Developing understanding: pre-service elementary teachers' changing conceptions of mathematics

Abstract

This descriptive case study explores how a conceptual understanding of fractions develops in pre-service elementary teachers enrolled in a reform-based, remedial mathematics skills course set at a middle school level, during their final year of an education program in a small Canadian university. I compared the learning trajectory of these adults to that of children and then developed a landscape of (re)learning fraction concepts within a constructivist-oriented environment. Fourteen prospective teachers completed pre/posttest content exams and interviews that followed-up each of these test instruments, as well as two paired problem-solving interviews that used a modified think aloud protocol, and open ended questionnaires. All interviews were videotaped using two cameras, capturing both the participant(s) and a zoomed in view of their hands and written work. Data collection took place at four points in the school year between September and March, providing snapshots of the developmental progression of fraction understanding in the pre-service teachers over time. Analysis occurred at multiple layers: first with the individual pre-service teachers, then with three clusters of participants grouped according to their mathematical abilities, and finally with the group as a whole. The theoretical framework utilized a constructivist view of learning and a particular landscape of learning that highlights strategies, big ideas, and models. The analysis revealed that the landscape of (re)learning fractions for adults is significantly different from that of children. Pre-service teachers moved from understanding fractions solely as procedures and worked to make sense of their fragmented knowledge around the intertwined concepts of fractions as relations, ratios, and operators. All participants demonstrated growth in their understanding of fractions; however, few participants developed a deep understanding of fractions as meaningful objects. The foundational concept of fractions as relations developed slowly over time and constrained the understanding of fractions as ratios and operators. Pre-service teachers experienced difficulties in understanding and identifying the appropriate referent unit, or fraction whole; this reflected the fragility of their understanding of fractions as relations. In order for pre-service teachers to understand fractions as meaningful objects and to shift to multiplicative thinking, modeling is a necessary but insufficient means.