Learning Mathematics Through Representations: Foundational Developmental Research

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

Abstract

[Figure in Session Summary]

Objective. We report findings from developmental research that identified learning trajectories that lead to generative understandings of integers and fractions on the number line. The work provided a foundation for the design of LMR curriculum and pedagogy.

Interview studies: Investigating student intuitions. Interview studies contributed to our understanding of the intuitions and difficulties that elementary students bring to lessons on integers and fractions on the number line (Saxe, Shaughnessy, et al. 2009). In one between-groups comparative study employing ‘open number line’ tasks, for example, we investigated the effects of context on students’ placements of 3 numbers on the line, with a focus on coordination of numerical units with linear units. For one group (n=24), the line was presented as a ‘racecourse,’ and, for a second group (n=24), as a conventional number line; the integers to be placed were either consecutive (e.g., 0, 1, and 2; or 5, 6, and 7) or non-consecutive (e.g., 9, 10, and 13; or 9, 12, and 13). Most students placed consecutive numbers at appropriate linear distances, but many placed irregular sequences without coordinating numerical and linear units -- for example, they placed numbers such as 9, 10, 13 at equal distances. Students in the thematic context group were more likely to place numbers accurately. Findings from interview studies like this one revealed that 5th grade students’ understandings and uses of the core principles of number lines are often neither robust nor flexible, but students demonstrate greater insight when they work with number lines representing everyday measurement contexts.

Tutorial study: Investigating learning trajectories. We investigated a tutorial approach designed to support students’ generative understanding of principles for the representation of integers on the number line (Saxe, Earnest et al., 2010). Students matched for classroom and pretest score were randomly assigned to tutorial (n=19) and control groups (n=19). Tutorial students played a communication game in which student and tutor each used CuisenaireTM rods to mark the same integer on a number line and then compare their solutions and reasoning (Figure 1). Over two sessions and 13 problems, tutors and tutees resolved discrepant solutions by progressively constructing “agreements” about number line principles and conventions, and these agreements guided solutions to additional problems – e.g., order (numbers increase from left to right) and unit interval (distances between consecutive numbers must be the same). In early problems, student and tutor used rods to represent linear distances on lines with only one number specified. Later the line became a self‐referential representation with two numbers specified – e.g., student and tutor marked the position of 9 on a line with 6 and 8 specified without using rods. The tutorial group improved over pre‐ to posttest (t(18)=7.92, p<.0001) whereas the controls did not.

Significance. Interview studies identified elementary students’ intuitions regarding the representation of integers and fractions on the number line. Tutorial studies identified the productive role of constructing mathematical definitions to support trajectories of learning toward generative ideas about the representation of number on the line.

Authors

• Darrell Earnest, University of Massachusetts - Amherst

• Yasmin A. Sitabkhan, Research Triangle Institute

• Geoffrey B. Saxe, University of California - Berkeley