Research Issues in the Learning and Teaching of Algebra

the Research Agenda for Mathematics Education, Volume 4

Sigrid Wagner, Carolyn Kieran

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781135434212
• Edition: 1
• Topic: Pedagogía
• Subtopic: Educación general

First Published in 1989. We clearly know more today about teaching and learning mathematics than we did twenty years ago, and we are beginning to see the effects of this new knowledge at the classroom level. In particular, we can point to several significant sets of studies based on emerging theoretical frameworks. To establish such a framework, researchers must be provided with the opportunity to exchange and refine their ideas and viewpoints. Conferences held in Georgia and Wisconsin during the seventies serve as examples of the role such meetings can play in providing a vehicle for increased communication, synthesis, summary, and cross-disciplinary fertilization among researchers working within a specialized area of mathematical learning. This monograph holds selected papers from four more recent conferences on Research Agenda in Mathematics Education.

Part I
Past Research and Current Issues

Algebra: What Should We Teach and How Should We Teach It?

John A. Thorpe
Department of Mathematics State University of New York at Stony Brook and National Science Foundation

The teaching of algebra in the schools is not significantly different today from what it was fifty years ago. Certainly, there have been some changes during those years. The "new math" movement of the 1960s attempted (and, briefly, succeeded) in introducing some new ideas and new approaches into algebra instruction. But the changes that have persisted into today's curriculum have been more cosmetic than substantial.

Meanwhile, mathematics and its applications have changed dramatically in the past fifty, even in the past ten, years. The emergence of sophisticated calculators and computers as tools in both computational and abstract mathematics have, in particular, changed the way that mathematicians do mathematics and the way that scientists, engineers, and social scientists use mathematics. It is time for algebra instruction in the schools to begin to reflect these changes. It is time for a major reevaluation of the content of the algebra curriculum and of the instructional strategies that are used in teaching algebra.

Goals for Algebra Instruction

Any such reevaluation must necessarily begin with an examination of the goals for algebra instruction. A number of goals have been proposed:

• To "develop student skills in the solution of equations, finding numbers that meet specified conditions" (Fey, 1984, p. 14);
• To teach students to use symbols to help solve real problems, such as mixture problems, rate problems, and so forth (A. Schoenfeld, remarks at the Mathematical Sciences Education Board conference, "The School Mathematics Curriculum: Raising National Expectations," UCLA, November 1986);
• To prepare students to follow derivations in other subjects, for example, in physics and engineering (H. Flanders, remarks at the annual meeting of the Mathematical Association of America, San Antonio, January 1987)¹;
• To enable students to become sufficiently at ease with algebraic formulas that they can read popular scientific literature intelligently (Mathematics Proficiency Committee, SUNY at Stony Brook, discussion, C.1975).

But, as Schoenfeld has pointed out, algebra should not be taught as a collection of tricks, as is common in textbooks—a trick for this, a trick for that. Students should see algebra as an aid for thinking rather than a bag of tricks. Whitney (1985) carries the thinking theme one step further: Students should grow in their natural powers of seeing the mathematical elements in a situation, reasoning with these elements to come to relevant conclusions, and carrying out the process with confidence and responsibility.

I am in complete agreement with these goals, especially the goal articulated by Whitney. All mathematics instruction, and algebra instruction in particular, should be designed to promote understanding of concepts and to encourage thinking. Drill and practice should be required whenever necessary to reinforce and automatize essential skills. But, whenever drill and practice are required, students should always have a clear understanding of why the particular skill is so important that its mastery is required.

The correctness of the goal of a "thinking curriculum" is, in my opinion, self-evident. This general goal, however, must be supplemented by more specific goals that provide guidance in the selection of topics to be taught. In my view there are three criteria, at least one of which must be met before any given topic merits inclusion in the curriculum:

• intrinsic value,
• pedagogical value, and
• intrinsic excitement or beauty.

Intrinsic Value

Some topics must be included in the curriculum because they are, or will be, important in the lives of the students. The distance formula, percents, graphs, and probability and statistics are examples of topics with clear intrinsic value.

As a more detailed example, consider the concept of function. Functions are at the very heart of calculus, and that is sufficient reason to justify the inclusion of functions in any algebra course for the college bound. But functions should be taught to all students, because the concept of function is one of the most important of mathematical concepts. An understanding of the function that translates interest rates into monthly payments on loans is of intrinsic value to most people several times in life. The function that assigns sales tax to purchases is encountered by most people on a daily basis.

But one caveat: Let us not define a function as a set of ordered pairs! The definition of a function as a set of ordered pairs is not only too abstract for an initial introduction, it is inconsistent with the way functions are viewed and used by professionals. A function is a rule—a rule that assigns to each member of some set a member of some other (or possibly the same) set. A function does have various representations—as a set of ordered pairs, as a graph, or as a point in a function space(!)—but we should teach the most intuitive and practical definition and not confuse our students with unnecessary abstractions.

As I was preparing this paper, I reviewed three of the currently most popular algebra texts, which I shall refer to as AW (Keedy, Bittinger, Smith, & Orfan, 1984), HBJ (Coxford & Payne, 1987), and HM (Dolciani, Wooton, & Beckenbach, 1983). One of my biggest disappointments was in observing that, in two of them (HBJ and HM), the ordered pair definition of function still persists. I recognize that this approach was popular among professional mathematicians 25 years ago, and that many of them advocated using this approach in the schools, but this was certainly one of the errors of the sixties and it is time that it were laid to rest.

At the risk of belaboring the point, I must mention that one of the best function machines around is the calculator. Use of the calculator in mathematics learning should begin in elementary school, and that is where the concept of function should be introduced. In that context, it should be patently clear that a function should be defined as a rule, or perhaps as a certain kind of machine, but certainly not as a set of ordered pairs!

Pedagogical Value

Some topics, which may or may not have intrinsic value, must be included in the curriculum because of their pedagogical value. By this I mean that some topics are important, not for their own utility, but rather because they form a necessary foundation or supporting structure for some other topic or topics that have intrinsic value.

As an example, consider the technique for completing the square. Although some might argue that this technique has intrinsic value (for example, in finding integrals of certain rational functions), the importance of the technique is difficult to argue for students who do not plan to study calculus. Even for those students who do plan to study calculus, the intrinsic value of completing the square is very limited. With the availability of symbol manipulation software, computers can calculate integrals of rational functions more rapidly and more accurately than any mathematician. The need to do such computations by hand is comparable to the need to do challenging long division problems by hand now that calculators are available. The need is not there!

Why, then, do we teach completing the square to algebra students? Because completing the square is a critical step in the derivation of the quadratic formula, and the quadratic formula is generally regarded as having intrinsic value.

Intrinsic Excitement

Some topics are just so interesting and exciting that their inclusion in the curriculum does not require any other justification. Examples abound in the sciences. Would anyone argue that a high school biology course should not contain some of the ideas of human genetics, that a chemistry course should not contain some information about laser chemistry, or that a physics course should not contain a discussion of elementary particles? These ideas, which are on the frontier of scientific research, are exciting to students and scientists alike. Teaching about these ideas communicates to students that the sciences are alive, fun to study, and worthy candidates for an interesting career.

In algebra, exponential growth and decay, especially if taught within the context of population dynamics or pollution control, can fit the criterion of intrinsic excitement. However, there is much more that is not now taught but could be, that would bring into the classroom not only the beauty and excitement of mathematics but also the flavor and vibrancy of current mathematics research.

For example, a student with access to a modest microcomputer and with a minimal understanding of quadratic functions and the complex plane can explore some of the ideas of fractals and chaos (see, e.g., Peitgen & Richter, 1986; Peterson, 1987). Chaotic dynamics is one of the most exciting areas of current mathematics research, and it is a source of some strikingly beautiful pictures. Students can create some of these pictures themselves, they can explore the consequences of small perturbations in parameters, and they can make and test conjectures.

There are other exciting areas of modern mathematics and its applications that can be at least discussed in secondary school mathematics classes. Tomography—the science of reconstructing images of the interior of an object, such as the human body, from shadow images, such as those obtained from CAT scans—is a mathematical science. Simulation—the technique by which the results of experiments that are too expensive, time consuming, dangerous, or otherwise impossible to carry out in practice are obtained theoretically using high speed computations on sophisticated computers— is a technique that not only uses mathematics but also stimulates the development of new mathematics.

Although these latter two examples are not algebra per se, they are nevertheless appropriate for discussion in an algebra class. As Steen (1986) has asked, "How many biology teachers would feel comfortable if they never mentioned DNA or viruses to their high school biology classes—simply because they were discovered after the teachers themselves were educated? Shouldn't mathematics teachers be just as embarrassed if they fail to introduce topics like tomography or simulation in their mathematics classes?" (p. 5).

Somehow, the fact that mathematics is a growing discipline, that current research in mathematics is yielding surprising, exciting, and valuable new results, that mathematics was not completed by the work of Euclid, or Newton, or Descartes—these facts must find their way into mathematics classrooms, from elementary school through high school and beyond. In particular, we must introduce more topics into our algebra classes on the basis of their intrinsic excitement.

Evaluating Standard Algebra Topics

The three criteria—intrinsic value, pedagogical value, and intrinsic excitement—can be used to measure the importance of including a given topic in a mathematics course. Let us examine from this perspective some of the topics that are standard fare in secondary school algebra courses.

The Real Number System

Surely there is no debating that some study of the real number system is necessary and appropriate, for both its intrinsic and its pedagogical value. High school algebra is, after all, primarily the study of polynomials (in one or several variables) with real number coefficients. The study of the solution sets of polynomial equations, even those with integer coefficients, requires a good understanding of real numbers, both rational and irrational. There is, however, room for debate on how much depth is optimal and which approach to representing the real numbers is most advantageous.

From the perspective of one who frequently teaches calculus to entering university students, I would prefer that students come to the university with a solid base of understanding about real number operations—including absolute value and exponentiation—and a good sense of numbers—positive and negative, rational and irrational. Moreover, students should be comfortable with inequalities, to the point that it is not a major task to determine which is larger:

Instruction a...

Table of contents

• Cover
• Title
• Copyright
• Contents
• Acknowledgments
• Series Foreword
• The Research Agenda Conference on Algebra; Background and Issues
• PART I. Past Research and Current Issues
• Part II. A Research Agenda
• Part III. Theoretical Considerations
• Participants