Personalized AI Practice Replicates Learning Rate Regularity at Scale

Jocelyn Beauchesne; Christine Maroti; Jeshua Bratman; Jerome Pesenti; Laurence Holt; Alex Tambellini; Allison McGrath; Matthew Guo

cs.CYMar 9, 2026

Personalized AI Practice Replicates Learning Rate Regularity at Scale

Authors: Jocelyn Beauchesne, Christine Maroti, Jeshua Bratman, Jerome Pesenti, Laurence Holt, Alex Tambellini, Allison McGrath, Matthew Guo,

Organizations: Campus AI

Abstract

Recent research demonstrated that students exhibit consistent learning rates across diverse educational contexts. We test these findings using a dataset of 1.8 million (366k post-filtering) student interactions from the digital platform Campus AI providing further evidence to the observation of regularity in learning rate among students. Unlike prior work requiring manual cognitive modeling, Campus AI automatically generates Knowledge Components (KCs) and corresponding exercises, both of which are validated by human experts. This one-to-many mapping facilitates the application of Additive Factors Models to measure learning parameters without complex cognitive modeling. Using mixed-effects logistic regression, we confirmed the core finding of prior work: students displayed substantial variation in initial knowledge ( $\text{IQR} = [2.78, 12.18]$ practice opportunities to reach 80% mastery) but remarkably consistent learning rates ( $\text{IQR} = [7.01, 8.25]$ opportunities). Furthermore, students using this fully automated system achieved 80% mastery in a median of 7.22 practice opportunities, comparable to the 6.54 reported for expert-designed curricula. These results suggest that automated, science-grounded content generation can support effective personalized learning at scale. Data and code are publicly available. https://github.com/Campus-edu-AI/learning-rate

Figures & tables

Type	Condition	Response
Category	Random variable types	Discrete or continuous
Fact	U wave on EKG	Late repolarization of Purkinje fibers
Rule/Plan	Eye focus during basketball shot	On the target (front or back of rim)
Concept	Contractual capacity	Legal ability to enter into a binding contract

Table 1: Examples of LLM-generated Knowledge Components across different types.

Clarity	%
Very clear	75.3
Somewhat clear	12.7
Not clear	11.9

Table 2: KC evaluation by instructors: clarity (n=361), relevance (n=359), difficulty (n=282).

Parameter	Population	Student Effect
	Effect	Mean	Q1	Median	Q3	SD
Initial Knowledge	0.686	-0.0169	-0.304	0.0221	0.314	0.461
Learning Rate	0.0657	-0.000262	-0.00555	0.000323	0.00516	0.0121

Table 3: Summary Statistics for Mixed-Effects Model Parameters

Figure 1: Observed learning curve versus statistical model predictions across practice opportunities on a Knowledge Component. The solid orange line shows empirical accuracy rates from student practice data. The dashed blue line represents the population-level prediction from a mixed-effects logistic regression model (iAFM framework). Students show consistent improvement from approximately 73% to 85% accuracy over 18 practice opportunities.

Panel of Figure 2 — Figure 2: Parameter distributions from the base mixed-effects logistic regression model (iAFM framework) for n=7161 students. Both distributions show the individual-level variation around population parameters, with initial abilities displaying greater heterogeneity than learning rates. Quartile boundaries (Q1, Q3) and 95% confidence intervals are marked for reference.

	Initial Knowledge			Learning Rate
Percentile	Log Odds	Percent Correct	Ops to 80% (Fixed Rate)	Log Odds per Opp	Percent Point Improvement	Ops to 80% (Fixed Knowledge)
25	$0.59$	$64.24$	$12.18$	$0.06$	$1.23$	$8.25$
50	$0.91$	$71.34$	$7.22$	$0.07$	$1.34$	$7.52$
75	$1.20$	$76.92$	$2.78$	$0.07$	$1.44$	$7.01$

Table 4: Median for initial accuracy and practice opportunities to reach mastery (

80\%

correct probability) for low (25) and high (75) percentiles of initial knowledge (assuming overall learning rate,

\delta

) and learning rate (assuming overall initial knowledge,

\theta

) for the base models.

	level	subject	kc type	$\theta_{\text{pop}}$	$\delta_{\text{pop}}$	$\mathbb{E}[\theta_{\text{s}}]$	$\text{SD}[\theta_{\text{s}}]$	$\mathbb{E}[\delta_{\text{s}}]$	$\text{SD}[\delta_{\text{s}}]$
model 0				$6.85e^{\text{-01}}$	$6.57e^{\text{-02}}$	$-1.69e^{\text{-02}}$	$4.61e^{\text{-01}}$	$-3.00e^{\text{-04}}$	$1.21e^{\text{-02}}$
model 1	✓			$6.94e^{\text{-01}}$	$6.82e^{\text{-02}}$	$-6.30e^{\text{-03}}$	$4.60e^{\text{-01}}$	$0.00e^{\text{+00}}$	$1.20e^{\text{-02}}$
model 2		✓		$6.86e^{\text{-01}}$	$6.56e^{\text{-02}}$	$-3.37e^{\text{-03}}$	$4.55e^{\text{-01}}$	$-1.00e^{\text{-04}}$	$1.20e^{\text{-02}}$
model 3			✓	$6.63e^{\text{-01}}$	$6.42e^{\text{-02}}$	$-1.97e^{\text{-03}}$	$4.59e^{\text{-01}}$	$-0.00e^{\text{+00}}$	$1.19e^{\text{-02}}$
model 4	✓	✓		$6.90e^{\text{-01}}$	$6.64e^{\text{-02}}$	$-1.93e^{\text{-03}}$	$4.55e^{\text{-01}}$	$0.00e^{\text{+00}}$	$1.19e^{\text{-02}}$
model 5	✓		✓	$6.86e^{\text{-01}}$	$6.63e^{\text{-02}}$	$-1.41e^{\text{-03}}$	$4.59e^{\text{-01}}$	$0.00e^{\text{+00}}$	$1.19e^{\text{-02}}$

Table 5: Results of ablation comparing population-level and student-level parameters across model variants. Each model includes different combinations of course level, subject, and KC type effects. Population effects (

\theta_{\text{pop}}

\delta_{\text{pop}}

) represent fixed intercept and slope, while student-level statistics show the distribution of random effects across n=7161 students. Initial knowledge values are in log odds; learning rates are in log odds per practice opportunity.

	$\mathbb{E}[\theta_{\text{kc type}}]$	$\mathbb{E}[\delta_{\text{kc type}}]$	$\text{SD}[\theta_{\text{kc type}}]$	$\text{SD}[\delta_{\text{kc type}}]$
KC type
association	$1.42e^{\text{-02}}$	$8.95e^{\text{-04}}$	$1.76e^{\text{-03}}$	$1.32e^{\text{-04}}$
category	$7.16e^{\text{-02}}$	$4.53e^{\text{-03}}$	$5.11e^{\text{-03}}$	$4.25e^{\text{-04}}$
concept	$1.83e^{\text{-02}}$	$1.16e^{\text{-03}}$	$5.39e^{\text{-03}}$	$3.67e^{\text{-04}}$
fact	$2.80e^{\text{-02}}$	$1.77e^{\text{-03}}$	$3.52e^{\text{-03}}$	$2.62e^{\text{-04}}$
principle/rule/model	$-6.89e^{\text{-02}}$	$-4.34e^{\text{-03}}$	$6.52e^{\text{-03}}$	$3.30e^{\text{-04}}$

Table 6: Comparing Intercepts and Learning Rates for the KC Types factor, expected value and standard deviation are computed over results from model 3, 5, 6 and 7.

Figure 3: Scatter plot the course subject factor effects, Average $\theta_{\text{course subject}}$ against $\delta_{\text{course subject}}$ averaged across ablation Models 2, 4, 6, and 7.

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How much have students' ordinary learning processes shifted in response to generative AI, and how does that affect their durable learning outcomes? Self-report surveys show little change, while small-scale behavioral studies report widespread AI use without the scale or duration to measure learning consequences. We address both questions using a ten-year panel of

3.2

million ALEKS learning interactions for investigating time-on-task, complemented by ALEKS PPL placement-assessment data for examining proctoring and learning outcomes, with a quasi-experimental design exploiting variation in tasks that are more susceptible to AI (text-based word problems) and less susceptible to AI (interactive graph-based problems). Learning time on AI-susceptible problems declines

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25\%

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Abstract

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