## Using Representations, Decomposition, and Approximations of Practices to Support Prospective Elementary Mathematics Teachers’ Practice of Organizing Discussions

Source: *Journal of Mathematics Teacher Education, Volume 17*, p. 463-487 (2014)

(Reviewed by the Portal Team)

This study examined the ways in which elementary mathematics methods course activities, designed as representations, decomposition, and approximations of practice, can be used to develop prospective elementary teachers’ ability to organize a discussion.

**Methods**

Data were collected from prospective elementary teachers from elementary mathematics methods courses at two university sites during the fall semester of 2011.

The authors employed an intervention which utilized the five practices as one way to represent and accomplish the decomposition of a complex practice (organizing a mathematical discussion) into smaller, more manageable practices. The authors asked prospective elementary teachers to complete an activity designed to approximate this practice. Prospective elementary teachers employed their conceptions of the purpose of a mathematical discussion, their knowledge of a Standards-based lesson on alternative multi-digit addition strategies, and their understanding of 10 samples of student work to set a goal for a discussion and engage in selecting, sequencing, and questioning to address it.

The authors interpret the participants' responses through a lens of the affordances of a mathematical discussion and describe the implications the results have for mathematics teacher education.

Prospective elementary teachers in this study wrote goals that varied according to their conceptions of the purpose of a mathematical discussion. All prospective elementary teachers’ goals intended to provide students opportunity to clarify and communicate their thinking. The authors found that the activities resulted in 21 prospective elementary teachers who in addition, focused their goal on extending student thinking and 12 whose goal also intended to address making connections within and between strategies.

Prospective elementary teachers’ justifications for their selecting of student work revealed an interesting result. The majority of prospective elementary teachers had set a goal for the discussion of introducing strategies or having students use strategies. However, in selecting students’ work to be shared, prospective elementary teachers frequently attended to the efficiency of students’ work, how easy they believed it would be for other students to make sense of the work, and the opportunity the work presented for helping students make connections within and across solutions. These types of justifications demonstrate although prospective elementary teachers’ explicit goals for the discussion did not always reflect all three affordances of mathematical discussions, these ideas were evident in many prospective elementary teachers’ selection process.

Furthermore, prospective elementary teachers’ sequencing of the three strategies demonstrated many intuitively understood how the strategies were related and built upon each other, even if their goals or approach to the discussion did not focus on making these connections explicit for students. Prospective elementary teachers largely wrote questions that supported their learning goals. Though many prospective elementary teachers were successful in writing clarifying and leading questions, there was a comparative dearth of questions providing students opportunity to make mathematical connections.

Prior to using this approximation of practice, employing separate activities specifically focused on questions to clarify student thinking as well as to scaffold mathematical connections within student work also seems germane.

In examining these findings from these components of organizing a discussion, the authors believe the use of decomposition of the practice of organizing a discussion was useful not only in supporting prospective elementary teachers’ development of particular facets of this complex practice, but also in allowing them to identify particular areas to target in their next iteration of their activities. Prospective elementary teachers in their subgroups were often successful in aligning their choices in the organizing a discussion with their identified goal for the discussion. This finding suggests that focused work in methods courses on identifying, developing, and discussing a variety of goals for a lesson discussion would be a beneficial approach to strengthen prospective elementary teachers’ development of the practice of organizing a discussion.