A Mathematical Vision...

Prompted by feedback from various people I thought I’d post up some more of my thoughts.  As I said a couple of weeks back when I started this blog, I had intended it to be something that would help me to reflect upon what I do.  I am questioning whether it is helping me to focus my reflective thoughts more successfully than I had managed before.   One thing that is for sure is that some of the feedback has really got me thinking - lots of thoughts about the big, deep issues of mathematics teaching.  I didn’t really know what I was searching for when I set out, but now I think I realise.  I want to be able to define my pedagogy and principles more clearly.  I want to have a clearer idea of who I am as a teacher, and whether what I do is as effective or worthwhile as I believe it to be?

I feel I am getting closer to my own teaching philosophy - it‘s amazing the revelations I‘ve had in the space of a few weeks.  I know I work in a results driven business - just like football managers.  But I’d like to think I’m more of an Arsene Wenger - trust the kids and get them doing some beautiful maths moves rather than a Sam Allardyce - rigid, boring but gets results.  Ok Arsene, might not always win the trophies, but I don’t think good exam results and true mathematical enlightenment should be mutually exclusive.

As you can no doubt tell, I am no great writer so in order to stop this turning into a completely unstructured rant I am going to use points made by other people to write about some more of the things I have been thinking about over the past week or so.

I want my students to experience maths rather than being 'taught' by me. However am I asking these kids to reinvent the wheel?   I also worry that perhaps we need to teach them a set of tools and then give them problems where they have to decide which tools to use.

I think there must be an element of re-inventing the wheel with any learning process.  If we were to withdraw this aspect  I think we would end up with purely a “here’s a result. Use it to do questions 1 to 30” approach to teaching.  If kids can’t reason what point is there in equipping them with tools they can’t use?  I can teach anybody to differentiate, but what good is that if they can only apply it to a few selected styles of problems they have learned off by heart?  What happens if something unusual appears in a question in the exam?  PANIC!!!  All kids need a set of tools to use to solve problems, I agree with this.  However, I also feel that we shouldn’t just hand kids these tools without having at least tried to get them to understand them.  I remember as a pupil at school - I wasn’t interested in the proofs our teacher would do as they weren’t examinable.  Well, what a shock I got at uni once proof became important - I was completely useless at it, never having had to properly reason before in my life.

For instance if I face a new maths problem I have a degree and PhD from which I can draw knowledge to help me solve a particular problem. What do the kids have to help them! 

The kids have lots of knowledge - we should be gently building upon things they already know.  A problem is something the kids haven’t got the answer to yet - but as long as the problems set are appropriate - they will almost certainly have the maths to tackle it.  Generally in a lesson there are only a couple of new facts, and as a rule at least one of these could be figured out by the kids as it is a follow on to something they already know!

Surely we should be equipping them to open up new areas of knowledge.

Definitely, but without having ever managed to open up an area of maths for themselves which is already well understood by others, how can they ever move forward to the cutting edge of our subject with confidence and more importantly - the ability and experience to do so?  For example getting kids to figure out simple results by themselves that are part of the course lets them practice the reasoning and logic skills that would be needed should they ever go on to be adults working on new areas of research.  If a pupil has never experienced trying to prove something and not succeeding at the first attempt but still going back to revise his ideas - if he has never had this experience - what hope is there of him ever opening up new areas of the subject later on?

People learn maths by thinking. 

Yes.   But I question whether lots of repeated applications of the same formula is “thinking”.  I loved the anology about a dancer learning to dance by dancing.  I think our kids should be solving problems to get good at maths - but I mean proper problems, not boring repetition.  Although I don’t dispute that in some cases I agree that lots of practice is required to deal with subtle algebraic differences between questions etc.

The only reason to have all of the V,A,K learning styles is to get their attention levels up by having some variety. The V,A,K learning styles are the medium. But the medium is not the message. 

I really liked this post.  Thanks for that input.  I suppose I’ve been guilty of trying to overly jazz up lessons to tick these boxes.

Random points:
I think we need to have confidence in our kids. Today with my new first year class I had them doing an investigation into interior and exterior angels of polygons. My ultimate aim was a formula for the sum of the interior/exterior angles of a n-sided polygon.  I gave them very little information other than what an interior and exterior angle was. I wasn't very prescriptive at all and didn't give them a lot of input into what to do. However, despite this most of them followed the same steps. Most of them chose a table to organise their data for example. These kids had a plethora of skills for solving these problems already, the only new thing that was required was a wee piece of knowledge which as the teacher I gave them.  After that I set them free to discover things for themselves.  I’m a great believer in the idea that if you discover something yourself you are more likely to remember it.

Sometimes we should maybe focus on how much of our "chalk and talk time" is actually delivering properly new content. As I said in my previous post, why should I stand at the board and do examples on adding fractions with x+ 3 and x + 5 as the denominators, when they already know how to add fractions. Why not challenge them first? Push them. All I would do at the board is be providing stabilisers for them. Which does seem a wee bit untrusting, as the kids already have all of the algebra and fractional skills to do the problems. They just haven't put it together before! Let them tackle it in pairs. Let them articulate ideas, let them come up with their own solutions

I know that sometimes when having these debates it’s easy to be idyllic.  When confronted by a class of 30 hormonal 15 year olds, it is something altogether different.  But I think we sometimes need to be brave and design lessons in a way so that pupils can explore for themselves, while at the same time, ensuring that if they don’t quite get there - they have the knowledge required for the exam.

I'll finish on a quote from Larkin’s rant - he asked how much has changed in the years since it was originally written?

“I was made to learn by heart: ‘The square of the sum of two
numbers is equal to the sum of their squares increased by twice
their product.’ I had not the vaguest idea what this meant and
when I could not remember the words, my tutor threw the book at
my head, which did not stimulate my intellect in any way.”


As usual, please post your own thoughts and reflections etc - and tell me if you think I'm wrong.