Process Standard

The goals of the NCTM's problem–solving process standard are that "in grades 9–12, the mathematics curriculum should include the refinement and extension of methods of mathematical problem solving so that all students can—

• use, with increasing confidence, problem–solving approaches to investigate and understand mathematical content;
• apply integrated mathematical problem–solving strategies to solve problems from within and outside mathematics;
• recognize and formulate problems from situations within and outside mathematics;
• apply the process of mathematical modeling to real–world problem situations."
  (NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 137)

Video Overview

Excerpts from ten classrooms show students learning mathematics through problem solving. Teachers use multilayered tasks to engage students in making conjectures, constructing meaning, and developing strategies. Students use paper and pencil, graph paper, graphing calculators, and models in their different problem–solving approaches. Specifically, students work individually and in groups:

• discussing enveloping functions
• considering different approaches to find a derivative
• applying experience solving a trigonometry problem to solve a more complex problem
• investigating patterns to develop a formula
• using multiple methods to solve two equations
• using scientific notation to compare the mass of two objects
• investigating properties of parallelograms
• collecting and graphing data from student-created models
• determining probabilities with tree diagrams and matrices
• solving a problem using multiple representations

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A Pre–Viewing Exploration for Teacher Workshops

Predicting Optimum Heart Rate from One's Age

Objective: To participate in and discuss problem solving using multiple representations of data.
Materials: Graph paper for each teacher.

Activity
Solve this problem individually.
Some people measure their heart rate at different stages of exercise to see if they have reached their optimum level. Use the data below to investigate the relationship between optimum heart rate and one's age.

AGE	OPTIMUM HEART RATE
31	117
42	107
24	122
61	99
55	104
15	127

1. Describe the data. Do you see a pattern? If so, describe the pattern.
2. What is the rate of change of the heart rate per year?
3. What other ways could this data be represented?
4. Use the graph or tabular data to determine a function that represents the data.

Exploring Content

1. Discuss how students could use the different representations of the data to investigate different elements of the problem.
2. Is it possible for students to solve the problem by using any of the representations and then finding the other representations? Explain.

Exploring Teaching Issues

1. Why would exploring multiple representations of data be important for students?
2. How could you promote students' ability to shift from one representation to another?

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Exploration Answers

Activity

1. Answers may vary. One possible answer is that the heart rate decreases with age.
2. The rate is not constant. It can be approximated. A good estimate is –5/8.
3. A graph or a function.
4. Answers may vary. One possible answer is: using (31,117) and (15,127), the slope = –5/8; using (24,122), the f(x) or y–intercept = 137. Function f(x) = (–5/8)x + 137.

Exploring Content

1. The tabular data makes noting differences between data easier for students. The graph makes determining the positive or negative relationship between age and heart rate easier. The function makes predicting rates for specific ages easier.
2. Yes. Students will have the same heart rate and age data, but their functions may vary because it is possible to find similar but different functions that closely reflect the same data.

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Topics for Discussion

Goals and Criteria

• What do you think are the most important skills for students to develop using problem solving?
• How can complexity, real–world applications, and student discovery be incorporated into your lessons? How can each of these lesson elements help students develop a deeper understanding of mathematical concepts?
• Is there usually an immediate application for mathematical concepts? Explain. If not, suggest how students could apply their learning from a mathematical problem that does not have an immediate application to future problems.
• What are the advantages and disadvantages of solving problems in groups? Cite examples from the video and your classroom.

Methods

• What are the advantages and disadvantages of considering multiple problem–solving approaches?
• Discuss how modeling real–world situations helps students understand and solve complex problems.
• How have you or might you use graphing calculators to encourage students' problem–solving skills?
• How could you use problem solving as an instructional method? A lesson goal?

Assessment

• Discuss when teachers in the video could assess students.
• Discuss assessment issues for cooperative group work in the video.
• How have you assessed students when observing them solving problems? How have you used the observations to make instructional decisions?

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