In two weeks time on October 13th I will be delivering a session that shares the title of this blog. The blog is meant to act as a preview to the session.
“Mastery”. Some people see it as the latest buzz-word to be shunned until we wait for the next “big thing”. For others it is central to teaching. For some it is a confusing term with no clear idea of what it actually means. And I can sympathise with all of these views…
The idea of “mastery” has been around for a long time. People much more knowledgeable have written about its provenance, its history and its progress to the modern day. Neither this blog nor my mathsconf session will be trying to reinforce or reinterpret any of this. I will not be attempting to explain the structure of a mastery curriculum (which is not exclusive to a mathematics curriculum). Better men than me have already done this, not the least of which is the LaSalle CEO Mark McCourt (if you haven’t read his blogs on mastery then you must). Saying that, it is important to understand certain aspects of its structure to understand where I hope my session fits in.
One of the central aspects of a mastery curriculum is teaching in a way that all pupils can access from their starting point, and then carefully assessing their understanding throughout the teaching process. A second is the use of correctives where the initial teaching isn’t successful – having different ways of approaching concepts when the first way falls short. The biggest aim of my session is to try and showcase some of the ways that teachers can approach this. Starting with what I see as important ideas to consider when thinking about structuring learning, I then aim to share practical examples of approaches that could be used either as part of the initial teaching or as a corrective approach. For those that know me, it won’t be surprising to hear that much (but not all) of this focuses on the use of representations to reveal the underlying structure of an idea (given that my book “” is entirely concerned with the use of representations and manipulatives to reveal underlying structure).
As an example, but not one I am using in the session, consider the “rule” that one negative number divided by another negative number results in a positive answer. Consider -15 ÷ -3:
One way of representing this is to use double sided counters; these usually appear with a yellow side (positive) and a red side (negative). Two different coloured counters can also work, and in fact to model this calculation we only need to consider negatives so a single colour of counter will suffice. The image above shows -15, and now we have to think about how we divide that by -3. One way of thinking about division is to think about creating groups, so a possible way of looking at this calculation is, “Start with -15 and create groups of -3.” These groups can be seen below:
When we think about division like this, the result of the division is “How many groups can we create?”. We can see that this process creates 5 groups, which means that -15 ÷ -3 = 5.
Often this “rule” is taught as an arbitrary rule, without any attempt to show where it comes from. In many classrooms, one could be forgiven if kids believed that the only reason this is a “rule” of maths is because teachers says so. But this rule is a necessary rule – if division works in the way we know it does then the answer to -15 ÷ -3 cannot be anything but 5. I finish my session with a discussion around other “rules” of maths, how appropriate representations can show where these rules actually come from, and also discuss how we can manage the transition from using representations/manipulatives to the abstract calculations. Hopefully I have whetted your appetite to hear more about teaching approaches that can support mastery in mathematics, and I look forward to seeing you (whether in my session or not) in Birmingham. Don’t forget to join us for the pre-drinks and networking the night before as well!