Understanding the pedagogical theories that underlie school mathematics reform
Research shows that most mathematics teachers, including prospective teachers, have strongly-held beliefs about student and teacher’s roles, desirable instructional approaches, students’ mathematical knowledge, how students learn and the purposes of schools (Thompson, 1992). These beliefs have mostly developed as a result of the teachers’ own schooling. Although rarely made explicit, the following views of knowledge, learning and teaching lie behind what takes place in most traditional classrooms:
Knowledge is a body of established facts and techniques that can be broken down and transmitted to novices by experts (positivistic view of knowledge).
Learning results from acquiring isolated bits of information and skills through listening, watching, memorizing and practicing (behaviorist view of learning).
Teaching is the direct transmission of knowledge from teacher to student; it takes place as long as the teacher provides clear explanations for the students to absorb (direct instruction view of teaching) (Borasi & Siegel, 1992, 2000).
In contrast, the teaching practices recommended by the NCTM Standards (NCTM, 2000) and illustrated in our classroom vignette are grounded in views of knowledge, learning and teaching informed by a constructivist perspective (e.g., Brooks & Brooks, 1999; Davis, Maher & Noddings, 1990; Fosnot, 1996). Although different interpretations of constructivism exist, current school mathematics reform efforts are generally characterized by the following constructivist assumptions:
Knowledge is socially constructed through human activity, shaped by context and purposes, and validated through a process of negotiation within a community of practice. Thus, it is always tentative rather than absolute. However, although knowledge is provisional in this paradigm, it does not mean that “anything goes.”
Learning is a generative process of making meaning that builds on personal knowledge and social interactions. This process may be stimulated by perceived dissonance. Prior knowledge, context and purpose play critical roles in the shaping of learning situations.
Teaching is facilitating students’ learning by creating a learning environment conducive to inquiry, setting up problem-solving situations to stimulate both student interest and cognitive dissonance about important mathematical ideas, and supporting students’ attempts to solve problems and make sense of mathematical concepts (Borasi & Siegel, 1992, 2000).
To fully appreciate the constructivist pedagogical approach recommended in the NCTM Standards, teachers need to identify and understand the non-traditional theories of teaching and learning mathematics and the research supporting such approaches.
Understanding students’ mathematical thinking
One of the main challenges that the teacher in our vignette experienced during her inquiry on area was interpreting her students’ thinking and responding appropriately, especially when students proposed new strategies or formulas for computing area and explained how they got their results. The teacher benefited considerably by having already investigated a range of possible strategies and solutions to the open-ended tasks she posed – although some of the students’ strategies still took her by surprise! Indeed, understanding students’ mathematical thinking is especially critical in any constructivist approach if teachers are to design instructional experiences that help students build on their existing knowledge (Confrey, 1991).
Research on Cognitive Guided Instruction (CGI) has provided both theoretical arguments and empirical evidence claiming that mathematics teachers benefit from knowing about their students’ prior knowledge and ways of learning specific mathematical concepts (Carpenter & Fennema, 1992; Fennema, Carpenter & Franke, 1997). Knowledge of child-constructed procedures is a crucial prerequisite for designing learning experiences that capitalize on, rather than override, the informal mathematical knowledge children bring to school. For example, many elementary teachers are surprised to learn that children often develop their own procedures for solving simple arithmetic problems before they enter school. Knowing this fact can help teachers rethink how arithmetic operations might be introduced.
Further empirical support for the value of teachers’ knowing how students think comes from the Integrating Mathematics Assessment (IMA) project. This project focused on making teachers aware of the key features of student thinking about fractions. As a result, students made significant gains in solving problems involving fractions (Gearhart, Saxe, & Stipek, 1995).
While the results of studies like CGI and IMA are compelling, it is reasonable to ask whether we should expect teachers to acquire research-based knowledge about student thinking in all the mathematical areas they will teach, especially when most topics taught in secondary school are not as well researched as basic arithmetic and rational numbers. Rhine (1998) suggests that rather than trying to create such a knowledge base among teachers, it may be more important to foster a new attitude, one that values analyzing student thinking as part of teachers’ everyday practice and provides strategies to help them do so.
Learning to use effective teaching and assessment strategies
One element that most distinguished the inquiry on area in our classroom scenario was the extensive use of teaching practices that are usually absent from traditional mathematics instruction. These included, for example, orchestrating group work using a variety of techniques, such as the initial “think-pair-share” activity; facilitating class discussions in which students shared results and jointly constructed new knowledge; using effective questioning techniques to synthesize key mathematical ideas; and assessing students’ learning in multiple ways, such as the performance assessment in which students created an area formula for a star.
The pedagogical recommendations articulated in the NCTM Standards (NCTM, 1991, 2000) call specifically for teaching practices like these that are not currently used by many mathematics teachers, especially at the secondary level (for comprehensive lists of such practices, see Koehler & Grouws  and Borasi & Fonzi [in preparation]). Non-traditional practices include not only facilitating what goes on in the classroom as lessons develop but also planning and assessing lessons effectively. Assessment has received special attention recently (e.g., Bright & Joyner, 1998; Lesh & Lamon, 1992; NCTM, 1995; Webb & Coxford, 1993) because determining what students know is necessary for teaching effectively within a constructivist paradigm. It is also critical for documenting the outcomes of reform efforts.
Learning to use novel teaching practices appropriately is not easy. Research on how people learn complex tasks may shed some light on what it takes teachers to adopt a new teaching practice. For example, Collins and his colleagues (1989) have suggested the following three-phase process for learning a complex task:
1. Modeling – The learner observes and examines how an expert engages in the task.
2. Scaffolded practice – The learner engages in the task himself/herself, but with the help of an expert and/or of other supporting structures.
3. Independent practice – The learner engages in the task without support.
Clearly, using new teaching practices effectively goes far beyond simply knowing they exist. While mathematics teachers should learn about a variety of teaching strategies to enrich their repertoire of resources, they should also have the opportunity to personally experience these practices in supported situations in order to evaluate fully their pedagogical potential. It is also critical for teachers to learn not only to use specific practices well but also to appreciate their strengths and limitations so they can choose practices most appropriate to an audience and to unique instructional goals.
Becoming familiar with exemplary instructional materials and resources
When reading about a well-designed, complex experience such as the inquiry on area described in our vignette, teachers might feel daunted by the prospect of creating similar lessons on their own. Fortunately, today’s mathematics teachers are not expected to always create innovative units on their own as they may take advantage of the many exemplary instructional materials informed by the NCTM Standards that have been produced in recent years. As argued by Russell, this by no means demeans the professionalism of teachers:
Curriculum materials, when developed through careful, extended work with diverse students and teachers, when based on sound mathematics and on what we know about how people learn mathematics, are a tool that allows the teacher to do her best work with students... . It is not possible for most teachers to write a complete, coherent, mathematically sound curriculum. It is not insulting to teachers as professionals to admit this. (Russell, 1997, p. 248)
Exemplary instructional materials may consist of replacement units, which are individual units designed to replace parts of the traditional curriculum while expanding the instructional goals and introducing some effective teaching practices or of comprehensive curricula. These consist of a sequence of units intended to totally replace the current mathematics curriculum at either elementary, middle or high school. Among the latter group, a set of instructional materials consistent with the NCTM Standards has been recently developed with support from the National Science Foundation (NSF) (see Figure 7 for a complete list of these comprehensive curricula and their websites’ addresses). Additional exemplary mathematics curricula have been identified in a study by the U.S. Department of Education (U.S. Department of Education’s Mathematics and Science Education Expert Panel, 1999).