Why is Mathematics Supposed to be so Difficult?

(This article was originally posted on blogger.com on Jul 7, 2005)

This post is a response to a recent article by Tine Byrckel in the Danish newspaper Information about “The Mystery of Mathematics”.

Tine’s main concern is two-fold. First of all why so few Danish women (at high-school level and up) are interested in mathematics and secondly, why math in high-school is presented as a mysterious subject, which definitions, theorems etc. you should just accept – as they are – since they are not motivated in any way, and that the core ideas in math and the important intuition about math are not given much time, if any at all.

Why women tend to show little interest in mathematics and physics, at least in Denmark and many other countries, I cannot really explain, but only guess about. One could imagine that this is an effect of both biological and cultural factors (there could be other reasons, but I will not discuss them here). Globally it is correct, that women are underrepresented in both math and physics (and in a number of other “hard sciences”). However, locally there can be big differences. I would estimate that at most 10% of those who are studying physics at the Niels Bohr Institute in Copenhagen are female, while the corresponding number in, for example, one of the biggest universities in Lisbon of Portugal is much closer to 50%. As far as I remember,  the percentage of women studying physics at the universities of Portugal and Italy is quite close to 50%. But the number of those female students, which end up graduation with a Ph.D., is probably much less than 50%. This could suggest that it is in many ways a cultural phenomenon. With respect to the biological side, one could imagine that the males and females different ways of thinking could manifest itself in the way that men seem to be more attracted to math and physics than women are. However, actually I don’t believe that this question is something that can be settled on the basis of science. What we are seeing is – in broad terms – only an instant view, and we don’t know how the situation is going to develop in the future, even with or without any governmental “regulations”. As an example, has the number of women attending medical school at the universities of Denmark increased quite tremendously in recent years, and there are now more females than males at these schools. 

I would like to add some comments about how math and physics (at least in Denmark) are taught in both high school and at the universities, but some of these comments might also apply to the situation in the US, for example. Generally I believe that the quality of the teaching of math and physics in most high schools is simply too low, and that could explain why many women tend to choose other fields of study before entering college. That there are, from my point of view, also many problems with the way math is taught at university, only makes the situations worse. But why should males react differently than females to the way math and physics is taught in high school?

I am criticizing the education in math and physics at high schools for several reasons. Generally, the teachers are simply not good enough. I suspect, among many other things, that the reason behind this is, that it is the lesser successful students at university, which later choose to teach in high school. I’m just saying this based on my own experience and based on my experience with private teaching of high school students while I was studying at the Niels Bohr Institute.

Here I would like to add a personal story. I’ve just recently taught a female first-year high school student in math and physics. The physics book was (not to my surprise) written by their own teacher, and contained a surprising high number of errors. Here I’m not talking about misspellings, misprints or the like, but about actual misconceptions of fundamental physics or wrong and imprecise definitions. In this book vectors are, as always, used in connection with Newton’s laws of motion combined with a plenty of fine illustrations of cars, bricks and much else, which are being affected by forces, which again are represented by vectors. But it is not explained any place in the book what a vector really is, only that you can add two or three of them, but not that you can multiply them with real numbers, subtract them and define an (inner) product of vectors. And there is not much help in the companion book from the math course, since here there are simply no chapters about vector calculus!

So the teaching is also poor since the books are too poor, and obviously not written or edited by experts. The books are not bad just because of the things I mentioned above, but also because they try to learn you lots of useless stuff. Classical geometry and trigonometry (about triangles etc.) might have been useful to be taught about around 50 years ago, but to teach about a huge number of theorems about the angles of triangles, the ratio between different angles, the ratio between the edges of similar triangles, cosine- and sine-relations and so on, simply has no or very little usability today, at least that is how the situation looks like for a student. Of course there are a few definitions and theorems, which could prove good to know (for example, you cannot escape knowing what cosine and sine means, if you are studying physics or engineering), but those fundamental things can be written down on a couple of pages compared with the actual 20-30 pages filled with silly theorems and definitions which no high school student possibly could remember.

I am of course aware, that there are an enormous number of applications of trigonometry. Fields which make use of trigonometry include astronomy (and hence navigation), optics, analysis of financial markets, medical imaging, architecture and so on, but the central point is not whether trigonometry is relevant, but rather what part of trigonometry you should choose to teach high school students about and how you should make them interested in the subject. I don’t know who is responsible for choosing the literature, which is to be taught from in high school, but I don’t think that those who are, have had any contact with the university community (or fundamental research) for many years, if ever.

All this said, I can understand why many students find mathematics mysterious. However, I don’t think that mathematics is so mysterious, no more mysterious than you make it look and feel like. This also depends a lot on how you actually teach math. It happens quite often at university that the teacher in a certain branch of mathematics (algebra, classical analysis, group theory and so on) present the subject as one huge pile of definitions (or axioms) and theorems which you are then supposed to spend a lot of time proving. This is simply not the way a mathematician thinks about math, or the way that the field develops. In my view, the good teachers are recognized by their ability to explain and visualize the central ideas behind the math.

 

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4 Responses to Why is Mathematics Supposed to be so Difficult?

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  2. appah pere says:

    i have a question
    my question is
    mention and briefly explain five (5) reasons why mathematics appears to be a difficult subject to the primary? i need the reply immidiatly
    thanks

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    […]Why is Mathematics Supposed to be so Difficult? « Thoughts on science and life[…]…

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