How Spaced Repetition Builds Long-Term Memory

The Forgetting Curve

In 1885, Hermann Ebbinghaus conducted the first rigorous experiments on memory decay. He memorized lists of nonsense syllables and measured his retention over time, producing what is now known as the forgetting curve - a mathematical description of how memories fade without reinforcement.

Ebbinghaus discovered that memory decay follows a predictable pattern: approximately 50% of newly learned information is forgotten within the first hour, 70% within 24 hours, and 90% within a week - unless the memory is deliberately reinforced. However, each successful retrieval strengthens the memory trace, making it more resistant to future decay.

This finding has a profound practical implication: the timing of your reviews matters as much as the reviews themselves. Review too early, and you waste effort on material you still remember. Review too late, and the memory has decayed so far that you are essentially re-learning from scratch. The optimal review happens just before the memory would have faded - maximizing both efficiency and retention.

How Spaced Repetition Works

Spaced repetition is a learning technique that schedules reviews at increasing intervals, timed to intercept the forgetting curve just before memory decay. After each successful review, the interval before the next review expands - reflecting the fact that each retrieval makes the memory more durable.

A concept you reviewed successfully yesterday might be scheduled for review in 3 days. If you recall it successfully then, the next review might be in 10 days, then 30, then 90. Each successful retrieval approximately doubles or triples the interval. Failed retrievals reset the interval to a shorter period, reflecting the weakened memory trace.

This produces an elegant efficiency: you spend the most time on concepts you find difficult (short intervals, frequent reviews) and the least time on concepts you have mastered (long intervals, rare reviews). Over time, the system converges on a maintenance schedule that keeps all concepts above a target retention threshold with minimal total study time.

The FSRS Model

The Free Spaced Repetition Scheduler (FSRS) is a modern, open-source algorithm that improves on earlier systems like SM-2 (used in Anki). FSRS uses a mathematical model of memory that tracks two key parameters per concept:

• Stability (S) - how many days until retrievability drops to 90%. Higher stability means the memory decays slower.
• Difficulty (D) - an inherent difficulty rating for the concept on a 0–1 scale. Some concepts are naturally harder to retain than others.

TrainRun implements an FSRS-inspired decay model in its game engine. The core formula that calculates retrievability (the probability of correct recall at time t) is:

// FSRS-inspired retrievability model (from TrainRun engine)
// R(t) = probability of correct recall at time t

function getRetrievability(state, nowMs) {
  const elapsedDays = (nowMs - state.lastReviewMs) / (1000 * 60 * 60 * 24)
  if (elapsedDays <= 0) return 1.0

  // Power-law decay: R(t) = (1 + t/(9*S))^(-0.5)
  // where S = stability (days until R drops to 90%)
  return Math.pow(1 + elapsedDays / (9 * state.stability), -0.5)
}

// After successful retrieval: stability grows exponentially
function onSuccessfulRetrieval(state, nowMs) {
  const r = getRetrievability(state, nowMs)
  // The lower your current retrievability, the more stability you gain
  // (retrieving a fading memory strengthens it more)
  const newStability = state.stability *
    (1 + Math.exp(5.5) * Math.pow(state.stability, -0.2) * Math.pow(1 - r, 1.2))

  return { ...state, stability: Math.min(newStability, 365) }
}

The key insight in this model: retrieving a concept that has partially faded (low current retrievability) produces a larger stability gain than retrieving a concept you still remember clearly. This mathematically captures the desirable difficulty principle - harder retrievals produce stronger memories.

How TrainRun Applies Spaced Repetition

In traditional spaced repetition systems (like Anki), you manually create flashcards and review them on a schedule. TrainRun applies the same memory science through gameplay:

• Per-concept tracking - the engine maintains a retrievability state for each concept encountered during a run, tracking stability, difficulty, reps, and lapses across a bounded 128-slot store.
• Collection as review- each time you collect a word pickup, the engine records a successful retrieval. Missing a pickup records a failed retrieval, which reduces stability and increases the concept's priority for future review.
• Replay as spacing - replaying a module after a delay introduces natural spacing. The model predicts which concepts have likely decayed, making the replay more effective than immediate repetition.

Why Spacing Works: The New Theory of Disuse

Bjork and Bjork's New Theory of Disuse (1992) explains why spacing works by distinguishing between two types of memory strength:

• Storage strength - how well the memory is encoded. This only increases; memories are never truly erased.
• Retrieval strength - how easily the memory can be accessed right now. This decays over time without use.

Spacing works because it allows retrieval strength to partially decay before the next review. The act of retrieving a partially-faded memory (effortful recall) increases both storage strength and retrieval strength more than retrieving an easily-accessible memory (effortless recall). This is why cramming fails: massed repetition keeps retrieval strength artificially high, producing the illusion of learning without building durable storage strength.

Practical Implications

The science of spaced repetition has clear implications for effective studying:

• Spread your study sessions out over days and weeks rather than cramming into one marathon session.
• Review material just before you would forget it - not immediately after learning it.
• Embrace the feeling of difficulty when recalling partially-forgotten material. That difficulty is the signal that real learning is happening.
• Track what you find difficult. The mastery system in TrainRun does this automatically, adjusting concept priority based on your collection success rate.

Combined with active recall, spaced repetition forms the foundation of evidence-based learning. TrainRun implements both - active retrieval through gameplay, scheduled review through the FSRS decay model - without requiring you to manage flashcard decks or review schedules manually.