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What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

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Digital Technologies in Designing Mathematics Education Tasks

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 8))

Abstract

There is a consensus among mathematics educators that in order to provide students with rich learning opportunities to engage with reasoning and proving, prospective teachers must develop a strong knowledge base of mathematics, pedagogy and student epistemology. In this chapter we report on the design of a technology-based task “What can you infer from this example?” that addressed the content and pedagogical knowledge of the status of examples in proving of pre-service teachers (PSTs). The task, originally designed and implemented with high-school students, was modified for PSTs and expanded to involve multiple components, including scenarios of non-descript cartoon characters to represent student data. The task was administered through LessonSketch, an online interactive digital platform, to 4 cohorts of PSTs in Israel and the US, across 4 semesters. In this chapter we focus on theoretical and empirical considerations that guided our task design to provide rich learning opportunities for PSTs to enhance their content and pedagogical knowledge of the interplay between examples and proving, and address some of the challenges involved in the task implementation. We discuss the crucial role of technology in supporting PST learning and provide an emergent framework for developing instructional tasks that foster PSTs’ engagement with proving.

The erratum of this chapter can be found at under DOI 10.1007/978-3-319-43423-0_17

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-43423-0_17

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Notes

  1. 1.

    These processes are the focus of our chapter. Discussion of the roles of examples in teaching mathematics is beyond the scope of this chapter.

  2. 2.

    To simplify communication we use the short term “the status of examples in proving” to denote both examples and counterexamples in both proving and refuting processes.

  3. 3.

    See Ko (2010) for an extensive literature review on teachers’ conceptions of proof.

  4. 4.

    The same task with minor variations was used in two different studies with similar foci and student populations.

  5. 5.

    The complete framework describes the status of examples with respect to two types of mathematical statements: universal and existential. Here we focus on universal statements only.

  6. 6.

    When presented in the form of a conditional statement (i.e., “if…then…”), the premises describe the domain of objects to which the statement refers and the conclusion describes the property that these objects have.

  7. 7.

    Based on student data from Buchbinder’s (2010) study.

  8. 8.

    All cartoon student characters in the scenario are named by the color of their shirt.

  9. 9.

    The Depict tool was not originally considered for implementation in the task as it was not available in Hebrew, however it has recently became available in languages other than English.

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Authors and Affiliations

  1. University on New Hampshire, Durham, NH, USA

    Orly Buchbinder

  2. Technion, Israel Institute of Technology, Haifa, Israel

    Iris Zodik

  3. Ohalo College, Katzrin, Israel

    Gila Ron

  4. University of Maryland, College Park, MD, USA

    Alice L. J. Cook

Authors
  1. Orly Buchbinder
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  2. Iris Zodik
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  3. Gila Ron
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  4. Alice L. J. Cook
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Corresponding author

Correspondence to Orly Buchbinder .

Editor information

Editors and Affiliations

  1. Department of Education Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

    Allen Leung

  2. Department of Mathematics “G. Castelnuovo”, “Sapienza” University of Rome, Rome, Italy

    Anna Baccaglini-Frank

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Buchbinder, O., Zodik, I., Ron, G., Cook, A.L.J. (2017). What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving. In: Leung, A., Baccaglini-Frank, A. (eds) Digital Technologies in Designing Mathematics Education Tasks. Mathematics Education in the Digital Era, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-43423-0_11

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  • DOI: https://doi.org/10.1007/978-3-319-43423-0_11

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