Coaching (3): maths is writing

If you want to learn maths, you need to learn how to write maths and how to explain maths. The reason is simple: the process of communicating is a process of assimilation. If you cannot explain something clearly, then something is not clear in your thinking, so practice explaining things to other fellow students and let others explain things to you (and reflect on how they communicate).

So far, so good. But there is an aspect that is often ignored: writing maths. And this is crucial. I did not learn to write maths properly until very late (and, actually, to write them really properly, until my post-doc). The reason? Two-fold: I did not give it so much importance (I was thinking that the important thing is to have the right answer), and nobody taught me (nor I knew how to go around learning it).

Writing and thinking are inseparable.

Fix that in your mind. Writing is the most powerful tool that you have to empower your thinking. Mathematical thinking is an iterative process of refining ideas and since we can only hold a finite number of them in our heads, paper becomes indispensable as a memory outsource; and as a wall to bounce off ideas and see them in a different light. 

You start with the draft of a strategy to attempt solving an exercise, maybe it is just a drawing, or maybe you write down the definition of each one of the concepts to have them in front of you. Then you iterate over what you have written, detecting where are the gaps, where are the points needing clarification, where things need to be written more precisely. And you keep iterating over and over. You try a path, it does not work, you try another.

In the end, you want to have a clean written answer that everyone would understand without having you to explain it. An answer without ambiguities and with every step specified. If you have that, you have a good answer, otherwise, something is missing, even if "inside your head, you have the answer". 

Many times you will "know" the answer and not know how to write it. Well, writing maths requires training. It is about structuring thoughts, and making thoughts precise. Nothing can escape the paper, if you really want to know if you have an answer to an exercise, write it down and be a hard judge of what is written: think that someone else has given it to you and be demanding.

So writing is not just writing. Writing is thinking. The better the writing, the better the thinking. 

In oral exams, the best students are clearly the ones that are able to write the answer better (even though the exam is an oral one!). 

Mind you, there is no course on writing maths, you will get no feedback on writing maths, but learn it. Look at examples, try always to improve yourself.