My favourite pedagogical principle: examples first! | Gowers's Weblog

]]>Clearly mark the point where you leave the example and start the general definition (or proof).

In your example, this would be the point “in other words”, and there should be a paragraph (and properties should be numbered and structured as a list, too).

I had a professor who would calculate with examples and suddenly claim to have proved the theorem. I started to understand *why* students could have been happy with the introduction of the “definition, theorem, proof” style.

]]>The fallacy, in my view, is a very simple one. If a clear explanation is provided, then no examples are needed, but they certainly help. (Strictly speaking, that’s consistent with what Oded says, but not with his implied conclusion that there’s no point in examples.)

A different way of disagreeing would be this: examples are often what make a clear explantion clear.

]]>“SHORT OPINION NR 1” reads:

“ON EXAMPLES.

If a clear explanation is provided, then no examples are needed;

otherwise no examples will help…” ]]>

(i) a,b in F then a+b in F

(ii)…etc

Whether this is put first or second is perhaps neither here nor there; the central thing, from my perspective, is to emphasize that we care about fields because they clarify a lot of things about N, Z, Q, and C (and perhaps even more to the point, they distinguish Z_n from Z_p, both of which can be defined without a recourse to abstraction). This may seem like an obvious statement, but I think a good deal of university mathematics is unfortunately written from the perpsective that this would ‘dumb-down’ the material.

So I can’t agree with the sentiment that “This is University, playtime’s over, it’s time for some real work.” In my personal experience I have been able to learn math much more quickly (and deeply) if I view it as play, rather than ‘real work.’ In the former instance, the subject itself motivates me; in the second, it’s not the math, but just a desire to impress people or just do what’s expected. And there are easier ways to do these thing than to learn about tensor products, etc. From an even broader viewpoint, surely less sterile math will be developed if people learn in their undergraduate days to pursue abstractions because they are useful and clarifying, rather than pursue the abstractions because someone of authority has written them down.

]]>Tim Gowers has two very interesting posts on using examples early on in a mathematical exposition of a subject. I can only second that and say that this is my favorite way of understanding mathematical concepts: Try to think through the simplest non-…

]]>Although my answer to the second question is very often yes, it’s really my answer to the first question that I’d like to persuade people of. I may even end up writing “Examples first III” to deal with it at greater length, but a brief summary of my view is this. It depends on a distinction between “direct” memory (the sort you’d need to remember digits of , say) and “derived” memory (the sort you’d need to reconstruct a proof of a theorem from one key idea using your years of mathematical training). One of the difficulties of reading mathematics linearly is that you sometimes have to rely on direct memory because you don’t yet have anything from which to derive a memory. This would be true if a proof had steps that appeared to be arbitrary, or a complicated lemma was stated and you didn’t know how it was going to be applied, or a definition was given that turned out later to be exactly what you needed. Putting examples (and other kinds of motivation) first is a way of reducing the reliance of the reader on direct memory, since it sometimes makes it possible to derive the memory instead. (E.g., in the fields example, one could derive the list of axioms, or at least get a long way towards doing so, from the simple idea that they are the properties that hold of addition and multiplication in the rationals.) It’s this factual point that I’m mainly trying to push. (By “factual” I mean that it makes a statement about the world that could be true or false.) The normative point (this is how we ought to present mathematics) is secondary — of course, it’s no secret what my views are here too, but I don’t hold them in a rigid way and can think of situations where presenting examples first would not be helpful.

]]>I think people should be especially skeptical about transforming their tastes into strong principles.

I second this. As one much wiser than I has said, “It is my firm belief that it is a mistake to hold firm beliefs.”

]]>Regarding ‘fields’ I like the way Amitsur do it basing the definition of a field on trying to examine properties of the real numbers; I see no problem in mentioning Q and C as “positive” examples and N and Z as “negative” examples before giving the formal definition provided the students already know these examples. I am not sure it makes a big difference. If the main objective is to define “fields” and the students need an introduction to the complex numbers I would not suggest to give this example first but to wait after fields are defined.

Actually my own style/taste of teaching is in the direction of examples-first plenty of preliminary chat, stories, philosophy, non linear development and plenty apropos and even dubious humor. It is perhaps a good pedagogical principle that people should follow, within reason, their taste and style. But I think people should be especially skeptical about transforming their tastes into strong principles.

Apropos Amitsur, while I do not remember him giving much chat and non linear stories at class I do remember we had many chats in our faculty club (Belgium house) and we even tried once to work on a problem: Take a non Papussian projective space (3-dimensional), say a projective space

over the Quaternion. Is it possible to find there seven pairs of lines e1, e2, … e7 ; f1 f2 …f7 so that ei is disjoint from fi for every i and ei intersects fj whenever i and j are different. Such a configuration is impossible for a Papussian projective space (over a field).

As far as I know, this problem is still open.

The fields you’re referring to (i.e. those in an ordered list) seem to be these, whereas Tim’s fields are those.

This is one of the many instances where people working in different fields (pun not intended) turn a word of common English (or any other language) into a technical term in different ways that are often unrelated to each other.

]]>I wonder if sequences could be taught that way — by first doing LimSup and LimInf and then then limits. That way the student is first introduced to non-decreasing or non-increasing sequences where the ‘closer to the limit’ intuition can work. ]]>

Quite strangely, I myself wrote a rather long post about what Deane Yang commented about! I feel that the lecturers tell us a story, and they each have their own unique way of doing so. And as with books, you sometimes prefer one over the other. One fantastic lecturer gives us definitions first and then examples straight after and I follow that. Hmm, I think each theorem, definition etc deserves its own attention. Had the Intermediate Value Theorem not been introduced by a diagram, I would have had great difficulty in understanding and following its proof.

]]>Of course, it would be possible to start with the formal definition and then say, “If you think that looks strange, let’s consider some examples that show why it is as it is.” In this case I don’t feel that one way round is significantly better than the other, though in some ways the motivation-first is a better reflection of how one comes up with a definition. And it also does what Deane Yang suggests in the second paragraph of this comment.

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