(A quick post from a delightful Mathematical Association of Tasmania conference in Burnie.)
There is a tendency for students, as they progress though the year levels, to think that being ‘good’ at maths means doing it quickly in your head. I often catch myself saying ‘don’t try to do all the steps in your head; write something down so I can see what you are thinking’.
A barrier to writing is that students want it to be ‘perfect’ the first time. (This was a big obstacle for me during my PhD, the first time that I truly had to constantly draft and refine on a large scale.) I tell my students (especially research students) that the only way to ‘awesome’ for most mortal academics is through many crappy drafts. Writing — making notes of any kind — helps us to organise our thoughts, to see connections, to expose flaws. Mathematical writing helps us to start problems that, at first glance, seem difficult. One prompt when students don’t know what to do is to suggest ‘write down everything you know/notice about the problem’.
When I assign undergraduate mathematical projects, I deliberately ask students for draft writing accompanied by at least ten pages evidence of ideas that didn’t work, rough notes and diagrams, and other signs of mathematical exploration. I want my students to understand that writing is a valuable part of the mathematical thinking process, not just of presenting the finished product.
It’s also perhaps because mathematics is largely seen as a highly abstract, intellectual activity that students become to feel that the only acceptable way is to do it in their heads. Primary school classrooms are rich in hands-on mathematical experiences for students. For reasons I don’t understand, we gradually withdraw this powerful experience from our classrooms until in universities predominantly only pen-and-paper remain. When I take simple hands-on activities (for example, cut-and-match card sorting tasks) into typical mathematics tutorial classes, some students look at me in amazement. (That is another post.)
Mathematical manipulation of all types of objects is how we develop mathematical intuition. In Tracy Zager’s upcoming book she quotes mathematician Reuben Hersh: “Intuition…is the effect in the mind/brain of manipulating concrete objects — at a later stage, of making marks on paper, and still later, manipulating mental images. This experience leaves a trace, an effect, in the mind.”
Physical manipulation should not be restricted to the primary grades. One of the greatest mathematical minds of our time, Fields Medallist Terry Tao (from my hometown of Adelaide!), isn’t reluctant to explore mathematical ideas in a variety of ways (as the extract below from an article last year in Australia’s Sydney Morning Herald demonstrates):
Terry Tao recalls the day his aunt found him rolling around her living room floor in Melbourne with his eyes closed. He was about 23. He was trying to visualise a “mathematical transform”. “I was pretending I was the thing being transformed; it did work actually, I got some intuition from doing that.” His aunt is likely still puzzled. “Sometimes to understand something you just use whatever tools you have available.”
Of course, the title of the post is misleading; it’s not ‘head or hands’ it is ‘head and hands’. If one of the world’s leading mathematicians isn’t afraid to build intuition by rolling on the floor, then neither should the rest of us be!