John C. Moyer

The teaching of equation solving: approaches in Standards-based and traditional curricula in the United States

This paper discusses the approaches to teaching linear equation solving that are embedded in a Standards-based mathematics curriculum (Connected Mathematics Program or CMP) and in a traditional mathematics curriculum (Glencoe Mathematics) in the United States. Overall, the CMP curriculum takes a functional approach to teaching equation solving, while Glencoe Mathematics takes a structural approach. The functional approach emphasizes the important ideas of change and variation in situations and contexts. It also emphasizes the representation of relationships between variables. The structural approach, on the other hand, requires students to work abstractly with symbols and follow procedures in a systematic way. The CMP curriculum may be regarded as a curriculum with a pedagogy that emphasizes predominantly the conceptual aspects of equation solving, while Glencoe Mathematics may be regarded as a curriculum with a pedagogy that emphasizes predominantly the procedural aspects of equation solving. The two curricula may serve as concrete examples of functional and structural approaches, respectively, to the teaching of algebra in general and equation solving in particular.

Examining Students’ Algebraic Thinking in a Curricular Context: A Longitudinal Study

This chapter highlights findings from the LieCal Project, a longitudinal project in which we investigated the effects of a Standards-based middle school mathematics curriculum (CMP) on students’ algebraic development and compared them to the effects of other middle school mathematics curricula (non-CMP). We found that the CMP curriculum takes a functional approach to the teaching of algebra while non-CMP curricula take a structural approach. The teachers who used the CMP curriculum emphasized conceptual understanding more than did those who used the non-CMP curricula. On the other hand, the teachers who used non-CMP curricula emphasized procedural knowledge more than did those who used the CMP curriculum. When we examined the development of students’ algebraic thinking related to representing situations, equation solving, and making generalizations, we found that CMP students had a significantly higher growth rate on representing-situations tasks than did non-CMP students, but both CMP and non-CMP students had an almost identical growth in their ability to solve equations. We also found that CMP students demonstrated greater generalization abilities than did non-CMP students over the three middle school years.

Longitudinally Investigating the Impact of Curricula and Classroom Emphases on the Algebra Learning of Students of Different Ethnicities

This paper explores how curriculum and classroom conceptual and procedural emphases affect the learning of algebra for students of color. Using data from a longitudinal study of the Connected Mathematics Program (CMP), we apply cross-sectional HLM to lend explanatory power to the longitudinal analysis afforded by growth curve modeling that we have reported elsewhere. Overall, we find that the achievement gaps tend to decrease over the course of the middle grades. However, differences in open-ended problem solving and equation-solving performance persist for African-American students. Classroom conceptual and procedural emphases also appear to differentially influence the performance of Hispanic and African-American students, depending on the aspect of algebra learning that is being measured.

Teaching Mathematics Using Standards-Based and Traditional Curricula: A Case of Variable Ideas

This chapter discusses approaches to teaching algebraic concepts like variables that are embedded in a Standards-based mathematics curriculum (CMP) and in a traditional mathematics curriculum (Glencoe Mathematics). Neither the CMP curriculum nor Glencoe Mathematics clearly distinguishes among the various uses of variables. Overall, the CMP curriculum uses a functional approach to teach equation solving, while Glencoe Mathematics uses a structural approach to teach equation solving. The functional approach emphasizes the important ideas of change and variation in situations and contexts. The structural approach, on the other hand, avoids contextual problems in order to concentrate on developing the abilities to generalize, work abstractly with symbols, and follow procedures in a systematic way. This chapter reports part of the findings from the larger LieCal research project. The LieCal Project is designed to investigate longitudinally the impact of a Standards-based curriculum like CMP on teachers’ classroom instruction and student learning. This chapter tells part of the story by showing the value of a detailed curriculum analysis in characterizing curriculum as a pedagogical event.

Learning to Notice Student Thinking About the Equal Sign: K-8 Preservice Teachers’ Experiences in a Teacher Preparation Program

In this chapter, we present our work and research related to preservice teacher (PST) noticing, describing how we provide PSTs with opportunities to notice student thinking about the equal sign and equality. We designed an instructional intervention in an integrated mathematics content and pedagogy course (with a field experience) to support PSTs in (1) learning about key mathematical ideas related to the equal sign and equality, and (2) rehearsing teacher noticing skills. Our PSTs rehearsed and reflected on their noticing skills by conducting two one-on-one clinical interviews with elementary students and participating in debriefing interviews with course instructors. Using this context, we examined (1) the extent to which PSTs attended to and further explored student understanding of the equal sign and equality, and (2) what PSTs perceived they learned about aspects of their teacher professional noticing skills and student thinking about the equal sign and equality. Our results indicate that the PSTs predominantly noticed the strategies students used to solve a task without focusing on student thinking about the equal sign and equality. In addition, our PSTs perceived that they strengthened either their own knowledge or student knowledge of the equal sign and equality while conducting their diagnostic clinical interviews.