A methodological framework and empirical techniques for studying the travel of ideas in classroom communities
Integers and fractions are challenging topics, and our goal is to design a curriculum unit that engages diverse students with the mathematics in sustained and coherent ways. Our approach to lesson design builds on socio-constructivist assumptions about meaningful learning: In the context of mathematical activities where students work with others and independently, students develop understanding as they produce, coordinate and adapt representations (number lines, area models, Hindu-Arabic arithmetical procedures) to serve mathematical functions (subtract negative numbers, compare fractions) in communicative and problem-solving activities. Lesson sequences should support students’ efforts to (a) extend their own sometimes idiosyncratic forms of mathematical representation for integers and fractions to serve new and important mathematical functions, and (b) incorporate new forms of representation valued in school to serve and extend mathematical functions that they already know. Thus we view learning as shifts in the relationships that students construct between mathematical forms and mathematical functions. Our principles for lesson design are grounded in our socio-constructivist assumptions about the ways that mathematical ideas travel and are transformed. 1. Lessons should target core mathematical ideas. Elementary mathematics textbooks in the United States contain too many mathematical topics and representations to be covered in depth, and students cannot be expected to develop rich connections within and across topics and representational forms. We are focusing on core generative ideas and setting aside more peripheral topics and representational forms. For fractions, we consider equivalence to be one core idea and the number line one core representational form. Equivalence is the basis for the principle that any particular representation of a rational number is not the number itself, and any rational number can be represented in an infinite number of ways. As a representational form, the number line can support elementary students’ understanding of equivalence, order and magnitude of rational numbers; it then becomes a critical tool for secondary school students’ later work with the Cartesian coordinate system and mathematical functions. 2. Lessons should engage all students. US classrooms are diverse, and we need
pedagogies that support the intellectual engagement of students who vary in mathematical understandings, interests and investment. This principle led us to the six-phase inquiry lesson structure depicted in Figure 13.1, a structure that encourages active participation, reflection and questioning about mathematics (Saxe et al. 2007). The lesson is organized around a Problem of the Day that is challenging and yet accessible – students must choose one answer among several alternatives that represent common patterns of student thinking. The multiplechoice format for the Problem of the Day is an adaptation of the Itakura method originally developed for science lessons (cf. Inagaki, Hatano & Morita 1998). To support student exploration of form-function relationships between fractions, representations and the functions they can serve, students revisit the Problem several times over the course of the lesson through individual work, small-group interactions, whole-class presentations and teacher-led discussions. Phase 1: Independent work and pre-assessment. Students work independently to solve a problem like the one illustrated in Figure 13.1, ‘How many fraction names for point A?’. To provide a scaffold for student engagement, students choose one of five multiple-choice alternatives and then justify their choice in writing. The answer choices are based on previous studies of students’ reasoning on similar problems (Saxe, Langer-Osuna & Taylor 2006; Saxe, Taylor, McIntosh & Gearhart 2005): (a) only one fraction name; (b) two fraction names; (c) between 3 and 10 fraction names; (d) between 11 and 20 fraction names; (e) more than 20 fraction names. Students’ answers are their initial forays into the mathematics as well as the teacher’s initial assessment of the range of ideas in the classroom. The teacher then chooses several students who represent that range to make presentations in Phase 2. Phase 2: Student whole-class presentations. The teacher invites several students to present their solutions. During the presentations, other students hear the solu-
Figure 13.1 An illustration of the six-phase inquiry lesson structure.