How many different... - a simple approach to depth and breadth

A long overdue change in maths teaching is taking place at the minute - a change from teaching techniques that work, with an appreciation of why the techniques work sometimes lacking, into teaching for a much greater understanding of the underlying concepts and a greater focus on comprehending why techniques work in the way they do to give the result they give. The term du jour is 'mastery', and with goes hand in hand the ideas of depth and breadth. No longer do we seek to provide our most able or 'high-fliers' with access to work that takes them beyond the maths they are studying; instead we seek to provide them with the opportunity to gain a deeper, more fuller understanding of the maths they are studying and experience a breadth of situations where that mathematics may apply. For many teachers this can be quite a challenge, particularly as many of us are the product of the previous approach, myself included, which means that any deeper understanding of this mathematics that we have come to, we have had to find ourselves. I mean no disrespect at all to my own teachers, most of whom I remember fondly, it was simply that the focus of the time was different. So now, with the need to stimulate a depth and breadth of understanding in pupils that we never achieved ourselves at the same point, the question remains how do we go about it. Well one way I have been using is to ask questions that immediately prompt pupils to explore similarities or differences, rather than just stop at recall. I am going to briefly outline two examples I have used recently, one with Year 7 and one with Year 10.

Like many schools we have a big focus on assuring basic numeracy. My Year 7 bottom set have a numeracy starter at the beginning of every lesson, and a starter I am using this week revolves around this image:

Now rather than simply ask what time is shown on the clock, or ask for times away from this time, my starter is simply "How many different ways can you find of writing the time on shown on this clock?". I have even put a hint up (as part of structuring the support for my bottom set) that reminds them to think about ways they might not write or say. The reason I feel that this has the potential to provide depth is that ultimately I have no idea what pupils will say. I have come up with about half a dozen responses, but the pupils could come up with more than I have, or simply different ones to me. Provided I am brave enough not to just reject wrong answers out of hand, but to explore answers and see how pupils have arrived at them, I am likely to be deepening both the pupils understanding of telling the time (a real problem for some) as well as my understanding of the pupils.

Again like many schools, we have our 'borderline' pupils working toward entry onto a reduced Higher tier (reduced in that we are realistic about the amount of the paper the pupils will focus their attention towards). Of course in the new GCSE a crucial part of this is the ability to reason proportionally, and a lot of our early work with these pupils is exploring different proportions and different ways of representing proportional relationships. I did this using Cuisenaire rods where pupils found the relationship shown in this image.

Most pupils were able to describe something like "5 purple bars = 2 yellows", but then I challenged them as to how many different ways of writing this relationship they could think of. There weren't many, but we did come up collectively with writing the ratio, the idea that there were 2.5 purple bars for every yellow etc. I also showed them a graphical representation and algebraically defining the relationship. The nice thing again though was the links we were then able to make in their previously studied maths.

There are a number of other opportunities to ask the question "How many different..." in maths lessons, a few others I will be using/have used:

• ways of calculating the perimeter/area?
• ways of writing this probability?
• ways of displaying this data?
• ways of changing the answer by putting in brackets?
• ways of drawing the factor tree.

I am sure other people will come up with others, but the point here is that if you need a relatively straightforward way of providing a bit of depth or breadth, try thinking about whether you can ask how many different ways of doing or seeing something your pupils can come up with.