**By Mark Dawes (June 2019)**

I spent an interesting day (7 June 2019) at the Cambridgeshire Maths Conference. Organised by the Cambridgeshire Maths Team, it followed a model of a full initial plenary, three smaller sessions and then a choice of final plenaries. This is brief overview of some of things I got out of the day.

**Reasoning – Mike Askew**

In the initial plenary session Mike Askew explored reasoning and how this can happen without carrying out calculations. He had us comparing the sizes of unfamiliar fractions by relating them to a scenario where groups of friends were sharing biscuits.

**Takeaways:**

- You don’t need to be able to calculate to be able to reason.
- Children’s reasoning starts with their understanding of the world.
- “I can directly teach you long multiplication, but I can’t teach you reasoning directly.” (Askew)
- Fractions start as a division model, where it is vital to know what ‘the whole’ is. (Perhaps this is why when I want to treat a fraction as a number on a numberline, some Yr 7 pupils want to know what it is a fraction _of_)
- It is lovely to have the opportunity to work with people from different schools and different key stages.
- CPA is not a journey from the less sophisticated to the more sophisticated. “I want echoes of the concrete still to be present in the symbolic” (Askew)

**Making Maths Make Sense – Ed Southall **

Ed had some lovely, interesting takes on why we do things the way we do, including what happens when we try to use the standard algorithm for subtracting big numbers, where the answer is negative.

There were some fascinating things with counting systems (base 4, base 5, a system that has 27 different number-words and associated parts of the body!).

When naming polygons I tend to use names of shapes up to dodecagon but then cop out and write 13-gon, etc. It was nice to see a full list of the Greek words for different polygons (and methods for combining them). Theoretically we could have gon(e) all the way up to a 999-gon (an ‘enneahectakaienneacontakaienneagon’, as I now know, thanks to Ed!).

**The power of intelligent variation – Craig Barton**

It was great to see one of Craig’s favourites from the variationtheory.com site. He showed us the one about means of sets of numbers and asked us to reflect on the links between the questions, what we liked about them, what surprised us, etc.

**Takeaways:**

- Lots of good presenting and teaching things that were almost incidental to the main point (but which had clearly been very carefully thought out!), such as asking for “2 things you like”, “2 things you would change”, etc, rather than just one.
- The different ways different teachers/schools had taken these questions on and were using them. For example, asking from one question to the next “how has the question changed and how has the answer changed”.
- The quality of the sets of questions is key. If you have a good set of questions, even if you aren’t skilled enough as a teacher to get into the full benefit of the variation with the pupils then the pupils’ experiences are no worse than if they had a traditional exercise with 15 unconnected questions.
- “The thinking continues after the answer has been revealed.” (Barton)
- “GCSE is obsessed with ratio at the moment.” (Barton)
- “Kids on the same table have different experiences with the same set of questions.” (Barton)
- Again – lovely to work with a Y6 teacher.

**Teaching Maths Words Well – Lisa Coe**

I knew all the other speakers (even those whose sessions I couldn’t attend) by reputation, personal experience or through their writing and tweeting. I hadn’t come across Lisa before, but I found her session excellent too, even though it was aimed at primary level (and I’m a secondary teacher).

The session began with a couple of important quotes:

“Disciplinary literacy is based on the idea that literacy and text are specialized, and even unique, across the disciplines” (Quigley, 2018).

“All names, within mathematics or elsewhere, are things which students need to be informed about” (Hewitt, 1999).

A new idea for me was that of having four different types of language in maths lessons (Monroe & Panchyshyn, 1995):

- Technical: words that are just used in maths, such as ‘integer’
- Subtechnical: words that have more than one meaning in different contexts, such as ‘volume’
- General vocabulary: everyday language that might be used in scenarios, questions and tasks, such as ‘vegetable patch’.
- Symbolic vocabulary: symbols for maths, such as 3, ÷

We then focused largely on subtechnical vocabulary, the range of different meanings a word can have (eg factor 30 suncream, X-factor, scale factor and a factor of a number) and used the Collins co-build dictionary to help us to devise some definitions.

**Takeaways:**

- All of the above!
- We might have different definitions of words at different stages in a child’s education.
- Lovely to work with a teacher of Y3 and an EY specialist!

**Problem Solving and Purposeful Practice – Craig Barton**

Craig focused here on three different activity types that will be familiar to those who have read his book. Hearing him talk about them was worthwhile and it was good to hear what he emphasised, such as the importance of having a few, general activity types that can be adapted and used in many different topics, so you don’t waste time trying to teach pupils more different types of activities.

After talking about SSDD problems and Goal-free problems (if you aren’t familiar with these do search for them) Craig finished by mentioning what he said was the least popular of his ideas, but which he likes: Maths Venns. I like these a lot!

Craig showed us this one: https://mathsvenns.com/factors-multiples-and-squares/

I like them for a number of reasons:

- They all have the same structure but can vary massively.
- A ‘nil return’ is often harder to explain. If it is possible to fill in a region then you can just put in the number, whereas if it is impossible you need to explain why.
- Regions might have no numbers that work, might have 1 number, might have a finite number of possibilities, or might have an infinite number of possibilities. Exploring these is interesting.
- This is a useful reminder that in Venn diagrams there is a region outside all of the sets.
- You can create these for very many topics, and for primary pupils up to sixth formers.
- You can think about what changes if the description of one of the sets changes. For example, if I change ‘multiple of 3’ into ‘multiple of 6’, what happens? What about if I change ‘multiple of 3’ into ‘multiple of 7’, what happens now?
- You don’t need to hand out printed resources – the diagrams are simple to draw.
- There is a vast amount of thinking required!

**Final thoughts**

A great day, with opportunities to catch up with friends and colleagues from different places, to hear excellent and thought-provoking sessions and to work with people with very different backgrounds, experiences and daily jobs from me.

Many thanks to Emma and her team for their excellent organisation and for having the foresight to create something that appealed so well to teachers from EY to Yr 13.

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