Exploring the effectiveness of interventions designed to deepen preservice teachers’ conceptual understanding of linear function and slope: A mixed methods study

A multitude of studies have revealed that considerable numbers of preservice teachers have impoverished understanding of many mathematics concepts and processes they need for teaching (Ball, 1988, 1990; Ball, Thames, & Phelps, 2008; Zaslavsky, Sela, & Leron, 2002). Ball, Lubienski, and Mewborn (2001) asserted, “Our insufficient understanding of the mathematical knowledge has meant inadequate opportunities for teachers to develop the requisite mathematical knowledge and the ability to use it in practice” (p. 433). A promising way to change teachers’ ingrained preconceptions in relation to mathematics is to improve their content knowledge level, which in turn impacts their practice actions. This study aimed to explore the effectiveness of hands-on or virtual manipulatives in advancing preservice teachers’ understanding of the algebraic concepts of linear function and slope. A mixed methods design was used to investigate and analyze the data. This study not only sought to examine the extent to which preservice teachers understood targeted mathematical concepts under the conditions with varied manipulatives, but also tried to provide reasonable explanations and insights as to their understandings. This study served as a facilitator to improve the quality of teacher preparation programs. Initial quantitative data reported that the participating middle school preservice teachers possessed a poor understanding of mathematical knoweldge in relation to linear function and slope, experiencing difficulties in four specific domains: identify points on the linear function, identify linear function (perpendicular), identify linear function (parallel), and properties of linear function and slope. A further qualitative analysis suggested that participants embraced varied misconceptions and knowledge deficiencies, which may account for their poor knowledge performance. Another set of quantitative data suggested that participants benefited from the virtual and hands-on manipulative interventions as to their understanding of linear function and slope. A further qualitative analysis engendered a surprising conclusion: the interventions had nothing to do with preservice teachers’ learning, but provided a context for peer discussion. Based on data from the after-intervention interviews and follow-up questionnaires, the results suggest that peer discourse maybe a more effective strategy in advancing preservice teachers’ grasp of specific points in mathematics.