International Approaches to Professional Development for Mathematics Teachers

Chapter 11: Assessing a Research/Professional Development Model in Patterning and Algebra

150 Chapter 11 Assessing a Research/Professional Development Model in Patterning and Algebra Ruth Beatty and Catherine D. Bruce Introduction Over the past fifty years, researchers have studied the importance of teacher professional development (PD). What teachers know and are able to do is a key factor influencing student learning (Fullan, Hill, & Crevola, 2006). However, there is a lack of empirical research on the effects of PD on teacher learning, particularly with respect to developing classroom knowledge and improving practice (Hill, 2004), which are crucial indicators of success (Fishman, Marx, Best, & Tal, 2003). Our goal was to develop a Research/PD Model that embedded professional development within a longitudinal research program for two reasons: (1) to continue to evaluate an innovative approach to teaching patterning and algebra in elementary grades; (2) to assess the effects of including teachers as collaborative partners in their development as mathematics educators. To this end, we examined teacher learning and its links to teacher participation (with a critical focus on teacher enactment), teacher knowledge, and classroom practice. Background to Patterning Patterning activities have long been recommended as a means of supporting students in developing an understanding of the relations among quantities that underlie mathematical functions. However, there is substantial evidence Assessing a Research/Professional Development Model || 151 suggesting that, with current instruction, identifying the link between patterns , generalized rules, and algebraic representations is difficult. Analysts (Lee, 1996; Mason, 1996) who have studied patterns and generalizing suggest that it might not be patterning problems per se that are difficult but the ways that they are presented to students and the limitations of the teaching approaches used. Although patterning problems might be presented initially in a variety of formats (geometric, tabular, and narrative), the tendency of most instruction is to prioritize the numeric/arithmetic aspect of patterning activities (Noss & Hoyles, 1996). This numeric approach to pattern learning diminishes the potential for students to recognize commonalities in mathematical relationships across multiple representations and obscures the underlying functional relationship of the pattern because the pattern rule becomes a sequence of arithmetic operations derived numerically in isolation from the context of the problem. Students do not perceive the need to understand the mathematical structures and relationships underpinning pattern rules, so these activities do little to expand their understanding, ability, or interest in finding and justifying meaningful general rules (functions). In response to this, Beatty conducted a series of experimental studies in grade 4, 5, and 6 classrooms with the goal of designing new approaches to pattern learning that emphasize multiple representations (geometric, graphic, and symbolic) as a means of supporting children’s developing understanding of mathematical functions. Based on Case et al.’s research methodologies (Moss & Case, 1999), the instructional sequences and curriculum materials were aligned with theoretically based learning trajectories and then tested in a variety of classrooms. The inquiry-based lesson sequence was designed to encourage meaning making through the use of mathematical discourse, multiple representations , manipulatives, and rich open-ended tasks. The three-year research program was iterative in that results from each year informed the design of subsequent instruction. Results have demonstrated positive outcomes based on the lesson sequence for this kind of instruction for all students, including those who have demonstrated very low math ability (Beatty & Moss, 2006). Linking Research, Professional Development, and Teacher Learning The study for this chapter took place in the third year of the research program. At that point, the content of the lesson sequences had been evaluated, but there still remained questions with respect to implementation of best practices. 152 || Professional Development of Mathematics Teachers Therefore, we invited experienced teachers to participate in the PD program component of the Research/PD Model as collaborative partners to further assess the quality and functioning of our experimental patterning and algebra lesson sequence. Simultaneously, the researchers tested this Research/PD Model to see whether it supported teachers in developing their pedagogical content knowledge, practice, and efficacy (confidence that they can promote student learning) (Bandura, 1977). Teacher efficacy is an important construct because it has been associated with variables such as teachers’ willingness to adopt innovations , teachers’ classroom management strategies, time dedicated to teaching challenging subjects and topics, and student achievement (Goddard, Hoy, & Woolfolk Hoy, 2004; Ross, 1998; Tschannen-Moran, Woolfolk Hoy, & Hoy, 1998). Fishman’s research (Fishman et al., 2003) on enactment details the importance of explicit teacher implementation of strategies modelled in PD sessions in order to positively influence teacher pedagogical content knowledge and...