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Regression toward the mean (“RTM”) is a phenomenon in which extremely high or low values—for example, and individual’s blood pressure at a particular moment—appear closer to a group’s average upon remeasuring. Although this statistical peculiarity is the result of random error and chance, it has been problematic across various medical, scientific, financial and psychological applications. In particular, RTM, if not taken into account, can interfere when...
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While the differential rate law relates the rate and concentrations of reactants, a second form of rate law called the integrated rate law relates concentrations of reactants and time. Integrated rate laws can be used to determine the amount of reactant or product present after a period of time or to estimate the time required for a reaction to proceed to a certain extent. For example, an integrated rate law helps determine the length of time a radioactive material must be stored for its...
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The standard deviation is the most common measure of variation. It is a value that tells us how far a data value is from the mean value in a dataset. Further, the standard deviation is always a positive value or zero.
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The rate of a reaction is affected by the concentrations of reactants. Rate laws (differential rate laws) or rate equations are mathematical expressions describing the relationship between the rate of a chemical reaction and the concentration of its reactants.
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Updated: Aug 6, 2025

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An astonishing regularity in student learning rate.

Kenneth R Koedinger1, Paulo F Carvalho1, Ran Liu2

  • 1Human-Computer Interaction Institute, Carnegie Mellon University, Pittsburgh, PA 15213.

Proceedings of the National Academy of Sciences of the United States of America
|March 20, 2023
PubMed
Summary
This summary is machine-generated.

Students show significant variation in initial knowledge but learn at remarkably similar rates, challenging existing learning theories. This study explores skill acquisition across diverse educational contexts.

Keywords:
deliberate practicelearning curveslearning ratelogistic regression growth modeling; educational equity

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Area of Science:

  • Cognitive Science
  • Educational Psychology
  • Data Science

Background:

  • Understanding individual differences in learning is crucial for effective pedagogy.
  • Existing research often focuses on overall achievement rather than specific learning dynamics.

Purpose of the Study:

  • To investigate the variability in student learning rates and initial performance.
  • To develop and apply cognitive and statistical models to analyze skill acquisition.

Main Methods:

  • Modeled student performance data from 1.3 million observations across 27 datasets.
  • Utilized online practice systems in math, science, and language courses (K-college).
  • Estimated initial correctness and learning rate per practice opportunity.

Main Results:

  • Students exhibit modest initial performance (approx. 65% accuracy) despite instruction.
  • Significant variation in initial performance exists (55% to 75% accuracy).
  • Learning rates are surprisingly consistent across students (approx. 2.5% accuracy increase per opportunity).

Conclusions:

  • Student learning is characterized by high initial performance variability.
  • Learning rate demonstrates remarkable consistency, posing a challenge to current learning theories.
  • Future research should explore factors contributing to initial performance differences.