Classification and Analysis of Pre-Service Teachers’ Errors in Solving Fermi Problems

Round 1

Reviewer 1 Report

Thank you for your contribution, improving the training of prospective teachers is always relevant.
The argumentation, organisation and structuring of the text are totally valid, as well as the theoretical underpinning of the article is correct. The statistical analysis is coherent and consistent.
The paper is well written although I suggest an extra revision, especially in some very long sentences and some use of comma.
The presentation of all the results is clear, I would only change the thickness of some lines in table 3 that do not facilitate the understanding that the types of errors (Simplification error, Mathematization error, Mathematical working error,...) are a summary of the upper errors.
Perhaps the number of sources (relevant papers) connecting the results in the conclusion should be expanded.

Regarding bibliographical references, you ought to:
-Revise italics in numbers 2, 13, ... 
-Unify the use of "In", "En", for example in numbers 9 and 11.

Author Response

We are grateful for the reviewer's comments and input, below we explain the changes made on the basis of their suggestions.

1) I suggest an extra revision, especially in some very long sentences and some use of comma.

We have split and simplified some long sentences into two short ones, also eliminating some parenthetical breaks that complicated the syntax of the sentences. For example: at the end of the introduction (p. 2, 54-57); at the end of subsection 2.3 (p. 4, 172-174); the sentence before Table 2 (p. 6, 267-269); second paragraph on p. 8 (303-309); the sentence after Figure 1 (p. 10, 424-426); the sentence above E8 on p. 11 (479-482); the sentence in the paragraph before Figure 3 (p. 12, 536-537); changed comma to full stop line 558; the sentences explaining the error E11 (p. 13, lines 569-571, 569-571, p. 568); the sentences explaining the error E11 (p. 13, lines 569-571, p. 569-571, p. 569-571, p. 569-571, p. 558). 13, lines 569-571, 572-574 and 578-581); the explanation of E13 on p. 13 (592-595); changed comma to semicolon in the enumeration of context features (p. 14, 642-645); shortened sentences and deleted brackets on p. 15 (687-688, 690-692 and 697-699); sentence parentheses deleted on p. 16 (738-739); parentheses deleted on p. 16, lines 743-745; sentence shortened on p. 16, 759-761.

2) I would only change the thickness of some lines in table 3 that do not facilitate the understanding that the types of errors (Simplification error, Mathematization error, Mathematical working error…) are a summary of the upper errors.

We have changed the thickness of the categories grouping error types in Table 3, as recommended by the reviewer.

3) Perhaps the number of sources (relevant papers) connecting the results in the conclusion should be expanded.

The conclusions section has been revised, relating the conclusions to the studies cited in the theoretical framework. A paragraph has also been included indicating the limitations of the work and possibilities for future research in relation to studies by other authors (two new references to the bibliography have been included).

4) Regarding bibliographical references, you ought to: Revise italics in numbers 2, 13, … Unify the use of "In", "En", for example in numbers 9 and 11.

The bibliography has been revised and errors have been corrected.

Reviewer 2 Report

Thank you for giving me the opportunity to review this timely and interesting research paper. Congratulate the authors for their valuable contribution. Overall, the paper offers a significant contribution to the research field. 

However, there are some minor aspects that need the authors attention:
a) In my opinion, there are too many in-text citations that seem to diminish the overall originality of the paper. I think the authors should reconsider their overall usage;

b) The Introduction part is too large, compared to the Conclusions, which should, in fact, be more detailed (specific categorization of errors for Fermi problems). You should elaborate on that in much more detail. This does not seem sufficient.

I am expecting much more about the possible implementation of your findings.

Thank you. 

Author Response

We are grateful for the reviewer's comments and input, below we explain the changes made on the basis of their suggestions.

1) In my opinion, there are too many in-text citations that seem to diminish the overall originality of the paper. I think the authors should reconsider their overall usage.

The use of citations seeks to highlight the relationship of our study with other research in the area. It should be noted that this article includes part of the results of a doctoral thesis and a broader study that addresses various aspects related to the resolution of Fermi problem sequences. References to other work by the authors are necessary to put into context the rationale, design and results of the present study. To improve the readability of the work, the names of the authors of the works cited have been included in those cases where the reference was direct.

In any case, some repeated or redundant quotations have been removed. For example, a repetition of a self-citation ([41]) on p. 10, when talking about the E5 error, has been removed. It is clear without the need to refer to our previous work, which has already mentioned above.

2) The Introduction part is too large, compared to the Conclusions, which should, in fact, be more detailed (specific categorization of errors for Fermi problems). You should elaborate on that in much more detail. This does not seem sufficient.

The conclusions section has been revised, relating the conclusions to the studies cited in the theoretical framework. A paragraph has also been included indicating the limitations of the work and possibilities for future research in relation to studies by other authors (two new references to the bibliography have been included).

Reviewer 3 Report

Thanks for the opportunity to review this paper. I found it stimulating reading, as a tertiary educator of pre-service primary mathematics teachers, very relevant. I believe it makes a valuable contribution. I have a few minor suggestions for improving the paper outlined below.

General comments

• It might be useful to, just after introducing the categorisation framework (lines 114-120) to provide an example of each in a table so that the reader has a firm idea of exactly what you are referring to. Ideally all examples could be presented in the context of a single problem context (if this is possible) to help clarify how they are similar and different as strategies.
• Another point of clarification around this classification system, are two-dimensional initial models considered a subset of ‘linear models’. If so, I think this should be clarified (line 230-231).
• Upon viewing Table 3, the fact that the error classifications within a category are independent of each other (i.e., each teacher can only be classified to one of these sub-categories) is not necessarily intuitive to the reader. Is it not possible for one teacher to demonstrate, for example, both E.12 and E.13? Can you clarify how this issue was resolved? (e.g., was some sort of hierarchical system used – such as E13 is a more fundamental error than E12, so if a teacher made both, they would only be classified to E12). It may be that you decided that, for any given problem, you could only classify the response to one of the errors (i.e., the ‘major error’). Again, if this is what you did, it needs to be more clearly stated. This could be clarified in the text or with some sort of note at the bottom of the table; however, if there is a conceptual reason why these categories are independent that is central to the classification scheme, it should definitely be addressed in the text.
• One potential limitation of the study that should be highlighted: undertaking the study using semi-structured interviews would have further fleshed out whether under-development plans were at least partly a consequence of some teachers not being sufficiently motivated to articulate themselves through the questionnaire medium. For example, in an interview situation, if someone stated “We need to know the total size of the porch first. To do this, we would have to estimate the total from the length and width” you would likely then ask them “What would you do next?”. If they said they would estimate how much area an individual occupies when standing, then they would be well on the way to generating a reasonable estimate (I had a similar thought when reading lines 441-443).

  Grammar/ Typos etc:

• Line 31 – problems should be singular (same as Line 106)
• Line 33-34 – delete ‘the’ in front of tasks
• Line 45: rephrase to “a sequence of Fermi problems”
• Line 103: ‘a mathematical model’
• Lines 114 to 120: italicise all the strategies
• Lines 149 to 151 – when using the author’s name actively, I think it should be included in the text, with the reference to the number of the article in parentheses. It is done in this instance, but not in other instances.
• Line 201: ‘associated with’
• Table 2: Consider revising: “Mathematical model incoherent with the initial one” (i.e., ‘with the initial one’. It is not clear what ‘one’ means in this context (does it refer to the model?).
• Line 284: ‘for future teachers’ instead of ‘in future teachers’
• Line 459 – should ‘measure’ be ‘measurement’
• Line 607 – complexity should be complex

Author Response

We are grateful for the careful review, the comments on the limitations of the study have given us the opportunity to reflect on lines of future work, improving and completing the conclusions section. The changes made based on each of the reviewer's suggestions are detailed below.

1) It might be useful to, just after introducing the categorisation framework (lines 114-120) to provide an example of each in a table so that the reader has a firm idea of exactly what you are referring to.

The description of the categories of analysis of the strategies has been revised and completed.

2) Another point of clarification around this classification system, are two-dimensional initial models considered a subset of ‘linear models’. If so, I think this should be clarified (line 230-231).

Indeed, as the reviewer points out, this section was not sufficiently clear, and the explanation of the initial one-dimensional and two-dimensional models has been completed. However, the reference to another work in which this aspect is dealt with in detail has been maintained.

3) Upon viewing Table 3, the fact that the error classifications within a category are independent of each other (i.e., each teacher can only be classified to one of these sub-categories) is not necessarily intuitive to the reader. Is it not possible for one teacher to demonstrate, for example, both E.12 and E.13? Can you clarify how this issue was resolved? (e.g., was some sort of hierarchical system used – such as E13 is a more fundamental error than E12, so if a teacher made both, they would only be classified to E12). It may be that you decided that, for any given problem, you could only classify the response to one of the errors (i.e., the ‘major error’). Again, if this is what you did, it needs to be more clearly stated. This could be clarified in the text or with some sort of note at the bottom of the table; however, if there is a conceptual reason why these categories are independent that is central to the classification scheme, it should definitely be addressed in the text.

Certainly it was not clear how the errors were computed: it has now been clearly indicated before table 3 that for each production all the errors that appeared were counted, without establishing any kind of hierarchy. We agree with the reviewer that it is possible that there is a dependence between errors, but this aspect is beyond the scope of this study, although it is certainly of great interest. That is why, in the conclusions section, this limitation has been pointed out and noted as a line of future work.

4) One potential limitation of the study that should be highlighted: undertaking the study using semi-structured interviews would have further fleshed out whether under-development plans were at least partly a consequence of some teachers not being sufficiently motivated to articulate themselves through the questionnaire medium. For example, in an interview situation, if someone stated “We need to know the total size of the porch first. To do this, we would have to estimate the total from the length and width” you would likely then ask them “What would you do next?”. If they said they would estimate how much area an individual occupies when standing, then they would be well on the way to generating a reasonable estimate (I had a similar thought when reading lines 441-443).

We fully agree with this assessment and have included it in the last part of the conclusions. We have presented lines of future work to overcome this limitation and related them to other recent research.

  Grammar/ Typos etc:

• Line 31 – problems should be singular (same as Line 106) OK
• Line 33-34 – delete ‘the’ in front of tasks OK
• Line 45: rephrase to “a sequence of Fermi problems” OK
• Line 103: ‘a mathematical model’ OK
• Lines 114 to 120: italicise all the strategies OK
• Lines 149 to 151 – when using the author’s name actively, I think it should be included in the text, with the reference to the number of the article in parentheses. It is done in this instance, but not in other instances. Revised.
• Line 201: ‘associated with’ OK
• Table 2: Consider revising: “Mathematical model incoherent with the initial one” (i.e., ‘with the initial one’. It is not clear what ‘one’ means in this context (does it refer to the model?).

It refers to the initial model, it has been corrected.

• Line 284: ‘for future teachers’ instead of ‘in future teachers’.

This part has been actually rewritten.

• Line 459 – should ‘measure’ be ‘measurement’ OK
• Line 607 – complexity should be complex OK

Thanks!