[Reposted from multijimbo.]

I have been a teacher for many years now; in fact, we’re now rapidly approaching the point at which I’m thrice the age of my students, rather than merely twice. I teach 2 very different things. On the one hand (which I can write without violating the basic tenets of the Number Liberation Front, as I’m using ‘one’ not as a number but as an adjective, I suppose), I teach mathematics to undergraduate university students, and on the other hand, I am 1 of the instructors in the University aikido club. I’ve been thinking recently about the commonalities and differences between the 2 types of teaching.

There is an obvious difference between the 2. To practice aikido requires physical contact. Someone grabs me, or attempts a strike, and I need to do something rather quickly. (Or is it quite quickly? Having lived in 2 countries where the use of ‘quite’ and ‘rather’ is different, I am now very confused and can’t remember which is the current local usage.) The end of a good, active aikido session can be sweaty. This physicality leads to a directness in teaching. When I’m being thrown by a student, I have the opportunity to feel exactly what they’re doing, right and wrong, which I can then feed back to them immediately.

Mathematics, on the other hand, can be done in isolation. (And to follow a random train of memories, this brings to mind Ms Shearer, my 6th grade teacher, with whom we spent a session listening to Simon and Garfunkel’s I am a rock, I am an island, and discussing how people cannot exist in isolation, as they remain part of the cultural in which they grew up.) Also, mathematics rarely involves physical combat. Not never, mind you, just rarely. In terms of teaching, though, mathematics teaching is a bit more at a distance than aikido teaching. Part of this is that mathematics classes tend to significantly larger than the aikido classes I teach. Also, a good, active mathematics class rarely ends in sweat.

Even so, there are for me some deep and significant similarities. These are things that no doubt are similar to the teaching of many things, but hey, this is my meditation. The similarity I would like to focus on here is the lead-a-horse-to-water phenomenon that is regularly, and sometimes almost brutally, brought home to me in both teaching fora.

In both aikido and mathematics, there are some basic, fundamental ideas that underlie everything that we do, and that I try to bring out and illustrate as much as I can through my teaching. This is after all, in my mind at least, what a teacher should do. I have spent time studying how to do particular things, learning from my contemporaries and those who have gone before, and I can use the miracle of language to take what I’ve learned and provide my students with some short cuts, so that they can get farther along the path a bit faster than me.

In aikido, 1 of these basic, fundamental ideas is that at any moment in a technique, I should understand where my balance is and what is happening within both my own centre and my partner’s centre. The way I like to try and embed this idea into my students’ brains is to have them go slowly through a technique, paying attention throughout. But this requires that the student is willing to do the technique slowly, and alas not all of them are. So I talk, I demonstrate, I cajole, but in the end, I cannot force. Ultimately, I cannot teach anything. All I can do is to provide guidance for my students on how they might learn and provide them with an environment within which they can learn.

In mathematics, the basic, fundamental idea on which I like to focus is that each statement, each assertion, needs to come from somewhere. With each question, we have to start with things we know to be true and work out from there. Part of an undergraduate mathematics education, and indeed mathematics education before university, is to provide students with a collection of facts, procedures and processes that we know to be true. Mathematics does not come from nothing. Mathematical facts do not spring full-grown from the head of Zeus. Rather, mathematical facts are the product of accretion and accumulation (and this is where the sweat comes from). We have just come to the end of the semester, and as in all previous years, I have the evidence that some of my students listened, and some didn’t.

So, what to do? There is nothing to do besides persist. Some students listen and some students don’t, but I have come to believe that it is these larger things, these fundamental ideas, that are by far the more important things that I teach, far beyond the individual techniques of aikido or the definitions, theorems and examples in mathematics. And so we persist. As Samuel Beckett once wrote, ‘Try again. Fail again. No matter. Try again. Fail again. Fail better.’