Spacing, Retrieval Practice, and Interleaving Applied to Mathematics

Findings

1. Interleaved practice in mathematics (Rohrer & Taylor)

Rohrer and Taylor’s lab studies had children practice four kinds of math problems either blocked (AAAA BBBB) or interleaved (ABCD ABCD). Interleaving impaired in-session performance yet doubled next-day test scores [1] — the signature pattern of a desirable difficulty.

Taylor & Rohrer (2010, Applied Cognitive Psychology) ran a classroom RCT: grade 7 (n=140) received blocked or interleaved practice over nine weeks, tested unannounced two weeks later. Interleaved-practice material scored higher [2][3].

Rohrer, Dedrick & Stershic (2015, J. Educational Psychology 107(3), 900-908) ran a dose-response RCT (n=126, grade 7): a higher dose of interleaving in the same worksheets raised scores at both ~2-day and 1-month delays, with zero extra practice time [4]. The benefit is not an artifact of superficial similarity between problem types — it holds even when interleaved problems look quite different, consistent with interleaving training strategy selection, not rote recall [5]. Teacher surveys rated interleaving highly feasible — it requires only resequencing existing problems [2][3].

2. Bjork’s desirable difficulties (UCLA)

Robert & Elizabeth Bjork (1994) coined “desirable difficulties”: conditions that slow acquisition often improve long-term retention and transfer, because performance-while-learning and learning-itself are dissociable [6]. Five difficulties have strong evidence: spacing, interleaving, retrieval practice, generation, and varied practice [7]. Instructional design optimized for smooth, error-free sessions (massed repetition, blocking, re-reading) produces learning that feels good but doesn’t last.

3. The testing effect (Roediger & Karpicke)

Roediger & Karpicke (2006) compared repeated studying vs. repeated testing of the same material. Immediately after, studiers looked better (~83% vs. ~71% recall); one week later the pattern reversed (~40% vs. ~61%) [8]. Retrieval-practice benefit grows with delay before the criterial test — the same signature as interleaving. Implication: an “answer, then feedback” loop should be the primary learning event, not an assessment bolted onto instruction.

4. Optimal spacing intervals — Cepeda et al.’s temporal ridgeline

Cepeda, Vul, Rohrer, Wixted & Pashler (2008, Psychological Science, >1,350 subjects) varied the study-to-relearning gap and tested retention up to a year later. The optimal gap is not fixed — as a fraction of the eventual test delay it runs ~20-40% for a 1-week goal down to ~5-10% for a 1-year goal [9]. Cramming before a quiz you need to remember for a year under-spaces; spacing reviews a month apart to remember something for a week over-spaces — exactly the tension adaptive schedulers (SM-2, FSRS, HLR) exist to solve.

5. Spaced-repetition algorithms used in real software

Leitner (1972). Cards live in boxes (classically 5) with fixed cadences (~1, 2, 4, 7, 14 days); a correct answer promotes, an incorrect one resets to box 1 [14].

SM-2 (Woźniak, 1987). Each item has an ease factor (EF), starting 2.5, floored at 1.3. Intervals: I(1)=1, I(2)=6, I(n)=I(n-1)×EF thereafter. A 0-5 quality rating adjusts EF via EF’ = EF + (0.1 − (5−Q)×(0.08 + (5−Q)×0.02)); Q<3 resets the item [15].

FSRS (Anki’s current default). Tracks three state variables per card: Stability S (days until recall probability decays to 90%), Difficulty D (1-10), and Retrievability R (0-1, decaying per a power-law curve, not exponential). A single knob, desired retention (typically 0.85-0.95, default ~0.90), drives the scheduler to invert the forgetting curve and pick the interval where predicted R hits that target. FSRS-6 fits ~19-21 weights per learner from review history via gradient descent, outperforming SM-2’s fixed ease factor once enough data exists (~1,000+ reviews) [10].

Half-Life Regression (Duolingo; Settles & Meeder, 2016, ACL). Models each item’s memory half-life h as a log-linear function of prior correct/incorrect counts; recall probability p = 2^(−Δt/h) — an explicit exponential curve (vs. FSRS’s power-law). Cut prediction error 45%+ vs. baselines and lifted daily engagement 12% in a live A/B [11].

Common thread. All four schedule the next exposure for the moment recall probability is about to cross a target threshold — not before (wasted repetition), not long after (already forgotten). They differ in whether the forgetting curve is fixed (Leitner, SM-2) or fitted per item/learner (FSRS, HLR), and exponential vs. power-law shape.

6. Mastery learning thresholds

Bloom’s mastery learning asks for ~80-90% accuracy before advancing, with remediation below threshold [12][13]. A common cheap proxy, especially in K-12 fact-fluency programs, is “N consecutive correct” (often 3), which resets cleanly on a wrong answer [13]. Recent adaptive-tutoring research found raising the mastery bar from ~0.95 to ~0.98 estimated-mastery probability improved performance on dependent subsequent lessons — the traditional threshold under-shoots for prerequisite content [12]. Fact-fluency literature stresses mastery must be assessed after a gap, not only in the training session, since immediate-recall accuracy overstates durable mastery [13].

7. Forgetting curves

Ebbinghaus’s classic curve: steep early loss (~42% forgotten within 20 minutes, ~67% within 24 hours) then a long flattening tail [16]. Ebbinghaus modeled this as roughly exponential, but modern consensus — and the reason FSRS replaced its own exponential model with a power-law curve in FSRS-4.5/6 — is that real forgetting decelerates faster than pure exponential decay predicts [16][10].

8. Procedural vs. conceptual math skill

Rohrer/Taylor’s work targets procedural skill: which method applies to which problem. Interleaving’s benefit is theorized to come from discrimination practice — noticing which strategy a problem calls for, something blocked practice never requires since the block gives the strategy away [1][2][5]. For conceptual understanding, spacing and retrieval practice still help via the same trace-strengthening mechanism, but interleaving adds transfer value — recognizing a concept’s applicability in a novel, mixed context [6][7][8]. In short: procedural fluency wants spaced and interleaved retrieval; conceptual understanding wants spaced retrieval and gains further from interleaving once multiple concepts are active.

Design implications for Math Challenge

1. Schedule per skill node, not per question. Track units like “2-digit subtraction with borrowing” as the schedulable entity — math skills generalize across many question instances, unlike flashcards.

2. Concrete recommended algorithm: FSRS-lite with a Leitner cold-start. New skills with <20 data points use a simple Leitner-like ladder (1 → 3 → 7 → 16 → 35 days, reset on wrong answer, no fitting required). Once enough attempts accumulate, switch to an FSRS-style model seeded with published FSRS-6 default weights, refit periodically offline. Expose one tunable knob: desired retention = 0.90 default, adjustable per grade band (0.85 for youngest ages to reduce frustration, 0.92+ for older/competitive users).

3. Two-stage mastery threshold. Require 3 consecutive correct at increasing difficulty within a skill as the “provisionally learned” signal (the common fact-fluency convention [13]), but don’t mark a skill “mastered” for scheduling until it also survives one correct spaced review at a ≥3-day gap — directly encoding the testing-effect lesson that immediate-recall streaks overstate durable learning.

4. Never block practice by skill once 2+ skills are in rotation. Once a second skill is due for review, interleave it with the current lesson within the same session (ABAB/ABCABC), rather than finishing one skill’s problems before starting the next — the single highest-leverage, zero-cost change the literature supports [1][2][4].

5. Session interleaving ratio: ~40-60% new/current-lesson mixed with ~40-60% due-for-review, drawn from 2-4 other skills, mixed at the question level (not sub-blocks of 3-4 same-type problems). For pre-K/K, bias toward 70/30 new/review and interleave at most 2 skills, given working-memory constraints the desirable-difficulties literature itself flags as a boundary condition [6][7].

6. Reviews are always retrieval, never passive re-exposure. A review event requires the child to produce an answer before any explanation shows, even for “already learned” material [8].

7. Scale review gaps to how long the skill needs to last, not a fixed calendar cadence. Tag skills as “unit-scoped” (tighter gaps, ~20-30% of the retention window) vs. “foundational” (progressively wider gaps, ~5-10% of a year-long horizon once well-established), per Cepeda’s ridgeline [9].

8. Track Difficulty separately from Stability per skill, as FSRS does, so a child who struggles gets both a shorter next interval and smaller stability gains per correct answer than one who found it easy — preventing a fixed-ease algorithm from treating a lucky guess the same as genuine mastery.

9. Model forgetting with a power-law curve, not pure exponential, for placement/adaptive-difficulty estimates of “how much has this child forgotten since last practice” — a pure exponential overstates forgetting at long delays and understates it shortly after learning [16][10].

10. Instrument both Rohrer signatures as internal metrics. Track same-session accuracy and delayed-recall accuracy (e.g., a short warm-up quiz on yesterday’s skills) as separate KPIs; expect interleaved-session accuracy to sometimes look lower than blocked while delayed accuracy is higher — don’t let a same-session accuracy dip trigger reverting to blocking.

11. Report “practiced” vs. “learned” separately to parents/teachers. Surface the two-stage mastery signal (item 3) rather than raw session accuracy, avoiding the trap where a same-day streak looks like mastery and then fails on the next unannounced review.

12. AI tutor feedback should prompt retrieval before revealing solutions. On a wrong answer, give a retrieval-scaffolded hint first (generation effect [6][7]); reserve full worked examples for a second wrong attempt.

Open questions for the project owner

1. Should scheduling state live per-child-per-skill only, or should we also maintain a population-level FSRS parameter fit to seed new children’s schedules before enough of their own data exists?
2. Should desired retention be a fixed 0.90 platform-wide, or a tunable knob for older/PhD-track users the way Anki exposes it to power users?
3. Should unit-scoped vs. foundational-skill tagging be authored manually per curriculum node, or inferred from prerequisite-graph depth?
4. Does the interleaving ratio interact with the anti-cheating behavioral-signal system — does mixing skill types make timing/pattern detection easier or harder to reason about?
5. Should the two-stage mastery bar (streak + delayed review) block progression to the next curriculum unit, or only affect review scheduling while progression stays on a separate accuracy threshold?