The problem was to develop a model of thought within the basic structure provided by general or "point-set" topology. To do this, it was necessary to make four basic assumptions. It was assumed that each individual possesses more than the classical five senses and that for each of these there exists a category of sensory data. Also, it was assumed that the Cartesian product of these categories formed a set M of thought elements for each individual, and that certain subsets of M provide support for cogitation. The relation, function, continuous function, and homeomorphism, which are used to relate sets in topology, are discussed as a possible ramification of the model for communication. The global properties of the homeomorphism and continuous function present each as a viable support for strong and meaningful communication between thought spaces of individuals.

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In this workshop we present an innovative teaching, learning and research setting that engages beginning teachers in mathematical inquiry as they investigate, represent and connect mathematical ideas through mathematical conversation, reasoning and argument. This workshop connects to the themes of teacher preparation and teaching through problem solving. Drawing on new paradigms to think about teaching and learning, we orient our work within complexity theory (Davis & Sumara, 2006; Holland, 1998; Johnson, 2001; Maturana & Varela, 1987; Varela, Thompson & Rosch, 1991) to understand teaching|learning as a complex iterative process through which opportunities for learning arise out of dynamic interactions. Varela, Thompson and Rosch, (1991) use the term co-emergence to understand how the individual and the environment inform each other and are “bound together in reciprocal specification and selection” (p.174). In particular we are interested in the conditions that enable the co-emergence of teaching|learning collectives that support the generation of new mathematical and pedagogical ideas and understandings. The setting is a one-week summer math program designed for prospective elementary teachers to deepen particular mathematical concepts taught in elementary school. The program is facilitated by recently graduated secondary mathematics teachers to provide them an opportunity to experience mathematics teaching|learning through rich problems. The data collected include questionnaires, interviews, and video recordings. Our analyses show that many a-ha moments of mathematical and pedagogical insight are experienced by both groups as they work together throughout the week. In this workshop we will actively engage the audience in an exploration of the mathematics problems that we pose in this unique teaching|learning environment. We will present our data on the participants’ mathematical and pedagogical responses and open a discussion of the implications of our work.