Tad Watanabe

A Developing Approach to Studying Students' Learning through Their Mathematical Activity.

We discuss an emerging program of research on a particular aspect of mathematics learning, students’ learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which students pass in the context of particular instructional approaches or learning environments. Generally missing from the literature is research that examines the process by which students progress from one of these conceptual steps to a subsequent one. We provide a conceptualization of a program of research designed to elucidate students’ learning processes and describe an emerging methodology for this work. We present data and analysis from an initial teaching experiment that illustrates the methodology and demonstrates the learning that can be fostered using the approach, the data that can be generated, and the analyses that can be done. The approach involves the use of a carefully designed sequence of mathematical tasks intended to promote particular activity that is expected to result in a new concept. Through analysis of students’ activity in the context of the task sequence, accounts of students’ learning processes are developed. Ultimately a large set of such accounts would allow for a cross-account analysis aimed at articulating mechanisms of learning.

Transformation of Japanese Elementary Mathematics Textbooks: 1958–2012

Quality of teaching is a major factor in students’ mathematics learning. Stigler and Hiebert (1999) showed that mathematics teaching in Japanese schools is significantly different from what is typically observed in US classrooms. However, Japanese mathematics educators claim that Japanese mathematics teaching has transformed significantly over the last 50 years. Although teaching is influenced by a variety of factors, textbooks play a significant role in what mathematics is taught and how it is taught. In other words, textbooks may significantly influence students’ opportunities to learn. Thus, six editions of a Japanese elementary school mathematics series since 1958 were analyzed to identify any change that might indicate the transformation of mathematics instruction in Japan. The analysis revealed that the features included in the series have changed over the years to support more explicitly the problem-solving-based mathematics instruction described by Stigler and Hiebert (1999).

Shiki : A Critical Foundation for School Algebra in Japanese Elementary School Mathematics

An analysis of the Japanese elementary school (Grades 1 through 6) mathematics curriculum materials reveals that the study of functional relationships (patterns) is a major emphasis in Japan, as is the case in curricula from other countries. However, the Japanese curriculum considers the ideas related to mathematical expressions, called “shiki” in Japanese, as the second pillar of elementary school algebra. This chapter elaborates how a Japanese textbook series attempts to realize this emphasis on writing and interpreting mathematical expressions.

Response to Part III: Challenges and Promises of Unchartered Water—Lesson Study and Institutes of Higher Education

When an educational practice is implemented in a location away from its origin, there are usually modifications required. The previous chapters in this part are examples of how Japanese lesson study was modified at three sites in the United States. In particular, they illustrate the use of lesson study as a research tool and lesson study in Higher Education.

Intended Treatment of Fractions and Fraction Operations in Mathematics Curricula from Japan, Korea, and Taiwan

In spite of extensive research efforts, teaching and learning fractions remain challenging throughout the world. Although students’ mathematics learning is influenced by many factors, one important factor is the learning opportunities afforded by their textbooks. Therefore, we examined how textbooks from Japan, Korea, and Taiwan—three high-achieving countries prominent in comparative studies—introduced and developed fraction concepts and fraction arithmetic. We used the content analysis method (National Research Council, On evaluating curricular effectiveness: Judging the quality of K-12 mathematics evaluations, 2004) to analyze the problems presented in the textbooks. Our analysis revealed that there were many similarities among the textbooks from these three countries, including the overall flow of the topics related to fraction concepts and fraction arithmetic. However, significant differences included how various fraction subconstructs were integrated in the textbooks and how fraction multiplication and division were discussed. These similarities and differences among high-achieving countries suggest fruitful directions for future research in the area of fraction teaching and learning.