I have been thinking a lot lately about the best way to address student misconceptions on the unit of work that they are currently studying. Ideally I hoped these misconceptions wouldn't develop in the first place, we spend a lot of time discussing and developing the understanding of the ideas before introducing any more formal procedure, and when we do introduce a procedure, I try to make sure it is theirs, not mine. The aim of doing this is to ensure that we have thrashed out all of the ideas and tested their veracity so that the misconceptions can be brought to the surface and addressed as the new content is introduced . However in looking at the work I receive from students from time to time it is clear that misconceptions creep in, despite my best efforts, I may not have covered all the bases. The other day I tried to address misconceptions in a different way to had previously. The misconceptions became clear to me as I was looking through their books I picked up some misconceptions that were common across multiple student's work, but instead of talking about these in class and trying to correct it myself, I wanted them to notice and to correct the errors. The hope in doing this was to get them to think more deeply about the errors. To do this I wrote up some solutions to questions like they were legitimate solutions to the problem, but in reality these solutions incorporated the errors that I saw in their books, these samples are shown in the images below. Each table was given a different proposed solution, these were put in the centre of the table and students were encouraged to discuss the solutions in their table groups. It was interesting for me to see that when we had the discussion about the questions every group was happy that their solution was correct, none of the groups believed there was an error in the solution. To me this was very interesting, these errors were prevalent in a number of students work, but they definitely were not there in all of them. Even those students who had answered questions similar to this correctly in their books were not able to identify the error in these solutions. So this was quite a surprise to me.
I was caught a little off guard by this, but I also had to think carefully about the way forward. At this point rather than telling them what the errors were for each problem, instead I just said "what if I was to tell you that every question on the table is incorrect. Knowing this, what is the error in the problem that you have in front of you". This seemed to surprise them quite a bit, being utterly convinced the solution was correct and raising the possibility that it is incorrect created that conflict in their mind, and stimulated a lot of discussion as they tried to find the error. They did not find this easy but eventually were able to notice the mistake. What I hoped they would get of this process is exactly what was achieved, they were then able to successfully examine their own work, and their own thinking, and correct any of the mistakes that had been previously made. They were able to look at their own work, which they thought was correct, and find the changes that they needed to make. I feel that it was a much more powerful way of looking at the errors they made, as the fix was not handed to them, they still had to work for it, they still had to own the learning.
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I was reading a post on Dan Meyer's blog the other day titled You can't break math. There were a number of aspects to that post that resonated with me, but I came across the passage below that really made me think about the subject I teach in a slightly different way, this passage is.. One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand. I started thinking about my own experience with my own mathematics education and in many ways the approach I took to my first few years of teaching. I realised that when I was learning maths in school, it was presented as magic rather than logic, and that is the way I presented it in my first years of teaching. My teachers didn't intend to present it in this way, and neither did I.
Part of the beauty in mathematics is in the patterns that emerge and in the certainty we have in our conclusions. However in the highly formulaic way that many classrooms still operate, more time is spent practicing how to use the formula under the guise "trust me it works", rather than spending the time to get students to develop a sound line of reasoning where that formula is the only logical conclusion. The formula is simply the highly refined end point of a lot of thinking about a particular mathematical idea. That formula is the point at which all the uncertainty in their argument has been stripped away and what remains is the pattern that has emerged. When we present that formula without the thinking behind it, without developing that understanding prior to presenting the formula, this is when when it appears that we have 'pulled a rabbit out of a hat', it is something that has come from nowhere. However it is important to remember that even magic is not magic to everyone, this has become abundantly clear to me through watching a lot of the TV show Penn and Teller Fool Us. For those who do not know the show, a range of aspiring magicians come on to a show and try to fool a world famous pair of magicians. Most of the time, these magicians cannot fool them. Even though the trick has been previously unseen, these magicians strongly understand the principles of magic, concepts such as misdirection and sleight of hand, so even if the trick is unseen, they can unpack the thinking that may produce that result. Their knowledge of these concepts is so strong, and so flexible, that they can apply them to any trick and to create new tricks of their own. This is what I want for my students in relation to their mathematical knowledge. More than just understanding the individual tricks, I want them to understand the underlying framework of the mathematics they are studying so that they can apply it in unfamiliar and sophisticated ways. 
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
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