Learning Mathematics Through Representations: Curriculum Design

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

Abstract

[Figures in Session Summary]

Objectives. In this presentation, we describe the Learning Mathematics through Representations curriculum and findings generated during iterative refinement of lessons.

Curriculum. Building on methods and findings from the developmental research, LMR lessons were designed through a process of iterative refinement in collaboration with two expert teachers (cf. Clements, 2007; Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Saxe, Gearhart, et al., 2009). Informed by design principles aligned with constructivist treatments of cognitive development, lessons support teachers’ efforts to elicit and build on student thinking, generate intellectual conflicts that can be resolved with mathematical definitions, and differentiate instruction in shared mathematical contexts. Pilot lessons were refined based on analysis of daily videorecordings, student work, and pre- and post-unit assessments containing items from a range of sources.

Iterative refinement led to a 19-lesson sequence. Figure 1 illustrates the scope and sequence for integers and the principles & definitions co-constructed over these lessons. As illustrated in Figure 1a, lesson tasks incorporate non-routine representations that challenge students’ understandings of the representation of integers and fractions on the number line; for example, students identify points on lines that are not partitioned equally. Lesson formats build on students’ initial ideas about task solutions by setting up contradictions for students to resolve through the construction and application of definitions/principles. As illustrated in Figure 1b, Cuisenaire RodsTM support students’ developing understandings of linear units, multi-units, and subunits (fractional parts of a unit); rods are used as tools for the construction of number lines, and location and interpretation of points.

Most lessons are organized in five phases (Figure 2). Students’ solutions to the opening problems surface disagreements and uncertainties that are addressed in the opening discussion, when the teacher guides the class toward partial resolution through the co-construction of definitions/principles and the use of CuisenaireTM rods to support reasoning about unitizing. During partner work and the closing discussion, principles/definitions and Cuisenaire rods continue to serve as resources for argumentation, shifting authority for knowledge from the teacher to the mathematics (Cobb, 2002). Closing problems serve as formative assessments of student learning.

Methods, data sources, and illustrative results: Learning gains. Figure 3 illustrates results of formative studies in partner teachers’ classrooms. Students in both 4th and 5th grades demonstrated marked gains on integers and fractions measures containing items from a range of sources.
Figure 3. Box plots showing gains of 4th and 5th grade classrooms.

Significance. A linear measurement interpretation of rational number is not supported in consistent ways in U.S. elementary curricula (Lewis, Perry, Friedkin & Baker, 2010; Murata, 2008). Formative studies supported the development of a new curriculum designed to support students’ generative understandings of the representation of integers and fractions on the number line.

Authors

• Maryl Gearhart, University of California - Berkeley

• Geoffrey B. Saxe, University of California - Berkeley