Teaching Problem-solving

While problems have been posed to students since antiquity (Boyer and Merzbach, 1991;

Robson, 2008), the modern problem-solving movement within mathematics education can be traced directly to the work of George Pólya. In How to Solve It, Pólya (1945) was the first to discuss problem-solving as a discipline in its own right and the first to offer a procedure for solving mathematics problems. Pólya suggests the following process for finding solutions:

1. Understanding the Problem.

2. Making a plan.

3. Carrying out the plan.

4. Looking Back.

See Appendix C for the expanded version of this advice. It is important to note that Pólya is not so much promoting a particular heuristic or algorithm, as one learns in mathematical topics, but a more general approach. Indeed, were we able to settle on a particular algorithm for solving a set of problems, they are no longer novel to us and so no longer problems. Instead, Pólya focuses more on powers that are used when we engage in mathematical thinking: specialising, generalising, conjecturing and convincing; he wanted not to teach ‘problem-solving’ but mathematical thinking.

Pólya expanded greatly on the ideas contained in How to Solve It in his two-volume work Mathematics and Plausible Reasoning (Pólya, 1954a,b)³, which contained many concrete

3Throughout we cite the single volume Pólya (1962), with a 1. or 2. to denote which volume in the original books is being referenced.

mathematical examples to demonstrate the processes involved. It is important to note that the final stage, Looking Back, refers to the point at which we decide whether we have been successful or not; whether we need to attempt to solve the problem again, or if there is a more elegant way of solving it than that we have found. It is not the only stage at which reflection should occur, as reflecting on past experience is important at all stages of the problem-solving process. Understanding requires us to relate a problem to what we know already, and making a plan without considering what has worked or not worked in the past is surely a bad idea.

Finally, when we carry out a plan we must continually reflect on whether the plan is working as we expect it to.

While Pólya’s ideas have been continually used and developed, notably by Mason et al.

(1982, 2010), and How to Solve It remains a set text for many mathematics education degrees, his strategies are not routinely taught to students. Begle (1979, p. 145) surveyed the empirical literature on these processes and came to the conclusion that:

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No clear-cut directions for mathematics education are provided by the findings of these studies. In fact, there are enough indications that problem-solving strategies are both problem- and student-specific often enough to suggest that hopes of finding one (or a few) strategies which should be taught to all (or most) students are far too simplistic.

In his review of the relevant research, Schoenfeld (1992) demonstrated three particular strategies useful in specific cases, such as when dealing with the roots of polynomials, and had the following to say:

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Needless to say, these three strategies hardly exhaust “special cases.” At this level of analysis—the level of analysis necessary for implementing the strategies—one could find a dozen more. This is the case for almost all of Pólya’s strategies. The indications are that students can learn to use these more carefully delineated strategies.

Developing strategies of more limited scope than those of Pólya demonstrate, Schoenfeld argues, a change from descriptive heuristics—that is to say, names for broad categories of processes—to prescriptive processes—what to do in specific cases—is a better approach to teaching students how to solve problems. Most recently, Lesh and Zawojewski (2007, p. 768) summed up the situation as follows:

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In mathematics education, Pólya-style problem-solving strategies—such as draw a picture, work backwards, look for a similar problem, or identify the givens and goals—have long histories of being advocated as important abilities for students to develop. Although experts often use these terms when giving after-the-fact explanations of their own problem-solving behaviors, and researches find these terms useful descriptors of the behavior of problem solvers they observe, research has not linked direct instruction in these strategies to improved problem-solving performance.

Briefly then, Pólya’s heuristic is too general to be prescriptive whilst Schoenfeld’s processes are too numerous to be decided between, effectively doing little to solve the problem at hand.

Perhaps as a result of the apparent difficulty in teaching problem-solving, there has been over the last several decades a “pendulum of curriculum change” (Lesh and Zawojewski, 2007, p. 764) swinging between problem-solving and basic technical skills. The pendulum currently appears to be swinging in the direction of problem-solving once again, though its teaching remains in something of a difficult position. The importance of problem-solving in developing mathematical thinking is appreciated by many, and the process that mathematicians go through when solving problems can be retrospectively described in common terms. One possible solution to the seeming impasse is to refer back to Pólya’s Problems and theorems in analysis (Pólya and Szegö, 1925), co-authored with Szegö, in which is found the following:

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The independent solving of challenging problems will aid the reader far more than the aphorisms which follow, although as a start these can do him no harm. — Pólya and Szegö, 1925, p. vii

This was written two decades before the publication of How to Solve It, and has an appealing clarity. If we want students to solve problems on their own, perhaps the best solution is to give them problems to solve with minimal concentration on strategy and process. In the following chapter we look at the rise of problem-based learning, a range of pedagogies aimed at teaching students through their own problem-solving and other mathematical enquiries.

Chapter 3

Problem-based Learning

In this chapter we draw a distinction between two broad pedagogical categories. The first, directed learning, describes the lecture based approaches traditionally used in university math-ematics courses. The second, discovery learning, refers to a set of student-centred pedagogies that became popular in the 1960s. A subset of the discovery pedagogies, called problem-based learning (PBL), is the focus of our discussion.

We stress that directed learning and discovery learning do not cover all pedagogies em-ployed in mathematics teaching, nor that each is necessarily distinct from the other.

3.1 Directed Learning

Direct instruction is a term coined by Siegfried Engelmann to describe teaching by lectures or the demonstration of material (Bereiter and Engelmann, 1966). We define directed learning to be learning where the majority of teaching time is handed over to direct instruction. We use the School of Mathematics at the University of Birmingham as our specific example; other U.K.

mathematics departments are considered in Section 3.1.3.

The first year, first semester, module MSM1Aa – Calculus and Algebra I is typical of Birmingham’s first year modules. In eleven weeks of study there are 44 lectures, 20 hours of computer lab work and 10 hours of example classes. Each lecture is 50 minutes and it is anticipated “that each ‘one hour’ lecture requires another two hours of private study”

(University of Birmingham, 2010, p. 21). Few other modules have this amount of computer work but the ratio of four lectures to one example class is usual.

In document Problem-solving in undergraduate mathematics and computer aided assessment (Page 29-34)