What-if-not – how to extend any problem, no matter how closed


I’ve been continuing my summer holiday reading in between, you know, having a holiday. The Art of Problem Posing is a book I’ve been dipping in and out off since it was recommended to me by Andrew Blair over at http://www.inquirymaths.co.uk and I finally got round to finishing it. The first edition was published in 1983 (the year I was born!) and it is still relevant and useful to my classroom today. This is a really readable book that presents a good model for posing problems both for teachers and students. It has lots of places to stop and engage meaningful with the content and, even though I probably skipped over those bits too much, these really helped to make the points more meaningful.

I was going to summarise and review the book but instead I thought I’d focus on the big take-away that I am going to be using a lot. I think I was already doing this but it is definitely going to feature more and more in my practise – namely, what-if-not.

What if hoax

When we solve problems, our first stage is to accept the problem as given – find the area of the triangle, discover the connection between Pythagorean’ triples, etc – and solve it. We can use the habits of mind and toolkit that we have developed (will blog on this but I think John has done a much better job than I will over here). But, what happens then? Or what if we can’t get started?

Well, we could think about what made that problem up (it was about the geometry of a triangle or it was a list of numbers where a connection seemed to exist) and start to play with it. We can extend any problem by asking the what-if-not question. What if it wasn’t a triangle but a square? What if one of the sides was curved? What if we were interested in perimeter instead of area? What if there wasn’t an obvious perpendicular height? How many other triangles will have the same area? What if the numbers were surds? Immediately, a whole vista of Mathematics opens up and any problem that was quite closed can become open and rich. We can either make the problem smaller and more manageable or larger and more all encompassing.

The book then looks at a couple of examples in detail and details some action of problem-solvers at work.

I’m going to make a poster of what-if-not (that I’ll share here) and encourage students to think about ways we could alter a problem – adding it to the toolkit. This could work really well as a plenary or extension with students of all abilities as we encourage Mathematical thinking. I think it could also help broaden the depth of questions I get from Inquiry lessons – what if something from the prompt wasn’t true or was different? If we are thinking about students having agency and ownership of the Mathematics being taught, I like helping them recognise that any problem that I present to them is one I have found to be interesting and useful but it is not sacrosanct. This could be another way to encourage students to engage Mathematically.

But then again, what if I didn’t do that?


Image from http://www.transitionnetwork.org/stories/mandy-meikle/2012-09/if-not-transition-then-wat