Integrating role-play and STEM-based collaborative learning in understanding permutations and combinations

Abstract

This article presents a STEM-based instructional approach for teaching combinatorics to 95 s-year students (72 IT engineering and 23 engineering management students) at a Serbian university of applied sciences. Although students had prior exposure to combinatorics, many relied on memorized formulas without conceptual understanding. To address this issue, we implemented a collaborative, constructivist learning design in which students collaboratively constructed combinatorial formulas through role-play, multiple representations, and real-world problem scenarios rather than receiving them in finalized form. Activities were connected to applications in computer science, reinforcing interdisciplinary relevance. Classroom data were collected through structured field notes, analysis of students’ written solutions, and follow-up interviews when necessary. Qualitative analysis focused on students’ reasoning processes and conceptual development, complemented by statistical comparison of individual test results with a previous cohort taught through traditional lecture-based instruction. The experimental group achieved significantly higher performance. The findings indicate that interdisciplinary, constructivist learning environments can foster deeper conceptual understanding and improved application of combinatorial reasoning. The study contributes to mathematics education literature by illustrating how collaborative knowledge construction in STEM contexts supports the transition from experiential activity to formal symbolic generalization in tertiary combinatorics instruction.

1 Introduction

Combinatorics is a fundamental area of discrete mathematics with applications in science, engineering, and everyday problem-solving (Modabbernia, 2025). It is often introduced early in higher education due to its potential to develop reasoning, problem-solving skills, and flexible thinking. Counting problems are widely used as an entry point (Lockwood et al., 2020) because they are broadly accessible, foster rich reasoning, support productive mathematical habits, and provide a natural context for computational thinking. Even students with limited prior knowledge can engage meaningfully, and such problems promote deep mathematical thinking (Desfitri, 2024).

However, research shows that students often struggle with combinatorial problems (Eizenberg and Zaslavsky, 2003; English, 2005; Godino et al., 2005), a pattern that persists at the university level (Desfitri, 2024). Many rely on memorized formulas without understanding their application, as observed at Subotica Tech—College of Applied Sciences, where combinatorics is taught in the second-year Statistics course. Prior studies have largely focused on traditional instruction, with limited exploration of collaborative, STEM-based, and role-play interventions in higher mathematics education.

This study addresses these gaps by implementing a constructivist, collaborative approach in which students actively construct combinatorial formulas through role-play, real-world scenarios, and interdisciplinary tasks. The intervention aims to enhance conceptual understanding, problem-solving skills, and engagement, demonstrating effective pedagogical strategies for higher mathematics education.

2 Pedagogical framework

2.1 Constructivist foundations

Constructivism views learning as an active process in which learners build understanding rather than passively receive information. As (Bodner 1986, p. 873) states, “Knowledge is constructed in the mind of the learner,” and von Glasersfeld (1995) similarly argues that the mind knows only what it has constructed. Learners connect new ideas with prior knowledge (Naylor and Keogh, 1999), while knowledge remains both personal and socially mediated (Sjøberg, 2010). It develops through cycles of comparing experiences with expectations and adjusting mental models (Taber, 2011). Teachers, therefore, create conditions that stimulate inquiry and dialogue (Brooks and Brooks, 1993), focusing on application rather than rote recall. Social interaction plays a central role: small-group collaboration and peer negotiation foster deeper understanding and prepare students for real-world problem solving (Good and Brophy, 1994).

2.2 Collaborative learning principles

Collaborative learning extends constructivist principles by emphasizing shared responsibility. For this study, we adopt a composite definition from Takači et al. (2015), integrating perspectives from (Laal and Ghodsi 2012, p. 486) and (Gokhale 1995, p. 22): “Collaborative learning is an instructional method in which students at various performance levels work together in small groups toward a common goal—that is, to solve a problem, complete a task, or create a product.”

Unlike individual constructivism, collaborative learning requires learners to support each other's progress, making group success interdependent (Gokhale, 1995). Research confirms its benefits: higher academic achievement, positive attitudes, and greater persistence (Springer et al., 1999). Early exposure also fosters critical-thinking skills (Loes and Pascarella, 2017). Socially, collaborative learning teaches students to respect differences and resolve conflicts constructively (Johnson and Johnson, 1985). Thus, it represents a natural extension of constructivism, highlighting the social dimension of knowledge construction.

2.3 Multiple representations and role-play

Representations mediate between abstract concepts and learners’ cognitive processes. Bruner (1966) distinguished three modes—enactive, iconic, and symbolic—which often intertwine in mathematics learning. In combinatorics, students may begin with physical selections (enactive), progress to diagrams (iconic), and finally construct symbolic formulas. This progression scaffolds understanding and reflects the approach adopted in this study, where role-play and visual models supported the transition toward symbolic reasoning.

Research underscores the significance of representations. Duval (1995) argued that mathematical knowledge cannot be mobilized without representational activity, while Kaput (1987) and Zazkis and Liljedahl (2004) emphasized their role in communication and conceptual understanding. Learners must develop fluency in translating between representational forms (Greeno and Hall, 1997; Mainali, 2021), and teachers should integrate representations purposefully (Goldin and Shteingold, 2001).

Role-play complements this approach by embedding mathematical ideas in interactive scenarios. Rao and Stupans (2012) define role-play as a method where students assume roles and solve problems within given contexts, enhancing learning across cognitive, psychomotor, and affective domains (Fu and Li, 2025). Studies confirm its benefits for engagement and performance (Ameyaw and Mante, 2025; Svensson and Regnell, 2017). In mathematics, role-play fosters creativity and problem-solving (Hye-Yun and Kyeong-Hwa, 2023) and has been applied successfully to combinatorics tasks involving permutations (Siligar et al., 2018). Combining role-play with multiple representations creates dynamic learning environments that promote deeper understanding and active participation.

2.4 STEM approach to learning

STEM education emphasizes integrating knowledge across disciplines to prepare students for real-world challenges. Fajrina et al. (2020) describe STEM as an interdisciplinary approach that connects academic concepts with authentic problems, enabling students to apply science, technology, engineering, and mathematics in meaningful contexts. This approach aims to develop critical thinking, creativity, communication, collaboration, and other 21st-century skills Dutchak (2021). highlights STEM's distinction from traditional models: it focuses on everyday life and real problems, ensuring integrated formation of scientific and practical knowledge through authentic experiences. This design supports lifelong learning and professional competencies. To achieve these goals, teachers must create activities that integrate content, provide authentic contexts, and encourage collaborative problem solving.

3 Learning environment

3.1 Setting and participants

The study was conducted at Subotica Tech—College of Applied Sciences, Subotica, Serbia, during the second semester of the 2024/25 academic year in the Statistics course. Ninety-five second-year engineering students participated voluntarily, including 72 future IT engineers and 23 future engineering managers. At the beginning of the semester, several students reported difficulty applying combinatorial formulas, often relying on memorization without conceptual understanding.

To address this, a constructivist, collaborative, STEM-based instructional design was implemented. Students were assigned to 23 heterogeneous groups of four and one group of three, based on prior performance in Mathematics 2 (maximum score: 100), following Takači et al. (2015). Each group included at least one IT engineer and one engineering management student to ensure interdisciplinary collaboration. Learning took place over three weeks, with four lessons per week, in two large classrooms.

3.2 Research question

The primary research question guiding this case study was:

How does a collaborative, constructivist, STEM-based instructional approach incorporating role-play and real-world applications influence second-year engineering students’ conceptual understanding of combinatorics, particularly in constructing and applying formulas for permutations and combinations, as well as their achievement and attitudes?

3.3 Learning objectives

3.4 Permutations without repetition: From decision trees to tree diagrams

The sequence of activities was designed to build on students’ prior knowledge from programming, algorithms, and business decision-making. The interdisciplinary integration explicitly connects students’ prior knowledge from computer science and management studies with combinatorial reasoning. Algorithmic thinking and decision-tree structures served as cognitive bridges, enabling students to transfer structured branching logic into mathematical counting processes.

All student groups consisting of four members approached the task of creating the required algorithm with great seriousness, and every group produced a correct solution. Figure 1 presents the four principal solution patterns. For publication clarity, Figure 1 was digitally redrawn from the students’ original hand-drawn diagrams and translated from Hungarian into English, preserving the original solution logic.

Figure 1

It took a little more time for the groups to complete this task, as most participants had not previously taken this course. In this group work, management students took the leading role. In most solutions, the decision trees were drawn horizontally. The following day, we inquired about this and learned that horizontal representations were also used in the Business decision making classes. Figure 2 was digitally redrawn from the students’ original hand-drawn diagram and translated into English while preserving the original decision structure.

Figure 2

The aim of the next task was to provide a gradual transition from using a decision tree, familiar from other subjects, to a tree diagram, which is used in combinatorics to represent and count all possible arrangements.

All student groups successfully completed the task. For part d), all groups except two gave the answer 6. One of the remaining two groups wrote the solution as 6 = 2 + 2 + 2, while the other wrote 3 × 2 = 6. All answers for part e) were similar to the following: “One possible way to connect resistors.” and for part f), all groups answered that there was no decision made. Five groups submitted the following response for part g): “We obtained information about the possible orders of connecting the three resistors.” In the remaining solutions, students also explicitly indicated the total number of possible orders.

Building on their hands-on experience with the four robots, students were able to generalize the concept. They observed that at each successive position in the line, the number of available choices decreases by one: 4 choices for the first position, 3 for the second, 2 for the third, and 1 for the last. This pattern corresponds exactly to the structure of the tree diagram, where each path in the diagram represents a unique sequence of arrangements.

From their observations, the students were able to derive a general formula for arranging n different elements in a sequence and answer sub-question (f) in the following way:The tree diagram provides a visual justification for this multiplication: the number of branches at each level corresponds to the number of remaining choices, and multiplying them together gives the total number of possible sequences. This illustrates how the combinatorial purpose of the tree diagram differs from the decision-making function of a decision tree while using the same underlying tree structure.

3.5 Permutations with repetition

After students mastered permutations without repetition, we introduced permutations with repetition through the following way.

3.6 Combinations without repetition

After exploring how repeated elements affect the number of arrangements in permutations, we turn to combinations without repetition, where the focus shifts from ordering to selecting subsets.

Although the primary purpose of this task was to pave the way toward the formula for combinations without repetition by means of permutations with repeated elements, it also offered students the opportunity to encounter an example of a bijective argument. In many counting problems in combinatorics, the goal is to determine the cardinality of a certain set. Instead of counting this set directly, it is often more convenient to formulate another problem, whose solution amounts to determining the cardinality of a different set. By establishing a bijection between the two sets, one shows that they have the same cardinality. This approach—referred to as a bijective proof—has been widely discussed in the combinatorics literature (for example, Stanley, 2009).”

Students divided roles among themselves. During the activity, they enacted the role-play: the employees took their seats in the reception room, the manager determined the assignments, the secretary placed the corresponding cards, and documented which employees worked with new clients and which with clients requiring support for their installed systems. In this way, one representative solution out of the ten structurally equivalent possibilities was recorded.

Upon analyzing the students’ notes and their solutions to Task 6, it became evident that nine groups (out of twelve) obtained a result corresponding to the number of permutations with repetition of five elements, indicating that they had successfully reached the intended conclusion. After performing several concrete divisions of work and making written notes, they wrote the solution to the problem, as shown in Table 2. Table 2 is a typeset reconstruction of the students’ original table. In the classroom, students marked the two job types with the initials of the Serbian color names—Z (zeleno, green) and Ž (žuto, yellow)—whereas Table 2 uses the English initials G (green) and Y (yellow) for clarity and consistency in the manuscript. They also calculated how many different ways the manager can select two of the five employees to work with new clients, and wrote the following formula:

Below is a typical explanation quoted in support of the given solution.

“Each division of work can be seen on cards that are on the desks of five employees, and there are 10 arrangements of the cards.”

The remaining three groups of students wrote out all ten possible assignments of work but did not notice the connection between this task and permutations with repetition.

The solutions of the first group are shown in Table 3 redrawn from the students’ table and translated into English. Group 1 originally marked the job types with the Serbian words “nov” (new) and “podrška” (support).

The students in each of these groups were interviewed the following day.

For the group whose solution is presented in Table 3, the teacher encouraged them to observe, for each possible work assignment, the corresponding sequence of words (listed top to bottom):

Teacher: “For each division of work, if we look only at the words next to the names of employees, what can we say about them?”

Student V: “There are two different words because there are two types of jobs.”

Student A: “In every case, there are three words: ‘new’ and two ’support’.”

Teacher: “If, for each division of work, we consider the corresponding sequence of words (from top to bottom), can we calculate how many such sequences can be written?”

Student B: “There are 5! divided by 2! times 3!.”

Student N: “That is the number of possible divisions of work. We didn’t think of that.”

With the other two groups, the discussion unfolded in a similar way, and through it, they realized how the task could be solved using permutations with repetition.

Students in one group drew a table similar to that in Table 3 and wrote radi (works) below the four letters (representing employees’ names) and ne radi (does not work) below the remaining three. They solved Task 8 quickly in the same way as Task 7.

The discussions in these groups lasted a bit longer than in the first task, and at one point, a student addressed the teacher, saying:

“The task says what four employees are doing, but it doesn’t say what the other three are doing.”

A response came from a student:

“They are not working on the financial project; they are doing something else.”

This comment was sufficient for the remaining groups to complete the task.

The next question required students to express the number three in the solution using the numbers seven and four. All groups quickly wrote the answer by representing 3 as 7–4.

Finally, the teacher summarized the discussion and formulated the general question:

In how many ways can k elements be chosen from a set of n elements? Students of all groups correctly wrote the formula, after which the teacher explained that it is denoted by and read as “n choose k.”

3.7 Data collection and analysis

4 Results

In this study, we distinguish between learning outcomes (conceptual understanding and procedural accuracy) and the development of structured problem-solving strategies. While test performance reflects mastery of combinatorial formulas, classroom artefacts document the evolution of students’ reasoning processes.

In the fourth week, students completed an individual written test consisting of four traditional combinatorics problems aligned with the instructional sequence. The average score for the current-year cohort was 82.4 points (SD = 9.2), compared to 74.1 points (SD = 10.0) for the previous-year cohort. An independent one-sided Welch's t-test confirmed a significant difference of 8.30 points, t(174.84) = 5.83, p = 1.29 × 10⁻⁸. The one-sided 95% confidence interval had a lower bound of 5.94 points. The standardized mean difference was Hedges’ g = 0.87, indicating a large effect.

Following the test, students completed an anonymous questionnaire. Most reported limited prior exposure to role-play or tree-based reasoning in mathematics classes. Nevertheless, 87% indicated that collaborative work and role-play supported their understanding of permutations and combinations, and 91% expressed interest in applying similar approaches in other topics. Open-ended responses emphasized the value of linking mathematical reasoning to programming and STEM-related contexts.

Directed content analysis of classroom dialogue, student artefacts, and questionnaire responses revealed a consistent representational progression. Tree diagrams made the decreasing number of available choices structurally visible in permutations without repetition. In programming-based tasks, students reconstructed permutations with repetition by reasoning about identical roles and overcounting.

This reasoning became pivotal in the transition to combinations. In the role-play task of selecting four managers from seven, students initially applied permutation logic but, through discussion, recognized that specifying four implicitly determined the remaining three (7−4). Across groups, students generalized the problem of choosing k elements from n and constructed the combination formula before formal notation was introduced.

Formulas were not provided in advance. Through coordinated role-play, diagrammatic representation, and structured task sequencing, students progressed from concrete enumeration to symbolic generalization.

The convergence of significant quantitative gains and consistent qualitative patterns indicates that the representationally rich, collaboratively structured design supported transfer from constructed understanding to formal individual problem solving.

5 Discussion

The findings provide converging quantitative and qualitative evidence that the instructional design supported students’ transition from concrete activity to formal combinatorial reasoning. The large effect size in cohort comparisons, together with consistent patterns in classroom discourse, artifacts, and post-task reflections, indicates that learning gains extended beyond procedural fluency to structural understanding.

The representational progression—enactive role-play, iconic tree diagrams, and symbolic generalization—aligns with constructivist principles emphasizing knowledge construction through activity and reflection. Students reconstructed permutation and combination formulas through coordinated representations and dialogue, rather than receiving them deductively, demonstrating structural abstraction from experiential engagement.

Collaborative processes played a central role. Students negotiated overcounting in permutations with repetition and shifted from arrangement-based to selection-based reasoning in combinations, illustrating how peer explanation, negotiation, and shared responsibility facilitated conceptual reorganization. These processes mirror principles of cooperative learning (Johnson and Johnson, 2017).

Tree diagrams functioned not merely as visual aids but as structural tools, making combinatorial constraints explicit and scaffolding the coordination of counting processes with symbolic expressions. The visual representations (decision trees and combinatorial diagrams) functioned as transitional modeling tools. Although initially focused on concrete problem-solving, these representations supported the gradual abstraction of structural relationships, facilitating the construction of general combinatorial formulas. Embedding problems in programming, electrotechnics, and management contexts strengthened relevance and transfer, as students connected mathematical reasoning to authentic STEM applications.

Overall, the study offers a replicable instructional model integrating role-play, structured representations, and collaborative reasoning to support abstraction in discrete mathematics. The design supported students in achieving the following objectives: conceptual understanding of permutations and combinations, representational fluency, applied STEM relevance, and collaborative competencies.

Several limitations should be acknowledged: the study was conducted in a single institutional context, over a short instructional period, and without longitudinal follow-up, which constrains generalizability. Future research could explore different contexts, other mathematical topics, and extended mixed-method or longitudinal designs.

6 Conclusion

This study investigated how role-play, collaborative learning, and structured task sequencing can support students’ transition from concrete activity to formal combinatorial reasoning. Students successfully constructed permutation and combination formulas through coordinated use of role-play, tree representations, and guided generalization, rather than receiving formulas in advance. Quantitative gains and qualitative evidence converged to show effective transfer to formal individual problem-solving.

The study contributes to constructivist mathematics education by demonstrating how multiple representations and collaborative dialogue can be deliberately sequenced to scaffold structural abstraction. Embedding combinatorial reasoning in STEM-relevant contexts and bridging permutations toward combinations illustrates a replicable pedagogical model for conceptual learning in discrete mathematics.

Several limitations should be noted. The study was conducted in a single institutional context, over a short instructional period, and without longitudinal follow-up. While cohort comparison and mixed data sources strengthen the findings, broader generalization requires caution.

Future research may test this instructional design in diverse educational settings, extend it to other mathematical topics, and employ longitudinal or expanded mixed-method approaches to examine long-term conceptual retention and transfer. Overall, the results suggest that carefully designed representational and collaborative environments can make combinatorics both conceptually accessible and pedagogically meaningful.

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