If you’ll allow me a brief aside, credit for this poster goes to Robin Bhattacharyya. Robin was the most knowledgeable person I have ever met and at the time, one of my heroes. Originally a University Challenge champion in 1995, he went back for more in 2011 Christmas special, in which he led his college to victory over a more recent hero of mine, Daisy Christodoulou. (strictly, it was Trinity vs Warwick but from memory, it may as well have been Bhattacharyya vs Christodoulou!)
Anyway, back to the maths… A recent tweet from Stuart Price reminded me of this poster.
Stuart was focussing in on one specific type of misunderstanding which features heavily in Robin’s poster: the over- or under-generalising of the distributive property. Trying to head off these problems early, I’ve been making a specific effort to teach year 8 about distributivity this week. It has gone better than in the past, so I thought I’d share what (I think) I’ve learned.
In the past, the phrase “multiplication is distributive over subtraction” sounds clunky and just saying it, let alone understanding it, has been trouble enough for my pupils. I’ve replaced it with “multiplication distributes over subtraction“. This small change has had a surprisingly large effect. I think this is partly because it relates more closely to the usual English usage of the verb ‘to distribute’; indeed, this also allows for visualisation: the multiplication is literally being distributed over the subtraction.
Secondly, Having checked whether several operations distribute over each other, I asked my pupils to generalise which operations distribute over others. In this section, Colombe introduced a metaphor which helped her peers to remember the general rule: Multiplication and Division are the government: they distribute (resources) to the citizens, Addition and Subtraction but not to themselves. This also ties in very nicely with Order of Operations.
This was my hook: how on earth did he do this? Did you notice that he didn’t actually know the intermediate numbers? This gained interest, and later on in the lessons I tried to set some questions in which pupils applied the distributive property to simplify some expressions, leading up to (318 x 75 – 50) / 25, but it didn’t go well. If I had used the strategy that John Mason suggested on Craig Barton’s podcast, of imagining what I was going to say and how the lesson would proceed, I feel like I may have anticipated this. It wasn’t clear that the equivalent of 318 x 3 – 2 was the ‘final answer’ in the simplification problems I had created. In my updated lesson plan, I have removed these questions, leaving just the explanation of the countdown genius as a teacher-led section of the lesson.
Another issue that came up as part of these lessons was a good question from my pupils: If it doesn’t distribute, what does happen? My answer at the time was fluffy… something like “it depends on the situation, usually you just apply the operation to one of the numbers”. I feel like I need to work on this, perhaps I need to add another sequence of lessons to my combining operations thread.
Overall though, I feel that this is the first time I’ve been even vaguely successful when teaching the distributive property, mostly because I managed to distill it, with the help of a pupil’s metaphor, to this:
One final reason why I think it went better than in the past: I was more committed. Previously, I would always question the value of teaching such lessons, because I suspected that the pupil’s future teachers wouldn’t make reference to the distributive property. Now I know that I will be teaching these pupils for several years and as HoD, I can better integrate this topic into our scheme of work, it is worth the investment.