In my final year of university I completed a physics experiment that involved the decay of a radioactive element. Theory predicted that paired emissions would be given out in a particular pattern and so we took readings with a couple of photomultiplier tubes to see if this was the case.
But radioactive decay is probabilistic. So it was necessary to make a large number of observations and then analyse this data statistically. This was the first time that I really got to grips with Microsoft Excel. I used a 486 PC that I had inherited from my sister and I saved my files on floppy disk. I point this out in order to emphasise that Excel and me go way back. I’ve solved problems on it many times and spent hours reading through the help files.
Yet I often can’t remember what I did. For example, I frequently want to add up the data in a row but only if there is data present. If there is no data in that row then I don’t want excel to return zero, I want it to return nothing (because zeros affect means and gaps in the data don’t). We use Excel sheets for analysing data all the time at my school and if you went and had a look at a few of the ones that I have created then you will see that I have solved this problem in a number of different ways. So I don’t know the answer, even now. Instead, I know that I can figure it out if I need to, which is a liminal kind of state to be in; a surprising result given the amount of experience I have with Excel.
I often don’t even bother with solving the problem at all. I just use one of my old Excel sheets and adjust it for a new purpose without going in to all those formulas and interacting with them.
This is a familiar feeling. I haven’t completed any computer programming for a very long time but I recall a similar pattern. I remember focusing my attention on developing a subroutine and then just calling upon that at will later, even if I couldn’t quite remember how the subroutine worked.
And I think we can extend this to maths. There are probably a lot of people out there who are doing a bit of calculus and making use of the result that without having a full conception of why this is the case. The variety of methods that teachers use to explain this idea speaks to the fact that there is no uniform, canonical understanding. Yet the useful result persists, doing its useful things on the pages of exercise books across the world.
I would argue that the strictly hierarchical view of mathematics that many idealists hold is useful but ultimately flawed. Maths is not some formal system where layers of understand fit neatly on top of one another. Nor is it the case that personal experience of problem-solving crystallises all of those understandings. It is far more iterative and roundabout than that. Sometimes we focus on the micro; on the details of each step of the process. Sometimes we zoom out and it feels a lot more like slotting black-box modules together.
There is some evidence to support this view of maths. A strictly hierarchical view would suggest that students need to gain conceptual understanding in order to then develop procedural understanding; so this must be the sequence of learning. Yet research suggests that cause and effect actually flows in both directions and that learning the how can precede and support learning the why.
Update: Following publication, I found myself in an argument as to whether maths is a formal system. It could be a formal system and yet we might not experience it as such. So arguing about this point, although fascinating, is not really relevant to the post.