Research Resources for Lesson Study: Principles and Rationale for a Linear Measurement Approach to Fractions

• In Event: Lesson Study Supported by Mathematical Resources: A Theoretical Framework, Randomized Trial, and Implications for Reform

Abstract

This paper presents design principles for selection of research-based resources for lesson study, and examines the specific rationale for focusing on a linear measurement model of fractions. Samples of the fractions resources will be presented.

Two research activities shaped the decision to focus on a linear measurement model of fractions. A review of 58 studies on students’ understanding of fractions suggested six major challenges in understanding fractions (e.g., understanding fraction as a number, seeing the multiplicative relationship in a fraction, keeping track of the whole, etc.). A comparison of the models used to introduce fractions in two U.S. and two Japanese textbook series was also conducted; the Japanese texts were examined as one example of a coherent curriculum that supports high achievement on international studies (http://nces.ed.gov/timss). Using categories from existing research (e.g., Lamon, 2005; Watanabe, 2002) all student materials in Grades 1-5 of the four texts were coded into 16 model types (e.g., circle area, part of a set).

We found that the U.S. texts introduce fractions one to three grades earlier than the Japanese texts, using many more models: a total of 15 in the U.S. texts, versus 4 in the Japanese texts. Both Japanese series use the same four models (linear measurement, liquid measurement, number line, and rectangle area). From our reviews, we developed a series of conjectures about how the linear measurement model might help students with the six major challenges, for example, help them to understand fractions as numbers that can be placed on a number line (not simply as pieces of a partitioned whole or parts of a set). Several U.S. studies (e.g., Dougherty & Zilliox, 2003; Saxe, 2007) provide evidence for these conjectures, yet the linear measurement model is rarely used in mainstream U.S. texts. Consequently, the resources focused on helping U.S. teachers investigate this model, through the use of mathematics tasks, student work, curriculum, research articles, and classroom video.

Overall design principles focused on maintaining the qualities of lesson study that build educators’ ownership–such as the opportunity to work with colleagues to actively investigate a problem of pressing interest–while also providing classroom-relevant pathways into research-based resources. Research on the qualities of effective professional learning provided many of the initial principles for selection and use of the research, such as provision of opportunities for teachers to solve mathematics tasks, to study student thinking, to connect research findings to their own classroom needs and goals, and to discuss ideas with colleagues in ways that would productively challenge current beliefs (Borasi & Fonzi, 2002; Borko, 2004; Carpenter, Franke, & Levi, 2003; Cohen & Hill, 2000; Desimone, Garet, Birman, Porter, & Yoon, 2003; Garet et al., 2001; Goldenberg & Gallimore, 1991; Hill & Ball, 2004; National Research Council, 2001; Wilson & Berne, 1999). We supplemented these ideas with several principles from Japanese and other curriculum, e.g., that teachers should work to solve problems individually before sharing and discussing solution methods with colleagues and should predict student solution methods (e.g., Hironaka & Sugiyama, 2006).

Authors

• Shelley Friedkin, Mills College

• Catherine C. Lewis, Mills College

• Rebecca R. Perry, Mills College