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Effects of reading-oriented tasks on students’ reading comprehension of geometry proof

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Abstract

This study compared the effects of reading-oriented tasks and writing-oriented tasks on students’ reading comprehension of geometry proof (RCGP). The reading-oriented tasks were designed with reading strategies and the idea of problem posing. The writing-oriented tasks were consistent with usual proof instruction for writing a proof and applying it. Twenty-two classes of ninth-grade students (N = 683), aged 14 to 15 years, and 12 mathematics teachers participated in this quasi-experimental classroom study. While the experimental group was instructed to read and discuss the reading tasks in two 45-minute lessons, the control group was instructed to prove and apply the same propositions. Generalised estimating equation (GEE) method was used to compare the scores of the post-test and the delayed post-test with the pre-test scores as covariates. Results showed that the total scores of the delayed post-test of the experimental group were significantly higher than those of the control group. Furthermore, the scores of the experimental group on all facets of reading comprehension except the application facet were significantly higher than those of the control group for both the post-test and delayed post-test.

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Acknowledgments

The authors wish to thank the participating teachers and students of this study. Although this paper is about part of a research project funded by the National Science Council of Taiwan (NSC 96-2521-S-018-004-MY3), 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 Taiwan Normal University, Taipei City, Taiwan

    Kai-Lin Yang & 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.

Appendices

Appendix A

The Task of Reading Proof with its Proposition

Believe me! You will be brought into a world which can make you say “Wow! How fun mathematics is! It’s quite simple!” after this class. Follow me, it’s time to start. GO!…Now, as clever as you are, please think carefully about the next questions of proof, and write down your thoughts in the following blank space:

figure a
  1. (1-1)

    What is “the given” in the above proof?

  2. (1-2)

    What is “the claim” in the above proof?

  3. (1-3)

    Thinking in the beginning of “the given” in the above proof, what can you infer in the next step?

  4. (1-4)

    Which sub-conclusion can you derive to apply “\( \overline {\text{BC}} = \overline {\text{AD}} \)” in the above proof?

  5. (2-1)

    In the above proof, which properties are written unclearly in the given?

  6. (2-2)

    In the above proof, which proof steps do you not understand? Please circle them.

  7. (2-3)

    Thinking! In the above proof, can you think which proof steps are redundant or unnecessary? Please delete them with straight lines in the above proof.

  8. (3-1)

    Now, please use brief sentences to write down the draft of the proof.

    figure b
  9. (3-2)

    Please discuss in groups. Write down the differences between drafts, and an agreed draft after the discussion.

  10. (4-1)

    If you were a teacher, which problem could you pose that would be answered by the arguments in the above proof?

  11. (4-2)

    Do you have the same answer according to the problem you devise in (4-1), (1-1), and (1-2)? What is the correct answer?

  12. (4-3)

    Please go back to check (1-3) and (1-4), and you can revise the conjecture of (1-3) and (1-4) if you need to.

  13. (4-4)

    Referring to (3-1), please describe the relationship between the given, properties or definitions, and the claim.

  14. (4-5)

    While applying the properties or definitions, what should you notice?

  15. (4-6)

    While reading mathematical proofs hereafter, how will you read for comprehending proofs?

The Statement-Posing Task

Jam read a mathematical book where some part of a page was torn to pieces. He found that a proposition was gone, but the proof of this proposition was there. Now, can you help Jam to think out what this proposition is?

figure c
figure d
  1. (1-1)

    According to the above proof, can you conjecture what this proposition is?

  2. (2-1)

    Think! In the above proof, which proof steps do you not understand? Please circle them.

  3. (2-2)

    Think! In the above proof, can you think which proof steps are redundant or unnecessary? Please cross them out in the above proof.

  4. (2-3)

    In the above proof, what do you think the key points are?

  5. (2-4)

    Which properties or definitions are used in the proof?

  6. (3-1)

    Now, please use brief sentences to write down the draft of the proof.

    figure e
  7. (3-2)

    Please discuss in groups, and write down the differences between drafts, and come up with an agreed draft after discussion.

  8. (4-1)

    Please go back to check (1-1) again. You can revise the conjecture of (1-1) if you need to.

  9. (4-2)

    Referring to (3-1), please describe the relationship between the given, properties or definitions, and the claim.

  10. (4-3)

    While applying the properties or definitions, what should you notice?

  11. (4-4)

    While you read mathematical proofs hereafter, how will you read to easily comprehend proofs?

Appendix B

figure f

*P# and C# respectively mean the corresponding premise and conclusion of each argument in the proof.

Answer the following questions on the basis of the proof process.

  1. (1)

    Do you agree that \( \overline {\text{BM}} = \overline {\text{CM}} \)? Explain why or why not.

  2. (2)

    Label ∠BMD in this figure as 1 and ∠CMD as 2.

  3. (3)

    If △BMD and △CMD are congruent, what is the corresponding side of \( \overline {\text{DB}} \)?

  4. (4)

    Besides the known conditions (the perpendicular bisector of \( \overline {\text{BC}} \), intersects \( \overline {\text{AB}} \) at D, and intersects \( \overline {\text{BC}} \) at M. And \( \overline {\text{DA}} = \overline {\text{DB}} \)), which conditions can be directly applied without any explanation?

  5. (5)

    If someone suggests that the proof process of lines 1, 2, 4, 3, 5, 6, 7, and 8 is correct, after lines 3 and 4 are interchanged, would you agree with his or her opinion?

  6. (6)

    If someone suggests that the proof process of lines 6, 1, 2, 3, 4, 5, 7, and 8 is correct, after the position of line 6 has been changed, 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 premises are necessarily required?

    2. (8-2)

      What final conclusion is derived from these premises?

  9. (9)

    Which statements can be validated from this proof?

  10. (10)

    From this proof process, it firstly derives an important result from the condition that the perpendicular bisector of \( \overline {\text{BC}} \), L, intersects \( \overline {\text{BC}} \) at M and other conditions.

    1. (10-1)

      What is this important result?

    2. (10-2)

      According to this important result in (10-1) and \( \overline {\text{DA}} = \overline {\text{DB}} \), one condition can be derived to confirm ∠DCA=∠DAC. What is this condition?

  11. (11)

    Choose the correct statements.

  12. (12)

    Do you agree that this proof process is correct?

  13. (13)

    Statement A: If L, the perpendicular bisector of \( \overline {\text{BC}} \) of △ABC, intersects \( \overline {\text{AB}} \) at D, and intersects \( \overline {\text{BC}} \) at M. And \( \overline {\text{DA}} = \overline {\text{DB}} \); then ∠DCA and ∠DAC must be equal.

    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 sometimes incorrect?

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

  1. (14)

    There is a circle with centre point P, radius \( \overline {\text{PS}} \) and \( \overline {\text{PQ}} \). If T is the midpoint of \( \overline {\text{PQ}} \), \( \overline {\text{ST}} \bot \overline {\text{PQ}} \), and S is the midpoint of \( \overline {\text{PR}} \), is △RSQ an isosceles triangle?

  2. (15)

    There are three points P, Q, and R. If S is the midpoint of \( \overline {\text{PQ}} \) and \( \overline {\text{ST}} \bot \overline {\text{PQ}} \), what conclusions can be derived?

  3. (16)

    If D is the midpoints of \( \overline {\text{AE}} \), and \( \overline {\text{BD}} \) and \( \overline {\text{AE}} \) are perpendicular to each other, and \( \overline {\text{AB}} = \overline {\text{BC}} \), then ∠AEC = 90°, is this correct?

Appendix C

figure g

*P# and C# respectively mean the corresponding premise and conclusion of each argument in this proof.

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Yang, KL., Lin, FL. Effects of reading-oriented tasks on students’ reading comprehension of geometry proof. Math Ed Res J 24, 215–238 (2012). https://doi.org/10.1007/s13394-012-0039-2

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