Knowledge and learning assessment is a popular topic. In existing models for constructing the knowledge structure of an individual, it is often considered whether an individual has mastered the skills to solve the corresponding item. However, the relationship between the number of skills an individual has mastered and the item is ignored. It is not reasonable to explain the phenomenon that individuals solve the same item but have different knowledge structures behind it. This paper introduces the concept of skill inclusion degree and constructs the variable precision α-models to delineate knowledge structures. The skill inclusion degree takes into account an individual’s mastery of the number of skills assigned to each item. Firstly, the concept of the skill inclusion degree is given, and some of its properties are discussed. Then, the variable precision α-model is constructed. Moreover, the relationship between knowledge structures delineated via the variable precision α-models by a skill map is studied, and the algorithm of knowledge structures delineated via these models by a skill map is designed. Finally, the experimental results on a real dataset demonstrate the feasibility and effectiveness of the proposed algorithm.
AlbertD. and HockemeyerC., Adaptive and dynamic hypertext tutoring systems based on knowledge space theory, Artificial Intelligence in Education: Knowledge and Media in Learning Systems39 (1997), 553–555.
2.
AnselmiP., StefanuttiL., de ChiusoleD. and RobustoE., The assessment of knowledge and learning in competence spaces: the gain-loss model for dependent skills, British Journal of Mathematical and Statistical Psychology70(3) (2017), 457–479.
3.
BottesiG., SpotoA., FreestonM.H., SanavioE. and VidottoG., Beyond the score: clinical evaluation through formal psychological assessment, Journal of Personality Assessment97(3) (2015), 252–260.
4.
CousoI., BustinceH., FernándezJ. and SánchezL., The null space of fuzzy inclusion measures, IEEE Transactions on Fuzzy Systems29(3) (2019), 641–648.
5.
CraigS.D., HuX., GraesserA.C., BargagliottiA.E., SterbinskyA., CheneyK.R. and OkwumabuaT., The impact of a technology-based mathematics after-school program using ALEKS on student’s knowledge and behaviors, Computers and Education68 (2013), 495–504.
6.
CosynE., UzunH., DobleC. and MatayoshiJ., A practical perspective on knowledge space theory: ALEKS and its data, Journal of Mathematical Psychology101 (2021), 102512.
7.
DoignonJ.P., Knowledge spaces and skill assignments, in: G. Fischer, D. Laming, (eds) Contributions to Mathematical Psychology, Psychometrics and Methodology. Springer, New York, 1994, pp. 111–121.
8.
De La TorreJ. and DouglasJ.A., Model evaluation and multiple strategies in cognitive diagnosis: An analysis of fraction subtraction data, Psychometrika73(4) (2008), 595.
9.
DoignonJ.P. and FalmagneJ.C., Spaces for the assessment of knowledge, International Journal of Manmachine Studies23(2) (1985), 175–196.
10.
DüntschI. and GedigaG., Skills and knowledge structures, British Journal of Mathematical and Statistical Psychology48(1) (1995), 9–27.
11.
DobleC., MatayoshiJ., CosynE., UzunH. and KaramiA., A databased simulation study of reliability for an adaptive assessment based on knowledge space theory, International Journal of Artificial Intelligence in Education29(2) (2019), 258–282.
12.
DaveyB.A. and PriestleyH.A., Introduction to lattices and order, Cambridge University Press (2002).
13.
de ChiusoleD., StefanuttiL., AnselmiP. and RobustoE., Modeling missing data in knowledge space theory, Psychological Methods20(4) (2015), 506.
14.
de ChiusoleD., StefanuttiL. and SpotoA., A class of k-modes algorithms for extracting knowledge structures from data, Behavior Research Methods49(4) (2017), 1212–1226.
15.
EppsteinD., FalmagneJ.C. and UzunH., On verifying and engineering the wellgradedness of a union–closed family, Journal of Mathematical Psychology53(1) (2009), 34–39.
16.
FalmagneJ.C., AlbertD., DobleC., EppsteinD. and HuX., Knowledge spaces: applications in education, New York: Springer–Verlag (2013).
FischerG.H. and FormannA.K., Some applications of logistic latent trait models with linear constraints on the parameters, Applied Psychological Measurement6(4) (1982), 397–416.
19.
GedigaG. and DüntschI., Skill set analysis in knowledge structures, British Journal of Mathematical and Statistical Psychology55(2) (2002), 361–384.
20.
HellerJ., A formal framework for characterizing querying algorithms, Journal of Mathematical Psychology48(1) (2004), 1–8.
21.
HellerJ., StefanuttiL., AnselmiP. and RobustoE., Erratum to: On the link between cognitive diagnostic models and knowledge space theory, Psychometrika81(1) (2016), 250–251.
22.
HellerJ., ÜnlüA. and AlbertD., Skills, competencies and knowledge structures, in: J. C. Falmagne, D. Albert, C. Doble, D. Eppstein, X. Hu, (eds) Knowledge Spaces: Applications in Education, New York: Springer-Verlag Berlin Heidelberg, 2013, pp. 229–242.
23.
KoppenM., Extracting human expertise for constructing knowledge spaces: An algorithm, Journal of Mathematical Psychology37(1) (1993), 1–20.
24.
Kickmeier-RustM.D. and AlbertD., 8 Competence-based knowledge space theory: options for the 21st century classroom, in: Measuring and Visualizing Learning in the Information-Rich Classroom. Routledge, 2015, pp. 125–136.
25.
KoppenM. and DoignonJ.P., How to build a knowledge space by querying an expert, Journal of Mathematical Psychology34(3) (1990), 311–331.
26.
LeyT., KumpB. and AlbertD., A methodology for eliciting, modelling, and evaluating expert knowledge for an adaptive workintegrated learning system, International Journal of Human–Computer Studies68(4) (2010), 185–208.
27.
LimL., LimS.H. and LimR.W.Y., Measuring learner satisfaction of an adaptive learning system, Behavioral Sciences12(8) (2022), 264.
28.
Líćzaro-CantabranaJ., Usart-RodríłguezM. and Gisbert-CerveraM., Assessing teacher digital competence: Theconstruction of an instrument for measuring the knowledge ofpreservice teachers, Journal of New Approaches in EducationalResearch (NAER Journal)8(1) (2019), 73–78.
29.
MinnS., AI–assisted knowledge assessment techniques for adaptive learning environments, Computers and Education: Artificial Intelligence (2022), 100050.
30.
MatsudaN., FurukawaT., BierN. and FaloutsosC., Machine beats experts: automatic discovery of skill models for data-driven online course refinement, International Educational Data Mining Society (2015).
31.
MahapatraT., GhoraiG. and PalM., Competition graphs under interval-valued m-polar fuzzy environment and its application, Computational and Applied Mathematics41(6) (2022), 285.
32.
MahapatraT. and PalM., An investigation on m-polar fuzzy threshold graph and its application on resource power controlling system, Journal of Ambient Intelligence and Humanized Computing (2022), 1–14.
33.
MahapatraR., SamantaS., PalM. and XinQ., RSM index: a new way of link prediction in social networks, Journal of Intelligent and Fuzzy Systems37(2) (2019), 2137–2151.
34.
PusungS., The effect of learning model and task assessment regarding student learning outcomes of science by controlling student prior knowledge, Research, Society and Development8(9) (2019), e29891290–e29891290.
35.
SpotoA., BottesiG., SanavioE. and VidottoG., Theoretical foundations and clinical implications of formal psychological assessment, Psychotherapy and Psychosomatics82(3) (2013), 197–199.
36.
StefanuttiL. and de ChiusoleD., On the assessment of learning in competence based knowledge space theory, Journal of Mathematical Psychology80 (2017), 22–32.
37.
SitthisakO., GilbertL. and AlbertD., Adaptive learning using an integration of competence model with knowledge space theory, in: 2013 Second IIAI International Conference on Advanced Applied Informatics. IEEE, 2013, August, pp. 199–202.
38.
SunW., LiJ.J., GeX. and LinY.D., Knowledge structures delineated by fuzzy skill maps, Fuzzy Sets and Systems407 (2021), 50–66.
39.
ShiY., MiY.L., LiJ.H. and LiuW.Q., Concurrent concept-cognitive learning model for classification, Information Sciences496 (2019), 65–81.
40.
SpotoA., StefanuttiL. and VidottoG., Knowledge space theory, formal concept analysis, and computerized psychological assessment, Behavior Research Methods42(1) (2010), 342–350.
41.
SpotoA., StefanuttiL. and VidottoG., An iterative procedure for extracting skill maps from data, Behavior Research Methods48(2) (2016), 729–741.
42.
SpotoA., VidottoG. and StefanuttiL., Skill map based knowledge structures: some considerations about their identifiability, Electronic Notes in Discrete Mathematics42 (2013), 73–80.
43.
WuR. and LinY.D., SOLO taxonomy-based knowledge structure with subjective items, Journal of Intelligent and Fuzzy Systems (Preprint)1–14.
44.
YangT.L., LiJ.J., LiZ.W., JinM., ZhouY.F. and LinY.D., Two kinds of variable precision models based on skill for constructing knowledge structures and skill subset reduction, Pattern Recognition and Artificial Intelligence35(8) (2022), 671–687.
45.
ZhangY. and FunkM., Learning path, in: Coding Art. Apress, Berkeley, CA, 2021, pp. 245–251.
46.
ZhiH.L. and LiJ.H., Granule description of incomplete data: a cognitive viewpoint, Cognitive Computation (2021), 1–12.
47.
ZhouY.F. and LiJ.J., Skill reduction and assessment in formal context, Journal of Frontiers of Computer Science and Technology16(3) (2022), 692–702.
48.
ZhouY.F., LiJ.J, FengD.L. and YangT.L., Learning paths and skills assessment in formal context, Pattern Recognition and Artificial Intelligence34(12) (2021), 1069–1084.
49.
ZhangT., LiH.H., LiuM.Q. and RongM., Incremental concept-cognitive learning based on attribute topology, International Journal of Approximate Reasoning118 (2020), 173–189.
50.
ZhouY.F., LiJ.J., WangH.K. and SunW., Skills and fuzzy knowledge structures, Journal of Intelligent and Fuzzy Systems (Preprint) (2022), 1–17.
