In recent years, with the continuous development of educational modernization, personalized teaching has become an important direction in current educational development. The traditional one-size-fits-all education model is no longer able to meet the personalized needs of students. Therefore, how to provide personalized learning path recommendations (how to arrange hierarchical and personalized assignments, completing consolidation learning recommendations) has become an urgent issues to be addressed. In current related research on personalized learning path recommendations, the purpose is to customize a learning path for individual learners that adhere to educational principles, is practical, feasible, and scientifically efficient. This approach assists learners in attaining their learning objectives to the fullest extent while minimizing associated learning costs.

Personalized learning path recommendation is a research topic that involves interdisciplinary studies in education, psychology, computer science, and other fields. Its research approaches and methods can be roughly categorized into three types: education theory-driven learning analytics (LA) methods, data-driven educational data mining (EDM) methods, and hybrid methods that combine the first two approaches (Aldowah et al., 2019, Siemens and Baker, 2012). In this context, learning analytics methods driven by educational theory primarily concentrates on the examination of educational principles, offering the benefit of robust interpretability. However, they also come with some drawbacks, including limited portability, elevated costs, and a substantial impact from researchers’ subjective factors. On the other hand, data-driven educational data mining methods aim to unveil inherent patterns in the personalized learning path recommendations from extensive data, providing advantages such as strong portability and cost-effectiveness. Nevertheless, some accompanied shortcomings should be considered, including low interpretability and challenges in convincing experts in fields such as education. Therefore, researchers combine the two to form a hybrid methods. It is undeniable that compared to data-driven educational data mining methods, a mature, systematic, scientific, and highly interpretable educational theory is more difficult to obtain, and the theoretical foundation of this paper — the knowledge space theory — is such a scientific and effective theoretical framework.

Furthermore, many scholars have already conducted relevant research: Doignon and Falmagne (1985) and Falmagne and Doignon (2011) proposed definitions for the outer fringe, inner fringe, and learning paths in knowledge space theory. Teachers can utilize the outer fringe for learning guidance or learning path recommendations and utilize the inner fringe for reinforcement learning recommendations (including assigning homework). Addressing the shortcomings of the theory, many researchers (Doignon, 1994, Düntsch and Gediga, 1995, Falmagne et al., 1990, Gediga and Düntsch, 2002, Heller et al., 2013, Korossy, 1997, Korossy, 1999) collectively proposed the competence-based knowledge space theory. Subsequently, Anselmi et al., 2022, Anselmi et al., 2024, Heller et al., 2017, Heller et al., 2015, Heller et al., 2016 and Heller, Steiner, Hockemeyer, and Albert (2006) proposed methods for obtaining competence state in competence-based knowledge space theory and using it to find learning paths. However, this method requires introducing additional information or changing the problem set to obtain a competence state. Stefanutti and de Chiusole (2017) proposed a method in competence-based knowledge space theory specifically targeting the conjunctive model for finding learning paths under the condition of not knowing the specific competence state. However, this method only considers the special case where the effective outer fringe is non-empty. Meanwhile, in subsequent research on the knowledge space theory, we observe that the majority of researchers tend to focus solely on the outer fringe, neglecting the inner fringe. This is because most properties of the inner and outer fringes have simple correspondences, but the application of the inner fringe should not be overlooked.

Nabizadeh, Leal, Rafsanjani, and Shah (2020) categorize personalized learning path recommendation methods into two main classes: Course Generation (CG) and Course Sequence (CS). The methods of CG generate and recommend the entire path to a user in a single recommendation, with learning assessment occurring only after completing the path. The methods of CS generate and recommend a path to a user step by step, taking into account the user’s progress, and the learning assessment happens as the user progresses along the path. The recommendation method based on knowledge space theory belongs to CS, requiring personalized learning guidance step by step until the learner achieves the learning goals. Hence, the focus of this paper’s research will be on how to provide learning guidance.

In knowledge space theory, teachers assess students’ current knowledge states, outer fringes, and inner fringes based on their responses to test questions. Teachers can propose the next problems for students to solve based on the outer fringe and guide their next steps of learning, thereby planning students’ learning paths. Teachers can also identify the problems that students need to consolidate based on the inner fringe and assign corresponding homework. However, the outer fringe of the knowledge state may be empty, meaning that students in this knowledge state cannot make progress by studying just one question. Similarly, the inner fringe of the knowledge state may also be empty, indicating that students in this knowledge state cannot consolidate their knowledge state by reviewing just one question. Therefore, many scholars confine their research to well-graded knowledge spaces, which are a type of knowledge structure where each knowledge state has a non-empty outer (inner) fringe. Additionally, Noventa, Spoto, Heller, and Kelava (2019) expanded the concept of the outer fringe, no longer limiting it to learning only one question at a time, which better reflects real-world situations. We refer to this as the outer master fringe, and correspondingly, we can define the inner master fringe.

Based on the above discussion, we can identify a limitation of knowledge space theory, which is that students should learn the skills and knowledge necessary to solve problems, rather than the problems themselves. Therefore, Falmagne et al. proposed competence-based knowledge space theory, elaborating on it extensively in Heller et al. (2013), which introduces skills into knowledge space theory. Similarly, this theory proposes the next skill for students to learn based on the outer fringe of competence state and identifies the skill for students to consolidate based on the inner fringe of competence state. However, since skills cannot be directly observed, the only way to determine whether a skill has been learned (consolidated) is by observing changes in knowledge state. However, in practical situations, students with varying competences might display identical knowledge states. Consequently, even if a skill is acquired or fails to be solidified (forgotten), the knowledge state may remain unchanged. Therefore, in the competence-based knowledge space theory, our emphasis lies on the effective (outer, inner) fringe of the competence state — comprising all skills capable of inducing alterations in the knowledge state. The change in knowledge state is inseparable from its fringes, and often, mastering a skill may solve multiple problems simultaneously. Therefore, whether introducing the concept of skills or not, acquiring the outer (inner) master fringe of the knowledge state is a crucial task.

However, Noventa et al. (2019) only define the master fringe, with its characterization limited to the performance level. That is, for a certain knowledge state K within the knowledge structure, its master fringe is determined by comparing it with adjacent states. Furthermore, there is no further research, let alone integration with competences. And this article provides a new characterization of the master fringe based on the perspective of competence. The new characterization here is mainly reflected in the fact that for a given knowledge state, corresponding skills can be pushed for different models to reach its adjacent state or consolidate the current state. Especially via the competence model, we identify a more reasonable conception of the master fringe. Specifically, for a knowledge state K, when considering competences, its next state may experience a jump situation, which means that it can directly reach a higher-level knowledge state K2 without passing through its adjacent knowledge state K1, where K is contained in K1 and K1 is contained in K2. This enables us to provide more detailed and accurate competence-based personalized learning guidance and reinforcement learning recommendations. Therefore, this paper aims, in the competence-based knowledge space theory, via the disjunctive, conjunctive, and competence models respectively, to propose a general method to directly compute the outer (inner) master fringe of the knowledge state based on the top or bottom of the equivalence class of competence state, and a general method for personalized learning path recommendation (reinforcement learning recommendation).

The organizational structure of this paper is as follows: Section 2 provides an overview of some basic concepts in knowledge space theory and competence-based knowledge space theory. Section 3, via the competence models, presents two characterization theorems (one characterizing the top or bottom of competence state using skill functions; another characterizing the master fringe of knowledge state using problem functions) and corresponding learning guidance (reinforcement learning recommendation) schemes. Additionally, special cases of these theorems under the disjunctive and conjunctive models are provided. Section 4 presents a real-world example. Section 5 concludes the paper.