I’ve come to the conclusion that there might be many things in mathematics that perhaps need not be studied. But algebra is not among the topics that can be dropped. I’ve come to this conclusion by thinking about some of the conversations I’ve had and some of the observations I’ve done. It’s a roundabout argument. I will divide it up in two posts.
Let me start with the First Year Seminar I went to with Mark Huibregste at Skidmore. This was not about algebra at all – but it made me think back on discussions I’ve had with a colleague in Sweden (Magnus Lundin). He says, sometimes, that it doesn’t really matter what you study as long as you do it seriously. One aspect of this is that whatever you study, if you don’t regularly use it, you will forget about all the details. But if you have studied it seriously there is something you do retain. I will leave open for the moment what it is that you retain.
The question about what you (or students) retain from a mathematical education (say at a Liberal Arts College) also came up in the discussion after my seminar at Bryn Mawr College. Thomas Hunter from Swarthmore said that it was an experience (of having learnt mathematics). Josh Sabloff from Haverford quite strongly argued that it must be something more than an experience. A heated but good-humored debate followed – with Paul Melvin as a mediator. Partly it was a question of the meaning of words like “experience”. I listened. Josh made up an example of a medical doctor that ten years after the last mathematics course had to read a scientific article about tests of some new drug. How would that person go about judging the evidence put forward in such an article? His answer was: using knowledge retained from a mathematical education.
Such retained knowledge [now I’m filling out the discussion in retrospect with things that were not explicitly said – but was implicit in the trains of thought as I understood it] could consist of specific things such as reading tables and diagrams, parsing formulas – but perhaps more likely abstracted knowledge such as analytical and logical thinking, discriminating between what’s important and not, et cetera.
This is similar to what I wrote about in an earlier post about algebra: U. Dudley’s argument that algebra [mathematics] teaches us to think. This is something we all want to believe. I said (at Bryn Mawr) that it would be very interesting to have some kind of evidence that such deep (and tacit) knowledge is indeed what results from a good mathematics education. Is there anyone out there who knows about research – perhaps case studies or deep interviews that has looked into this?
But granted that mathematics teaches us quantitative, analytical and logical skills that are retained (perhaps tacitly) long after the details are forgotten, it can perhaps be contended that it does not really matter what parts of mathematics is studied as long as the teaching is good and the studies are serious.
Axiomatic Euclidean geometry is almost certainly in itself entirely useless knowledge for most people. But it may very well teach how to think!
So what about algebra? At Bennington College, Andrew McIntyre, had a course called “Introduction to Pure Mathematics”. Initially it was centered on geometry. It is a course whose aim is to teach mathematical thinking. But then he changed subject matter to algebra. Starting from the axioms for a field (in practice the real numbers, but the supremum axiom is not needed) he found it easier to do the proofs in detail in algebra than in geometry (whether from Euclid or Hilbert). As a bonus it turned out that an axiomatic approach to algebra lead to more reliable algebra knowledge for the students.
Is this the argument for algebra?
Not really – but it’s to late to continue – so you just get a cliff-hanger here.