Human Experimenting (but not what you think)

I've been reading all these fabulous, exciting articles about getting students online and connecting and interacting etc..., and been feeling more and more like this is something I want to try. But doing something so different is SCARY. No-one else at my school had tried anything like this before as far as I could make out. And my syllabus time is pressurised, and and and... But at the end of the day, I felt like these were just excuses. On Friday I decided to book the computer room, in the hopes that this would force me to just do it!

I then spent most of the weekend madly planning and trying to work out how to make the best use of the internet/geometer's sketchpad time. The lesson itself was an introduction to the sine/cosine/area rules in Trigonometry, with a very strong group of grade 11s. So I planned the first half of the hour and a half lesson to be a "collaborative research" style lesson, where we all searched for and posted links (under a twitter hashtag) about sites which we thought were interesting/useful/funny around the topic of applications of trigonometry, and proofs of the laws.

This went okay, though by the time the kids had logged in, and sorted Twitter out and so forth there wasn't that much time to actually get the interactions going properly. Keeping the class on-task was also tricky, but I'd hope that that would improve with time and practice (on my part as well as the class').  I also found it tricky to "sell" the value of the whole exercise to the kids. My most frequently asked question was "Is this for marks?"!

The second part of the lesson was an investigation of the sine rule, followed by a guided proof. The idea was that they would follow instructions to form a conjecture regarding the sine rule, and then prove their own conjecture (again, following instructions). Well, that went...okay...but you'll gather from my hesitation that it didn't go quite as I intended.

It was mainly slow. My instructions needed a certain amount of streamlining, but there were decent, especially since I was there to facilitate. But I underestimated how difficult it is to PROVE stuff when you're in grade 11. Even if you're very good at Maths.

To summarise:

• Advantages - Not a total disaster. I saw some potential for developing the ideas and activities. I think it started to develop some (non-curricular) skills which are important to acquire.
• Disadvantages - Time-consuming, and not necessarily always directly relevant to the (curricular) abilities that the kids need to acquire. Inefficient (but is real learning ever efficient?).

All in all, I think I will try this again...not every lesson or for every topic, but I will definitely try it again. In the immediate context, I think we will return to the computer lab on Wednesday to complete our proofs!

Proof and Imagination

A few years ago I saw a movie called Proof. It's about an elderly Maths professor and his daughter. He spends his whole life trying to prove a single theorem, and eventually gets Alzheimers or something like that. His daughter sacrifices her academic career to care for him... Anyway, I won't give the plot away, but the idea of the film centres around the great value of proof.

Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.

Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.

It's imagination that has a shape.