Recently, there has been an increased interest in learning trajectories (LTs) by the mathematics education research, assessment, curriculum, and policy communities. Widely regarded as promising tools for improving instruction and student learning, LTs are “descriptions of children’s thinking and learning in a specific mathematical domain, and a related conjectured route through a set of instructional tasks designed to engender those mental processes or actions hypothesized to move children through a developmental progression of levels of thinking, created with the intent of supporting children’s achievement of specific goals in that mathematical domain” (Clements and Sarama, 2004, p. 83). Much of the current work with LTs exists at the macro-level, such as the expressed “research-based learning progressions” basis for the Common Core State Standards for Mathematics (CCSSO, 2010, p. 4). It is reasonable to assume the work with LTs will influence the work of the new assessment consortia and of curriculum developers. Yet it is imperative for the mathematics education community to remember that systemic innovations rely on classroom teachers’ capacity to bring these innovations into action, transforming ideas into teaching reality. Teachers adapt innovations using the knowledge they have about children’s mathematical thinking and their pedagogical skills for eliciting, understanding, and responding to students’ thinking.

The recent CPRE report (2011) entitled Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction calls on our community to “translate the available learning trajectories into usable tools for teachers” (p. 57). Researchers have begun to investigate teachers’ understandings of LTs as frameworks for making sense of and responding to student thinking. Early results suggest that when teachers learn about LTs, they experience a shift in their views of students’ mathematical thinking from correct or incorrect to a gradual gain of competence and adequacy. Additionally, many come to focus on students’ strategies and underlying thinking. Such promising outcomes necessitate a better understanding of the learning contexts that may support teachers in coming to understand and use LTs in their practice. Therefore, we propose a symposium to bring together scholars from four different research groups working to understand teacher learning of LTs. These researchers will share and discuss their emerging findings related to three guiding questions:

(1) What is the process through which teachers come to understand and use LTs in their teaching?

(2) In what ways do LTs support teachers’ attention to student thinking?

(3) What are characteristics of professional learning contexts that promote teacher learning of LTs?

Session Outline

The proposed symposium is focused on the conference theme, Linking Research and Practice, and addresses the Professional Learning priority area of the 2012 Research Pre-Session. The session will begin with a brief introduction on learning trajectories and the literature on teacher learning, followed by an overview of the guiding questions that serve as organizers for the symposium (5 minutes). Then, each of the four research groups will be allotted approximately 12 minutes to describe the context of their project and to respond to one or more of the questions.

Presenter and Presenter will ask: How do teachers develop knowledge about a LT on length measurement to support classroom instruction and improve professional knowledge? They will describe a year-long learning experiment introducing and using an LT on length measurement developed by Clements, Sarama and Barrett (2009; CPRE 2011, p. 25). In a district-based model for professional development, 24 teachers met for 10 days of summer workshop and four days of research lessons during the academic year. The teachers engaged in video analysis to categorize student strategies along the LT, conducted clinical teaching with one student and conducted action research. Data included written and video records. We analyzed levels of sophistication for teachers’ pedagogy (Fraivillig, Murphy & Fuson, 1999). Preliminary findings highlight critical stages of incorporation of the LT, shifting from object to formative assessment tool.

As part of Presenter’s NSF-funded Cognition Based Assessment (CBA) project, 27 elementary teachers studied, and used in their classrooms, an integrated system of research-based learning progressions, assessment tasks, and instructional suggestions for various mathematical topics. Presenter will describe the various ways that teachers learned, understood, and used CBA materials—from the beginning of the project, to, for some teachers, 5 years later. His descriptions will focus on both the "macro" level—how teachers used CBA in their teaching, and on the "micro" level—how teachers conceptualized students’ thinking (mathematically and psychologically) and how these conceptualizations affected teachers’ conceptualizations of related instruction.

Presenter and Presenter will report on a large-scale professional development project to examine the effectiveness of their curriculum built on LTs.  Reporting findings from a randomized trial design, they will will share that teachers who learned about LTs developed richer mathematical classroom environments, had significantly greater gains from pre- to post-tests compared to students in control classrooms, and were more responsive to their students’ mathematical thinking. 

Presenter and Presenter will share findings from a year-long design experiment investigating the ways in which teachers adopted and appropriated knowledge of Confrey and colleague’s equipartitioning LT. In a school-based professional development setting, 24 K-5 teachers met for 60 hours throughout the school year to engage in designed professional learning activities to learn about the LT and to enact their understandings in their classrooms. Video and written records of teachers’ participation in selected learning activities were analyzed using constant comparison methods. Preliminary findings suggest that activities most effective in supporting teacher learning of LTs are those based on artifacts that present children and children’s work in ways that challenge teachers’ deficit perspective about students’ abilities and stimulate teachers to be more generative in interpreting students’ behaviors.

The remaining 30 minutes will be dedicated to audience questions and discussion related to the guiding questions.