Cognitive Obstacles in Engineering Students’ Mathematical Modeling of Derivatives: Insights from Skippy, Switcher, and Floater

Abstract

Mathematical modeling competency is essential for engineering students, yet significant cognitive obstacles impede their ability to apply theoretical concepts like derivatives to real-world optimization problems. This study investigates the cognitive processes and obstacles encountered by Industrial Engineering and Management students when solving applied derivative problems, utilizing the Mathematical Modeling Cycle (MMC) and Duval’s theory of semiotic registers as analytical frameworks. A qualitative case study design was employed, analyzing students’ written exam responses to an applied optimization task involving tour organization with variable pricing structures. Three representative cases were examined in detail, revealing distinct patterns of cognitive engagement. Results identified specific cognitive obstacles including misunderstanding of variables and domains, weak connections between mathematical and economic contexts, difficulties in graphical representation of constraints, and deficits in validation and critical thinking. While students demonstrated procedural fluency in symbolic manipulation and mathematical work, they struggled to coordinate between different semiotic registers (verbal, algebraic, graphical, and contextual) and failed to complete the full modeling cycle, particularly in the crucial validation stages. These findings suggest that cognitive obstacles stem from representational gaps rather than general learning difficulties, indicating the need for targeted pedagogical interventions that explicitly address transitions between semiotic registers and emphasize the iterative nature of mathematical modeling in engineering contexts.

1. Introduction

Mathematics is an indispensable tool for engineers, underpinning virtually every aspect of modern engineering practice and innovation (Pepin et al., 2021). Its critical role in engineering education is widely acknowledged, serving as the foundational language for analysis, design, and problem-solving in complex, real-world scenarios. There is a growing body of research exploring the mutual connection between mathematical modeling (MM) ability and learning mathematics among engineering students (Pepin et al., 2021). This research generally shows that engaging in MM not only helps students apply mathematics to real-world problems but also deepens their conceptual understanding and mathematical competencies (Blomhøj & Jensen, 2007). The integration of MM into pedagogical approaches provides a structured mechanism through which this deepening of understanding and enhanced engagement can occur.

Calculus concepts are particularly important for engineers to master. They provide the quantitative language for understanding dynamic systems, optimization, and change. Throughout their foundational calculus courses, students become familiar with the engineering language of derivatives, Riemann sums, and integrals, enabling them to solve applied word problems and translate them into mathematical language. Within the Industrial Engineering and Management academic program, calculus is directly linked to professional relevance. For example, the “extrema problem” illustrates how derivatives are applied to optimize profit, cost, or other variables in business and management contexts. Such applications demand not only computational skill but also critical thinking, careful interpretation of problem statements, clear variable definition, identification of relevant constraints, and contextual validation of results. This highlights the inherent complexity of mathematical modeling in engineering, where theoretical knowledge must be flexibly adapted to diverse and often ambiguous real-world situations.

However, the pedagogical approach to mathematics for engineering students necessitates a distinct methodology compared to that for pure mathematics students. Engineering disciplines demand that mathematical concepts be understood not merely as abstract theories but as practical instruments (Novikov et al., 2019). This requires a pedagogical focus on translating abstract mathematical concepts into practical applications and interpreting results within an engineering framework, thereby developing competencies that directly support professional practice. The Mathematical Modeling Cycle (MMC) (Leikin et al., 2025) directly embodies this pedagogical philosophy by providing a structured, cyclical process for precisely this translation and interpretation. This framework is not merely a supplementary tool but a formal representation of the very approach engineering disciplines requires for mathematical understanding. Despite its undeniable importance, the acquisition of mathematical proficiency often presents significant challenges for engineering students. This sets the stage for a deeper investigation into the specific difficulties encountered in this crucial educational domain.

Solving applied problems in calculus—particularly those involving derivatives—further requires students to coordinate between multiple representational forms: symbolic, graphical, verbal, and contextual. This dimension is captured by Duval’s theory of semiotic registers (Duval, 2017), which explains how mathematical comprehension depends on the ability to operate across and transition between such registers. When students fail to coordinate representations—for example, between symbolic manipulation of derivatives and their graphical or contextual meaning—conceptual obstacles emerge.

This study therefore adopts a dual theoretical perspective, drawing on the MMC as a process lens and Duval’s framework as a representational lens. Together, these perspectives enable a comprehensive analysis of both where students encounter difficulties in the modeling process and why these breakdowns occur by tracing the representational transitions that hinder engagement with applied derivative problems.

1.1. The Research Gap and Problem Statement

While the importance of mathematical modeling is widely recognized, there is a critical need to understand how students genuinely engage with mathematical modeling beyond superficial or procedural application. This raises fundamental questions: What does “engaging” truly mean in this context? And how can such engagement be effectively fostered? The specific cognitive processes, challenges, and particularly the cognitive obstacles engineering students encounter when applying these models (especially in complex scenarios like derivative problems) are not yet fully understood.

It has been posited (see next section for the details) that when students encounter specific cognitive barriers that impede meaningful comprehension and problem-solving, their capacity for deep, conceptual engagement with mathematical content becomes constrained. Superficial engagement, such as memorizing formulas without understanding their underlying principles, may occur, but genuine, transformative engagement is hindered. Therefore, identifying and analyzing these cognitive barriers provides crucial diagnostic information, which can inform the design of interventions aimed at fostering genuine engagement by enabling students to move beyond superficial interactions toward profound conceptual understanding.

1.2. The Paper’s Structure

The remainder of this paper is structured as follows: Section 2 presents the theoretical background, primarily focusing on Duval’s theory of semiotic registers and its relevance to understanding student learning in calculus. Section 3 details the methodology employed in this research, including participant selection and data collection procedures. Section 4 presents the findings from the analysis of students’ responses to applied derivative problems. Section 5 discusses these findings in light of Duval’s framework, identifying specific cognitive obstacles. Finally, Section 6 offers conclusions and proposes educational implications derived from the study.

1.3. Contribution of the Study

As part of a broader framework, this research investigates how Industrial Engineering and Management students approach and solve applied problems involving the concept of a function’s derivative, identifying the key challenges and obstacles they encounter. By analyzing students’ methods of tackling verbal and applied calculus problems, the study seeks to deepen understanding of the cognitive and didactic processes involved in learning these concepts. The research aims to bridge mathematical theory with practical applications, offering insights into improving calculus education, and proposes strategies to enhance students’ mathematical and applied skills in engineering contexts.

2. Theoretical Background

This section delineates the theoretical foundations underpinning the study. It first examines the role of mathematics within engineering education, with particular attention to mathematical modeling as a central professional and pedagogical construct. Subsequently, it considers the broader educational significance of mathematical modeling, emphasizing its function in fostering authentic problem-solving and the development of higher-order competencies. The discussion then introduces Duval’s theory of cognitive processes and semiotic registers as an analytical framework for investigating students’ engagement with mathematical concepts. Finally, a distinction is drawn between general learning challenges and specific cognitive obstacles, thereby establishing the conceptual basis for interpreting the empirical findings of the study.

2.1. The Role of Mathematics in Engineering Education: A Focus on Mathematical Modeling

Mathematics has long been a fundamental element of engineering education, serving both as an entry criterion and as a cornerstone of engineering knowledge (Reisel et al., 2012). Historically, the inclusion of mathematics in engineering curricula reflected the profession’s 18th- and 19th-century origins, when mathematical competence was considered essential for addressing practical technical challenges and underpinning scientific and industrial progress. Early engineering programs frequently mirrored mathematicians’ curricula in their emphasis on analytical rigor and abstraction (Burlbaw et al., 2013).

Over time, the distinction between mathematics for mathematicians and mathematics for engineers became more pronounced. While pure mathematics emphasizes abstraction, formal proof, and general frameworks, engineering mathematics is oriented toward applying mathematical concepts to physical systems, design processes, and real-world problem solving (Murakami, 1988). This distinction has been reinforced by increasing reliance in engineering on computational tools, simulation, and visualization. In this frame, mathematics is positioned not as an academic pursuit per se, but as an instrumental tool for modeling and interpreting real-world phenomena.

2.1.1. Perspectives on Mathematics Instruction for Engineers

The unified mathematics view perceives mathematics as an intellectual domain whose theoretical coherence and abstraction merit study in their own right. Proponents argue that theoretical depth affords engineers the flexibility to confront novel technological challenges beyond the scope of formal curricula. This outlook significantly influenced early engineering curricula, wherein mathematics education was nearly identical to university-level pure mathematics. In this case, the intellectual habits forged by abstract mathematical thinking are seen as transferable to engineering practice.

In contrast, the engineering mathematics perspective asserts that engineering mathematics instruction should aim primarily at enabling students to apply mathematical tools effectively in engineering contexts. The emphasis is on numerical methods, approximations, computational techniques, and model development directly tied to engineering problems (Murakami, 1988). As Cardella (2010) notes, engineering mathematics is best understood as a technical language for communication and problem-solving rather than as an abstract discipline (Cardella, 2010). This applied orientation became increasingly prominent in the 20th century, as engineering education specialized and integrated technology-driven content. These two perspectives reflect a longstanding curricular tension between theoretical depth and practical relevance.

2.1.2. Mathematical Modeling as a Core Aspect of Engineering Mathematics

Mathematical modeling (MM) encompasses the translation of real-world issues into mathematical representations, analysis by theoretical or computational means, and interpretation of results within their original context (Cardella, 2010). It is inherently iterative: assumptions are refined, models modified, and analysis repeated as empirical or experimental data become available. Modeling thus mirrors the cognitive and procedural reasoning central to engineering.

Modeling also combines creativity with logical rigor: it requires not only technical fluency in manipulating mathematical structures, but also the ability to abstract and conceptualize essential features of physical systems (Kilpatrick, 1918). This duality elevates modeling beyond mere calculation into a primary mode of engineering thought.

Engagement with MM has been shown to enhance conceptual understanding, problem-solving abilities, and motivation by situating mathematics within meaningful engineering contexts (Etemi et al., 2024; Xu et al., 2025). Modeling tasks require students to interpret and validate quantitative results, assess model limitations, and make decisions—activities aligned with professional engineering practice (Cardella, 2010).

Modeling also encourages interdisciplinary integration, drawing together mathematical, physical, computational, and systems thinking (Cardella, 2010). Historical scholarship emphasizes that abstraction from practical problems has been an enduring characteristic of engineering practice, from applied mechanics to structural design (e.g., Blum, 2015).

Engineering-specific empirical research supports these claims: for instance, Merck et al. (2021) found that undergraduate engineering students perceived online hydrological modeling modules as improving their modeling skills, conceptual understanding, and use of modeling technologies—though technological challenges sometimes impeded learning (Merck et al., 2021). Czocher et al. (2021) also discusses how integrating MM into engineering curricula can support student persistence and understanding. Finally, Titu et al. (2024) developed an aerospace-industry-specific mathematical model connecting human resources expertise to deliverable quality, exemplifying real-world modeling applications.

The centrality of modeling aligns with evolving scientific methodology; wherein computational modeling now complements theory and experimentation as a core mode of inquiry. In engineering, numerical simulation and algorithmic reasoning increasingly substitute for physical prototypes, accelerating design cycles and innovation and further reinforcing the importance of modeling-oriented mathematical preparation.

Engineering curricula around the globe have increasingly integrated MM through dedicated courses, computational labs, and problem-based learning approaches (Kerr, 2015; Sanchez-Lopez et al., 2023). Analyses of curricular evolution reveal that modeling and computational methods became mainstream in engineering programs following the advent of digital computing in the mid-20th century. More recent literature indicates that modeling-focused curriculum practices support the transfer of mathematical knowledge into engineering contexts and foster a balanced development of conceptual and practical competence (Merck et al., 2021; Wieman & Freeman, 2014; Xu et al., 2025).

Considering MM as a crucial element in engineering education, the subsequent section addresses didactical strategies and pedagogical frameworks designed to integrate modeling into engineering students’ learning processes, supporting their progression toward competence in applied problem-solving and design.

2.2. Mathematical Modeling and Its Role in Education

Mathematical modeling has emerged as a central element of mathematics education worldwide, reflecting the growing emphasis on equipping students with the ability to apply mathematics to authentic, real-world situations. Its importance stems not only from its direct application in fields such as science, technology, engineering, and mathematics (STEM), but also from its contribution to cultivating critical thinking, creativity, and flexible problem-solving abilities essential in 21st-century life and work (Monteiro & de Carvalho, 2021; Niss, 2012).

Recent years have witnessed significant growth in MM research and its inclusion in educational curricula. Bibliometric studies reveal a rapid increase in publications related to MM, particularly after 2017, with a peak in 2021, reflecting its establishment as a mature field of inquiry within mathematics education. This expansion reflects not only heightened academic interest but also policy-level recognition, as many countries explicitly include modeling competencies within national mathematics curricula (Borromeo Ferri, 2018; Kaiser & Stender, 2013).

MM offers a transformative perspective on mathematics teaching and learning. Traditional instruction often emphasizes procedural fluency and abstract problem-solving, frequently relying on simplified, decontextualized word problems (Nolen et al., 2023; Tall & Razali, 1993). These approaches tend to produce mechanical solutions, limiting students’ ability to transfer knowledge to unfamiliar contexts and to engage in higher-order reasoning (Duval, 2006). In contrast, MM situates mathematics within authentic and often ill-defined contexts, requiring students to make assumptions, negotiate constraints, and develop mathematical tools that may be adapted or iteratively refined (Gainsburg, 2006; Monteiro & de Carvalho, 2021). This redefinition of problem-solving positions MM not merely as an application of known mathematics but as a generator of new mathematical insights and innovative solutions.

Nevertheless, MM research remains theoretically fragmented, with no single, universally accepted modeling framework (Borromeo Ferri, 2018). Researchers often adopt diverse perspectives—applied, educational, sociocritical, epistemological, pedagogical, or conceptual—leading to varied interpretations and implementations (Kaiser & Stender, 2013). While this diversity enriches the field, it poses challenges for cumulative knowledge-building and practical application. Addressing this issue requires integrating multiple perspectives into coherent frameworks that can inform both curriculum design and classroom practice (Niss, 2012).

2.2.1. Mathematical Modeling in Engineering Education

Engineering practice inherently involves modeling to understand real-world systems, create mathematical representations, analyze designs, and make informed decisions under constraints (Gainsburg, 2006; Hoyles et al., 2001). Therefore, engineering education places particular emphasis on mathematical modeling, not simply as an academic exercise but as a professional competency that enables students to transition from theoretical learning to professional practice, where problems are frequently ill-defined, open-ended, and multidisciplinary (Blum & Leiss, 2007).

The Mathematical Modeling Cycle (MMC) is widely adopted as a pedagogical tool to structure this learning process. It commonly comprises five stages: (1) understanding and simplifying a real-world situation, (2) developing a conceptual or “real” model, (3) translating it into a mathematical model, (4) performing mathematical analyses and obtaining results, and (5) interpreting, validating, and, if necessary, refining results within the real-world context (Blum & Leiss, 2007; Kaiser & Stender, 2013).

Recent research extends the role of modeling beyond solving predefined problems, emphasizing problem posing as an integral and iterative element of the modeling process (Leikin et al., 2025; Silver, 2013). Problem posing involves formulating new questions or reframing existing ones, a skill particularly valuable in engineering, where complex systems often resist straightforward solutions. Studies indicate that expert instructional designers naturally align their problem-posing activities with the stages of the modeling cycle, demonstrating the close relationship between contextual understanding and mathematical reasoning (Leikin & Ovodenko, 2024).

Furthermore, effective modeling requires two complementary literacies: contextual literacy (understanding the physical or professional context) and conceptual literacy (the ability to manipulate mathematical concepts effectively) (Leikin & Ovodenko, 2024). Engineering education, therefore, increasingly emphasizes authentic, cognitively rich modeling tasks that reflect professional engineering practice, integrating both contextual interpretation and mathematical reasoning (Gainsburg, 2006; Monteiro & de Carvalho, 2021).

2.2.2. Digital Tools and International Perspectives

Recent international research has expanded the understanding of representational coordination within digital learning environments, further supporting Duval’s theoretical claims regarding the role of semiotic registers in mathematical reasoning.

Digital environments such as GeoGebra, MATLAB, or other dynamic simulation tools provide new opportunities to support these representational transitions. They enable learners to manipulate algebraic expressions, visualize corresponding graphs, and analyze numerical data interactively, thus facilitating the conversion processes that are central to meaning-making in applied calculus. From this perspective, digital tools not only serve as visualization aids but also as mediating artifacts that promote cognitive flexibility and cross-register fluency. Such affordances are particularly relevant for identifying and addressing representational obstacles that may hinder conceptual development, thereby linking empirical findings on digital environments to the theoretical foundations of semiotic representation proposed by Duval.

Digital technologies have become central to teaching and learning in science and mathematics education. Hillmayr et al. (2020) conducted a comprehensive meta-analysis of 92 studies from multiple countries and found a medium, statistically significant positive effect of digital tools on students’ achievement in secondary education. Their analysis revealed that the effectiveness of digital integration is strongly influenced by contextual factors. Teacher training, for instance, served as a critical moderator: interventions preceded by professional development demonstrated higher gains compared to those without training. The type of digital tools also mattered, with intelligent tutoring systems and dynamic simulations yielding substantially higher effects than hypermedia systems. These findings suggest that digital tools are most effective when used to complement, rather than replace, traditional instruction. Data from the International Computer and Information Literacy Study (ICILS) further showed that while 87% of teachers believe digital tools help students learn at appropriate levels, 37% consider them distractions, underscoring the need for teacher confidence, pedagogical digital literacy, and institutional support in technology implementation.

The expansion of digital learning has simultaneously intensified global attention toward measuring and promoting digital literacy. Chetty et al. (2018) argued that the digital divide persists, particularly in low- and middle-income countries, due to infrastructural deficits and limited digital skills. They conceptualized digital literacy as a multidimensional construct encompassing information, computer, media, communication, and technology literacies, assessed through technical, cognitive, and ethical dimensions. However, the lack of a unified, internationally comparable measurement framework constrains policymakers’ ability to address inequities effectively. Chetty and colleagues thus advocated for a standardized global digital literacy index, supported by agile benchmarking systems capable of adapting to technological change. This recommendation aligns with global strategies proposed by the OECD (2015, 2016) and the G20 Leaders (2015), who emphasized that human capital development through digital skills is as crucial as physical infrastructure for economic growth and educational equity.

In mathematics education, digital tools have also advanced research into cognitive processes. Casalvieri et al. (2023) employed eye-tracking technology to explore how students at varying expertise levels conceptualize derivatives. They found that experts focused analytically on textual and symbolic representations, while novices concentrated on graphical elements, often conflating the function’s value with its derivative. These results align with Sahin et al. (2015), who argued that relational understanding of the derivative—integrating procedural and conceptual reasoning—depends on connecting key “big ideas,” such as rate of change, slope, and limit. Without such integration, students tend to rely on instrumental reasoning, applying differentiation rules mechanically. Digital diagnostic tools like eye-tracking thus offer valuable insights into students’ reasoning processes and misconceptions, enabling educators to design more targeted interventions.

Collectively, these studies demonstrate that digital technologies, when implemented thoughtfully, enhance learning outcomes, bridge intercultural divides, and illuminate cognitive processes. Yet they also emphasize that successful digital integration requires pedagogical expertise, institutional support, and sustained attention to equity and digital literacy at both local and global levels.

2.3. Duval’s Theory of Cognitive Processes in Mathematical Learning: Semiotic Registers

In the context of our research, Duval’s theory of cognitive processes in mathematical learning provides a powerful framework for further insight. Duval posits that understanding mathematical concepts requires the ability to transition between different semiotic registers, such as algebraic, graphical, and numerical representations. These registers are distinct systems of representation for mathematical objects, each requiring specific cognitive skills.

The ability to transition, convert, and coordinate between these different semiotic registers is not merely helpful but crucial for deep mathematical understanding and effective problem-solving, especially in applied contexts. Duval argues that the development of deep mathematical understanding is often hindered when students struggle to coordinate and translate between these registers. Furthermore, individual differences in cognitive schemas associated with these registers can significantly impact students’ ability to fully grasp and internalize calculus concepts, thereby creating cognitive obstacles to meaningful learning. These obstacles arise when students cannot fluidly connect registers or when their internal schemas within a register are incomplete. For instance, one hypothesis suggests that each learner possesses different schemas related to registers, influencing how students interpret and approach calculus concepts, with varied understanding levels creating cognitive obstacles.

Cognitive obstacles stemming from semiotic register issues can arise from at least two distinct underlying causes, each potentially requiring different pedagogical approaches. One cause is the presence of incomplete schemas within a specific semiotic register, meaning a deficit in the internal cognitive structure or understanding of concepts within that particular representation. The other cause is a lack of experience in cross-representational thinking, which suggests that even if individual schemas are relatively complete, students may not have had sufficient practice or explicit instruction in the act of converting or coordinating between these different representations.

The application of Duval’s theory provides a robust framework for pedagogical intervention, moving beyond a simple identification of learning difficulties to a precise diagnosis of actionable cognitive obstacles. This research therefore uses Duval’s theoretical framework to examine how students handle mathematical registers and identify specific points where cognitive obstacles arise.

Table 1. Duval’s semiotic registers and the associated cognitive skills required (Duval, 2017, Chapters 1.1.1, 2.1.4; 2006, Chapters 1–2; Azevedo Montenegro et al., 2021, pp. 589–592).

Semiotic Register	Description/Purpose	Key Cognitive Skills Required
Verbal/Natural Language	Everyday language used to describe problems, concepts, and real-world scenarios.	Interpretation, comprehension, abstraction, context recognition.
Algebraic/Symbolic	Formal mathematical expressions, equations, and formulas.	Manipulation, calculation, formalization, symbolic reasoning.
Graphical	Visual representations such as graphs, diagrams, and charts.	Visualization, interpretation of trends, estimation, spatial reasoning.
Numerical	Discrete values, tables, and specific calculations.	Computation, pattern recognition, approximation, data interpretation.
Applied Contextual (Economic/Engineering)	The specific real-world domain (e.g., economics, physics, engineering) in which mathematical concepts are applied.	Contextual interpretation, application of theoretical constructs to practical scenarios, real-world validation.

summarizes Duval’s semiotic registers and the associated cognitive processes and obstacles relevant to mathematical learning.

2.4. Distinction Between Challenges and Cognitive Obstacles

To illustrate the crucial distinction between challenges and cognitive obstacles, particularly when applying Duval’s framework, several examples from an initial analysis of student responses to applied derivative problems can be considered:

• Misunderstanding of Variables and Domains (see also Duval, 2017, Chapter 1.3.3): Students may encounter an obstacle when they struggle to correctly define variables and their domains. This represents a critical aspect of aligning symbolic (algebraic) understanding with contextual understanding in the problem. According to Duval’s framework, a failure to transition accurately between registers (e.g., from verbal problem descriptions to algebraic formulations) can lead to this obstacle. This difficulty stems from a fundamental misalignment that prevents coherent understanding of the problem context, rather than a general challenge. In the context of the MMC, this corresponds to difficulties in the initial stages of “understanding and simplifying a real-world situation” and “developing a conceptual or ‘real’ model”.
• Application of Theoretical Concepts to Practical Contexts (see also Duval, 2017, Chapter 2.2): Integrating theoretical constructs, such as the derivative, into real-world applications poses a challenge that can become an obstacle when students struggle with managing multiple semiotic registers, for instance, interpreting derivatives within an economic context. If students fail to connect symbolic representations with applied meanings, comprehension becomes limited. This difficulty indicates an obstacle because it reflects an inability to fully harmonize theoretical and practical knowledge, rather than a mere challenge of applying familiar concepts. This obstacle is evident in the MMC stages of “translating it into a mathematical model” and “interpreting, validating, and, if necessary, refining results within the real-world context.”
• Graphical and Algebraic Representation of Constraints (see also Duval, 2017, Chapter 3.1): When creating graphs or solving equations, students may overlook constraints, such as maximum and minimum values. In Duval’s terms, coordinating between graphical and algebraic registers is crucial. Ignoring these constraints represents an obstacle, not merely a challenge, as it indicates a specific barrier to interpreting the graph within real-world limits, rather than a general difficulty in drawing or calculating values. Such issues can arise during the “performing mathematical analyses and obtaining results” stage of the MMC, particularly if the validation step is overlooked.
• Critical Thinking in Optimization Problems: Optimization tasks require students to connect multiple registers (algebraic, numerical, and contextual). Difficulties in validation and critique of results within feasible boundaries reveal obstacles in coordinating registers, according to Duval. This reflects more than a general challenge, as it obstructs students’ ability to interpret and validate results meaningfully in context—a cognitive obstacle that limits accurate application. This directly corresponds to the final “interpreting, validating, and, if necessary, refining results within the real-world context” stage of the MMC.
• Economic Context and Function Interpretation: Translating functions like cost and revenue into economically relevant relationships involves navigating multiple semiotic registers. When students fail to establish these connections, it highlights obstacles in bridging abstract mathematical functions with economic applications, as Duval’s framework suggests. This inability to coordinate representations reflects an obstacle rather than a challenge, as it prevents students from achieving functional understanding across contexts. This relates to the translation and interpretation stages of the MMC.

The distinction between challenges and cognitive obstacles is not merely a theoretical refinement but carries significant implications for both research methodology and the design of effective educational interventions. Not all observed difficulties necessarily constitute cognitive obstacles in the strict Duvalian sense (Duval, 2017). Some problems—such as inaccuracies in graphing constraints or incomplete application of domain limits—may instead represent didactical challenges, that is, difficulties arising from limited prior exposure, insufficient guided practice, or instructional design gaps (Brousseau, 1997/2002) rather than from deep cognitive barriers. In contrast, a cognitive obstacle denotes a more fundamental conceptual blockage that persists even under appropriate instructional support because it originates from the learner’s internal representational system—for example, an inability to coordinate symbolic and graphical registers or to maintain consistent variable meanings across representations (Duval, 2006, 2017; Ferretti et al., 2024). Thus, while difficulties in graphing constraints may initially appear as obstacles, they can be reinterpreted as didactical challenges if students’ reasoning improves through scaffolding or practice.

Recognizing this distinction enables more precise diagnostic and pedagogical responses. When a difficulty is understood as a general challenge, a broad intervention—such as providing additional opportunities for guided practice—may suffice. However, when a difficulty is identified as a cognitive obstacle rooted in a specific semiotic register gap, the intervention must be more targeted. The structured perspective offered by the Mathematical Modeling Cycle (MMC) helps clarify such situations: difficulties that emerge in particular stages of the cycle can be directly linked to representational gaps (Blum & Leiss, 2007; Shahbari & Tabach, 2020). This diagnostic precision allows educators to move beyond generic instructional strategies toward focused interventions that address the specific points of cognitive breakdown. In doing so, they can strengthen students’ ability to transition between registers and thereby enhance comprehension and problem-solving in applied calculus contexts.

The present study constitutes an initial step in examining this distinction and, as such, does not permit strong empirical claims regarding the essential differentiation between cognitive obstacles and didactical challenges. Within this framework, cognitive obstacles are treated as hypothetical constructs—analytically inferred from students’ reasoning patterns rather than empirically verified through extensive qualitative data—whereas didactical challenges are examined theoretically, drawing on insights from prior research and classroom-based evidence reported in the literature. To advance this line of inquiry, a subsequent qualitative study is currently being designed to gather empirically grounded evidence that will enable a finer-grained and evidence-based justification of this distinction. Such an approach is expected to produce a more comprehensive understanding of how cognitive and didactical dimensions interact in shaping students’ reasoning processes when engaging with complex mathematical representations.

3. Materials and Methods

The core aim of this study is to bridge mathematical theory with practical applications within the field of Industrial Engineering and Management. This highlights the practical relevance and potential impact of the research, offering insights into improving calculus education and proposing strategies to enhance students’ mathematical and applied skills in engineering contexts.

3.1. Research Objectives and Questions

Guided by this focus, the study pursued two objectives:

• To characterize the cognitive processes reflected in students’ solutions to applied derivative problems. This objective focuses the investigation on understanding how students analyze and solve problems requiring the application of derivatives, with particular attention to the logical and cognitive processes involved in this activity.
• To identify the challenges and obstacles students encounter when applying derivatives in engineering and management contexts. This objective aims to determine the specific difficulties and challenges students face when required to apply theoretical concepts in solving applied problems, especially those related to the professional fields of Industrial Engineering and Management.

From these objectives, two research questions were formulated:

RQ1: Which cognitive processes are reflected in the written solutions of Industrial Engineering and Management students when addressing applied derivative problems?

RQ2: What challenges and cognitive obstacles emerge from students’ written solutions when applying derivative concepts in engineering and management contexts?

3.2. Research Design and Theoretical Framework

To address these questions, the study employed a qualitative case study design, drawing on students’ written exam responses to an applied optimization problem. The analysis was framed by two complementary theoretical perspectives: the Mathematical Modeling Cycle (MMC) (as conceptualized by Leikin & Ovodenko, 2024, Figure 1) and Duval’s theory of semiotic registers (2017).

Given its recognized value for bridging theoretical mathematics and practical engineering applications, we adopted this dual conceptual framework to analyze students’ exam performance on applied derivative problems. This decision aligns with the growing emphasis on authentic and iterative problem-solving processes in engineering education (Blum & Leiss, 2007; Kaiser & Stender, 2013). Our research investigates how students approach mathematical problems in exams, focusing not only on the correctness of their final answers but also on their reasoning processes across the stages of the modeling cycle and their representational fluency. Specifically, the study analyzes how students

• Interpret and simplify real-world-inspired exam problems;
• Develop conceptual and mathematical models;
• Execute appropriate mathematical procedures;
• Validate and contextualize their results.

To support this investigation, a task-based instrument was designed to function as a research tool. The “Organizing a Tour” problem (Figure 2) integrates contextual, symbolic, and graphical elements to engage students in a complete modeling cycle. It was purposefully constructed to elicit transitions between modeling stages (the MMC) and semiotic registers (Duval), making it a suitable probe for identifying students’ modeling behaviors and reasoning strategies within an applied calculus context.

In this study, the MMC was used as a process lens, identifying where in the modeling cycle students encountered breakdowns, while Duval’s framework served as a representational lens, explaining why these breakdowns occurred by revealing difficulties in transitioning between registers (verbal, symbolic, graphical, and contextual). Together, these perspectives provide a comprehensive analytical framework for understanding students’ engagement with applied derivative problems.

3.3. Task Design

The applied task presented to students required the use of the derivative to solve a real-world optimization problem relevant to engineering economics. The problem involved planning a guided tour with variable pricing based on the number of participants. Figure 2 presents the task given to students, as follows:

Figure 2. The “Organizing a Tour” task.

Figure 2. The “Organizing a Tour” task.

This problem integrates contextual, symbolic, and graphical elements, designed to elicit a full modeling cycle from problem understanding through formalization, analysis, and validation. It was also expected to prompt different but mathematically valid approaches (e.g., defining the variable as “participants above 50” or as “total participants”), creating opportunities to observe both successful strategies and potential obstacles.

3.4. Data Analysis

This section focuses on students’ engagement with the MMC in the context of the “Organizing a Tour” task. The analysis follows the framework depicted in Figure 1, based on Leikin and Ovodenko (2024), which distinguishes between the stages (nodes) of the modeling process and the cognitive efforts (connections) that facilitate transitions between them. These efforts—understanding, simplifying, mathematizing, mathematical work, interpreting, and validating—frame the analysis of how students progress from contextual understanding to formal mathematical reasoning. The flow below illustrates this progression through modeling activity.

Contextual Situation: In this stage, students begin by engaging with the problem scenario: organizing a tour with a participant-based pricing structure. The task specifies that the ticket costs $200 for 50 participants and decreases by $2 for each additional participant, up to a maximum of 80. Costs include a fixed fee of $6000 and a variable cost of $32 per participant. The objective is to determine the number of participants that will yield maximum profit.

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Efforts: At this stage, students focus on understanding the scenario and simplifying the information by identifying key quantities—such as the number of participants, pricing rules, and cost structure—and the overarching goal of profit maximization. Students may notice qualitative patterns, such as the relationship between the number of participants and ticket price. These initial observations lay the foundation for structuring relationships in the next stage of the modeling cycle.

Contextual Model: In this stage, students begin to relate the quantities identified in the scenario. They recognize that the ticket price decreases linearly with the number of participants and that profit depends on the balance between revenue and cost. Some students informally introduce a varying quantity—such as “extra people above 50” or “total number of participants”—but do not yet express it symbolically.

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Efforts: Through simplification and preparatory reasoning, students begin to move toward mathematization. They describe contextual dependencies and anticipate how relationships might later be represented algebraically. The idea of a variable starts to take shape, as students ask, “What should we vary?” This stage supports the transition toward variable definition and functional representation.

Mathematical Model: In this stage, students construct formal mathematical representations of the situation. Two modeling paths are common:

Option 1: Let $x$ be the number of participants above 50 (Total: $50+x$, Domain: $0\le x\le 30)$.

• Ticket price per person: $y\left(x\right)=200-2x$
• Revenue: $R\left(x\right)=\left(50+x\right)\left(200-2x\right)$
• Cost: $C\left(x\right)=6000+32\left(50+x\right)=7600+32x.$
• Profit: $P\left(x\right)=R\left(x\right)-C\left(x\right)=\left(50+x\right)\left(200-2x\right)-(7600+32x)$

Option 2: Let $x$ = total number of participants (Domain: $50\le x\le 80).$

• Ticket price per person: $y\left(x\right)=200-2\left(x-50\right)=300-2x.$
• Revenue: $R\left(x\right)=x\cdot y\left(x\right)=x(300-2x)$
• Cost: $C\left(x\right)=6000+32x$
• Profit: $P\left(x\right)=R\left(x\right)-C\left(x\right)=x\left(300-2x\right)-(6000+32x)$

➢

Efforts: Through mathematization, students transition from informal reasoning to formal algebraic modeling. Defining $x$ marks a pivotal shift to symbolic representation, enabling the use of functions and preparing for analytic work such as graphing, differentiation, or optimization.

Mathematical Outcomes: In this stage, students analyze the symbolic profit function to determine its maximum value. This involves algebraic manipulation—expanding and simplifying expressions—and, for some, applying calculus to locate the vertex of the quadratic function.

➢

Efforts: A key component of this mathematical work is constructing or interpreting a graph of the profit function. The graph, typically a downward-opening parabola, visually represents the relationship between the participant count and profit. It helps students identify the maximum and understand the function’s behavior within the domain. The analysis yields a critical point at $x=17$ (Option 1) or at $x=67$ (Option 2), indicating the optimal value. These results constitute the mathematical output of the modeling activity and serve as a bridge to the next stage, where students interpret them in context.

Contextual Meaning of Outcomes: In this stage, students interpret the mathematical results in relation to the original problem context. They recognize that the critical value obtained at $x=17$ corresponds to 67 participants and yields a ticket price of $166. The computed profit of $2978 is evaluated considering the tour’s financial goals. The graph produced during mathematical work further supports this interpretation by confirming that the maximum lies well within the valid domain.

➢

Efforts: Through interpretation, students assess the feasibility and realism of the result. They begin connecting mathematical outcomes back to the real-world scenario and determining whether the solution offers a reasonable basis for decision-making.

Return to Contextual Situation: In the final stage, students revisit the original constraints and objectives of the problem. They verify that their solution respects the participation range, pricing logic, and cost structure, and use it to formulate a concrete recommendation.

➢

Efforts: Through validation, students conclude that the optimal decision is to organize the tour for 67 participants, each paying $166, resulting in a profit of $2978. This confirms that the model is both mathematically sound and contextually meaningful, completing the modeling cycle.

Student responses were first examined holistically to identify how they moved through the modeling stages. Then, selected solutions were mapped in detail, identifying specific evidence from their written work and associating it with stages and efforts. This analysis allowed us to explore how modeling competencies emerged or broke down throughout the problem-solving process.

To illustrate the range of student modeling approaches, three representative cases were selected. Each case is referred to by a metaphorical nickname—“Skippy,” “Switcher,” and “Floater”—to emphasise the central characteristics of their modeling behavior.

3.5. Participants

The research population included 30 Industrial Engineering and Management students. All participants had previously studied elementary calculus, including the concept of extrema, and were familiar with functions, graphing, and symbolic manipulation. Participants ranged in age from 25 to 35 and engaged willingly and enthusiastically with the modeling activity. Three students were selected for in-depth analysis, each highlighting different strengths and obstacles in navigating the MMC and coordinating semiotic registers, providing rich evidence for exploring the research questions. The selection of the three students was purposeful rather than random. They were chosen not to represent the full diversity of students’ approaches but to illustrate typical reasoning patterns observed within the broader group, while still allowing for some variation in the types of conceptual difficulties encountered. This strategy supports an in-depth qualitative examination of reasoning processes that reflect common trends rather than extreme or idiosyncratic cases. Consequently, the findings are not intended to be statistically generalizable but to provide analytic generalization (Yin, 2018), contributing to a deeper understanding of how mathematical meaning develops within collaborative learning contexts. This methodological decision also seems appropriate in light of the planned next stage of the research, namely, the development of an instrument—grounded in the present analytic generalizations—that will enable the examination of a much larger population of students. The intended instrument is expected to support the analysis of responses from several hundred participants, thereby extending the qualitative insights of the current study into a broader, quantitatively supported framework.

4. Findings

This section presents an analysis of students’ engagement with the modeling cycle in response to the “Organizing a Tour” task. Drawing on the framework outlined in the Section 3.4, we examine students’ modeling activity through both the modeling stages (contextual situation; contextual model; mathematical model; mathematical outcomes, interpretation, validation, and refinement) and the efforts that connect these stages (understanding, simplifying, mathematizing, mathematical work, interpreting, and validating).

To illustrate the diversity of modeling behaviors, we selected three representative student solutions that typify distinct patterns of cognitive engagement:

• Skippy demonstrates strong procedural fluency in mathematical work but “skips” over critical modeling efforts related to contextual understanding, interpretation, and validation.
• Switcher shows flexible but inconsistent movement between modeling representations, frequently “switching” between meanings of the variable x without clear alignment or justification.
• Floater begins with contextual grounding but ultimately “floats” between modeling components without consolidating the model or validating the result within the problem context.

The following tables provide a structured analysis of each student’s solution, aligned with the modeling cycle and cognitive efforts framework. For each student, we present the evidence drawn from their written solution and an analysis of their modeling efforts at each stage. To enhance clarity and reflect the structure of our theoretical framework, the analysis is organized into three separate tables per student, corresponding to: (1) Contextual Literacy, (2) Mathematical Literacy, and (3) Interpretation and Validation.

To support a more nuanced analysis of the final modeling stage, we differentiate among three types of validation observed in student work:

• Minimal validation involves superficial or procedural checks (e.g., rejecting negative values) without deeper reflection.
• Structural validation refers to internal mathematical consistency checks, such as verifying a maximum using the second derivative.
• Contextual validation assesses the real-world plausibility of results and indicates a full return to the task context.

This typology informs our interpretation of students’ reasoning in the “Interpretation and Validation” tables and is referenced where relevant throughout the case analyses.

4.1. The “Skippy” Case

Skippy begins the modeling process abruptly, focusing immediately on symbolic expressions without restating or interpreting the problem in natural language (

Table 2. “Skippy”—Contextual Literacy in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Contextual Situation	Evidence: No evidence.
Efforts: Understanding and Simplifying: Missing. No attempt to interpret or restate the problem; no unpacking the practical constraints or goals.
Contextual Model	Evidence: Student introduces $x$ as the number of participants without defining its domain (e.g., $0\le x\le 30$); relationships between price, participants, and profit are not described before the algebraic formulation.
➢ Efforts: Simplifying → Mathematizing: The transition is incomplete. The student skips explicitly describing contextual relationships. While the variable $x$ appears, it is introduced within algebraic expressions, not as part of contextual reasoning. The modeling effort is largely symbolic, not grounded in structuring real-world dependencies.

).

A primary cognitive obstacle for “Skippy” is a lack of contextual grounding and underdeveloped understanding and simplifying efforts within the MMC. The student’s entry into the modeling process is abrupt, focusing immediately on symbolic expressions without explicit interpretation or restatement of the problem in natural language. This indicates a difficulty linked to Duval’s “Verbal/Natural Language” register for initial problem comprehension and abstraction.

More specifically, “Skippy” encounters the cognitive obstacle of “Misunderstanding of Variables and Domains.” The variable x is used without explicit definition of its domain (e.g., 0 ≤ x ≤ 30) or clear description of its contextual relationships, introducing x directly within algebraic expressions. This reflects a specific representational gap in transitioning from verbal problem descriptions to coherent algebraic formulations.

While Skippy’s engagement with the contextual situation is limited, they proceed confidently into the mathematical modeling phase. The next table examines the student’s symbolic and procedural work in greater detail (

Table 3. “Skippy”—Mathematical Literacy in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Mathematical Model	Evidence: Correct formulation of cost, price, revenue, and profit functions using $x$ (Option 1). See Equations (1)–(4) in Figure A1 (“Skippy”) in Appendix A.
➢ Efforts: Mathematizing: (a) Defining Functions: The student shows procedural fluency in setting up the symbolic model. However, there is no evidence of interpreting or justifying these expressions based on the scenario. The mathematization effort is formal but disconnected from context.
Mathematical Outcomes	(1) Evidence: (b) Maximize profit: (1) Derives profit function, solves $P^{\prime }\left(x\right)=0$ and obtains critical points $x=17.$ Uses maximum tests (see derivative sign table at right). (2) Calculates 67 participants and a profit.
	(c) Graphing (see graph at right): (3) Notes domain $x\ge 0$ (first mention). (4) Finds intercepts: $x_1=55.58,x_2=-21.58$, rejects $x_2$. (5) Sketches graph with max: (17, 2978). Applies: $P^{″}\left(x\right)=-4$. (6) Graph sketched with incorrect domain (e.g., x > 30).
➢ Efforts: Mathematical work: The student displays high procedural fluency and differentiates, finds critical points, interprets derivatives, and sketches the graph of the function. They follow a school-taught sequence for quadratic graphing. Graphing is algebraically accurate but contextually weak: domain constraints are added reactively and inconsistently. Partial interpreting effort is visible (e.g., rejects $-21.58$ as unrealistic), but validating is incomplete. Graph includes invalid values. The function is treated as abstract, not as a contextual model.

).

“Skippy” demonstrates strong procedural fluency in mathematical work but limited contextual grounding. This student is capable of correctly formulating cost, price, revenue, and profit functions using symbolic (algebraic) representation (Option 1). They show proficiency in the mathematical work stage of the MMC by differentiating the profit function, finding critical points, and performing tests to confirm a maximum. Furthermore, “Skippy” exhibits the cognitive skill of graphical visualization and interpretation, sketching a parabola for the profit function. There is also evidence of partial interpretation, as the student correctly translates the optimal x value (17) back into the number of participants (67) and calculates the corresponding profit ($2978).

Furthermore, “Skippy”’s graphical representation, while algebraically accurate, includes values outside the valid domain (e.g., x > 30) and domain constraints are added reactively and inconsistently. This highlights a cognitive obstacle in “Graphical and Algebraic Representation of Constraints”, indicating a barrier to interpreting the graph within real-world limits. The student treats the function as abstract rather than as a contextual model, leading to incomplete interpreting and validating efforts. Although an impossible solution (x = −21.58) is rejected—demonstrating minimal validation—and a second derivative test is applied (structural validation). However, a systematic check against all contextual constraints (like the maximum group size or realistic pricing) is absent, indicating a lack of contextual validation and preventing a full return to the contextual situation and comprehensive validation. This limited validation reveals difficulties in coordinating registers for critical thinking in optimization problems (

Table 4. “Skippy”—Interpretation and Validation in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Contextual Meaning of Outcomes	Evidence: Correctly interprets $x=17$ as 67 participants. Calculates $P\left(17\right)=2978$. Graph used to confirm maximum. Graph exceeds domain (e.g., x > 30), and domain constraints not discussed.
➢ Efforts: Interpreting: The student recognizes the contextual meaning of the symbolic result (e.g., 67 participants) but does not elaborate. There is no justification of feasibility (e.g., whether $2978 is realistic) and the graph remains out of sync with constraints.
Return to Contextual Situation	Evidence: Rejects $x=-21.58$ as invalid. No verbal check against full constraints (50–80 participants, max group size, realistic pricing). No recommendation is written.
➢ Efforts: Validating: The student begins validation by rejecting an impossible solution but does not apply this reasoning systematically. The graph includes points that exceed the upper bound of 30 (Option 1). No explicit decision or evaluation of model accuracy. The solution remains mathematically valid but not contextually verified.

).

4.2. The “Switcher” Case

While “Skippy” exemplifies a student with strong procedural fluency but limited contextual grounding, “Switcher” presents a contrasting modeling profile. This student demonstrates flexible movement between contextual and mathematical representations but struggles with consistency. Their modeling is marked by frequent shifts in the meaning and use of the variable $x$, which undermines coherence across modeling stages. The analysis below highlights how this instability in representational coordination impacts both mathematical accuracy and contextual interpretation (

Table 5. “Switcher”—Contextual Literacy in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Contextual Situation	Evidence: Correctly identifies fixed cost = $6000, variable cost = $32 per participant. Recognizes that the goal is to maximize profit.
➢ Efforts: Understanding: The student understands the setup and goal but does not rephrase or interpret the problem explicitly.
Contextual Model	Evidence: (1) Assigns $x$ as “number of participants above min”. (2) Constructs a table and price function sketch showing price drops by $2 per extra participant [Shift in meaning: $x$ changes from “above min” to “total participants” without warning.]
➢ Efforts: Understanding → Simplifying: The student begins relating quantities. Though not formally defined, the use of $x$ and the sketch reflect awareness of linear dependence between price and participants. Simplifying → Mathematizing: Transitions into symbolic modeling without formal redefinition of $x$. The student makes initial attempts to represent contextual relationships (e.g., decreasing price), but fails to sustain coherent variable use. There is a mismatch between informal reasoning and symbolic representation—conceptual confusion about the meaning of $x$ and its role undermines the integrity of the contextual model.

,

Table 6. “Switcher”—Mathematical Literacy in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Mathematical Model	Evidence: Defines: (1) Domain $50<x\le 80$, reflecting total participants. (2) Correct formulation of cost, price, revenue, and profit functions using $x$ (Option 2). See Equations (1)–(4) in Figure A2 (“Switcher”) in Appendix A.
➢ Efforts: Mathematizing: (a) Defining Functions: “Correct” symbolic functions, BUT inconsistency in the modeling process: the variable $x$, initially introduced as the number of participants above 50, is now used as the total number of participants without clear redefinition or domain adjustment. This results in a mismatch between the contextual model and the symbolic model, undermining conceptual coherence
Mathematical Outcomes	Evidence: (b) Maximize profit: (1) Defines $P\left(x\right)=-2x^2+268x-6000$ with domain $50<x\le 80$ (secondly). (2) Derives profit function and Solves $P^{\prime }\left(x\right)=0$, obtains critical points $x=67.$ Uses maximum test.
	(3) Writes, “The number of people needed to maximize profit is 67.” (c) Graphing: (6) Finds intercepts: $x_1=28.41,x_2=105.58.$ (7) Plots $x_1$, $x_2$, and rejects them as unrealistic. (8) Plots max: (67, 2978). Graph includes “holes” and values beyond domain.
➢ Efforts: Mathematical work: Correct technique applied to model, correct derivative and solution. Graphical representation is mathematically accurate but contextually inconsistent: includes values outside domain and lacks justification for domain ends.

and

Table 7. “Switcher”—Interpretation and Validation in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Contextual Meaning of Outcomes	Evidence: Interprets result: “67 participants maximize profit.” Calculates P(67) = 2978.
➢ Efforts: Interpreting: The student makes a correct numerical interpretation but does not connect back to all context constraints (e.g., price reasonableness, feasibility).
Contextual Situation	Adds a comparison table, calculates $P\left(50\right)=2400$ and $P\left(67\right)=2978$. No verbal justification or decision, but comparison is present.
Efforts: Validating: Demonstrates a basic validation effort by comparing profit values at two participant counts. However, there is no interpretation or decision based on this comparison. Constraints and pricing realism are still unexamined. Partial return to context, but the cycle is not fully completed.

).

“Switcher” demonstrates an initial understanding of the problem setup and goal by correctly identifying fixed and variable costs and the objective of profit maximization. The student makes initial efforts to relate quantities, constructing a table and a price function sketch that reflect an awareness of linear dependence between price and participants. This shows an emerging cognitive process of simplification and preparatory reasoning in the “contextual model” stage.

While the contextual phase shows early engagement, the mathematical modeling stage reveals inconsistencies in variable usage that impact symbolic coherence, as shown in Table 6.

Mathematically, “Switcher” exhibits correct procedural technique, applying derivatives accurately to find the critical point and solving for the optimal number of participants. A numerical interpretation is also correctly made, stating that 67 participants maximize profit. Furthermore, a basic validation effort is present through comparing profit values at two participant counts (P(50) and P(67)).

The most prominent cognitive obstacle for “Switcher” is the inconsistency and conceptual confusion regarding the meaning and use of the variable x. Initially, x is assigned as the “number of participants above min”, but it later shifts to represent “total participants” without clear redefinition or domain adjustment. This mismatch between informal reasoning and symbolic representation undermines the integrity and coherence of the contextual model and symbolic model, creating a significant representational gap. This directly exemplifies the cognitive obstacle of “Misunderstanding of Variables and Domains”, specifically the failure to sustain accurate transitions and coherent variable use between verbal, contextual, and algebraic registers. These inconsistencies in mathematical modeling affect not only the formulation and solution but also the interpretation and validation of results, as analyzed in the following table (Table 7).

Switcher’s validation phase reflects minimal validation (e.g., rejecting implausible values) and limited contextual validation through numerical comparisons of profit outcomes. However, structural validation—such as graphing the function to confirm the maximum or systematically checking feasibility—is absent. This partial return to the contextual situation suggests an incomplete execution of the modeling cycle.

Despite correct symbolic functions being formulated (Option 2), the underlying inconsistency in the modeling process due to the variable’s changing meaning limits deep conceptual understanding. This struggle to coordinate and translate between different semiotic registers, particularly between the initial contextual definition of x and its inconsistent algebraic application, hinders meaningful learning.

Like in “Skippy”’s case, “Switcher”’s graphical representation is mathematically accurate but contextually inconsistent, including values outside the valid domain without clear justification. This again points to difficulties with “Graphical and Algebraic Representation of Constraints”. Although there is a basic validation, the student fails to fully connect back to all context constraints, such as the reasonableness of the price or overall feasibility of the solution within the real-world scenario, indicating an incomplete validating effort within the MMC.

4.3. The “Floater” Case

Unlike “Switcher,” who oscillates between contextual and symbolic representations, “Floater” begins with a clear contextual focus but struggles to stabilize the modeling process. The student’s engagement with the modeling task reflects an intuitive grasp of the scenario, yet this understanding is not consistently translated into formal mathematical modeling. The following table (

Table 8. “Floater”–Contextual Literacy in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Contextual Situation	Evidence: Student lists task data ($200 per person, max 80 persons, $6000 fixed cost, variable cost = $32 per participant and participant limits). Recognizes that the goal is to maximize profit.
➢ Efforts: Understanding: Student reads and records all key quantities.
Contextual Model	Evidence: Student verbally defines $x$ as the number of participants above the minimum.
➢ Efforts: Understanding → Simplifying: Student correctly introduces $x$, but no domain (i.e., $0\le x\le 30$) is stated yet. Begins transitioning into modeling. Shows initial structuring of the situation. No intermediate representations (e.g., tables or verbal relationships) are used to explore or justify the dependencies before moving to formal notation. Simplifying → Mathematizing: The student begins with a contextual understanding and defines $x$ meaningfully, showing emerging contextual literacy. However, inconsistencies appear as they move into formal modeling.

) presents Floater’s efforts in the early stages of the modeling cycle, highlighting the interplay between contextual literacy and the challenges of sustaining representational coherence.

“Floater” begins by engaging with the contextual situation, listing all task data, and recognizing the goal of profit maximization. The student makes an initial attempt at defining the variable x verbally as the “number of participants above the minimum”, reflecting some initial contextual understanding.

Procedurally, “Floater” is capable of differentiating the profit function correctly and finding a critical point, demonstrating skills within the algebraic/symbolic register. There is also a partial numerical interpretation, converting the symbolic result (x = 42) into a contextual statement (“total of 92 participants”).

However, “Floater” encounters severe cognitive obstacles related to an inconsistent alignment between the defined variable and its use in mathematical expressions, indicating a profound disconnection between the contextual model and symbolic formulation. For instance, the cost function assumes x is the total number of participants, while the ticket price function aligns with x as added participants, and the revenue model contains a fundamental flaw. This highlights a critical lack of variable stability and is a clear instance of the cognitive obstacle of “Misunderstanding of Variables and Domains”. This problem also extends to “Economic Context and Function Interpretation”, as the student fails to bridge abstract mathematical functions (cost, revenue, profit) with their specific economic applications, preventing functional understanding across contexts.

A significant challenge for “Floater” is the incomplete mathematizing effort, where the initial contextual understanding is not robustly translated into a coherent mathematical model. This results in modeling errors that undermine the validity of subsequent mathematical outcomes, even if the differentiation is performed correctly (

Table 9. “Floater”—Mathematical Literacy in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Mathematical Model	Evidence: Construct: The cost of the tour, $C\left(x\right)=6000+32x$. The ticket price, $y\left(x\right)=200-2x$. The expected revenue, $R\left(x\right)=200·50+x(200-2x)$. The profit from the tour, $P\left(x\right)=R\left(x\right)-C\left(x\right)$. Figure A3 (“Floater”).
➢ Efforts: Mathematizing: (a) Defining Functions: There is no consistent alignment between the defined variable and its use in expressions. This signals disconnection between the contextual model and symbolic formulation, and a lack of variable stability. The cost function $C\left(x\right)$ assumes $x$ is the total number of participants (50–80). The ticket price function matches the interpretation of $x$ as added participants. The revenue model suggests that only added participants pay a reduced price, while 50 pay full price, indicating confusion about the uniform pricing model in the task. The mathematizing effort is incomplete and not well grounded in the contextual model.
Mathematical Outcomes	Evidence: (b) Maximize profit: $P(x)=-2x^2+168x+4000$ $P^{\prime }(x)=-4x+168=0$ $4x=168$ $x=42$ (c) Graphing:
➢ Efforts: Mathematical work: Student differentiates the profit function correctly, computes $P^{\prime }\left(x\right)=0,$ and finds a critical point correctly. BUT, (1) no sufficient argument is given to confirm the point is indeed a maximum; (2) the modeling errors in previous stages undermine the validity of this result; (3) the graph lacks domain constraints and includes unrealistic values (e.g., negative x, values beyond x = 30). The graph is accurate algebraically but not contextually meaningful.

).

Crucially, “Floater” completely lacks validation and critical thinking regarding the obtained results. The calculated optimal number of participants (92) explicitly exceeds the maximum allowable (80), yet the student does not reflect on the reasonableness or feasibility of this outcome or whether other contextual constraints are satisfied (

Table 10. “Floater”–Interpretation and Validation in the Modeling Process.

Modeling Stage ➢ Efforts’ Analysis	Evidence (from Student’s Solution)
Contextual Meaning of Outcomes	Evidence: Student calculates $x=42$ and writes, “To maximize profit, 42 additional participants are needed, leading to a total of 92 participants.”
➢ Efforts: Interpreting: The student converts the symbolic result ($x=42$) into a contextual statement (total participants = 92) using the relationship $50+x$, indicating partial interpretation. However, they do not reflect on whether this result is reasonable or whether other contextual constraints (e.g., ticket price, domain) are satisfied. No link to profitability, logistics, or pricing strategy is made.
Return to Contextual Situation	Evidence: No evidence.
➢ Efforts: Validating: No validation appears. The student does not confirm whether the result is within acceptable participant bounds, nor reflect on whether a group of 92 people is feasible under the problem constraints. No reasoning about the realism or decision-making implication of the result is shown.

).

This demonstrates a severe cognitive obstacle in “Critical Thinking in Optimization Problems”, as the student fails to interpret and validate results meaningfully within the context. The absence of domain constraints in the graph and the inclusion of unrealistic values further illustrate the inability to connect graphical representations with real-world limits. This indicates that “Floater” does not complete the MMC, failing to return to the contextual situation to verify the solution’s soundness and contextual meaning. Specifically, all three forms of validation are absent: minimal validation (e.g., domain checks), contextual validation (feasibility and real-world reasoning), and structural validation (mathematical consistency and justification).

Collectively, these three cases illustrate how students may demonstrate procedural competence while still struggling with representational coherence and contextual validation. The patterns observed in “Skippy”, “Switcher”, and “Floater” highlight recurring challenges in navigating the Mathematical Modeling Cycle and coordinating semiotic registers, providing a foundation for the broader discussion of pedagogical implications in the next section.

5. Discussion

This study specifically set out to investigate the cognitive processes and obstacles encountered by Industrial Engineering and Management students when solving applied derivative problems, aiming to bridge mathematical theory with practical applications in engineering and management. Utilizing the Mathematical Modeling Cycle (MMC) and Duval’s theory of semiotic registers as key analytical frameworks, the research provides detailed insights into how students engage with complex, real-world optimization tasks. The analysis of cases like “Skippy,” “Switcher,” and “Floater” revealed distinct modeling patterns of cognitive engagement, highlighting both procedural strengths and significant representational gaps that ultimately impeded deeper conceptual understanding.

5.1. Cognitive Processes in Student Solutions

The written solutions consistently demonstrated that students possess procedural fluency in mathematical work, particularly within the algebraic/symbolic register. Students were generally proficient in formulating mathematical functions such as cost, price, revenue, and profit using algebraic expressions, provided a variable was already defined. This proficiency extended to correctly differentiating these functions to identify critical points for optimization and performing tests, such as the second derivative test, to confirm maximum values.

Beyond symbolic manipulation, students also exhibited partial engagement with the graphical and numerical registers. For instance, both “Skippy” and “Switcher” attempted sketching parabolas for profit functions, reflecting an ability to visualize relationships graphically, even if the contextual meaning was sometimes lost. Numerical interpretation was also evident, with students translating symbolic results, such as x = 17 or x = 67, into contextual statements about participant numbers and calculating corresponding profit values. “Floater”, despite earlier modeling errors, correctly performed differentiation, showcasing a specific strength within the symbolic register. Initial cognitive processes like understanding and simplifying were observed to varying degrees, with “Switcher” and “Floater” making efforts to relate quantities and identify goals in the early stages of the MMC.

The findings indicate that students are not entirely disengaged from contextual reasoning but often struggle to sustain it across the full modeling process. These observations align with previous research highlighting the central role of semiotic coordination in students’ understanding of mathematical models, as emphasized by Duval (2006, 2017). Specifically, our finding that students struggled to connect graphical and algebraic representations echoes the results of Casalvieri et al. (2023), who found that students’ difficulties often indicate gaps in inter-register coordination rather than purely procedural errors. Similarly, Sahin et al. (2015) demonstrated that fostering relational understanding—integrating procedural and conceptual reasoning—is dependent on connecting these key representations, emphasizing the pedagogical potential of representational fluency, which our students lacked.

5.2. Cognitive Obstacles in Applied Contexts

Despite these procedural strengths, significant cognitive obstacles—defined as “specific representational gaps that hinder meaningful learning”—emerged from students’ difficulties in coordinating, converting, or translating between different semiotic registers. These obstacles often manifested as breakdowns at various stages of the MMC, preventing a complete and coherent engagement with the modeling process:

• Misunderstanding of Variables and Domains. Across all three cases, variable definition proved unstable. “Skippy,” for example, introduced the variable x abruptly within algebraic expressions without explicit definition or domain specification, treating the function as an abstract entity rather than as a contextual model. Domain constraints were often added reactively and inconsistently. “Switcher” demonstrated a critical inconsistency in variable definition, initially assigning x as the “number of participants above min” but later implicitly switching its meaning to “total participants” without clear redefinition or domain adjustment, leading to a mismatch between informal reasoning and symbolic representation. “Floater” exhibited an even more profound lack of variable stability, with x inconsistently aligned across cost, ticket price, and revenue functions. This difficulty, exemplified by “Switcher’s” implicit switching of the variable’s meaning, reflects a fundamental failure to accurately transition between the verbal problem description and algebraic formulations. According to Duval’s (2017) framework, this constitutes a critical representational gap that directly impacts the foundational understanding and simplifying efforts in the early stages of the MMC, which are necessary for “developing a conceptual model” (Blum & Leiss, 2007; Kaiser & Stender, 2013).
• Weak Link Between Mathematical and Economic Contexts. Students struggled with the application of theoretical concepts to practical contexts and economic function interpretation. For instance, “Floater’s” modeling error related to uniform pricing indicated a failure to connect abstract mathematical functions (revenue, cost) with their specific economic meanings. This difficulty in coordinating the symbolic register with the applied contextual register is a known barrier to effective engineering practice, where mathematics is positioned as an instrumental tool for modeling and interpreting real-world phenomena (Cardella, 2010). Such obstacles primarily hindered the crucial “translating into a mathematical model” and “interpreting, validating, and refining results” stages of the MMC.
• Graphical and Algebraic Representation of Constraints. While “Skippy” and “Switcher” produced algebraically accurate graphs, these often included values outside the valid real-world domain without clear justification or discussion of domain constraints (e.g., x > 30 or x < 50). “Floater”’s graph similarly lacked domain constraints and included unrealistic values. This obstacle stems from a specific barrier to interpreting the graph within real-world limits, indicating a breakdown in coordinating the graphical and algebraic registers during the “performing mathematical analyses” stage when the crucial validating step is overlooked. From the MMC perspective, this indicates incomplete mathematical work and absent validating efforts; in Duval’s framework, it signals difficulties in coordinating graphical and contextual registers. This finding aligns with the research of Casalvieri et al. (2023), whose eye-tracking data suggested that students often focus on graphical elements while conflating conceptual meanings. Furthermore, as argued by Sahin et al. (2015), the inability to integrate symbolic and graphical reasoning limits the development of a relational understanding of the derivative.
• Deficits in Validation and Critical Thinking. Across all three cases, students failed to complete the modeling cycle by fully returning to the contextual situation. “Skippy” only partially validated the solution by rejecting a negative x value but failed to systematically check against all contextual constraints (e.g., maximum group size, realistic pricing). “Switcher” performed a basic comparative validation but did not interpret this comparison to make a concrete decision or reflect on pricing realism and other constraints. Most critically, “Floater” exhibited a complete lack of validation, failing to recognize that the calculated “total of 92 participants” explicitly exceeded the maximum allowable 80 participants. This oversight reveals an inability to reflect on the reasonableness or feasibility of the result within the problem context. This represents a severe cognitive obstacle in interpreting and validating results meaningfully within the context, directly corresponding to the final “interpreting, validating, and, if necessary, refining results within the real-world context” stage of the MMC. This failure to interpret and validate results meaningfully corresponds directly to the final and critical “interpreting, validating, and, if necessary, refining results within the real-world context” stage described by Blum and Leiss (2007) and Kaiser & Stender (2013). From Duval’s (2017) perspective, this represents a failure to re-coordinate symbolic outcomes with verbal and contextual registers.

In essence, these findings underscore that while Industrial Engineering and Management students often possess the necessary procedural skills, their deep engagement with applied derivative problems is frequently hindered by their inability to fluidly coordinate and translate between different semiotic registers. The “abrupt entry” into symbolic expressions (“Skippy”) and the “inconsistent alignment” or “lack of variable stability” (“Switcher”, “Floater”) exemplify how crucial the early MMC steps of understanding and simplifying and mathematizing are. Moreover, the pervasive weakness in validation across all cases signifies that students often treat the mathematical model as an end in itself, rather than as an iterative tool requiring continuous contextual verification. These identified obstacles are not merely general learning difficulties; they are precise cognitive breakdowns rooted in specific semiotic register gaps, preventing students from developing the contextual and conceptual literacy essential for authentic engineering problem solving. The identification of obstacles related to inter-register coordination is consistent with Casalvieri et al. (2023), who used eye-tracking data to show that students’ visual attention patterns reflect underlying semiotic difficulties. Together, these studies point to the persistence of representational barriers across different instructional settings, confirming that such obstacles are not limited to procedural gaps but reflect deeper cognitive constraints on meaning-making.

In this way, the study contributes to both theory and practice: theoretically, by showing how the MMC and Duval’s framework complement one another in diagnosing cognitive obstacles, and pedagogically, by identifying targeted areas—such as variable definition, domain reasoning, and contextual validation—where instruction can be designed to foster deeper conceptual understanding and more authentic modeling practices.

6. Conclusions: Implications for Engineering Math Education

The findings of this study carry significant implications for the pedagogical approaches to calculus within engineering and management education. Engineering disciplines demand that mathematics be understood as a practical instrument for analysis, design, and problem-solving (Pepin et al., 2021), necessitating competencies in translating abstract concepts into practical applications and interpreting results within an engineering framework. The observed cognitive obstacles highlight a gap between current instructional practices and these essential demands.

To bridge this gap, pedagogical approaches must move beyond generic practice and explicitly target the cognitive obstacles identified in this research. To foster genuine engagement and deeper conceptual understanding, educational interventions must be highly targeted. Rather than generic practice, teaching strategies should explicitly aim to enhance students’ abilities to transition fluidly between semiotic registers.

First, instruction should foreground the importance of clear and consistent variable definitions, ensuring that students practice stabilizing the meaning of variables across verbal, algebraic, and contextual representations. This includes making domains explicit and treating them as integral to problem formulation rather than as afterthoughts.

Second, educators should also emphasize the iterative nature of the Mathematical Modeling Cycle. Students require structured opportunities to practice all stages of the MMC, particularly the initial understanding and simplifying efforts and the crucial final interpreting and validating stages. This aligns with the MMC structure proposed by Blum and Leiss (2007), emphasizing the need for critical reflection on whether mathematical results are reasonable, feasible, and economically sound within the real-world context.

Third, instructional strategies should address representational coordination directly. For instance, when students struggle with graphical constraints, interventions should focus on the specific cognitive act of connecting the abstract graph to its real-world domain and limitations. Similarly, to overcome challenges with the economic context, instruction should explicitly link mathematical functions to their practical economic meanings and implications.

By differentiating between general learning challenges and specific cognitive obstacles, educators can move from merely observing what students cannot do to understanding why they cannot do it, leading to more effective and impactful teaching. This precise diagnostic approach, informed by Duval’s theory, will likely improve students’ comprehension and problem-solving skills in applied calculus contexts, better preparing them for professional engineering practice.