Development here follows Gagné-like honesty: aptitudes, practice and catalysts. Mathematical talent is not a finished essence.
Windows differ by domain. A closed sport window is not a closed writing window. Read the clocks.
Stages
- Play with quantity
Games, not drills only.
Number sense.
- Lift the ceiling
Next year’s sheet.
SMPY spirit.
- Write reasons
Not only answers.
Proof begins.
- Find a peer or mentor
A circle, camp, or teacher.
Catalyst.
- Pick a flavour
Contest, modelling, proof, teaching.
Not one maths.
- Review load
Camps can harvest rather than teach.
Ethics.
Levers
- Above-level tasks — Without them, measurement is fiction.
- Proof talk — Language for being stuck.
- Peer density — IMO cultures are social.
- Spatial cross-training — For some topics.
- Who hears about CTY — The equity lever.
Windows
- Early number sense — Preschool magnitude play.
- Middle-school search — SMPY’s classic ages.
- Specialist secondary — 과학고, физмат, maths lyceums.
- Adult conversion — Proof can start late; some contest windows do not.
Supports
- Johns Hopkins CTY / SMPY-style search — Above-level tests plus courses.
- Korea 과학고 and 영재교육 — State science high schools.
- China USTC 少年班 and olympiad schools — Early university and contest pipes.
- Russia/Soviet физмат lineage — Specialist maths-physics schools.
Mathematical talent: catalysts and windows do the daily work. Destiny talk is the failure mode.