Against introducing precise definitions first

Question

After introducing eight different ways of viewing the derivative of a function (infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic), Thurston, in his famous essay,

Thurston, William P. "On proof and progress in mathematics." New Directions in the Philosophy of Mathematics (1998): 337-55.

says

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His last sentence above suggests that, in many circumstances, one should not introduce precise definitions as a start to explaining a mathematical concept, but instead sketch out the different ways of thinking about that concept, and only then show how these different ways are reflected in a formal definition.

My guess is that this is not a widely applied teaching technique, and that many teachers would disagree, and believe you cannot convey a mathematical concept until those definitions are in place.

Two questions:

Q1. Is my hunch correct? Is there a widespread belief among math educators that definitions should precede the concept (to put it in a nutshell)?

Q2. If so, what are some counter arguments to delaying the definitions so they don't collapse the different ways of viewing a concept to one way?

Answer 1

At the secondary level, students have not yet mastered formal mathematics and most will need to continue learning concepts before definitions in many cases. The van Hieles (the Dutch educators who developed the Van Hiele Model of how students learn geometry) insisted that definitions should never come first. They claimed that students should first have informal experiences with a new idea until the class has developed a shared language around a concept before introducing the formal vocabulary that goes with the idea. Some educators disagree, insisting students need experience with analyzing new definitions.

I think the decision of when to introduce formal definitions will often need to be made on a case-by-case basis, and may depend on both the abstractness of the definition and the maturity of the student. Many ideas will need some exposure before the formal definition so that a context exists in which the definition can make sense. But some ideas are transparent enough that students could tackle them without preparation. For example, geometry students can probably understand the formal definition of a quadrilateral without any introduction because they will have already had exposure to that idea in previous classes. Students might be able to understand the definition of a logarithm if it is unpacked immediately, but a bit of preparation beforehand to introduce the definition of logarithm would probably be helpful. Defining a limit or a derivative for the first time will definitely require a considerable amount of conceptual preparation before introducing the formal definition.

Don't forget that introducing definitions first may be common to mathematics courses, but in the "real world," definitions almost always come about after mathematicians have wrestled with an idea for some time. Early definitions are often vague and imprecise. Allowing students to approach new ideas in a similar way is therefore an intellectually honest approach to real mathematics.

Answer 2

One of my most vivid memories of graduate school was working on a problem concerning the Gelfand-Kirillov dimension of certain rings. I had been puzzling over the definition (which was rather new to me) and had reached the conclusion that for a certain class of rings the GK-dimension was greater than 1. My advisor told me that I could not possibly be correct, because the dimension of such rings had to be 1. I insisted that per the definition of GK-dimension, my result was correct, and began to outline the proof, but he cut me off. "If what you say is right," my advisor said, "Then you have the definition wrong." But (I protested) that was the definition in the textbook! "If the definition in the textbook leads to that conclusion," said my advisor, "then the definition in the book is wrong, because that conclusion is unacceptable, and you need to figure out what the definition ought to be."

The point is that while from a purely logic-deductive perspective definitions are a priori stipulations from which conclusions follow, this perspective in many ways fundamentally misrepresents what mathematicians actually do. In practice definitions are instrumental: we construct them so that the properties we want to follow as conclusions will be entailed by them. (See also: Imre Lakatos, Proofs and Refutations.) For this reason I think it's always important for students to first see some examples of the thing to be defined, build some intuition of what kinds of properties they hold, and then present a definition that actually captures those properties.

(Related: my answer at Why would you teach Calculus before teaching Real Analysis?)

Answer 3

Teaching undergrads and graduate students, by this point I do not like "precise definitions" at the outset, for the sort of reasons given by W. Thurston, and also for reasons given by I.M. Gelfand. Namely, a suite of examples of phenomena and of causal mechanisms to illustrate various aspects of the topic at hand. Arguably, only after looking through the immediate examples can we understand what our terminology needs to cope with and refer to.

Also, I've found that most undergrads have cognitive or semantic troubles with "definitions"... since very little else in the world works this way. They often react by "memorizing" the definitions, rather than integrating them with phenomena. Artifacts of formatting and wording are often indistinguishable to them from genuine issues. This seems bad to me.

By the time students are in grad school, they seem to be "reconciled to" checking definitions without questioning the virtues of them. But, indeed, just as with dictionaries of human languages, definitions of complicated things are often not infinitely precise, and meanings or references drift with time.

The notion of "function" exemplifies some of these troubles. First, it would be silly to deny that at entry level most functions are given by formulas. Second, yes, we want a notion of "function" that does not depend entirely on formulas. Third, yes, the set-theoretic epiphany to define a function $f:X\to Y$ as a subset of the cartesian product $X\times Y$ with certain properties was a very nice umbrella. "(E.g., Lebesgue) measurable" as a qualification was/is very useful,... and/but leads to a slight loss of pointwise sense, since now "almost everywhere" turns out to be the useful degree of precision. And, indeed, $L^2$ spaces of "functions" do not have pointwise values, yet are the right things to look at for many purposes. Similarly, "generalized functions" (a.k.a. "distributions") usually do not have pointwise values, so are certainly not specifiable by "graphs".

... though, yes, the Schwartz Kernel theorem does recover many of the lost items!

And so on.

In particular, "pointwise convergence" is not usually the real issue, to my mind! Convergence in $L^2$, or in Sobolev spaces, or uniformly pointwise, sure.

So, then, one of my first talking points in teaching graduate-level real analysis this year, with an eye to updating the course content, is to illustrate the various troubles with "pointwise" notions, arguing that we need/want a broader outlook on the notion of "function", at the very least to be able to accommodate actual practice!

From the viewpoint of exposition, too, I do not like definitions. If/when they appear without motivation or examples, why should I accept them? Why should the students? An implicit justification "by authority" is not the kind of reason I want to promote. I want to see phenomena first, and any discussion of modifications of word-usage afterward.

That is, although I realize that quite a few people like the other-worldliness of mathematics, a great many students have difficulties, and I think with good reason, operating in a context where they are essentially required to suspend their own judgement and play by rules whose significance and origin is effectively kept secret. There may be virtues in the formal exercise of playing such games, but it is not clear to me that this is really the main selling point of mathematics.

Yes, I do like actual proofs, rather than merely heuristics, and this does require precise references and referents, but that's not the same thing as "definitions". Also, proving boring, superficial things very precisely is an odd sort of exercise, of limited interest to many people, or only of interest at very particular moments. A more interesting kind of thing is, for example, legitimizing the mathematical aspects of things physicists (Dirac, Bethe, et al) have done, with observable physical corroboration, to repurpose the mathematics more broadly. In that sort of circumstance, it is not within our power to "define" anything into or out of existence. Instead, to my perception, the issue is of grappling with examples and phenomena. Yes, there is the secondary issue of mutual intelligibility, but it does seem to me secondary.

Answer 4

In the comments I mentioned using a game called Definitions war. Before presenting a given definition, ask students to come up with their own definitions of an object. Then work on improving it, with some fun. Here's an example of a class attempting to come up with a good definition of 'rectangle.'

If we are to 'teach' the definition of derivative, we can start with thinking about a tangent line to a curve (students generally have intuition about this), and explore why that's so hard to 'define.'

Answer 5

I agree with Scott Eberle that the decision should be made on case-by-case basis, but I would like to argue that most of the established concepts would actually benefit from giving the precise definition first.

The basic reason is that it gives students something concrete to think and reason about, thus avoids any vague hand-wavy limbo state.

My rule of thumb is to check if there is one, well-established canonical definition of a concept. If there is, I would state it first, and only then consider:

• different intuitions and understandings,
• how they compare to the given formal definition,
• how we could use some other definition which currently we have as theorems,
• how some intuitions fail, i.e., should not be used as definition (but can be possibly still useful, e.g. they work for some smaller class of objects).

One the other hand, if there is no such clear canonical answer, i.e., different areas use different definitions, or the actual definition is bootstrapped from simpler cases, then I find prefer to start with laying out the various intuitions and generalizations (partial definitions).

To give some examples:

• There is one, very simple definition of a function: a collection of pairs; only after that I would say anything about representing it as a function-graph or creating such pair collections from formulas.
• Regular languages can be defined using monoids, automata, grammars/regular expressions; each approach has its own set of intuitions, and I find it beneficial to state those first, before picking option as the definition and proving others as theorems.

I have never taught derivatives, but I think I would like to state the formal definition first, and then show how it accommodates various intuitions we would like it to have.