Hierarchical Bayesian Knowledge Tracing in Undergraduate Engineering Education

Yiwei Sun

REVIEW 5 major objections 7 minor 10 references

Hierarchical Bayesian Knowledge Tracing in Undergraduate Engineering Education

T0 review · 5 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A hierarchical Bayesian logistic model fitted to 446,844 Statics answers produces stable, interpretable rankings of skill difficulty and student ability that a non-hierarchical baseline cannot match.

desk verdict A cleanly written but fundamentally flawed paper: the Statics difficulty rankings are built on mislabeled skills, and the model is just regularized logistic regression with in-sample evaluation. read the letter →

arxiv 2506.00057 v1 pith:FLRWHV47 submitted 2025-05-29 cs.CY cs.LGstat.APstat.ML

Yiwei Sun This is my paper

classification cs.CYcs.LGstat.APstat.ML

keywords hierarchicalBayesianmodelingknowledgetracingitemresponsetheorystudentabilityestimationskilldifficultyStatics2011datasetlearninganalyticsmodelcalibration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that a hierarchical Bayesian version of a two-parameter logistic (item-response-style) model can turn raw answer logs from a large engineering course into interpretable, stable measures of skill difficulty and student ability. Applied to the Statics2011 dataset (446,844 responses from 4,151 students on 110 skills), the model yields a hardest/easiest skill ranking, a trimodal ability distribution, and individual learning trajectories. Its predictions modestly beat a non-hierarchical one-hot logistic baseline (AUC up to 0.831 vs 0.760; log-loss down to 0.440 vs 0.515) and are well calibrated across probability deciles. The deeper point is that interpretable parameters, not just accuracy, are what make a student model usable for deciding who needs remediation and what content needs re-teaching.

What carries the argument

The central object is the hierarchical two-parameter logistic (IRT-style) model, $P(y_i=1)=1/(1+\exp[-(\theta_{s_i}-\beta_{k_i})])$, where each student has one ability parameter and each skill one difficulty parameter. Weak Gaussian priors on both sets of parameters implement Bayesian partial pooling, which is what stabilizes the MAP estimates on subsamples and prevents extreme values unless the data strongly support them. Estimation minimizes the negative log-likelihood plus prior penalties using L-BFGS-B, and the resulting $\hat\theta_s$ and $\hat\beta_k$ are then used directly for rankings, histograms, calibration checks, and trajectory plots.

What would settle it

Re-fit the model on a dataset with ground-truth multi-KC labels using a conjunctive multi-KC model and compare the resulting hardest-skill ranking to the single-KC ranking; a material change would show the reported rankings depend on the unverified single-skill attribution. Alternatively, fit a fully Bayesian version with a stronger prior and check whether the trimodal ability distribution and the extreme difficulty logits survive.

Watch

Extended reading notes

Core claim

The paper's central claim is that fitting a hierarchical logistic model with per-student ability $\theta_s$ and per-skill difficulty $\beta_k$, via maximum a posteriori estimation under weak Gaussian priors, recovers stable and intuitively plausible structure from the Statics2011 dataset. The model defines $P(y_i=1)=1/(1+\exp[-(\theta_{s_i}-\beta_{k_i})])$ and shares information across the class through $\theta_s \sim N(0,100)$ and $\beta_k \sim N(0,100)$, allowing estimates to remain stable across random subsamples. The estimated parameters rank free-body-diagram-related skills as hard, place students with stronger physics backgrounds higher, and split the ability distribution into three bands (low, middle, high) that the author connects to different instructional needs. A byproduct is a relative mastery index $\Delta = \bar\theta - \bar\beta = 2.49$ logits, indicating the average student's proficiency slightly outpaces average item difficulty.

Load-bearing premise

The model attributes every response to one knowledge component $k_i$, so a wrong answer on a multi-concept problem is blamed entirely on that one skill; the paper acknowledges this credit/blame ambiguity and yet does not test how sensitive its difficulty and ability rankings are to it.

Editorial extensions

If this is right

Instructors can use the skill-difficulty ranking to concentrate remediation on the few tail-end skills (in this dataset, Percent Discount and Quadratic Formula) instead of spreading effort across all topics.
The trimodal ability distribution translates into differentiated instruction: foundational support for the low band, scaffolded practice for the middle band, and enrichment for the high band.
Because calibration curves lie within a few percentage points of the identity line, the model's predicted probabilities can be reported to instructors as trustworthy success odds.
The model's stability on 20,000-record subsamples suggests the same analysis is feasible for smaller courses that lack hundreds of thousands of responses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct testable extension is to replace the single-skill-per-response assignment with a conjunctive multi-KC model; the paper's own credit/blame discussion implies that single-KC attribution may bias difficulty estimates for skills that routinely co-occur in multi-concept problems.
If the trimodal ability bands are real subpopulations rather than MAP artifacts, a fully Bayesian fit with a stronger prior should preserve the three modes while shrinking the extreme logits; that experiment would separate the pedagogical claim from the estimation procedure.
The observed negative correlation between practice frequency and estimated difficulty is correlational; an intervention study that randomizes practice amounts could test whether repeated exposure actually drives the difficulty drop.
Applied to other courses, the same hierarchy could be extended with question-level difficulties nested inside skill difficulties, which would let instructors see whether a hard skill is hard everywhere or only on certain item formats.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This paper is a regularized logistic regression presented as hierarchical Bayesian knowledge tracing, and the main pedagogical findings are built on a data-labeling error. The five easiest and hardest skills in Table II are Congruence, Percent Discount, Quadratic Formula, etc., none of which are engineering Statics concepts. The paper claims these come from the OLI Statics2011 dataset, but they look like a middle-school math item bank. That internal inconsistency invalidates the central claim that the model yields difficulty estimates aligning with intuitive expectations for Statics.

To be fair, the paper is clearly written. It uses a public dataset, describes the model (a two-parameter IRT with fixed Gaussian priors, MAP estimation), and reports sensitivity checks across subsample sizes and seeds. It also explicitly acknowledges that weak priors push extreme logits and suggests a stronger prior or full Bayes as a fix. That honesty is worth noting. But the positives end there.

The model has no temporal hidden state, so calling it knowledge tracing is misleading. It is a one-hot logistic regression with L2 regularization. The prior variance is fixed rather than estimated, so the hierarchical Bayesian claim is overstated. Evaluation in Table III is in-sample—the hierarchical model is fit and evaluated on the same subsamples, while the baseline is fit on the full data, making the comparison unfair. Calibration and observed-vs-predicted plots are also in-sample. The relative mastery index Δ is just the difference of the means of θ and β; because the logistic model is invariant to an additive shift, Δ is not identifiable. The trimodal ability distribution is interpreted as distinct subgroups without any clustering test or mixture model. No code is shipped, and no comparison with prior analyses of the same dataset is provided.

The core problem is that the paper does not establish its claims. The predictive improvement is not demonstrated out-of-sample, and the difficulty ranking is nonsense because the skill labels are wrong. This is not a matter of methodological taste; the central pedagogical conclusions rest on a mislabeled dataset.

My recommendation is to reject. The paper does not deserve a serious peer-review cycle in its current state. If the author corrected the skill labels, added out-of-sample evaluation, and renamed the model, a resubmission could be considered. But as it stands, the contribution is a parameter fit of a standard model to a public dataset with an unverified data join.

Best,

Referee Report

5 major / 7 minor

Summary. The paper applies a hierarchical Bayesian logistic model to the OLI Statics2011 dataset, estimating student ability parameters θ_s and skill difficulty parameters β_k via MAP estimation under weak Gaussian priors. It reports skill difficulty rankings, a relative mastery index, ability histograms and calibration curves, and compares predictive performance against a non-hierarchical logistic baseline. The stated contributions are interpretable measures of skill difficulty and student ability for engineering education, plus evidence for distinct student subgroups that could guide differentiated instruction.

Significance. If the results held, the paper would offer a useful, interpretable application of hierarchical Bayesian IRT to a large public educational dataset, and its emphasis on interpretability over raw predictive accuracy is well placed. The paper is clearly written and engages with a real teaching need. However, the validity of the central claims is undermined by several load-bearing problems: the performance evaluation is in-sample, the skill labels reported in the results are not Statics concepts, the relative mastery index is an artifact of the prior specification, and the treatment of multi-KC items is not justified. Because these issues affect the main conclusions rather than only the presentation, the current form does not support the paper's claims.

major comments (5)

[Table III; Figures 4 and 5] The reported AUC, log-loss, and calibration curves are computed on the same subsamples used to fit the hierarchical model, and the baseline is also evaluated on its training data. No held-out set or cross-validation is described anywhere in the paper. Because the hierarchical model includes a separate ability parameter for each student, in-sample evaluation will overstate its predictive performance relative to a simpler baseline. The claim that the model 'improves predictive accuracy' is therefore not supported. The authors must re-evaluate on a held-out test set or via k-fold cross-validation.
[Table II; Section II] The five easiest and hardest skills reported in Table II are 'Congruence', 'Distributive Property', 'Probability Concepts', 'Nets of 3D Figures', 'Equation Selection', 'Percent Discount', 'Quadratic Formula', 'Rotations', 'Percents', and 'Reflection'. None of these is a concept from an undergraduate Statics course, and the Introduction states that skills like free-body diagram analysis emerge as more difficult, which is not reflected in this table. This internal inconsistency suggests that the skill column used in the analysis is not the Statics knowledge-component model described in the Methods, or that the dataset was misjoined. Because every downstream result (difficulty ranking, relative mastery index, subgroup interpretation, remediation recommendations) is derived from these labels, the central claim that the estimates align with intuitive expectations for Statics is invalid unless the label provenance is corrected and the analysis is repeated.
[Section III, 'relative mastery index'] The relative mastery index Δ = mean(θ) - mean(β) is not identifiable from the likelihood, because adding any constant c to every θ_s and every β_k leaves all predicted probabilities unchanged. The MAP objective in Section II includes the prior penalties Σθ_s²/(2σ²) + Σβ_k²/(2σ²); at the optimum the derivative with respect to such a shift forces sum(θ) + sum(β) = 0. With the reported counts N_s = 2621 and N_k = 107 and mean(β) = -2.40, this constraint yields mean(θ) = -(107/2621)(-2.40) ≈ 0.098, matching the reported mean of 0.09. Thus the statement that student proficiency slightly outpaces item difficulty is a mathematical consequence of the zero-mean priors and the ratio of student to skill counts, not an empirical finding about the course. This quantity should be dropped or reparameterized with an identifiable intercept.
[Section II, model equation] The model assigns each response to a single knowledge component k_i, but the paper acknowledges that the Statics2011 dataset contains multi-concept problems and cites the credit/blame assignment problem. The Methods never explain how multi-KC responses were mapped to one skill: no filtering step, no KC-selection rule, and no justification for treating multi-KC items as if they were single-KC is given. For a multi-KC item, a wrong answer may be due to any of several missing skills, so attributing the response to a single skill biases both the difficulty and ability estimates. The authors must either restrict the analysis to single-KC responses or use a model that explicitly handles conjunctive or disjunctive combinations of knowledge components.
[Figures 1 and 3; Section III] The paper claims the class separates into 'distinct student subgroups' based on the trimodal ability histogram and the three horizontal bands in the ability-versus-attempts plot. These figures are visual interpretations of MAP point estimates from a model with weak priors (σ² = 100), and no uncertainty quantification, posterior intervals, or formal clustering or mixture analysis is provided. Students with few responses can receive extreme MAP estimates, so the apparent modes and bands may reflect sparse data rather than genuine subpopulations. The authors should report the distribution of responses per student, examine the uncertainty in individual ability estimates, and validate the subgroup claim with a formal statistical test or mixture model before drawing pedagogical conclusions.

minor comments (7)

[Introduction] The Introduction claims that 'students who had more physics background show higher ability estimates,' but no physics-background variable appears anywhere in the Methods or Results. This claim should be removed or substantiated with an analysis.
[Section II] The Methods state that 'incomplete or ambiguous records' were removed, but no counts or criteria are provided. The number of removed records and the exact filtering rule should be reported.
[Table I] The dataset is said to contain 4,151 students and 110 distinct skills, yet Table I reports N = 2,621 students and N = 107 skills after fitting. The reasons for these exclusions should be stated explicitly.
[Figure 2] The caption attributes the negative correlation between difficulty and practice frequency to 'repeated practice drives mastery,' but the direction of causality is confounded: easier or more frequently assigned items may naturally receive more attempts. The wording should be tempered to a descriptive association.
[Section II and Table III] The term 'log-loss' is used without a definition or formula, and the metric is not explained in context. A standard definition should be supplied.
[References] Several references are incomplete (e.g., the Pavlik et al. entry lacks a publication venue) and the citation style is inconsistent between author-year in the text and numbered entries in the reference list.
[Equation (1)] The model equation appears with missing parentheses/brackets in the displayed form; the typesetting should be corrected so that the denominator is clearly 1 + exp[-(θ - β)].

Circularity Check

2 steps flagged · score 6.0 of 10

The reported predictive-accuracy gains (Table III, Figs. 4–5) are in-sample fits, making the main quantitative contribution a restatement of the MAP objective.

fitted input called prediction [Section II (MAP objective) and Section III, Table III and Figure 4]
"Parameters were estimated using MAP estimation, which involves maximizing the posterior probability by minimizing the negative log-likelihood combined with the priors' penalties: −Σ_i [y_i log p_i + (1−y_i) log(1−p_i)] + ... . ... We fitted our hierarchical two-parameter logistic model via MAP estimation on four different subsamples—20,000 and 40,000 records each ... For each model, we computed the AUC to measure discriminative ability and the log-loss to quantify the accuracy of the predicted probabilities."

The MAP objective minimized is exactly the log-loss (the negative log-likelihood) computed on the training subsample. The paper then reports log-loss, AUC, and calibration curves on the same subsamples used for fitting, with no held-out set or cross-validation described. Therefore the hierarchical model's 'better' log-loss and calibration are in-sample training-fit values, not out-of-sample predictions; the reported predictive-accuracy gain is a restatement of the fitting objective by construction.
fitted input called prediction [Section III, Figure 5]
"we compared each skill’s empirical success rate in the 20,000-record subsample to the average probability predicted by our two-parameter logistic model. ... Although the blue (observed) and orange (predicted) bars align closely—demonstrating that our hierarchical MAP estimates correctly order items at the tails—one should note that pure MAP estimation with weak priors can push logits to extreme values (and hence predicted probabilities very near 0 or 1) when the data are overwhelmingly one-sided."

The observed proportions are computed on the exact subsample used for MAP fitting, and the 'predicted' probabilities are the same model's fitted values on that subsample. The close observed-versus-predicted alignment is therefore a measure of training-data fit, not independent confirmation of the difficulty ranking. Moreover, the paper itself acknowledges that weak-prior MAP drives extreme one-sided skills' predictions toward 0 or 1, so the reported agreement is partly forced by the estimation procedure.

full rationale

The paper's central derivation chain is a standard two-parameter logistic (IRT-style) model estimated by MAP; I find no load-bearing self-citation, ansatz-smuggling, or uniqueness-imported-from-authors circularity, since all references are external and the model is a conventional parameterization. The main circularity is that the quantitative support for 'predictive accuracy' (Table III, Figures 4–5) is computed on the same subsamples used for fitting; the MAP objective is exactly the reported log-loss, so the reported gains are in-sample fit rather than prediction. I do not score the Table II skill-label mismatch (Congruence, Distributive Property, Probability Concepts, etc.) as circularity: that is a data-integrity/correctness concern external to the derivation chain, not a derivation that reduces to its inputs. Likewise, the trimodal ability histogram and the Figure 2/3 associations are descriptive readings of fitted parameters; they are weak evidence but not circular in the definitional sense. Because one or more 'predictions' reduce to the fitting objective, partial circularity applies: score 6.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The inferential chain relies on standard logistic assumptions plus several unverified domain choices: single-skill attribution for multi-KC items, fixed prior variance, static ability, and representative subsampling. The key parameters θ and β are estimated, not independently derived, and the evaluation does not leave the training data.

free parameters (1)

σ² (prior variance) = 100
Fixed weak Gaussian prior variance for student ability θ and skill difficulty β. Chosen by hand to balance bias and extremes; not estimated from data, and directly controls shrinkage of all reported estimates.

assumptions (5)

standard math Responses are independent given ability and difficulty
The likelihood sums log probabilities over individual responses with no dependence structure, so every answer is treated as an independent Bernoulli trial.
domain assumption Ability and difficulty are static during the analyzed period
Stated in the Conclusion: 'our approach inherently assumes stable abilities and skill difficulties within the analyzed timeframe.'
domain assumption Each response is associated with a single skill
The model equation indexes each response to one skill k_i, despite Section I acknowledging multi-KC 'credit/blame assignment' problems in the same dataset.
ad hoc to paper The fixed prior σ²=100 adequately regularizes the MAP estimates
No data-driven choice or sensitivity analysis for σ² is provided; the paper admits at the extremes that weak priors push logits to extreme values.
domain assumption The 20,000 and 40,000 observation subsamples are representative
The paper does not describe how subsamples were drawn or whether their skill and student coverage matches the full dataset.

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Cite this review

Pith. "Pith review of Hierarchical Bayesian Knowledge Tracing in Undergraduate Engineering Education." pith.science (2026). https://pith.science/paper/FLRWHV47

@misc{pith2026250600057,
  author       = {Pith},
  title        = {Pith review of: Hierarchical Bayesian Knowledge Tracing in Undergraduate Engineering Education},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLRWHV47}},
  note         = {Machine review of arXiv:2506.00057}
}

read the original abstract

Educators teaching entry-level university engineering modules face the challenge of identifying which topics students find most difficult and how to support diverse student needs effectively. This study demonstrates a rigorous yet interpretable statistical approach -- hierarchical Bayesian modeling -- that leverages detailed student response data to quantify both skill difficulty and individual student abilities. Using a large-scale dataset from an undergraduate Statics course, we identified clear patterns of skill mastery and uncovered distinct student subgroups based on their learning trajectories. Our analysis reveals that certain concepts consistently present challenges, requiring targeted instructional support, while others are readily mastered and may benefit from enrichment activities. Importantly, the hierarchical Bayesian method provides educators with intuitive, reliable metrics without sacrificing predictive accuracy. This approach allows for data-informed decisions, enabling personalized teaching strategies to improve student engagement and success. By combining robust statistical methods with clear interpretability, this study equips educators with actionable insights to better support diverse learner populations.

Figures

Figures reproduced from arXiv: 2506.00057 by the authors.

**Figure 2.** Relationship between skill difficulty βˆ k and logtransformed practice frequency per skill. Each point represents one skill; the negative trend indicates that skills attempted more often tend to have lower difficulty estimates, suggesting that repeated practice drives mastery [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Estimated student ability ˆθs plotted against the base-10 logarithm of each student’s total number of problem-solving attempts. Each point represents one of the 2,621 students active in the subsample [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Observed versus model-predicted success rates for the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Predicted mastery trajectory for the lowest-ability [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Reference graph

Works this paper leans on

10 extracted references · 8 canonical work pages

[1]

Worsley and P

M. Worsley and P. Blikstein, ``Analyzing engineering design through the lens of learning analytics,'' Journal of Learning Analytics, vol. 1, no. 2, pp. 151--186, 2014. [Online]. Available: https://lemanncenter.stanford.edu/paper/analyzing-engineering-design-through-lens-learning-analytics

work page 2014
[2]

D. B. Knight, C. Brozina, and B. Novoselich, ``An investigation of first-year engineering student and instructor perspectives of learning analytics approaches,'' Journal of Learning Analytics, vol. 3, no. 3, pp. 215--238, 2016

work page 2016
[3]

K. R. Koedinger, P. I. Pavlik Jr, J. Stamper, T. Nixon, and S. Ritter, ``Avoiding problem selection thrashing with conjunctive knowledge tracing,'' in Proceedings of the 4th International Conference on Educational Data Mining (EDM), 2011, pp. 91--100. [Online]. Available: https://pact.cs.cmu.edu/pubs/Koedinger,

work page 2011
[4]

A. T. Corbett and J. R. Anderson, ``Knowledge tracing: Modeling the acquisition of procedural knowledge,'' User Modeling and User-Adapted Interaction, vol. 4, no. 4, pp. 253--278, 1994. [Online]. Available: https://link.springer.com/article/10.1007/BF01099821

doi:10.1007/bf01099821 1994
[5]

M. V. Yudelson, K. R. Koedinger, and G. J. Gordon, ``Individualized bayesian knowledge tracing models,'' Proceedings of the 6th International Conference on Educational Data Mining, pp. 171--178, 2013

work page 2013
[6]

P. I. J. Pavlik, L. G. Eglington, and L. M. Harrell-Williams, ``Logistic knowledge tracing: A constrained framework for learner modeling,'' ERIC, Tech. Rep., 2021. [Online]. Available: https://files.eric.ed.gov/fulltext/ED618076.pdf

work page 2021
[7]

Piech, J

C. Piech, J. Spencer, J. Huang, S. Ganguli, M. Sahami, L. Guibas, and J. Sohl-Dickstein, ``Deep knowledge tracing,'' in Advances in Neural Information Processing Systems, vol. 28, 2015, pp. 505--513. [Online]. Available: https://stanford.edu/ cpiech/bio/papers/deepKnowledgeTracing.pdf

work page 2015
[8]

Abdelrahman, Q

G. Abdelrahman, Q. Wang, and B. P. Nunes, ``Knowledge tracing: A survey,'' ACM Computing Surveys, vol. 55, no. 11, pp. 1--37, 2023

work page 2023

Show all 10 references

[9]

Yeung, ``Deep-irt: Make deep learning based knowledge tracing explainable using item response theory,'' arXiv preprint arXiv:1904.11738, 2019

C.-K. Yeung, ``Deep-irt: Make deep learning based knowledge tracing explainable using item response theory,'' arXiv preprint arXiv:1904.11738, 2019. [Online]. Available: https://ar5iv.labs.arxiv.org/html/1904.11738

1904 arXiv
[10]

S. Shen, Q. Liu, Z. Huang, Y. Zheng, M. Yin, M. Wang, and E. Chen, ``A survey of knowledge tracing: Models, variants, and applications,'' IEEE Transactions on Knowledge and Data Engineering, 2024. [Online]. Available: https://ieeexplore.ieee.org/document/10494775

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Worsley and P

M. Worsley and P. Blikstein, ``Analyzing engineering design through the lens of learning analytics,'' Journal of Learning Analytics, vol. 1, no. 2, pp. 151--186, 2014. [Online]. Available: https://lemanncenter.stanford.edu/paper/analyzing-engineering-design-through-lens-learning-analytics

work page 2014

[2] [2]

D. B. Knight, C. Brozina, and B. Novoselich, ``An investigation of first-year engineering student and instructor perspectives of learning analytics approaches,'' Journal of Learning Analytics, vol. 3, no. 3, pp. 215--238, 2016

work page 2016

[3] [3]

K. R. Koedinger, P. I. Pavlik Jr, J. Stamper, T. Nixon, and S. Ritter, ``Avoiding problem selection thrashing with conjunctive knowledge tracing,'' in Proceedings of the 4th International Conference on Educational Data Mining (EDM), 2011, pp. 91--100. [Online]. Available: https://pact.cs.cmu.edu/pubs/Koedinger,

work page 2011

[4] [4]

A. T. Corbett and J. R. Anderson, ``Knowledge tracing: Modeling the acquisition of procedural knowledge,'' User Modeling and User-Adapted Interaction, vol. 4, no. 4, pp. 253--278, 1994. [Online]. Available: https://link.springer.com/article/10.1007/BF01099821

doi:10.1007/bf01099821 1994

[5] [5]

M. V. Yudelson, K. R. Koedinger, and G. J. Gordon, ``Individualized bayesian knowledge tracing models,'' Proceedings of the 6th International Conference on Educational Data Mining, pp. 171--178, 2013

work page 2013

[6] [6]

P. I. J. Pavlik, L. G. Eglington, and L. M. Harrell-Williams, ``Logistic knowledge tracing: A constrained framework for learner modeling,'' ERIC, Tech. Rep., 2021. [Online]. Available: https://files.eric.ed.gov/fulltext/ED618076.pdf

work page 2021

[7] [7]

Piech, J

C. Piech, J. Spencer, J. Huang, S. Ganguli, M. Sahami, L. Guibas, and J. Sohl-Dickstein, ``Deep knowledge tracing,'' in Advances in Neural Information Processing Systems, vol. 28, 2015, pp. 505--513. [Online]. Available: https://stanford.edu/ cpiech/bio/papers/deepKnowledgeTracing.pdf

work page 2015

[8] [8]

Abdelrahman, Q

G. Abdelrahman, Q. Wang, and B. P. Nunes, ``Knowledge tracing: A survey,'' ACM Computing Surveys, vol. 55, no. 11, pp. 1--37, 2023

work page 2023