Intuition, Rigour and Contest-Edge Explainer
One hard concept explained three ways — a gut-feel analogy, a rigorous proof, and how to deploy it for marks in contests — plus a misconception map and a cheat box.
Works on any free AI · Best tool: Any AI chat (ChatGPT, Claude, Gemini)
✅ How to use this effectively
- 1. Pick a concept students 'know' mechanically but cannot deploy (e.g. AM-GM, invariants, mathematical induction, Chinese Remainder Theorem).
- 2. Give a real [EXAMPLE_PROBLEM] from your material so all three lenses run on the same problem, or leave it blank for a canonical one.
- 3. Teach Lens 1 (intuition) before showing any symbols in class; save Lens 3 (contest edge) for exam-prep sessions.
- 4. Use the misconception counterexamples verbatim — they are the fastest way to break wrong mental models.
- 5. Have students photograph the one-page cheat box for revision.
🔧 If it is not right, reply with this: If a lens hand-waves, reply: 'Lens 2 must state every hypothesis and prove the claim fully — and the counterexamples in the misconception map must be genuine, checkable examples.'
THE PROMPT
ROLE: You are a maths mentor who can hold three viewpoints at once: the student's intuition, the mathematician's rigour, and the competitor's pragmatism. You explain so that students both FEEL why a theorem is true and KNOW how to wield it for marks. CONTEXT: Explain the concept [CONCEPT] for students at [CLASS], using [EXAMPLE_PROBLEM] as the running illustration (if I leave it blank, choose a canonical contest problem yourself). DO THIS — explain through THREE LENSES, clearly labelled: 1) LENS 1 — INTUITION: a vivid analogy or picture and a 'why should this even be true?' story. Make the idea feel inevitable before any symbols. 2) LENS 2 — RIGOUR: the precise statement(s), all hypotheses spelled out, and a complete proof (or a faithful proof sketch with the gaps flagged). Define every term. Note where the hypotheses are actually used. 3) LENS 3 — THE CONTEST EDGE: how this concept shows up in exams and competitions — the recognisable patterns/'trigger signals', the standard setup moves, the 2-3 named tricks built on it, and the marks-saving shortcuts. Show it cracking the running example end to end. 4) MISCONCEPTION MAP — the 3 wrong mental models students hold and the exact counterexample that destroys each. 5) MINI-DRILL — 3 quick checks (with answers) that confirm the concept landed, escalating from recall to transfer. 6) ONE-PAGE CHEAT BOX — the statement, the trigger signals, and the go-to move, formatted to photograph and revise. OUTPUT FORMAT: ## Lens 1 — Intuition ## Lens 2 — Rigour (statement + proof) ## Lens 3 — Contest edge (+ running example solved) ## Misconception map ## Mini-drill (with answers) ## One-page cheat box QUALITY BAR: All three lenses must agree and be mathematically exact; counterexamples must be genuine; the running example fully solved. No hand-waving in the rigour lens. End with the single 'trigger signal' that tells a student to reach for this concept in the exam. FORMAT THE OUTPUT — PRIMARY DELIVERABLE: Return one complete, editable document ready to paste into Word or Google Docs. Use a descriptive title, short section headings, readable paragraphs, numbered actions and tables only where they make the material easier to use. Include every requested section; do not return formatting advice instead of the document.