Maths is as much about using words as it is about manipulating numbers. And this is nowhere more apparent than in the reasoning strands of the curriculum and the Maths Hubs programme.
Reasoning distinguishes mathematics from other academic subjects: constructing sequences of logically connected statements that, cumulatively, have explanatory power is what mathematicians, uniquely, do. Developing mathematical reasoning in our pupils and students is therefore more than a curriculum-motivated requirement: it’s our duty if we are going to ensure the next generation of mathematical thinkers.
It shouldn’t be a surprise, then, that one of the Maths Hubs programme’s national collaborative projects this year is devoted to just that aim. So let’s dive in and consider an activity that we might use with pupils. Which of these number sentences is true? And which false?
It’s not especially interesting that these statements are all false; the richness of the activity lies in the pupils identifying the reasons why they might appear to be true, and then explaining why they’re not despite appearances. We want our pupils to be confident that if they double a number and then halve the answer, they will get the same number they started with – so why is the green box false? Is the purple box false in the same way as the blue one is? Might a statement with the structure of the ones in the purple or the red boxes ever be true, if numbers other than 5, 8 and 2 were chosen?
So how does this fit in with the Maths Hubs project? Well, the aim is to improve teachers’ confidence in developing their pupils’ reasoning. The Work Group Leaders in each Maths Hub will be supporting five mathematics departments as they experiment with different strategies for increasing the amount and the depth of their pupils’ reasoning, and then – crucially – as they reflect together on the impact of these strategies on the pupils’ reasoning and also on their own understanding and practice. Professional development through professional collaboration is therefore at the heart of the project; the “unit of change” is the department not the individual. The participating teachers will each increase their repertoire of techniques for nurturing pupils’ mathematical thinking, but together they will deepen their understanding of why these techniques are effective, and how best to use them: reasoning about reasoning.
For more ideas on how to build more reasoning activities into maths lessons, read the It Stands To Reason section in the monthly Secondary Magazine on the NCETM website.