Learning the Language of Mathematics

Just as everybody must strive to learn language and

writing before he can use them freely for expression of his

thoughts, here too there is only one way to escape the weight

of formulas. It is to acquire such power over the tool that,

unhampered by formal technique, one can turn to the true

problems.

— Hermann Weyl [4]

I want to show how making the syntactical and rhetorical structure of mathematical language clear and explicit to students can increase their understanding of fundamental mathematical concepts. I confess that my original motivation was partly self-defense: I wanted to reduce the number of vague, indefinite explanations on homework and tests, thereby making them easier to grade. But I have since found that language can be a major pedagogical tool. Once students understand HOW things are said, they can better understand WHAT is being said, and only then do they have a chance to know WHY it is said. Regrettably, many people see mathematics only as a collection of arcane rules for manipulating bizarre symbols — something far removed from speech and writing. Probably this results from the fact that most elementary mathematics courses — arithmetic in elementary school, algebra and trigonometry in high school, and calculus in college — are procedural courses focusing on techniques for working with numbers, symbols, and equations.         

Although this formal technique is important, formulae are not ends in themselves but derive their real importance only as vehicles for expression of deeper mathematical thoughts. More advanced courses —such as geometry, discrete mathematics, and abstract algebra — are concerned not just with manipulating symbols and solving equations but with understanding the interrelationships among a whole host of sophisticated concepts. The patterns and relationships among these concepts constitute the “true problems” of mathematics. Just as procedural mathematics courses tend to focus on “plug and chug” with an emphasis on symbolic manipulation, so conceptual mathematics courses focus on proof and argument with an emphasis on correct, clear, and concise expression of ideas. This is a difficult but crucial leap for students to make in transitioning from rudimentary to advanced mathematical thinking. At this stage, the classical trivium of grammar, logic, and rhetoric becomes an essential ally.

There is, in fact, a nearly universally accepted logical and rhetorical structure to mathematical exposition. For over two millennia serious mathematics has been presented following a format of definition-theoremproof.

Euclid’s Elements from circa 300 BC codified this mode of presentation which, with minor variations in style, is still used today in journal articles and advanced texts. There is a definite rhetorical structure to each of these three main elements: definitions, theorems, and proofs. For the most part, this structure can be traced back to the Greeks, who in their writing explicitly described these structures. Unfortunately, this structure

is often taught today by a kind of osmosis. Fragmented examples are presented in lectures and elementary texts. Over a number of years, talented students may finally unconsciously piece it all together and go on to graduate school. But the majority of students give up in despair and conclude that mathematics is just mystical gibberish.

I have been working for several years now on developing teaching strategies and developing teaching materials for making the syntactical and logical structure of mathematical writing clear and explicit to students new to advanced mathematics. The results have been gratifying: if the rules of the game are made explicit, students can and will learn them and use them as tools to understand abstract mathematical concepts. Several years ago, I had the opportunity of sharing these ideas with the Occasional Seminar on Mathematics Education and now through this article, I hope to share them with a wider audience.

One should NOT aim at being possible to understand, but at

being IMPOSSIBLE to misunderstand.

-----— Quintilian, circa 100 AD

The use of language in mathematics differs from the language of ordinary speech in three important ways. First it is nontemporal — there is no past, present, or future in mathematics. Everything just “is”. This presents difficulties in forming convincing examples of, say, logical principles using ordinary subjects, but it is not a major difficulty for the student. Also, mathematical language is devoid of emotional content, although informally mathematicians tend to enliven their speech with phrases like “Look at the subspace killed by this operator” or “We want to increase the number of good edges in the coloring.” Again, the absence of emotion

from formal mathematical discourse or its introduction in informal discourse presents no difficulty for students.

The third feature that distinguishes mathematical from ordinary language, one which causes enormous difficulties for students, is its precision. Ordinary speech is full of ambiguities, innuendoes, hidden agendas,

and unspoken cultural assumptions. Paradoxically, the very clarity and lack of ambiguity in mathematics is actually a stumbling block for the neophyte. Being conditioned to resolving ambiguities in ordinary speech,

many students are constantly searching for the hidden assumptions in mathematical assertions. But there are none, so inevitably they end up changing the stated meaning — and creating a misunderstanding. Conversely,

since ordinary speech tolerates so much ambiguity, most students have little practice in forming clear, precise sentences and often lack the patience to do so. Like Benjamin Franklin they seem to feel that mathematicians spend too much time “distinguishing upon trifles to the disruption of all true conversation.”

But this is the price that must be paid to enter a new discourse community. Ambiguities can be tolerated only when there is a shared base of experiences and assumptions. There are two options: to leave the students in the dark, or to tell them the rules of the game. The latter involves providing the experiences and explaining the assumptions upon which the mathematical community bases its discourse. It requires painstaking study of details that, once grasped, pass naturally into the routine, just as a foreign language student must give meticulous attention to declensions and conjugations so that he can use them later without consciously thinking of them. The learning tools are the same as those in a language class: writing, speaking, listening, memorizing models, and learning the history and culture. Just as one cannot read literature without understanding the language, similarly in mathematics (where “translation ” is not possible) this exacting preparation is needed before one can turn to the true problems.

This article is a report on my efforts to make the rhetorical and syntactical structure of mathematical discourse explicit and apparent to the ordinary student.As such it is about teaching and learning the tool of language in mathematics and not about grappling with the deeper problems such as the discovery of new mathematics or the heuristic exposition of complex mathematical ideas or the emotional experience of doing mathematics. As important as these deeper problems are, they cannot be approached without first having power over the tool of language. Mastering the trivium is necessary before the quadrivium can be approached.

Mathematics cannot be learned without being understood

— it is not a matter of formulae being committed to memory

but of acquiring a capacity for systematic thought.

---------— Peter Hilton [3]

Systematic thought does not mean reducing everything to symbols and equations — even when that is possible. Systematic thought also requires precise verbal expression. Since serious mathematics is usually communicated in the definition-theorem-proof format, the first step in learning the formal communication of mathematics is in learning definitions. For this reason, and because it requires the least technical sophistication, I will illustrate my general methodology with definitions. Although the examples below are kept elementary for the sake of the general reader, the principles they illustrate become even more critical the more advanced the material. This is sometimes a difficult point for students, who may not understand the need for meticulous precision with elementary concepts. But to have the technique needed to deal with complicated definitions, say the definitions of equivalence relations or of continuity, it is necessary to first practice with simple examples like the definitions .