Generating Examples for Generalised Rules: #collabomaths

I was at the National Maths Conference on Friday in Sheffield and could easily spend a blogpost on summarising the many things I learned. Happily, many people have already done so (and took more photos). Instead, I’d like to focus on something I’m going to try doing differently as a result of the conference and invite you to join a very small, very geeky, Twitter party.

geek party 2

I was struck by what was shared in the Shanghai session, when those who visited showed examples of how teachers create progression in their examples for procedures. In particular, the design they employ appears to really build up concepts of underlying structures, by showing how varied they are. James Pearce gives an excellent summary here.

Specifically, Shanghai teachers seem to prepare their examples and explanations to help students see a broader range of applications for a rule. Here is an example for multiplying indices, in terms of the examples we might show to our students in the UK:

indices UK

The focus in Shanghai is on a broader range of applications, in order to make it easier for students to generalise the rule. Here is a rough example:

indices Shanghai

This would not have been instinctive to me, thinking about the cognitive load on my students and the risk that struggles with directed numbers or non-integers would cloud what was happening. However, few of the examples are inherently harder and it creates more opportunities for interleaving (in addition to illuminating the broader rule).

Here is another example, for difference of two squares:

difference of two squares

I particularly liked the final one, and how that would be so much better a preparation for the new GCSE spec! I’m wondering if, in my efforts to make sure that work is scaffolded and students’ working memories aren’t overwhelmed, I’ve presented too narrow a range of applications at the outset and thus made it harder for them to see how to apply it outside of that narrow structure.

With this in mind, Richard White and I thought we would use the approach we learned in Luke’s session to generate ideas, whereby there is a ‘splurge’ of initial ideas and we later sort them to decide the best range of examples to show to students.

Admittedly it was an odd way to spend the later part of a Saturday night, but we found it wonderfully, geekily enjoyable to focus on a narrow piece of the curriculum and think about how we could create more demanding examples that better exemplify a rule. Here is what we created in about half an hour:

surds 1surds 2surds 3surds 4surds 5surds 6surds triangle

It’s far from exhaustive, but is a much better basis for planning work on (simple cases of) multiplying surds and bringing rigour to a SOW (and supporting new or struggling teachers, as well as non-specialists). It gives a clearer goal in terms of “What should they be capable of by the end?” and “What examples will I share to get these ideas across?” Richard has since used the approach in NQT mentor meetings to help those teachers to think about planning in a more focused way (i.e. to move away from resources towards exposition). As a professional exercise, it was really enjoyable as it made for a happy marriage of focus and creativity.

We are planning our next topic and, due to living in different cities, are going to see if it’s possible to try generating examples via Twitter. We’re going to have our first attempt this Wednesday (30th September) from 4.30-5.30 using #collabomaths as the hashtag (better suggested will be accepted!). I am also trying to corral my maths teaching hero (the man who taught me almost all I know, in my first school, but who thinks using MS Word is the height of tech sophistication). We will probably go with expanding single brackets, but it’s TBC. If you would like to join the teeny party, you would be very welcome 🙂

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