Teaching Math by Discovery

Robert and Ellen Kaplan’s latest, Out of the Labyrinth, describes the joys of pure discovery learning as applied to mathematics. The current system emphasizes the power of aptitude and centers around a slow accumulation of knowledge that takes place over 12 years. Instead, they recommend their own Math Circle, where children who were not screened for prior knowledge or aptitude are nonetheless caused to invent irrational numbers at the age of 5.

Most of the middle of the book is theoretical and explains analogies for math learning; as is the usual for books that weave together theory and practice, the theory can be safely skipped. The important parts of the book are the parts at the beginning that explain the setup of the Math Circle, the chapters on the problems with aptitude-based teaching, and the chapters toward the end about math curricula.

Chapter Three, “The Myth of Talent,” advocates ideas interesting and enlightening enough that it could be expanded into several books. The Kaplans don’t quite claim that all people are equally talented at math, which would be just plain wrong. Instead, they show how the American educational system emphasizes aptitude too much.

Much later in the book, they write about stereotype threats, which cause women (and minorities) to perform worse on tests when they’re told that women (or minorities) perform worse than men (or white people). I’ve heard that this research has been generalized to tiered learning: students in general perform at about the level they’re expected to, so that tiering causes lower-ranked students to underperform.

For sure, the USA’s problems in math education aren’t purely the result of fourth-grade aptitude tests. Singaporeans are segregated into tiers beginning in first grade and into different schools beginning in seventh, and yet routinely get the highest TIMSS scores in the world. The difference is mainly that Singaporean students are told, “You’re a failure, so you have to study hard,” which reduces performance less than the American version, “You’re a failure, so math is too hard for you and you needn’t care about it.”

So it’s partly cultural. However, the educational system can’t change culture, so focusing on telling people that test scores say very little about mathematical talent and deemphasizing standardized tests early on will work better. The Kaplans don’t talk about that entirely, but it’s the natural conclusion one draws from reading Chapter Three.

The Math Circle is designed to counter that problem. It’s not selective, although there’s a limited element of self-selection among its students. It stimulates discussions among students, which then promote the (re)discovery of mathematical facts, such as the existence of irrational numbers. It gets students to think for themselves, thereby reaching proofs of theorems on their own. And it has no tests or graded assignments.

Once the foundation of that is laid down, the instruction proceeds extremely rapidly; a whole topic takes ten weeks to cover. Among the list of topics for children ages 14 to 18 is algebraic geometry, which is usually only taught in the first year of graduate school to students repeatedly selected for interest in and talent for math.

Although within each topic the instruction is fairly linear, in general the topics don’t really depend on earlier topics. This counters the problem in traditional education, wherein students who have a problem with one area of instruction can’t keep up afterward.

Where the book’s thesis breaks down is toward the end, where the Kaplans generalize from their overwhelmingly positive experience with the Math Circle to general curricula. Standardized curricula are designed to be “Teacher-proof,” they say, requiring teachers to teach to the national average rather than to the class. Direct instruction should be replaced with more discovery, they imply.

One of the main problems with current education is that it’s developed and evaluated by teachers, who by and large were diligent A students in school. Not surprisingly, the system works well for diligent A students. Everyone else – creatives, uninterested students, students with specialized interests, B-F students – gets shafted.

Likewise, one of the main problems with educational reformers is that by and large they’re extraordinary teachers. Attempts to use the Math Circle’s principles in a normal classroom setting routinely confound most teachers.

Michel Thomas could teach people a language in a week – or at least make them believe they knew the language after a week, considering his record of creativity with the truth. He never explained how his system works, even when UCLA contacted him in order to use his method to teach their language courses. When it was finally reverse-engineered and fitted for use in a classroom setting, it no longer performed any better than other intensive language courses.

The same principle applies to discovery learning. In science education, studies have shown that direct instruction is superior to discovery learning in teaching not only scientific facts but also experiment design. In math education, American schools are increasingly using discovery learning hand in hand with overemphasis on calculators, without any improvement in results.

In other words, the Kaplans can make students excited about mathematics when they teach by discovery. The other couple hundred thousand math teachers evidently can’t. Not surprisingly, in low-income schools, which aren’t under immense pressure from parents to be decent, teacher-proof methods increase test scores.

It’s neither obvious nor shown in the book that the Math Circle requires discovery learning to be successful. For all I know, an equally enthusiastic teacher with a different teaching philosophy could achieve the same results by proving theorems on the blackboard and only asking students to think of ways of generalizing the results or the method of proof.

Not surprisingly, the apparent rarity of teachers who can teach in ways similar to those of the Math Circle is why application to schools is so limited. In elementary school, I had a series of classes that were supposed to teach creative thinking and worked in ways that were similar to the Math Circle. But I only had access to those classes because I was in a gifted class, which got perks other classes didn’t.

It’s entirely possible sending the same teachers to more classes but at less frequent intervals, or at the same frequency but to a regular class, would’ve helped even more. But there were neither the funds nor the number of teachers required for universal coverage, so instead, the city allocated these enrichment programs to just one class.

Implementing the easy reforms, such as making math classes less calculator-dependent, is likely to only make math education less dismal. Something more fundamental is needed to make it satisfactory, let alone good. And by all means, experimenting is good, as is supporting good enrichment programs like the Math Circle. Getting children excited about what the school system turns them off of is always a positive thing. But the Math Circle’s principles are as generalizable to the school system as the USA’s victory in Korea was to Vietnam.

One Response to Teaching Math by Discovery

  1. rocket spanish

    Spanish is an important languag and you can go far with it

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