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The nature of prospective mathematics teachers’ pedagogical content knowledge: the case of multiplication of fractions

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Abstract

The purpose of this study was to investigate prospective mathematics teachers’ knowledge of common conceptions and misconceptions that sixth and seventh grade students had about multiplication of fractions. In addition, prospective teachers’ knowledge of the sources of these misconceptions and strategies they knew to overcome these misconceptions was examined. Data were collected from 17 Turkish prospective teachers at the end of the spring semester of the 2004–2005 academic year. A case study design was used in which data were collected through the Multiplication of Fractions Questionnaire and semi-structured interviews. The prospective teachers suggested many difficulties that elementary grade level students may have and stated that these difficulties stemmed from students’ lack of formal knowledge and rote memorization of the algorithms. In addition, the prospective teachers suggested many strategies that could be used to overcome these misconceptions or difficulties. These strategies could be grouped under three headings: strategies based on teaching methods, strategies based on formal knowledge of fractions, and strategies based on psychological constructs.

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Acknowledgments

We are grateful to the anonymous reviewers for their insightful comments and suggestions on an earlier version of this article.

Author information

Authors and Affiliations

  1. Faculty of Education, Department of Elementary Education, Middle East Technical University, 06531, Ankara, Turkey

    Mine Isiksal & Erdinc Cakiroglu

Authors
  1. Mine Isiksal
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  2. Erdinc Cakiroglu
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Correspondence to Mine Isiksal.

Appendices

Appendix A: the multiplication of fractions questionnaire (MFQ)

  1. 1.

    Mert has 7 chocolate bars. He decided to give one-third of these chocolates to his close friend Emre. How many chocolates will Emre get?

    1. a.

      Write a mathematical expression for the problem

    2. b.

      Find the answer to the problem

    3. c.

      List two common mistakes students in sixth or seventh grade may make while performing (a) and/or (b)

    4. d.

      Describe possible sources for each of these mistakes depending on students’ thinking

    5. e.

      How will you overcome these difficulties?

  2. 2.

    Elif bought a bottle of milk. She gave \( \frac{1}{2} \) of it, which was \( 1\frac{3}{4} \) lt to her grandmother. How much did the bottle of milk originally contain?

    1. a.

      Write a mathematical expression for the problem

    2. b.

      Find the answer to the problem

    3. c.

      List two common mistakes students in sixth or seventh grade may make while performing (a) and/or (b)

    4. d.

      Describe possible sources for each of these mistakes depending on students’ thinking

    5. e.

      How will you overcome these difficulties?

  3. 3.

    Consider the following multiplication of fraction problems and answer the following questions for each of them.

a. \( \frac{2}{3} \times \frac{3}{5} \) b. \( 1\frac{1}{2} \times \frac{1}{3} \),

  1. 1.

    List two common mistakes students in sixth or seventh grade may make while performing the operations

  2. 2.

    Describe possible sources for each of these mistakes depending on students’ thinking

  3. 3.

    How will you overcome these difficulties?

Appendix B: sample items from the semi-structured interview protocol

The researchers ask the following questions based on the answers given in the MFQ.

  • In general, the phrases that are used during the interview include:

  • “What do you mean by…..”

  • “Here you mentioned that ….’

  • “Tell me more on……” (if there is something that is not clear to the researcher on the questionnaire)

  • “Why do you think so…..” (if there is something that is not clear to the researcher on the questionnaire)

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Isiksal, M., Cakiroglu, E. The nature of prospective mathematics teachers’ pedagogical content knowledge: the case of multiplication of fractions. J Math Teacher Educ 14, 213–230 (2011). https://doi.org/10.1007/s10857-010-9160-x

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  • DOI: https://doi.org/10.1007/s10857-010-9160-x

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