Despite my fifteen years of teaching experience, I still feel slow1 to realise what I’m sure is obvious to others — just because students get the answer right, it doesn’t mean that they understand. I was sharply reminded of this just now.
This year I’ve been making a concerted effort in my Mathematics for Primary Educators course2 to have students articulate and justify their thinking, including on problems that I used to think were ‘straight forward’ (like this one from our most recent quiz).
The breakdown of responses from 43 students are as follows:
(a) 3 students
(b) 31 students
(c) 7 student
(d) 1 student
(e) 1 student
There are 12 students who clearly don’t know how to manipulate negative exponents yet3. Am I right to conclude that there are 31 students who do? Have a look at three of the responses (rewritten in my handwriting).
Whoa. Their misconceptions are just as interesting as the 12 students who circled an incorrect answer but I would never have known if I hadn’t asked them to explain their thinking.
 Remind me again why a PhD is deemed sufficient for teaching at university level?
 A content course for pre-service maths teachers, currently across primary and secondary school.
 At least, as demonstrated on this problem.