Mathematics Teaching and Learning as a Mediating Process: The Case of Tape Diagrams

Abstract

This article examines how visual representations may mediate the teaching and learning of mathematics over time in Japanese elementary classrooms. Using the Zone of Proximal Development Mathematical Learning Model (Murata & Fuson, 2006; Fuson & Murata, 2007), the process of mediation is explicated. The tape diagram, a central visual representation used in Japanese mathematics curriculum, is explored for its roles and the student learning that is intended to be mediated over time, illuminating aspects of the process. The study argues that the consistent and coherent use of one representation can bridge student understanding over time, focusing on mathematical relationships and problem-solving processes. The study also suggests different instructional approaches between U.S. and Japanese curricula that are reflected in the uses of representations.

Notes

1. Many researchers consider language as a primary tool for such instances. For this article, I am using the term “tools” to more broadly discuss any mediating objects (concrete or non-concrete) that are used in cultural activities. In mathematics classrooms, mathematical symbols (e.g., + and = ), visual representations, and concrete manipulatives are examples of the tools.

2. Japanese schools adhere to a national curriculum. The Japanese Ministry of Education, Culture, Sports, Science, and Technology determines and publishes a Course of Study that includes general education guidelines for schools (including broader national goals and guidelines for different subject areas and grade levels). This Course of Study is revised approximately once every ten years. Publishing companies then closely follow the guidelines to design their textbooks. There are six elementary mathematics textbook publishers in Japan, and all share a common structure, organization, and presentation of mathematics topics for each grade and thus use tape diagrams. The grade 1 to 6 textbook series analyzed for this study, Study with Your Friends Mathematics (Gakkotosho, 2005) is one of the most commonly used textbook series (Japanese Ministry of Education, 2006). It is also available in English.

3. In the sections that follow, text at some places may seem to suggest student learning, however this study did not collect empirical data on student learning. What the article intends to communicate is that the Japanese curricular approaches found in the study tend to support certain student learning. These claims on student learning are also grounded in the high achievement of Japanese students in international comparative studies as well as the author's prior studies based on empirical data (Murata, 2004; Murata, Hattori, Otani, & Fuson, 2004).

4. While U.S. research identifies three major story problem types for students' addition-subtraction learning (join, change, compare; see Carpenter, 1999), in Japanese textbooks, there is the fourth category: order. In “order” problems, objects/people are typically lined up, and by using the information of the order of particular object/people in the line, students find answers: “Some children are standing in a line. Takeo is the fourth from the front; Yoko is the fifth person behind Takeo. What is the order of Yoko from the front?”

Boaler, J., Cleare, N., Dieckmann, J. Fiori, N., Segunpta-Irving, T., & Shahan, E. (2008). Making mathematics “colorful & bright”—how students became more engaged and successful during five weeks of mathematics class. (Manuscript under preparation).

Murata, A. (2008). Assisting mental math methods through visual supports, problem sequencing, and learning of conceptual prerequisites. Manuscript in preparation.