Practice-based pedagogy in mathematics and science teaching methods: Challenges and adaptations in contextJust over five years ago, conversations between mathematics and science teacher educators in our faculty led to considerations of how to develop a more consistent pedagogical approach to teacher education across subject areas and programs. The conversations included a discussion of Cycles of Enactment and Investigation (Kazemi, Franke, & Lampert, 2009), an approach to teacher education which had already shown promise in terms of supporting novice teachers in understanding the complexities of teaching (Lampert et al., 2013). In what follows, we describe our own practices as mathematics and science teacher educators, and our collaborative efforts to develop novice teachers' pedagogies of practice in three complex and varied teacher education programs via Cycles of Enactment and Investigation. More specifically, we present some of the challenges we have encountered with respect to implementing Cycles in our own academic context, along with how we have adapted the Cycle to support novice teachers within the context of our own programs. We offer this work as an introduction to Cycles of Enactment and Investigation in a Canadian setting, as well as an example of how programs developed elsewhere might be adapted to more local contexts.Research indicates that novice teachers frequently leave teacher education programs with a great deal of theoretical knowledge about effective mathematics or science teaching, but scant opportunities to practice these pedagogies in context (Grossman, Hammerness, & McDonald, 2009;Windschitl, 2006).The result in the field is that novice teachers enact a mixed bag of pedagogical approaches (Windschitl, 2006). These can range from inquiry-based methods, to rote methods formulated through the "apprenticeship of observation" -the lengthy process of enculturation into beliefs about teaching that novices have engaged in throughout their lives as students (Lortie, 1975). Over the past five years we have been developing a program of mathematics and science teaching that requires novice teachers to enact core practices (McDonald et al., 2014) that elicit and respond to students' thinking. To develop This is a post-peer-review, pre-copyedit version of an article published in 'Canadian Journal of Science, Mathematics and Technology Education'.
We examined the codevelopment of mathematical concepts and the mathematical practice of defining within a sixth-grade class investigating space and geometry. Drawing upon existing literature, we present a framework for describing forms of participation in defining, what we term aspects of definitional practice. Analysis of classroom interactions during 16 episodes spanning earlier and later phases of instruction illustrate how student participation in aspects of definitional practice influenced their emerging conceptions of the geometry of shape and form and how emerging conceptions of shape and form provided opportunities to develop and elaborate aspects of definitional practice. Several forms of teacher discourse appeared to support students' participation and students' increasing agency over time. These included: (a) requesting that members of the class participate in various aspects of practice, (b) asking questions that serve to expand the mathematical system, (c) modeling participation in aspects of practice, (d) proposing examples that create contest (i.e., monsters), and (e) explicitly stating expectations of and purposes for participating in the practice.
Reasoning about mechanisms is one of the hallmarks of disciplined inquiry in science and engineering. Despite the central importance of mechanistic reasoning, its origins are not well understood. Numerous curricular efforts involve simple machines and related physical systems, but these do not yet build toward a systematic and longer-term vision for promoting the development of reasoning about mechanisms. The research we describe here was developed in partnership with a team of engineers and science educators who aim to support the early development of mechanistic reasoning through a curriculum that challenges children to design kinetic toys called MechAnimations. Our research aims to characterize the intellectual resources available to children as they engage in design challenges and to describe the process by which these design activities may promote development of mechanistic reasoning. This paper provides an in-depth look at children's prior understandings of a key aspect of MechAnimation designthe mechanics of linkages and levers. In a flexible interview, 9 children at grades 2 and 5 were asked to predict and explain the motion of mechanical linkages. Children explored contrasting pairs of mechanisms, chosen to highlight components of the system important to its functioning (such as the location of the fulcrum in relation to the input). As one might expect, many student responses focused on aspects of the mechanical system that were not oriented toward its function. For example, "it looks like a plus sign." However, children also exhibited more sophisticated thinking, such as describing the parts and structure of the mechanisms. The most sophisticated student responses included mechanistic descriptions of how the parts and structure worked to: constrain motion, affect the direction of rotation, coordinate the direction of motion for input and output levers, coordinate the movement of lever arms, and affect the magnitude of motion. Overall, children who more readily tended to relations between input and output seemed better able to predict mechanism motion. All children demonstrated at least some elements of mechanistic thinking, but many of their responses lacked coordination of multiple elements. When children coordinated multiple elements, they were also more likely to successfully predict the motion of one or more outputs, given an input. Children who predicted incorrectly tended to exhibit mechanistic reasoning only after observing the mechanisms move, if at all. Although designed to ascertain children's' naïve ideas about mechanisms, certain aspects of the interviews seemed to support the development of elements of mechanistic reasoning. For example, comparing the motion of contrasting mechanisms helped some children notice relevant variables not apparent before the contrast. These results suggest methods for characterizing mechanistic reasoning as well as potential resources for supporting its development. We anticipate that the latter may profitably be incorporated into design challenges.
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