In the 1970's and early 1980's, there was great interest in the ‘processes’ by which things were done, and thinking mathematically is a fine example. However, while interest in delineating processes of thinking and creativity have become of renewed interest recently, the language in which they are described has changed considerably. We found that it made more sense to us, and to people with whom we engaged mathematically and pedagogically, to think in terms of natural powers that learners bring to the classroom. The task of teaching then becomes one of provoking learners to make use of and to develop those powers in the context of mathematical thinking.
We follow Caleb Gattegno in seeing awareness as the basis for action; without awareness, there is no action. However, some awareness may be so integrated into our functioning that we are not consciously aware of them operating. This is certainly the case when we suddenly find ourselves acting automatically out of habit. Again following Gattegno, mathematics as a discipline only arises when people become aware of the actions they are performing in certain contexts (relationships and properties in number and space) and articulate
these awarenesses to produce ‘mathematics’. So mathematics as a body of knowledge in books can be seen as formal recognition, expression and study of awarenesses that inform mathematical actions in problematic situations. To become a teacher requires becoming aware of the awarenesses that generate mathematical actions, because these are what trigger pedagogical actions. Consequently it is vital to educate one’s awareness by engaging oneself in mathematical tasks which bring important mathematical awarenesses to the
surface, so that they can inform future action.
Awareness is closely related to cognition; action is closely related to behaviour. An often overlooked aspect of human psyche is the emotions or the domain of affect. The original book addressed this through suggesting that being stuck is an ‘honourable state’ from which it is possible to learn, and that expressing emotion-laden observations about being stuck and having an insight (AHA!), however transitory, releases energy which enables progress to be made. It celebrated positive emotions: the pleasure of making sense through use of your own powers, the excitement of discovery, the aesthetic pleasure in an interesting result, and the satisfaction of finding a resolution. I add here that developing a disposition to recognize problematic situations in the material world as well as in the world of mathematics, the ‘questioning attitude’ is also a significant contribution to the effective domain.
The current emphasis on collaborative activity as a necessary component of mathematics learning is a development from the value recognized and promulgated through the original book, that working together can be stimulating and can open up avenues that no single individual might have recognized by themselves.
At the same time, it is vital to have periods of ‘own thinking’ during which possibilities are considered and either pursued or dropped. Some people like to start individually, and then, after a period, exchange possibilities; others like to have a period of collective idea-generation followed by own thinking before coming together again. Certainly it is helpful to have communal reflection as a force to bring to the surface and articulate insights and observations about salient moments in the exploration, even though these will often have occurred during individual thinking. The presence of significant others is an effective contribution to stimulating the impulse to express and so clarify your own thinking, as well as to connect it to the thinking of others.
I also take the opportunity afforded by this blog to introduce persistent and ubiquitous mathematical themes which imbue mathematics. A brief description of powers, themes and related notions can be found.
No comments:
Post a Comment