Some Thoughts on Math Education

I've been reading this article, "A Brief History of American K-12 Mathematics Education in the 20th Century" by David Klein, and I wanted to write some of my own (basically uninformed) thoughts on math education.

As in some other academic areas, there is a sort of conflict in math education between the rigorous teaching (through repeated practice, drilling, etc.) of basic math skills and a more learner-centered, discovery-driven style of teaching, where students are encouraged to think critically and find their own paths forward, often by inventing their own ways of doing math or by solving novel, difficult problems.

Obviously this is not an either/or proposition, and unfortunately no style of math teaching is known to result in a uniformly well-educated population. I am ignorant of actual empirical findings in this area (and in any case skeptical of many of the types of empirical findings that I suspect exist, on methodological grounds), so I do acknowledge that everything I'm saying is essentially just based on my opinions from introspection and observation of the world around me.

In order to be any good at math, in order to solve new (to you) types of problems, and so on, you do need to practice the skill of working on an unknown new problem which isn't susceptible to a specified set of skills (as opposed to most of the exercises at the end of the section of any math text, which are usually basic applications of the techniques taught in the section). I think that this type of work is also what math is, on some level, about, and that discovery-centered learning (or whatever it's called) can lead to a great appreciation of math as a field.

On the other hand, if you can't reliably add fractions (as many of my precal students cannot), your ability to explore new problems will be severely constrained. If you never learned long division because your teachers think it is boring and obsolete in the age of ubiquitous computing devices, you'll find it harder to learn polynomial long division, and when you encounter a more novel problem later, you may not even imagine it as a way forward. If your notion of a limit is only vague, you won't be able to write an analysis-style proof to solve a problem in a metrizable topological space. If you can't compute a double integral you may never understand the unique properties of the normal curve. If you can't mechnically process the symbolic logic behind a proof by contradiction, you may introduce logical errors even when you understand the argument you're trying to make.

All of which is to say that doing anything interesting at a given level usually relies on the boring techniques of previous levels.

I had a funny moment the other day when I needed to compute some zeroes of a function for my own work, because I realized I was using an exact skill I had just taught in our precal class. I think my precal students probably imagine that my work is a lot more like theirs than it actually is (almost none of my work involves computational "problems"), yet here was an elementary technique from their class which I absolutely needed to use in order to proceed.

Returning to my uninformed ramblings about younger students, there is also this. Little kids should not be bored into submission by having to do pages and pages of long division problems while being forbidden from exploring their own math interests (or discouraged from ever having those interests). At the same time, some kids (and adults) enjoy the part of math where you learn to do something neatly and properly and then execute that skill over and over. (I enjoy this aspect of proof-writing, so this enjoyment can also exist on a level well past arithmetic.) That enjoyment is not wrong or somehow antithetical to the spirit of mathematics, and the frustrations a kid may feel with never being taught a correct algorithm for doing anything and being expected to derive and explain her own methods for every new thing are also legitimate. (On a practical level, a ton of jobs are ideally suited to people who enjoy being methodical and careful, and cultivating that habit is a proper function of education.)

In my ideal world, people would have some appreciation for the abstract qualities of math, and at the same time, would feel comfortable doing the kinds of manipulations they find helpful. They'd be able to double a recipe either by using reasoning to develop an ad hoc method or by relying on a trusty algorithm for fraction multiplication. Would-be engineers would show up to college with at least the basic skills required to study calculus, and kids predisposed to be mathematicians would arrive with some experience having fun working on hard or weird problems.

Bad Week

I usually post cheerful things, but not today. I'm not sure whether to organize this post by subject or by calendar, but let me start by subject.

Topology

We had an in-class midterm Thursday morning. I left 3 out of the 5 questions completely blank. This exam also has a take-home portion, originally due Tuesday but now technically due Thursday. I have answered three of the questions so far and I have worked several hours on each of the other two questions with no results.

One of those two was originally wrong, which I pointed out to our professor (gingerly, since I am usually mistaken about such things), and so he fixed it (hence the new due date). The new question is similar to a question we had in our last homework which I couldn't answer then, either. I can get 10 bonus points for turning in a counterexample to the original statement, which I have written up.

So basically I am up to 60/100 points for this exam, and the maximum I can get (considering the lost points from the in-class part) is 80/100. I have no reason to believe that low scores are OK in this class and I think most people will be able to get the problems I couldn't get.

Algebra

We had an in-class exam Friday afternoon. I wrote correct (I believe) answers for 2.5 out of 5 problems, and a woefully incomplete answer for 1/2 of another problem. I'm expecting around a 50% on this exam. We also have an assignment due Thursday which I haven't even looked at.

Probability

I would have had a meeting with my professor for this reading course on Thursday, but I asked him to postpone it, so we meet tomorrow morning instead. I have four assigned problems this week, of which I have completed 1.5. There is another question that I can tell is extremely easy but I can't find the math to express the question and so I can't possibly give an answer. I haven't even looked at the fourth question.

Normally we would meet again on Thursday by which time I would have read another section of our book and answered the questions for that section, but I can't see that happening at all. I have no idea how much of this the professor will tolerate; he doesn't have a particularly tolerant demeanor.

So basically I feel completely doomed right now. Everything I work on turns to shit. I should note that, unlike in most of my life so far, I am not in this position as a result of slacking off. I've been working hard the entire semester. I studied for both exams and felt reasonably well-prepared for each one. I've taken about three Sundays off this entire semester.

Anyway, that's my life at the moment.