Multi-objective Clustering Algorithm Applied to the MathE Categorization Problem

Active learning encompasses strategies where students actively engage in activities beyond passive listening to lecturers (Alhawiti, 2023; Meyers & Jones, 1993). Students participate in the discovery process, information processing, and application tasks during active learning. Active learning is based on two fundamental assumptions: first, that learning inherently involves active participation, and second, that individuals who prefer diverse learning styles should be supported (Meyers & Jones, 1993). Research has consistently demonstrated enhanced learning outcomes when students actively participate in the learning process (Meyers & Jones, 1993). Nevertheless, it is crucial to acknowledge that traditional lecturers are significant and that active learning should always be accompanied by well-defined content and learning objectives (Meyers & Jones, 1993).

The development of learning skills and the learning process are strongly dependent on the active involvement of students in their education (Alhawiti, 2023). The case study research shows that active teaching strategies increase lecturers’ attendance, engagement, students’ acquisition of expert attitudes towards the subject, and engagement improves students’ cognitive, learning, and skills while also improving engagement (Alhawiti, 2023). Besides, students expressed a high level of satisfaction with active learning methods (Alhawiti, 2023; Deslauriers et al., 2019).

Individuals differ in their profiles of strengths and weaknesses across different types of intelligence, and educational and professional success depends on the ability to enhance their strengths and compensate for weaknesses (Gardner, 1999, 2011). The recognition of individual learning styles and the implementation of personalized learning resources has been shown to improve educational experiences and results in an increasing number of studies (Lwande et al., 2021). Nonetheless, there are still notable differences in the data about how well this method works for various learning styles (Villegas-Ch et al., 2024). There is a gap in the thorough understanding of its application and advantages because some current research only concentrates on a particular style or in limited contexts (Pritalia et al., 2020). The work (Villegas-Ch et al., 2024) presents the implementation of a machine learning system designed to identify students’ learning styles and adapt educational content accordingly. Additionally, the authors assess the tangible effects of these customized approaches on students’ engagement with the subject matter and academic achievement. The findings imply that personalized learning is a potent and successful instrument that can enhance both the educational experience and student performance at the same time. As a result, the study (Villegas-Ch et al., 2024) provides a workable model for the effective application of artificial intelligence-supported educational personalization while reiterating its importance.

Under this scenario, the MathE platform emerges as an innovative, dynamic, and intelligent digital tool for teaching and learning Mathematics. MathE is a collaborative e-learning platform that enhances students’ mathematical learning processes in higher education. Its core objective is to cultivate virtual learning and foster knowledge exchange (Azevedo et al., 2021, 2024). MathE is a pioneering platform that significantly departs from traditional learning, ushering in an interactive and engaged learning paradigm. MathE operates as a nonprofit initiative, offering unrestricted access 24 hours a day, serving as a valuable resource for anyone eager to increase their mathematical knowledge and comprehension.

Currently, the MathE platform has 1824 multiple-choice questions, organized into two difficulty levels: basic and advanced (Azevedo et al., 2024). This rating is assigned by the registered lecturer who inserted the question on the platform. Nevertheless, previous research (Azevedo et al., 2021) has shown that the two-level categorization is inadequate for effectively distinguishing the available content. Additionally, it has been verified that most of the students who use the MathE platform only perform the basic level questions; they rarely attempt to answer the advanced level ones, causing demotivation in using the platform. Moreover, from previous work (Azevedo et al., 2024) and students’ feedback, it is known that the opinion of students and lecturers about the complexity of the questions is often divergent: while for a lecturer, a question may be very basic, for a student, the same question may be considered very difficult, and vice versa. To group the questions and obtain a more accurate distribution by difficulty level, it is important to consider both perspectives: students and lecturers.

The dataset used in this work is based on the answers stored on the MathE platform system, which is stored in terms of correct or incorrect answers. Therefore, the historical data for each question is evaluated, taking into account the number of correct or incorrect answers to define two different variables (the error rate and the cumulative score for each question), which will define the difficulty of the question from the students’ point of view. After that, the students’ opinion is combined with the lecturers’ instruction through a weighted average, defining a score for each question. This score will be used to group the questions into different difficulty levels. To perform this procedure, the MCA is proposed, and the classical k-means algorithm (Arthur & Vassilvitskii, 2007) is used for results comparison. This work presents an improved version of the MCA in Azevedo et al. (2024), which was dependent on an initial parameterization and used the NSGA-II in the optimization process. In this paper, a refined mathematical problem formulation is put forth, which does not require parameterization, and for optimization procedure this work advocates the adoption of MOPSO (Coello-Coello & Lechuga, 2002), deeming it to be more robust and more aligned with the proposed approach.

The MCA proposed in this work is a bio-inspired clustering algorithm that uses a multi-objective strategy to provide a set of optimal solutions, allowing the decision maker to select the most fitting solution for the problem since it recognizes that the decision maker possesses vital information essential to addressing the issue effectively (Azevedo et al., 2025). The MCA intends to minimize an intra-clustering measure and simultaneously maximize an inter-clustering measure to automatically define the optimal number of centroids and the element’s optimal distribution. The MCA is detailed in Section 5. This paper represents a noteworthy contribution by introducing an automatic bio-inspired clustering algorithm. Through an optimization process, the algorithm autonomously determines the number of cluster centroids and the distribution of elements. This addresses major concerns commonly associated with clustering algorithms documented in the literature (Azevedo et al., 2025). Since the MCA is a multi-objective algorithm, different from traditional clustering algorithms, it defines a set of optimal solutions. This provides decision-makers with various solutions, introducing flexibility to handle the complexities of the problem. It is often challenging to incorporate certain knowledge about the data into the mathematical model. Therefore, providing the decision-maker with a set of optimal solutions is valuable, allowing them to use their understanding of the data to choose the best solution for the problem. Additionally, one of the main advantages of MCA is the ability to customize the algorithm according to specific problem constraints, such as the minimum and maximum number of clusters and the quantity of elements in each cluster, for example. A realistic scenario is presented and discussed to demonstrate the added value of the approach.

This paper is organized as follows: after the introduction, Section 2 presents some concepts of active learning and research involving the development of algorithms and digital systems involving this concept. After that, the MathE platform is defined in Section 3. Section 4 presents the problem statement and the methods used to evaluate the question score, combining the lecturers’ and students’ opinions regarding the question’s difficulty level. The same section also defines the considered dataset. Section 5 describes the MCA and the presentation of the results and discussion achieved with the proposed methodology is shown in Section 6. Finally, this research’s main conclusion and future direction are described in Section 7.

There are many differences in the way each student learns. Some of them are more logical and learn more using their logical abilities. Others prefer to use their bodies, hands, and sense of touch to learn (Gardner, 1999). According to Gardner (1999, 2011), people exhibit varying combinations of strengths and weaknesses in different areas of intelligence, and educational and professional success depends on the ability to leverage their strengths and compensate for weaknesses (Gardner, 1999, 2011). Hence, achieving effective teaching tailored to each student’s unique learning style is a formidable challenge.

The conventional teaching approach, often called lecturer-centered or passive learning, may not be the most effective method for all students. In the passive learning model, the lecturer is the primary authority figure, while students act as passive recipients (Rodríguez, 2012). Consequently, they passively acquire information through lectures and direct instruction, with the primary objective being positive outcomes in tests and assessments. On the other hand, active learning emphasizes the active role of students in the learning process, although the lecturer remains an authority figure. In active learning, the lecturer’s primary duty is to guide and facilitate students’ learning and understanding of the material, assessing their progress through various assessment methods, including group projects, student portfolios, and class participation. In this educational model, the classroom, teaching, and assessment are interconnected, as students’ learning is consistently assessed during lecturers’ instruction. This approach fosters a more equitable relationship between the lecturer and the student, each playing a crucial role in the learning process (Alhawiti, 2023; Farrow & Wetzel, 2021).

The Principles and Standards for School Mathematics, published by NCTM in 2000 (NCTM, 2000), outlines the essential components of a high-quality school mathematics program relevant to 21st century skills. The document lists six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology), which, although defined more than 20 years ago, continue to make perfect sense in the present. Among the principles, Equity and Technology stand out. Equity does not imply uniformity in instruction for every student. Rather, it requires providing reasonable and appropriate adaptations and including appropriately challenging content to ensure access and success for all students. High-quality Mathematics instruction can enable all students to learn Mathematics while respecting their individual characteristics, backgrounds, or physical challenges (NCTM, 2000). Technology is essential in teaching and learning since the students can develop a deeper understanding of mathematics with the appropriate use of technology (NCTM, 2000) and teacher support.

Recently, there has been a significant increase in the integration of emerging technologies in education, particularly during the COVID-19 pandemic, which lasted for over two years (Treve, 2021; Al-Kumaim, 2021). This transformation has profoundly impacted conventional educational practices, leading to greater diversity in teaching and learning methods and fundamentally altering the trajectory of higher education in the future (Rangel-de Lázaro & Duart, 2023).

With the pandemic, the discussion about e-learning, active learning, and related terms has been accelerated dramatically (Engelbrecht et al., 2023). Thus, it is recommended that the educational system and learning environments should adapt their teaching approaches to address the increasing demand for students to engage in critical thinking, problem-solving, and skill development. This can be accomplished by emphasizing active learning strategies, fostering students’ personal growth, and enhancing their professional abilities (Alhawiti, 2023; Rangel-de Lázaro & Duart, 2023; Lara-Lara et al., 2023).

Several cases of using e-learning and active learning have emerged in this context. Duolingo, one of the world’s largest language learning platforms (Portnoff et al., 2021), employs these techniques to personalize the user experience and achieve more satisfactory results. Through gamification and adaptive learning, this platform makes the process of learning a new language easier, addressing individual difficulties and customizing the level of challenge for each user.

In the healthcare field, it has become evident that the application of active learning is beneficial for exercise recommendations and user adaptation (Mahyari et al., 2022). Personalization, particularly for new profiles when the training dataset is unavailable, represents a challenge that can be overcome with an active methodology. The concept of specialized and real-time active learning has proven to be more accurate, using feedback to estimate the system’s uncertainty, and involving an expert when it falls below a certain threshold.

Furthermore, in research-oriented teaching, the use of e-learning has shown significant advantages, as highlighted in the study carried out on the e-learning course Mathematical Analysis at Borys Grinchenko University of Kiev (Astafieva et al., 2020). This example illustrates the potential of e-learning to promote Mathematical competence and highlights the importance of appropriate pedagogical and methodological approaches to take full advantage of its benefits. In addition to these advantages, it is worth highlighting that the effective development of mathematical competence is only achievable through the active participation of students and collaborative interaction, as demonstrated by the Moodle platform used in the Mathematical Analysis subject. This emphasizes the importance of promoting active engagement and partnership between students and lecturers. Consequently, it is clear that, despite the advantages of e-learning, there are challenges associated with its effective use. As a result, the study exposes the need for further research to explore additional e-learning opportunities in the development of mathematical skills among students.

Despite being in different domains, the three reported cases in references (Portnoff et al., 2021; Mahyari et al., 2022; Astafieva et al., 2020), use e-learning tools for individual enhancement and adaptability, making the learning experience more rewarding and tailored to the user. Consequently, applying these learning techniques, even in diverse fields, has proven highly advantageous, indicating significant potential and the need for further research and development.

MathE is an international platform with the goal of enhancing the quality of teaching, learning, and assessment methods in higher education Mathematics content. This platform is a non-commercial tool, being completely free for those interested in improving their knowledge and understanding of Mathematics. MathE has been online since 2019 at mathe.ipb.pt. The platform is currently being used by a significant number of users; to be more precise, there are currently 1468 students enrolled on MathE from 21 nationalities: Portuguese, Brazilian, Turkish, Tunisian, Greek, German, Kazakh, Italian, Russian, Lithuanian, Irish, Spanish, Dutch, Slovenian, Swedish, Ukrainian, French, Finnish, Bulgarian, Algerian and Romanian. Additionally, there are 119 lecturers from 21 countries and 60 higher education institutions registered.

Moreover, MathE has a YouTube channel (youtube.com/@matheproject4778) where all videos available on the platform are linked to each corresponding topic. It should be mentioned that there are two types of videos available on the platform; those carefully selected from the internet for the MathE lecturers’ team members and those exclusively produced by the MathE partnership to meet the platform’s specific needs. For more details about the platform, refer to Azevedo et al. (2024, 2021).

At its current stage, the platform covers fifteen topics and twenty-two subtopics, among those that constitute the classical core of graduate courses: Analytic Geometry, Complex Numbers, Differential Equations, Differentiation (including 3 subtopics: Derivatives, Partial Differentiation and Implicit Differentiation and Chain Rule), Discrete Mathematics (with 2 subtopics: Recursivity and Set Theory), Fundamental Mathematics (with 2 subtopics: Elementary Geometry and Expressions and Equations), Graph Theory, Integration (with 5 subtopics: Integration Techniques, Surface Integrals, Triple Integration, Definite Integrals and Double Integration), Linear Algebra (including 5 subtopics: Matrices and Determinants, Eigenvalues and Eigenvectors, Linear Systems, Vector Spaces, and Linear Transformations), Numerical Methods, Optimization (with 2 subtopics: Linear Optimization and Nonlinear Optimization), Probability, Real Functions of a Single Variable (with 2 subtopics: Limits and Continuity and Domain, Image and Graphics), Real Functions of Several Variables (with 1 subtopic: Limits, Continuity, Domain and Image) and Statistics, as presented in Fig. 1.

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When selecting a topic and subtopic, students can answer a set of questions related to their chosen subject. At the end of the test, they receive feedback, and the questions are accompanied by supporting material such as videos and teaching materials produced by experts collaborating on the platform’s development. Figure 2 illustrates an example of the MathE supporting materials system. In this particular case, since the student provided an incorrect answer to the question, the platform showed the correct answer and also suggested a video and written material related to the content addressed in the question.

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This section presents more details regarding the problem addressed in this paper and the methods applied to combine the lecturers’ and students’ opinions on the questions’ difficulty level. Initially, the procedures to evaluate the question’s score are defined. After that, the dataset utilized is presented.

Multi-objective Clustering Algorithm is an unsupervised learning method that aims to divide the dataset into groups (clusters) based on the similarities and dissimilarities of the elements, focusing on discovering underlying patterns in an unsupervised manner (Rehman & Belhaouari, 2022). The following sections begin by presenting the bi-objective problem and the notation used in the algorithm, followed by the clustering measures and the Multi-objective Clustering Algorithm.

The MathE platform aims to offer a dynamic and compelling way of teaching and learning Mathematics, relying on interactive digital technologies that enable autonomous study (Azevedo et al., 2021). This work focuses on developing a strategy to group the questions considering students’ and lecturers’ opinions. For this, a score for each question is assigned, and the MCA algorithm is used to group the questions according to their difficulty level. Besides, the k-means is used to compare the clustering results.

The parameters used in the question score evaluation are: \(NA_{min} = 3\), and \(NA_{max} = 30\), \(\alpha = 1\) and \(\beta = 1\). Regarding the number of samples to be used, \(NA_{min}\) and \(NA_{max}\) were respectively defined equal to 3 and 30, since these values are the most appropriate values to obtain CS values and represent the students’ opinions, according to the statistical test and evaluation presented in Azevedo et al. (2024).

Regarding the choice of \(\alpha\) and \(\beta\) equal to 1, as previously mentioned, these parameters represent the lecturers’ and students’ weight, respectively, in the question categorization. In this work, both parameters were considered equal to 1 due to the dataset restrictions in terms of the number of answers computed per question.

The MCA main parameters are: population size and maximum of iterations equal to 100, the MOPSO repository size equal to 30, minimum number of centroids \(k_{min} = 3\), and maximum number of centroids \(k_{max} = \left[ \sqrt{m} \right]\). The \(k_{min} = 3\) is defined based on some previous research that reports two levels are not enough for the platform organization Azevedo et al. (2021, 2024).

The following sections present the results for each subtopic considered: Linear Transformation (Section 6.1), Fundamental Mathematics (Section 6.2), and Vector Space (Section 6.3).

It is essential to explicitly state that each cluster generated by the algorithms signifies a distinct difficulty level for the questions within the dataset. Cluster 1 denotes the lowest difficulty level, with subsequent clusters progressively increasing in complexity. The question score represents the complexity, thus, the higher the question score, the greater its complexity, as defined in Section 4. Thus, the hierarchical arrangement ensures a clear understanding of the difficulty progression within the clusters.

The use of e-learning platforms has become more prominent in recent years. Technological tools play a key role in facilitating teaching and learning, particularly in the realm of Mathematics, a subject often perceived as challenging. Thus, the students can develop a deeper understanding with the appropriate use of technology (NCTM, 2000) and the support of their lecturer. This work was addressed to solve the MathE questions categorization problem. Initially, all the questions available on the platform are divided into two difficult levels (basic and advanced), but the student reports several complaints (Azevedo et al., 2021, 2024) about using the platform, considering only two difficult levels that were defined only by the lecturer knowledge.

The MCA offers a notable advantage through its application of multi-objective optimization, presenting a diverse set of optimal solutions. This versatility empowers decision-makers to select the most fitting solution for a given problem. When confronted with challenges related to the MathE platform, certain information proves challenging to encapsulate within a mathematical model. Consequently, the incorporation of multi-objective solutions becomes particularly compelling. This approach grants decision-makers the flexibility of choice and facilitates the integration of crucial insights through a human-in-the-loop collaborative system.

From the range and variability of the Hybrid Pareto front generated, it is possible to perceive the impact of combining different measures to solve a problem. In this way, if only one pair of measures was considered in the model, the solution would be restricted to the optimum provided by one combination of measures and could be inappropriate for the decision-maker. So, the Hybrid Pareto fronts strategy enriches the model’s final solution.

Furthermore, the MCA does not require the prior indication of the cluster number, which is a common complaint in the literature regarding k-means and other partitioning clustering algorithms (Azevedo et al., 2024). In general, the results presented by the MCA are very promising. Regarding the solutions proposed by k-means, the MCA achieved a better balance in terms of the number of questions within the clusters, as shown by the results presented in Fig. 5c in relation to Fig. 6b, and also, Fig. 11b in relation to Fig. 12b.

As far as future developments are concerned, the goal is to optimize and determine the weight assigned to students and lecturers when calculating question scores. Furthermore, with regard to MCA, it is intended to investigate alternative multi-objective strategies that go beyond MOPSO, to refine the optimization of the proposed model further. Additionally, it is necessary to implement and assess the question categorization strategies using other MathE datasets.

Finally, future approaches involving hybrid algorithms are also necessary. The existing literature has focused on comparing bio-inspired or hybrid methods with traditional ones through mathematical analysis of runtime, convergence, and parameter configurations. Few of these studies have systematically compared the performance of different bio-inspired algorithms in machine learning tasks or different machine learning techniques in bio-inspired optimization algorithms. This leads to a lack of experimental results in selecting the most suitable method for a particular combination. The unavailability of such surveys may be due to the lack of publicly available source code, variation of encoding techniques, different objective functions, and evolutionary operators. As a result, there is a vast amount of published work since numerous metaheuristic algorithms can be combined with machine learning. Nevertheless, it remains difficult to point out which combinations are most appropriate or even why one is more advantageous than the other. Thus, additional research effort is necessary to facilitate the integration between deep learning, increased explainability, meta-learning, cross-domain applications, real-time adaptation, and human expertise. These trends aim to improve hybrid algorithms’ performance, efficiency, and versatility in tackling complex real-world problems.