Communicating mathematics

Question

As a TA, my ratings have been fairly mediocre. There's nothing to suggest I'm a problem, but I have room to improve.

What's more disturbing to me is that I even struggle to communicate math with people who know a lot more than I do. It's no surprise that they struggle to communicate with me. They know more, probably forgot what it's like to know less, and generally do not give the details on everything, sometimes even neglecting to mention that a particular detail needs to be filled in. But when I communicate, I try to leave out as few details as possible and clearly enunciate when I am leaving them out, and to minimize the number of mistakes I make. I am extremely nitpicky about this.

So there must exist other obstacles to communication other than just misrepresentations. I suspect that what I'm lacking is a pedagogical element, which would also explain my interaction with students. How can I go about finding out what is lacking, and then fixing it? What are some common teaching mistakes made by people with the attitude that mathematics should be precise and airtight? It's not that I'm such an adherent of my own philosophy that I refuse to "dumb things down." That used to be the problem, but it appears that when I attempt to convey vague ideas, people do not understand me.

Answer 1

As an experienced teacher, let me offer a few bits of advice.

By all means, do go to talks and see what makes some speakers effective (in general, perhaps not always to you) and what makes other speakers less so.

One of the arts of teaching is to anticipate where the audience will have difficulties. In large part, this comes with experience. But it's important to think about it when you're preparing your own lectures. If you've already seen that students have had issues understanding certain concepts or certain types of computations, try to address that head-on in your presentations.

Another mistake that many teachers make (sadly, even those with lots of experience) is to get mired in proofs/derivations before the students even have a context or any reason to care. Well-chosen examples done before the general theory help a lot. Indeed, I would say the same thing holds for you to understand a theorem: You really don't understand it until you can give me one or two examples (perhaps one of them trivial) to illustrate its significance and its application. This is part of the motivation to which others have referred. :)

Last, as far as teaching is concerned, letting the students know that you care about their learning and progress is huge. It's ok, once you establish rapport, to be honest about your disappointment in their test scores, etc., but they have to know that you want them to learn. And, Jeff, particularly in relatively low-level courses, they do not want to sit through your doing pedantic proofs that would satisfy you as a student. It's much more important for you to teach your precalculus and calculus students how to approach problems and how to write up understandable solutions of the problems. They don't need to know (let alone understand) the proof of the Maximum Value Theorem in Calc I; they do need to know how to give a solid application of it in the context of an applied max/min problem. They need to know how to set up functions in a logical way, think about their domains, understand why the function should have a maximum (other than just "the problem asked for it"), and then find it. (Many students these days have such weak algebra skills that those weak skills get in the way of so much more conceptually interesting stuff.)

And, last, I would concur with the folks on here encouraging you to answer more questions by way of learning to practice explaining things at different levels. Sometimes on here it's very hard to assess the level of the asker. I've noticed several graduate students (at least I think they are) giving explanations that would be appropriate for people at their level, rather than for the one who asked (who was clearly in an introductory undergraduate course).

OK, so my notion of "a few bits" is askew. :)

Answer 2

Most people, when presented with a claim, need a certain amount of time and/or contextual information to understand not why that claim holds, but what it means and why it matters.

Simple example: if you were to present to me a theorem about simple groups, having defined them, I would be staring at you wondering why I should care, because I have no concept of what makes simple groups important.

Answer 3

It is very hard to know when precision is going to trade off with clarity and when precision is going to be necessary for clarity. I don't pretend to know. However, two defaults:

1. at all levels of math, you should probably go a little slower than your instincts tell you to go. Do not worry about insulting an expert by going too slow. If the expert wants you to speed up, the expert will indicate that. Then you can speed up. Similarly, most students appreciate a slow clear explanation and are not insulted (in fact are happy) when you explain something they already know.

2. pausing to let the other person talk is extremely helpful. I find I have to force myself to do this.

Answer 4

A fairly random example of a great pedagogical technique for mathematics - examples first.

For example, suppose I was trying to explain to you what a ring is in algebra. I could start by telling you that a ring is a set $R$ together with two binary operations $+$ and $.$ such that:

• $r+s=s+r$ and $(r+s)+t$=$r+(s+t)$ for all $r,s,t\in R$
• There exists $0\in R$ such that $r+0=r$ for all $r\in R$
• etc. ...

However, you are much more likely to understand what a ring is if I give you some concrete examples. For example, I could say, 'Rings are like $\mathbb{Z}$. If we consider the integers, then we have two important operations $+$ and $\times$, which have the following interesting properties...' Then I could point out that other interesting sets like the integers $\mod n$, which also have two operations which behave in a similar way. Only then would I give you the formal axiomatic definition of a ring. The next step would be showing that many interesting things that are true for the integers are true for all rings, and that many other interesting things about the integers (like unique factorization) are true for rings with certain properties. Then you would gain some understanding of what rings are and why they are worth introducing.

Similarly, I don't think anyone would find the following definition at all meaningful if they hadn't studied topology before:

A topological space is a set $X$ together with a collection $\tau$ of subsets of $X$ (called the open sets) such that:

• $\varnothing,X\in\tau$.
• $\tau$ is closed under taking unions: for all (possibly infinite) collections of sets $(U_\alpha)_{\alpha\in A}$ with $U_\alpha\in\tau$, the union $\bigcup_{\alpha\in A}U_\alpha$ is in $\tau$.
• $\tau$ is closed under taking finite intersections: for all $A, B\in\tau$, $A\cap B\in\tau$ (and therefore all intersections of finitely many members of $\tau$ are contained in $\tau$).

Much better is to start off by introducing the more concrete idea of metric spaces (by first showing that a lot of concepts in real analysis, like convergence and continuity, can be expressed entirely in terms of the distance between points, and showing that more abstract ideas of distance can be useful) and then showing that the definition of a continuous function between metric spaces can be expressed entirely in terms of the open sets in that space, and then introducing topology from there.

Having given you some examples, I'll now state the general principle: if you want to explain some mathematics to someone, always start by telling them some motivating examples. This serves two purposes. Firstly, it shows them why they should care about the idea you're introducing, since all mathematical concepts were originally created because they were interesting in some way. Secondly, it helps them to understand the ideas that you're telling them about, since they have some concrete examples to link them to in their mind.

Of course, that's just one useful pedagogical technique, and it's not necessarily useful in all contexts. But it is often very useful.

For a much better exposition of this idea, see this blog post by Timothy Gowers.