Problem-Solving Strategies in Mathematics

This book is a guide and classroom resource that can also assist an independent student looking to broaden his or her problem-solving strategies. The level is appropriate for secondary education. The will serve both the student looking for a self-contained overview to take at any pace and the educator looking to interject fresh material into a lecture or lesson. Each of the ten strategies include contrasting solutions in a format meant to provide insight into the problem. Only elementary mathematics is used, as the goal is to improve the student’s procedure, not enlarge on mathematical techniques or even increase mathematical sophistication per se.

I feel the authors missed three opportunities: helpful duplication for contrasting, editing out unnecessarily similar examples, and including analytic geometry techniques appropriate for the target audience.

Some duplication can be helpful, such as the same problem in different chapters subjected to contrasting techniques. Within a chapter, problems that may not seem to the student to be analogous can be shown to be isomorphic for a technique. This is a missed opportunity here, however, in multiple places. Problems 4.8 and 4.15 in the chapter “Adopting a Different Point of View” are both solvable by the “Chicken McNugget Theorem.” This is clearly stated as the “exemplary solution” in the first case, while the second case urges listing possible combinations until one encounters an impossible value, which was the dismissed “common approach” in 4.8. The authors missed the chance to show that considering the least total score that cannot be obtained in 3- and 7-point football scores is identical to the problem in boxing chicken nuggets.

The ten techniques discussed are approached in this order: logical reasoning, pattern recognition (both of these have material I can use even for my college students), working backwards, point of view, extreme cases (I feel some hints to the concept of limits would improve this), analogous problems (featuring the non-standard “analgous” spelling), organizing data, visual representation, “accounting for all possibilities”, and “intelligent guessing and testing.”

Up to and including the extreme cases approach I feel the material is strong, well-organized, and lucidly enlightening for the target audience. In the extreme cases chapter, the reader is guided to solving the Monty Hall problem by considering not three doors, but the case of a thousand doors. Readers can easily understand the problem and I feel it would be an improvement to revisit this and other problems with contrasting techniques in other chapters.

The analogous problem section has much overlap with “Adopting A Different Point of View” and perhaps these chapters could have been merged. It is also at this point of the book that some weaker problems appear. Problem 6.7 begins “Given a randomly drawn pentagram…” and the Exemplary Solution begins “…since the pentagram was not specified as to shape or regularity, we could assume that it is a pentagram that is inscribed in a circle…” I would not encourage that chain of reasoning in my students!

I think this text is best used as an educator resource to present to students curated and altered selections to bolster coursework. This could even mean using the material to move student up Bloom’s taxonomy toward more active learning by asking them to compare and contrast the solutions given as well as developing their own, improved solutions, possibly using techniques from one chapter on problems stated in another.

Tom Schulte is by day a software architect for Plex Systems and by night teaches mathematics at Oakland Community College in Michigan.