Even more...

As I said, the UK government wants "harder sums". It wants rigour. Picture phalanxes of Roman centurions - very comfortable with their Roman numbers - marching rigorously up very straight roads, their shields held close together. Nothing gets past them.

My strategy for conquest is different. I've made a list of some of its components. Like Caesar's Gaul, so far it's got three parts:

First part:

I. Pick subjects that give power.

II. Find the subjects where kids can be creative,

III. Go into history and biography

Second part:

IV. Enthusiasm

V. Discussion

VI. Presentation

Third part:

VII. Climb up and down the ladder of abstraction 

- § - § - § - § - § - § - § - § - 

But now I'm going to add another part:

VIII. Estimation

I asked my friend Charlie who works as an engineer what maths he thought really needed to be taught at school. Estimation, he said, you need that all the time. And luckily it's become more accessible and more engaging than ever with Andrew Stadel's estimation180.com . I've used this with my Y4 class last year, and we're going to start again next week. What's so great about it, is that the kids are interested in the estimation and they like the challenge. They especially like it when there's a video "reveal" at the end.

I have this idea that if we like it, we'll start creating estimation challenges in the Year 4 classrooms, maybe begin an estimation blog, perhaps begin to find estimations to do at home too, photo or video.After that we get other classes to have a go.  Nothing too ambitious. First we take Manhattan, then we take Berlin.

My trial challenge was not a complete success, but it's helped me to get the measure of what's involved:



Anyway, here's Mr Stadel talking about what he does:



IX. The real world

To be honest, this is something I know I don't do enough of. Using real things, real places, things you might find at home. We've just been looking at reading scales, and for the first time we've got the classes to look for dials and scales at home this week. The range is amazing: weighing scales, a barometer, pressure gauges on pumps, a metronome, the rpm and the speedometer in car, a clock... and some things from an aeroplane: how level you are and speed. There is just so much to talk about.

In fact I can't stick to the real world. We did a bit of not-so-real world with our creation of meters to measure things not normally measured. The idea was to create a bit more attachment to our dial by investing more in it than usual.

It gave us a good chance to talk about what kind of units you might invent, as well as looking what the un-numbered marks represented. It also meant we could spend a bit more time on dials and still be doing fresh things.

X. Modelling!

This as Turtle Gunn Toms says in a comment on Graham Fletcher's excellent post on modelling, means taking a situation and mathematising it.

It's another thing I really don't do enough of. Probably none of us do enough of it! The ideal is a situation where you have a question, you put numbers to it, out comes some kind of answer.

It's a lot harder to find good examples for the primary / elementary classroom. I'd like to have a collection of this kind of mathematical modelling question.

Here's one I'm thinking of trying soon: I was talking to the kids: "Here I am, standing in the middle of the room..." and it occurred to me, "Where  exactly is the centre of the room? How would you work that out?" I said my thoughts out loud of course.

Not a very natural question perhaps, but I'd be interested to see how the class go about answering that. Some estimation first of all of course...

The ladder of abstraction

I'm interrupting my list to write about this. It's sort of in reply to a tweet from Tracy Johnston Zager, though I suspect I haven't answered the question in exactly the way she intended:

@mpershan @MathMinds @Zakchamp Maybe @Simon_Gregg could chime in about how he handles/introduces manipulatives?
— Tracy Johnston Zager (@TracyZager) September 12, 2014

No, actually this chime can be part of the list.

7. Climb up and down the ladder of abstraction

I've been reading Roy Peter Clark's entertaining and instructive book Writing Tools: 50 Essential Strategies for Every Writer over the summer (link to free podcasts of this on iTunes). One of the 50 strategies is "climb up and down the ladder of abstraction". Here's a snippet:

The ladder of abstraction remains one of the most useful models of thinking and writing ever invented. Popularized by S. I. Hayakawa in his 1939 book Language in Action, the ladder has been adopted and adapted in hundreds of ways to help people ponder language and express meaning.
The easiest way to make sense of this tool is to begin with its name: the ladder of abstraction. That name contains two nouns. The first is ladder, a specific tool you can see, hold with your hands, and climb. It involves the senses. You can do things with it. Put it against a tree to rescue your cat Voodoo.
The bottom of the ladder rests on concrete language. Concrete is hard, which is why when you fall off the ladder from a high place, you might break your foot. Your right foot. The one with the spider tattoo.
The second noun is abstraction. You can’t eat it or smell it or measure it. It is not easy to use as a case study. It appeals not to the senses, but to the intellect. It is an idea that cries out for exemplification.

Here's Hayakawa's illustration of his ladder:

This fits very well with the way I like to teach. Say in maths, I want the kids to use stuff they can touch or be physically part of. To take an example that I think works really well, I'll get them using the Cuisenaire rods to make their own patterns, and then I want them to take the numbers inherent in those patterns, then beyond that, the relationship between the numbers, and if possible to abstract that relationship into algebraic form. I'm quite prescriptive about what I want them to do, but hopefully the limitations are creative, because I want them to be creative.

If you haven't read Richard Feynman on why there is no science education in Brazil, do. He was astonished at how little teachers went down the ladder again from abstract to concrete.

"I didn't see how they were going to learn anything from that. Here he was talking about moments of inertia, but there was no discussion about how hard it is to push a door open when you put heavy weights on the outside, compared to when you put them near the hinge – nothing!"

bench by Zaha Zahid

The architect Zaha Hadid is designing the new maths gallery for the London Science Museum, a building I spent a lot of time in as a kid. She has a maths degree, and she uses maths as a place to find new abstract forms for her buildings. She was also inspired by the abstract mathematical paintings of Kazimir Malevich.

Black square by Kazimir Malevich

So, she's using abstract mathematical forms and ideas, and, quite literally, making them concrete:

"When I came to do architecture people said you must know how to add. There is that aspect to maths, of course. But there is another that was of interest to me and that was abstract thinking, and that was when I realised how important that degree was."

Zaha Hadid's design for the Mathematics Gallery

To me, that's thought-provoking.