Here’s a gem of a paper from Jonathan Tuminaro and Edward Redish.
The authors have carried out a detailed analysis of the discussions a group of physics students had when solving a particular problem. They’ve worked hard (the researchers, as well as the students) – the first case study they chose was a conversation 45 minutes long.
While tackling the problem, the students have ‘played’ several epistemic games – or, put more simply, have used different ways of thinking. There are six different games identified – corresponding to six distinctly different ways of thinking about the same problem. Students don’t stick to one game though, they can flip between several. Very quickly, they are:
1. Mapping meaning to mathematics. This is where the students work out what is going on (or what they think is going on) and put it into a mathematical form (e.g. to make an equation) – then the equation can be used for things.
2. Mapping mathematics to meaning. Kind of the reverse of (1). Here the students start with a mathematical expression they know, and work out what it might mean in practice.
3. Physical Mechanism Game. In this game the students try to draw sense from their own intuition of the physical principles involved.
4. Pictorial Analysis Game. Here diagrams are used as the major step.
5. Recursive Plug-and-Chug. I’ll quote from the authors here, because they do it so well: "[here the students] plug quantities into physics equations and churn out numeric answers, without conceptually understanding the physical implications of their calculations." (The emphasis is mine.)
6. Transliteration to mathematics. Here the students draw from a worked example of another, similar problem, and try to map quantities from problem A onto quantities of problem B.
Now, I ask myself, which methods do I see my students doing in my classes, and using in the assignments I set. I have to say that in many cases I’m not sure – and probably my teaching is the worse for it. I can say which games I would like to see students using (1 to 4) and which games would make me shudder (5 and 6 – in which the students develop no physical understanding about what is happening), but do I know? There are certainly ways of getting students to use the ‘right’ games, notably setting the right kind of assessment questions.
OK, so which games do I play most in my research? I’d say probably 1, 2 and 3. I do a lot of physical modelling, in which I represent a problem (e.g. how do neurons in the brain behave in a certain environment) through a series of equations (game 1) and then work out the implications of those equations (game 3). I also draw a lot from my intuition about physics (e.g. if you increase the pressure across a pipe, you’ll get more flow, regardless of what shape the pipe is) – that’s game 3.
Finally, those physicists among you might like to know what problem the students had to solve. It was this. Three electrical charges, q1, q2 and q3 are arranged in a line, with equal distance between q1 & q2, and q2 & q3. Charges q1 and q2 are held fixed. Charge q3 is not fixed in place, but is held in a constant position by the electrostatic forces present. If q2 has the charge Q, what charge does q1 have?
o q1 o q2 o q3
The authors say that most experienced physics teachers can solve this problem in less than a minute. I solved it in about five seconds, using game 3, with a tiny smattering of game 2. The students concerned (3 of them together) took 45 minutes – this massive difference is perhaps interesting in its own right.
Reference (it’s well worth a read if you teach physics at any level): Tuminaro, J. and Redish, E. F. (2007) Elements of a cognitive model of physics problem solving: Epistemic games. Physical review special topics – Physics education research (3) 020101. DOI: 10.1103/PhysRevSTPER.3.020101.