getting students to function in algebra

GETTING STUDENTS TO FUNCTION IN ALGEBRA

Judah L. Schwartz Massachusetts Institute of Technology & Harvard Graduate School of Education

&

Michal Yerushalmy University of Haifa

This is a newly revised version of a paper prepared for the Proceedings of the Third International Conference on Mathematics Education held at the University of Chicago in November 1991. When the paper was originally written, the authors thought of it as a kind of manifesto about a desirable approach to the mathematics of function in secondary school. Now, more than a decade later - recognizing much, but insufficient progress - we still feel justified in putting forward these arguments. JLS/MY May 2003

Introduction We1 have been doing research on the nature of the cognitive map secondary mathematics teachers have of their subject, particularly in the areas of algebra, trigonometry and precalculus. We find it remarkably confused and incoherent, reflecting with great fidelity the confusion and the incoherence of the curricular materials that the teachers are asked to use in their teaching. Whatever else needs to be done to improve the teaching of mathematics in the secondary schools of this country, and indeed there is much to be done, we believe that the content of the subject must be made coherent and pedagogically workable. It is with this goal in mind and the experience of working closely with secondary teachers for the past several years that we present the following outline of a course on the mathematics of function.

Outline of An Extended Course on the Mathematics of Function We organize primary and secondary mathematics along two dimensions. These are mathematical objects, and the mathematical actions of modeling, transforming and comparing. Throughout this two dimensional space we continually attend to a theme of "mathematical big ideas".

Mathematical Objects The objects of school mathematics are numbers and quantities, shapes, patterns and functions, data and arrangements. We relegate the mathematics of number to the elementary grades and devote most of our attention in the secondary grades to the mathematics of function and the mathematics of shape. While we recognize that there are important connections between the mathematics of function and the mathematics of shape, primarily in the area of analytic geometry, we focus here primarily on the mathematics of function. We take the position that mathematically and pedagogically the primitive and fundamental object of the school subjects of algebra, trigonometry, probability and statistics, pre-calculus and calculus is the function2 . In fact, we take a stronger position we maintain that the function is the only pedagogically necessary and desirable object in these subjects. Other constructs in the various secondary school mathematics subjects such as equations, inequalities, identities, relations, formulae, etc. can all be understood in terms of the fundamental object, i.e. the function. We have considered the following classes of functions as mathematical objects;

1

The authors are deeply indebted to their colleague Daniel Chazan, and to the many teachers and students with whom they have worked during the past several years. 2

The concept of function entails the concept of variable and vice versa

       

point functions (functions on discrete domains) linear functions absolute value functions quadratic functions power functions (including integer & fractional powers) rational functions periodic functions (including trigonometric functions) transcendental functions (e.g. exponential & logarithmic functions)

The function has many representations - two of the most powerful being the symbolic and the graphical. These two representations are cognitively complementary. Each representation offers access to insights that is available only with difficulty in the other representation. We believe that given the growing availability of small computers students should always be in environments where both of these representations of the functions they are working with are always available. Moreover, it should be possible to manipulate either representation of a function sui generis and see the consequences of one's action in both representations. All plotting software allows users to manipulate the symbolic representation symbolically and see the consequences graphically. Some of the software environments that we have developed and disseminated are currently the only available software environments that allow users to manipulate representations of functions graphically as well as symbolically.

Mathematical Actions - Modeling The fundamental mathematical object of algebra, trigonometry, pre-calculus and calculus courses is the function. Everything that one does in these areas of mathematics one does with functions. When we express in mathematical form relationships between quantities in the world around us we are modeling - this is the central creative act of all science and engineering. In the mathematical subjects we are dealing with in this outline, the mathematical expressions of these relationships are in the form of functions, either fit to measured data, or expressing the consequences of some posited mechanism at work. It hardly needs to be pointed out that the traditional way in which this kind of mathematical action is addressed in current curricula, i.e. through the solution of "word" or "story" problems is a caricature of the kind of activity we hope our youngsters will learn to do with some agility. If one analyzes the modeling a student is asked to do in a typical word problem one is led to the realization that the student is being asked to devise two functions that refer to the same quantity and then to constrain these two functions to be related to one another via an equality or an inequality. This is a complex task under the best of circumstances, all the more so when students do not really understand the nature of the objects, i.e., the functions, they are manipulating. Moreover, it has severe detrimental consequences for the subsequent mathematical development of the student. By asking students early on to solve simple equations, we encourage them to make the reasonable inference that in algebra and similar mathematical subjects the symbol x stands for an unknown number that they are being asked to find. When they subsequently encounter this symbol as a

variable in a function such as x 2 + 2 or Sin x, they are frustrated at their inability to find the value of x. We feel that we need to emphasize a kind of modeling activity that is relatively uncommon, i.e. writing single functions that describe the relationship between dependent quantities. We think this ought to be done at the outset without any concern for the need to constrain two functions to be related to one another. The logic of this approach is simple. It is easier to write one function that describes a situation than it is to write two functions that describe the same situation as well as the constraining relation between these two functions. Further we feel that students ought to have a small catalogue in their heads of the sorts of situations the modeling of which might involve, say quadratic functions (areas and uniformly accelerated motion), absolute value functions (distance on a number line, or in a lattice) or exponential functions (growth and/or decay at rates proportional to the amount present). This mental catalogue should be visual as well as symbolic, with paradigmatic graphs accompanying the more usual symbolic expressions. Needless to say, we want students to be able to address more standard sorts of modeling tasks. Given, however, that standard word problem modeling tasks require students to formulate pairs of functions with each pair describing the same attribute of the situation being modeled, we feel that such tasks should be delayed until the students understand well the nature of the function and what is meant by the comparison of functions.

Mathematical Actions - Transforming and Comparing Most of the instructional time in secondary mathematics subjects is spent in transforming and comparing functions, although it is rarely recognized that that is the nature of the activity. It is clearly the case that all problems of the form "simplify", "factor", "expand", "collect similar terms" are instances of students being asked to transform the symbolic form of a function and re-express it in a different but equivalent form. We think the following kinds of transformations of functions are important for students to become agile with. Some of these transformations involve acting on the symbolic representations of functions and others involve acting on the graphical representations. We assume that these transformations are carried out in software environments in which both representations are present and are dynamically linked.       

representing symbolic representations (factor, expand,...) representing graphical representations (sketching, graphing, scaling,...) modifying symbolic representations (vary coefficients & exponents) modifying graphical representations (translation, dilation, reflection) binary combination (+, -, *, /, composition - graphical & symbolic) rate of change (graphical & symbolic) accumulation (graphical & symbolic)

All problems of the form "solve" or "find the solution set of" are instances of a student being asked to write a given comparison of functions, i.e., an equation or inequality, as an

equivalent comparison. For example the comparison of functions (an equation in this case) 9x - 5 = x2 + 3 can be transformed into a variety of equivalent forms one of which is | x - 9/2 | = 7/2 which in turn leads directly to the solution set x = {1,8}. This manipulation of comparisons of functions usually happens in a context in which the equation as a comparison of two objects called functions is, at best, ill- understood. What does it mean to say 9x - 5 = x2 + 3 Clearly 9x - 5 is not the same as x 2 + 3. A functions approach in an environment in which both symbolic and graphical representations are present allows students to understand more readily what is meant by a "solution set" and to be able to view equations, identities and inequalities in a coherent fashion all as examples of different kinds of comparisons of functions. The power of this approach is made further evid ent by the fact that all of the relations of analytic geometry can be understood as comparisons of functions of two variables. Thus, for example, the relation x2 + y2 = x + 4 which is normally depicted as an ellipse in the xy plane is now seen to be the comparison of two functions of two variables, i.e. F(x,y); x 2 + y2 and G(x,y); x + 4 The graphical representation of the function F is a parabolic bowl [with circular crosssection], while the graphical representation of the function G is a plane [tha t intersects the xy plane along the line y = -4]. The projection of the intersection of these two surfaces on the xy plane is the ellipse that is usually plotted. It is not necessary to tell the usual story of the ellipse being "two functions" 3 3

The ellipse (x/a)2 + (y/b)2 = 1

is normally regarded as a relat ion involving two variables. In order to plot this relation in the xy plane students are taught to solve for y, thereby obtaining the two functions of one variable b[1 - (x/a)2]1/2 and -b[1 - (x/a)2]1/2 It should also be noted that the traditional graphical representation in analytic geometry of the slicing of the cone by a plane involves not the quadratic function F(x,y): x2 +y2 but rather the function F(x,y): [x2 +y2 ]1/2 .

In summary, the comparing of functions allows us to add two items to the list of mathematical actions. These are  

comparing functions of one variable which introduces the composite objects called equations, inequalities and identities, and comparing functions of more than one variable which introduces the composite objects called relations.

We can summarize this discussion of mathematical objects and actions in the following matrix.The mathematical objects, i.e., functions, are listed across the top and indicate the columns. The mathematical actions done with these objects are listed down the left side and indicate the rows 4 . A Conceptual Map of The Mathematics of Function point functions

linear functions

quadratic functions

|x| functions

polynomials

representing symbolically

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

representing graphically

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

mani pulating symbolically

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Algebra I

Calculus

Calculus

Calculus

Calculus

Calculus

Calculus

mani pulating graphically translation, dilation, reflection bi nary operations +- * / comparing functions equations, inequalities, & relations how do f’ns change differentiation how do f’ns accumul ate integration

rational functions

peri odic functions

exp/log functions

Calculus

Calculus

Calculus

Calculus

Calculus

Calculus

Calculus

Calculus

Algebra I

4

We have attempted to show where the beginnings of subject, i.e. A lgebra I, lie by marking those cells that are found in most (probably all) A lgebra I courses in red. There are other cells in wh ich Algebra I appears in blue to indicate that there are many Algebra I courses that cover this material. The right most four columns (except for the bottommost two rows) are usually covered in Algebra II, Trigonometry and Pre-calc classes.

We can think of this organization of the subject then in the following form; - every mathematical "occasion" is an occasion for attention to mathematical objects, the operations that can be carried out on and with those mathematical objects and the "meanings" we wish to ascribe to those objects and those operations.

Mathematical Big Ideas There is an important collection of mathematical "big ideas" that are ordinarily not present in mathematics courses until students get to be quite far along and quite committed to mathematics or to a science that uses mathematics extensively. Normally these big ideas are thought of as heuristics and are not generally taught in any formal or consistent way. Here is a list of some of these ideas;           

representation conjecture proof invariance symmetry boundedness "between"ness continuity successive approximation frame of reference scale

We believe that these and other equally important mathematical ideas can be introduced early on in the mathematical education of all students if it is done in the context of interesting and powerful exploratory software environments. In many respects, it is in the articulation of these mathematical big ideas that the aesthetics of mathematics lies. While we do not have recipes for how to introduce these heuristic strategies comprehensively throughout the course, our experience of several decades with exploratory software environments strengthens our conviction in both the importance and the feasibility of this position.

On dynamically linked multiple representations - A Small Atlas of Images The function has different, and complementary representations, most importantly symbolic and graphical representations. Given that there are some students who are more comfortable with images than symbols and some who prefer symbols to images, the prospect of allowing different students to approach the subject in ways that they are more comfortable with is indeed a promising one. The growing availability of personal computers makes it possible, and increasingly feasible, for secondary students and their teachers to do their mathematics in environments in which both these representations are simultaneously available and linked to one another. Changing a coefficient or an exponent in the symbolic expression of a

function triggers a simultaneous change in the graph of the function. Similarly, deforming the graph of a function by stretching or dragging it triggers a simultaneous change in the symbolic expression of the function being deformed. We conclude this paper with a small atlas of illustrations from some of the software environments we have made to instantiate these ideas. The intent of these examples is to show the power of working simultaneously with both symbolic and graphical representations of functions.

Families of functions Consider any family of functions of one variable of the form F(U,V,x) where U and V are parameters. For particular values of U and V, say U0 and V0 , the function is plotted. At the same time, if the functional form of F is known then the point U0 ,V0 in a UV plane is an equally good description of the particular function. Imagine a software environment in which the user, having chosen a functional form, controls the motion of a point in the UV plane, thereby generating a family of functions that is plotted in the standard Cartesian plane.

Figure 1 A family of straight lines Ux + V depicted conventionally on the right in the Cartesian plane as well as on the left in the UV parameter plane (fro m THE FUNCTION FAMILY REGIST ER)

Re-expressing symbolic representations of functions Any problem that asks a student to "factor" or "expand" or "simplify" is asking the student to rewrite the symbolic representation of a function in an equivalent form. Imagine an environment in which every symbolic expression is accompanied by the corresponding graphical representation. Should the student make an error, the graph corresponding to latest expression will differ from the graph of the previous expression. This visual information can be used to guide students' symbolic manipulation, particularly if a graph of the difference between the current and previous expressions is also available.

Figure 2 The user is asked to re-express the function (x+3)(4x-1) in the form A(x + B)2 + C. As a first step the function 4x2 -3 is entered. The graph window shows the original function (x+3)(4x-1), the trial function 4x2 -3 as well as the difference between these two functions. The graphs make clear that the identity of the function was not preserved. The difference function makes clear that the discrepancy lies in the linear term. (fro m Calcu lus Unlimited)

Modifying symbolic representations of functions In order to enable students to understand better how functions depend on the parameters they contain, it is useful to have an environment in which families of functions can be built and explored.

Figure 3 Co mparing and contrasting functions of the form xn and 1/xn . Note that the two families have some interesting points in common. (fro m Calcu lus Unlimited)

Modifying graphical representations of functions Modifying functions by translating, stretching and squeezing, and reflecting their graphical representations provides a rich collection of operations to gain insight into the nature and properties of families of functions.

Figure 4 Bu ild ing a family of parabolas by stretching (fro m Calculus Unlimited).

Binary combination of functions Imagine an environment that is logically a calculator which operates with functions rather than numbers. There are five kinds of binary operation of functions, i.e. addition, subtraction, multiplication, division and composition.

Figure 5. Multip lying a quadratic function by a linear function. Note the positions of the roots of the resulting cubic. (fro m Calculus Unlimited)

Comparing functions of one variable Equations, inequalities and identities involving functions of one variable are all comparisons of two functions of the form F(x) compared via < or = or >to G(x). They can all be represented graphically by plotting the left and the right sides of the comparison on the same set of axes. The solution set, if any, of the comparison is then immediately evident. Moreover, in such an environment, it is possible to explore easily operations that may be carried out on the symbolic representation of the two functions being compared. Needless to say, some of these operations will preserve the solution set and some will not.

Figure 6a A quadratic inequality displayed graphically. The solution set of the inequality is displayed below the graph. The table indicates truth or falsity of the comparison at a variety of values of x. (fro m Calculus Unlimited).

By the same token, we can imagine an environment in which the two functions being compared are manipulated graphically rather than symbolically. All the while the solution set is preserved. What does solving an equation mean in this environment?

Figure 6b. The inequality explored in an environ ment in which the graphs are manipulated and the symbols respond to the graphical manipulat ion. (fro m UNSOLVING...)

Comparing functions of two variables Relations [ of the form F(x,y) < or = or > G(x,y) ] can be viewed as comparisons of functions of two variables. The traditional graphical represe ntation of relations as curves (or regions) in the xy plane are simply the projections of the intersections of the surfaces F(x,y) and G(x,y). { The reader will note that the analytic geometry of conic sections is a special case in which F(x,y) is (x 2 + y2 )1/2 and G(x,y) is Ax + By + C)

Figure 7a. The relation x2 - y2  9 looked at as a quadratic inequality relating two functions of two variables. The figure shows lattice points in the xy plane that lie in the solution set of the inequality. The figure also shows the level set of the function x2 - y2 . (fro m unpublished draft software)

Figure 7b. The quadratic inequality of the previous figure d isplayed as two intersecting surfaces. The solution set appears as a projection on the xy plane.

Conclusion In this paper we have tried to argue for an approach to the mathematics of function that takes as fundamental three premises. The first premise is that the function, and its entailed concept - the variable, can serve as the root construct on which all of the mathematical objects of algebra can be built. The second premise is that the function has different, and complementary representations, most importantly symbolic and graphical representations. Given that there are some students who are more comfortable with images than symbols and some who prefer symbols to images, the prospect of allowing different students to approach the subject in ways that they are more comfortable with is indeed a promising one. The third premise is that the growing availability of personal computers makes it possible, and increasingly feasible, for secondary students and their teachers to do their mathematics in environments in which both these representations are simultaneously available and linked to one another. One way to characterize the essence of what we mean by understanding is to say that people who understand something can represent it to themselves in a variety of different ways. Being able to move nimbly across these representations is surely an e mpirical test of that understanding. It is this goal of mathematical understanding for students and teachers that drives us.