Abstract

During the last few years I lead two development teams that created and experimented 6 years curriculum in algebra and calculus for the middle and high school (grades 7 to 12). The innovative curriculum-- "Visual Mathematics" (CET 1995, 1997) -- is based on prominent principles of the current mathematics reform. It considers the function to be the fundamental object of secondary school mathematics and suggests mathematics studies that are based on learning by posing problems, conjecturing, learning within a community of learners by creating and sharing ideas and learning with intensive support of technology that is especially crafted to provide learning and thinking tools. Mapping the curriculum around the concept of function and the use of technology allowed a shift from mere technical manipulations towards the semantics of the manipulations. It also offered students the possibility to connect important mathematical ideas even prior to mastering the symbolic language. Turning the mathematical representations of functions to be the habit of mind and the natural choice for problem solving requires a long learning process. Thus, research of the opportunities and the complexities requires a long term investigation. Having a full curricular sequence of algebra centered around these concepts allowed us to plan longitudinal studies and follow the evolution of algebraic knowledge during three years. The methods of included classroom observations, students? and teachers portfolios and personal interviews.

We interviewed the same 12 pairs of students twice a year during three years. Each interview consists of mathematical problem solving tasks that were not directly addressed in the curriculum but rather involved conceptual mathematical thinking. The data from the long term interviews helps: to discuss the nature of problem solving in algebra when using a function approach with graphic technology, to challenge the traditional model of algebra problem solver, and specifically, to portrait a profile of the less able students? performance in algebra in the technological environment.

We videotaped various teaching and learning episodes, mainly group discussions or whole classroom discussions that involve non-traditional classroom discourse and actions. During these discussions students often carry out what seems as mathematician habits of mind. While confronting with new problems, they are able to visualize in flexible ways, they see functions in unexpected places, they reason using deductive and inductive methods, they experiment with the problem, question about the problem and critique its solution. The language they use is precise and is organized around mathematical objects and processes. Thus, they are in fact practicing habits of the mathematics community while dealing with school algebra content.

We documented a single classroom with the same teacher and students for three years. Documents include weekly observations, and a collection of the teacher and students portfolios. We are using this data, mainly as seen and analyzed by the classroom teacher herself, to understand the impact of the function approach on the standards of the algebra curriculum, on definitions of algebraic thinking, and the practice of the algebra classroom.

In my talk I will outline the curricular agenda, the rational behind the design of the software that supports this agenda, and will concentrate on the research analysis methods. The theoretical foundations for the analysis of the data will be offered as a scheme to analyze and assess learning within technological environments for modeling and manipulating in algebra and calculus. © Asian Technology Conference in Mathematics, 1998.