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A model of reading comprehension of geometry proof

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Abstract

This study aims to investigate a construct of reading comprehension of geometry proof (RCGP). The research aims to investigate (a) the facets composing RCGP, and (b) the structure of these facets. Firstly, we conceptualize this construct with relevant literature and on the basis of the discrimination between the logical and the epistemic meanings of an argument, then assemble the content of RCGP from literature and propose a hypothetical model of RCGP. Secondly, mathematicians and mathematics teachers are interviewed for their ideas on reading mathematical proof in order to enrich the content of RCGP. Adapting the phases of reading comprehension in language, the content of RCGP is classified into six facets. Lastly, these facets are structured using the hypothetical model and then justified by students’ performance in the facets of RCGP using the multidimensional scaling method. The results sustain that the structure of facets can be characterized by this conceptualized model.

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Acknowledgements

The authors wish to thank anonymous reviewers and the editor, Norma Presmeg, for commenting on earlier drafts. We also thank Jenny Chang for proofreading our English manuscript. This paper is part of a research project funded by the National Science Council of Taiwan (NSC 93-2521-S-003-003 and 93-2511-S-033-002). The views and opinions expressed in this paper are those of the authors and not necessarily those of the NSC.

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Authors and Affiliations

  1. Department of Mathematics, National Changhua University of Education, #1, Jin-De Road, Changhua, 500, Taiwan

    Kai-Lin Yang

  2. Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan

    Fou-Lai Lin

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  1. Kai-Lin Yang
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  2. Fou-Lai Lin
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Correspondence to Kai-Lin Yang.

Appendix A

Appendix A

Answer the following on the basis of this question and the proof process.

  1. (1)

    Label∠AMC of this figure as 1 and ∠MAC of this figure as 2.

  2. (2)

    Do you agree that ∠AMC =∠MAC? Explain why or why not.

  3. (3)

    If ▵AMC and ▵BMD are congruent, what is the corresponding angle of ∠MAC?

  4. (4)

    Besides the known conditions (\( \overline{{AB}} \) and \( \overline{{CD}} \) intersect at the point M, \( \overline{{AM}} = \overline{{BM}} \), \( \overline{{CM}} = \overline{{DM}} \)), which conditions can be directly applied without any explanation?

  5. (5)

    If someone suggests that the proof process of line 1, 3, 2, 4, 5 is correct after line 2 and 3 are interchanged, would you agree with his or her opinion?

  6. (6)

    If someone suggests that the proof process of line 1, 2, 4, 3, 5 is correct after line 3 and 4 are interchanged, would you agree with his or her opinion?

  7. (7)

    Which properties are applied in this proof?

  8. (8)

    On the basis of the question and the proof,

    1. (8–1)

      Which conditions are necessarily used?

    2. (8–2)

      What is derived from this proof?

  9. (9)

    From this proof process, it firstly derives an important result from \( \overline{{AM}} = \overline{{BM}} \), \( \overline{{CM}} = \overline{{DM}} \) and other conditions, and then derives a condition used to confirm \( \overline{{AC}} //\overline{{DB}} \).

    1. (9–1)

      What is this important result?

    2. (9–2)

      What is this condition used to confirm \( \overline{{AC}} //\overline{{DB}} \)?

  10. (10)

    Which statements can be validated from this proof?

  11. (11)

    Choose the correct statements.

  12. (12)

    Do you agree that this proof process is correct?

  13. (13)

    Statement A: If \( \overline{{AB}} \) and \( \overline{{CD}} \) intersect at the point M, \( \overline{{AM}} = \overline{{BM}} \), \( \overline{{CM}} = \overline{{DM}} \), then \( \overline{{AC}} \) is parallel with \( \overline{{DB}} \).

    1. (13–1)

      Do you agree that this proof process can prove that Statement A is always correct?

    2. (13–2)

      Do you agree that this proof process can prove that Statement A is sometimes correct and sometime incorrect?

    Answer the following questions on the basis of what you know.

  14. (14)

    If a quadrilateral PURV has two diagonals \( \overline{{PR}} \) and \(\overline{{UV}} \), and Q is the midpoint of both \(\overline{{PR}} \) and \(\overline{{UV}} \),then is this quadrangle a parallelogram?

  15. (15)

    If \( \overline{{PR}} \) and \( \overline{{UV}} \) intersect at a point Q, and Q is the midpoint of both \( \overline{{PR}} \) and \( \overline{{UV}} \), which conclusions can be derived?

  16. (16)

    If \( \overline{{XY}} = \overline{{YZ}} \), \( \overline{{MY}} = \overline{{YN}} \) and ∠XYM=∠ZYN, then are \( \overline{{XM}} \) and \( \overline{{NZ}} \) parallel with each other?

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Yang, KL., Lin, FL. A model of reading comprehension of geometry proof. Educ Stud Math 67, 59–76 (2008). https://doi.org/10.1007/s10649-007-9080-6

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