Knowledge structures delineated by fuzzy skill maps

doi:10.1016/j.fss.2020.10.004

Fuzzy Sets and Systems

Volume 407, 1 March 2021, Pages 50-66

https://doi.org/10.1016/j.fss.2020.10.004 Get rights and content

Abstract

Skills represent the underlying cognitive abilities, which consist in methods, algorithms or tricks etc. Skills can be used to interpret individuals' knowledge states. A skill map assigns a subset of skills to each item q. The disjunctive model assumes that at least one of the assigned skills is acquired for the mastery of the item q, and the conjunctive model states that all of the assigned skills are necessary for the mastery of the item q. The latent cognitive abilities of different individuals are distinct. Different levels of proficiency in skills may be required to solve different items. It's natural to use a fuzzy skill map or a fuzzy skill multimap to represent the levels of proficiency in skills for solving items. A fuzzy skill map $(Q, S, τ)$ assigns a fuzzy set of skills to each item q, which represents the level of proficiency in skills needed to solve the item. A fuzzy skill multimap $(Q, S, μ)$ assigns a collection of fuzzy sets of skills to each item q, any fuzzy set C of skills in $μ (q)$ can be viewed as an approach to solve the item q. A knowledge space and a simple closure space can be delineated by a fuzzy skill map via the disjunctive model and the conjunctive model respectively. A knowledge structure can be delineated via the competency model by a fuzzy skill multimap. Algorithms for delineating knowledge structures via the disjunctive model and the conjunctive model by a fuzzy skill map are only concerned with skills in S. Delineating the knowledge structure by a fuzzy skill multimap just depends on the competencies and the weak competencies. For any fuzzy skill map $(Q, S, τ)$ or fuzzy skill multimap $(Q, S, μ)$ , the knowledge states delineated by some skills can be replaced by knowledge states delineated by some other skills. Having removed these skills, the relationship between items and skills can be described more concisely without changing the knowledge structure. A minimal set of skills can faithfully summarize the relationship between items and skills, which results in a substantial economy of storage in a computer memory and a high efficiency of building, testing and searching for knowledge structures.

Section snippets

Introduction and preliminaries

Knowledge space theory (KST) establishes a valuable and accurate framework of mathematical psychology for knowledge assessment and further learning [5], [6], [7], [11]. The knowledge state of an individual is represented by a finite subset of items that he is capable of answering correctly in ideal conditions [5], [7]. KST was developed as a theory for predicting observable responses to a given collection of problems, thereby operating on the performance level [5], [12], [17], [32], [33].

The

An overview of KST and fuzzy set theory

We envisage a field of knowledge that can be parsed into a set of items Q, each of which has a correct response. The knowledge state of an individual is represented by the subset of items in the domain that he is capable of answering correctly in ideal conditions. This means that he is not working under time pressure or impaired by emotional turmoil of any kind. The collection of all knowledge states will be referred to as knowledge structure when it includes ∅ and Q.

The knowledge structure $(Q, K$

The disjunctive model of fuzzy skill maps

The approach for delineating knowledge structure via a skill map or a skill multimap is different from the method of query-routine [3], [12], [14], [19], [21], [24]. Falmagne et al. sketched a first approach that links the observed solution behavior to some underlying cognitive constructs by assigning to each item a subset of skills that are relevant for mastering it [12]. Skill map was introduced into KST by Doignon in 1994 to describe knowledge structure from the perspective of latent ability

The conjunctive model of fuzzy skill maps

Definition 4

Let $(Q, S, τ)$ be a fuzzy skill map and $T \in F (S)$ . The knowledge state K delineated by T via the conjunctive model is specified by $K = {q \in Q | τ_{q} \subseteq T}$ .

Remark 2

For the conjunctive model, solving an item is required to reach level of proficiency in $τ_{q}$ , that is $τ_{q} (s) \leq T (s)$ for any $s \in S$ . Note that $T = \emptyset$ delineates the state ∅, since $τ_{q} ⊈ T$ for any $q \in Q$ , and $T = {\frac{1}{s_{1}}, \frac{1}{s_{2}}, \dots, \frac{1}{s_{m}}}$ delineates Q since $τ_{q} \subseteq {\frac{1}{s_{1}}, \frac{1}{s_{2}}, \dots, \frac{1}{s_{m}}}$ for any $q \in Q$ . Thus, the family of knowledge states delineated via the conjunctive model by fuzzy skill map $(Q, S, τ)$

The competency model of fuzzy skill multimaps

For any item $q \in Q$ , we assign a collection $μ (q)$ of fuzzy sets of skills to q. Any fuzzy set C of skills in $μ (q)$ can be viewed as an approach to solve the item q. Different fuzzy sets in $μ (q)$ represent different solution paths to solve the item [7], [15].

Definition 6

A fuzzy skill multimap is a triple $(Q, S, μ)$ , where Q is a nonempty finite set of items, S is a nonempty finite set of skills, and μ is a mapping from Q to $2^{F (S)} ∖ {\emptyset}$ such that each $μ (q), q \in Q$ , is a nonempty family of $F (S)$ .

Remark 3

Each fuzzy set C of skills

Minimal sets of skills

For any fuzzy skill map or fuzzy skill multimap, the knowledge states delineated by some skills may be replaced by the knowledge states delineated by some other skills, that is, some skills in S are redundant. Having removed these skills, the relationship between items and skills can be described more concisely without changing the knowledge structure. A large number of such superfluous skills will lower the efficiency of building, testing and searching for knowledge structure and waste the

Conclusions

A fuzzy skill map or a fuzzy skill multimap may be conceived as generalizations of a skill map or a skill multimap. The knowledge state of an individual can be delineated by a fuzzy set in $F (S)$ via a fuzzy skill map $(Q, S, τ)$ or a fuzzy skill multimap $(Q, S, μ)$ . A fuzzy skill map $(Q, S, τ)$ assigns a fuzzy set of skills to each item q. Here, Theorem 2, Theorem 4 prove that a knowledge space and a simple closure space can be delineated via the disjunctive model and the conjunctive model by a fuzzy

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful suggestions and comments. This work is supported by the National Natural Science Foundation of China (No. 11871259, 11971287, 11701258, 61379021) and the Natural Science Foundation of Fujian Province (No: 2019J01748, 2020J02043).

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Forward-graded and backward-graded structures of knowledge are two important classes of knowledge structures. Spoto and Stefanutti (2020) establish necessary and sufficient conditions for skill maps to delineate these structures. We introduce fuzzy skills to describe varying levels of proficiency in skills and extend the theoretical results of Spoto and Stefanutti (2020) for delineating forward- and backward-graded knowledge structures using fuzzy skill maps. The paper establishes necessary and sufficient conditions for fuzzy skill maps to delineate a backward-graded simple closure space, a forward-graded knowledge space, and a forward-graded simple closure space. Furthermore, the competence-based local independence model (CBLIM) with fuzzy skills is introduced and its unidentifiability is discussed.
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Citation Excerpt :
It's natural to construct polytomous knowledge structures from the perspective of latent cognitive abilities. Based on the works of Stefanutti et al. [6], Heller [4] and Sun et al. [10], we propose the concept of fuzzy skill function on polytomous items. The relationship between polytomous items and fuzzy skills can be established by means of fuzzy skill functions on polytomous items.
The dichotomous knowledge structure theory was introduced to polytomous items, which is proved meaningful in representing individuals' partial mastery of items. Given that different proficiencies in skills may be required to solve polytomous items, fuzzy skills can be applied to represent latent cognitive abilities of individuals. Then we consider how to construct polytomous knowledge structures by fuzzy skills. To characterize relationships between polytomous items and fuzzy skills related to solving these items, we introduce the concept of fuzzy skill function on polytomous items. Given a set $Q \times L$ of polytomous items and a set S of skills, a fuzzy skill function $(Q \times L, S, σ)$ assigns each response category $(q, l) \in Q \times L$ with a family $σ_{(q, l)}$ of fuzzy sets on S, where each element in $σ_{(q, l)}$ represents an approach for solving q to the level l. Notice that there are two special kinds of fuzzy skill functions: one is disjunctive, while the other is conjunctive. The former delineates polytomous knowledge spaces, while the latter delineates polytomous closure spaces. Based on these facts, we design algorithms for delineating the two kinds of polytomous knowledge structures. For any fuzzy skill function on polytomous items, all competencies and all weak competencies are enough to delineate a polytomous knowledge structure. Thus, we design an algorithm for delineating polytomous knowledge structures via fuzzy skill functions on polytomous items. This algorithm only depends on competencies and weak competencies of fuzzy skill functions on polytomous items. Besides, we take “Addition and subtraction of fractions” as knowledge domain to illustrate the process of constructing polytomous knowledge structures from fuzzy skills.
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Heller (2021) and Stefanutti et al. (2020) provided the mathematical foundation for the generalization of knowledge structure theory (KST) to polytomous items. Based on their works, the well-gradedness can be extended to polytomous knowledge structures. We propose the concepts of discriminative polytomous knowledge structure and well-graded polytomous knowledge structure. Then we show that every well-graded polytomous knowledge structure is discriminative. The basis of any polytomous knowledge space is formed by the collection of all the atoms. We discuss the sufficient and necessary conditions of polytomous knowledge structures to be well-graded polytomous knowledge spaces. Moreover, we provide an example to illustrate that a well-graded polytomous knowledge space is not necessarily a polytomous closure space.
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Concept-cognitive learning has been widely used in simulating the human brain to learn concepts. However, the existing concept-cognitive learning models mainly focused on how to acquire knowledge and its properties, but not how to solve the problems. The underlying skills for solving problems are ignored in the cognitive learning process. Indeed, a concept-cognitive learning process is always accompanied with problem solving and skill learning, and the skills are necessary for solving problems. Knowledge space theory is an effective mathematical analysis approach for knowledge assessment. Nevertheless, the existing learning paths for skill were evaluated by constructing the concept lattice, which is a NP hard problem. To overcome these limitations and problems, a novel concept-cognitive learning model from a perspective of competences is proposed. Firstly, knowledge and skills can be represented by item sets and skill sets. And a good semantic explanation between knowledge and skills is provided by a competence-based concept, which presents that one can acquire the most knowledge with the least amount of skills. Secondly, a competence-based concept-cognitive learning model and its properties are put forward to describe the concept-cognitive learning through skills. Moreover, by analyzing the sufficient and necessary relationship between skills and knowledge, a competence-based information granule structure is constructed. Finally, a transformation method of information granules is proposed to convert a general information granule into sufficient and necessary competence-based information granules (i.e., competence-based concepts). And the experimental results of UCI data sets show that the competence-based concept-cognitive learning model is feasible and effective.

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^☆: This work is supported by National Natural Science Foundation of China (No. 11871259, 11971287, 11701258, 61379021), by Natural Science Foundation of Fujian Province (No. 2019J01748, 2020J02043).

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Knowledge structures delineated by fuzzy skill maps☆

Abstract

Section snippets

Introduction and preliminaries

An overview of KST and fuzzy set theory

The disjunctive model of fuzzy skill maps

The conjunctive model of fuzzy skill maps

The competency model of fuzzy skill multimaps

Minimal sets of skills

Conclusions

Declaration of Competing Interest

Acknowledgements

Inf. Sci.

J. Math. Psychol.

Int. J. Man-Mach. Stud.

J. Math. Psychol.

J. Math. Psychol.

J. Math. Psychol.

J. Math. Psychol.

Int. J. Hum.-Comput. Stud.

J. Math. Psychol.

J. Math. Psychol.

J. Math. Psychol.

Inf. Sci.

J. Math. Psychol.

J. Math. Psychol.

J. Math. Psychol.

Inf. Control

The assessment of knowledge and learning in competence spaces: the gain-loss model for dependent skills

Br. J. Math. Stat. Psychol.