what is maths? reality confronts inspiration

I thought this anonymous comment on my Simplicio-Salviato blog said quite a lot:

... The students I teach are quite difficult at most times, and are generally quite behind in maths. They can often remember ways of doing things by rote, but have troubles generalising and get completely stuck if they find themselves in an unexpected situation. Probably for this reason, they really like worksheets and textbooks. They can get the answers (one way or another) - in fact there _is_ an answer. They have no idea of what this answer means, or, for that matter what the question means or why the hell they're doing it. But that doesn't really matter if they can get the answer. On the other hand, whenever I have tried more open, exploratory or play based activities, the students tend to revolt (in a manner of speaking). Not always, but often. There's no clear end in sight, and they have difficulties working independently, so things start going awry. A lot of this, no doubt, comes from their lives outside of school, and earlier schooling.

It's a conundrum - I can work through the state's curriculum via a textbook and tick all the boxes, and the administrators will be happy and so will the students (though they won't actually have learnt anything useful), or I can tear my hair out trying something different where the students get confused, I don't cover the expected material, and the outcome might not be much better. I'm tending towards the latter option, in the hope that over time the students will become more responsive...

All this is true and says a lot succinctly, written by a real teacher in a real school - and that counts for something.

(In saying that I'm not forgetting that Paul Lockhart is also a real teacher in a real school and an inspirational maths teacher and polemicist)

The dialogue between reality and inspiration needs to continue, both sides of this discussion need to be listened to.

Anonymous points out there is tremendous and real institutional inertia on both teachers and students to perpetuate an uninspiring "plug and chug" pseudo maths methodology. This inertia is fueled by uninspiring textbooks (devoid of maths history and open ended challenges) and that many students actual prefer to do maths that way than face the discomfort and struggle of real thought and effort. It is also true that some open ended discovery learning or inquiry approaches achieve even worse results than simple old fashioned plug and chug.

These traditional approaches exist and are perpetuated for real reasons - even though they are rotten, boring and deserve to die.

There is risk involved in new innovative approaches. They can work but they have to be well thought out, based on sound theories and implemented in a flexible, not dogmatic manner.

Hegel: all that is real is rational; all that is rational is real
Engels: reality proves to be necessity; all that exists deserves to perish