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What Are the Needs of Teachers Who Are Engaged in
School Mathematics Reform?

Knowing the needs that teachers engaging in school mathematics reform today are likely to experience is critical to help providers set goals for a specific professional development initiative and to evaluate the potential contributions of any given program. To highlight the extensive changes in teaching practices called for by the current reform and the challenges that these changes are likely to present teachers, we will first present an image of a reform-oriented mathematics classroom. The vignette that follows describes an actual classroom experience (see Callard, 2001, for a more detailed account of this experience). We will refer to this vignette throughout the chapter to illustrate key points about the learning needs of teachers engaged in school mathematics reform as identified in the research on teacher development and reform.

An image of a reform-oriented mathematics class

The instructional unit captured in this vignette was developed by Mrs. Callard, the classroom teacher, based on a set of instructional materials created to support an illustrative inquiry unit on area for middle school students (Borasi, 1994a). While these illustrative instructional materials provided an overall design for the unit, Mrs. Callard had to make a series of pedagogical decisions to adapt the unit to her own goals and to the constraints of her 8th grade mathematics class. For example, since she knew that her 6th grade colleagues had already worked with students on developing the concept of area and had introduced area formulas for rectangles and triangles, she decided to focus the unit on developing area formulas, drawing from the second part of the instructional resource materials.

Mrs. Callard began her four-week unit on developing area formulas with an activity that would invite students to review what they already knew about area with the goal of building on their prior knowledge and also identifying gaps and misconceptions in their understanding. The activity required students to find the area of a complex figure drawn on graph paper – the “fish” reproduced in Figure 1.

Figure 1 The fish

this is a picture of a geometric fish on graph paper

 

Mrs. Callard handed out a copy of this figure to each student, instructing the class to work on this task individually for a few minutes and then to share their preliminary results with a partner prior to a whole class discussion (so as to invite collaboration and scaffold their sharing in front of a large group). To further support the student’s mathematical thinking and suggest alternative approaches, the teacher also made available a variety of tools, such as rulers, compasses, scissors, calculators, string, and tape, and even additional copies of the “fish.” As the students worked, the teacher moved around the class for about 15 minutes observing, encouraging and supporting the students.

When most pairs reached solutions that satisfied them, the teacher asked volunteers to show their solution/strategies to the rest of the class. This sharing enabled students to appreciate the variety of approaches that could be used to solve this problem. These included strategies such as breaking the fish into rectangles and triangles and then adding the areas of these simpler figures, “boxing” the fish and then taking away the extra pieces, or simply counting the whole squares in the fish and approximating the partial ones. As each pair shared its solution, the teacher asked the students to articulate the strategy they used, and she recorded it on newsprint, so as to make each strategy explicit and to enable the class to later examine the strengths and weaknesses of the alternative strategies for computing the area of a complex figure. In this discussion, the teacher also pointed out the key role that the area formulas for rectangle and triangle played in several of the strategies identified.

One of the teacher’s main goals was to have her students appreciate that area formulas are not mysterious things to be memorized, but rather they are a short-hand notation that summarizes an effective strategy for computing the area of figures with certain common characteristics. This idea was further highlighted in the next activity, where students had to compute the area of different kinds of “kites” (see Figure 2).

 

Figure 2 "Kites"

this is a picture of 5 various shaped kites

 

Under the teacher’s direction, the entire class attempted to develop an area formula that would work for all kites. Different students, by focusing on different characteristics of the kites in Figure 2, proposed the various procedures and formulas summarized in Figure 3.

 

Figure 3

Alternative area formulas for “kites” and their graphical explanation

this is a picture of 4 kites with seperate area formulas

 

As the class critically examined these potential solutions, the teacher carefully facilitated the discussion, making sure that nobody was left out and everybody’s contribution was seriously considered. She also occasionally asked questions to highlight important mathematical points, noting, for example, that students developed different yet equally acceptable area formulas depending on what they chose to measure and how they named their variables.

Mrs. Callard also took advantage of the controversy that erupted when one student observed that formula D “may not work for all kites.” Instead of resolving the student’s concern, she asked the class how they could decide whether something was a kite or not. Since a “kite” is not one of the standard figures usually defined in mathematics textbooks, this apparently simple question led the class to grapple with the challenging task of creating their own definition for kite and then defending it! Eventually, the class voted to define a kite as “a quadrilateral with perpendicular diagonals.” Based on this definition, the class concluded that formulas A and B were acceptable area formulas for kites, while formulas C and D worked only for special kinds of kites. This activity enabled the students to experience first-hand the power and excitement of “creating” mathematical formulas and definitions and also provided them with a deeper understanding of these fundamental mathematical concepts. To help the students reflect on and better appreciate the significance of what they had learned, the teacher then asked students to write answers to a few questions about mathematical definitions.

To help students synthesize and generalize what they had learned so far, Mrs. Callard led the class through a careful review of the steps they had followed over several class periods to come up with an area formula for a kite. She recorded each step on newsprint and later distributed this list (reproduced Figure 4) to the students as a reference for developing other area formulas in the future.

 

Figure 4

Key steps in developing an area formula

1. We started with examples of the figure and computed their area.
2. Shared strategies, ideas – discussed.
3. We checked to see if one strategy would work on all of the figures.
4. Tested the strategy.
5. Wrote a formula defining variables carefully.

6. Explained why the formula works.

As a culminating activity for the unit and as a form of performance assessment, Mrs. Callard asked students to create an area formula for a given “star.” She carefully assigned partners for this project, taking into account students’ different mathematical strengths, weaknesses and unique learning styles. Because students worked on part of the project in class, the teacher also provided more scaffolding for some pairs of students as needed. In the students’ poster presentations at the end of the project, most of the pairs showed remarkable mathematical thinking abilities, and they communicated the results of their work effectively (see Figure 5 for an example).

To gather feedback about the learning achieved by individual students, the teacher also assigned two traditional take-home tests, one on developing area formulas and the other on applying known area formulas in practical situations. Students’ grades for this unit also took into account their performance on homework and in-class assignments, so as to provide a comprehensive assessment of the students’ learning based on data gathered from a variety of complementary tools.

Figure 5

Example of a “star” poster

this is a picture of a 10 pointed geometric star

 

 

 

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