Working With Practitioners and Learning Trajectories: Sharpening the Focus on Mathematical Reasoning in Grades 5–9

• In Event: Partnering With Practicing Teachers to Strengthen Inclusive Instruction Using Learning Trajectories

Abstract

From our perspective a learning trajectory is an integrated, empirically based learning and assessment framework that is focused on the development of one or more big ideas in mathematics and the links between them (Siemon, 2019). Consistent with more familiar definitions of learning trajectories (e.g. Clements & Samara, 2004; Confrey, & Maloney, 2010), the framework incorporates an evidence-based learning progression that serves as a model of how students’ thinking evolves over time, validated diagnostic assessment tools that identify where students are in relation to the learning progression, and targeted teaching advice to help teachers progress students’ learning from one level/zone of the progression to the next.

One of the reasons for the focus on big ideas was to make learning trajectories “more comprehensible and useful to practitioners“ (Baroody et al, 2004; p, 253) thereby increasing the likelihood that teachers would use them to make more informed decisions about the next steps in instruction than might otherwise have been made in the absence of evidence of student reasoning (Wiliam, 2011). This is particularly relevant in Australia, where teachers of mathematics are expected to make their own instructional decisions based on a relatively small number of nationally determined content descriptors organised by grade level and strand and written in a form that can be observed or measured in practice. As a consequence, the descriptors provide little or no indication of the key underpinning ideas needed to support qualitative changes in the way students think about particular aspects of mathematics and the relationships between them (e.g. Simon, 2006) and student reasoning is rarely, if ever, a focus of instruction.

This paper will report on the role of practitioners in a recent four-year study on the development and use of learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years (Siemon & Callingham, 2019). The decision to focus on algebraic, spatial, and statistical reasoning was ambitious but felt necessary to provide the sort of evidence and resources needed to support a significant and sustained change in practice away from low-complexity, procedural exercises to teaching based on a deeper understanding of the big idea and the connections between them (Sullivan, 2011). Another reason for this focus was that despite the demonstrated efficacy of working with an evidenced-based learning and assessment framework for multiplicative thinking (e.g. Siemon, 2016), many teachers found it difficult to see the connection between multiplicative thinking and the mathematics content descriptors for their grade level. In this paper we will illustrate how teachers were involved in the iterative research design, specifically in relation to task development and the trial and refinement of partial credit scoring rubrics. We will also provide a rationale for and the development of targeted teaching advice in this context and consider some of the challenges involved in implementation.

Authors

• Dianne E. Siemon, Royal Melbourne Institute of Technology

• Rosemary Callingham, University of Tasmania

• Lorraine F Day, University of Notre Dame Australia

• Marj H. Horne, The Australian Catholic University