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Can a number of diagrams linked to a proof task in 3D geometry improve proving ability?

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Abstract

The current study attempts to investigate whether using multiple diagrams attached to a geometry task could reduce difficulties in the proving process regarding concepts in three-dimensional geometry. Ninety 12th-grade students participated in the study. The research included the use of a questionnaire whose items invited open-ended responses regarding two tasks. Interviews were also conducted with some participants. Qualitative and quantitative analyses were performed. The results revealed a correlation between the students’ responses and the number of diagrams they received. Students who were given several diagrams gave more correct responses and correct and complete proofs than students who were given only one diagram presenting several diagrams attached to the proof task reduce the visual obstacles that students face in constructing geometrical proofs. The findings from the interviews support the results from the questionnaires; interviews revealed presenting further diagrams allow students to adapt their proving processes.

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Authors and Affiliations

  1. The Arab Academic Institute for Education, Faculty of Education, Beit Berl College, Kfar-Saba, Israel

    Aehsan Haj-Yahya

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  1. Aehsan Haj-Yahya
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Appendix Interview

Appendix Interview

Interviewer: Here is the task: Given cube ABCDEFGH and its edge AB = 5 cm, calculate the angle AHF. (The interviewer presented Diagram 3.)

Rami: Yes, it’s 90 degrees.

I: Why?

R: I think it's 90 … I don’t know … you know, because the two diagonals in the faces are angle bisectors.

I: Are you sure?

R: Yes, I think … the faces are perpendicular … also because of the net of the cube.

I: I want to show you another diagram for the same task. (The interviewer presented Diagram 1.)

R: You know, we have an isosceles triangle … Yes, because the diagonals are equal, we have congruent faces.

I: Do you want to change your answer from before? Is it 90 degrees?

R: We have √50 and √50 … We are missing information needed to calculate the angle…

I: I want to present to you another diagram for this problem. (The interviewer presents Diagram 4.)

R: Mm … I think we have an equilateral triangle.

I: So …

R: All the angles are equal … The angle is equal to 60 degrees.

I: Here the second task, given cube ABCDEFGH and its edge AB = 5 cm, I is the med point of AE, J is the med point of CG. Which quadrilateral is HJBI..(The interviewer presented diagram 4).

R: I remember it. Its from the questionnaire.

I: You claimed that its rectangle, please look at your response!

R: Yes I proved that its rectangle, I proved that every two opposite sides are equal, and the angles are 90 degrees.

I: Are you sure?

R: Maybe there is no 90 degrees as we reveled in the previous task. Yes its only parallelogram.

I: I want to present to you another diagram for the same task. (The interviewer present diagram 1) are you want to change your answer?

R: Maybe it is rhombus?

I: Please think about it!

R: I think it’s a rhombus, it’s a cube?

I: Yes it’s a cube.

R: O.K. it’s a rhombus, all the right angle triangles in the faces are congruent.

I: Why?

R: Because all the edges are equal, and we have midpoints in the edges.

I: You are right.. can it be a square?

R: No, because there is no right angle.

I: Why?

R: …I don’t know.

I: What if I draw the diagonal in the quadrilateral (researcher draw it), is it equal to The diagonal AC on the cube base?

R: I think it equal to the diagonal AC.

I: So what. Is it proved that there is no 90 degrees?

R: I don’t know.

I: Could you calculate it?

R: Yes I can calculate all the sides of the quadrilateral and its diagonal.

I: So.

R: If we can apply Pythagorean theorem there is 90 degrees and if not there is no 90 degrees.

I: O.K.

R: (Calculating) we cant apply the theorem, there is no 90 degrees..

I: So.

R: We proved only that it’s a rhombus.

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Haj-Yahya, A. Can a number of diagrams linked to a proof task in 3D geometry improve proving ability?. Math Ed Res J 35, 215–236 (2023). https://doi.org/10.1007/s13394-021-00385-8

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