P. Holt Wilson

Learning Trajectory Based Instruction: Toward a Theory of Teaching

In this article, we propose a theoretical connection between research on learning and research on teaching through recent research on students’ learning trajectories (LTs). We define learning trajectory based instruction (LTBI) as teaching that uses students’ LTs as the basis for instructional decisions. We use mathematics as the context for our argument, first examining current research on LTs and then examining emerging research on how mathematics teachers use LTs to support their teaching. We consider how LTs provide specificity to four highly used frameworks for examining mathematics teaching, namely mathematical knowledge for teaching, task analysis, discourse facilitation practices, and formative assessment. We contend that by unifying various teaching frameworks around the science of LTs, LTBI begins to define a theory of teaching organized around and grounded in research on student learning. Thus, moving from the accumulation of various frameworks into a reorganization of the frameworks, LTBI provides an integrated explanatory framework for teaching.

Teachers’ Uses of a Learning Trajectory in Student-Centered Instructional Practices

The growing national attention to students’ learning trajectories (LTs) renews the opportunity to explore the ways that teachers may use students’ thinking in their instruction. In this article, we examine teachers’ learning of two frameworks, one for students’ thinking in a particular domain and one for broad student-centered instructional practices, in the context of an elementary grades mathematics professional development setting. As a part of a retrospective analysis of a design experiment, we analyzed 19 lessons of teachers who participated in 60 hr of professional development designed to support their learning of one LT and one framework about student-centered instructional practices. Our findings describe the ways in which teachers brought together these frameworks to enact instructional practices that elicit and use students’ mathematical thinking in classroom instruction. We conclude by arguing that LTs can serve as a referent for student-centered instructional practices, bridging guidelines for student-centered instruction with domain-specific understandings of students’ thinking for teachers.

Teachers, Attributions, and Students’ Mathematical Work

This chapter reports on a design experiment within a professional development context purposefully planned to teach teachers about students’ mathematical thinking and learning through studying a mathematics learning trajectory. Through an examination of teachers’ discourse in 60 h of school-based professional development, we identify eight attributions that participating elementary teachers use to discuss students’ mathematical successes or failures when engaging with the project’s professional learning tasks. Our findings suggest that teachers used a variety of attributions to explain students’ mathematics and incorporated their knowledge of the learning trajectory into their repertoire of attributions.

Changes in Teachers’ Discourse About Students in a Professional Development on Learning Trajectories

This study examines teachers’ discussions in a professional development setting to understand the ways in which learning a mathematics learning trajectory may change aspects of their discourse about students as learners. Using mixed methods, we bring together two theoretical frames that use a Vygotskian perspective on learning to analyze professional discussions among 22 elementary-grade teachers participating in a yearlong, 60-hour mathematics professional development program. Results indicate that over time, some discursive patterns for explaining students’ academic performance changed to incorporate the trajectory, while others remained unaffected. Whereas this change transformed one of the patterns in a way that led to new explanations for student performance, another pattern changed only slightly and was still used to express the same explanations for performance.

Examining aspects of elementary grades pre-service teachers’ mathematical reasoning

ABSTRACT This mixed methods study investigated undergraduate elementary grades pre-service teachers’ (PSTs) mathematical reasoning. In the context of combinatorics, we examined two aspects of mathematical reasoning, namely, the processes of enumerating and generalizing, and sought to understand the relationship between them through analyses of 150 PSTs’ solutions to a set of combinatorics tasks. A subset of eight PSTs participated in follow-up interviews to further understand these aspects of reasoning when completing similar tasks. Findings suggest that the processes of reflecting and recalling prior knowledge mediated the relation between these two processes of PSTs’ reasoning. An analysis of PSTs’ written work suggested that many did not draw on their process of enumeration to generalize. Subsequent opportunities for reflection during interviews allowed PSTs to refine their enumeration process. For some, recalling prior knowledge assisted them in connecting their observations with an underlying mathematical structure, enabling them to generalize in ways that extended their understanding to explain contextual features of the problem. For others, this recollection shifted their focus and prevented their use of structure to generalize. For instructors of mathematics courses in teacher preparation programs, these results suggest that PSTs benefit from opportunities to reflect on their reasoning and that a careful consideration of how PSTs use their prior knowledge is essential in developing mathematics knowledge for teaching.