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.
[1] Remind me again why a PhD is deemed sufficient for teaching at university level?
[2] A content course for pre-service maths teachers, currently across primary and secondary school.
[3] At least, as demonstrated on this problem.
Wonder in Mathematics 
The things students learn are often not what we thought we taught. Reminds me of research by Ken and Nerida from over 20 years ago on the limits of paper and pencil assessment (e.g. https://www2.merga.net.au/documents/RP_Clements_Ellerton_1995.pdf). If I remember correctly, they also found that it can work in the opposite direction – some students who get it wrong on paper do understand the concept being assessed. The challenge remains with what to do with the information (correct answers from incorrect ‘methods’).
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I’m asking students how they know a lot too.
eg
It’s interesting, isn’t it, how you can get the right answer for the wrong reasons.
Are students getting it right for the ‘wrong reasons’ more frequently than would be the case randomly? For me this was definitely the case (they said they were guessing) which made me wonder what is going on.
I’m curious to see some of the answers where students gave valid explanations too.
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