Patterns of Reasoning about Mathematical Models: A Case Study of High School Mathematics Teachers

Article excerpt

This case study illustrates the results of instruction in a mathematical modeling course for high school mathematics teachers. In this course the teachers were introduced to the process of mathematical modeling. They explored various problems and data sets. The study examines the course outcomes in terms of two teaching goals: (1) helping participants learn to build, and to reach consensus about plausible mathematical models, (2) influencing participants' thinking about appropriate mathematics curriculum and instruction. The development of participants relative to each of the course goals was traced. Most participants were successful in achieving both goals. The findings of the study highlight the need for design of professional development activities that take into account the diverse mathematical backgrounds of teachers.

Calls for restructuring learning environments that build around genuine mathematical inquiry have increased substantially over the course of the past few years. Explicit in the current recommendations for reform in school mathematics is the notion of helping learners develop the disposition to engage in activities similar to those of mathematicians (NCTM 2000). Current convictions shared by many mathematicians and mathematics educators about the nature of mathematical knowledge and how it is best learned include inquiry into real life questions and planning problem and data driven curriculum that motivate the search for meaning and the use of mathematics (National Research Council 2000). There is also the belief that mathematical concepts and theories are tools that are based on our collective experience in the world, and that we use these tools to make sense of our experience (Bishop 1988; Davis & Hersh 1981).

Mathematics teachers have traditionally recognized the dialectic between mathematical theories and concepts and using mathematics to solve problems by affirming two types of goals in their instruction. The first goal involves helping students to control the mathematical theories and facts. The second goal involves helping students use these algorithms in solving application problems. This controlled approach to teaching is problematic because it transforms the complex, socially embedded process of mathematical inquiry into traditional school tasks--bits of curriculum that can be easily managed in teacher-centered classroom.

Many reports of recent efforts to reform mathematics teaching have described classroom environments which are less teacher centered and which model many of the ideals and processes of the mathematics community (Borasi 1992; Manouchehri & Enderson 1999; Wood, Cobb & Yackel 1991; Yackel & Cobb 1996). Significant feature of this kind of classroom include valuing of student ideas as currency for classroom interactions, the teacher taking the role of co-learner, and evolving standards for collective validation of ideas. Often called a learning community (Lampert 1990), this kind of classroom has been recently described by Cobb and Bauserfeld (1995) as a setting in which the goal is to create a community of validators. Their description of interactions focus on argumentation, group discussions, and co-learning:

 
  The standards of argumentation established in an inquiry classroom are 
  such that the teacher and students typically challenge explanations 
  that merely describe the manipulation of symbols. Further, acceptable 
  explanations appear to carry the significance of acting on taken-as 
  shared objects. Consequently, from the observer's perspective, the 
  teacher and the students seem to be acting in a taken-as-shared 
  mathematical reality, and to be elaborating that reality in the course 
  of their ongoing negotiations of meaning (pg. 2-3) 

Transforming the learning environment to resemble a community of validators is primarily dependent upon the teacher's initiative. …