations present in typical Algebra One courses,
but critiqued by the current National Council of Teachers of Mathematics
Standards reform movement. In the 1989 NCTM Curriculum and Evaluation Standards’ summary of changes in content and emphases in 9–12
mathematics, it is suggested that there be decreased attention to some traditional symbolic manipulations, for example “the simplification of radical
expressions” and “the use of factoring to solve equations and to simplify
rational expressions” (p. 127).
I find this specific recommendation fascinating; it reflects a lack of
comfort with high school Algebra courses which focus solely on having
students master a myriad of symbolic manipulations (Fey, 1989; Lacampagne et al., 1995). It also represents an important and specific statement of
values. In an era when symbol manipulators make skills related to simplifying less important than in the past, when students in Algebra are soon
taught other methods to solve equations, and when students need more
opportunities to see a purpose for the study of algebra, the authors of this
Standards document are specific; they indicate the manipulations which
they find less worthy than others.9 Yet, besides calling for decreased attention to these particular manipulations, the curriculum standards give little
direction on ways to reconceptualize the symbol manipulation work which
remains in the curriculum.
Therefore, in attempting to capture how the view of algebra offered a
teacher by a “functions-based” approach differs from a standard approach,
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
131
I think it is useful to tackle the issue of symbol manipulation. I have
chosen to focus in particular on the solving of quadratic equations. First,
this content appears late enough in an Algebra course to have connections
with a large part of the content of the course. Second, solving quadratic
equations is one of the contexts in which solving by factoring appears in
the traditional curriculum.
FINDING TRUE STATEMENTS VS FINDING INPUTS
WHICH GIVE EQUAL OUTPUTS
To begin this exploration, it is useful to contrast the meaning of equations
and solving equations in a standard and a “functions based” approach.
A standard approach
What is an equation? “a sentence about numbers.” p. 24
“a pattern for the different statements
– some true, some false – which you
obtain by replacing each variable by
the names for the different values of
the variable” p. 44
What does it mean To find “the set consisting of the
to solve an equation members of the domain of the variof a single variable? able for which an open sentence is
true.” p. 54
A “functions based”
approach
A comparison of two functions which
share the same domain. One seeks to
find elements of the domain for which
the two functions will produce the
same output.
To find the values in the shared
domain for which the two functions
will produce the same output.
While, on the surface, these two views are different, it is initially
unclear whether the differences are substantial. But, it seems to me that
there are substantial differences between the two views. To illustrate this
difference, imagine the situation of students who have not been taught
methods for solving equations, but who, like students in Lampert’s (1990)
class, are presented with an equation and a definition of an equation and
are asked to solve the equation using any methods at their disposal.10
In my dayschool teaching, I did not give students tasks for which I
had not taught an algorithm; if I had not introduced an algorithm, my
assumption was that students could not solve the problem. As a result, I do
not know how I would have described an unfamiliar problem to students
without doing an example; since I always taught algorithms first, I was not
used to describing the properties of a solution. Now, I wonder about the
sorts of resources my students would have had for finding numbers which
would make x 2 − 8x = 5 a true statement. They might have chosen a
number (almost randomly) and tested whether that number worked or not.
132
DANIEL CHAZAN
By choosing a series of numbers they might have see that they were getting
closer and farther from a true statement, but they would have no notion
about the potential number of solutions. In order to work on a problem
like this, I suspect my dayschool students might have invented a tabular
representation of this sort.
Value of variable
(x)
Resulting equation
(x 2 − 8x = 5)
True/false
5
6
7
8
9
−15 = 5
−12 = 5
−7 = 5
0=5
9=5
f
f
f
f
f
But, we would have not made tables of this kind before in class. We did
not usually evaluate expressions for a range of different values (and notice
that this is a push in the direction of a functional approach). And, I do not
know if they would have recognized that between 8 and 9 one might find
a solution.
By contrast, I was regularly able to give my students at Holt problems
for which they did not have an algorithm; I became practiced at describing
the properties of a solution, rather than giving an example of a solution of
a problem of a given type. Having access to a variety of representations
and technological tools which operated on these representations somehow
made this possible. The students could both understand the goal of the
problem and had resources with which to tackle the problem. Given the
same “solve an equation” type of problem (find the shared inputs which
will generate equal outputs), even in the absence of technology, students
who have been making tables and graphs of functions and who are given
the functions based definition of an equation can make a table with three
familiar columns.
Value of variable
(x)
f(x) = x 2 − 8x
g(x) = 5
5
6
7
8
9
−15
−12
−7
0
9
5
5
5
5
5
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
133
In analyzing such a table, my Holt students had previous experience with
problems of the form: here is a function f (x), say f (x) = 3x − 2, find the
input for which the output is 0. Thus, they had experience using a calculator’s capabilities (and implicitly using arguments based on continuity)
to find decimal approximations for answers involving a nonterminating
decimal. Thus, I could expect that they would find a solution between
8 and 9. With their experiences with graphs, they could potentially use
the graphical representation’s depiction of points of intersection to hone
in on that solution, to speculate about the number of possible solutions,
and perhaps to search for more than one solution. Knowledge about the
geometrical behavior of graphs of particular families of functions might
even allow for arguments justifying an expectation for two solutions.
So, these approaches seem to differ in the way they define “solving an
equation” and the resources they provide to students for solving specific
types of equations before being taught an algorithm.11 But, there are other
differences as well.
STANDARD FORM VS DIFFERENCE FUNCTION
These approaches also differ in how they conceptualize the process of
solving equations. In comparing the two approaches’ conceptualizations of
the solving of quadratic equations, I’ll begin with their rationale for transforming equations into equivalent equations, particularly transforming
quadratic equations into the standard form.
In Dolciani and Wooton’s text, students are taught to solve quadratic
equations in three ways. But, each of these methods assumes that the equation is in “standard form,” or has as its first step rewriting the equation in
standard form. The transformations for doing such rewriting are the core
of chapter four (though there they are carried out on linear equations). The
purpose of such transformations is to help one solve an equation by transforming it into “an equivalent equation whose solution set can be found
by inspection” (p. 113). Since equations are about numbers, properties of
numbers can be used to prove that these transformations do not change the
solution set, though no such proof is expected from students.12
By contrast, if solving equations is finding an equation whose solution
set can be found by “inspection,” then the approach Sandy and I used
has some difficulty in justifying transformations of equations. Using the
tabular and graphical capabilities of graphing calculators, one can arrive at
a rational number approximation to the real solutions of any polynomial
equation with rational coefficients by inspection. Although, as a teacher, I
might try to make some sort of appeal for exactness in an answer, one no
134
DANIEL CHAZAN
longer has the same rationale for transformation of equations into equivalent equations.13 Of course, one can still explore operations on one or
both sides of an equation which preserve the solution set (Yerushalmy and
Gilead, 1997). In an added complication, when justifying such operations,
one must determine in what way the properties of numbers now apply to
functions as well.
However, in a functions-based approach, there is a very different sort
of rationale available for the desire to transform a quadratic equation, for
example, into Dolciani and Wooton’s “standard” form. Such transformations change a problem about the comparison of two functions into a
problem about the x intercepts of one function; they move us from the
third strand of our course back into the second strand. I will illustrate
with reference to a problem type that has developed out of a mathematical
model which the Michigan Department of Commerce teaches to potential
entrepreneurs planning to start up a new business.
The Michigan Department of Commerce proposes a simplistic model
for evaluating the viability of a business, one which our Holt students
were able to use before they had learned to solve equations symbolically. The model assumes that one can conceptualize costs and revenue
as dependent variables of number of items sold.14 It then suggests that
entrepreneurs estimate their fixed and variable costs on a monthly basis,
determine their monthly revenue as a function of number of items sold,
and compare these two quantities to find a break-even point, the number of
items which they will need to sell in order to break-even. The department
then suggests that entrepreneurs use a series of methods to assess the feasibility of selling this number of items. Thus, one way of conceptualizing
this procedure is to suggest that the department is asking entrepreneurs to
solve:
revenue (no. of items sold) = costs (no. of items sold)
As I have watched students work on these problems, one often finds
them changing this equation into a different sort of equation. Rather than
examine revenue and costs separately, they often prefer to combine the two
into a new function, using subtraction:15
profit (no. of items sold) = revenue (no. of items sold) − costs (no. of items sold)
Then solving this equation for no. of items sold turns into solving the
equation:
profit (no. of items sold) = 0
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
135
But, this sort of problem was referred to earlier, it is a type of problem
we give to students early in their investigation of single functions, find
inputs for which the output will be zero.
FACTORING TO SOLVE VS WRITING IN
AN “X INTERCEPT” FORM
In the Dolciani and Wooton text, expressions with x’s and y’s are variable
expressions, as opposed to numerical expressions; they are ways to express
a number. The early part of the book introduces students to ways of operating on these sorts of expressions for numbers. Thus, based on analogy
between factoring of the integers over the set of the integers, Dolciani
and Wooton use eleven sections of chapter seven to introduce students to
the factoring of quadratic polynomials. In Section 7-13 of Dolciani and
Wooton’s text, as an application of (and motivation for) factoring quadratic
polynomials and the zero-product property of the real numbers, students
are taught to solve polynomials by factoring. If one can write a polynomial
as a series of factors, one can solve an equation in the standard form by
examining each factor.
From my dayschool recollections, this chapter represented a substantial
chunk of the course. I had my students spend a tremendous amount of
time learning to factor quadratics. I remember students being frustrated by
the techniques associated with factoring. On p. 273, there is a gray box
with a seven step algorithm which is substantially vaguer than most of the
algorithms presented in the text. The last step is to multiply the factors
to check that the product produces the original expression. Unfortunately,
students having difficulties with factoring also had difficulties multiplying
and thus could not reliably check their own work, even when disposed to
do so. Traces of my recollections of this frustration are found in Dolciani’s
and Wooton’s teachers’ guide:
Sometimes students may suspect that you know a “secret weapon” capable of revealing the
factors directly. Assure them that the teacher’s only advantage is experience, and they will
be acquiring that. (p. 37)
If at some point in the chapter the pupils ask, ‘why are we learning to factor?’ the answer
may be given in terms of problem solving. Some problems lead to equations that cannot be
solved by the techniques learned to date. (p. 35)
At the same time, I did not feel that the text helped them understand
what factoring was for and what it told you. If factoring is something
that is done to numbers, what does factoring an expression tell you
about the number the expression represents? Alternatively, if the integers
136
DANIEL CHAZAN
are factored over the prime numbers and there is analogy between that
factoring and the factoring of polynomials over the set of polynomials with
integral coefficients, in what way is the set of polynomials with integral
coefficients prime? These were not questions that the book explored.
Traces of my frustrations as a teacher are found in the teacher’s guide as
well:
you must maintain a proper balance of emphasis on technique and on the ideas behind it.
Students who manipulate symbols without understanding do not learn algebra. Those who
understand the basic ideas, but who are slipshod in applying them are not doing satisfactory
work. Thus, students must have the triple goal of learning “what you do,” “how you do it,”
and “why you do it.” (p. 37)
I, of course, did not feel that I was maintaining a proper balance.
By contrast, factoring has a different flavor in a functions-based
approach. Rewriting expressions in different forms is an example of an
operation on a representation of a single function which does not change
the function. When coordinating graphs and algebraic expressions, there
is an opportunity to see relationships between the coefficients in different
ways of writing an expression and aspects of its graph. For example, a
factored form of a linear expression – a(x − r) – has the x intercept – r –
as a coefficient; it allows one to determine the x intercept of the function
by inspection of the expression. Similarly, writing a quadratic expression
ax 2 + bx + c in the form a(x − h)2 + k makes the coordinates of the vertex
of the parabola – (h, k) – available by inspection.
Thus, a functions based approach which links tables, graphs, and
algebraic expressions, again, supports an understanding of the goal of
a type of problem without suggesting that the sole reason to engage in
the problem is to outfit oneself with a tool to solve as of yet unfamiliar
future problems.16 This sort of an approach even allows students to begin
to explore this task before being taught an algorithm. If given a linear
expression in an ax + b form and asked to write it in a factored form,
students can use their ability to approximate the x intercept of a graph to
make a first attempt at a solution to this task. When they have made a guess
for the appropriate x intercept, they can then test their guess by comparing
the tables created by the two expressions. They could even automate this
method of testing conjectures by creating a difference graph which will
compare the two expressions (as in Yerushalmy, 1991; Yerushalmy and
Gafni, 1992).
Tasks, like asking students to use the graphical and tabular feedback to
correct mistaken factoring and to explain the nature of the error, can then
challenge students to begin to identify patterns between the coefficients of
expressions in different forms. In fact, having mastered factoring of linear
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
137
expressions, my Holt students were intrigued by the new level of difficulty
and the more complex pattern posed by quadratic expressions. Patterns
which students identify can be tested by application to new expressions.
Subsequently, properties of numbers can be used to justify patterns which
seem to work consistently.
Finally, if students have also built functions out of other functions using
binary operations, then writing quadratics in a factored form can take
on another meaning. The factored form emphasizes two linear functions
which have been multiplied to create the quadratic. Such a perspective
is useful when tackling problems about relationships between areas of
rectangular figures and the lengths of their sides.
COMPLETING THE SQUARE AND A FORMULA FROM THE
THEORY OF EQUATIONS VS RELATIONSHIPS BETWEEN
THE Y INTERCEPT, VERTEX, AND X INTERCEPT AND A
FUNCTION ON FUNCTIONS
Having learned to solve quadratic equations by factoring, in Dolciani and
Wooton’s text, this type of problem is set aside for five chapters. Then, in
Section 13-3, after learning about trinomial squares, students are taught to
solve quadratic equations by completing the square. This method suggests
a different form of the equation which is solvable by inspection. There
are two parts. First, one rewrites an equation in the standard form as an
equation in a form where a trinomial square is equal to a constant. Then,
one can “use the property of square roots” and solve the resulting two
linear equations.
This method is then quickly generalized in the next section. Comparing
a “standard quadratic equation” (with coefficients a, b, and c) and a special
quadratic equation (with coefficients 2, −5, 1), Dolciani and Wooton show
that the quadratic formula can be derived from the procedure for solving
by completing the square. They then suggest that students use this formula
to find the numbers which solve a quadratic equation in the standard form.
−b ±
√
b2 − 4ac
.
2a
By way of contrast, in a “functions based” approach, rather than introducing an algorithm justified in terms of the properties of real number
and generalizing it, the quadratic formula can be developed as a statement
of geometrical relationships between the y intercept, vertex, and x intercepts of parabolas. Students bring much background knowledge to such an
138
DANIEL CHAZAN
exploration late in the year. Based on work creating new functions out of
pairs of functions, they may have observed that there are quadratic functions which cannot be created by the multiplication of two linear functions
with rational coefficients. Thus, solving a quadratic equation (over the real
numbers) is equivalent to finding the x intercepts of the graph of the related
quadratic function. If these x intercepts do not exist, then there are no
solutions over the real numbers.
For example, one year, I had a student who was had an interesting (and
wrong) conjecture about the relationship between the vertex and y intercepts of a parabola. He claimed that the distance between the x intercepts
of a parabola and between the vertex and the x axis had to be the same.
In exploring his conjecture, we came to realize that to find the x intercepts
of a parabola one moves from the vertex (h, k) the square root of the
quantity k/a to the right and left (the equivalent of solving a an equation
in the form of trinomial square minus a constant is equal to zero).17
In other words, we had written a new quadratic formula. For functions in
the form, a(x − h)2 + k, to find their x-intercepts, evaluate:
!
k
h±
.
a
This way of thinking about the quadratic formula suggested to me that
the formula describes how the x intercepts of a parabola depend on the
coefficients of the vertex form of the function’s expression. The x inter-
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
139
cepts depend on the location of the vertex and the steepness of the opening
of the parabola. In terms of the standard quadratic formula, the remaining
step is to connect the ax 2 + bx + c form of the function’s expression with
its vertex form.
Thus, instead of thinking of the quadratic formula as a “formula” from
the theory of equations for determining the two numbers which “solve”
a quadratic equation, the quadratic formula became a function on the
coefficients of a quadratic function in the standard form. This function
on quadratic functions returns the values of the x intercepts of the input
quadratic function. With the graphical representation of functions in mind,
we can see this “function” as making connections between the vertex point
of a quadratic function and its x intercepts. The outputs of this “function”
can be used to rewrite the original function in a factored form which
highlights its x intercepts.
WHY STUDY SCHOOL ALGEBRA?
Dolciani and Wooton’s text makes an attempt at the beginning of each
chapter to state its purpose. The first chapter has a statement which is
an introduction to the course as a whole. These statements often try to
justify the study of school Algebra. They remind me of my frustrations
teaching at the dayschool. The rationale is always future directed. Algebra
is a foundation towards larger goals; or, it provides techniques which are
useful in solving interesting problems; it is not justified in and of itself.
Here are some of the statements which appear:
Chapter 1: Mathematics, the language of science, is the language of dreamers who plan to
achieve their dreams . . . The algebra that you will learn in this course is one of the essential
foundations for the theories on which space travel is based (p. 1).
Chapter 4: In this chapter you will study how to solve an equation by transforming it
into a simpler equivalent equation. You will then be able to solve a number of interesting
problems (p. 111).
Chapter 6: You are now ready to learn how to perform operations with expressions called
polynomials. You will then use these new techniques in the solution of problems more
complicated than those you have solved up to now (p. 205).
Chapter 13: As in the study of a language, mathematics becomes more interesting after
the basic skills have been acquired. You will find this to be true in this chapter where you
will study about quadratic equations and inequalities. With such open sentences you will
increase your power to solve problems (p. 495).
Chapter 14: As you look into your own future, can you see the role mathematics may
play in it? Space engineers require a knowledge of mathematics greater than that you now
possess. They did not learn their mathematics as part of their jobs. They learned it in order
to get their jobs. Since many occupations which are challenging require a knowledge of
mathematics, you should plan to include it in your education (p. 523).
140
DANIEL CHAZAN
The theme of the final chapter, that of Algebra as a requirement for
challenging employment, is one often used by teachers in justifying school
Algebra. Yet, in U. S. policy debates surrounding recent moves to provide
greater access to college by requiring Algebra for graduation from high
school, such rationales are offered by mathematicians and mathematics
educators and questioned by others. Here are some comments which
followed the adoption of Algebra as a graduation requirement in the
Washington DC school district.
Too many of us were forced to take algebra when the time and energy could have been
devoted to subjects that truly were beneficial . . . Would millions of high school students
trudge into their algebra classes if they weren’t a gate through which they were forced to
pass to enter college? (McCarthy, April 20, 1991, p. A21)
Mathematics is not just another science; it is the language through which all of science and
much of management science is taught . . . The student who closes the door on high school
algebra (and so on all of mathematics) closes the door on much more . . . (Roberts, April
27, 1991, p. 15A)
A 1992 analysis of 1,400 jobs by the New York Department of Education found that
78 percent of them required no algebra, and only 10 percent required more than a little
. . . (Bracey, June 12, 1992, p. C5)
Such justifications were the best that I had to offer when teaching at the
dayschool, but luckily my students did not often ask this sort of question.
They all were college intending and understood that school in general and
algebra in particular where stepping stones towards the futures they desired
and thought they could have. My teaching at Holt was quite different. For
these students, school and algebra were not outfitting them for the futures
that they saw available. My students were quite skeptical about the value
of school knowledge.
Thus, besides identifying the objects of study in Algebra for myself,
I felt I needed to be able to find those objects of study in the worlds of
my students; if not in their experience then in the experience of people
who they knew. In Dewey’s terms, I needed to psychologize the subject
matter, to view it “as an outgrowth of (my students) present tendencies
and activities” (p. 203). Only then would I begin to have a response to the
question of “What is Algebra all about? Why would anyone want to know
Algebra?”
A typical response to the second of these questions is for the teacher to
seek the “relevance” of school Algebra to students’ lives. Under Sandy’s
guidance, we took a different approach. Rather than assume the complete
burden of generating “relevance,” we asked our students to share this task;
we asked them to find relationships between quantities in the world around
them; we enlisted their aid in exploring connections between the mathematics studied in school and their lives.18 Exploration of the subject matter,
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
141
in this case school Algebra, became one avenue for having students share
with us their experience of the world around them, for them to educate us.
Involving students in the question of “relevance” and the choice of
“relationships between quantities” as the central mathematical object of
school Algebra have changed my response to questions about the meaning
and purpose of school Algebra, which I always used to dread. Since
“relationships between quantities” are mathematical objects which my
students can find in the world around them, the choice of structuring the
course around these objects provides me with an alternative to looking for
“relevance.”
I will illustrate the ways in which school Algebra has provided us with
opportunities to learn about the context in which our students live. In my
third year of teaching at Holt, we organized a project which involved interviews with local business people about “rules of thumb” used in their work;
we sent our students out into their community to do an interview project
and job shadowing (Chazan and Bethell, 1998). As one student said, we
asked the class to go on a hunt for math in the workplace. The project
asked pairs of students to find the mathematical objects we were studying
in the workplace of their community sponsor. Students would visit the
sponsors workplace four times during the year – three after school visits
and one day-long excused absence from school. In these visits, the students
would come to know the workplace and learn about the sponsors work.
Based on these visits, we would ask students to write a report describing
the sponsor’s workplace and answering the following questions about the
nature of the mathematical activity embedded in the workplace.
Quantities
Measured/counted
vs computed
Computing
quantities
• What quantities are • When a quantity is
measured or counted by computed, what information
(the people you interview)? is needed and then what
computations are done to get
the desired result?
• What kinds of tools are • Are there ever different
used to measure or count?
ways to compute the same
thing?
• Why is it important to
measure or count these
quantities?
• What quantities do they
compute or calculate?
• What kinds of tools are
used to do the computing?
• Why is it important to
compute these quantities?
Representing quantitites
and relationships
between quantities
Comparisons
• How are quantities kept • What kinds of comtrack of or represented in parisons are made with
this line of work?
computed quantities?
• Collect examples of
graphs, charts, tables,
. . . that are used in the
business.
• How is information
presented to clients or to
others who work in the
business?
• Why are these
comparisons important
to do?
• What set of actions
are set into motion as a
result of interpretation
of the computations.
142
DANIEL CHAZAN
Though the technical details of the project were sometimes overwhelming, the students came back with useful information. The school
building was being renovated and expanded. So, for example, Rebecca
worked with a carpeting contractor who in estimating costs read the dimensions of rectangular rooms off an architect’s blueprint, multiplied to find
the area of the room in square feet (doing conversions where necessary),
then multiplied by a cost per square foot which depended on the type of
carpet to compute the cost of the carpet. The purpose of these estimates
was to prepare a bid for the architect where the bid had to be as low as
possible without making the job unprofitable. Rebecca used the following
chart to explain this procedure to the class.
Inputs
Length
Width
10
20
15
x
35
25
30
y
Area of the room
Output
Cost for carpeting
room
Though many of the sponsors initially indicated that there were no
mathematical dimensions to their work, students often were able to show
sponsors places where the mathematics we were studying was to be
found. For example, Jackie worked with a crop and soil scientist. She was
intrigued by the way in which measurement of weight was used to count
seeds. First, her sponsor would weigh a test batch of 100 seeds to generate
a benchmark weight. Then, instead of counting large number of seeds, an
amount of seeds were weighed and a computation was done to indicate the
number of seeds which such a weight would contain.
Joe and Mick, also working in construction, found out that in laying
pipes, there is a “one by one” rule of thumb. When digging a trench for the
placement of the pipe, the nonparallel sides of the trapezoid have a slope
of 1 foot down for every one foot across. This ratio guarantees that the dirt
in the hole will not slide down on itself. Thus, if at the bottom of the hole,
the trapezoid must have a certain width in order to fit the pipe, then on
ground level the hole must be this width plus twice the depth of the hole.
Knowing in advance how wide the hole must be avoids lengthy and costly
trial and error.
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
143
SUMMARY
In order to illustrate the qualitative – rather than quantitative – ways in
which teachers’ subject matter knowledge might differ in the resources it
provides for the support of student-centered instruction, I have reflected on
my own subject matter knowledge and the ways in which it has influenced
my Algebra One instruction. I have examined how my capacity to support
student exploration differed when I taught Algebra as conceptualized in
two different ways for teaching. I have argued that a functions-based
approach provided me with important resources for supporting student
exploration that were missing when I taught with Dolciani and Wooton’s
text and that this approach helped me address frustrations in my earlier
number-based teaching.
Specifically, the function-based approach to algebra’s identification of
objects and processes which can be done to these objects helped me
communicate with novices in Algebra in at least two ways. From the
start of the course, it allowed me to help students appreciate what the
course is about and how it is related to the world around them. Identification of the central objects of study was central in helping students
see algebra in the world around them. This understanding also helped
make algebra less mysterious by helping students understand mathematical tasks in terms of the characteristics of desired solutions. This sort
of understanding is precisely what I felt I was not able to provide to
students when the teaching of algebra felt like slogging through a list of
disconnected techniques. Thus, when using the functions-based approach,
I was able to help students understand the goals of problems for which
they did not have solution algorithms and to work productively on such
problems.
Second, the various canonical representations of functions which Sandy
and I used provided students with important resources for solving problems on their own. As Lampert (1989) argues, such representations make
it “possible to shift the locus of authority in the classroom – away from the
teacher as a judge and the textbook as a standard for judgment, and toward
the teacher and students as inquirers who have the power to use mathematical tools to decide whether an answer or a procedure is reasonable”
(pp. 223–224).19 In addition, representations can also support students in
generating ideas. For example, while the graphical representation of functions is challenging for students (see, for example, Goldenberg, 1988),
it also provides students with an important tool. Geometric intuitions,
like those associated with continuity, are then available for reasoning
about the behavior of inputs and outputs to functions. Since the invention
of such graphical representations, access to these sorts of intuitions has
144
DANIEL CHAZAN
proved quite powerful for mathematicians, and I saw the power of these
representations in the work of students as well.
CLOSING THOUGHTS
Teachers, for the purposes of instruction, can conceptualize the same
mathematical subject matter in different ways. Alternative conceptualizations of subject matter offer different sorts of support to teachers and
students.20 However, my purpose in this article is not to argue for one
curricular approach to algebra over another. I am not suggesting that a
functions-based approach is the “only” approach to teaching algebra in
an exploratory way. Perhaps others could develop an approach based on
number as object, or on algorithm as object (Cuoco, 1990) that would
support student exploration as well.
Instead, in closing, I would like to understand better why a particular approach to a subject matter might provide a teacher with resources
for supporting student exploration which another does not. If supporting
student exploration is more than simply severing the connections between
problem and solution, on the one hand, and solution method, on the other,
or deciding to listen to students, what is it that an approach to the subject
matter can provide? How can we come to distinguish approaches that
have the potential to support student exploration from those which do
not? Understanding such differences, I believe, is one productive way to
approach the question of the nature or quality of teachers’ subject matter
knowledge, rather than its quantity.
My analysis of the experience with a functions-based approach suggests
two general characteristics for the evaluation of a conceptualization of
mathematical subject matter for teaching. First, does this conceptualization
identify the central mathematical objects of study in the curriculum. Does
it indicate the sort of mathematical processes in which these objects are
involved? Second, does it help students come to experience these mathematical objects as objects? What sorts of representations of these objects
do students have to work with in order to “see” the effect of a process on
an object? I propose that conceptualizations which identify central objects
and processes and which help students represent the objects and the impact
of processes on them will be conceptualizations which provide the teacher
with rich resources for supporting student exploration.
Such a conclusion suggests that technology – especially tools designed
specially for educational tasks (See Yerushalmy, this issue) – has a role to
play in supporting teachers’ understandings of mathematics. In particular,
technological tools which are written from a pedagogical perspective and
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
145
are developed around central mathematical objects and processes seem
especially important. They can provide students access to informative
representations of mathematical objects and actions taken on them that
would be quite tedious and time-consuming to construct by hand.
NOTES
1. Calls of this kind are found under the banner of “core curricula,” “less is more,”
“depth over breadth,” “essential skills,” “teaching for understanding,” “coherent intellectual story lines,” and organization of curriculum around “big ideas” (California
State Department of Education, 1992; Cohen, 1993; Hirsch, 1996; National Council of
Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991;
Sizer, 1992).
2. Ball (1992) argues that: school reformers assume these sorts of teaching will demand
deep subject matter understandings of teachers and that most teachers do not have such
understandings, yet, are vague about the sorts of subject matter knowledge teachers
actually require. In an effort to move the field forward, she distinguishes between
teachers’ knowledge of the substance of mathematics, knowledge about the nature and
discourse of mathematics, knowledge about mathematics in culture and society, and
capacity for pedagogical reasoning about mathematics. In this paper, I will concentrate
on the first category – the teachers’ knowledge of the substance of mathematics.
3. To be clear, neither faithfully represents algebra as a field of research. Dolciani and
Wooton, for example, argue that “the approach of this textbook is mathematically
correct, but informal and intuitive rather than axiomatic” (1970/73, p. 3). Similarly,
the functions-based approach that I used is representative neither of the Algebra or
Analysis taught at the university level. It is instead a pedagogically driven approach.
4. One indication that this development represents change is that such explorations have
come under fire by critics as fundamentally misrepresenting algebra (see, for example,
Lacampagne, Blair and Kaput, 1995; Pimm, 1995). Another indication is that this
theme is reflected in the way different projects describe themselves (see the different
groupings of the approaches in Bednarz, Kieran and Lee, 1996).
5. This text represents the fruit of the School Mathematics Study Group. The two authors
and two consultants were involved in School Mathematics Study Group and the book
identifies ways in which it explicitly builds on the work of this group.
6. Use of functions-based approaches to the teaching of algebra seems to be more popular
in North America than in Europe. For example, Hewitt (1995) does not address any
such approaches in discussing “Imagery as a tool to assist the teaching of algebra.”
7. Interestingly, I could not have written a similar description for the previous approach I
took to school Algebra. Any description I would have written at that time would have
been technique based.
8. I hesitantly label this approach a functions-based approach for the following reasons:
we do not use a modern definition of function but instead concentrate on relationships
between quantities; as a result we do not often use the word function with students;
and we do not examine the structure of the real numbers in detail, we mainly use the
rational numbers and do not carefully define transcendental functions.
9. Interestingly, these recommendations, and the general issue of symbol manipulation
form a flash point between K-12 mathematics educators committed to the Standards
146
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
DANIEL CHAZAN
movement and some university mathematicians. For some mathematicians, these skills
are important prerequisites for successful participation on college calculus courses.
In suggesting this thought experiment, I am not suggesting that students must invent
or discover every algorithm in school mathematics. Instead, I am trying to capture an
aspect of the two approaches which seems different to me.
Ball and Lampert, influenced by the notion of pedagogical content knowledge, discuss
the representations which they choose and develop with students (see for example Ball,
1993a; Lampert, 1989). Here the introduced representations are standard ones.
Since the equations that are dealt with in chapter four are linear equations, the equations which can be solved by inspection are ones in which the variable is isolated
on one side of the equation. In Chapter 7, when polynomial equations are solved
by factoring, in order to create equations that are solvable by inspection, one seeks
equations with zero on one side and the standard form of the polynomial on the other.
Thus, the difference between the linear and the quadratic cases is technique driven.
There may be others like understanding that certain kinds of comparisons of functions
are equivalent.
This assumption is clearly problematic. For an example of a students’ questioning of
this assumption, see (Chazan, 1996).
Our Holt students are familiar with the notion of difference functions from a couple
of places in the curriculum. They have explored the creation of functions by binary
operations on other functions. And they have used difference functions to get feedback on their attempts to rewrite the same function in different forms (see below and
Yerushalmy, 1991; Yerushalmy and Gafni, 1992).
I’m not convinced that this provides a strong rationale for this type of problem
however. Why is it useful to have a symbolic form of this type? I am still not quite
happy. The information is still available in other representations. I think the argument
more has to be made that this is learning about the symbolic representation, the same
way in which students have earlier learned about the other representational systems.
Then students must be given problems for which it is useful to know how to write
expressions in different forms.
This was a new realization for me at the time (see Chazan, 1992).
Functions may be easier to find than some of the other objects suggested earlier.
Similarly, Ball (1993) suggests that “weaving what I term a representational context
in which students can do – explore, test, reason, and argue about – and, consequently,
learn about particular mathematical ideas and tools is at the heart of the difficult work
of teaching for understanding in mathematics . . . ” (p. 160).
As my experience with Dolciani and Wooton’s approach suggests, contra Dewey,
approaches to the subject matter that are designed for teaching are not immune to
the evils which he ascribes to views of the subject matter driven by the scientist.
REFERENCES
Ball, D. L. (1992). Teaching mathematics for understanding: What do teachers need to
know about the subject matter. In M. Kennedy (Ed.), Teaching Academic Subjects to
Diverse Learners (pp. 63–83). New York: Teachers College.
Ball, D. L. (1993a). Halves pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. Carpenter and E. Fennema (Eds), Learning,
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
147
Teaching, and Assessing Rational Number Constructs (pp. 157–195). Hillsdale, NJ:
Lawrence Erlbaum.
Ball, D. L. (1993b). With an eye on the mathematical horizon: Dilemmas of teaching
elementary school mathematics, The Elementary School Journal 93(4): 373–397.
Van Barneveld, G. and Krabbendam, H. (Eds) (1982). Conference on Functions: Report 1.
Enschede, The Netherlands: Foundation for Curriculum Development.
Bednarz, N., Kieran, C. and Lee, L. (Eds) (1996). Approaches to Algebra: Perspectives for
Research and Teaching. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Bracey, Gerald (1992, June 12). If you ask me – cut out algebra! Mostly it’s a useless,
impractical exercise. Washington Post.
California State Department of Education (1992). Mathematics Framework for California
Public Schools. Sacramento: Author.
Chazan, D. (1992). Knowing school mathematics: A personal reflection on NCTM’s
teaching standards, Mathematics Teacher 85: 371–375.
Chazan, D. (1993). F(x) = G(x)?: An approach to modeling with algebra, For the Learning
of Mathematics 13(3): 22–26.
Chazan, D. (1996). Algebra for all students? Journal of Mathematical Behavior 15(3):
455–477.
Chazan, D. and Bethell, S. (1998). Working with algebra. In Mathematical Sciences
Education Board (Ed.), High School Mathematics at Work: Essays and Examples
from Workplace Contexts to Strengthen the Mathematical Education of All Students.
Washington, DC: National Research Council.
Cohen, D., McLaughlin, M. and Talbot, J. (Eds) (1993). Teaching for Understanding:
Challenges for Policy and Practice. San Francisco: Jossey-Bass.
Computer-Intensive Curricula in Elementary Algebra (1991). Computer-Intensive
Curricula in Elementary Algebra. The University of Maryland and The Pennsylvania
State University.
Confrey, J. and Smith, E. (1995). Splitting, covariation, and their role in the development
of exponential functions, Journal for Research in Mathematics Education 26(1): 66–86.
Cuban, L. (1993). How Teachers Taught: Constancy and Change in American Classrooms
1890–1990, 2nd edn, New York: Teachers College.
Cuoco, A. (1990). Investigations in Algebra. Cambridge, MA: MIT.
Dewey, J. (1902/1990). The School and Society; The Child and the Curriculum. Chicago:
University of Chicago.
Dolciani, M. and Wooton, W. (1970/73). Modern Algebra, Book One: Structure and
Method. Boston: Houghton-Mifflin.
Dugdale, S. and Kibbey, D. (1986). Interpreting Graphs. Pleasantville, NY: Sunburst.
Fey, J. (1989). School algebra for the year 2000. In S. Wagner and C. Kieran (Eds),
Research Issues in the Learning and Teaching of Algebra. Hillsdale, NJ: Erlbaum.
Goldenberg, P. (1988). Mathematics, metaphors, and human factors: Mathematical, technical, and pedagogical challenges in the educational use of graphical representations of
functions, Journal of Mathematical Behavior 7: 135–173.
Hewitt, D. (1995). Imagery as a tool to assist the teaching of algebra. In R. Sutherland and
J. Mason (Eds), Exploiting Mental Imagery with Computers in Mathematics Education.
Berlin: Springer-Verlag.
Hirsch, E. D. (1996). The School We Need and Why We Don’t Have Them. New York:
Doubleday.
Jaworski, B. (1994). Investigating Mathematics Teaching: A Constructivist Enquiry.
London: Falmer.
148
DANIEL CHAZAN
Kieran, C., Boileau, A. and Garancon, M. (1996). Introducing algebra by means of
a technology-supported, functional approach. In N. Bednarz, C. Kieran and L. Lee
(Eds), Approaches to Algebra: Perspectives for Research and Teaching (pp. 257–293).
Dordrecht, The Netherlands: Kluwer Academic Publishers.
Lacampagne, C., Blair, W. and Kaput, J. (Eds) (1995). The Algebra Initiative Colloquium.
Washington: U.S. Department of Education.
Lampert, M. (1989). Choosing and using mathematical tools in classroom discourse. In J.
Brophy (Ed.), Advances in Research on Teaching, Vol. 1 (pp. 223–264). Greenwich, CT:
JAI Press.
Lampert, M. (1990). When the problem is not the question and the solution is not the
answer: mathematical knowing and teaching, American Educational Research Journal
27(1): 29–63.
Lee, L. (1996). Algebraic understanding: The search for a model in the mathematics
education community. Unpublished Dissertation, Universite du Quebec a Montreal,
Montreal.
Lester, F. and Garofalo, J. (Eds) (1982). Mathematical Problem Solving: Issues in
Research. Philadelphia: Frankline Institute Press.
Love, E. (1988). Evaluating mathematical activity. In D. Pimm (Ed.), Mathematics,
Teachers and Children (pp. 249–263). London: Hodder and Stoughton.
Mason, J. (1996). Expressing generality and roots of Algebra. In N. Bednarz, C. Kieran and
L. Lee (Eds), Approaches to Algebra: Perspectives for Research and Teaching (pp. 65–
86). Dordrecht, The Netherlands: Kluwer Academic Publishers.
McCarthy, C. (1991, April 20). Who needs algebra? Washington Post.
Morgan, C. (1998). Writing Mathematically: The Discourse of Investigations. London:
Falmer.
Moschkovich, J., Schoenfeld, A. H. and Arcavi, A. (1993). Aspects of understanding:
On multiple perspectives and representations of linear relations and connections among
them. In T. A. Romberg, E. Fennema and T. P. Carpenter (Eds), Integrating Research on
the Graphical Representation of Function (pp. 69–100). Hillsdale, NJ: Erlbaum.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA.
National Council of Teachers of Mathematics (1991). Professional Standards for Teaching
Mathematics. Reston, VA.
National Council of Teachers of Mathematics (1994, September). A Framework for
Constructing a Vision of Algebra: Working Draft. Reston, VA.
Peterson, P. L., Fennema, E. and Carpenter, T. (1991). Using children’s mathematical
knowledge. In B. Means, C. Chelemer,and M. S. Knapp (Eds), Teaching Advanced Skills
to At-Risk Students: Views from Research and Practice (pp. 68–101). San Francisco, CA:
Jossey-Bass.
Pimm, D. (1995). Symbols and Meanings in School Mathematics. London: Routledge.
Rojano, T. (1996). Developing algebraic aspects of problem solving within a spreadsheet
environment. In N. Bednarz, C. Kieran and L. Lee (Eds), Approaches to Algebra:
Perspectives for Research and Teaching (pp. 137–147). Dordrecht, The Netherlands:
Kluwer Academic Publishers.
Roberts, Wayne (1991, April 27). Algebra is not just math, its the language of science,
Minneapolis Star Tribune.
Schoenfeld, A., Smith, J. and Arcavi, A. (1990). Learning: The microgenetic analysis of
one student’s understanding of a complex subject matter domain. In R. Glaser (Ed.),
Advances in Instructional Psychology, Vol. 4. Hillsdale, NJ: Erlbaum.
TEACHERS’ MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION
149
Schwartz, J., Yerushalmy, M. and Educational Development Center. (1989). The Function
Supposer: Explorations in Algebra. Pleasantville, NY: Sunburst.
Schwartz, J. and Yerushalmy, M. (1992). Getting students to function in and with algebra.
In G. H. and E. Dubinsky (Eds), The Concept of Function: Aspects of Epistemology and
Pedagogy. Washington: Mathematical Association of America.
Schwartz, J. and Yerushalmy, M. (1995). On the need for a bridging language for
mathematical modeling, For the Learning of Mathematics 15(2): 29–35.
Sizer, T. (1992). Horace’s School: Redesigning the American High School. Boston:
Houghton Mifflin.
Thompson, P. (1993). Quantitative reasoning, complexity, and additive structures, Educational Studies in Mathematics 25: 165–208.
Tikhomirov, O. K. (1972/1981). The psychological consequences of computerization. In
J. V. Wertsch (Ed.), The Concept of Activity in Soviet Psychology (pp. 256–278). New
York: M.E. Sharpe.
Usiskin, Z. (1987). Why elementary algebra can, should, and must be an eight-grade course
for average students, Mathematics Teacher 80(6): 428–438.
Wheeler, D. (1993). Knowledge at the crossroads, For the Learning of Mathematics 13(1):
53–55.
Wood, T., Cobb, P., Yackel, E. and Dillon, D. (Eds). (1993). Rethinking elementary school
mathematics: Insights and issues, Journal for Research in Mathematics Education 6.
Yerushalmy, M. (1991). Effects of computerized feedback on performing and debugging
algebraic transformations, Journal of Educational Computing Research 7: 309–330.
Yerushalmy, M., and Schwartz, J. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In E. Fennema, T. Romberg, and T. Carpenter
(Eds), Integrating Research on the Graphical Representation of Function. Hillsdale, NJ:
Erlbaum.
Yerushalmy, M. and Gafni, R. (1992). Syntactic manipulations and semantic interpretations
in algebra: The effect of graphic representation, Learning and Instruction 2: 303–319.
Yerushalmy, M. and Gilead, S. (1997). Solving equations in a technological environment:
Seeing and manipulating, Mathematics Teacher 90(2): 156–163.
Yerushalmy, M. and Schwartz, J. (1997). A Procedural Approach to Explorations in
Calculus. Haifa, Israel: Haifa University.
Michigan State University
116L Erickson Hall
East Lansing, MI 48824
U.S.A.