Navigating Mathematical Transitions: Students' Adjustments to Fundamental Changes in Curriculum and Pedagogy Principal Investigator: Jack Smith, Michigan State University
Students can experience mathematical transitions when they move from classrooms where one perspective on doing and thinking about mathematics is illustrated and practiced to others where quite different perspectives dominate. Such discontinuities in expectations can seriously challenges students, who must make sense of and adjust to them. Mathematical transitions are now more likely given the uneven implementation of curricula and teaching practices inspired by the NCTM Standards. Two important places to search for and examine these transitions are introductory algebra (in middle and high school) and calculus. Curricular materials in these content areas and the teaching practices suggested by them can vary dramatically according to: (1) the fundamental objects and processes of study (equations and symbol manipulation procedures vs. functions, multiple representations, and solutions in each representation); (2) the typical problems presented to students (commands to manipulate symbols vs. questions about varying quantities in context); and (3) the organization of classroom learning (teacher presentation and individual problem-solving vs. these in combination with small group work and whole class discussion). This project will develop a working conceptualization of mathematical transitions (i.e., major factors and how they interact) and analyze high school and college students' passage through them. Three main questions will be addressed: (1) What are the characteristics of successful (and unsuccessful) mathematical transitions?; (2) How do students navigate them?; and (3) What kinds of resources ("internal" and "external" to the student) support more successful transitions? Four school sites (two high schools and two universities) will participate. At two sites (1 high school and 1 college, curricula and teaching practices shift from the study of equations and their solution, practice with many problems, individual study, and teacher presentation to the study of functions in multiple representations, verbal and numerical "answers," work on contextualized problems, and small group and whole group work. At the other two sites, these conditions are reversed. The project will select 100 students (25 per site) and track their progress through 2+ years of mathematics. It will assess and analyze the change in their emerging career and educational goals, mathematics achievement, content learning, beliefs about mathematics and themselves as learners, daily experience and reactions to their work, and strategies for adjusting to different expectations. The project will produce a composite portrait of the experience and learning of the entire sample and a representative set of detailed case studies to illuminate the dynamics of students' experience. It will attempt to present its findings to teachers, parents, and administrators, as well as researchers.