The authors helped to establish the Math Circle, a mathematics learning program. They came up with the idea in 1994 while having dinner with a friend Tomás Guillermo:
The academic year was beginning, and what had long been bothering each of us inevitably came up: how could the teaching of math be in such bad shape — especially in a city known for mathematical research? All that our students, whether college freshmen or ninth graders, knew about math was that they had to take it, and that they hated it.
We had chicken with cream and complaints, but the chocolate mousse that followed was too light for such gloom, and one of us said: “Why don’t we simply start some classes ourselves?” (p. 3)
Over the ensuing years, they have taught mathematics, not to “young geniuses, hand-picked for special training” (p. 3), but to participants from a cross-section of both ages and social classes in the Boston area. Each course takes about ten hours. “We’ve kept the style informal, the bureaucracy minimal — and we couldn’t have had more fun” (p. 4). In a course, a problem is chosen for the students to tackle, e.g., finding, as a function of the number of points on the circumference of a circle, the number of regions formed by chords connecting all pairs of these points. (See pp. 71–74 for a description of student engagement in this problem.) A good deal of time is then devoted to brainstorming. The leader of each course plays a subtle, indirect role, on the whole letting students find their own way to solutions and to proofs. Problems are chosen from a range of topics. Among them are clock arithmetic for five-to eight-year-olds, solution of polynomials by radicals for nine- to thirteen-year-olds, and algebraic geometry for fourteen- to eighteen-year-olds (pp. 200–1).
They sum up the content of their book in this way: “What has stood in the way of enjoying and mastering math as one does music, how we have removed these barriers in the Math Circle, and how such circles may now spread, is the subject of this book” (p. 4). Preschoolers are generally enthralled with aspects of mathematics, but often become disheartened in school. The reason, the authors maintain, for this state of affairs lies not with the native talent of young people. Learning mathematics comes as naturally as learning a native language. Success at learning mathematics has a lot to do with exposure to its pleasures and mysteries — what the Math Circle attempts to provide.
Many impediments stand in the way of learning mathematics — the topic in the chapter “The Great Barrier Reef”. The difficulty of learning the significance and meaning of symbols is first looked at. A pedagogical principle is employed in this and other contexts: “If you want a student to master something — call it R — and R is a means to S, then work on S; R will slip in unobtrusively under the radar, whether it is a symbol, a technique, a lemma, theorem, theory, or point of view” (p. 87). Students also have problems with learning to deal with equations: “If the symbols don’t get you, the equations will” (p. 88). Various aspects of equation apprehension are covered, ending with a suggestion that an equation (like distance = rate xtime) should not be “told to you and then imposed on reality” but should show through “at the end like ribs of rock that make sense of a landscape” (p. 93). Other barriers discussed include the impersonal language of mathematics and the problem of students alienated from mathematics who just give up.
The chapter “How mathematics has been taught” addresses pressure to routinize mathematics education, never-resolved policy disputes, recurrent cycles of sweeping reforms, and the plight of teachers not trusted by policy makers intent on offering “teacher proof” programs.
Another chapter, “How mathematicians actually work”, examines the process of coming to an insight — frequently involving an effortless and seeming unconscious realization preceded by agonizing struggle over considerable time. They lament pressure to produce results within rigid time frames:
Working with converging walls of time may funnel intensity down, but makes what it works on an arbitrary goal rather than an entrée into significant vistas. The chance to profit from being wrong has been squeezed away, and we’re left with the illusion that solutions are always sailed at head on, rather than tacked around. How easy it is to forget that mathematics is a humanist rather than a mechanical enterprise. (p. 156)
The Kaplans discourage an atmosphere of excessive competition, and instead foster cooperation in working on a problem. “A roomful of egos becomes absorbed in It rather than I, and surprising pleasures open up of forgetting yourself in the play” (p. 162). They also remark:
Since the majority of suggestions that come up are faulty or imprecise or incomplete, fear and embarrassment disappear in a mutual mulling over and reshaping of questions and answers. [Students] like the image we’ve passed on to them from the mathematician Barry Mazur: we are all very small mice gnawing at a very big piece of cheese; no shame, then, in having bitten into a hole, nor any need to hoard up precious crumbs. (p. 12)
Mathematics competitions, especially the Putnam, come under scrutiny (pp. 189-97). The Kaplans’ assessment coheres with their approach to teaching given in their book:
Why should we hand over the way we learn mathematics to the less attractive parts of our nature (greed for conquest, a shallow view of the self and of mathematics, and a dislocated sense of the good)? Why not let math evolve with us, gaining in imagination and scope, and ever more buoyant in the freedom which is its essence? Let’s treat it seriously, and the people doing it as equals. Let’s play with their strengths, rather than at their weaknesses. Let’s invent our way together toward discovery. (p. 197)
Many quotations embellish and strengthen the book. For reasons not given, a minimalist citation style is adopted in which only the author of a quotation is given and no bibliography is provided.
Out of the Labyrinth offers a lucid, firsthand account of unfolding mathematical insight and growth of mathematical knowledge. Many are likely to find this book useful and insightful, including students, teachers, philosophers, psychologists, and those who might wish to follow in the Kaplans’ footsteps.
Dennis Lomas (firstname.lastname@example.org) has studied computer science (MSc), mathematics (half dozen, or so, graduate courses), and philosophy (PhD). He resides in Prince Edward Island (Canada).