I recall that I was interested in mathematics from a relatively early age, probably from primary school. I was never a particularly talented mathematician — that is, I was certainly not a Wunderkind who won Olympiads, nor did I accelerate through school — but I always did well in maths, and most importantly, I have always had great curiosity and interest in the subject. More than likely my good marks were the result of my curiosity and passion, not the other way round. Oh alright, stereotypical Asian parenting also plays a role, I’ll happily admit. You may have read news articles or psychological studies about the phenomenon of math anxiety. I was rather skeptical about this when I first heard of it, but it seems to be a real psychological condition! You can read this rather dramatic article about this so-called phobia of numbers. Needless to say, this was not a problem for me. I enjoy maths, and I generally do well in maths, which only serves to increase my interest and confidence. When I don’t do well, there is still the enjoyment of a good challenge, of having grappled with a difficult problem or concept, even as I fall miserably short of the desired solution. Incidentally, if you ask a musician why they devote so much of their daily routine to practice and rehearsal, usually the response is along these lines: I enjoy it; music inspires me and I hope to inspire others; it is rewarding and fulfilling; this is my passion, and so on. In both cases, music or mathematics holds intrinsic interest for certain individuals, and they are motivated to master the skills or understand the concepts in their area of interest, and perhaps go as far as to extend current knowledge by exploring and developing new techniques and ideas. In short, I would like to propose that creativity is essential in mathematics. Obviously this is not the same kind of creativity necessary for artistic development, but nevertheless some form of it is needed. Now this is not a theorem which can be proved rigorously, but I hope you will allow me to conjecture a little!
I begin by lamenting the fact that, in general, the mathematics taught in schools [*1] does not foster a creative approach. You might recall having to learn abstruse formulas and “laws”, do annoying arithmetic, wrestle with weasel-worded problems which bear no resemblance to anything concrete and practical, and, worst of all, having long and tedious pages full of repetitive exercises for homework. [*2] This is rote-learning at its finest: there is a body of information (a syllabus) which some authority (e.g. Board of Studies) has deemed to be required knowledge, it is then taught (some might say “dictated”) to us, and we more or less regurgitate this dot-point knowledge in the exams. (Incidentally, I remember having to use a Chemistry exercise book literally titled Dot-Point Chemistry in Year 11). If this was your experience of the subject, you could be forgiven for hating maths. However, if you are lucky, your maths teacher may be enthusiastic and be willing to explain concepts clearly and in an engaging fashion — not merely dictating what you should know, but how these formulas work, why they are important, perhaps even taking the class through a proof or two. For example, you are probably familiar with Pythagoras’ Theorem for right-angled triangles. But when we learn about this fundamental theorem in school, it is usually given to us as a fact, and we take for granted the following process: a problem is posited or discovered, which mathematicians then attempt to resolve; a theorem may be proposed; a proof is attempted. (By the way, here is a nifty graphic which illustrates a proof by dissection). It is absolutely not an arbitrary nor obvious process. The Ancient Greeks were able to prove some extraordinary results using only the basic properties of straight lines, angles, and shapes. It is hard to imagine a maths class today going like this: “here are a few basic facts about lines and angles, now play around and investigate for a bit, see if you can deduce the formula for the area of circle.” (In doing so, you will have also defined/invented some notion of what we now know as the number pi.) Of course, as a more mathematically-inclined reader might have realised, this was more or less what Archimedes did. If the Egyptian pyramids are a wonder of human engineering and architecture, then Ancient Greek mathematics is surely one of the wonders of human intellect.
Furthermore, any serious mathematician should realise that a formula or a proof, no matter how elegant, is not the end of the road. As is customary and indeed necessary in science, an extraordinary result leads to more novel questions and further investigation and research. In our simple example with Pythagoras’ Theorem, a^2 + b^2 = c^2 turns out to have many profound consequences. One particular consequence was deeply troubling for the Ancient Greeks: irrational numbers. Construct a right-angled triangle with two of the sides being 1 unit in length. Then by Pythagoras’ Theorem, the third side (the hypotenuse) has length √(2), the square root of 2, which is an irrational number: it cannot be expressed as a ratio of whole numbers. (If you’re curious, we can show this using a proof by contradiction. It’s rather lovely.) The existence of such a quantity completely contradicted the Pythagoreans’ view of a universe constructed of whole numbers and their ratios. Legend has it that the guy who made the discovery was either exiled, or taken out to sea to be drowned.
In our modern times, other less drastic situations in which the Pythagorean identity a^2 + b^2 = c^2 appears include the following:
- The distance between any two points on a (2D) plane (but also see point number 5).
- The Cartesian formula for a circle of radius r centred at the origin (0, 0) is in fact x^2 + y^2 = r^2 (taking a hint from the point above, why is this obvious?)
- This fundamental relationship in trigonometry: (sin x)^2 + (cos x)^2 = 1
- The modulus (absolute value) of a complex number z = a + bi (where i = √(-1), the square root of minus 1) is written |z|, and can be defined as |z|^2 = a^2 + b^2. Doesn’t that look familiar?
- The length of a vector (this actually generalises the Pythagorean identity for all dimensions greater than 2, so that you can, for instance, find the distance between two opposite corners of a 9-dimensional hypercube if you wanted, even though we obviously have no clear idea what such a monstrous thing would look like!)
- Combined with integral calculus, the identity appears in the formula to find the length of a curve.
These may or may not be the most inspiring of examples for the average reader, but hopefully my point is getting clearer. It is one thing to learn a formula, or an important result such as Newton’s Laws of Motion. If you recall the typical high school situation, it does not take any imagination to reproduce the formula F = ma, plug in some numbers, and answer an exam question. It is another thing entirely to be able to understand the ramifications of a proof or result, extend the concept to other areas of research, or apply the result to solve real-life problems, and so on. This second and more profound type of problem-solving requires genuine creative thinking, an aspect which I believe should be more prominent in the teaching and learning of science and mathematics. Of course, not everyone will end up studying mathematics at a university level, and even fewer will continue to study mathematics beyond 2nd year. Moreover, I recognise some truth in the common objection: “but we don’t need this in our daily lives!”, or “how will this theorem help me in the future?” But in raising such concerns, we have already fallen into a trap. It is not about the theorems per se, because what is more important than memorising formulas and theorems is being equipped with mental discipline, skills in logical deduction, and creative thinking: in short, that we have powerful tools for problem solving in any given situation and for the appreciation of phenomena of all sorts — physical, social, digital, etc. — regardless of what career path we choose.
 – The situation is obviously quite different in university courses.
 – However, there are certain “routines” in mathematics that are worth practising by repetition. For example, being able to solve quadratic equations quickly, and differentiation and integration of common functions are all important fundamental skills in the study of higher-level maths.
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