Engaging in Mathematical Experiences-as-Learners
In the type of professional development experience we describe in this chapter, teachers engage as genuine learners in mathematical learning experiences. While the nature, content and duration of these learning experiences may vary considerably, they all model effective instructional and/or learning practices promoted by school mathematics reform. Reflection is a critical part of these activities because it helps teachers analyze the experiences in light of their own beliefs and practices.
Theoretical rationale and empirical support
The benefits of teachers experiencing mathematics as learners go well beyond the important, rather obvious one, that teachers learn more mathematics. Research shows that teachers’ beliefs about mathematics and about teaching mathematics are formed mostly as a result of having been students in traditional mathematics classrooms (Thompson, 1992). Since traditional mathematics is informed by pedagogical beliefs and practices that are radically different from those promoted by the current reform efforts, many teacher educators argue that before classroom teachers can change their beliefs, they must have personal experience of alternative pedagogical approaches (Brown, 1982; Schifter & Fosnot, 1993).
Further support for the value of experiences-as-learners for teachers comes from research on the learning of complex tasks. As we discussed in Chapter 1, Collins and his colleagues (1989) identified modeling as the first of three phases in the process of learning a complex task. When the complex task is learning a novel approach to teaching mathematics, we believe that facilitated “experiences as learners” activities offer an especially effective vehicle for such modeling. First, teachers observe an expert mathematics teacher educator teach mathematics in a non-traditional way. Second, because teachers participate in this instructional experience as learners themselves, they are in a unique position to examine how their students may feel about the new approach. As a result, they are in a better position to evaluate its potential advantages and drawbacks.
Simon’s “learning cycles” model of teacher learning, which we described in Chapter 3, clarifies further the multiple roles that this type of activity can play in a professional development program. In Simon’s first phase of the learning cycle, teachers must participate in situations that engage them actively as learners and that evoke cognitive dissonance. In this way, they are stimulated to construct new meanings. In the second phase, through sharing and discussing these constructions with a group, teachers come to consensus and make generalizations. This model suggests to us that good mathematical learning experiences for teachers need to invite active engagement, provoke cognitive dissonance, and encourage social as well as individual construction of meaning. Simon’s model further claims that what is learned in one cycle can be used to stimulate another cycle of learning. We suggest that reflecting on these mathematical learning experiences can become the catalyst for teachers to begin yet another “learning cycle,” this time focusing on the nature of mathematics as a discipline, how people learn and what can best support such learning.
Research corroborates the benefits of teachers experiencing mathematics as learners articulated above. This type of professional development experience plays a central role in several professional development programs with documented success (Simon & Schifter, 1991; Schifter & Fosnot, 1993; Borasi, Fonzi, Smith & Rose, 1999). A systematic study conducted by Simon and Schifter (1991) in the context of one of these programs has specifically shown changes in teachers’ beliefs and practices toward a more constructivist approach to teaching mathematics. Since mathematical experiences-as-learners were not the only kind of professional development experience employed in these professional development programs, the results may not be considered conclusive. However, case studies and anecdotal evidence (Schifter & Fosnot, 1993; Borasi, Fonzi, Smith, & Rose, 1999) further confirm that experiences-as-learners were a critical element in changing the beliefs and practices of several participants in these programs.
Illustration 1: A facilitated inquiry on area for teachers
We derive the illustration in this section from one of the Introductory Summer Institutes in the Making Mathematics Reform a Reality in Middle School (MMRR) project described in Chapter 2. This experience-as-learners was designed to help teachers analyze how an inquiry approach to teaching mathematics involves a radical rethinking of both mathematical content and pedagogical practices. It was also intended to introduce teachers to an “illustrative inquiry unit” they might be teaching in their own classes later -- a unit on area formulas designed for middle school students (the same unit featured in the classroom vignette included in Chapter 1). This experience-as-learners thus engaged participants in an inquiry similar to one they might be using with students.
The participants in the implementation described here included elementary teachers, secondary mathematics teachers, and special education teachers at the middle school level. It took about seven hours over three consecutive days to complete.
The instructor began by asking participants to take off their “teachers’ hats” and become learners in a series of activities about the concept of area. The instructor warned participants that this was not going to be a simulation in which they should pretend to be elementary or secondary students. Rather, the content would challenge everyone at their own level of expertise, so they should participate as genuine learners and use all they knew to deal with the tasks presented to them.
The first task was to find the area of a “fish” similar to the one middle school students worked with in the classroom vignette (see Figure 8).
Each teacher worked on this task first individually, then with a partner. The pairs then shared their results with the whole class. Most secondary mathematics teachers broke the fish into simpler figures, computed their areas using formulas they knew, and then added up those areas. A special education teacher had used a similar approach, yet made more efficient by using the symmetry of the fish and folding the figure in half. An elementary teacher showed instead how she had “boxed” the fish and then subtracted the area of the “extra pieces.” Another elementary teacher “admitted” that she had simply “counted the squares,” matching partial squares as best as she could to form whole squares.
Everybody was surprised by the variety of these approaches and by the fact that non-mathematics specialists had proposed the most creative solutions. A lively discussion surrounded this sharing, and participants came to appreciate the value of alternative strategies for finding the area of complex figures and the role that area formulas played in some of these strategies.
Next, the instructor challenged the participants to develop some area formulas on their own. First, she modeled this novel process by creating, together with the participants, an area formula for “diamonds.” Later in the activity, she defined a diamond as “a quadrilateral with perpendicular diagonals.” This task, and the reflection that followed it, highlighted important elements in the process of developing area formulas.
Participants then worked independently in small groups to develop area formulas for “regular” stars. The next day, they shared the area formulas they had created and explained the process they had used to derive them. Once again, everyone was amazed by the variety of area formulas thus created and by the creativity shown by several class members who had little mathematical background.
To help participants further appreciate the complexity of the mathematical concept of area, the instructor asked them to grapple with some thought-provoking questions for homework:
Why are squares chosen as the “unit” to measure areas? Could other shapes be used? Why or why not?
How do we choose the “size” of the squares to be used as units? Can this choice affect the value of the area of a given figure?
Area formulas essentially enable us to compute the area of a two-dimensional figure by taking only linear measures (i.e., the length of the height, base, radius, etc.). How is this possible? Does this mean that you can measure area with a ruler?
Can we ever find the area of a curved figure EXACTLY? For example, does A=π r2 give us the exact value for the area of a circle or just a good approximation?
The difficulty they encountered responding to these apparently simple questions astounded the teachers. In all of their years as students of mathematics, not even the secondary mathematics teachers had been asked to think about questions like these, because learning about area had been reduced to memorizing and applying area formulas.
In the next session, the group discussed these questions in depth. At the end of this discussion, the facilitator handed out a mathematical essay on area as a follow-up reading assignment, to both validate some of the conclusions the group had reached and expand them further.
A number of follow-up activities encouraged the participants to reflect on this unusual learning experience and to analyze it from different perspectives. Participants listed “what they had learned” about area from this experience. This list was quite detailed and complex. Interestingly, although the teachers included a few technical facts, such as learning area formulas for diamonds and stars, they primarily identified elements related to mathematical processes and the nature of mathematics. For example, they highlighted the importance of learning to develop area formulas on their own, of understanding the role played by the choice of unit in measuring area, and of recognizing that mathematical problems could have more than one acceptable solution. Several participants also mentioned gaining increased confidence as learners of mathematics as a result of this experience.
The facilitator then began a discussion on the instructional goals that should inform a unit on area for their students. Not surprisingly, the group established quite different goals for their students than it is traditionally the case, such as: Students should understand the concept of area (how it is useful, what is actually measured); students should understand the concept of scale; students should discover that there is more than one formula for a given figure; students should be able to derive formulas.
A day later, as a culminating experience for the whole Summer Institute, the facilitator asked participants to reflect on this experience-as-learners on area and another experience-as-learners on tessellations they had engaged in a few days earlier. This time, the participants were asked to identify the teaching practices that the institute instructors had modeled in these experiences. As individuals shared their reflections with the whole group, the facilitator probed responses to elicit the rationale for, and potential effects of, these practices in their own classroom instruction.