For example, when someone identified the think/pair/share technique for the “fish” activity, a teacher pointed out how helpful it had been for her to work individually on the task first. Others corroborated this observation, noting the value of getting personally engaged in a task before interacting with others. In contrast, one participant expressed his relief at knowing that this individual stage would last only a few minutes, since he initially believed he would never be able to compute the area of the “fish” alone. This discordant opinion invited some considerations about differences in individuals’ preferences and learning styles. Other people then commented on the power of the whole group discussion and how it had helped them go well beyond what they had achieved working with just one partner. The group agreed on the value of being able to explain one’s strategies and solutions to another person first, and all the participants felt that this stage had been beneficial not only for gathering courage to report their ideas to the whole group but also for clarifying and expanding ideas by talking with a partner.
This reflective session also enabled the participants to recognize and discuss the role of less evident yet equally important pedagogical decisions, such as starting the unit with the complex, open-ended task of finding the area of the “fish.” Participants noted the marked contrast between this decision and the traditional practice of assigning complex problems only after students have learned specific procedures that are presumed to be prerequisites for solving problems efficiently. This insight led to discussing the different assumptions about learning that distinguish constructivist/inquiry-based mathematics from traditional practices grounded in behaviorist learning theories.
Illustration 2: Working alongside mathematicians in a real-life setting
We adapted the illustration in this section from the Growth in Education through a Mathematical Mentorship Alliance Project (GEMMA) (ENC, 2000; Farrell, 1994).
As part of the GEMMA project, teachers participated in an eight-week summer internship in local businesses heavily involved in the use of mathematics and science, such as consumer marketing companies, scientific consulting firms, and automobile and other manufacturing companies. Each teacher was assigned a mentor in a company, and they worked together solving authentic problems that confronted the business. These projects included analyzing market surveys, testing fan blades for engines, researching the operation of a microwave that was being installed on a factory production line, determining and graphically displaying the relationship among molecules in a new material, and creating a computerized model of transportation systems. The companies expected teachers to be fully contributing members of the problem-solving team. In doing so, teachers had to learn about current industry practices for solving problems and to identify where and how mathematics was used.
During the internship, teachers attended a series of seminars where they discussed what they were doing, what mathematical applications they were learning, and what new instructional practices they were generating from their experiences with industry. By the end of the summer internship, teachers were expected to have designed some applied mathematical problems that they would pilot in their own classrooms. The project goal was to create a booklet of such “applications problems” to share with the other mathematics teachers.
The outcomes far surpassed the GEMMA project directors’ expectations. They hoped the teachers would discover applications for the kind of mathematics they taught, which they did. However, the directors found that the internship experiences also introduced and/or reinforced many of the current reforms in pedagogy. In their final papers, for example, teachers wrote that they teach with a greater purpose and that they feel a need to integrate mathematics and science. They also wished to create collaborative learning environments in their classrooms and to give students much more responsibility for their learning.
Main elements and variations
The previous illustrations highlight several of the elements we believe need to be a part of any high-quality experience-as-learners.
Some of these elements have to do with the nature of the mathematical learning experience for the teachers. In order to be effective, we believe that these mathematical experiences need to accomplish the following:
Challenge the participants intellectually, regardless of their mathematical backgrounds or the grade levels they teach. Only under these conditions can teachers be genuine learners and benefit fully from participating in these instructional experiences.
Be mathematically sound and address key concepts. In order to strengthen teachers’ knowledge of mathematics and invite them to rethink the goal of school mathematics, these experiences must offer opportunities to learn worthwhile and significant mathematics.
Allow for mathematical reflection and discussion in addition to mathematical problem-solving. Doing so is essential to ensure that teachers revise and enhance their current understanding of key mathematical concepts and procedures, and do not just engage in “activities for activity sake.”
Model non-traditional ways of learning and/or teaching mathematics. Participants must experience alternatives to traditional school mathematics in order to appreciate their potential for student learning.
Another set of characterizing elements involves the reflections that follow the mathematical learning experience itself. As both illustrations show, these reflections are critical to the success of any experience-as-learners in initiating teachers’ rethinking of their views of mathematics, teaching and learning. The following list captures the characteristics of optimal reflective activities:
Reflective activities should occur after the learning experience is over, not during it. In this way, participants may find it easier to abandon their teacher roles as they engage in the mathematical learning experience and be genuine learners in it.
There should be opportunities for individual reflections as well as group discussion. Participants need to make personal sense of the experience as well as hear other people’s insights and perspectives.
Despite these common characteristics, successful experiences-as-learners can also differ in substantial ways, as reflected by our two illustrations. Important variations can occur along any of the following dimensions:
Duration and complexity of the mathematical experience. Both of our illustrations included intense mathematical experiences – a 7-hour inquiry on area in Illustration 1, and a summer-long project in Illustration 2. In contrast, there are examples in the literature of shorter mathematical experiences, involving the solution of a problem or other isolated mathematical tasks.
Diversity of participants. Participants may be a rather homogeneous groups of mathematics teachers teaching at the same level of schooling or they may include mathematics specialists and non-mathematics specialists at different grade levels (as it was the case in Illustration 1).
Facilitator’s role. The facilitator may purposefully model some innovative teaching practices (as in the inquiry on area reported in Illustration 1) or simply work alongside teachers in a joint task (as expert mathematicians did in Illustration 2).
Scope and structure of follow-up reflections. Reflective activities may be open-ended or focused explicitly on specific aspects of the learning experience. For example, facilitators may ask teachers to reflect on the teaching practices modeled, the reactions of different learners to the experience, or their views of mathematics. Leaders may also elicit individual reflections in different ways, such as asking teachers to respond in writing to written prompts, to write in journals or to brainstorm ideas with a partner before having teachers share and discuss them.
Experiences-as-learners can also take place in a variety of professional development formats. They can be part of an after-school workshop, a summer institute, a university course, an on-site study group, or even an immersion situation in which teachers become mathematics-learners and problem-solvers alongside mathematicians in real-world settings. In many cases, part of the participants’ mathematical experience may require projects or other assignments that are undertaken by each teacher independently.
Experiences-as-learners can be conducted by facilitators with a variety of backgrounds. Although mathematicians might seem to be ideal facilitators for this type of professional development, they may need to work collaboratively with experienced teachers or mathematics educators who can complement their subject matter expertise with experience in instructional innovation. Conversely, experienced teachers playing the facilitator’s role may benefit from coaching on the differences between teaching adults and K-12 students and from readings about the “big mathematical ideas” that form the core of any experience as learners. Regardless of their affiliation, facilitators leading experiences-as-learners need both a strong mathematical background and the ability to model innovative teaching practices.
Teacher learning needs addressed
Experiences-as-learners have the potential to address many of the teacher learning needs we identified in Chapter 1, yet the extent to which they do so depends on how the activity is implemented. In this section, we discuss what specific variations of experiences-as-learners can best help meet the needs of teachers who are interested in pursuing school mathematics reform and how.
Developing a vision and commitment to school mathematics reform. Mathematical experiences-as-learners can be powerful to help teachers understand what school mathematics reform really mean and why it should be promoted. When a skilled mathematics teacher educator designs the activities to demonstrate the kind of mathematics instruction promoted by the reform movement, teachers can appreciate the vast difference between traditional and constructivist-based practices. For example, the inquiry on area reported in Illustration 1 allowed the teachers themselves to learn about a traditional mathematical topic by focusing on big mathematical ideas, solving problems through inquiry and constructing knowledge with others. It also illustrated concretely the new roles that teachers and students must play when a constructivist view of learning informs mathematics instruction.
The personal success and enjoyment that participants experience in novel mathematical activities are powerful motivators toward instructional innovation. Committed teachers want their students to experience the same positive emotions about mathematics. We have observed this happen, especially with teachers who have bad memories of being students in traditional mathematics classes. Even teachers who were successful students in traditional settings, however, can experience vicariously their colleagues’ delight when they share such thoughts as “I never knew I could do mathematics! If only I had been taught this way!” This kind of response is especially common when the group includes non-mathematics specialists.