Peter Krautzberger on the web

The single most important subject

Recently, I had a few conversations about teaching mathematics varying from elementary school to undergraduate level. These are not refined, in depth analyses, just some quick thoughts that I think worthwhile to ponder every once in a while.

The single, most important subject in school is language

The single, most important subject in school is the native/primary language — especially, for mathematics. This seems to strike people as odd. But I find this statement trivial. Too many people seem to think that everybody understands language anyway and that’s that. As a mathematician and science fan I often found that the lack of proficiency in languages compromises people’s abilities to deal with everyday concepts in our scientific world. The complexity of language is a challenge not to be taken lightly. Even professional journalists are frequently overwhelmed by the complexity of scientific writing — over at scienceblogging.org [Wayback Machine] you will probably find some science blogger crying out against a piece of misunderstood and/or abused science everyday.

In mathematics, it is however even more vital than in the sciences and the humanities. The sciences have empirical facts and the humanities have both emotional and factual aspects that allow us to understand their concepts with information outside of the language being used. If you don’t know what I mean, just look up to the stars, look at some original historical documents or listen to a poem read aloud.

In mathematics the situation is different, language is both alpha and omega. The initial step in understanding a new concept in mathematics is alike to Galileo’s problem: his first telescope was of such primitive nature that if you didn’t not know what you were looking at, you could easily get confused and not see what’s there. Written mathematics is often the only way to make a first step, maybe blindly following the formalism, maybe trying to find understanding by meditating over prose. Once this initial step has been taken we have diagrams, we have computer animation and most of all fruitful discussions; we have all sorts of helpful tricks to add to this initial understanding. But that is not enough, one cannot stop there. In the end, when push comes to shove, we only trust a detailed proof in written or spoken language (and if Doron Zeilberger ever reads this: code is valid for me). There are no other facts, no independent empirical facts, no historical facts that can significantly support a mathematical thought. The abstract thought that gives rise to mathematics can, it seems, only be exchanged neutrally using language.

At school level, it might seem that language is only needed for word problems. But these are often the worst examples — head over to Dy/dan to understand that. Language is used uselessly; lots of words for no mathematical content. At university on the other hand you often find only our highly specialized language without any supposedly superfluous understandable language. For example, you will encounter the tradition that refuses to find a way to write proofs as they are discovered (which actually often makes them more accessible); the university variant of dy/dan’s pseudocontext are epsilon-delta arguments written the ‘flawless’ but inaccessible way: let \epsilon > 0 ; set \delta to \frac{ 2 \sqrt{ln(\epsilon) – 5.2346}}{3\pi} .

So, teach language more! It’s the one and only subject that guarantees a) life long, self-governed education, b) citizens that can understand complicated (political) issues and c) students that have a better chance of excelling at mathematics (and in the long run produces better mathematicians, hooray)!