Mathematical Modeling

Mathematical Modeling, is the process of expressing, analyzing, and resolving complex real-world situations, events, or problems using mathematical terms, structures, and relationships. This concept functions as a bridge between real life and the mathematical world, forming a cyclical methodology that involves translating authentic scenarios into the language of mathematics and reinterpreting the resulting mathematical outcomes back within their real-world context.

Theoretical Foundations

In the literature, mathematical modeling is defined as a process in which students apply mathematical knowledge to solve real-world problems, while the concept of a “mathematical model” refers to the symbolic products—such as formulas, graphs, and equations—that emerge from this process and represent reality. Thus, modeling is not a static body of knowledge but a dynamic competency domain in which the functioning and structure of a situation are expressed through the symbolic language of mathematics to achieve understanding.

At the core of mathematical modeling lies the requirement that the situation under consideration exhibit a “stable” or “partially stable” structure. Not every situation in nature or social life can be modeled; only those with a discernible pattern, relationship, or rule can be expressed mathematically. For instance, predictable and mathematically describable (stable) phenomena such as physical laws or population growth form the subject of modeling, whereas chaotic and entirely random (unstable) situations fall outside this scope.^【1】 Modeling is not limited to natural phenomena such as those in the natural sciences; it also plays an effective role in defining the rules of human-made social systems such as electoral systems or pricing tariffs.

The Modeling Process

Mathematical modeling has an iterative and cyclical structure rather than a linear progression. The process begins with encountering a real-world problem situation and continues with its mental structuring. The key stages include understanding the real-life problem, simplifying the situation and identifying variables, translating the relationships among these variables into mathematical terms (mathematization), generating a solution through mathematical operations, interpreting the result within the real-world context, and validating the model’s validity.

Various cyclical models have been proposed in theoretical frameworks to explain this process. Early approaches suggested more linear steps, while contemporary models—such as those by Borromeo Ferri or Blum and Leiß—offer flexible structures that allow for feedback loops and revisions between stages.^【2】 In these cycles, the individual moves from the real world (real situation) to the mathematical world (mathematical model), arrives at a conclusion through mathematical operations (mathematical result), and then returns the result to the real world (real outcome/interpretation) to assess whether the problem has been resolved. If the model is found inadequate during validation, the process returns to its beginning, where assumptions are reviewed and the model revised.

Competency Dimensions

Mathematical modeling competency refers to an individual’s capacity to express real-life problems mathematically, develop appropriate models, and apply these models within their context. This competency is not limited to cognitive skills alone but constitutes a multicomponent structure encompassing metacognitive, affective, and social dimensions. Within the cognitive dimension, sub-competencies include understanding the problem, simplifying it, mathematizing it, working mathematically, interpreting results, and validating the model.

The metacognitive dimension involves planning, monitoring, and evaluating one’s own thinking processes during modeling; the affective dimension encompasses factors such as motivation, willingness, and self-efficacy. Moreover, since modeling activities typically require group work, social competencies such as communication, discussion of ideas, and collaborative decision-making are integral to the process. Research indicates that students generally perform better in the problem understanding stage but struggle with higher-order cognitive processes such as simplification, mathematization, and validation.

Relationship with Related Concepts

Mathematical modeling is closely related to problem solving and abstraction, yet it is structurally distinct from them. While problem solving is a general umbrella concept, mathematical modeling is regarded as a specialized and more comprehensive form of problem solving, grounded in real-life situations and bound by the principle of “reality.” Traditional word problems typically present ready-made data, have a single correct answer, and involve artificial constructs; modeling problems, by contrast, involve scattered or incomplete data, require assumptions, and allow for multiple solution pathways, making them open-ended in structure.

In relation to abstraction, modeling supports the process of identifying common features of real-life situations and transforming them into mental constructs. Since modeling a situation requires recognizing its invariant elements and relationships—such as a function or pattern—the modeling process is fully aligned with empirical abstraction.

Design of Activities and Principles

The quality of mathematical modeling activities used in educational settings is decisive in helping students acquire these skills. An effective modeling activity must be non-routine, connected to real life, and provide opportunities for students to reveal their own thinking processes.

In designing such activities, six fundamental principles identified by Lesh and colleagues stand out:

1. The activity requires students to construct a structure or system (Model Construction),

2. The context must be a realistic situation aligned with students’ experiences (Reality),

3. Students must be able to evaluate their own solutions (Self-Assessment),

4. The solution process must be documented in writing (Model Documentation),

5. The produced model must be adaptable to similar other situations (Model Generalization),

6. The constructed model must be simple yet functional (Effective Prototype).

Use in Educational Contexts

Teaching mathematical modeling requires a student-centered and process-oriented approach. The teacher’s role shifts from direct knowledge transmission to acting as a guide (facilitator) who supports students’ thinking during the modeling process. Assessment must focus not only on the accuracy of the final product but on the overall quality of the modeling process. For this purpose, the use of multiple assessment tools such as rubrics that measure students’ assumptions, strategies, and validation processes, observation forms, peer assessments, and project portfolios is recommended. Research reveals a positive relationship between modeling competencies and mathematical achievement, although this achievement is not always directly correlated with attitudes toward mathematics.^【3】