Cognitive Obstacles in Engineering Students’ Mathematical Modeling of Derivatives: Insights from Skippy, Switcher, and Floater
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
This manuscript addresses the cognitive obstacles faced by engineering students when engaging with applied derivative problems. By combining the Mathematical Modeling Cycle (MMC) with Duval’s theory of semiotic registers, the study provides a theoretically grounded and pedagogically analysis. Below, I provide major and minor recommendations.
1) The review of empirical studies could be expanded. Recent contributions on students’ use of digital tools and dynamic software in modeling (e.g., GeoGebra, MATLAB) could enrich the discussion of representational transitions. Similarly, more international comparative studies in engineering mathematics education could situate the findings within a broader context:
- Casalvieri, C., Gambini, A., Spagnolo, C., & Viola, G. (2023). The concept of derivatives through eye-tracker analysis. In Proceedings of the 15th International Conference on Computer Supported Education-Volume 2 (Vol. 2, pp. 378-385). Science and Technology Publications.
- Sahin, Z., Yenmez, A. A., & Erbas, A. K. (2015). Relational understanding of the derivative concept through mathematical modeling: A case study. Eurasia Journal of Mathematics, Science and Technology Education, 11(1), 177-188.
2) The empirical base is limited to three detailed cases out of 30 students. While the depth of analysis is commendable, the representativeness of these cases should be further problematized. The authors should clarify whether the three students were chosen for typicality or for diversity of errors, and what this implies for the generalizability of the findings.
3)The distinction between “challenges” and “cognitive obstacles” is important, but at times the boundaries remain blurred. For instance, difficulties in graphing constraints are framed as obstacles; however, these may also be interpreted as didactical challenges arising from insufficient practice rather than strictly cognitive barriers. A clearer justification for classifying each as an obstacle would strengthen the argument.
4) The authors may consider balancing these narrative devices with more neutral descriptors in the main text, reserving the metaphors for illustrative purposes.
5) While the paper highlights students’ limited engagement in validation, the analysis would benefit from more fine-grained attention to what counts as “validation.” For instance, distinguishing between minimal validation (rejecting negative solutions) and robust contextual validation (assessing feasibility within real-world constraints) would enhance the pedagogical insight.
The manuscript is a valuable contribution to research on engineering mathematics education. Before publication, the authors should (1) expand the literature review, (2) refine the discussion of sample representativeness, (3) sharpen the distinction between challenges and cognitive obstacles, and (4) streamline the theoretical background.
Author Response
Comments 1: Expanding the Literature Review
The review of empirical studies could be expanded. Recent contributions on students’ use of digital tools and dynamic software in modeling (e.g., GeoGebra, MATLAB) could enrich the discussion of representational transitions. Similarly, more international comparative studies in engineering mathematics education could situate the findings within a broader context:
Casalvieri, C., Gambini, A., Spagnolo, C., & Viola, G. (2023). The concept of derivatives through eye-tracker analysis. In Proceedings of the 15th International Conference on Computer Supported Education-Volume 2 (Vol. 2, pp. 378-385). Science and Technology Publications.
Sahin, Z., Yenmez, A. A., & Erbas, A. K. (2015). Relational understanding of the derivative concept through mathematical modeling: A case study. Eurasia Journal of Mathematics, Science and Technology Education, 11(1), 177-188.
Response: Thank you for this constructive comment. We agree that expanding the empirical context enriches the discussion. Therefore, we have integrated a dedicated section to address recent contributions concerning students’ use of digital tools and dynamic software in mathematical modeling and to situate our findings within international comparative studies in engineering mathematics education.
The expansion is detailed in the new Subsection 2.2.2, titled “Digital tools and International Perspectives,” which can be found on pages 6–7 of the revised manuscript.
This new subsection serves to:
Discuss how digital environments support the representational transitions critical to Duval’s (2017) theory, aligning with the suggestion to incorporate dynamic software studies.
Specifically reference the two suggested papers: Casalvieri et al. (2023) is discussed in relation to eye-tracking and the conceptualization of derivatives, and Sahin et al. (2015) is cited to support the concept of relational understanding and connecting "big ideas."
Introduce international perspectives, citing works on digital literacy and global trends in STEM education (e.g., Hillmayr et al., 2020; Chetty et al., 2018), which broadens the context of our findings on cognitive obstacles.
We believe this addition significantly strengthens the theoretical and empirical foundation of the paper.
Comments 2: Clarification Case Selection and Representativeness
The empirical base is limited to three detailed cases out of 30 students. While the depth of analysis is commendable, the representativeness of these cases should be further problematized. The authors should clarify whether the three students were chosen for typicality or for diversity of errors, and what this implies for the generalizability of the findings.
Response: Thank you for raising this important point regarding case selection and representativeness. We fully agree that the methodology should be clarified to properly frame the generalizability of our findings. Therefore, we have added a clarification in the Materials and Methods section, specifically within Subsection 3.5 (Participants), which can be found on page 14 of the revised manuscript. We believe this clarification addresses the comment by explicitly detailing the rationale for the case selection and its implications for the study's generalizability.
Comments 3 The distinction between “challenges” and “cognitive obstacles
The distinction between “challenges” and “cognitive obstacles” is important, but at times the boundaries remain blurred. For instance, difficulties in graphing constraints are framed as obstacles; however, these may also be interpreted as didactical challenges arising from insufficient practice rather than strictly cognitive barriers. A clearer justification for classifying each as an obstacle would strengthen the argument.
Response: Thank you for this insightful comment. We recognize that the distinction between "challenges" and "cognitive obstacles" is crucial to our argument, and we agree that a clearer justification is needed to avoid blurring the boundaries. Therefore, we have significantly revised the final part of Subsection 2.4 (Distinction Between Challenges and Cognitive Obstacles), which can be found on pages 8-10 of the revised manuscript.
Specifically, the clarification now includes:
Conceptual Distinction: We explicitly define a didactical challenge as a difficulty arising from limited prior exposure, insufficient guided practice, or instructional design gaps (e.g., simple inaccuracies in graphing). In contrast, a cognitive obstacle is defined as a more fundamental conceptual blockage that persists because it originates from the learner’s internal representational system (e.g., an inability to coordinate symbolic and graphical registers).
Framing the Findings: We clarify that within this study’s framework, the obstacles are treated as hypothetical constructs—analytically inferred from students’ reasoning patterns (their inability to transition registers)—rather than empirically verified.
Future Research: We acknowledge that our study constitutes an initial step and explicitly state that a subsequent qualitative study is being designed to gather empirical evidence to provide a finer-grained and evidence-based justification of this distinction.
MMC Link: We emphasize how the Mathematical Modeling Cycle (MMC) helps clarify the distinction, as difficulties emerging in specific stages can be directly linked to the semiotic register gaps highlighted by Duval’s (2017) framework.
This revision refines the theoretical framing and acknowledges the need for further empirical work to substantiate the distinction, thereby strengthening the foundation of our argument.
Comments 4: Balancing narrative devices
The authors may consider balancing these narrative devices with more neutral descriptors in the main text, reserving the metaphors for illustrative purposes.
Response: Thank you for this suggestion. We agree that relying heavily on the metaphorical nicknames ("Skippy," "Switcher," and "Floater") throughout the main body of the paper could detract from the objective, academic tone. Therefore, we have conducted a revision throughout Section 4 (Findings) and Section 5 (Discussion) (starting on page 23 and continuing through page 26) to balance the narrative devices. The specific change implemented is:
Retention for Introduction: The metaphorical nicknames are retained only for the first mention of each case in Section 3.4 (Data Analysis) and Section 4 (Findings) to serve their intended purpose as memorable, illustrative descriptors of the student's primary modeling behavior.
Shift to Neutral Descriptors: In all subsequent analysis paragraphs within the body of Section 4 and throughout the entire Section 5 (Discussion), we now primarily use the neutral, descriptive term "the student" when referring to the individual case while clearly attributing the findings to the specific case (e.g., "The student's entry into the modeling process (Skippy)..." or "Across all three cases, variable definition proved unstable. Skippy, for example, introduced the variable x...").
This ensures the paper maintains a formal tone in its analytical sections while still leveraging the descriptive power of the metaphors for initial introduction.
Comments 5: Attention to what counts as “validation.”
While the paper highlights students’ limited engagement in validation, the analysis would benefit from more fine-grained attention to what counts as “validation.” For instance, distinguishing between minimal validation (rejecting negative solutions) and robust contextual validation (assessing feasibility within real-world constraints) would enhance the pedagogical insight.
Response: Thank you for this excellent comment. We agree that a more fine-grained attention to what counts as "validation" will significantly enhance the pedagogical insights derived from our findings. Therefore, we have introduced a typology of validation early in the Findings section and applied this enhanced language to the case analyses and the subsequent Discussion. Specifically, we have made the following changes:
New Validation Typology: A new paragraph has been inserted at the beginning of Section 4 (Findings), found on page 15, which defines three distinct types of validation observed:
- Minimal validation: Superficial or procedural checks (e.g., rejecting negative values).
- Structural validation: Internal mathematical consistency checks (e.g., using the second derivative).
- Contextual validation: Assessing the real-world plausibility and coherence of results within the task context (the most robust form).
Application in Case Analyses: We have revised the "Interpretation and Validation" tables (e.g., Table 2.3 for Skippy) and the accompanying narrative to explicitly use these terms when describing each student's effort (e.g., noting that "Skippy" performed minimal validation and structural validation, but lacked contextual validation).
Enhanced Discussion and Conclusion: This refined terminology strengthens the argument in Section 5 (Discussion) and Section 6 (Conclusion) by allowing us to more precisely frame the cognitive obstacle as a deficit in the Contextual Validation stage of the MMC.
We hope this clarification significantly enhances the quality of our analysis and the pedagogical relevance of the study.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for Authors
A brief summary
The main objective of this article is to characterise the cognitive processes and challenges and obstacles encountered by Industrial Engineering and Management students when solving applied derivative problems, using the Mathematical Modelling Cycle (MMC) and Duval's semiotic register theory as analytical structures.
This is a qualitative investigation with a case study design. The students' written responses to questions from an applied optimisation exam were analysed. The description of the aspects to be considered in the data analysis is very clear and detailed.
The solutions presented by three students who typify different patterns of cognitive engagement were studied in detail.
One of the strengths of this study is related to the contribution it can make to the field of education, helping to understand what patterns of cognitive engagement can be expected and what factors may constitute an obstacle to learning. In this sense, it allows teachers to anticipate appropriate strategies that can help students overcome these difficulties and, consequently, help students develop fundamental mathematical skills.
General comments
Overall, the article is well structured, beginning with an introduction, followed by the theoretical framework focused on three topics of extreme importance to the study: (1) The Role of Mathematics in Engineering Education: A Focus on Mathematical Modelling; (2) Mathematical Modelling and Its Role in Education; Duval's Theory of Cognitive Processes in Mathematical Learning: Semiotic Registers; the methodology used in the study; the results; the discussion and the conclusion.
The topic is relevant and the study is pertinent as it can contribute to teaching practices that are better suited to needs and help students to become more proficient in mathematics and learn more effectively.
The literature review seems adequate, consistent and highlights aspects directly related to the questions that the authors set out to answer in this study.
Some of the bibliographical references are current, with about half being less than 10 years old and a third less than 5 years old.
The theoretical framework section, particularly the subsection “Duval's Theory of Cognitive Processes in Mathematical Learning”, should include references, especially to the author's works, including in the table. The following section, “Distinction Between Challenges and Cognitive Obstacles”, should also include relevant and current references on the subject.
The tables and figures used in the article are important visual elements, facilitating consultation and understanding of the information by the reader. However, some tables are too large and, in addition to detracting from the aesthetics of the article, are split across more than one page, ultimately having the opposite effect to that expected when these visual elements are used in a text. Therefore, it is suggested that the tables in the results section be subdivided into smaller tables.
The results are presented in an organised manner and aligned with aspects that are the subject of analysis to answer the research questions.
The discussion is also very well organised, based on the two defined objectives: to characterise the cognitive processes evidenced by students and to identify the cognitive obstacles that hinder quality learning.
However, both in the discussion and in the conclusions, there is a lack of cross-referencing of the results of this study with data obtained in similar studies and with what is mentioned in the literature on this subject. To improve the quality of the article, particularly the discussion of the results and conclusions, it is suggested that this aspect be considered in the reformulation.
The references used in the text do not follow the same standards as the references used in the final list. It is suggested that adjustments be made to ensure consistency and compliance with the journal's guidelines for authors. In addition, although it is a minor detail, it is suggested that when there is more than one reference in parentheses to support an idea, the authors of these references should appear in alphabetical order.
Author Response
Comments 1: The theoretical framework section, particularly the subsection “Duval's Theory of Cognitive Processes in Mathematical Learning”, should include references, especially to the author's works, including in the table. The following section, “Distinction Between Challenges and Cognitive Obstacles”, should also include relevant and current references on the subject.
Response: Thank you for your constructive feedback. We agree entirely that the theoretical framework and the distinction section required stronger, more explicit referencing to ground our analysis. We have addressed this comment through the following specific revisions in Section 2:
- Duval's Theory of Cognitive Processes in Mathematical Learning (Section 2.3)
We have significantly strengthened this subsection by explicitly including core references to the author's key works in the text and in the accompanying table.
- Distinction Between Challenges and Cognitive Obstacles (Section 2.4)
We have heavily revised this section to integrate relevant and current references that underpin the proposed distinction, including the references we discussed previously.
We hope that these revisions ensure that both the theoretical foundation and the analytical distinction are rigorously supported by authoritative and current literature in mathematics education. The corresponding full references have also been verified and updated in the final reference list (Section 7).
Comments 2: The tables and figures used in the article are important visual elements, facilitating consultation and understanding of the information by the reader. However, some tables are too large and, in addition to detracting from the aesthetics of the article, are split across more than one page, ultimately having the opposite effect to that expected when these visual elements are used in a text. Therefore, it is suggested that the tables in the results section be subdivided into smaller tables.
Response: Thank you for this essential comment. We agree that the large tables in the results section significantly compromise the article's aesthetics and readability, particularly when they are split across pages. The goal of using visual elements is to enhance comprehension, and we acknowledge that the current format is counterproductive.
Therefore, we have subdivided the original comprehensive tables for each case study (e.g., the original Table 2 was too large) into three smaller, focused tables. This change is implemented throughout Section 4 (Findings), starting on page 16 of the revised manuscript.
For each of the three case studies (Skippy, Switcher, and Floater), the single large table has been replaced by the following three smaller, logically grouped tables:
- Contextual Literacy in the Modeling Process (Focusing on Contextual Situation and Contextual Model stages)
- Mathematical Literacy in the Modeling Process (Focusing on Mathematical Model and Mathematical Outcomes stages)
- Interpretation and Validation in the Modeling Process (Focusing on Contextual Meaning of Outcomes and Return to Contextual Situation stages)
We hope that this subdivision not only improves the aesthetics and flow but also enhances the analytical presentation by clearly separating the cognitive efforts related to contextualization, mathematical work, and validation, making the findings easier for the reader to consult and understand.
Comments 3: The discussion is also very well organised, based on the two defined objectives: to characterise the cognitive processes evidenced by students and to identify the cognitive obstacles that hinder quality learning. However, both in the discussion and in the conclusions, there is a lack of cross-referencing of the results of this study with data obtained in similar studies and with what is mentioned in the literature on this subject. To improve the quality of the article, particularly the discussion of the results and conclusions, it is suggested that this aspect be considered in the reformulation.
Response: Thank you for this very important comment. We fully agree that cross-referencing the study's results with existing literature in the Discussion and Conclusions is essential for strengthening the academic quality of the manuscript and situating our findings within the broader field. We had previously addressed this by revising both sections to incorporate the necessary external references. Specifically, the changes were made, can be found in Section 5 (Discussion) (starting on page 23) and Section 6 (Conclusion) (starting on page 26) of the revised manuscript.
Comments 4: The references used in the text do not follow the same standards as the references used in the final list. It is suggested that adjustments be made to ensure consistency and compliance with the journal's guidelines for authors. In addition, although it is a minor detail, it is suggested that when there is more than one reference in parentheses to support an idea, the authors of these references should appear in alphabetical order.
Response: Agree. Thank you for pointing out the inconsistency in our in-text citation formatting and the need for alphabetical ordering when citing multiple references—these are important details for journal compliance. We have implemented a thorough review and correction process to ensure full consistency with APA 7th edition standards and the journal's guidelines throughout the manuscript.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for Authors
Accepted in present form
