The list goes on

Another item for the list:

XI. Computing

When I was a kid at school I got into programming in Basic. It was mathematical in ways that maths lessons weren't. For instance, the only time I ever asked a teacher how to do something that hadn't been taught was when I needed a bit of maths for a program I was writing.

"How do I stop this weather data looking so spiky?"
"You need a moving average."

When I did my teacher training at the Roehampton Institute back in the early 80s, computing was as you know in its early days, not quite as early as when I was at school (there was a screen for instance!) but early days:

But there was a book I really enjoyed, and found captivating, Seymour Papert's Mindstorms. The idea of using Logo in the classroom to program a turtle was a really inspiring one, where you really needed to get to get to grips with maths to get a lot of things done. This short video from the time gives you some idea of what was starting:

Over the years, I've used Logo a lot. You know the kind of thing: forward 10 right 90...
I used the Microworlds  environment quite a bit. Now I use Scratch. It can have too many knobs on, too many attractive features, and I think the simple maths of Logo is the best; it continues to be a great way of really needing a lot of maths to achieve an end. I never seem to get quite far enough with it to get on to the wealth of exciting possibilities, but I think this might be changing with the new emphasis on coding in the computing curriculum. I'm using code.org's great resources and courses with my kids, and encouraging the whole school to develop this kind of coding, and although the finely-grained steps of the courses are mainly convergent, I think they provide a great preparation for wonderful divergent work.

It's this sort of blank slate tool that I like best of all. I use other packages that aren't blank slate (Education City, mymaths.co.uk) and there are lots of brilliant websites for developing skills, but the tools I like best are the powerful ones where it's down to the students to make something of them. (Geogebra is another brilliant example, one I used today for instance to get the students to generate lots of different kinds of hexagons.)

Here's a great tribute to Papert:

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...

Even harder sums...

La Rentrée, as the French call it, is fast approaching. I'd better get on with my list of good ways to get to the challenging stuff. These are of course obvious to many of us teachers, but then again, they are not at all as universal as they could and should be, so they're worth reiterating.

Last time I gave my three ways of making maths harder - without the useless drudgery:

1. Pick subjects that give power. 
2. Find the subjects where kids can be creative
3. Go into history and biography.

So, three more:

4.  Enthusiasm

5.  Discussion 

6.  Presentation

alder trees in the New Forest

4. Enthusiasm - the teacher's, that is. This can be squeezed out by an over-structured curriculum, and pressure to get numerical results. But when it's there it can make school something more than just school. Take this case: Roger Deakin in his brilliant book Wildwood describing what his biology teacher set up:

…Barry infected us all with his wild enthusiasm. 

Although he would modestly deny it, Barry Goater was the instigator of an extraordinary educational experiment. In a quiet corner of the New Forest he established a camp for the detailed study and mapping of the natural history of a stretch of the wild forest woodland, bog and heath surrounding Beaulieu Road by his Biology sixth form. The camp became something of an institution at our school in the relatively treeless Cricklewood. It was traditional for each generation of us sixth form naturalists to return there again and again and taste the intoxicating pleasure of exploration and discovery in the wild for ourselves. Each of us had a particular project, literally a field of inquiry, and the work we were doing was genuinely original. We learnt the scientific disciplines of botany, zoology and ecology, and we kept our eyes open as all-round naturalists. What we discovered was particular to the place, and, best of all, it belonged to us. 

Beaulieu Road was our America, we were pioneers, and the map we jointly drew and refined through gradual accretions of personal observation represented not only the complex natural ecology of the place, but also an ambitious and entirely novel cooperation between several generations of the sixth form botanists and zoologists of our school. Through our cumulative endeavours we were charting the relationships between the plants and animals of the place. But the records we kept were also a testament to our own human relationships as naturalists, biologists and zoologists. We were learning at first hand how exploration and scholarship can evolve and progress in time through cooperation and the free exchange of ideas. Small wonder that the experience influenced so many of our lives so profoundly.

There's so much in this description. But isn't it interesting how a teacher's enthusiasm can lead to "the intoxicating pleasure of exploration and discovery in the wild"for themselves, to  a world that the students discover that belongs to them?!

4. Discussion: Roger Deakin talks about cooperation and the free exchange of ideas. Ideally, it comes naturally when there's some big project that the class or group is working on. Sometimes it needs to be structured. Lots of teachers naturally use a variant of the think-pair-share strategy: students think about a question on their own a little, then they talk about their ideas with someone else, then they might share what they've arrived at with the bigger group. It gets away from the teacher questions-pupil answers routine (which is useful some of the time) where only at most one student is getting to put their ideas into words at any one time. This is important, because to know something really well, it's best if you can explain it too, and hear other people's explanations of it. And even better if you can modify your understanding as you discuss, refining what you first thought.

5. Presentation - the students that is. As I say, it's important to be able to explain something, and you get to know it more deeply in doing so. When we made factor trees last year we explained our factor forest display to the other classes who would see it.  When four girls created a beautiful mathematical square, they explained it the other classes. 

To be continued.

Harder sums...

So the UK government wants "harder sums". (Part of a drive to raise standards - see Michael Tidd on The Level 4b myth for thoughts on this.) And they want to have kids learning their eleven and twelve times tables, instead of the tables from one to ten.

To me, this is not the way to go. Not because I don't want a challenge. I do.

Steep paths - even rocky cliffs - are fine, if they lead somewhere. If they are just a demanding rock face that leads to... more demanding rock face, without opening up onto a fertile and beautiful landscape, then maybe children develop grit or obedience or something, but they're not making the most of their maths learning.

Take the twelve times table. Not a big thing. But really, is that taking us somewhere?? Read Jon McLoone on Is There Any Point to the 12 Times Table? for his interesting thoughts on this.

Creating their own pattern

So how would I like there to be more challenge? I really want kids to go further, rather than that their work is harder. I'm trying to make it as easy as possible to learn as much as possible. Anyway I'm trying to get my ideas on this spelled out, so here's a beginning of a list:

1. Pick subjects that give power. The same sums with more digits, rarely-used algorithms like long division (on this, see Owen Elton's Why Gove is Wrong about Long Division) or dividing fractions don't seem to me to lead anywhere much. Beginning algebra (in a fun and appropriate way - see my Year 4 lessons this year for example) on the other hand gives a really powerful tool for making generalisations.
2. Find the subjects where kids can be creative, make something of their own. Get them up on the top of Bloom's taxonomy. An example is getting kids to generate their own patterns with manipulatives, and then describe and explain the pattern with numbers (example). As Keith Devlin said in his recent blog post, Most Math Problems Do Not Have a Unique Right Answer.  In the real world, creativity is going to be very useful.



Rolling like Galileo

3. Go into history and biography.The new maths curriculum for England has added Roman numbers . This could be just a dull dead-end, or it could be part of a sequence of lessons looking at how number systems developed that could really grab some children. Telling the story of a maths idea by talking about its discoverer will help the kids to go further with it. Take our work where we talked about Euler. The kids are prepared to go further because there's a narrative to engage them. (I'll probably do the Euler work again next year, but add something on graphing,  maybe using Joel David Hamkins' great booklet on graph coloring, chromatic numbers, and Eulerian paths and circuits) In Primary we have a bit more freedom - we teach the whole curriculum, so it's easier to make links - between maths and history, or,as in the case of our work on Galileo with science and English too.

That's a start. I'll  post more of my personal list later.

I'd really love to hear other people's ideas on all this, whether it be connected with the first three items in my list, or about any successful ways to extend children's learning.