Teaching children mathematics

The program (what a school should actually do)

Math is the domain where the general learning-science findings (explicit instruction for novices, worked examples, retrieval and spacing) have their best subject-specific causal evidence — and where a central extra fact emerges: math sub-skills are modular. Fluency doesn't create problem-solving; problem-solving doesn't create computation; neither creates algebra readiness. A math program is a set of deliberately taught strands, not one skill that radiates.

Core instruction (all ages).

• Explicit teaching as the default: model → worked examples → guided practice → extensive independent practice → feedback. Non-negotiable for new content and struggling students (the most replicated finding in math intervention research, three federal review cycles). (evidence: explicit vs reform math)
• Never adopt a minimal-guidance constructivist curriculum — the one stable causal loser (0.22 SD behind every rival in the only multi-curriculum RCT). Among mainstream structured texts, differences are now small — choose a coherent practice-heavy one and invest in implementation instead of re-adoption. (evidence: math curricula)
• For older students with solid foundations, add structured problem-first work (productive failure) for conceptual depth — explicit-first for novices, struggle-first for the prepared.

Ages 5–12 — build the foundations deliberately.

• Fact fluency through brief, low-stakes, timed retrieval practice (1–5 min, frequent, on taught facts, with feedback), layered on conceptual instruction. The anti-timed-test claims have no experimental support; keep practice private and low-pressure and the surviving critique is fully addressed. (evidence: fact fluency)
• Manipulatives as guided representation, not play: plain objects, high guidance, deliberate concrete → pictorial → abstract fading; especially fractions. (evidence: manipulatives/CRA)
• Word problems as an explicit strand: teach problem TYPES (schemas) with diagrams, train transfer across surface features on purpose. (evidence: word problems & schema instruction)
• Teach concepts and procedures in alternation — concepts-first dogma has no experimental warrant.

Ages 11–18 — placement by readiness, support by dosage.

• No uniform algebra timing in either direction (acceleration-for-all and delay-for-all both demonstrably fail). Place by objective readiness scores — which also erases demographic placement gaps that subjective gatekeeping creates.
• Accelerate students above the bar; give students below it double-dose algebra (a second scheduled support period at the right level, in place of an elective) — the standout result: +0.18–0.24 SD on independent tests and +3.3pp BA attainment on a 7% base twelve years later. (evidence: algebra timing)

What to refuse to spend money or time on

• Minimal-guidance discovery/constructivist curricula (the RCT loser).
• Dropping timed practice on anxiety grounds — the causal claim is unsupported and the competence→anxiety direction dominates in childhood.
• "Conceptual understanding first, procedures later" as a sequencing rule.
• Rich, toy-like manipulatives and unguided hands-on time — they hurt transfer.
• Algebra-timing mandates (either direction) and subjective placement gatekeeping.
• Chasing textbook rankings among mainstream structured texts (~0–0.05 SD at stake) or trusting WWC curriculum labels at face value.
• Generic "problem-solving skills" training — teach specific schemas; transfer must be trained, not hoped for.

The hereditarian bottom line for a founder

Math achievement is ~55–65% heritable by adolescence, driven by the same "generalist genes" as g — and that makes the causal record more impressive, not less: every effective lever here moves the mean of a heritable trait through domain skills, and none of it moves g. Three lens lessons: (1) the correlational case for early algebra was pure ability selection (able kids take algebra early; forcing others in harms them); (2) objective ability measurement beats both ability-blind mandates and subjective judgment — for equity as well as efficiency; (3) the double-dose result shows the ceiling on fatalism: instruction that never touched g still moved graduation and BA attainment by double digits relative to base rates. Test-score fadeout is the norm (early-math effects fade >60% in a year, mostly reverting to pre-existing differences), so durable value comes from cumulative curriculum, maintained fluency, and credential-relevant course progression — not one-shot boosts.

Confidence

Open questions a founder should watch

• Durability: almost nothing in math instruction has ≥1-year experimental follow-up except the algebra-policy literature; assume fadeout and build maintenance in.
• Whether double-dose replicates outside Chicago (half of large districts adopted it; none have causal evaluations).
• Whether schema instruction moves independent standardized scores (never yet demonstrated).