Blog: Ian Davies, Director of Secondary
Mathematics Mastery: challenging our secondary students
Posted: 5/03/15Here’s a place value question for you to ponder:
How many ten thousands are there in eighty-five million?
Have you worked it out yet? It’s not that tricky if you have a good “number sense” and know the connections between “columns” in place value charts, but my hunch is it will have taken you a few moments of reflection to come up with an answer – it won’t have been as automatic as, say “How many 10s are there in 85 000?” even though the level of understanding required is pretty much the same. I think that what adds to the challenge here is the relative unfamiliarity of working with the numbers involved, rather than their size. “How many millions are there in four hundred and fifty million?” is even more straightforward and the numbers there are even bigger.
Now think about this series of questions (please “do the maths” if you like!):
How many tens are there in eight hundred and fifty?
How many hundreds are there in eight thousand five hundred?
How many ways can you find to show your answers are correct?
What connections can you see?
Can you generalise?
What else can you ask/show/prove..?
These questions are examples of what we at Mathematics Mastery encourage teachers in the partnership to use to challenge students who grasp content quickly. They are designed to deepen conceptual understanding and develop reasoning within a topic area – place value here clearly – rather than to “extend” students by giving them more maths, perhaps the next “topic” in the scheme of work or even the next perceived step in maths, perhaps standard index form in this case. This latter strategy can sometimes (often?) lead to a superficial coverage of a lot of maths, again sometimes (or often?) with little understanding of any of it or any ideas of how it connects with other areas of our beautiful subject.
There’s been a lot of reaction on the web to the recently published OfSTED report “The most able students: An update on progress since June 2013” which you can find here. One key thing we try to do at Mathematics Mastery is to be very careful with our language around this issue. We talk about “high attainers” rather than “more able”, not in any way to diminish the success they’ve already had in mathematics – we’re always keen to celebrate achievements – but at least as a reminder that sometimes even high attainment is not necessarily a reflection of deep understanding. Have you ever taught ‘A’ level to a student with a very high GCSE grade but who finds adjusting to Year 12 maths extremely challenging? It’s not as if they’ve lost “ability” over the summer holiday! Even more so, we shy away from using the term “low ability” as a label for students who are currently low attaining; low ability implies difficulties cannot be overcome and we passionately believe that low attainers with the right teaching and mindset can, and will, get better at mathematics.
We agree with the concerns that OfSTED have quite rightly raised about lack of challenge, especially at KS3, and fully support their recommendations to tackle the issues surrounding the curriculum and its delivery. We too believe in the importance of great teachers and great teaching at KS3 as well as KS4 because consistent high quality teaching is the key to long-term impact. As a professional development collaboration, our response is to keep supporting our partner schools to provide appropriate challenge for all their students, including those who are already high attaining. We want to deepen their understanding, to apply their maths in different contexts, to solve rich and sophisticated problems, to pose and to investigate their own conjectures. We believe this approach will help them to develop into mathematical thinkers much more powerfully than by doing seemingly more advanced maths at a shallow level. Here’s another example of a task from one of our Year 7 units, this time on area:
Go on – have a go, you know you want to!
Did you spot that there was more than one possible solution? This realisation open ups a whole series on follow up questions: Why is there more than one solution? Is the diagram helpful or could it perhaps be misleading? Will there always be a number of alternative solutions no matter what numbers we choose? How many? Why? Can you think of ways of adapting this question? Could you challenge students to come up with more questions/generalisations/ideas/problems?
That’s got to be a better challenge than going straight to using/repeating the area of a parallelogram or a trapezium…it’s certainly more fun! We believe that providing this sort of challenge and giving students the chance to explore and create mathematics is a crucial aspect of developing their learning, helping all students to succeed and the vast majority to excel at least at GCSE, but also to be properly ready for further mathematical study at A level and beyond.