Chapter 8 - Gathering and Making Sense of Information

We described a number of creative and novel learning experiences for teachers in the previous chapters, but some traditional learning experiences still have much to contribute to teacher learning. Indeed, in several of the illustrations reported in the previous four chapters, participants read articles or listened to presentations. In this chapter, we show how teachers can benefit from these as well as other forms of data gathering and sense-making, including action research, as a main venue for learning. More specifically, we will examine ways in which teacher education informed by a constructivist paradigm can facilitate teachers’ learning from and with texts, videos, presentations, and even data they have gathered in their own research.

Having teachers listen to experts’ presentations and doing assigned readings has been the preferred mode of professional development so far at both pre-service and in-service levels. Interestingly, however, not much research documents the effects of these learning modes on teachers’ knowledge, beliefs or practice.

Nevertheless, gathering and making sense of information continues to be a valuable tool for teachers and any other learners. This mode of learning can become an integral part of constructing a personal understanding of issues and theories that are at the core of school mathematics reform. Indeed, readings, presentations, and data collection and analysis can all contribute to teacher education although they may take on different forms and purposes when informed by a constructivist perspective.

Recent research on reading, in particular, can help us begin to reconceptualize how making sense of information can become an active and socially constructed process. Reading researchers have argued that reading does not need to occur as an isolated, or even individual, activity (e.g., Harste & Short, 1988). First, reading should be purposeful. In other words, teachers should read either to address questions that they feel the need to know more about or because their concerns could not be resolved through discussion. Reading can also be a catalyst for other experiences. Indeed, reading can fulfill many functions while teachers inquire into any topic (Siegel, Borasi & Fonzi, 1998). Readings can provide background information, raise questions for further inquiry about a topic, synthesize different points of view, and offer models for teachers’ own practice. Research also teaches us that reading is not a passive or straightforward matter of decoding or extracting information from text (e.g., Pearson & Fielding, 1991; Rosenblatt, 1994). Rather, readers always construct meaning in interaction with the text, their own background and interests, and their purposes for reading the text. Furthermore, such construction of meaning can be even more productive when it is augmented by interactions with other learners, so that different interpretations can be shared and discussed.

Reading researchers also argue for expanding our notion of what constitutes a text (e.g., Bloome & Egan-Robertson, 1993; Green & Meyer, 1991), noting that the principles of reading outlined above also hold true for other “texts,” such as videos, presentations or electronic media. Indeed, teachers can benefit from actively constructing and negotiating meaning not only through written texts but also videos they watch together or independently, information they gather on the Internet or presentations made by an expert or a colleague.

In addition to benefiting from information others provide, teachers can gather their own data to illuminate issues of particular interest to them. Teachers can gain from participating in many forms of research, but “action research” is especially promising as a form of professional development (Holly, 1991; Eisenhower National Clearinghouse, 2000). Action research is defined as “an ongoing process of systematic study in which teachers examine their own teaching and students’ learning through descriptive reporting, purposeful conversation, collegial sharing, and reflection for the purpose of improving classroom practice” (Eisenhower National Clearinghouse, 2000, p.18). Action research thus offers an ideal way for teachers to learn more about teaching and learning mathematics and to apply the results immediately to their own practice, although conducting full-blown action research studies is not the only way that teachers can benefit from gathering and analyzing classroom data.

The experience captured in this illustration took place in the Leadership Seminar that was one of the components of the Making Mathematics Reform a Reality (MMRR) project we described in Chapter 2. After several teachers had participated in the first year of the program, they wanted to make more radical changes in their teaching. During the first year, they had attended a Summer Institute introducing them to an inquiry approach to mathematics instruction and then implemented an illustrative inquiry unit on either tessellations or area in their own classrooms. Their experiences with the tessellation and area units made them aware of the inadequacy of traditional approaches to teaching geometry in the middle school curriculum. Although they felt that the next logical step would be to revise their school’s geometry curriculum, they were not sure how to proceed. In the usual process for rewriting curriculum, teachers sat around a table, and based on the current textbook, discussed what contents should be covered at each grade and how. The teachers suspected that this process might at best eliminate some repetition in the existing curriculum, but that it was not likely to help them reconceive the entire middle school geometry curriculum.

After some lead teachers shared these concerns in the Leadership Seminar, the facilitators decided to use this opportunity to lead the group in a systematic rethinking of the teaching and learning of geometry in middle school. Such an experience could serve as a model for lead teachers interested in replicating a similar process with colleagues in their own school. An even more important goal for this experience, however, was to familiarize the lead teachers with the resources offered by relevant research studies and exemplary instructional materials, so they could use these resources well in the future.

The group inquiry started with a few readings about geometry. As a homework assignment, participants read two mathematical essays from the book On the Shoulders of Giants (Steen, 1990). One essay focused on the concept of “Shape” (by Senechal) and the other on “Dimension” (by Banchoff). As part of the same assignment, participants reviewed the NCTM Standards (1989) for geometry in middle school.

In the group discussion that resulted, the lead teachers analyzed the meaning and rationale of each of the NCTM geometry standards in light of the “big ideas” of geometry presented in the two essays. This discussion enabled participants to enhance their understanding of the mathematical concepts presented in the two essays and to consider implications for instruction. For example, some teachers said they found it very helpful to think of geometry as the study of “shapes,” especially as they had come to realize the connection between the geometric properties of a shape and its possible functions. This realization helped them frame in a more meaningful way the study of geometric figures for their students. It also helped them change their instructional goals because they agreed that students should learn strategies for identifying the attributes of any geometric figure, not just memorize a pre-established set of properties for a few standard figures.

Although very helpful, this activity did not immediately result in a plan for what to teach about geometry, and how, at different grade levels in middle school. The facilitators then suggested that the group look at the choices made by two of the comprehensive middle school math curricula funded by the National Science Foundation, the Connected Mathematics Project and Mathematics in Context. In both cases, groups composed of mathematicians, mathematics educators, and teachers had grappled for years with the same question: What should students learn about geometry in middle school? The facilitators argued, then, that the group should capitalize on all the thinking that had gone into the development of these exemplary curricula.

However, it turned out to be difficult to extract from the curricula the choices that the authors had made about what geometry content to cover and how, and the rationale for these decisions. Although the background materials accompanying each of these curricula did address, to some extent, these choices and how they were made, the information was not specific enough for the group. It soon became clear that the group needed to examine the individual geometry units in each curriculum.

To make this task less daunting and time-consuming, the group divided up the responsibilities. Each participant, including the facilitators, agreed to review one or two units from each curriculum to identify what was taught and how and to present their findings to the group. To ensure consistency, the facilitators proposed some guidelines for the review and report on each unit and then modeled a presentation.

A 3-hour session was then devoted to the geometry unit presentations. To get a sense of how topics in each curriculum were sequenced, participants presented the units in the order they were intended to be taught. As each unit was presented, a facilitator recorded on newsprint the key ideas about geometry that the unit addressed. At the end of the presentations, the teachers had a detailed list of the geometry content that each curriculum covered.

The group then compared these lists to identify similarities and differences between these two Standards-based curricula and the traditional middle school geometry curriculum. Many teachers were amazed at the richness of the lists describing the new curricula when compared with the traditional middle school math curriculum. They were struck especially by the emphasis in both of the new curricula on three-dimensional geometry and spatial visualization, topics they rarely covered but that were highlighted in the geometry essays they had read. On the other hand, they were puzzled by the presence of some new topics, such as Euler’s formula and graph theory in the Mathematics in Context curriculum.

The facilitators then suggested they seek a mathematician’s help to examine further the relative importance of the topics on the lists. The facilitators met independently with Dr. Sanford Segal, a research mathematician on the faculty at the University of Rochester, to share the group’s lists and ask whether he felt comfortable commenting on the mathematical significance of the topics listed. They also shared some information about the group’s background and goals to help him prepare his contribution.

Dr. Segal then joined the group for a 2-hour session in which he presented his comments on the relative importance of items on the lists from a mathematical stand-point, and then he answered questions. His presentation and the follow-up discussion further confirmed the critical role of spatial visualization in mathematics, and hence the importance of developing this skill in middle school through appropriate learning experiences. On the other hand, Dr. Segal’s personal position on the relative importance of graph theory and transformation geometry challenged the need to introduce these topics at the middle school level.

Overall, all participants, facilitators included, emerged from this inquiry with a much deeper understanding of what the “big ideas” in geometry are and a greater appreciation for the complexity of making good choices about mathematics content at any grade level.

Continued




	Gathering and Making Sense of Information We described a number of creative and novel learning experiences for teachers in the previous chapters, but some traditional learning experiences still have much to contribute to teacher learning. Indeed, in several of the illustrations reported in the previous four chapters, participants read articles or listened to presentations. In this chapter, we show how teachers can benefit from these as well as other forms of data gathering and sense-making, including action research, as a main venue for learning. More specifically, we will examine ways in which teacher education informed by a constructivist paradigm can facilitate teachers’ learning from and with texts, videos, presentations, and even data they have gathered in their own research. Theoretical rationale and empirical support Having teachers listen to experts’ presentations and doing assigned readings has been the preferred mode of professional development so far at both pre-service and in-service levels. Interestingly, however, not much research documents the effects of these learning modes on teachers’ knowledge, beliefs or practice. Nevertheless, gathering and making sense of information continues to be a valuable tool for teachers and any other learners. This mode of learning can become an integral part of constructing a personal understanding of issues and theories that are at the core of school mathematics reform. Indeed, readings, presentations, and data collection and analysis can all contribute to teacher education although they may take on different forms and purposes when informed by a constructivist perspective. Recent research on reading, in particular, can help us begin to reconceptualize how making sense of information can become an active and socially constructed process. Reading researchers have argued that reading does not need to occur as an isolated, or even individual, activity (e.g., Harste & Short, 1988). First, reading should be purposeful. In other words, teachers should read either to address questions that they feel the need to know more about or because their concerns could not be resolved through discussion. Reading can also be a catalyst for other experiences. Indeed, reading can fulfill many functions while teachers inquire into any topic (Siegel, Borasi & Fonzi, 1998). Readings can provide background information, raise questions for further inquiry about a topic, synthesize different points of view, and offer models for teachers’ own practice. Research also teaches us that reading is not a passive or straightforward matter of decoding or extracting information from text (e.g., Pearson & Fielding, 1991; Rosenblatt, 1994). Rather, readers always construct meaning in interaction with the text, their own background and interests, and their purposes for reading the text. Furthermore, such construction of meaning can be even more productive when it is augmented by interactions with other learners, so that different interpretations can be shared and discussed. Reading researchers also argue for expanding our notion of what constitutes a text (e.g., Bloome & Egan-Robertson, 1993; Green & Meyer, 1991), noting that the principles of reading outlined above also hold true for other “texts,” such as videos, presentations or electronic media. Indeed, teachers can benefit from actively constructing and negotiating meaning not only through written texts but also videos they watch together or independently, information they gather on the Internet or presentations made by an expert or a colleague. In addition to benefiting from information others provide, teachers can gather their own data to illuminate issues of particular interest to them. Teachers can gain from participating in many forms of research, but “action research” is especially promising as a form of professional development (Holly, 1991; Eisenhower National Clearinghouse, 2000). Action research is defined as “an ongoing process of systematic study in which teachers examine their own teaching and students’ learning through descriptive reporting, purposeful conversation, collegial sharing, and reflection for the purpose of improving classroom practice” (Eisenhower National Clearinghouse, 2000, p.18). Action research thus offers an ideal way for teachers to learn more about teaching and learning mathematics and to apply the results immediately to their own practice, although conducting full-blown action research studies is not the only way that teachers can benefit from gathering and analyzing classroom data. Illustration 9: Using a variety of resources to rethink the teaching and learning of geometry in middle school The experience captured in this illustration took place in the Leadership Seminar that was one of the components of the Making Mathematics Reform a Reality (MMRR) project we described in Chapter 2. After several teachers had participated in the first year of the program, they wanted to make more radical changes in their teaching. During the first year, they had attended a Summer Institute introducing them to an inquiry approach to mathematics instruction and then implemented an illustrative inquiry unit on either tessellations or area in their own classrooms. Their experiences with the tessellation and area units made them aware of the inadequacy of traditional approaches to teaching geometry in the middle school curriculum. Although they felt that the next logical step would be to revise their school’s geometry curriculum, they were not sure how to proceed. In the usual process for rewriting curriculum, teachers sat around a table, and based on the current textbook, discussed what contents should be covered at each grade and how. The teachers suspected that this process might at best eliminate some repetition in the existing curriculum, but that it was not likely to help them reconceive the entire middle school geometry curriculum. After some lead teachers shared these concerns in the Leadership Seminar, the facilitators decided to use this opportunity to lead the group in a systematic rethinking of the teaching and learning of geometry in middle school. Such an experience could serve as a model for lead teachers interested in replicating a similar process with colleagues in their own school. An even more important goal for this experience, however, was to familiarize the lead teachers with the resources offered by relevant research studies and exemplary instructional materials, so they could use these resources well in the future. The group inquiry started with a few readings about geometry. As a homework assignment, participants read two mathematical essays from the book On the Shoulders of Giants (Steen, 1990). One essay focused on the concept of “Shape” (by Senechal) and the other on “Dimension” (by Banchoff). As part of the same assignment, participants reviewed the NCTM Standards (1989) for geometry in middle school. In the group discussion that resulted, the lead teachers analyzed the meaning and rationale of each of the NCTM geometry standards in light of the “big ideas” of geometry presented in the two essays. This discussion enabled participants to enhance their understanding of the mathematical concepts presented in the two essays and to consider implications for instruction. For example, some teachers said they found it very helpful to think of geometry as the study of “shapes,” especially as they had come to realize the connection between the geometric properties of a shape and its possible functions. This realization helped them frame in a more meaningful way the study of geometric figures for their students. It also helped them change their instructional goals because they agreed that students should learn strategies for identifying the attributes of any geometric figure, not just memorize a pre-established set of properties for a few standard figures. Although very helpful, this activity did not immediately result in a plan for what to teach about geometry, and how, at different grade levels in middle school. The facilitators then suggested that the group look at the choices made by two of the comprehensive middle school math curricula funded by the National Science Foundation, the Connected Mathematics Project and Mathematics in Context. In both cases, groups composed of mathematicians, mathematics educators, and teachers had grappled for years with the same question: What should students learn about geometry in middle school? The facilitators argued, then, that the group should capitalize on all the thinking that had gone into the development of these exemplary curricula. However, it turned out to be difficult to extract from the curricula the choices that the authors had made about what geometry content to cover and how, and the rationale for these decisions. Although the background materials accompanying each of these curricula did address, to some extent, these choices and how they were made, the information was not specific enough for the group. It soon became clear that the group needed to examine the individual geometry units in each curriculum. To make this task less daunting and time-consuming, the group divided up the responsibilities. Each participant, including the facilitators, agreed to review one or two units from each curriculum to identify what was taught and how and to present their findings to the group. To ensure consistency, the facilitators proposed some guidelines for the review and report on each unit and then modeled a presentation. A 3-hour session was then devoted to the geometry unit presentations. To get a sense of how topics in each curriculum were sequenced, participants presented the units in the order they were intended to be taught. As each unit was presented, a facilitator recorded on newsprint the key ideas about geometry that the unit addressed. At the end of the presentations, the teachers had a detailed list of the geometry content that each curriculum covered. The group then compared these lists to identify similarities and differences between these two Standards-based curricula and the traditional middle school geometry curriculum. Many teachers were amazed at the richness of the lists describing the new curricula when compared with the traditional middle school math curriculum. They were struck especially by the emphasis in both of the new curricula on three-dimensional geometry and spatial visualization, topics they rarely covered but that were highlighted in the geometry essays they had read. On the other hand, they were puzzled by the presence of some new topics, such as Euler’s formula and graph theory in the Mathematics in Context curriculum. The facilitators then suggested they seek a mathematician’s help to examine further the relative importance of the topics on the lists. The facilitators met independently with Dr. Sanford Segal, a research mathematician on the faculty at the University of Rochester, to share the group’s lists and ask whether he felt comfortable commenting on the mathematical significance of the topics listed. They also shared some information about the group’s background and goals to help him prepare his contribution. Dr. Segal then joined the group for a 2-hour session in which he presented his comments on the relative importance of items on the lists from a mathematical stand-point, and then he answered questions. His presentation and the follow-up discussion further confirmed the critical role of spatial visualization in mathematics, and hence the importance of developing this skill in middle school through appropriate learning experiences. On the other hand, Dr. Segal’s personal position on the relative importance of graph theory and transformation geometry challenged the need to introduce these topics at the middle school level. Overall, all participants, facilitators included, emerged from this inquiry with a much deeper understanding of what the “big ideas” in geometry are and a greater appreciation for the complexity of making good choices about mathematics content at any grade level. Continued