AgentSkillsCN

math-by-reference-tutor

生成完整的数学学习模块，以物理和/或金融为根基：包含讲座、分层练习、基于常见误区的巧思问答、附带 pytest 测试的 Python 实现脚手架，以及精简版的解答手册。当用户说“教我[数学主题]”、“制作讲座+练习”、“给我一些练习题”、“逐步提升难度”、“出一些巧思题目”、“添加代码脚手架+测试”、“制作解答手册”时，尤其是当他们希望在正式数学学习的同时，获得物理或金融领域的直观理解时，此技能便能派上用场。

SKILL.md
--- frontmatter
name: math-by-reference-tutor
description: Generates a complete maths learning module anchored in physics and/or finance: lecture, tiered exercises, misconception-driven trick questions, Python implementation scaffold with pytest tests, and a compact solution manual. Use when user says "teach me [math topic]", "make a lecture + exercises", "give me practice problems", "ramp up difficulty", "give trick questions", "add coding scaffold + tests", or "make a solution manual", especially when they want physics/finance intuition alongside formal maths.

Math-by-Reference Tutor (Physics + Finance)

Inputs (defaults)

  • •topic: required
  • •level: intro | intermediate | advanced (default intermediate)
  • •track: physics | finance | both (default both)
  • •num_exercises_per_tier: default 6
  • •num_trick_questions: default 6
  • •deps: stdlib + numpy only (unless user explicitly allows more)

If topic is ambiguous, choose the most standard interpretation and note it in README.md as "Assumption: …".

Module bundle

Output these files using === /module/<path> === headers. No code fences around markdown; use code fences for Python files.

code
/module/README.md
/module/lecture.md
/module/exercises.md
/module/trick_questions.md
/module/solution_manual.md
/module/rubric.md
/module/implementation/scaffold.py
/module/implementation/tests/test_scaffold.py

Anchoring (R1)

Every core idea must be anchored in the selected track(s) using concrete applications, not analogies.

Physics anchors — use at least one of: dimensional analysis, limiting cases, conservation/invariants, geometric meaning, differential equation interpretation.

Finance anchors — use at least one of: no-arbitrage/replication, convexity/monotonicity, conditioning/filtration, tails/stress, optimisation constraints.

Lecture (R2)

Structure: motivation → core definitions → two key properties/theorems (with proof sketches appropriate to level) → worked example(s) per track → failure modes + invariant checks.

Keep concise (~900 words max).

Exercises (R3)

Three tiers (A/B/C) with monotonically increasing difficulty. Each exercise states expected answer format.

Exercise types to mix across tiers: compute, prove/justify, translate (physics↔finance), diagnose (spot errors), approximate (limits/asymptotics), numerics (implement + compare), interpret (domain meaning).

Trick questions (R4)

Each includes the question, "Most people do wrong:" (1 sentence), and "Minimal correction:" (1 sentence). No duplicates; each maps to a distinct failure mode.

Implementation + tests (R5)

scaffold.py: core functions/classes with TODO markers + runnable __main__ demo.

test_scaffold.py: tests that fail before TODOs are completed. Include identity/invariant tests, numerical sanity checks, limit-case tests, and one random property-style test (stdlib random + numpy only, no external property libs). Use pytest.

Solution manual (R6)

Compact (~1200 words max). Include compressed derivations, reusable templates ("if you see X, do Y"), a verification checklist, and a "when to use what" decision guide.

Rubric (R7)

Mastery gates per tier: A = mechanical competence, B = conceptual transfer, C = synthesis/novel problems.

Workflow

  1. •Parse inputs and set defaults.
  2. •Draft prerequisites and scope boundary.
  3. •Build minimal concept order (internal).
  4. •Write lecture.md.
  5. •Write exercises.md.
  6. •Write trick_questions.md.
  7. •Write scaffold.py with TODOs + demo.
  8. •Write test_scaffold.py with fail-first behaviour.
  9. •Write solution_manual.md + rubric.md.
  10. •Emit bundle with path headers.

Complexity bump (when user asks "harder")

Adjust along 1–2 axes only and state them in README.md:

  • •algebraic complexity
  • •abstraction/proof depth
  • •numerical stability
  • •modelling realism (physics: non-idealities; finance: tails/constraints)
  • •generalisation (dimension, nonlinearity)

Regenerate only affected files.

Common issues

Topic too broad: pick a standard slice, state it in assumptions, list "next slices."

Prerequisites too advanced for level: add a prereq mini-bridge (≤200 words) at the start of lecture.md.

Tests too confusing: ensure tests fail only at TODO-related points with clear messages indicating what to implement next.

Validation script

Validate a generated module bundle: python scripts/validate_bundle.py