Perceptions of professional growth: A mathematics teacher educator in transition

School Science and Mathematics, Mar 2003 by Timmerman, Maria A

Second, I concur with Baroody and Coslick's (1998) ideas about mathematical thinking. Mathematical thinking refers to a learner's development of three processes-problem-solving, reasoning, and communicating-that are necessary to engage in mathematical inquiry. To understand and extend knowledge about the world, both teacher and student users of mathematics need to develop these processes. When I refer to mathematical thinking, my ideas reflect Baroody and Coslick's description of thinking as "reorganizing our thinking-broadening our perspective-rather than merely accumulating information" (p. 1-11). For mathematical thinking to occur, learners should be actively engaged in tasks that prompt reflection, rethinking, and a potential reorganization of their ideas.

Third, I see mathematical understanding as the building of connections among various types of information. An assumption is that mathematical ideas can be represented internally. Mathematical relationships are formed in the developing mental structures of the mind. According to Hiebert and Carpenter (1992), "Mathematics is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and the strength of the connections" (p. 67). In order to build understanding through the network of ideas, concepts, principles, facts, or procedures, both internal and external representations need to be connected. Reflective thought facilitates the construction of mathematical relationships. These relationships can be described as "networks of knowledge," which may take on two different structures: (a) webs or Co) vertical hierarchies (Hiebert & Carpenter, 1992). Also, a variety of different knowledge structures may appear when the web and hierarchical models are mixed.

Lastly, the meaning I make of a constructivist view of learning obligates me to provide prospective teachers with a challenging student-centered learning environment. The prospective teachers need to participate in mathematical tasks and dialogue that connect to and build upon their prior knowledge and experiences. I view constructivist thinking as a way of knowing the world such that understanding is determined by an individual's own mental interpretations that give meaning to his or her experiences. One way teacher educators may make reflective thinking explicit is to create a situation in which cognitive conflict or ambiguity arises. As prospective teachers reflect on any dissonance in their thinking, a reorganization of ideas may occur. Networks of knowledge are dynamic, and understanding increases when connections emerge between new ideas and existing disconnected pieces of information.

Still, constructing mathematical ideas is often difficult when individuals work alone. I perceive the process of teaching as an interactive support for learning. According to Cobb (1994), mathematical learning is both an individual and collective activity; that is, "a process of active individual construction and a process of enculturation into mathematical practices of wider society" (p. 13). From a social constructivist perspective, I argue that teacher educators and prospective elementary teachers support the teaching/learning process through listening to and making sense of each other's mathematical thinking through discussion.