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Investigating further preservice teachers’ conceptions of multidigit whole numbers: refining a framework

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Abstract

This study was designed to investigate preservice elementary school teachers’ (PSTs’) responses to written standard place-value-operation tasks (addition and subtraction). Previous research established that PSTs can often perform but not explain algorithms and provided a four-category framework for PSTs’ conceptions, two correct and two incorrect. Previous findings are replicated for PSTs toward the end of their college careers, and two conceptions are further analyzed to yield three categories of incorrect views of regrouped digits: (a) consistently as 1 value (all as 1 or all as 10), (b) consistently within but not across contexts (i.e., all as 10 in addition but all as 1 in subtraction), and (c) inconsistently (depending on the task).

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Notes

  1. For the purpose of this paper, I use PSTs to refer to preservice elementary school teachers in the USA. Even though the literature has shown that preservice elementary school teachers struggle in various countries, the current study was conducted in the USA, and, as such, claims made about preservice elementary school teachers as a population will be restricted to those in the USA. The ideas brought forward by the refinement of the framework are, however, generalizable as an explanatory framework (Steffe & Thompson, 2000).

  2. Note that students in the USA often term regrouping in the context of addition carrying and regrouping in the context of subtraction borrowing.

  3. A course at this institution is typically three credits for a course that meets for 3 h each week for a 16-week period (15 weeks of instruction and 1 week of final exams). Undergraduate students are expected to take a minimum of 12 credits for full-time status; graduate students are expected to take a minimum of nine credits for full-time status.

  4. Thanheiser (2009) examined PSTs at a different institution and at a different point in their education, and whereas the framework from that study was derived from interviews with PSTs before their first mathematics content course (typically in their first or second year at the university), the current study was conducted with PSTs after they had completed all their content requirements (typically in their 4th year at the university).

  5. All names are pseudonyms.

  6. The survey used for the study used this instead of the. For further use of this survey I suggest a consistent use of the for both 3.1 and 3.2.

  7. From The effects of professional development on the mathematical content knowledge of K-3 teachers by Philipp et al. (2008), presented at the annual meeting of the American Educational Research Association, New York, NY. Reprinted with permission of the authors.

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Acknowledgments

This report is partially based on a paper that was presented at the annual meeting of the American Research Association (AERA), San Diego, April 2009. The analysis of this study was supported in part by a Portland State University Faculty Enhancement Grant. I thank Dr. Keith Weber for his thoughtful comments on initial discussions and drafts of this manuscript. I also thank Dr. Randy Philipp, who, with his continued interest in this work, motivated my looking at these data in the way described in this paper. In addition, I thank Bonnie Schappelle and the anonymous reviewers for their thoughtful comments and Briana Mills for helping with the coding of the data.

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Authors and Affiliations

  1. Portland State University, Portland, OR, USA

    Eva Thanheiser

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  1. Eva Thanheiser
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Correspondence to Eva Thanheiser.

Appendix

Appendix

Survey questions

  1. 1.

    One of the main topics in 2nd grade is place value. What in your mind are the essential elements of place value a 2nd grader should understand? Please be as specific as you can be.

  2. 2.

    Another main topic in 2nd grade is addition and subtraction of 2-digit numbers. What in your mind does a 2nd grader need to understand to be able to add and subtract 2-digit numbers? Please be as specific as you can be.

  3. 3.

    Please consider the regrouped ones in the problem below:

    figure a
    1. 3.1.

      What does the 1 above the 8 represent?

    2. 3.2.

      What does theFootnote 6 1 above the 3 represent?

    3. 3.3.

      Compare the two 1s. Are they the same or are they different? Please be as specific as you can be.

  4. 4.

    Please answer the questions below:Footnote 7

    figure b
    1. 4.1.

      Does the 1 in each of these problems represent the same amount? Please explain your answer.

    2. 4.2.

      Explain why in addition (as in Problem A) the 1 is added to the 5, but in subtraction (as in Problem B) 10 is added to the 2.

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Thanheiser, E. Investigating further preservice teachers’ conceptions of multidigit whole numbers: refining a framework. Educ Stud Math 75, 241–251 (2010). https://doi.org/10.1007/s10649-010-9252-7

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  • DOI: https://doi.org/10.1007/s10649-010-9252-7

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