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Integrating Content to Create Problem-Solving Opportunities

Using a Journal Article as a Professional Development Experience
Problem Solving 

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Title:           Integrating Content to Create Problem-Solving Opportunities
Author:       Darrin Beigie
Journal:      Mathematics Teaching in the Middle SchoolIssue:         February 2008, Volume13, Issue 6, pp. 352-360

Rationale for Use
Problem Solving is one of the Process Standards in Principles and Standards for School Mathematics, NCTM 2000.  The Grades 6-8 Problem Solving Standard (page 256) states that “Problem solving in grades 6-8 should promote mathematical learning. Students can learn about, and deepen their understanding of mathematical concepts by working through carefully selected problems that allow applications of mathematics to other contexts.”  The author in this article describes how this idea can become a reality in the classroom.

This article describes how traditional exercises can be integrated to form challenging problem–solving situations, to foster higher–order thinking skills and a better understanding and mastery of mathematical content.

Procedure
Session One: 

  1. Ask participants to read the quote from Principles and Standards for School Mathematics at the beginning of the article: “Problem Solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned (NCTM,2000, p.182) .”
    • Working in either large or small groups, have participants, based on their experiences, share:
      • What it means to say problem-solving is a process;
      • That problem-solving should provide a context in which concepts and skills are learned; and
      • The challenges presented by the message in the above quote for classroom instruction.
  2. Assign Problem 1, Leaning Ladder Problem, Figure 1, page 354 of the article, for participants to work on together (groups of 2 to 4).
    • Share with the whole group solutions to the problem along with a discussion of the following:
      • The different strategies they used, even those that did not lead directly to a solution.
      • The mathematical content in this problem.
      • The challenges this problem may present to students.
      • Why did the author considered this a problem-solving task?  How does it meet the description of problem-solving stated in the quote discussed in #1?
      • Why did the author say that this problem is the combination of two mathematical procedures, and hence the result is a non-routine problem for most student?
  3. Read the section titled: Integrating Content to Create a Problem Solving Opportunity on pages 354 – 359, then discuss the following (large or small groups):
    • Choose a statement from this Section that you find interesting. Share why you find it so.
    • Examine the examples of student’s work included in this Section, Figures 2 and 3:
      • How do these solutions compare to what you did?
      • Based on the students’ written work, how would you compare these two students’ problem-solving abilities?  Relate it to the statement in the article from NCTM (2000), “ students’ problem-solving failures are often due not to a lack of mathematical knowledge but to the ineffective use of what they do know.” P.54
      • What questions could you ask the student whose work is included in figure 3, page 355 of the article, to help him or her sort through their ideas to possibly lead to a more successful solution.  Is there evidence that this student could integrate content to solve a problem?
    • The author says that this article describes, “one seventh-grade teachers’ classrooms efforts to integrate traditional exercises from different content areas to form more robust questions that provide genuine problem-solving opportunities for students” P. 352. Share your thoughts on this statement in reference to the problem and the section from the article discussed above.

Session Two: 

  1. Assign Problem 4, Pythagorean Triple Problem, Figure 9, page 358.

    Pythagorean Triplets Problem - Give all examples you can determine of three positive integers satisfying the Pythagorean equation: a^2 + b^2 = c^2 

    NOTE: This is an example of an open-ended problem that has infinite solutions.  This problem challenges students to find and realize that some problems have more than one solution, and most importantly, asks them to make a generalization to describe these solutions. In this case the generalization can be described by an algebraic equation. It also challenges students to go the next step and justify this generalization.
    • Ask participants to solve the problem and as they do consider the following:
      • What is the mathematics children use when doing this problem?
      • Why is this task considered problem-solving? Why is it an open-ended problem?
      • Do you think your students would know that an essential ingredient of problem-solving with open-ended solutions is to make and justify a generalization about the solution. Is it clear in this problem?
      • While students are solving this problem, what clarifying questions might you ask students who are struggling with finding a solution or getting started to find a solution?
    • Share different approaches to how participants solved the problem and discuss questions above.
  1. Ask participants about the opportunities in their mathematics program for students to respond to open-ended problems.
  2. Examine one student’s solution to this problem, in Figure 10. Page 358. 
    • What do you notice about the solution?
    • What strategies did the student use?
    • Has the student made an appropriate generalization for the solution? Why or why not?
    • What question(s) might you ask the student to have him or her consider other Pythagorean triplets, such as, 5, 12, 13.

Next Steps 

  1. Ask participants to consider assigning  at least one of the four problems  posed by the author in Figures 11, 12, 13, 14, pages 358 -359 to their students. Then bring to next session student solutions and reflections on the following:
    • Why is that problem, chosen from Problem 5 - 8, considered a problem- solving experience rather than just an exercise, as described by the author, page 352?  How was it a problem-solving experience for your students?
    • What mathematical content is used or explored in more depth in this problem?
    • What strategies did the children use to solve the problem?
    • What challenges did this problem present to the students?
    • Share your role as teacher while students worked on solving this problem. In what way did it present challenges for you?
  2. Provide each participant with a copy of the Problem Solving Standard, Grades 6 -8, Principles and Standards for School Mathematics, NCTM, 2000. Ask them to read the Standard, considering the following questions to direct discussion:
    • What ideas in this Standard do you find particularly interesting? Why?
    • What idea from the Standard has special implications for you when using a problem solving approach to instruction?
    • What idea(s) in this Standard would you like to explore further or know more about?
    • Share an idea in this Standard, you have not used already, but would like to try in your classroom. What support would you need?
    • What are some of the challenges you anticipate in using a problem solving- approach to teaching mathematics, as advocated in the Standard and in the article “Integrating Content to Create Problem-Solving Opportunities”, Darrin Beigie, MTMS, Vol 13 No 6 Feb 2008.
  3. Read the article Bay-Williams, Jennifer M., and Margaret R. Meyer. “Why Not Just Tell Students How to Solve the Problem?” Mathematics Teaching in the Middle School 10 (March 2005):340-41. This article is included in the bibliography of the enhanced article above.

    Suppose at your school you are asked to make a presentation to parents to help them understand why the school is taking a problem solving approach to teaching and learning mathematics.  What ideas from this article could you use in your presentation to convince parents or caregivers that their child is more effectively learning and retaining content, along with developing a confidence that they can solve new, unfamiliar problems?

Connections to other NCTM Publications  

 

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