You want what actually works for learning hard things—methods with causal evidence, not folklore. Here’s the short list that consistently moves the needle, with RCTs/meta‑analyses behind it and notes on where each shines (and where it backfires).
- Retrieval practice (“testing effect”), ideally with feedback
What it is: Actively recall (without notes), get the answer, then check/correct.
Evidence: Big, durable gains across hundreds of experiments and classroom trials; practice tests outperform restudy and other controls. Effects are larger when feedback is provided and when retrieval is spaced. Meta‑analyses with college‑level materials included. Use for: all levels; concepts, proofs, derivations, formulas. SAGE Journalseducation.wsu.edu
Upgrade: Successive relearning. Practice to correct retrieval in a session, then re‑achieve correct retrieval in later, spaced sessions—combines retrieval + spacing and beats either alone in college classes. Use for: core definitions, theorems, canonical derivations, algorithm steps. SAGE JournalsOvid - Spacing with calibrated intervals (not cramming)
What it is: Distribute study of the same material over time.
Evidence: Large synthesis + a long‑horizon RCT: the optimal first review ≈ 10–20% of the time until your target exam/usage, with the fraction shrinking as horizons get longer (e.g., 3–7 days before a 30‑day exam; weeks before multi‑month targets). This beats both massed study and poorly timed spacing. Use for: everything you want to retain beyond next week. Learning Attention and Perception LabAugmenting Cognition - Interleaving problem types (not blocking)
What it is: Mix problem types (e.g., eigenvalue problems, perturbation, variational methods; or ML optimization, generalization, generative modeling) rather than doing long runs of the same kind.
Evidence: Multilevel meta‑analysis shows a moderate overall benefit; strongest for discriminating similar categories; small‑to‑moderate for mathematics. Interleaving helps learners identify the deep feature that dictates the method. Use for: problem sets where the solver must choose the right tool. psychologie.uni-wuerzburg.de - Worked examples → example‑problem pairs → fading
What it is: Study fully worked solutions with self‑explanations, then do a similar problem yourself; gradually remove steps (“fading”).
Evidence: Meta‑analyses show clear gains in mathematics and other structured domains, especially for novices; adding self‑explanation prompts beats examples alone. Use for: heavy‑symbolic or procedural topics (tensor derivations, backprop variants, KKT conditions). Dana Miller-Cotto, PhDScienceDirectmrbartonmaths.com - Self‑explanation prompts (explain each step to yourself)
What it is: While reading examples or proofs, generate “why” and “how” explanations in your own words (not summaries).
Evidence: Meta‑analyses (math and beyond) show small‑to‑moderate, reliable gains, especially for procedural + conceptual understanding; quality of explanations matters. Use for: proofs, derivations, algorithm internals. ERICGwern - Active learning (do things during study, not just watch/read)
What it is: Frequent checks for understanding, clicker‑style questions, group problem‑solving, short writing, etc.
Evidence: Massive STEM meta‑analysis: exam scores up and failure rates down (21.8% → 33.8% under lecture). Caveat: it can feel worse; RCT shows students think they learn less while actually learning more. Use for: any course or self‑study session—engineer activities every 8–12 minutes. PubMedPNAS+1 - Pretesting / errorful generation (then feedback)
What it is: Try to answer before instruction/study; you will be wrong, then get correction.
Evidence: Review + multi‑experiment papers: pretests boost later learning even with many initial errors—provided corrective feedback follows. Use for: new chapters/papers; before lectures; before reading proofs. SpringerLinkSC PanPMC - Analogical comparison / contrasting cases
What it is: Compare 2–3 solved problems that share deep structure but differ on surface details; explicitly list what’s invariant.
Evidence: Meta‑analysis: case comparisons improve learning and transfer; “time‑for‑telling” studies show problem‑solving/contrasting cases before instruction can prime deeper understanding (with the right guidance). Use for: mapping Schrödinger ↔ diffusion analogies; maximum‑entropy ↔ regularization; different variational formulations. lrdc.pitt.eduAAALab - Generative visuals (drawing / multiple representations)
What it is: Sketch systems, diagrams, data flows, state spaces, energy landscapes; translate among equations, graphs, code.
Evidence: Syntheses show drawing‑to‑learn yields sizable gains when guided; broad reviews highlight benefits for sense‑making. Use for: circuit diagrams ↔ Hamiltonians; model graphs ↔ loss surfaces. SpringerLinkSAGE Journals - Self‑regulated learning (SRL) training (planning, monitoring, error‑logging)
What it is: Short, explicit training in setting goals, planning spaced/retrieval cycles, monitoring calibration, and analyzing errors.
Evidence: Meta‑analyses in higher ed (including online/blended) show moderate gains in achievement and in SRL behaviors themselves. Use for: keeping a complex study plan on the rails. ScienceDirectTaylor & Francis Online
- Expertise reversal effect: Heavy guidance (worked steps, dense scaffolds) helps novices, but the same scaffolds can hinder intermediates/experts—they induce redundancy and extra cognitive load. Fade scaffolds as mastery rises; increase open‑ended problem solving. Taylor & Francis OnlineScienceDirect
- Feedback isn’t one-size-fits-all: Formative, task‑focused feedback is generally beneficial; overly generic or ego‑focused feedback can be useless or harmful. Immediate vs delayed timing both work; recent syntheses suggest both can be effective, with slight advantages for delayed in some contexts—what matters is specificity and consistency plus corrective information. Andy Matuschakmrbartonmaths.comSpringerLink
- Interleaving scope: Biggest benefits when problems are confusable (similar but require different methods). If items are unrelated, interleaving adds noise; if items are trivially distinct, blocking may suffice. psychologie.uni-wuerzburg.de
- Rereading/highlighting as a primary strategy: lower utility compared to retrieval/spacing. PubMed
- “Learning styles” tailoring (visual/auditory/etc.) has no causal support for improving outcomes. Use the representation that fits the content (e.g., visuals for spatial/structural info), not a self‑label. PubMed
- Speed reading claims (triple speed, same comprehension) don’t survive scrutiny; there’s a speed–accuracy tradeoff. Skim for triage; read normally for mastery. faculty.cas.usf.edu
Weekly cadence (repeat):
- Pretest (10–15 min). Before a new unit/paper, answer 5–8 concept questions cold; commit to guesses; then reveal answers. Log misconceptions. (Pretesting + feedback.) SpringerLink
- Example study with self‑explanations (30–45 min).
- QM: Work through 1–2 fully worked examples (e.g., bound state in a finite well; time‑independent perturbation) and write a one‑sentence “why” for each step.
- ML: Do the same for a derivation (e.g., Adam’s update, ELBO derivation); explain each algebraic move and assumption. (Worked examples + self‑explanation.) Dana Miller-Cotto, PhDERIC
- Example‑problem pairs (30–60 min). Immediately solve a near‑isomorphic problem without the worked steps; check against solution; annotate your error log. (Fading support.) Dana Miller-Cotto, PhD
- Interleave (30–60 min). Mix 3–5 problems spanning confusable types:
- QM: boundary‑value vs. scattering vs. perturbative;
- ML: convex vs. non‑convex optimization, classification vs. regression variants, generative vs. discriminative. (Interleaving.) psychologie.uni-wuerzburg.de
- Retrieval sessions (10–20 min), 3–4×/week.
- Build cards or a checklist that require output: state the theorem/definition, re‑derive a key step, name assumptions, sketch a diagram.
- Successive relearning rule: practice to correct today; re‑achieve correct after a gap; retire only after two spaced, correct retrievals. (Retrieval + successive relearning.) SAGE Journals
- Spacing math: If your exam/goal is 30 days away, first review at ~3–6 days; if 180 days, first review at ~1–3 weeks; lengthen thereafter. (Ridgeline result: optimal gap is ~10–20% of the retention interval, shrinking with longer horizons.) Learning Attention and Perception Lab
- Analogical comparison (15–20 min). For each topic, compare two solved cases and explicitly list: deep structure, key invariants, and what changes.
- QM: harmonic oscillator ↔ small‑oscillation classical system;
- ML: L2‑regularized logistic regression ↔ MAP with Gaussian prior. (Analogical learning.) lrdc.pitt.edu
- Draw it (10–15 min). Sketch potentials, wavefunctions, loss surfaces, computational graphs, or causal DAGs; label. (Generative drawing.) SAGE Journals
- Active learning checkpoint (5–10 min). Micro‑quizzes, clicker‑style prompts, or quick proofs/derivations—no notes. (Active learning.) PubMed
Novice → intermediate → advanced transitions (avoid the flip):
- Start heavy on worked examples; fade steps weekly.
- By mid‑course, shift to more open problems and fewer scaffolds; if performance collapses, re‑insert minimal scaffolds. (Expertise reversal effect.) Taylor & Francis Online
- Build a retrieval deck with derivation prompts, not trivia. Example (ML): “Re‑derive Adam’s bias‑corrected updates; state the exact moments corrected and why.” Example (QM): “Derive ⟨x⟩ and ⟨p⟩ time‑evolution for a Gaussian wavepacket; state approximations.” (Retrieval.) SAGE Journals
- Interleave by decision, not topic. Make sets where the first move is to choose the method. If that choice is obvious, your set is poorly designed. (Interleaving.) psychologie.uni-wuerzburg.de
- Error log > solution manual. After each session, write why your error made sense at the time and the cue that would have prevented it; revisit these during spaced reviews. (SRL + feedback best practices.) ScienceDirectAndy Matuschak
- Compare solutions you’d confuse. Put two derivations side‑by‑side and mark the invariant principle; force yourself to articulate the discriminant. (Analogical comparison.) lrdc.pitt.edu
- “It feels worse, so it’s not working.” Active methods feel harder yet learn more. Trust delayed performance on criterion tasks, not vibes. Track weekly no‑notes quizzes. PNAS
- “I’ll just keep reading until it clicks.” That’s fluency illusion. Replace with retrieve→check→space cycles. PubMed
- “More hints can’t hurt.” Past a point and with rising expertise, scaffolds can backfire. Fade them. Taylor & Francis Online
- Practice testing / retrieval: medium–large; classroom meta shows robust advantages vs. restudy. Confidence: high. SAGE Journals
- Spacing (with calibrated gaps): medium–large for long‑term retention. Confidence: high. Learning Attention and Perception Lab
- Worked examples (+ self‑explanations): small–to–moderate (bigger for novices). Confidence: high. Dana Miller-Cotto, PhD
- Interleaving: small–to–moderate overall; larger in confusable categories. Confidence: medium‑high. psychologie.uni-wuerzburg.de
- Active learning in STEM: failure rates down markedly; exam gains. Confidence: high. PubMed
- Pretesting: small–to–moderate, reliably positive with feedback. Confidence: medium‑high. SpringerLink
- Generative drawing / multiple representations: small–to–moderate with guidance. Confidence: medium. SAGE Journals
- SRL training: moderate in higher ed/online. Confidence: medium‑high. ScienceDirect
- Take the pretest for your next topic (10 min). SpringerLink
- Work one fully worked example with self‑explanations (20–30 min). ERIC
- Do an example‑problem pair (20–30 min). Dana Miller-Cotto, PhD
- Build 8 retrieval prompts from that material; schedule reviews at ~10–20% of your target horizon. Learning Attention and Perception Lab
- Tomorrow: mix that problem type with two similar, confusable types. psychologie.uni-wuerzburg.de
If you want me to turn this into a one‑page, fill‑in‑the‑blanks template for your current topic (QM or ML), say so and I’ll draft it. Otherwise, go do the unglamorous thing that actually works: retrieve, space, interleave, compare, and explain—then repeat.
—Jeeves
(No‑nonsense footnotes):
Active learning meta (failure rates, exam gains): Freeman et al., 2014; perception vs learning: Deslauriers et al., 2019. PubMedPNAS
Retrieval practice meta: Adesope et al., 2017. SAGE Journals
Spacing “ridgeline” (10–20% rule of thumb): Cepeda et al., 2008. Learning Attention and Perception Lab
Worked examples meta (math; with self‑explanation): Barbieri et al., 2023; Renkl et al., 2003. Dana Miller-Cotto, PhDmrbartonmaths.com
Interleaving meta: Brunmair & Richter, 2019. psychologie.uni-wuerzburg.de
Pretesting review: Pan, 2023; experiments: Pan & Sana, 2021. SpringerLinkSC Pan
Analogical comparison meta: Alfieri et al., 2013. lrdc.pitt.edu
Drawing/generative visuals: Ainsworth, 2021; Cromley et al., 2020. SAGE JournalsSpringerLink
SRL meta: Theobald, 2021; Xu et al., 2023. ScienceDirectTaylor & Francis Online
Expertise reversal: Kalyuga, 2003; meta‑analysis 2025. Taylor & Francis OnlineScienceDirect
Feedback syntheses: Shute, 2008; Kluger & DeNisi, 1996; recent timing evidence. Andy Matuschakmrbartonmaths.comSpringerLink
Low-utility + myths: Dunlosky et al., 2013; Pashler et al., 2008; Rayner et al., 2016. PubMed+1faculty.cas.usf.edu