Very often students studying for maths GCSE or SATs are forced by their teachers to work neatly. Everything must be laid out nicely with a title and date, and credit is given as much for presentation as the actual thinking that goes on.
I believe this is a big mistake. Sure, onece the messy, non-linear, experimental business of real mathematical thinking has really been attended to, then a neat and tidy presentation of the findings is great – but it should be understood that this is the wrapping to the actual gift which is the thinking itself.
I always encourage my students to feel free to make as much mess as they want while thinking about maths. I encourage scribbled sketches, and scrawled notes made on rough paper or a mini-whiteboard. I want my students to take up as much space with their thinking as they need, and only when this creative process has yielded genuine understanding (which it so frequently does), do I suggest they focus on presenting their results neatly.
This is not just some maverick idea. Professional mathematicians have a word for it – it’s called scratch work. As a practical suggestion, I would encourage every child to either draw a line down each page of their exercise book and allow everything in one column to be as messy as they please, or to use the back of the book for rough work.
Of course, if our focus is on correctness and order at every step of the way, as is probably expected by the inspectors and administrators, then this approach may not be approved of. However, it is how mathematical thinking actually happens and to reject it is to reject essence of the subject and the very processes that make it worth studying beyond the most basic level.
NB There is one important exception to the above suggestion: Place value is an incredibly powerful piece of human technology that underpins everything that makes use of mathematics in the modern world (which is a whole lot of things from cars to smartphones and much much more).
Because of this, Students should always make sure that when laying out numbers above or below each other for the purpose of performing a sum, the columns must line up – units above units, tens above tens etc. Without this, it becomes much too difficult to keep track of what is happening and mistakes are often made.
In summary – maths is messy until it isn’t. Enjoy it!