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What's the first thing you learn about mathematics? No it's not “1 + 1 = 2”, it's the first thing that you were confused by since the moment you could remember. Few of us remember the first time we looked into a mirror and were confused by the human-looking figure that seems to move when we move. Of those of us who do, even fewer were lucky enough to immediately divine that the most consistent way of explaining this, i.e. a theory, was to regard the figure in the mirror somehow as some sort of an optical alias of ourselves.
Experiences like these at early stages of our lives shape our understanding of mathematics, and indeed anything else, as we grow older. I still remember the first time I played a video game on the classic NES (Nintendo Entertainment System), in which the main character would go off one side of the screen and re-appear from the other, and to this day this is still how I imagine a donut. Similarly, I still associate my knowledge that the Earth is round, with the memory of my confusion at a young age as to why ships disappear into the Earth as they sail far away.
Some experiences however, seem counter-intuitive at first sight. I've known many cases of children, including myself, becoming perplexed by the fact that 2 divided by 0.5 is 4, which is larger than 2 itself. How can something divided by another thing be larger than itself? Of course this fact is only made counter-intuitive by our earlier experience, say in 1st and 2nd years of school, when division usually results in a smaller number e.g. 9 / 3 = 3 < 9. When we are too used to something, we are reluctant to change. I’ve known a child as old as 16 who is still perplexed by 2 / 0.5 = 4, and this is only because every time she runs into something like this she would avoid thinking about it, and not bother changing the way she thinks to accommodate this fact. Instead, she would take out a calculator every time and complain about how confusing it is. However the more inquisitive of us soon find out that there are many ways of thinking about numbers and proportions that give this fact intuitive sense, and change our mindset to accommodate a new way of thinking. It is the inquisitive nature, the willingness to think, that gives you mathematical wisdom, and not just accept that “I can’t make sense of this”.
Should this inquisitive behaviour prevail, it will guide you through A-level and even Advanced Extension mathematics, and lead you to success. In university however, as workload increases exponentially alongside social obligations and complexities, substantially more mathematical counter-intuitions will appear. The experience of our inquisitive behaviour may lead us to our own failure at this time. It is very important to know what not to remember. To explain this I will digress briefly.
Our survival as an individual relies on many qualities and inherent skills that we possess. Perhaps the most important of these is our ability to adapt to new environments, and a particular case of this is our natural ability to compress information. Humans pride ourselves with this miraculous skill, and use it continuously throughout our lives with such fluency that we hardly realise or even think about it. For example, when you’re reading this article, what you will, or at least ought to remember, is not each and every letter or word that is written here. Almost none of us will be able to write it out word-for-word after reading it once, but almost all of us will be able to write something that conveys some information very similar to what this articles does.
This is a childishly simple fact, however it does bring me to my original point. We are in constant danger of forgetting to compress information in this manner, especially when we are doing something we are afraid of, e.g. degree level mathematics. Those of us who remember doing A-level or high school equivalent mathematics will recall that there were a great many sine and cosine formulae to memorise for exams. Of course they could all be derived very quickly from the fact that the square of a sine plus the square of the cosine is equal to 1, and maybe a couple of properties of trigonometry that we ought to already know by that stage. This is much less information to memorise than to learn all the trigonometric formulae, but some of us still choose to learn and memorise all of those formulae. Situations like this will occur relentlessly through university mathematics, and what’s worse, when too many of these occur in a short period of time, we panic and our brains lose the ability to keep track of the important things to remember, i.e. the fundamental definitions and theorems that will derive the rest “less” important stuff. Instead, the brain will choose almost arbitrarily the information it wants to remember, and thus cause more panic as time goes by and workload increases, creating a vicious cycle. In these circumstances, while retaining our inquisitive nature, it is important to sit back and take some time to think about what needs to be left out of our memory for now. The important things worth memorising are definitions, maybe some important theorems, and most importantly the method of doing things.
Mathematics is one of the hardest, most challenging subjects at university, and yet it is among the subjects which require the least pieces of information to be learnt. As mathematicians, the very challenge for us is the development of our ability to compress information. Not just mathematical information – any information. This ability is our talent, and our inquisitive nature is its seed.
