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A Three-Phase Lesson Format 49

● Provides formative assessment data. As students discuss ideas, draw pictures or use manipulatives, defend their solu-tions and evaluate those of others, and write reports or explanations, they provide the teacher with a steady stream of valuable information. These products provide rich evi-dence of how students are solving problems, what miscon-ceptions they might have, and how they are connecting and applying new concepts. With a better understanding of what students know, a teacher can plan more effectively and accommodate each student’s learning needs. ● Is a lot of fun! Students enjoy the creative process of problem solving, searching for patterns, and showing how they figured something out. Teachers find it exciting to see the surprising and inventive ways students think. Teachers know more about their students and appreciate the diversity within their classrooms when they focus on problem solving.

When students have confidence, show perseverance, and enjoy mathematics, it makes sense that they will achieve at a higher level and want to continue learning about math-ematics—opening many doors to them in the future. In the following section, a three-phase lesson format is explained. This lesson structure engages students in learning through problem solving.

A Three-Phase Lesson FormatIn a non-problem-based lesson, teachers typically spend a small portion of a lesson explaining or reviewing an idea and then go into “production mode,” where students wade through a set of similar exercises. When this explain-then-practice pattern is used, students are conditioned to wait for the teacher to tell them how to do something, rather than try to apply their own knowledge. The mathematical proficiencies described in Chapter 2, in particular adaptive reasoning, strategic competence, and productive disposi-tion, are not developed in such a lesson; rather students are imitating what the teacher is modeling and replicating it. After a teacher explanation, teachers find themselves going from student to student to reteach the lesson, because it didn’t meet the students where they were or engage students.

In contrast, teaching through problem solving, also called problem-based teaching, does start where the stu-dents are, engage students in the mathematics, and involve students in justifying their thinking. A problem-based lesson is often taught in three phases—before, dur-ing, and after (see Figure 3.7). The lesson may take a full day or even longer. Each phase of the lesson has a specific goal. How you attend to these goals may vary depending on the class, the problem itself, and the purpose of the lesson.

The Before Phase of a LessonThere are three related agendas for the before phase of a lesson:

1. Activate prior knowledge. This means to begin by pulling in what students’ have previously learned, as well as connect to their personal experiences.

2. Be sure the problem is understood. This does not mean to explain how to solve it, just to be sure the task at hand is clear.

3. Establish clear expectations. This includes both how they will be working (individually, in pairs or small groups) and what product you expect to demonstrate their understanding of the problem.

These before phase agendas need not be addressed in the order listed. For example, for some lessons you will do a short activity to activate students’ prior knowledge for the problem and then present the problem and clarify expecta-tions. Other lessons may begin with understanding the problem and then having students brainstorm their own experiences related to the topic of the problem.

Teacher Actions in the Before PhaseWhat you do in the before portion of a lesson will vary with the task. For example, if your students are used to solving story problems and know they are expected to use words, pictures, and numbers to explain their solutions in writing, all that may be required is to read through the problem with them and be sure all understand it. On the other hand,

FIGURE 3.7 Teaching through problem solving lends itself to a three-phase structure for lessons.

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How does the teacher activate prior knowledge and connect the task to prior student experiences?

How does the teacher ensure that the students understand the problem, and how does she decide it is time to move on to exploring the task?

Are expectations clear? How so?





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4. Provide worthwhile extensions.

How does the teacher let go?

How does she ensure that students are working independently in groups or individually?

What does she do to monitor and assess students’ thinking?



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How does the teacher listen actively without evaluation?

How does he promote a community of mathematical thinkers?

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Divide the circle according to what seems reasonable to you—in terms of degrees—for the sector of time allowed for each of the three phases.

Explain your thinking. Provide your rationale regarding time allotments.

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Sampler - assets.pearsonglobalschools.comassets.pearsonglobalschools.com/asset_mgr/current/...lesson. The Before Phase of a Lesson There are three related agendas for the before phase

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