Today's post is mainly about two (accidentally complementary) pieces I read recently about math learning and teaching (something I became re passionate about after last semester spent with first years students at King's -- this semester I have second and third year students).
The first one is In Math You Have to Remember, In Other Subjects You Can Think About It. I remember that when I first saw this title, my heart missed a beat, while my mind was screaming "What The F*** !". Turns out that this entry of the Mathematica Association of America, written by Keith Devlin (currently at Standford University and who recently wrote Introduction to Mathematical Thinking [amazon.com]), was delightful (the title was quoting a high school student talking about maths) and introduced me to the brilliant work of a female British mathematician, Jo Boaler, who (as I have just noticed on the front of her Standford page) was once told "not to talk about [her] research results in America as American [school] teachers are 'too weak' to be able to work in the ways shown to be effective". I can't say I am very surprised..., and for those of you with too much time on their hands, the related Hacker News thread (probably where I first heard about the piece) makes an interesting reading.
The second piece is an article by David Tall (Warwick university), who came to similar conclusions as myself about the (computational) processes underlying the mental mathematical activity, the article is called The chasm between thought experiment and formal proof [warwick.ac.uk], on the struggle of first year maths students in mathematics. I didn't learn anything new in reading it, having experienced, both as a student and a teacher, all experimental observations of the paper. But then, if any, this paper did re-enforce my belief that there are three things that can be done to significantly increase the level (and success) of math students starting university. The first, should actually be done before university and simply consists in learning (the basics of) programing, say, using a simple scripting language. The second (which unfortunately has apparently been dropped in the UK, and partially explains why the UK students are so weak), is to teach Euclidean geometry at school (a student I was talking to the morning was surprised to hear that I learnt geometry, and learnt to write geometry proofs, from middle school). The third is simple, but not done in the UK, at least not done at King's College: from the beginning of the first year, write formal proofs correctly, perfectly.
The purpose of those exercises, is really to help the student to build, maintain and reinforce two components of their mental activity. On one side the mental (mathematical) intuition, grounded in both our personal experiences since we were born and also some results of our evolutionary past, and on the other side the mechanistic, logical, procedural perfection of formal reasoning. Both of them are strong in any talented maths student, but the real fun starts when those two parts working together, continuously reinforce each other and form what I call mathematical understanding.
Quoting from Devlin's piece "Mathematics is a way of thinking about problems and issues in the world. Get the thinking right and the skills come largely for free". I have been claiming this to fellow students for a while, pointing out (because I never miss an opportunity to slam religious thinking) that living in a mental world driven by religious myths, really doesn't help one to be able to think as is required for modern mathematics. Sadly, without being understood. Which is unfortunate, because religious indoctrination has a tendency to kill any form of mathematical intuition (*), and, in general, real religious people don't feel well with mechanistic formal reasoning (which applied to their faith would invariably lead to contradictions).
(*) Not at all because there is any sort of intrinsic incompatibility between one and the other, but because religious indoctrination definitively kills a deeper and important mental faculty of the human mind: the ability for personal extraction of data and models from one intimate physical experimentation (and this covers thought experiments), which is the precursor of the corresponding mathematical mental ability. I have actually noticed myself that the more one is religious the less they have any form of pure mathematical intuition.
Of all the above, and reflecting on my own education (both school and personal childhood education), I realized (even more) that my mathematical talents at less the result of genetic configuration and much more the result of having been lucky enough to have had a good childhood learning environment and activities. Many little events of my childhood story all contributed together to make me the person/student that I am. Of course, I knew all that, but I had never really realized how much I should treasure it, simply because I could have been not so lucky.