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How to learn mathematics

This short essay is mainly aimed at an adult audience, but I have tried to make it (and others on this site) accessible to as many groups of teachers and learners as possible. I am always interested in new material and improving old material. If you have suggestions/edits which would make things on here, more accessible, please let me know. Another thing to note about this short piece, is that it is about the ways that I respond to stuff I don't understand and I am sure this is not universally applicable. I put it out there just in case somebody might find it useful. (A bit like Gottlob Frege did after the letter from his chum Bertrand Russell effectively tore down much of his life's work. I should add that this is where our similarities end. Frege was one of the greats in logic and I am not even particularly good.)

Learning mathematics is unlike learning any other subject and the reason for this is that mathematics is by its very nature, abstract. It is often possible to couch mathematical ideas in practical problems, but the techniques required to solve the problem is usually still going to be very abstract and inevitably difficult to solve. This requires the learner to acquire pictures in their heads, with which to navigate the difficulties which mathematics throws up.

The very nature of number is weird. Numbers, on their own, are adjectives without nouns. If you are trying to describe a jigsaw of 1000 pieces or a bag of sugar which has a mass of 1000g or a desired item which costs £1000, it is clear that the 1000 is the bit common to each situation, so 1000 must have some independent meaning, completely separate from the contexts in which it occurs. It is this abstraction from collections of things to viewing the collection as a thing in and of itself which makes matters difficult and have sent many great mathematicians (Frege comes once more to mind) into a major tailspin. This study of numbers as the study of collections is the basis for set theory, which is, for now, beyond the scope of this set of resources. However, if this material is useful to people, then who knows?

So how do you learn mathematics? It is not easy, that is certain, but nor is it as formidable as many people think. What it requires is a lot of patience and the humility to acknowledge when you do not understand something. You need to ask for help, which involves admitting to yourself (this is hard for many people) that you do not understand the problem you are struggling with.

Many people think that being unable to understand or solve a problem is a simple binary: you either can or you cannot. In my view, this is the worst and most dangerous gibberish perpetrated on learners by teachers and parents. You only have to think back to the last time you stared at a problem, which you are invested in solving and which was driving you up the wall. I guarantee that you can find, in your past, if you think hard about it, a situation where you were stuck, and subsequently, you became unstuck.

Henry Ford, not a thinker with whom I have a great deal in common usually, said

"Thinking is the hardest work there is. That is why so few people engage in it."

And he's not wrong. It is hard, but all too many give up trying way too soon. Every morning, before I get up, I like to play a silly word-search game on my phone (Retirement rules!). The game gives you 7 letters and you have to find the words that fill the grid just using those letters. It is not a highly intellectual game but it has taught me some fallacies in my thinking. Many mornings, I become utterly convinced that I have found every single 4 letter word ending with a 'e', but still having the blank space sitting there in front of me, and the inductive knowledge (induction is the idea that something which happens every day without fail, will probably happen tomorrow as well.) that, given that I succeed in finishing the puzzle every day, I am also going to solve this one. But there is an unhelpful part of my brain which insists that "this time is different", and "you are never getting this one" and so on and so forth. The secret to finishing the puzzle, is to stop listening to these unhelpful "head voices" and concentrate. Run through categories of words in your mind, don't forget the ones that start with a vowel (this is something I constantly forget), think about how letters are put together, the rules of etymology (word derivation; where they come from) and on and on. If you persist, you will usually win out over your own doubts. The structure of how words work helps to overcome the voices of gloom.

Learning mathematics is similar, but more so. Many of us have gremlins sitting on our shoulders, whispering in our ear about how stupid and slow we are. If you do not have these thoughts then good luck to you, but in my experience, an awful lot of people do and it is to them that I am addressing myself. Once you have a deep-seated handle on the structures of mathematics, it goes from being something opaque and difficult to being interesting and beautiful. Even a simple acrostic like BIDMAS (Bracket, Indices, Division, Multiplication, Addition, Subtraction) can teach you things about life and decision making, if you think about it in the right way.

BIDMAS tells us the order in which operations should be applied, so that everybody does the same thing and hopefully gets the right answer. $6p \div 2p$ anybody? Is the answer $3$, or might it be $3p^2$?

If I rewrite that problem differently, the difficulty becomes easier to see, even if we are no closer to solving it:

\[\begin{align} &6p \div 2p \\ &= 6 \times p \div 2 \times p \\ &= 3p^2 \end{align}\]

If we apply BIDMAS strictly in this example, we do end up with $3p^2$, which is probably not what was originally meant. I would argue that most often when someone writes $6p \div 2p$ what they really mean is $6p \div (2p)$, to which the answer is, unambiguously, $3$. Here's the thing: this kind of thinking makes me happy. I am perpetually thrilled that even facile and seemingly obvious problems contain ambiguity and uncertainty. Most people, who have read a little maths, think that uncertainty is absent from mathematics and this is to misrepresent the subject as though every problem in the world is a binary yes/no decision. Right or wrong. Mathematics is the way to get a solid handle on a priori (logical) thinking. Of course, with the arrival of statistics, data analysis and game theory, mathematics also takes a big run on a posteriori (empirical) thinking as well. There is something here for everyone.

Difficult things become easier with familiarity. If you practise a piece of music, your playing becomes more fluent and fun. if you read a lot, your writing becomes better and is more enjoyable for others to read. If you do a lot of mathematics, then it too becomes more accessible, understandable and fun. All that is required is determination and a little bit of self-belief. If you don't particularly want to learn Maths, then this is what psychologists call a self-fulfilling prophesy. You will be right, and that also, is okay. There are many things I have given up on trying to learn due to there not being enough time to learn everything. Just be honest about what you want. Don't try to learn mathematics because you think you ought to, or because someone else says you should, because you will find it a depressing and difficult and thankless task.

This site represents a bringing together of some of the things I have learned about learning over the past 60 odd years and I am putting it out there so that others can benefit from the few things I have gleaned. If this is of no interest to you, please move swiftly onward, if not, then I hope you find it helpful.