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Mythmatics

Using a Journal Article as a Professional Development Experience
Problem Solving  

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Title:              Mythmatics
Author:          Larry E. Buschman
Journal:         Teaching Children Mathematics
Issue:            October 2005, Volume 12, Issue 3, pp.136-143.

Rationale 

NCTM’s recommendation to make mathematics more problem-centred has been challenging for a number of reasons. The author of this article suggests that the main reason for this is that several myths have grown up around problem solving. This article deals with eleven of these myths and suggests ways, based on research, they can be overcome. In this session participants will have an opportunity to reflect on these myths and then make suggestions, based on a sharing of experiences with teaching problem  solving, on how to overcome these myths.

Procedure 

Session One 

  • Provide a copy of the article to each participant and ask them to read the quote at the beginning of the article: “Principles and Standards (NCTM 2000) recommends that classroom instruction be more problem-solving centered- children need to be given the opportunity to engage in genuine problem solving by answering questions to which the answer is not apparent or the solution method is not known in advance” (p. 136).
  • Working in either large or small groups, have participants, based on their experiences, share what it means:
    • To recommend that classroom instruction be more problem solving centred;
    • That genuine problem solving involves answering questions to which the answer is not apparent or the solution is not known in advance.
     
  • At the end of the discussion ask participants what they see as the greatest challenges in implementing a problem-solving centered instructional approach.
  • The article addresses eleven myths that are associated with problem solving.
  • Ask participants to share some myths or misconceptions about problem solving that they have met while teaching mathematics to elementary school children.
    • These myths could be recorded on chart paper for future reference.
     
  • Review with participants the headings of the eleven myths identified in the article. Are there any that were not identified above?  Are there any identified above that are not included in the article?
  • Divide the group into four groups. Inform participants that in this session all of the eleven myths identified in the article will not be discussed at this time and proceed to assign selected Myths to the groups for reflection and discussion. This will happen first in the small groups, followed by whole group sharing and discussion.
  • Group A - Myth 2:  Problem Solving is best taught as a separate subject
    • Read the section, then discuss the author’s comments relative to your own instructional program that “problem solving is a way of understanding and doing mathematics” and the two statements from Principles and Standards for School Mathematics (2000) quoted in the section.
    • What are the challenges in using problem solving to help children understand and do mathematics? What ways have you overcome these challenges?
    • What would you like to know about in relation to this myth?
     
  • Group B – Myth 4: Problem Solving is best taught as heuristics and problem solving strategies
    • Read the section, then discuss the following:
      • What problem solving strategies are you and your students most familiar with? What approach does your curriculum take to the teaching of problem solving strategies?
      • The author references the latest research on problem solving which shows that children are natural problem solvers and are quite capable of inventing their own strategies. Share experiences you might have had with children inventing their own problem solving strategies.
      • What would you like to know about in relation to this myth?
       
     
  • Group C – Myth 6: Before young children can do problem solving, they must first learn the “basics”, including computational facts and algorithms
    • Read the section, then reflect together on the following:
      • The author says that “children can learn the basics through problem solving, and in so doing, acquire factual knowledge that is useful and useable”.  Share examples of problem solving experiences where this can happen.
      • The author states that this is a myth that appeals to many adults. Why do you think this is so? What can you do to dispel this myth? Do you think these myths have been transferred to children?
      • What would you like to know about in relation to this myth?
       
     
  • Group D – Myth 9 and 10: Getting the answer is still what matters most when doing problem solving and the process is more important than the answer
    • Read the section, then reflect together on the following:
      • There has been an ongoing discussion among educators about what is more important- the answer or the process. What are your views on the answer-process debate?
      • The author states that problem solving, “is about learning to think mathematically and realizing that problems are not really solved until one understands what one has done and why the actions were appropriate”.  How can you help make this happen in the classroom? What strategies have you used in your classroom to assess whether children understand what they have done?
      • What would you like to know about in relation to these two myths?
       
    • Have each group make a short presentation summarizing their discussion. Then invite the whole group to discuss how they can help overcome these myths and allow children, as the author states (p. 142), to experience the feeling of personal satisfaction and empowerment that comes from solving challenging problems.
    • Invite participants to reflect on how instruction would have to change in their classrooms (or classrooms they are associated with) in order to counteract the myths about problem solving.
     

Next Steps 

  • Invite participants to consider one of the myths discussed and explore the implications of it in their teaching situation.  If possible, bring the group back (or provide an alternative means of sharing), for participants to share their experiences and discuss the myths further.
     

Session Two 

The Problem Solving Standard for grades Prek-2 and grades 3-5 (PSSM, 2000, p 119, 183) gives examples of problems that are likely to prompt children to use particular strategies, allow for the development of certain mathematical ideas, and provide a context for using skills.  Invite participants to discuss these assertions by NCTM based on their childrens’ experiences solving one of the problems below.

Example 1: “I have pennies, dimes, and nickels in my pocket.  If I take three coins out of my pocket, how much money could I have taken?”  (PreK - 2 Problem Solving Standard, NCTM, 2000)

Example 2 : “Show all the rectangular regions you can make using 24 tiles (1-inch square), You need to use all tiles. Count and keep a record of the area and perimeter of each rectangle and then look for and describe any relationships you notice.” (Grades 3 – 5 Problem Solving Standard, NCTM, 2000)

  • If possible have participants come together and share samples of children’s work, records made of children’s conversations when justifying their solutions, and reflections by the teacher alone or in groups about their children’s learning or changes in instructional strategies if doing it again.
  • Ask participants to consider what mathematical ideas are developed when engaged in solving these problems.

Connections to other NCTM Publications 

 

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