This is a book about solving problems in mathematics. The author shares methods he has developed over many years teaching high school in Greece, and he illustrates these methods on a plethora of worked examples. The methods have been a success in the author’s classes. They also square with cognitive psychology and were met with approval by those he worked with while he was a Fulbright scholar in the United States.
The first method, called the description method, is related to brainstorming. The purpose is to eliminate the time spent staring at a problem and making no progress. A preparatory phase involves reading the problem several times for understanding and then making a few initial attempts at a solution. Next comes the heart of the method: After isolating an important portion of the problem statement, the student writes out a list containing short notes about as many ideas that come to mind concerning that portion of the problem. The author’s first example is a list that might be written if the problem statement contains the phrase “odd numbers:”
- They have the form 2n + 1
- odd + odd = even
- odd + even = odd
- odd × even = even
- odd + even ≠ 0
- the difference of two successive odd numbers is always 2
- so the sequence 1, 3, 5, … is an arithmetic sequence
- properties of arithmetic sequences?
and so on.
From time to time, the student should look back and consider whether any of the ideas that have been generated would help solve the problem. (For maximum impact, a list of ideas should be generated for each important word or phrase in the problem statement.)
While constructing these lists, the mind must be allowed to roam free. “Be a young child again who writes, or sings a naïve poem, or song with no set patterns and no fear of critical reactions. The difficulties in generating new ideas are often due to psychological reasons.” The author gives additional tips such as: use simple symbols; draw diagrams; have a convenient chair and lighting. If the student continues allowing related ideas to pour out onto the paper, often the crucial idea(s) in the problem solution will be among them. It is easier to recognize than to recall. The method is thus beneficial not only by bringing forth key ideas, but also by placing them before the eyes.
The author mentions experiments showing that would-be problem solvers are more likely to miss a key idea when they do not generate the list of short notes. Groups of people excel individuals at problem solving because of the various perspectives proffered by different members of the group. The free-flowing creation of the list of short notes allows the individual to similarly muster various perspectives (rather than being fixated on just one and therefore probably failing to make progress). “If you cannot solve your problem, don’t go straight at it. Go around it.”
Another method the author puts forward is called getting out of loops. In trying to solve a problem, students often repeatedly cycle through methods they are familiar with. To get out of this loop, they are encouraged to create a two-column table. In the left column, the student writes a list, in short note form, of all attempts so far made to solve the problem. In the right column, the negation is taken of each statement in the left column. This appears to be a powerful technique in at least two ways: 1) In creating the left column, students are forced to make explicit their assumptions, which can then be critiqued in various ways. 2) More immediately, the right column provides fresh, indeed contrary, solution approaches.
The author relates an experiment that he did in class. Chapter 5 of a Greek high school textbook of Euclidean geometry discusses parallel lines and the sums of the angle measures of convex polygons. Chapter 6 discusses parallelograms. After studying chapter 5, one group of students received a problem: “Assuming that opposite angles in a quadrilateral are congruent, prove that its opposite sides are parallel.” Most students were able to start with the fact that all four angles summed to 360 degrees, and using the given congruences, deduce that adjacent angles are supplementary. From properties of parallel lines, the result follows.
Another group of students was given the same problem after they had studied chapter 6. They found the task much more difficult! They tried to apply the same method they had been using throughout chapter 6: drawing diagonals, comparing triangles, etc. After being asked to look back at chapter 5, most students quickly solved the problem. It was that hard for students to marshal the correct perspective, even when they had been using this perspective so recently!
So, toward the goal of looking at problems from as many perspectives as possible, the author supplies a list of tips to augment the practice of listing solution attempts and their negations. Among the tips are “Draw auxiliary lines — then more auxiliary lines,” “What other descriptions could have been made?” and “Where else could you have started from?”
The author next introduces the spiral method. This is repeated application of the description method, the method of getting out of loops, and means-end analysis. Means-end analysis he briefly describes as studying the difference between the given and the goal, and then bridging the two. The bulk of this chapter is a sequence of examples illustrating the spiral method.
The next chapter is a hodge-podge of various problem solving approaches, each of which is allotted one or more examples. “Intensive Use of the Data,” “Working Backwards,” and “Setting Subgoals” are three of the approaches treated. The final sections of the chapter are short forays into combinatorics, calculus, and matrix algebra.
The author next considers two problems that he faces in his classroom. First, he would like his students to retain in their memory proofs that contain one or more nontrivial steps. “A similar problem is that of students who take the difficult countrywide university entrance examination in Greece. Beyond problem solving techniques, they have to memorize the solutions to over 700 problems.”
To help memorize proofs, the author has his students study a proof and then condense it into a summary of two to five lines that capture the main points of the proof. Each line is numbered, and space is left between the lines. The students can them fill in the remaining details in the blank spaces.
To memorize the solutions to large numbers of problems, the author advocates grouping the problems according to their topic and method of solution. Most of the chapter is spent illustrating this approach via a large number of calculus results and problems. This is the least original part of the book, being essentially a digest of single-variable differential calculus. It is partially redeemed, however, by the injection of one of the author’s methods here and there.
The final short chapter is an attempt to formalize, in the form of computer algorithms, some of the methods the author has introduced. The list of properties that is generated via the description method, for example, becomes a list of if-then statements that, roughly speaking, serve as transformations that produce either a solution to the given problem or at least more material to work with.
The author acknowledges that problem solving cannot be fully reduced to a mechanical procedure. But this chapter is in line with perhaps the book’s main thrust: that the creativity required for problem solving can be prompted, even guided, by mechanical procedures. Most of the author’s methods involve cycling between applying mechanical transformations and stepping back, taking stock, and letting the mind flow free.
This book is thus a helpful contribution to problem solving. Fully-worked examples take up the bulk of its pages, so that readers can spend most of their time practicing the methods and checking their work. In addition, each chapter except the last ends with a list of problems. A 37-page appendix provides solutions to these. Many of the examples and problems are rather advanced for high-school students, some even for lower-division undergraduates. (The examples referred to in this review are among the very easiest.)
The author’s stated audience is teachers and students. Additionally, math team coaches and members of problem solving groups should find these methods promising. Anyone who applies these methods to solve the many examples and problems in the book cannot help but become a better problem solver.
David A. Huckaby is an assistant professor of mathematics at Angelo State University.