Thursday, 30 June 2011

The yin and yang of maths education - part 2: My journey into this thinking.

File:Yin and Yang.svg
I took my GCSE in maths in the very early days of that exam - when it was supposed to be a revolutionary change from O-level.  And what a revolution it was!  Previously my set would have done O-level in a year (at 15) and then AO-level in the next year (age 16) so that the most able could go on and do A-level the next year and Further Maths the year after that.

All of a sudden we were doing GCSE.  Over 2 years so we had time to grow.  With barely any defined content apart from some odd stuff on shears I seem to remember.  Instead we had to do 6 extended projects which we designed ourselves.

It was complete and utter chaos.  Our poor teacher had no experience in running classes in this way and suddenly he had to supervise 34 students all doing their own independent research but being marked to a core scheme.  The intensity of the strain on him meant that those of us who had completed two pieces of work to a decent standard (only two were graded) we left to our own devices while he struggled to help the others through.  We played a lot of cards.  It was fun.  It made me think, a lot, about what the heck was going on?

After 2 years of that the school thought we simply couldn't do A-level in a year.  We hadn't done AO-level- how on earth would we cope with a whole A-level in 9 months (Sept to May)?  They didn't know what to do with us.  We told them to let us try and, for reasons no-one could fathom, we coped.  Instead of one or two completing further maths, 7 of us did and instead of 1 or 2 going to Oxbridge, 6 of us got in that year.  Weird!

What we'd been through wasn't sensible.  It wasn't sustainable.  It wasn't fair to teachers!  But it was interesting.  I can still so clearly remember the contents of one of my independent investigations.  I developed a great deal through doing it.  But how much time did we waste which we could have spent studying stuff?

It was no wonder the traditional teachers complained and there was a swing back to knowledge and skills based learning.

When I entered teaching I came to understand the frustrations of the teacher who was great at teaching kids mathematical knowledge and skills but who felt overwhelmed and frustrated by the idea of teaching them how mathematise the real world or to construct mathematics for themselves.  Until you've seen people doing the latter well and have experimented with it for yourself, it's really difficult to imagine, especially if you're working in a teaching environment which is structured with the assumption that traditional teaching will be taking place. I've watched teachers try and fail to make the shift in their teaching.  I've watched some come to demonise constructivist teaching because it hasn't worked for them.

On the other hand I've seen the most incredible and inspiration teacher from the constructivist paradigm lock themselves in a world of beliefs which denigrate traditional modes of instruction.  "Exams are bad", "you should never test kids", "you shouldn't grade kids and so on."  For me these arguments have some meaning and validity.  I have always tried to be aware of the contexts and particular students where such actions can be counterproductive.  But on the other hand I've seen how most kids love to collect badges, to tick boxes, to achieve closure and completeness.  How if you ask them to climb a ladder they will say 'how high?'  And I don't see that this is a bad thing. 

Most teachers clearly belonged either in one paradigm or the other and I deeply admired teachers from both, provided that they were at ease talking about why they were as they were, their motives were sound, they were sensitive to childrens' needs and they were at ease with and, when they had the energy, interested in teachers who were different to them and ideally keen to see if there was anything worthwhile for them they could learn from them. 

Some heads of department lay in the traditional paradigm but tried to make as much time as they could (without compromising the teaching of core skills) in curriculum planning for activities which focused on developing students' investigational and thinking skills.  I understood exactly where these teachers were coming from.

Other departments put the constructivist approach first (in a wide variety of ways) and signficantly subordinated the teaching of core skills and exam preparation.  Now where this worked well it worked brilliantly and I think this was because teachers who constantly teach in this way are constantly educated and developed through their own teaching.  They therefore become exceptionally skilled teachers.  If they can combine this level of skill with secure exam preparation you get an incredible maths department that students and educated or aware parents love.  But if less skilled teachers try to imitate this structure of teaching without proper support it can all go rather wrong.  So wrong, it seems, that we needed an inspection regime which abolished all departments of this type, good or bad (my apologies to those departments which struggle on against the odds and are, in my opinion, the exceptions which proves the rule).

So that was the background to my thinking.  The interactive teaching technology with personalised tracking came along..... and I'll write about how that changed things in my next post.

Saturday, 25 June 2011

The yin and yang of maths education - Part 1.

File:Yin and Yang.svg
Many authors have written about the need to move from a curriculum which teaches an established curriculum to one which encourges students to mathematise their experiences of the world for themselves.

I see things a little differently.

I believe that both forms of mathematics education are justified and of great importance.

Modern technology offers students a wide variety of ways by which they can efficiently and effecitvely acquire orthodox knowledge.  Teachers who utilise this technology can create more time in the classrooms on the types of tasks which effectively develop students' abilities to interpret real life situations, struggle with connected and extended task and to learn to express and develop their own thinking. 

Modern technology can also enhance the quality and relevance of the experience a students has when they explore mathematics for themselves because they can rapidly interact with established thinking, notation and invididuals who are interested in their area of enquiry.

So it is natural both that more constructivist teaching can and will occur and also that such teaching will lead to a greater coverage of core concepts.  But I prefer not to think of the outcome as being a shade of grey although that's how my lessons might appear to others. I prefer the visualise black and white, yin and yang.

Here is an analogy.
Which is better - pop music?  classical music?  something which is halfway between the two?  or something which is unashamedly both, black and white, yin and yang with no compromise to either?  Here's a video to illustrate my preference.

More orthodox justifications for and desriptions of this perspective will follow as and when I have time.

Sunday, 19 June 2011

How do the Chinese do it? Chapter 8 - A vision of much more

Last night I attended an e-seminar with Alexandre Borovik.  I struggled to follow much of it as the conversation was heavily accented and the sound quality poor.  But then, in the last few minutes of the hour, something tremendous.  I opened a draft of his book:
Shadows of the Truth: Metamathematics of Elementary Mathematics
and in it, muddled among many other paths of thought, I could see that he has this incredible insight into the deeper concepts of mathematics that only children and highly eduated pure mathematicians can see.  Insights of which I have only begun to scratch the surface in my teaching.

Thank you Alexandre.

How do the Chinese do it?  Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6

Sunday, 5 June 2011

How do the Chinese do it? Part 7: The Last Chapter?

After the initial blast (maybe 6 lessons) of starter questions of this type, I simply come back to this starter activity whenever I feel like it for one-off questions.  It's worked well for me right up throught topics like surds and complex numbers.

There's nothing whizz bang or whistles and bells about it.  Students aren't inspired as they walk into the room, there's usually a bit of grumbling (oh no not this again) but their engagement builds gradually as they have to think for themselves mathematically.

It's effortless to set up (just 4 letters and 2 numbers on the board). I like the fact that I can spend my time listening to students and helping them give voice to their personal journeys into understanding primitive structures for mathematics rather than on class management. 

This task rests easily with there being a little off task chat and it taking some students a minute or two to properly engage.  As I've said before I've not taught in schools where there have been high standards of behaviour and I have never had the luxury of expecting all my students to come to my classes bright-eyed, bushy-tailed and well behaved.  I've had to learn to take my students from being badly behaved to being fully engaged and therefore on task through the way I teach and the type of task I use rather than through demanding that they behave well before I start to teach.

It's important to understand that this task will become more powerful for you as you use it with more classes.  You will gradually become aware of a wider and wider variety of structures that students use to support their thinking and as you become aware of them you become better at spotting what it is that a students is struggling to explain and at helping them to express it clearly to their peers. I think you will be surprised how your own thinking expands and takes you in the direction of confidently using flexibly the wide range of structures the teachers in Liping Ma's book used for calculation.

Is this the last post on this topic?  Maybe not.  Maybe you or I will come back with new ideas stimulated by this task.  I hope so.  Thanks to all who have asked questions.  Please keep them coming!

How do the Chinese do it?  Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6

Saturday, 4 June 2011

A little light relief

Many thanks to the teachers at our local school for putting the spirit I know so well onto Youtube.

This was there way of livening up a year 11 leavers' assembly which would have been otherwise weighed down too much by thoughts of the loss of a student.

Cockermouth School Teachers do Glee 2011!!!!

The news article

How do the Chinese do it? Part 6

I always ask students to describe their thinking on these problems.  To compare and contrast the structures they are using. 

Sometimes they work individually and I pick people at random to talk to the class, somtimes they work in pairs or groups. 

Then I'll 'go at little crazy'.  We'll do an SDPQ with two 1cm lines.   Or two lines with obviously different but unmarked lengths.  It sounds bizarre but the conversations are wonderful.  Structural insights are everywhere.  The students are really thinking hard.  By now they know I'm serious when I say this is all about the journey and that I'm only a tiny bit interested in the answers.

When they're ready we'll move on to algebra.  We can use letters, constants, linear term, quadratic terms and so on.  Remember learned tricks only get half marks.  They need to be able to explain structurally what's going on.  They will need to take those structures they've developed with numbers and transfer them to the algebra.  It's powerful, it's challenging, it's rewarding and its deep.  You should expect the unexpected, namely that when students describe their thinking you will be hearing things you have never heard before and will need to go away and think about.  Don't worry - if you get stuck you'll probably find the student who came up with it has had deeper thoughts themselves after a day or two.  Those who say something innovative often think about it a lot after the lesson.

How do the Chinese do it?  Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6

How do the Chinese do it? Part 5

If I was teaching a lesson on long multiplication or long division, I would set an SDPQ starter which contained one of the toughest questions they would face.  It was fascinating to see them trying to wrestle out an answer.  The method I was teaching that lesson would then become closely contextualised into their existing thinking as this existing thinking was now active, rather than being developed somewhere separate in their mind. 

They didn't object to doing a lesson on long multiplication or division even if they could do the initial example, because by now they understood the validity and power of having multiple methods and of revisiting structural methods.  But, of course, in general I found they needed far less teaching on basic operations, because we were covering it very effectively through these starters.

One structural lesson I would never miss would be counting the squares in rectancles which leads to grouping them in hundreds tens and unit in a visual representation of the grid method for multiplication, because this so beautifully scaffolds the expansion of quadratic brackets and more.

How do the Chinese do it?  Links to my other blogs in this series.