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Multiple Solutions: More Paths to an End or More Opportunities to Learn Mathematics

Using a Journal Article as a Professional Development Experience
Problem Solving 

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Title:           Multiple Solutions: More Paths to an End or More Opportunities to Learn Mathematics
Author:       Rose Mary Zbiek and Jeanne Shimizu
Journal:      Mathematics TeacherIssue:         November 2005, Volume 99, Issue 4, pp 279-287

Rationale for Use 

This article exemplifies the following ideas from the Problem Solving Standard:

  • build new mathematical knowledge through problem solving; 
  • solving problems that arise in mathematics and in other contexts.

While the content is geometry, the lesson on selecting good problems based on the opportunities to deepen content knowledge is one that goes across content strands. Coupled with a discussion of the Problem Solving Standard, a rich professional development session around the nature and importance of problem solving within all content is possible for in-service or pre-service secondary teachers.


Procedures/Discussion Questions 

Goal: Participants will compare problem solving strategies for real world problems.

Overview: In order to facilitate the discussion of the article, participants in the professional development session should first engage in the math in the article. However, due to the complexity of the two problems, it's suggested that participants work on the "Skin Problem" on page 279 for about 15-20 minutes, participate in a facilitated discussion and then work on the "Melon Problem" on page 283 for about the same amount of time and participate in a facilitated discussion. Let participants know they may not have time to completely finish either of the problems, but that you are interested in the strategies they use.

Post the following questions to consider when doing math in this article with colleagues:

  • Solve the problems for yourself, keeping track of your method in your notes. Why did it make sense to you to solve it this way?
  • What are some of the ways students might solve it? What misconceptions might they bring? What will justifications sound like?
  • What are the mathematical goals? What is the trajectory of the mathematics (Where do these problems fit in the scope and sequence)?

The Skin Problem:

  1. Post the questions and give participants the "Skin Problem" (figure 1 on page 279). Allow some private think time for each person to consider the problem individually; next invite them to share their thinking with a partner.  Ask one person to share while the other one listens and then switch roles before they begin discussion and/or pursue the problem together. Partners then join another pair forming a group of four. Each group of four will continue to discuss the problem and the questions. Each group should produce a chart detailing all of the strategies their group has pursued and as much of the solution as they have had time to consider. You will need to limit the time depending on your professional development session length.
  2. Facilitate a “walk-about” so each group can consider the solutions of others. Groups take a note pad with them and place notes where they have questions about another groups work. Ask groups to share solutions that are unique as well as address those places where questions arose. All groups need not share their solutions and arriving at a complete solution to the problem is not necessary.
  3. Reproduce the three pieces of student work from the article (figures 2, 3, and 4) - one packet per pair of participants. Allow time to review the student work privately, making notes about student understanding and misconceptions and pressing for evidence of their inferences. Share in their group of four using a go-round protocol that assures each person has an opportunity to share their thinking about the work. Ask them to look for similarities and differences among the student solutions and those of other participants from the walk-about.
  4. Facilitate a brief discussion that includes: (record comments on chart paper)
    • What mathematical goals does this problem address?
    • What is the core mathematics – the mathematics that is essential to the problem?
    • What is the potential mathematics – the mathematics that might come out of the discussion of the problem?
    • What are the characteristics of the different solutions to this problem (moving toward the emphasis of the article that the solution characteristics are all similar and therefore limit the depth of mathematics in the discussion)?
  5. Read the article from page 279 through page 282 stopping at Example 2.  Add any ideas to the previous discussion of mathematical goals and solution characteristics. Ask, "What one idea stood out for you in this part of the reading and why?" Share with a partner. Ask participants at random to share what their partner said.

The Melon Problem:

  1. Repeat the procedure for the "Melon Problem" (figure 5 page 283). Post the questions for doing math with colleagues. Allow private think time, partner share, group discussion and presentation using charts.
  2. Facilitate a walk-about to allow groups to look for similarities and differences in participant solutions.
  3. Reproduce the student work shown in figure 6 "The Volume Solution", figure 7 "The Surface Area Solution" (figures 8, 9a, and 9b support the Surface Area Solution), and figure 10, "The Weight Solution."
  4. Facilitate a brief discussion focusing on the same questions
    • What is the core mathematics – the mathematics that is essential to the problem?
    • What is the potential mathematics – the mathematics that might come out of the discussion of the problem?
    • What are the characteristics of the different solutions to this problem?
  5. Read the remainder of the article from page 282 Example 2 to the end. Add any ideas to the previous discussion of mathematical goals and solution characteristics. Ask, "What one idea stood out for you in this part of the reading and why?" Share with a partner. Ask participants at random to share what their partner said.

Summary:

  1. Distribute copies of the Problem Solving Standard for Grades 9-12 from the NCTM PSSM (available on the NCTM website at http://standards.nctm.org  ).

    Note: This reading might be assigned for participants to read between segment 2 and 3 of the professional development session rather than spend time reading during this segment. If the reading is assigned between segments, provide participants about 5 minutes to review their notes on the reading.
  2. Facilitate a discussion of issues around teaching problem solving. You may want to use the following questions or questions from participants that became apparent during the previous work on the article.
    • What are the issues that surround problem solving?
    • How does the use of the two problems deepen student conceptual and procedural understanding of mathematics? How are the problems alike and how are they different?
    • The article says, “…choosing challenging problems to facilitate student learning of new mathematics is not an easy task.” What are some considerations that need to be addressed around task selection?
    • Though we have only worked with problem solving in the content area of geometry, how do these strategies apply in other content areas?

Next Steps 

Have participants reflect on lessons they currently use (or would like to use) that include problem solving. At the next meeting, have participants bring a lesson they are planning to use and modify, if needed, the lesson according to the considerations around task selection.

Ask participants to teach the lesson, select 3 samples of student work and bring those to the next session. At the next session, use a Looking at Student Work protocol to determine student understanding and misconceptions around problem solving. Use that information to plan the next problem solving lesson.

Connections to Other NCTM Publications 

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