This afternoon, I went to a seminar given by Brent Davis (http://brentdaviscalgary.appspot.com/); currently visiting the UK from the University of Calgary. He was speaking about complexity thinking – and ways in which this approach can be applied to the study of the knowledge that mathematics teachers need for teaching.
The main point of interest for me was the idea that teachers do not necessarily know what they know. Even the very best teachers, who create the most valuable learning opportunities for chilfdren, do not draw on their mathematical knowledge in a conscious or systematic way. This is partly due to the fact that teachers of maths think in a very different way to children learning maths for the first time.
Brent gave some example of work he has done with teachers using Concept Studies – a means of revealing tacit knowledge about mathematical concepts. One example given of a Concept Study involved mulitplication. There are a number of ways to conceive of multiplication; as repeated addition, as grouping, as an area or array, as the compression or stretching of a number line, as a linear function, and so on. Different conceptions lead to very different implications for other aspects of mathematical thinking. One exmaple given was that the number 1 is a prime number under some conecptions but not under others.
So part of the difficulty of teaching and learning mathematics comes from the fact that, while teachers might be drawing on a variety of conceptions of mathematical objects and processes (even if not in a conscious way), children are often not aware either of what these conceptions are, which are appropriate for the task at hand, and what implications each might have. Brent was saying that the conduct of Concept Studies can transform a teacher’s practice by revealing these aspects of tacit knowledge – a teacher’s awareness of the metaphors being used, and an awareness of the extent to which particular conception are or are not consistent with particular tasks can tranform the experience of a learner.
This reminded me of my work on interdisciplinarity. Part of the difficulty of interdisciplinary research consists in the tacit assumptions, conventions and traditons of the discipline that a researcher carries with them. The reason that this is problematic is that it is often the case that different researchers carry tacit content that is not compatible with that of other researchers. Successful interdisciplinary work must therefore involve some revelation of this tacit content.
