Learning Mathematics Through Representations: Overview

• In Event: Learning Mathematics Through Representations: A Research-Based Curriculum

Abstract

[Figures are provided in Session Summary.]

The first presentation provides an overview of the Learning Mathematics through Representations (LMR) curriculum design research project. We describe the mathematics that is the focus of LMR research and development, as well as the three research strands featured in the symposium: foundational developmental research (presentation 2), curriculum design (presentation 3), and experimental study of curriculum effectiveness (presentation 4).

In most elementary curriculum materials, the topics of integers and fractions are treated as unrelated, and students may construct partial and unintegrated understandings. Two premises of the Learning Mathematics through Representations (LMR) project are that (a) the number line affords an integrated treatment of rational number and (b) a well-orchestrated curriculum sequence supports students’ construction of learning trajectories that support generative understanding of these core domains. These two ideas seeded the design project.

A core representation in mathematics, the number line is a geometric representation of number, and, as shown in Figure 1, order and interval are central ideas (Lehrer, 2003; Piaget, Inhelder, & Szeminska, 1960; Saxe, Shaughnessy, et al., 2007). Order includes relations of ‘less than’ and ‘greater than,’ expressed on the number line as spatial direction. Three kinds of intervals on the line are the target of LMR research and curriculum. Unit intervals, here defined as the distance between 0 and 1 (or the equivalent), are the basis for locating integers on the line; subunit intervals are equally partitioned distances of a unit interval and the basis for locating fractions; multiunit intervals are multiples of unit intervals and the basis for locating numbers in a patterned multiunit series (e.g., by 2’s, 3’s, 10’s etc.). The numerical label for a point depends upon its distance from zero, a distance understood in terms of units, subunits, and/or multiunits. Related ideas (e.g., symmetry and absolute value) are also illustrated in Figure 1.

The initial years of the LMR project were organized as two coordinated strands of work: foundational developmental research and curriculum design with iterative refinement of lessons. Developmental research (presentation 2) included interview and tutorial studies on elementary students’ understandings of number line representations, and the findings and methods informed the design of curriculum content, curriculum tasks, and lesson pedagogy. Building on the developmental research, curriculum design (presentation 3) emerged through a process of iterative refinement in collaboration with two expert teachers (cf. Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Cobb & Gravemeijer, 2008; Lehrer & Schauble, 2004; Saxe, Gearhart, et al., 2009). In 2010-11, we added a third component to the project – an experimental study in 21 classrooms to evaluate the effectiveness of the curriculum (presentation 4).

Author

• Geoffrey B. Saxe, University of California - Berkeley