| This mixed-method study investigated prospective K--8 teachers' understandings of two-factor multiplication with integers and positive fractions and its implications for their pedagogical content knowledge (PCK) development. The theoretical framework for the study weaves together APOS: Action-Process-Object-Schema theory (Dubinsky, Dautermann, Leron, & Zazkis, 1994), Pirie and Kieren's (1994) dynamic model of mathematical understanding, and the ideas of relational understanding (Skemp, 1976). Beliefs about mathematics and mathematics education were surveyed among all 267 preservice K--8 teachers enrolled in their last semester of mathematics at the university through the Teachers' Conceptions of Mathematics and Mathematics Education instrument (Romberg and Middleton, 1994). Descriptive statistical results are presented graphically and discussed in terms of measures of central tendency, variability, and clustering. The primary focus in analysis of interviews with twelve of the 267 prospective teachers was the de-encapsulation of mathematical objects to processes, then to actions, to investigate relational understanding of multiplication. To catalyze de-encapsulation, the interview protocol called for computation, problem-posing, and problem-solving.; Interviewees were presented seven simple numerical multiplication prompts. For each prompt the participant was asked to compute the product and to pose and visualize solving a problem, based on the prompt, that would be appropriate for working with schoolchildren. Participants' computational work, drawings, explanations and justifications were analyzed using the multi-layered theoretical framework. Several participants had difficulty completing computations involving fractions and none of the twelve pre-service teachers could pose an effective and meaningful problem for prompts involving two fractions or two negative numbers.; Most problematic were the conceptions interviewees associated with properties they noticed: the property of negative number, of fraction, of multiplier, of negative number as multiplier, and of fraction as multiplier. Discussion of the twelve women's understandings of multiplication is followed by implications for teacher preparation and suggestions for research and practice. In particular, problem-posing and visualization of problem-solving can serve as research tools to explore schematic mathematical understanding and can provoke prospective teachers' self-examination of their conceptual understanding of elementary mathematics topics. |
