We reject a trajectory approach to teaching that classifies “good” and “bad” teaching actions and seeks to move teachers’ practices from one of these poles to the other. In this article we offer instead a conceptualisation of mathematics teaching actions as a “landscape of possibilities”. We draw together terms commonly used in the literature to describe teaching strategies, and add our own, to offer an expanded view of teaching actions. We illustrate each with data extracts drawn from our various studies of mathematics teachers and classrooms, and explain how a range of teaching actions can be woven into a coherent teaching practice. Note that we are not talking about growth in teaching in this paper, nor about change in teachers’ practice over time. We aim to simply talk about and conceptualise teaching in ways that can broaden our understanding of it.
Excerpt: The main purpose of the present paper is programmatic. That is to say, its core is an outline for a research programme which projects well into the future. The main objective of this research is to intensively expand on an alternative understanding of the fundamental mechanisms of cognition, how they operate in living ‘systems, and how can they be manufactured by man, ultimately giving the possibility of truly intelligent artifacts. The main emphasis of this paper is in situating this way of approaching the cognitive issues in its conceptual roots.
Excerpts: Since this unruliness and chaotic structure are there, and since we are here, that is, we are alive, it follows another platitude, that such chaotic regimes are not incompatible with life, and, in fact, they are probably doing something interesting. So, we had better find out what is that. Those two are, as it were, platitudes. There is a counter platitude, which is that, once you try to pursue these questions, they turn out to be not easy to pursue. It is very hard to do research in this area. I’m trying to counterbalance enthusiasm for trying to discover something that is obviously there, with a concern about its difficulty.
Excerpt: The main purpose of this background paper is to provide an X-ray picture of the current state of affairs of cognitive science, in regards to perception and the origin of meaning. Now, like anybody who has ever examined a scientific discipline with any proximity, I have found the cognitive sciences to be a diversity of semi-compatible visions, and not a monolithic field. Further, as any social activity, it has poles of domination so that some of its participating voices acquire more force than others at different periods of time. This is strikingly so in the modern cognitive science revolution which was heavily influenced through some lines of research, particularly in the USA. My bias here is to emphasize diversity.
Excerpt: It is […] clear that there are a number of epistemological options for diverseresearch programs in [Artificial Life]. […] [My] purpose here is to sketch the options that I have been cultivating for some 20 years and why.