Ok, I’m taking some time off from my technology chronicles and ramblings to perform a complete mind core dump regarding something else that I’m interested in: Mathematics and more particularly the way it is taught.First, some background: millions, probably billions of students around the world take high school math, many Indian students included. While many of us like maths, few of us actually love it, primarily because the way mathematics taught, in most places around the world and by most teachers, is, in my opinion, fundamentally flawed and leaves math students unable to grasp the true essence of mathematics. First off, mathematics is not so much a science or a subject, as it is a language. In the famous sci-fi movie Contact, Jodie Foster’s character, Dr. Ellie Arroway calls mathematics “the only universal language”. Know what? She’s right. Mathematics in its most basic form is simply a language (or more precisely, a notation) used to describe various logical constructs and processes. However, unlike most human languages, mathematics is inseparably and closely tied to whatever ideas and logic it conveys and has few ornamental components. In this way, mathematics is closer to computer languages than it is to human languages. Both strive to achieve maximum clarity and efficiency and both encourage the creation of new structures and algorithms to deal with new problems. Both evolve much faster than human languages and both are much richer than any human language in existence. However the most fundamental difference between these two and human languages, is that while all human languages allow for a certain amount of ambiguation, mathematics and computer languages reject ambiguity entirely, without losing richness i.e., There are multiple ways of expressing an idea or achieving a goal, but any statement can mean one and only one thing. Assuming that you can read the language properly.
Which brings me to my argument: mathematics is not taught like a language, at least, I have never been taught that way. The way mathematics is taught (in Indian schools certainly) is an inefficient and inconsistent jumble of ideas, formalisms, methods, and “problem solving tricks”. The result is that we learn some topics (set theory, Euclidean Geometry, Trigonometry, Calculus) , but we are simply unable to tie everything together into a consistent and usable conceptual/practical framework. We tend to see mathematics as a set of separate components which have only a few overlaps (like co-ordinate geometry) and consequently are utterly incapable of appreciating the true beauty of it all. In his book Hyperspace, Michio Kaku said that Mathematics is the set of all possible logical structures whereas physics is the set of those logical structures that actually exist. Unfortunately we see maths as just a set of problems to be solved. Mathematics, from the high school level onwards should be taught like computer programming: you are actively encouraged to use different branches of mathematics to solve problems and it is irrevelant what you use as long as you get the job done concisely and efficiently. The better mathematics teachers that I’ve seen actually do this: seamlessly melding together Calculus and Trigonometry or Geometry and Complex Numbers with all the dexterity of an eagle riding a rising thermal.
Of course, teaching or even using mathematics in this way would be completely different from what is the “standard” now. Many would say that maths like this too hard and not neccessary or is above the “standard”, once again completely missing the point. It is not that mathematics like this is harder than what is taught, it’s just different. Just as there’s no point in comparing a thoroughbred race-horse to an Olympic sailing boat, there’s no point in comparing the modern pattern with what I’m suggesting. We would require an army of excellent teachers who are not only good teachers, but regularly use mathematics like this themselves. They would have to be the Hackers of mathematics, who do things that only people like them have the capacity to truly understand, but yet hold us all spellbound by mere glimpses of their intellectual prowess. Unfortunately there are all too few of such people. At this moment I can think of only about 5 or 6.
How would one teach maths like this? Well, don’t ask me for details, but I do have an idea or two. The first step would be taking a leaf out of the book of one of mathematics most popular children: computer science. In my opinion, computer scientists got it right where mathematics teachers failed miserably: their whole world revolves around describing actions and data in machine-readable languages. They keep logic and representation essentially separate and so can easily use different representations in a concerted way. (This failure is unfortunate as most early computer scientists like Alan Turing and Jon von Neumann were originally mathemticians) So all math students should first be taught some amount of computer programming. Secondly Mathematics should be taught in breadth rather in depth. That is, students should be taught about the basics of a wide number of areas and then be allowed to choose the areas that they find most useful and interesting and be allowed to delve deeper into those areas as and when they need. Maths books and problems should also require the use of numerous different areas of maths to be solved properly. Importance should be given to visualizing the problem and the data given. An equation of a line in co-ordinate geometry should immediately cause students to have at least a rough idea of what it should look like and where on the plane it should be (though some students I know do this, their number is amazingly few). Students should then use visual intuition as much as mathematical analysis to solve the problem.
All that being said, I really don’t think that math education (especially at the high school level where it matters most) will change fundamentally, so all this is probably a waste of time and bandwidth and I might be very wrong as well. At least I have something to rant about.
