Using the Van Hiele Theory to Explain Pre-Service Teachers’ Understanding of Similarity in Euclidean Geometry

Round 1

Reviewer 1 Report

I find this paper really interesting and promising. I am truly sorry but I cannot accept it until the major issues (1 and 2) are fixed. I know it requires conducting a new study with another group of respondents, but I strongly recommend the Author to take this effort.

Issue 1

The author refers to van Hiele's levels, but as (s)he does not give the content of all the tasks and does not present how (s)he decided to assign van Hiele's levels to each task, I am unable to judge whether I agree with this assignment or not. As a reader, I would like to first see all the tasks, learn about the Author’s a priori analysis of these tasks, and only then read about the study results. The way the tasks are presented is rather chaotic. It does not give the full picture of the whole study in advance. And what happened to the item 1.6? Why is it missing?

Issue 2

Tasks 2.4 and 2.5 presented in the text raise some serious concerns.

These tasks seem to be a poor choice for this study, because in addition to the PQR triangle, only two other triangles are visible in the picture. It is impossible to give a wrong answer. Well, unless you impose the condition, which is in the task, that the vertices must be given in the correct order. But does the order in which the vertices are given affect whether the triangles are similar or not?

And here is the major issue that can hardly be fixed: person A who sees the similarity of the triangles in the task but does not give the vertices in the correct order and person B who simply gives the names of the other two triangles in the picture, with the vertices not written in the expected order, will have answers considered equally bad. Things get even worse when we realize that person C who does not see the similarity of the triangles but gives the names of the other two triangles and just by chance gives the vertices in the correct order will get a full credit for that task.

The answers obtained in this task do not allow inferences to be made about the respondents' perception and understanding of the similarity of triangles.

I agree that it is better to keep the right order of vertices when talking about similar triangles, but I disagree that it is necessary to do so. It is a matter of elegance and convention, not the correctness.

Issue 3

In the lines 166-167 the Author writes: “The question for Item 1.2 was: What are the minimum conditions required to show that two  polygons are similar?” . Item 1.2 is assigned VH level 2.

Meanwhile, in his/her former descriptions, the Author states:

Level 2: At this stage a person can state the properties of similar shapes and can investigate properties of similarity by translations, reflections and rotations (lines 109-110)

Level 3: A person can use ratio to see if the sides of a figure are in or not in proportion and can use informal deduction to calculate angles to check if the corresponding angles of two figures are equal or not. However, they may not grasp which conditions are necessary and which could be sufficient for similarity.(lines 115-119)

I would expect some explanation for assigning VH level 2 to the 1.2 item.

Issue 4

I would also question the judgement made on PST7 response presented in Figure 3. My guess is that someone reasoned like this: OE is perpendicular to AB. A line perpendicular to the chord of the circle drawn through its center divides the chord into halves - it does not have to be a reason for the former statement, but rather a second argument for stating that AE=EB. Maybe the first line was meant to provide a justification for AE=EB, and what is written on the right hand-side in the second line was meant to be a summary of one’s reasons for the claim? It would be good to hear PST7’s explanations. What is missing in the reasoning for sure is the argument for OE being perpendicular to AB, but it’s not necessarily the problem of circular reasoning.

Minor issues

·         The text mentions one task that was solved by the respondents, while the presentation of the examples from the research instrument shows that it included several tasks. This should be unified.

·      At the beginning of the manuscript, the Author states that “The aim of this study was to explore pre-service teachers’ (PST’s) understanding of similarity and proofs in Euclidean geometry”, but at the end we read that: “In this study we set out to explore the understanding of similarity in Euclidean geometry (…)”. I agree that the paper explores various issues related to the concept of similarity, but the notion of proof is being brought to the readers’ attention only with respect to similarity. It is too broad and thus unjustified to say that the paper deals with PST’s understanding of proof in Euclidean geometry.

·       Lines 88-89: According to the van Hiele model [23] as students get instruction in geometry, the ways in which they reason about a concept changes - the ways change or the way changes;

·       Lines 135-137: The study was conducted with 34 PSTs from one South African university. PSTs of mixed abilities and mixed gender who voluntarily participated in the study. - The study was conducted with 34 PSTs from one South African university who participated voluntarily. Respondents were of mixed abilities and mixed gender.

·        Least challenging items: I would rather describe these items in one of the orders: 3.3, 3.2 and 1.2 or 1.2, 3.2 and 3.3. The order proposed by the Author does not reflect the tendencies found in the difficulty of the items.

 

Author Response

Please see attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

For the Authors:

Title

The title of the article seems appropriate.

Keywords

The keywords of the article are considered appropriate.

Abstract

-You do not need to provide the method information.

- Offer a specific recommendation.

Introduction

-Reinforce this section with international literature over the past five years.

-Add transition sentences between paragraphs.

Method

The method of the study is not clearly stated. The validity and reliability information of the data collection tool should be given.

Findings-Results

-Comments after tables and figures should be strengthened.

Discussion and Conclusion

-Give the reasons for your results.

- Make suggestions about your results.

- Talk about your limitations.

References.

Technical deficiencies in this section need to be corrected to APA 7.

Author Response

Please see attachment

Author Response File: Author Response.pdf

Reviewer 3 Report

I think this paper, which is correctly written overall, deals with an interesting  subject and has an adequate methodology of investigation to answer its research questions. Despite the limitations of the study (some of them are pointed out by the authors), I think it is a good contribution to the study of misconceptions about the concept of similarity in geometry, and can be a good starting point for further research on the topic.

However, I have some comments about changes I strongly suggest to be included in the final version of the paper.

1- A complete careful read of the paper searching writing mistakes should be made. I detected some of them but probably the author/s can find some more and polish the paper in this regard.

2- In the abstract, lines 15-16, the sentence should be revised. "This study recommends that PST'S need more opportunities to make..." should have the verb changed, since as it is it sounds confusing.

2- Line 175, an "of" is missing before "this process".

3- In Table 1, items 1.1 and 1.2 should not end with a question mark, since they are not phrased as questions.

4- In page 7, Item 3.2 includes a description in the targeted VH level column that talks about showing the corresponding angles are equal. In fact, inspecting the figure I agree this is the fastest way to show the similarity of the triangles in question. This way of doing it seems independent of item 3.1. However, in page 8, when discussing about this item 3.2 (lines 212-218), item 3.1 is presented as "the most important result they needed" to prove item 3.2. This looks like a contradiction. In fact, after trying, I don't think it is trivial at all to show the similarity of those triangles without using the angle-angle-angle criteria and using instead the angle-side-angle or the side-side-side criteria. In any case, I think the author/s should clarify how knowing item 3.1 was used to derive item 3.2 or if this was actually not needed.

5- For aesthetic reasons, I would suggest making the images that include the answers of the students less wide. For example, figures 1 and 7 adjust nicely to the width of the text, but the rest don't. I don't think there would be problems to read their text if their size was reduced in this sense.

 

If these changes are included, I recommend the paper to be published in Education Sciences.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Abstract

-In the abstract, tell us about your sample group.

-Talk about your method.

Method

-Use academic language when writing headlines. For example: Method, Data collection tools, analysis of data…

Discussion and Conclusion

-Give the reasons for your results.

References.

Technical deficiencies in this section need to be corrected.

for example: references on 3,4,5,6,7,...

Author Response

Thank you for your suggestions- here is our response: 

-In the abstract, tell us about your sample group.

We have explained that we had 34 preservice high school maths teachers who gave written responses and 7 of these participated in interviews as well.

-Talk about your method.

We have added details about the data collection and the data analysis

Method

-Use academic language when writing headlines. For example: Method, Data collection tools, analysis of data…

We have revised the headings in the method section as suggested

Discussion and Conclusion

-Give the reasons for your results.

 We have added explanations for each of our results

References.

Technical deficiencies in this section need to be corrected.

for example: references on 3,4,5,6,7,...

We have corrected and made sure journal titles, titles of conference papers and books are in italics