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Developing an Interactive Instrument for Evaluating Teachers’ Professionally Situated Knowledge in Geometry and Measurement

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International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 7))

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Abstract

In this study, we propose a content specific, short, interactive, on-line, scenario-based instrument that incorporates virtual manipulatives developed in GeoGebra, as one of the many ways for evaluating and describing teachers’ professionally situated knowledge (PSK) in the domains of geometry and measurement. To define PSK of mathematics teachers, we use a combination of Shulman’s Pedagogical Content Knowledge (PCK) and its corresponding mathematical knowledge. We describe the methodology used to develop the instrument as well as the corresponding rubrics. The study design followed a concurrent mixed-methods approach, in which the quantitative and qualitative phases of data collection were intermingled to explore the research questions related to identifying components of the PSK. As a case study, we used PSK of the area of a trapezoid, since (a) this topic is familiar to most middle school and secondary mathematics teachers; and (b) a narrow focus was most advantageous when using Delphi methodology to draft an instrument, which was then fully developed using methods of grounded theory. This comprehensive approach led to a deep investigation of multiple and diverse data sources collected from practicing teachers, which we used to create their PSK profiles in the area of trapezoid.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Radford University, Radford, VA, USA

    Agida G. Manizade

  2. Faculty of Education, University of Windsor, Windsor, ON, Canada

    Dragana Martinovic

Authors
  1. Agida G. Manizade
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  2. Dragana Martinovic
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Corresponding author

Correspondence to Agida G. Manizade .

Editor information

Editors and Affiliations

  1. Utah State University, Logan, Utah, USA

    Patricia S. Moyer-Packenham

Appendices

Appendix 1

Virtual manipulatives (presented as the work of Whitney, Paul, Adam, and Donna) together with guiding questions to help teachers evaluate the students’ work, explain their thinking, and provide alternative approaches and proofs.

Item W: Whitney’s Approach

When presented with the task of developing a formula for the area of any trapezoid in her high school geometry class, Whitney developed the diagrams as a strategy for deriving the formula for the area of a trapezoid described by the sketches below. She decomposed a trapezoid into a rectangle and two congruent triangles. Then, she added the areas of all three shapes to calculate the area of the trapezoid.

  1. (a)

    Based on the diagram above, describe Whitney’s thinking. If she were to complete the formal derivation of the area formula using her diagrams, would her method work for any trapezoid? Why, or why not?

  2. (b)

    If Whitney’s approach presents a challenge or misunderstanding, what underlying geometric conception(s) or understanding(s) might lead her to the error presented in this item?

  3. (c)

    If Whitney’s approach presents a challenge or misunderstanding, how might she have developed them?

  4. (d)

    What further question(s) might you ask Whitney to understand her thinking?

  5. (e)

    What instructional strategies and/or tasks would you use during the next instructional period to address Whitney’s challenge(s) (if any presented)? Why?

  6. (f)

    If applicable, how would you use technology or manipulatives to address Whitney’s challenge or misunderstanding?

  7. (g)

    How would you extend this problem to help Whitney further develop her understanding of the area of a trapezoid?

Item X: Paul’s Approach

When presented with the task of developing a formula for the area of any trapezoid in his high school geometry class, Paul developed the diagrams as a strategy for deriving the formula for the area of a trapezoid described by the sketches below. He decomposed a trapezoid into a rectangle and a right triangle. Then he added the areas of these shapes to calculate the area of the trapezoid.

  1. (a)

    Based on the diagram above, describe Paul’s thinking. If he were to complete the formal derivation of the area formula in his diagrams, would his method work for any trapezoid? Why, or why not?

  2. (b)

    If Paul’s approach presents a challenge or misunderstanding, what underlying geometric conception(s) or understanding(s) might lead him to the error presented in this item?

  3. (c)

    If Paul’s approach presents a challenge or misunderstanding, how might he have developed them?

  4. (d)

    What further question(s) might you ask Paul to understand his thinking?

  5. (e)

    What instructional strategies and/or tasks would you use during the next instructional period to address Paul’s challenge(s) (if any presented)? Why?

  6. (f)

    If applicable, how would you use technology or manipulatives to address Paul’s challenge or misunderstanding?

  7. (g)

    How would you extend this problem to help Paul further develop his understanding of the area of a trapezoid?

Item Y: Adam’s Approach

When presented with the task of developing a formula for the area of any trapezoid in his high school geometry class, Adam developed the diagrams as a strategy for deriving the formula for the area of a trapezoid described by the sketches below. He created a midsegment FS of trapezoid ABCD. This allowed him to create a new trapezoid, FABS. He rotated this trapezoid around point S to create a parallelogram, FE’A’D. He calculated the area of parallelogram FE’A’D to find the area of the original trapezoid, ABCD.

  1. (a)

    Based on the diagram above, describe Adam’s thinking. If he were to complete the formal derivation of the area formula in his diagrams, would his method work for any trapezoid? Why, or why not?

  2. (b)

    If Adam’s approach presents a challenge or misunderstanding, what underlying geometric conception(s) or understanding(s) might lead him to the error presented in this item?

  3. (c)

    If Adam’s approach presents a challenge or misunderstanding, how might he have developed them?

  4. (d)

    What further question(s) might you ask Adam to understand his thinking?

  5. (e)

    What instructional strategies and/or tasks would you use during the next instructional period to address Adam’s challenge(s) (if any presented)? Why?

  6. (f)

    If applicable, how would you use technology or manipulatives to address Adam’s challenge or misunderstanding?

  7. (g)

    How would you extend this problem to help Adam further develop his understanding of the area of a trapezoid?

Item Z: Donna’s Approach

When presented with the task of developing a formula for the area of any trapezoid in her high school geometry class, Donna developed the diagrams as a strategy for deriving the formula for the area of a trapezoid described by the sketches below. She created a line CF parallel to the side AD of the trapezoid ABCD. She then extended side AB. This allowed her to create a parallelogram, AFCD. Then she subtracted the area of triangle BFC from the area of parallelogram AFCD to calculate the area of the original trapezoid ABCD.

  1. (a)

    Based on the diagram above, describe Donna’s thinking. If she were to complete the formal derivation of the area formula in her diagrams, would her method work for any trapezoid? Why, or why not?

  2. (b)

    If Donna’s approach presents a challenge or misunderstanding, what underlying geometric conception(s) or understanding(s) might lead her to the error presented in this item?

  3. (c)

    If Donna’s approach presents a challenge or misunderstanding, how might she have developed them?

  4. (d)

    What further question(s) might you ask Donna to understand her thinking?

  5. (e)

    What instructional strategies and/or tasks would you use during the next instructional period to address Donna’s challenge(s) (if any presented)? Why?

  6. (f)

    If applicable, how would you use technology or manipulatives to address Donna’s challenge or misunderstanding?

  7. (g)

    How would you extend this problem to help Donna further develop her understanding of the area of a trapezoid?

Appendix 2

Components of the rubric used to identify teachers’ knowledge of student challenges and conceptions at Level 4 (Table 14.1) and Level 1 (Table 14.2).

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Manizade, A.G., Martinovic, D. (2016). Developing an Interactive Instrument for Evaluating Teachers’ Professionally Situated Knowledge in Geometry and Measurement. In: Moyer-Packenham, P. (eds) International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives. Mathematics Education in the Digital Era, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-32718-1_14

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  • DOI: https://doi.org/10.1007/978-3-319-32718-1_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32716-7

  • Online ISBN: 978-3-319-32718-1

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