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.