Here are high-level descriptions of novel algorithms that I've developed.

Fractional Implicit Repetition (FIRe)
  • Generalizes discrete spaced repetition on independent flashcards to fractional implicit spaced repetition on knowledge graphs of interconnected skills and concepts.
  • Estimates student knowledge profiles and selects personalized learning tasks that optimize knowledge persistence over time, striking an optimal balance between learning new topics (maximizing knowledge gain) and reviewing already-seen topics (minimizing knowledge loss).
  • Algorithm structure has analogies to biology: topics = cells, spaced repetition latent state = chemical concentrations within cells, knowledge graph = brain, correct answers = stimulants of cell growth, incorrect answers = inhibitors of cell growth, learning tasks = stimuli to brain.
  • Speeds up learning by a factor of 4x and improves mastery: students learning via FIRe on Math Academy's personalized learning platform can complete AP Calculus BC in just 35 minutes per school day with improved AP exam scores as compared to an instructor-led course consisting of 12 hours per week (1 hour class and 1 hour homework per school day, plus an amortized 2 hours per week of studying for quizzes/tests and the AP exam itself).
  • Has enabled sufficiently motivated 6th grade students to progress from prealgebra to AP Calculus BC over the span of just 2 semesters.

XP Penalty System
  • Context: Students are routinely served an handful of learning tasks to choose from, and they earn XP for completing tasks with satisfactory performance. In the absence of a penalty system, adversarial students will complete tasks that they feel are easy and then submit a bunch of random guesses to intentionally fail out of tasks that require more effort.
  • Applies a penalty (negative XP) when it detects that a student is failing tasks as a result of being unwilling to put in effort. Tracks the amount of "anger" that would build up in a tutor or guardian sitting next to the student, and then translates that anger into an XP penalty.
  • Effectively shuts down adversarial behavior while simultaneously not impacting cooperative students. Many adversarial students' pass rates jumped from 50% to over 90%.

Fast Diagnostic
  • Performs inference during diagnostic exams to massively reduce the number of questions that must be asked to characterize a student's knowledge profile.
  • Efficiently searches a sequence in which topics are conventionally covered in standard math classes, while simultaneously using a prerequisite graph for causal inference and filtering.

Expression Equivalence
  • Determines whether a free response mathematical expression matches the answer key expression.
  • Constructs a sample of numerical substitutions such that the free response answer is almost certain to be correct if it matches the answer key on the sample.
  • Intelligently handles not only numerical overflow but also details like mathematical ambiguity and context-dependence of mathematical rigor just like a human grader would.