Keith Devlin is a mathematician with an interest in maths education. He has recently written a couple of blog posts for the Mathematical Association of America where he advances his views on the mathematics curriculum. In the first post, he describes how he has worked in two very distinct fields of mathematics and that, as he moved from one to the other, he forgot a lot of the terminology and specifics of the earlier field. He also claims:
“…essentially none of the concepts, definitions, or methods I learned (and practiced) either in school or in my undergraduate mathematics degree played any role in any of that professional work.” [Original emphasis]
Instead, the construct that Devlin claims he was able to apply to all situations was mathematical thinking.
This oddly absolute statement is somewhat contradicted in the second, longer post. Here, Devlin proposes that the school maths curriculum should consist of arithmetic, geometry, some ‘elementary’ algebra but definitely no calculus. He describes the kind of task that he would like to see: students use Google to find out about how UPS deliver their parcels. Seemingly in response to criticism he has previously received, he then goes on to recognise the value of mathematical knowledge:
“Sure, you can’t conduct that kind of investigation if you haven’t mastered some mathematical topics well. As I noted earlier, there is no by-passing that step. (Many readers of my articles and social media posts mysteriously seem to skip over that part and get angry at the straw man that results.) But the three topics I listed above (arithmetic, algebra, geometry) do just fine for preparing the groundwork.”
So none of the maths concepts Devlin learnt in school helped him as a professional mathematician, but you need a good grounding in arithmetic, geometry and elementary algebra in order to complete a project about delivering parcels. This does not appear to be consistent.
Throughout the two posts, Devlin sets up an interesting dichotomy between mathematical thinking and learning sets of procedures to be used as a ‘toolkit’. The former apparently relies on understanding whereas the latter is about rote mastery. And Devlin is also convinced that something fundamental changed around 1990. Since then, technology has advanced to such an extent that it can perform any mathematical procedure for us. This argument downgrades the importance of procedural knowledge and upgrades the importance of the mathematical thinking construct.
This is pretty much orthodox educational progressivism. However, Devlin does depart slightly from the script. He does not think that mathematical thinking can be taught as a set of strategies (my mind goes to George Polya when I think of attempts to do this) but instead is developed implicitly through interacting with mathematical content. However, as I have noted, Devlin gives mixed messages as to whether this content is foundational or essentially interchangeable.
Devlin cannot help having a dig at those ultracrepidarian maths teachers who have taken him to task on Twitter and simply do not understand problem solving like he does:
“I suspect those contributors to that Twitter thread had little experience in real-life mathematical problem solving. (Which is why those of us who have should devote time and effort into keeping teachers informed about current praxis.)”
There is plenty to unpack in Devlin’s two posts, but I first want to start with one of Devlin’s asides. By removing calculus and other fiddly bits from the maths curriculum, Devlin believes that this will remove ‘barriers to entry’ for young people and make maths more like a regular subject. He then suggests, “There has never been an educational problem of “English anxiety” or “Art phobia”, right?”
I think you will see English anxiety in many classrooms around the world stemming from kids not being taught how to read properly. This may manifest in what educational psychologists euphemistically refer to as ‘externalising behaviours’ – the kids will misbehave – or ‘internalising behaviours’ such as a withdrawal from class tasks (e.g. see this study from Finland).
Given the largely anecdotal nature of Devlin’s claims, I am also tempted to respond with an anecdote of my own. I was taught Art largely by what might be called the ‘talent sorting’ approach. Students who demonstrated natural talent attracted teacher attention, whereas those of us who did not were largely left alone to draw and paint without much instruction. I am naturally good at representing shape and proportion, but I had a problem using shading and texture to represent depth. As I progressed through school, I became more aware of my deficiencies through comparing my work with peers until I developed a frame of mind that could accurately be described as ‘art phobia’. I suspect I am not alone.
However, to return to Devlin’s claims, I am aware of no evidence to suggest that there is a dichotomy between knowing lots about maths – all those concepts, procedures and terminology – and Devlin’s concept of mathematical thinking. Mathematical thinking, such as it exists, is probably composed of knowing lots of stuff about maths. Problem solving is not about absorbing general problem-solving principles, it is about holding knowledge relevant to solving that particular problem.
Cognitive load theory suggests that the two main components of the mind that we use to solve mathematical problems, working memory and long-term memory, work together. We can only process about four items in working memory at any one time, but we can activate entire schema in long-term memory and effortlessly bring them in to working memory as if each schema is a single item. We therefore get better at problem solving, not by learning problem-solving strategies or developing some abstract mathematical thinking ability, but by having lots of stuff to draw upon from long-term memory. Really, that’s it. Knowledge is what you think with and you can’t think with knowledge that is sitting in a calculator or on the internet.
It may be the very effortlessness that convinces so many people that they are not using previously learnt knowledge when they solve a problem. The curse of knowledge is a well-known effect where experts, such as mathematicians, underestimate just how much they know and how much they draw upon this knowledge as they work. They take it for granted. It is this effect that led to a trainee science teacher entering my science class when I was sixteen, writing ‘surfactants and surface active agents’ on the board and speaking for about five minutes before it gradually dawned on him that he had lost every single student in the room.
Devlin is right to suggest that campaigners have, for many years, argued for taking calculus out of the school curriculum. However, I am not sure that this is the solution people think it is. Maybe students would persist with maths for longer but they are still going to have to encounter the tough stuff at some point. As a maths teacher, I am faintly amused at the prospect that kids are going to be far more engaged by a project about delivering parcels. I suspect most would find it extremely boring, but these are the kinds of suggestions we end-up making when we decide that maths has to be real-life and relevant.
Perhaps a better solution is to ensure that we have qualified maths teachers teaching students at every level of school education and that they are using the most effective approaches such as interactive explicit teaching.
Apparently, in his next post, Devlin will be writing about cognitive science. I look forward to reading and critiquing that piece. Given that he is out of his field, he will want to avoid the fate of those Twitter maths teachers.