Pre-Service Teachers’ Strategies in Solving Absolute Value Equations and Inequalities

Round 1

Reviewer 1 Report

Review

Pre-service Teachers’ Strategies in Solving Absolute Value Equations and Inequalities

The study aims to investigate strategies that pre-service teachers used while solving absolute value equations and inequalities from the perspective of symbol sense. In a qualitative case study through two written tests followed by interviews involving 36 pre-service mathematics teachers in Bandung, Indonesia. The results revealed three different solution strategies for solving either absolute value equations or inequalities and difficulties occurred during the solution processes of equations and inequalities were addressed. The researchers conclude that the symbol sense characteristics are fruitful for interpreting the selection of solution strategies used by the participants.

The researchers make clear from literature that students and teachers struggle with absolute value equations and inequalities. They account for this struggle that lack of symbol sense (see line 45), i.e., a lack of giving meaning to and see structures of equations and inequalities. From the other mentioned characteristics of symbol sense only, I think, checking the result is used in this research. I think the symbol sense lens might be used in a study like this, but in the way it is done at the moment. Drijvers et al (2012) described symbol sense with strategic work, taking a global view, and algebraic reasoning, as complementing basic skills, with its focus on procedures, local focus and calculations. In this study considering strategies and reasoning might be indicators of symbol sense. However, the tasks and solutions portrayed in this study, suggest that it is more about procedural skills. In one of the interviews, it is said that the properties strategy seems easier (to solve inequalities). This suggest that the different strategies have be taught before (in secondary school?). Also, the written examples suggest that these tasks have been studied before and have a more procedural character for these pre-service teachers. And of course, the pre teachers might have used symbol sense, but more have to be done in a study to show this symbol sense.

From a study of literature, one would expect more details about what difficulties students might have with absolute equations and inequalities. A section about theory is missing.

In the method section, some support for the choice of these equations/inequalities is needed. I would like to have some information about the teaching of this subject in Indonesia secondary school and in the teacher program.

In the results section, one would like to see what the first strategy was, the second strategy, and the third. Maybe also, the relation between correct and strategy. And maybe, correlation between the choice the first strategy with equations and with inequalities.

From the interviews the researchers might say something about the use of the graphs (did teachers who used this strategy have to correct their answer from other strategies. We also need some prove for the statement that checking was done through substitution (line 269), and that the teachers had considered the best strategy (line 289).

In the discussion the role of education in this subject should be discussed. How much training did these teachers had in this subject? The same is true for graphing these absolute value functions.

In the conclusion some issues are claimed without really proof from the results of the study (see line 307, 310, 312)

In the article some times more or less the same is told about equations and about inequalities. Then, for me, it is enough to tell it once. In this way a link between equations strategies and inequalities strategies is automatically established.  

Summary and suggestions.

Given the method and results I wonder whether the symbol sense lens will help. Some aspects from it might help: what strategies do these teachers choose first, second, third? How many strategies do they know? What arguments do they have for these choices (form interviews?)? What triggers them to choose for a particular strategy. Do they choice with equations other strategies then with similar inequalities? What kind of mistakes do they make? Also, discuss the procedural aspects of these tasks. And what does this say about the education of this subject?

I think that with these data something can be said about all these questions, and such a study is interesting for many.

Detail: link to references is difficult now.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Thank you for the paper. I found it interesting. I hope that you find the comments below useful in refining and developing your research and the papers that come from it. I will begin with some comments relating to specific lines and conclude with some broader comments.

Lines 20-27: I can see that your referencing is in line with the journal's requirements but the list of referenced papers at the end should be numbered to help the reader connect the cited reference with the details in the reference list. 

Line 28: Make it clear that you are talking about secondary level pre-service teachers. Are these maths teachers?

Line 30: ‘Most’ in the sample in the cited paper? Where was this? Not sure that you can make a global claim as you are here.

Line 32: What is ‘many’? Some more detail would be useful here.

Line 36-37: The claim that the research in this field is limited seems out of place here. You don’t explain how you have arrived at this conclusion. The fact that you follow this statement by citing more papers on the subject seems to undermine your claim.

Line 46: It might be helpful to the reader to explain why you are using the term ‘symbol sense’ rather than ‘number sense’.

Lines 47-48: Is this your definition? If not, it would be useful to cite the source of the definition directly.

Lines 70-71: There seems to be some paradox here. You are claiming that they have enough mathematical knowledge to take part in the study but are also claiming that previous studies with similar students have revealed a lack of ‘symbol sense’ (i.e. mathematical knowledge) when solving equations and inequalities. I think that your contention here needs further explanation.

Some additional information about the students might be helpful (i.e. their sex)

Line 78: Omit the word ‘one’ at the end of the line.

Lines 78-82: More detail is needed about the nature of the tasks and how you determined the degree of difficulty of each.

Line 86: Could the exact questions be provided as an Appendix?

Line 105-117: It would be useful to have an example to illustrate this solution strategies here. While this may be obvious to an audience of maths experts, others might really benefit from seeing what these different solution strategies ‘look like’. This occurs later (around Line 150 onwards) but I think it would have been useful to see this exemplified earlier.

Line 271: Friendliness with symbols and numbers; earlier it has been stated that symbol sense and number sense are synonymous. This suggests that they are different?

Line 274: This feels like an interesting finding but has been rather dismissed. While I appreciate that the author(s) suggest that this might be something to be explored in future research, it feels strange to dismiss it here. Equally, it would have been interesting to know more about the data that led to this conclusion; there isn’t much data about the success rate of the different solution strategies for example.

Line 279: I wonder what these ‘more appropriate instruments’ might be and why they were not used in this study.

Line 303: I wonder whether there are other solution strategies that the participants didn’t use, or whether there really are only three viable strategies. If so, this isn’t an especially surprising finding. It might be more interesting to explore why, of these three strategies, particular items on the test seemed to elicit the use of one strategy over another i.e. what were the features of the equation, or inequality to be solved that lent it to a particular solution strategy.

I also wonder how much of this finding is largely due to the teaching these participants had.

 

Here are some more general comments about your paper:

The lit review is rather brief. This may indicate a lack of available lit to review but if this is the case it should be stated.

It would be useful to situate the study more explicitly i.e. to say that it is being done in Indonesia.

It would be interesting to have heard from the participants about why they picked a particular strategy in a particular case (e.g. Task 2? In the equations condition). In your presentation of the findings a more detailed analysis beyond the percentage of answers using a particular strategy would have been interesting. It strikes me that a potentially interesting aspect of this data is that certain test items seemed to lend themselves to a particular solution strategy.

I wonder whether the provision of graph paper would have yielded different results i.e. would have made graphing easier and therefore encouraged the graphing strategy. It would be good to acknowledge and discuss this in the paper.

During the presentation of the data there is no mention of ‘symbol sense’. While I can see that using this as a lens to explore the participants’ solution strategies, it needs to be more integrated into the discussion and into the data presentation. There didn’t seem to be any measuring of ‘symbol sense’ in any way.

I wonder if some of the findings (e.g. that the participants mainly used the graphing strategy as a means of checking) are due to the ways that this particular group of participants has been taught. Would a different group of participants having been taught in a different way have produced different results? It would have been interesting to know whether the authors of the paper were also responsible for teaching the students the mathematics that was on the test?

There is very little discussion about the selection of the sample i.e. was it self-selecting. Did they volunteer? Did they receive any kind of course credit for participating? Why were only 5 interviewed? Some additional information about the sample and about the process for recruiting participants would have been useful.

The section on the analysis of the data is rather limited. There is no real discussion about how the interview data was analysed. Did more than one person analyse the data? Were there particular themes that emerged from the discussion? What questions were asked? The paper would really benefit from some additional transparency in this regard. 

I hope that the comments do not sound overly harsh. There are positive features to this paper and the research in general is worthy of pursuing. I hope that the comments help you to refine and develop your work further. 

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

I thank the authors very much for submitting this interesting work to Education Sciences. I believe that the article is in a good shape in general. However, there is one aspect that I would like to highlight: I miss a more profound integration of your research against the backdrop of mathematics education. In particular, I would like you to contextualize your study within related research into teaching and learning about algebraic concepts. In recent time, there has been a lot of research into teaching and learning algebra, for example on algebraic thinking, abstract thinking, abstract algebra, group theory and so on. Results on these topics may impact your research and vice versa. Hence, to make things complete, you should state as to how it fits into similar lines of mathematics education research, both in terms of methods, in terms of results and in terms of implications for teaching and learning mathematics. 

Author Response

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Round 2

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Thank you for resubmitting your article. 

I can see that you have made some revisions and these have improved the paper. However, I feel that there are a number of areas that require some further refinement and explanation. 

1. The interview data is still rather weak. The paper would benefit from some further discussion about the questions you asked and how the responses were analysed. 

2. Most significantly, the discussion about symbol sense is not yet fully integrated into the findings. This means that the whole paper is rather lacking in a solid theoretical underpinning. The choice of symbol sense is a good one but needs to be explained in more detail and integrated throughout the discussion. You say that symbol sense is 'analogous' to number sense but then state that the two are different. This needs unpicking further for the theoretical underpinning of the paper to be stronger. 

I do wish you all the best with your research work and with your highly valuable work in training the next generation of teachers. 

Author Response

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Reviewer 3 Report

My earlier concerns also hold true for this version of the manuscript. The authors have not accordingly addressed my issue: I see many parallels with the results of this research to earlier research in mathematics education that has not been referred to here. I ask the authors to consider recent research into algebra education. In particular, I would ask them to associate research they conducted for the specific case of inequalities to the overarching and more general question of dealing with algebra problems. Learning difficulties that students encounter when being taught abstract algebra in general may inform future research also for the specific cases, e.g., regarding inequalities. 

Hence, I must ask the authors to take into account and discuss recently published research into t/l abstract algebra to contextualize their research. The manuscript might also profit from a sharper connection of this article to earlier research published in the same journal - which would support why the authors submitted their article to Educ. Sci!

Author Response

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Round 3

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

Thank you for this submission. I think that the paper has evolved considerably since it was first submitted. I think that the inclusion of more specific reference to symbol sense in the discussion of the data is very helpful. Just a couple of very minor points:

Line 86 – Remove the word ‘one’.

Line 112 – Provide more detail about the criteria used to select the participants for the interview. You say that it was ‘based on their written work’ but have not stated what the criteria were. This need only be a couple of sentences. 

Line 397 – ‘we view the use of a graph strategy …’

Author Response

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Reviewer 3 Report

the paper has been improved according to my comments.
Congratulations to the authors. 
the paper is now ready for publication. 

Author Response

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