A new tool for modeling uncertainty in educational programs in the form of phase categories is proposed. Uncertainty naturally exists in the reflections between the topics, outcomes, and competencies of educational programs. A review of the literature exploring the formalization and evaluation of educational programs is conducted. A diagram of the relationships between the elements of the model within an educational program is provided. Approaches to applying category theory to describe educational programs in the form of a rigorous mathematical structure are described, where objects are interpreted as topics or outcomes, and morphisms are interpreted as logical dependencies between them. A number of approaches to the reasonable determination of the values of weight coefficients of mutual influences between functor mappings are proposed. Several lemmas are formulated and proved, which mathematically justify the authors' proposed toolkit for modeling uncertainty. Examples of the application and interpretation of the described phase models are given. The prospects for further research and the possible development of uncertainty modeling in educational programs using phase-category tools are considered.
The higher education system in Ukraine is undergoing profound reforms related to the transition to a competency-based approach and the updating of state standards. The new standards no longer define program learning outcomes (PLOs), but only outline integral competencies and a list of general and professional competencies. The formulation of SLOs is entrusted directly to the developers of educational programs, which, on the one hand, increases the autonomy of educational institutions, but on the other hand, creates the challenge of ensuring transparent and reasonable correspondence between SLOs and the competencies defined by the standards.
In such conditions, the content of educational disciplines becomes crucial. Disciplines are carriers of knowledge and skills that, through the formation of SLOs, ensure the achievement of graduate competencies. Gaps or duplications in course topics or imbalances in their structure directly affect the quality of training, even if formally the program covers all the necessary PRN and competencies. In this regard, it is important to find formalization tools that can assess the quality of educational content not only at the level of an individual course, but also in the context of the entire educational program.
Traditional tools, such as knowledge graphs and ontological models, effectively model semantic relationships, but are limited in expressing compositional and complex transformations between elements of the educational process. Category theory, which emerged as a mathematical apparatus for describing abstract structures and relationships, is increasingly being used in computer science, logic, and system modeling. Its potential lies in its ability to formalize compositions, mappings between structure levels, and consistent transformations, making this approach promising for the educational sphere. The use of the categorical apparatus opens up opportunities for accurate modeling of dependencies between the topics of the educational component, program learning outcomes, and competencies, as well as for identifying critical nodes and gaps in the educational content.
The issues of formalization and evaluation of educational programs are actively researched in
education, computer science, and applied mathematics. Modern approaches use network models,
ontologies, and knowledge graphs, which allow structuring educational content and reflecting the
connections between its components [
The use of mathematical methods for modeling and structuring educational content is one of
the leading areas of contemporary research. Examples include the analysis of educational programs
through network models [
The following abbreviations are used in Figure 1: TAD is an academic discipline topic; CLO is a learning outcome; PLO is program learning outcome; Comp is competency.
In this context, ontological models and knowledge graphs, which have proven to be effective
tools for organizing educational content [
Despite this, in the context of a competency-based approach, ontologies and knowledge graphs reveal a number of limitations. They are mostly declarative and do not describe the procedural aspect of the formation of results and competencies, nor do they take into account the sequence or alternatives of educational trajectories. Graph structures do not provide a strict compositional logic, which makes it impossible to formally describe educational paths. The relationships between the different levels of the educational structure, shown in Figure 1, are established by experts and cannot be verified for consistency at. In addition, ontologies do not have built-in mechanisms for quantitative assessment of program coverage or balance.
These limitations necessitate the use of more formalized models capable of describing
compositional relationships and mappings between different levels of the educational process. Such
possibilities are provided by category theory, which provides a rigorous mathematical apparatus
for modeling structures and relations. Its fundamental principles are set out in the classic works of
Mac Lane [
The key constructs of category theory—categories, functors, and natural transformations—allow for the systematic modeling of multilevel educational structures. Subjects can be interpreted as objects, logical dependencies between them as morphisms, and reflections in learning outcomes and competencies as functors. Natural transformations, in turn, open up opportunities for comparing different teaching strategies or learning scenarios.
Of particular importance is the concept of Olog [
Thanks to this, category theory expands the possibilities of analysis: it allows not only to formally describe the structure of educational content, but also to quantitatively assess the coverage of competencies by topics, identify duplications or gaps, and explore the variability of educational trajectories. Thus, the categorical approach emerges as a promising direction for systematic assessment of the quality of training at the level of disciplines and educational programs.
We will apply category theory to describe educational programs in the form of a strict mathematical structure, where objects are interpreted as topics or results, and morphisms as logical dependencies between them. This approach allowed the authors to perform a formal check of the consistency and coverage of the programs. Despite its novelty, this model operates exclusively with binary relations: the presence or absence of a connection. It does not take into account the variability of influences inherent in the real educational process.
To account for the fuzziness in modeling educational systems, methods based on fuzzy set
theory are gaining increasing attention [
A promising direction for combining structural and fuzzy approaches is the use of fuzzy
categories. The idea was first proposed in works that generalize classical category theory by
introducing degrees of membership for morphisms [
Despite the existence of developments in the field of mathematics and computer science, there
have been no attempts to apply phase categories directly to the modeling of educational programs
[
Thus, analysis of the literature shows that: 1. Classical categorical models provide a rigorous apparatus for describing educational structures, but do not take into account uncertainties. 2. Fuzzy logic methods allow working with fuzzy estimates, but do not integrate into the structural description of programs. 3. Phase categories have the potential to combine both approaches, creating a generalized model for analyzing educational programs. 3.1.
Category. A category is a fundamental structure of category theory that generalizes the concepts
of sets and functions, focusing on the relationships between elements [
Definition 1. Category C consists of: 1. A set of objects Obj (C ) 2. A set of morphisms HomC ( A , B ) for each pair of objects A , B ∈ Obj (C ) 3. The composition operation of morphisms: ∘:HomC(B,C)×HomC(A,B)→HomC(A,C) 4. Identity morphisms idA ∈ HomC(A,A) для кожного A∈Obj©.
Associativity: for all f∈Hom(A,B), g∈Hom(B,C), h∈Hom(C,D): h∘(g∘f)=(h∘g)∘f.
Unity: for every f∈Hom(A,B): idB∘f=f=f∘idA.
In a pedagogical context, objects of a category are interpreted as topics or modules of a discipline, and morphisms as logical or educational dependencies between them. The composition of morphisms reflects the possibility of constructing educational trajectories through the sequential mastery of topics.
For example, let’s imagine a simple category for the discipline “Mathematics”: objects— “Arithmetic,” “Geometry,” “Algebra”; morphisms—“dependency: Arithmetic → Algebra” (because basic operations are necessary for equations). Composition allows us to create the trajectory “Arithmetic → Geometry → Algebra,” which models logical progress in learning. This approach helps to identify “critical nodes”—topics on which many others depend—and optimize the program to avoid gaps.
Classical category theory operates with objects and morphisms that either exist or do not exist. This ensures rigor, but in the case of educational programs, it proves insufficient: the influence of a subject on learning outcomes cannot always be described as binary. For example, one topic may almost completely shape a certain outcome, another may only partially shape it, and a third may shape it indirectly. 4.1.
To describe such situations, the apparatus of fuzzy sets [
The phase category Cf generalizes the classical category by introducing a degree of membership for each morphism. Thus, instead of the statement “there is a morphism from A to B,” we have “there is a morphism from A to B with degree μ. Example 1. Interpretation in education. 1.
Objects: subject topics, learning outcomes (CLO), program outcomes (PLO), competencies (Comp).
Morphisms: relationships “provides,” “contributes,” “corresponds,” with an attached degree μ showing the intensity of influence. 4.3.
Composition in the phase-category preserves the logic of the classical category, but takes into account weaker links. A typical method is the minimum rule: μ(f∘ g)=min(μ(f), μ(g)).
This means that the contribution of a topic to competence through intermediate results cannot exceed the weakest link in the corresponding chain.
Example 2. If the topic “Agile methodologies” with an intensity of μ=0.9 forms the CLO “uses Scrum/Kanban,” and this CLO with an intensity of μ=0.7 corresponds to PLO11, then the total contribution of the topic to PLO11 is 0.7. 4.4.
Phase category Cf generalizes the classical category by allowing morphisms to have degrees of
membership μ ∈ [
Assessing the quality of educational programs in modern higher education is a complex, multilevel process that involves verifying the compliance of programs with state standards, professional requirements, and stakeholder expectations. To formalize the relationships within education, we will apply category theory according to the scheme shown in Figure 1.
This model allowed us to present educational content in the form of a rigorous mathematical structure and enabled a quantitative analysis of its consistency.
However, this approach has limitations related to the binary nature of representations: each relationship between elements of the educational program was interpreted as either present or absent. In real life, relationships are rarely so clear-cut. The impact of a single subject on learning outcomes can be significant or only partial; the formulation of outcomes often contains elements of uncertainty; expert assessments of the alignment of program outcomes with competencies are subjective and vague. As a result, the classical categorical model does not fully reflect the complexity and variability of the educational process.
To overcome this limitation, the article proposes using fuzzy categories, which generalize
classical categories by introducing degrees of membership of relations μ∈ [
In such a model, each morphism is assigned not only the fact of existence, but also a numerical indicator of the strength or reliability of the connection. This allows formalizing the fuzziness inherent in both the description of learning outcomes and expert assessments of their achievement.
The purpose of this work is to develop a phase-categorical model for evaluating educational programs that: 1. Preserves the compositional and rigorous nature of the categorical approach. 2. Takes into account the vagueness and unevenness of the connections between educational elements. 3. Creates a basis for the development of software with support for analytics and visualization. 4.4.1.
Let Cf be a phase-category consisting of:
1. A set of objects Obj(Cf ), which in an educational context are interpreted as topics (Ci),
learning outcomes (CLOj), program outcomes (PLOm), or competencies (Compm).
2. The sets of phase morphisms HomCf(A,B)⊆ [
These elements must satisfy the axioms of the category: associativity of composition and unity of identity morphisms.
Lemma 1: Associativity of composition. For any morphisms f: A→B, g: B→C, h: C→D in the phase category Cf, the following holds
h∘ (g∘ f)=(h∘ g)∘ f, i.e.
μ(h∘ (g∘ f))=μ((h∘ g)∘ f).
Proof: Consider the composition of morphisms in a phase category. According to the composition rule:
μ(g∘ f)=min (μ(f),μ(g)).
Then for the left side of the axiom:
h∘ (g∘ f):A→D, μ(h∘ (g∘ f))=min (μ(h),μ(g∘ f))=min (μ(h), min (μ(f),μ(g))).
Since the min operation is associative on [
min (μ(h), min(μ(f), μ(g)))=min (min (μ(h), μ(g)), μ(f)).
For the right side:
(h∘ g)∘ f:A→D, μ((h∘ g)∘ f)=min (μ(h∘ g),μ(f))=min (min (μ(h), μ(g)), μ(f)).
Since both sides are equal in terms of associativity min, then:
μ(h∘ (g∘ f))=μ((h∘ g)∘ f).
Therefore, the associativity axiom is satisfied.
Example 3 in a pedagogical context: Let the objects in category CTopics be topics of the discipline “Software Product Development Technology” (C1: Software Life Cycle, C2: Development Methodologies, C3: Architectural Solutions), and the morphisms be: 1. f: C1→ C2, μ(f)=0.9 (the life cycle strongly influences methodologies). 2. g: C2→ C3, μ(g)=0.7 (methodologies partially determine architecture). 3. h: C3→ C4 , μ(h)=0.8 (architecture affects documentation). 4. Composition g∘ f:C1→ C3, μ(g∘ f)=min (0.9,0.7)=0.7. 5. h∘ (g∘ f):C1→ C4, μ(h∘ (g∘ f))=min (0.8, 0.7)=0.7.
6. (h∘ g)∘ f:C1→ C4, μ(h∘ g)=min (0.8, 0.7)=0.7, μ((h∘ g)∘ f)=min (0.7, 0.9)=0.7.
The equality μ(h∘ (g∘ f))=μ((h∘ g)∘ f)=0.7 confirms associativity, and the value μ=0.7 shows that the strength of the relationship is limited by the weakest link (methodology → architecture).
Lemma 2: Uniqueness of identical morphisms. For any morphism f: A→ B in phase category Cf, the following holds
idB∘ f=f, f∘ idA=f, that is
μ(idB∘ f)=μ(f), μ(f∘ idA )=μ(f).
Proof: By definition, μ(idA) = 1 for any A ∈ Obj(Cf). Consider the composition: μ(idB∘ f)=min (μ(idB), μ(f))=min (1,μ(f))=μ(f). μ(f∘ idA )=min (μ(f), μ(idA))=min (μ(f), 1)=μ(f). Since μ(idB∘ f)=μ(f) and μ(f∘ idA )=μ(f), the unity axiom is satisfied.
Example 4 in a pedagogical context: Let in category CCLO objects are learning outcomes (CLO1: Applies Scrum/Kanban, CLO2: Develops technical specifications), and the morphism f: CLO1→CLO2, μ(f)=0.6 (Scrum partially affects technical specifications). Then: idCLO2∘ f: CLO1→CLO2, μ(idCLO2∘ f)=min (1, 0.6)=0.6=μ(f); f∘ idCLO1 : CLO1 →CLO2, μ(f∘ idCLO1 )=min (0.6, 1)=0.6=μ(f).
This confirms that identical morphisms act neutrally, preserving the degree of connection. 4.4.2.
μ(f)=1 corresponds to the presence of a connection (classical morphism) μ(f)=0 corresponds to its absence.
The composition μ(g∘f)=min (μ(f), μ(g)) reduces to the classical one: if μ(f)=1, μ(g)=1, then μ(g∘f)=1; if at least one μ=0, then μ(g∘f)=0. Identical morphisms with μ(idA )=1 correspond to classical idA. Thus, the phase category is a generalization of the classical one, preserving its axioms.
The proposed model aims to formalize the relationships between subject topics, learning outcomes, program outcomes, and competencies, taking into account the vagueness and unevenness of their influences. It generalizes the previous categorical model by using phase categories, which allows describing not only the structure but also the strength of the relationships. 5.1.
CTopics—category of discipline topics.
CCLO—ategory of course learning outcomes.
CPLO—category of program learning outcomes.
CComp —category of competencies defined by higher education standards.
The objects in each category correspond to specific elements of the curriculum, and morphisms describe their internal dependencies (for example, logical transitions between topics or interrelationships between competencies). 5.2.
To establish correspondences between the levels of the model (described in subsection 4.1: CTopics,
CCLO, CPLO, CComp), phase functors are introduced, which generalize classical functors by taking into
account the degrees of membership μ ∈ [
Definition of a phase functor. Let Cfand Dfbe two phase categories. The phase functor Ff: Cf→ Df is defined as: 1. 2.
Ff(f): Ff(A)→Ff(B) with μ(Ff (f))∈[
Composition preservation: for f: A→B, g: B→C, μ(Ff(g∘f))=min(μ(Ff(f)), μ(Ff(g))) =min(μ(f),μ(g)). 4. Preservation of identity morphisms: μ ( F f (id A))= μ (id F f ( A))= 1.
1. F0: CTopics→ CCLO, which reflects the topics of disciplines in the learning outcomes of disciplines (CLO). 2. F1: CCLO→ CPLO, linking learning outcomes to program outcomes (PLO). 3. F2 : CPLO→ CComp, which reflects program outcomes in competence (Comp).
Each mapping Fi is assigned a degree of membership μ(Fi(f)) ∈ [
Phase functors preserve category axioms, as shown in Section 3.4 for composition of morphisms. Here we prove that the phase functor Ff preserves associativity and unity under transfer.
Lemma 3: Preservation of associativity. Let f: A→ B, g: B→ C, h: C→ D in Cf. Then Ff(h∘ (g∘ f)) = Ff((h∘ g)∘ f).
Proof:
On the left side: μ(Ff(h∘ (g∘ f)))=μ(h∘ (g∘ f))=min (μ(h), min(μ(g), μ(f)))=min(μ(h), μ(g), μ(f)). From the right: similarly, min(min(μ(h), μ(g)), μ(f))=min(μ(h), μ(g), μ(f)). The equality holds by the associativity of min.
Lemma 4: Preservation of unity. Let f: A→ B.
Then Ff(idB∘ f)=Ff(f) and Ff(f∘ idA)=Ff(f).
Proof: μ(Ff(idB∘ f)) = min (1,μ(f)) = μ(f) = μ(Ff(f)). Similarly, for f∘ idA. ■
These proofs confirm that phase functors are a correct generalization of classical ones, preserving mathematical consistency.
Example 5 of application in an educational context. Consider the phase functor F0: CTopics→ CCLO for the discipline “Software Product Development Technology.” Objects: topics C1 (Software Life Cycle), C2 (Development Methodologies); CLO: CLO1 (Understanding the Life Cycle), CLO2(Applying Agile). Morphism f: C1→C2, μ(f)=0.8 (the life cycle partially determines the methodologies). Then F0(f): CLO1→CLO2, μ(F0(f))=0.8, which reflects a fuzzy influence. 5.4.
To visualize phase-functor mappings, a diagram is provided (Figure 2), where edges with weights μ illustrate the strength of connections between model levels. The diagram shows the complete chain from topic C2 to competency SC10 via CLO and PLO, with composition μ(F)=min(μ(F0), μ(F1), μ(F2)).
Figure 2 illustrates phase-functor mappings between model levels. Each functor Fi is labeled with a weight μ indicating the strength of the link (e.g., μ(F0) = 0.9 for a strong influence of the topic on CLO). The composition F(dotted arrow) shows the overall path from topic C2 to competency SC10, where μ(F)=0.7 is limited by the weakest link (F1. This allows for visual analysis of imbalances; for example, if μ(F1)<0.5, then the entire chain is considered weak, signaling a gap in the program.
Phase functors provide flexibility to the model, allowing for the integration of expert assessments or automated calculations of μ (e.g., based on NLP analysis of syllabi). 5.5.
Determining the weight of edges is a separate important and complex task. Its solution requires the use of different approaches or the development and justification of new methods. Various approaches can be proposed to determine the weight of edges:
2. A posteriori 3. Test 4. Axiomatic 5. Combined—as a justified combination of the four approaches mentioned above.
We will briefly describe the content and directions of research related to the above approaches to determining the numerical values of weight coefficients of mutual influences between functor mappings, schematically shown in Figure 2.
The expert approach is the most acceptable today, since this area of research is insufficiently
developed and the best tool in such cases is expert judgment and adequately developed expert
technologies [
The a posteriori approach consists in a posteriori determination of the level of influence on
Comp competencies through CLO and PLO based on the known results of applying the
phasecategorical model of the educational process proposed by the authors to specific educational
programs implemented in specific higher education institutions [
The test approach can be applied to already implemented and operating educational programs by
testing students who have mastered the relevant programs and formally received documentary
confirmation of this [
Note: Testing of students can be carried out both during their studies and for graduates of
higher education institutions [
The axiomatic approach is based on the a priori use of expert judgments and consists in the
preliminary determination of the level of influence depending on the “weight” of the components,
i.e., the number of results, disciplines related to the educational program in force at the higher
education institution [
The combined approach is a reasonable combination of the four approaches mentioned above, or the use of only some of these four. 5.6.
Interpretation of phase reflections 1. If μ is close to 1, the relationship almost completely determines the correspondence between the elements (for example, the topic directly shapes a specific learning outcome). 2. If μ has an intermediate value, the relationship is partial or depends on additional conditions (e.g., the topic contributes to the outcome only in combination with others). 3. If μ is low (e.g., μ < 0.3), the influence is weak or indirect. 5.7.
Thanks to the compositional nature of the categories, it is possible to trace the contribution of topics to the final competencies through all intermediate levels. The corresponding degree of belonging is calculated as a minimum across all links:
μtopic→Comp= min(μtopic→CLO, μCLO→PLO, μPLO→Comp).
This reflects the principle that “the strength of a chain is determined by its weakest link.” Thus, even if the connection between the topic and the learning outcome is strong, but the subsequent reflection in competence is weak, the overall contribution of the topic to competence will be limited. 5.8.
The result of building the model is a set of “weight maps” between all levels of the educational process. They can be represented as adjacency matrices with weights μ or as graphical diagrams where the thickness of the edges is proportional to the strength of the connection. Such maps provide a transparent visualization of the coverage of competencies and allow you to identify:
To demonstrate how the phase-categorical model works, let us consider the program learning outcome PLO11 of the educational program “Computer Science.” In the classical category model, this outcome is linked to a number of subject areas through learning outcomes (CLO). However, the links are interpreted as binary—either present or absent. The use of phase categories allows each link to be given a numerical assessment of its strength of influence. 6.1.
The model uses three subject areas that are part of different educational components (EC):
2. EC28 “Software Documentation.” 3. EC31 “Requirements Engineering.”
Each topic is reflected in the corresponding learning outcomes (CLO), which in turn are linked to PLO11. For each reflection, experts determined the degree of relevance μ, which reflects the strength of influence. 6.2.
Topic “Agile methodologies” → CLO “applies Scrum/Kanban” (μ=0.9).
Topic “Software Documentation” → CLO “develops technical specifications” (μ=0.6). Topic “Requirements engineering” → CLO “specifies requirements” (μ=0.8). CLO “uses Scrum/Kanban” → PLO11 (μ=0.7).
CLO “develops technical specifications” → PLO11 (μ=0.5).
CLO “specifies requirements” → PLO (μ=0.9). 6.3.
The total contribution of the topic to PLO11 is determined as the minimum of the values μ on the path “topic → CLO → PLO11”. Results obtained:
Agile methodologies: min (0.9, 0.7)=0.7.
Software documentation: min (0.6, 0.5)=0.5.
Requirements engineering: min (0.8, 0.9)=0.8.
Thus, the topic “Requirements engineering” has the greatest contribution to the formation of PLO11, while “Software documentation” plays a supporting role. 6.4.
Next, we will consider the reflection of PLO11 in the competence of the educational program. For example, PLO11 can form: 1. SC10 “Ability to apply software engineering methods” (μ=0.8). 2. SC15 “Ability to engage in project activities” (μ=0.6).
Then the contribution of topics to competence is calculated using the same logic: 1. 2. 3. 4. 5. 6.
Agile methodologies → SC10: min (0.7, 0.8)=0.7.
Agile methodologies → SC15: min (0.7, 0.6)=0.6.
Software documentation → SC10: min (0.5, 0.8)=0.5.
Software documentation → SC15: min (0.5, 0.6)=0.5.
Requirements engineering → SC10: min (0.8, 0.8)=0.8.
Requirements engineering → SC15: min (0.8, 0.6)=0.6. 6.5.
To illustrate the results, a “phase map” was constructed in the form of a weighted graph, where the nodes correspond to topics (Ci), learning outcomes (CLOj), program outcomes (PLO11), and competencies (SC10, SC15), and the thickness of the edges is proportional to μ (Figure 3). An alternative representation is an adjacency matrix, where the cells reflect the strength of the connection.
Testing the phase-categorical model using the example of PLO11 allowed us to draw a number of important analytical conclusions that cannot be identified in the classical binary model. 7.1.
The calculated membership values showed that the topic “Requirements Engineering” has the greatest contribution to the formation of PLO11 and related competencies (μ=0.8). This information is valuable for curriculum design, as it allows us to identify the disciplines and topics that play a key role in achieving the target results.
The topic “Software Documentation” showed a weak influence on PLO11 (μ=0.5), which may indicate the auxiliary nature of this content block. In the classical model, this connection would be marked as “present,” but the phase approach allows us to record its low intensity and classify it as secondary. 7.3.
Comparing the weight contributions between topics allows us to assess the balance of the program. It was found that PLO11 is formed mainly due to the topic “Requirements Engineering,” while other topics have a less pronounced influence. This indicates an imbalance, which can be both positive (focus on key competencies) and negative (risk of underestimating auxiliary skills).
Table 1 shows the adjacency matrix for the discipline “Software Product Development Technology,” which quantitatively describes the relationships between topics, CLOs, PLOs and competencies. For example, the value μ=0.8 for C4→SC10 indicates a strong contribution of the topic “Requirements Engineering” to the competency “Software Engineering Methods,” while μ=0.5 for C3→SC10 reflects a weaker connection for “Software Documentation.” In both cases, the model allows for a clear identification of strong and weak contributions, which is the basis for analyzing balance and identifying gaps in the program (see subsection 6). 7.4.
The aggregated values of μ for competencies showed that some of them have low total coverage (<0.5). This indicates potential gaps in the program that need to be addressed by introducing additional topics or revising existing learning outcomes. For example, in the case of SC15 (“Ability to engage in project activities”), the value μ=0.5–0.6 indicates insufficient development of this competency. C2: Agile methodologies
specifications
methods SC15: Project activities 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
In the classical categorical-binary model, all three topics are equally considered to “ensure” PLO11. In contrast, the phase model showed a qualitatively different picture:
2. it showed weak and secondary connections. 3. it made it possible to quantitatively assess the balance and identify gaps.
This confirms that the phase-categorical model is a significant improvement on the classical approach and allows for more accurate and relevant analytics to ensure the quality of educational programs.
The proposed phase-categorical model creates new opportunities for the analytics of educational
programs, combining the rigor of the categorical approach with the flexibility of fuzzy set theory
[
However, the use of phase categories is associated with a number of challenges. The key issue is
the subjectivity of expert assessments when assigning weight coefficients, as different experts may
interpret the significance of topics or results differently. Also, the lack of uniform methodological
recommendations for scaling within the interval [
An important area of application is the support of adaptive learning. If weight coefficients are
updated based on individual student performance, it becomes possible to form personal educational
trajectories aimed at strengthening poorly developed competencies. This opens up the prospect of
using phase categories not only in the context of auditing educational programs, but also in the
design of adaptive learning technologies [
The article presents a generalization of the categorical model of educational programs through the application of phase categories. The proposed approach allows for the consideration of the vagueness and varying intensity of the connections between topics, learning outcomes, program outcomes, and competencies.
The study showed that: 1. Phase categories provide a formalized apparatus for describing uncertainty in representations of the educational process, while preserving the compositional and rigorous nature of classical categorical logic. 2. Using the example of the PLO11 program outcome, it was demonstrated that the model allows identifying critical topics with the greatest contribution to the formation of competencies, revealing weak and secondary connections, and performing gap analysis. 3. A comparison with the binary categorical model confirmed that the phase approach provides a much more accurate and flexible representation of the educational program. 4. The developed model creates a basis for practical implementation in the form of software that can support the audit of educational programs, visualization of competency coverage, and the construction of adaptive educational trajectories.
Further research directions include: 1. Developing algorithms for automated determination of weighting coefficients based on student performance statistics and NLP analysis of syllabi. 2. Integration of the model into learning management systems (LMS) and accreditation platforms. 3. Conducting comparative studies of different educational programs and international standards to assess the universality of the approach.
Declaration on Generative AI While preparing this work, the authors used the AI programs Grammarly Pro to correct text grammar and Strike Plagiarism to search for possible plagiarism. After using this tool, the authors reviewed and edited the content as needed and took full responsibility for the publication’s content.