Teacher learning needs addressed
At first, the activity of analyzing student thinking might seem to relate only to the teacher learning need we have called “understanding student thinking.” While this is indeed a main goal of this kind of professional development experience, our two illustrations show that analyzing student mathematical activity can achieve much more than that. In this section, we discuss how this type of professional development experience can contribute to most of the teacher learning needs we identified in Chapter 1:
Developing a vision and commitment to school mathematics reform. Although teachers focus on what students do and think in this type of experience, the act of examining students’ mathematical activity in innovative learning situations can also contribute to teachers developing a vision and commitment to school mathematics reform. In this case, teachers can develop images of school mathematics reform in action from the instructional context that generated the student samples. The samples themselves can also show evidence of what students can accomplish when offered the kind of learning opportunities promoted by reform. This may then lead teachers to challenge traditional learning goals and practices and to experience a felt need for instructional change. The potential for this type of experience to engender a vision of reform, however, depends on the artifacts chosen and the structure and facilitation of the experience. If participants are to draw larger implications for the teaching and learning of mathematics, facilitators must help them move beyond the specifics of the learning situation they are analyzing and encourage the discussion to develop in that direction.
Strengthening one’s knowledge of mathematics. As our examples illustrate, analyzing student thinking can lead teachers to a better understanding of mathematical ideas. This is especially true when the facilitator carefully selects and sequences artifacts around a “big mathematical idea” and then focuses part of the conversation on uncovering and examining that idea. Teachers’ learning of new mathematics can further be enhanced through presentations or follow-up reading assignments on the mathematical idea examined.
Understanding the pedagogical theories that underlie school mathematics reform. Analyzing student thinking can also introduce teachers to the constructivist theories of learning that inform the current recommendations for school mathematics reform. However, in order to truly meet this teacher learning need, the analysis of students’ artifacts should be supplemented by readings and/or presentations about the theoretical foundations and empirical research supporting a constructivist perspective. This component is missing in both our illustrations.
Understanding students’ mathematical thinking. Understanding students’ mathematical thinking is obviously at the core of this kind of professional development experience. As both examples illustrate, examining specific examples of students’ mathematical activity in depth gives teachers valuable insights about the many different ways in which students at different grade levels approach problems or develop specific concepts or skills. Even more importantly, it can help teachers learn to conduct a similar analysis of their own students’ work, to both understand where students might be in their development of key mathematical ideas and to devise learning experiences to best help them progress. This second goal, however, calls for teachers to collect and analyze artifacts from their own classes.
Learning to use effective teaching and assessment strategies. While learning new teaching practices is not an explicit goal of this kind of professional development experience, there are two notable exceptions. First, teachers can learn strategies for encouraging students to share their thinking and approaches to solutions. Second, teachers can learn to interpret students’ work. We argue that both these strategies are at the core of school mathematics reform.
Supporters of this kind of professional development experience would also argue that these practices are likely to result in better instruction. Knowing how their students’ think can empower teachers to make informed instructional decisions and to devise effective assessments. As the vignette on examining the results of a test on area (Illustration 4) shows, even well-designed assessment tools can prove ineffective unless teachers learn to interpret the results and use them to inform instruction.
Finally, we should not forget that teachers, whenever they examine student thinking that takes place in reform mathematics classrooms, are exposed to other teachers’ worthwhile teaching practices.
Becoming familiar with exemplary instructional materials and resources. Becoming familiar with exemplary instructional materials and resources is not typically a goal of analyzing student thinking. One exception occurs when teachers examine student work in lessons adapted from exemplary instructional materials. In this case, the analysis of the students’ work can become an effective vehicle to examine the potential outcomes and goals of the materials.
Understanding equity issues and their implications for the classroom. Analyzing student thinking can be powerful for exploring issues of equity in learning mathematics in schools. Teachers have reported being surprised by the reasoning skills that students from disadvantaged backgrounds and students with disabilities reveal when given the opportunity to explain their solutions. These experiences can challenge teachers’ biases against students with different learning styles or cultural backgrounds. At the same time, knowing how differently students may approach a task alerts teachers to the influence that race, class, gender and disability may have on students’ mathematical performance. We need to keep in mind, however, that to capitalize on this potential, the selected artifacts must represent a wide-range of abilities and socio-cultural backgrounds.
Coping with the emotional aspects of engaging in reform. While coping with the emotional aspects of engaging in reform is not an explicit goal of experiences that analyze students’ thinking, some teachers may need help dealing with the discomfort and frustration this kind of professional development activity may generate. It is not uncommon for teachers to feel overwhelmed as they realize how powerful, yet time consuming, it is to examine the thinking process of each of their students in-depth. Therefore, facilitators should watch for and be ready to address these feelings. Although there is no easy way to resolve the time constraints teachers must live with, facilitators can discuss realistic expectations for analyzing students’ thinking as part of everyday practice and suggest some concrete strategies to make it a possibility.
Developing an attitude of inquiry towards one’s practice. As we mentioned earlier, one of the most desirable outcomes of examining student thinking is that teachers develop the habit of paying careful attention to students’ work. Teachers can then determine what students already know and do not know and make better instructional decisions. In other words, developing an attitude of inquiry toward students’ work is a central goal of this type of professional development experience, although it may not necessarily invite teachers’ inquiry on other aspects of their practice.
Although analyzing students’ thinking might seem at first to be a rather narrowly focused strategy, our analysis reveals that this type of professional development experience is complex and powerful. The analysis of students’ thinking can take a number of different forms, depending on what kind of artifacts are examined and who provides them. The implementation of this activity also depends on how the facilitator focuses the process of analysis, the specific tasks that enable the analysis, and the role the facilitator plays in both the design and the implementation of the professional development experience. The choices that the facilitator makes on each of these dimensions determines which different teacher learning needs can be met.
Suggested follow-up resources
If you are interested in learning more about exemplary professional development materials that can help teacher educators plan and facilitate the analysis of student thinking, we recommend the following resources:
Fennema, E., Carpenter, T., Levi, L., Franke, M.L., and Empson, S.B. (1999). Children’s mathematics: Cognitively guided instruction. Professional development materials. Portsmouth, NH: Heinemann. (videotapes available from the University of Wisconsin at Madison).
The creators of CGI offer a detailed and varied set of materials to support teacher educators in implementing a professional development program based on this approach. These materials provide first of all a description of the research model for studying students’ thinking about numbers and operations that informs the program. They also include suggestions for planning a comprehensive professional development program designed to introduce this research model, invite teachers to examine their own students’ thinking, and help them make instructional decisions accordingly. Facilitators of such program can also find examples of lesson plans for specific sessions, problems sets and students’ work to use with participants, and tips about various implementation issues. Videotapes of students’ problem solving are not included in the published materials, but they are available directly from the authors.
Schifter, D., Bastable, V., and Russell, S. J. (1999). Developing mathematical ideas (DMI) (casebooks + facilitator’s guides + videos) Parsippany, NJ: Dale Seymour.
This set of materials for teacher educators supports the implementation of an entire professional development program for elementary teachers who want to focus on numbers and operations. The sixteen 3-hour sessions that comprise this program have the analysis of students’ thinking at their very core – whether the analysis is conducted through a written “case,” video images of students engaged in mathematical activities, or student work the participants collect from their own classes. In each session, the Facilitator’s Guide provide concrete suggestions about how to analyze the student artifacts and develop productive discussions about them.