My general practice with abstractmath.org has been to write about the problems students have at the point where they first start studying abstract math, with some emphasis on the languages of math. I have used my own observations of students, lexicographical work I did in the early 2000’s, and papers written by workers in math ed at the college level.
A few months ago, I finished revising and updating abstractmath.org. This took rather more than a year because among other things I had to reconstitute the files so that the html could be edited directly. During that time I just about quit reading the math ed literature. In the last few weeks I have found several articles that have changed my thinking about some things I wrote in abmath, so now I need to go back and revise some more!
In this post I will make some points about definitions that I learned from the paper by Edwards and Ward and the paper by Selden and Selden
I hope math ed people will read the final remarks.
Peculiarities of math definitions
“When I use a word, it means just what I choose it to mean–neither more nor less.” — Humpty Dumpty
A mathematical definition is fundamentally different from other sorts of definitions in two different ways. These differences are not widely appreciated by students or even by mathematicians. The differences cause students a lot of trouble.
List of properties
One of the ways in which a math definition is different from other kinds is that the definition of a math object is given by accumulation of attributes, that is, by listing properties that the object is required to have. Any object defined by the definition must have all those properties, and conversely any object with all the properties must be an example of the type of object being defined. Furthermore, there is no other criterion than the list of attributes.
Definitions in many fields, including some sciences, don’t follow this rule. Those definitions may list some properties the objects defined may have, but exceptions may be allowed. They also sometimes give prototypical examples. Dictionary definitions are generally based on observation of usage in writing and speech.
Imposed by decree
One thing that Edwards and Ward pointed out is that, unlike definitions in most other areas of knowledge, a math definition is stipulated. That means that meaning of (the name of) a math object is imposed on the reader by decree, rather than being determined by studying the way the word is used, as a lexicographer would do. Mathematicians have the liberty of defining (or redefining) a math object in any way they want, provided it is expressed as a compulsory list of attributes. (When I read the paper by Edwards and Ward, I realized that the abstractmath.org article on math definitions did not spell that out, although it was implicit. I have recently revised it to say something about this, but it needs further work.)
An example is the fact that in the nineteenth century some mathematicians allowed $1$ to be a prime. Eventually they restricted the definition to exclude $1$ because including it made the statement of the Fundamental Theorem of Arithmetic complicated to state.
Another example is that it has become common to stipulate codomains as well as domains for functions.
Student difficulties
Giving the math definition low priority
Some beginning abstract math students don’t give the math definition the absolute dictatorial power that it has. They may depend on their understanding of some examples they have studied and actively avoid referring to the definition. Examples of this are given by Edwards and Ward.
Arbitrary bothers them
Students are bothered by definitions that seem arbitrary. This includes the fact that the definition of “prime” excludes $1$. There is of course no rule that says definitions must not seem arbitrary, but the students still need an explanation (when we can give it) about why definitions are specified in the way they are.
What do you DO with a definition?
Some students don’t realize that a definition gives a magic formula — all you have to do is say it out loud.
More generally, the definition of a kind of math object, and also each theorem about it, gives you one or more methods to deal with the type of object.
For example, $n$ is a prime by definition if $n\gt 1$ and the only positive integers that divide $n$ are $1$ and $n$. Now if you know that $p$ is a prime bigger than $10$ then you can say that $p$ is not divisible by $3$ because the definition of prime says so. (In Hogwarts you have to say it in Latin, but that is no longer true in math!) Likewise, if $n\gt10$ and $3$ divides $n$ then you can say that $n$ is not a prime by definition of prime.
The paper by Bills and Tall calls this sort of thing an operable definition.
The paper by Selden and Selden gives a more substantial example using the definition of inverse image. If $f:S\to T$ and $T’\subseteq T$, then by definition, the inverse image $f^{-1}T’$ is the set $\{s\in S\,|\,f(s)\in T’\}$. You now have a magic spell — just say it and it makes something true:
- If you know $x\in f^{-1}T’$ then can state that $f(x)\in T’$, and all you need to justify that statement is to say “by definition of inverse image”.
- If you know $f(x)\in T’$ then you can state that $x\in f^{-1}T’$, using the same magic spell.
Theorems can be operable, too. Wiles’ Theorem wipes out the possibility that there is an integer $n$ for which $n^{42}=365^{42}+666^{42}$. You just quote Wiles’ Theorem — you don’t have to calculate anything. It’s a spell that reveals impossibilities.
What the operability of definitions and theorems means is:
A definition or theorem is not just a static statement,it is a weapon for deducing truth.
Some students do not realize this. The students need to be told what is going on. They do not have to be discarded to become history majors just because they may not have the capability of becoming another Andrew Wiles.
Final remarks
I have a wish that more math ed people would write blog posts or informal articles (like the one by Edwards and Ward) about what that have learned about students learning math at the college level. Math ed people do write scholarly articles, but most of the articles are behind paywalls. We need accessible articles and blog posts aimed at students and others aimed at math teachers.
And feel free to steal other math ed people’s ideas (and credit them in a footnote). That’s what I have been doing in abstractmath.org and in this blog for the last ten years.
References
Things. University of Chicago Press, 1990. See his discussion of concepts and prototypes.
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