A few years ago, I read a book on misconceptions in mathematics. The authors, Julie Ryan and Julian Williams, contend that many misconceptions result from overgeneralising. Ever since, I have noticed these kinds of errors on an almost daily basis in my teaching.
What do they look like? I might be teaching function identities, for example, and demonstrate that for the function
The following is true
However, a student might then assume that this is true for all functions or for a larger class of functions.
Similarly, all maths teachers have encountered students making the following mistake
And this can be understood as an overgeneralisation of the distributive property of multiplication, something that is true i.e.
I have recently noticed a similar misconception. The distributive property is only true for a linear factor, yet some students may write e.g.
I used to deal with these through explanation alone. I would discuss with a student why the maths does not work, maybe by expanding out the terms. I would focus, as many maths teachers would, on students understanding the mistake. Usually, when confronted with the logic of the error, most students recognise it as an error pretty quickly.
I still do this, but I have come to realise that I am perhaps engaging the wrong part of my students’ minds. Let’s borrow the model of ‘System 1’ and ‘System 2’ thinking that Daniel Kahneman outlines in Thinking Fast and Slow. The kinds of errors listed above are usually made in the middle of a more complex maths problem that is consuming attention and so it is likely that students are conserving limited resources and using instinctive System 1 thinking when they make these mistakes. By appealing to a logic argument, I am engaging their System 2 thinking, but this is not the kind of thinking that is making the errors.
The solution is partly lots of practice. However, drawing on the work of Engelmann and others in the way they structure units of Direct Instruction, I think a key part of the solution is also the use of non-examples. By explicitly and repeatedly teaching what is, and what is not, we give students a more instinctive feel for avoiding these misconceptions. And the easiest way to build up a bank of these non-examples is to look at the mistakes students make.