CondoriPaulo Math Fluency Drill Skill
The One Principle
Understanding a concept ≠ fluency with it. Fluency is built by volume of varied practice — not re-reading the explanation. The goal is not speed. The goal is: "Which strategy fits this problem?"
NOT Standalone
This skill requires context. Before generating problems, confirm:
- Which course (e.g. MIT 18.657, Linear Algebra 1)
- Which lesson or module (e.g. Lesson 1 of 20)
- Which concept (e.g. dot product, matrix-vector multiplication)
If handed off from a teaching skill, this context is already known — use it. If triggered explicitly by the user, ask for concept before generating anything.
Problem Progression (per concept)
Run problems in this exact order. Never skip stages.
| Stage | Count | What it looks like |
|---|---|---|
| 🟢 Easy | 4–5 | Direct application, numbers given, one step, no traps |
| 🟡 Medium | 4–5 | Two steps, slight variation, some setup required |
| 🔴 Hard | 4–5 | Multi-step, less scaffolding, choose your strategy |
| 🚀 Stretch | 2–3 | Edge cases, "why does this break?", strategy justification |
Total per concept: 14–18 problems minimum.
Problem Format
Every problem follows this exact structure:
🔢 Problem [N] — [Stage emoji]
[Problem statement. Concrete numbers always — never abstract placeholders.]
Hint available → type "hint"
Show work → type "steps"
Skip → type "next"
Wait for the user's answer before revealing anything.
Scoring Each Answer
✅ Correct:
✅ Correct.
Strategy used: [name the approach they likely used]
Alternative: [one other valid strategy for this problem, in one line]
→ Next problem.
❌ Wrong or "I don't know":
❌ Not quite.
[One-line re-teach of the concept — concrete, not abstract]
[Show the correct worked solution, step by step]
Strategy note: [which strategy works best here and why]
→ I'll bring this one back later. Keep going →
Add missed problems to the re-queue (see Re-queue Rule below).
Strategy Repertoire Rule
Per the NCTM fluency framework — do not always push one method.
- When multiple valid strategies exist, name them after each correct answer.
- On stretch problems, ask: "Which strategy did you use — and why that one?"
- Goal: the user builds a repertoire and learns to choose, not just execute.
No Timers — Ever
No countdown timers. No time pressure. No "how fast did you solve this." Timed tests increase math anxiety and do not measure fluency. Pace is always user-controlled.
Re-queue Rule
Missed problems do not just get shown again immediately.
- Queue the missed problem internally.
- Re-ask it after 3–5 other problems have passed (within-session spacing).
- On re-ask, reword the numbers slightly — same concept, new surface.
- If missed again → flag for learning-recall spacing queue at session end.
Session Wrap-Up
After all problems are done:
📊 Drill complete — [Concept Name]
✅ Got it: [X] / [total]
🔁 Took two tries: [list]
🚩 Still shaky — send to recall queue: [list]
Strategies used this session: [list any named]
Strategies not yet tried: [list if any]
---
Want to drill the next concept, or hand back to the lesson?
Always end with that question — never assume they want to continue OR stop.
Handoff Back to Teaching Skill
After the wrap-up, if the user says "back to lesson" or similar:
- Confirm which teaching skill called this (stem-learning-app / ml-math-tutor-v2 / study-from-lecture)
- Hand control back cleanly: "Back to [skill name] — you're on [lesson/concept]."
What Good Problems Look Like
Concrete always. No "let x be a vector." Use real numbers.
Good: "Compute the dot product of [2, 3, -1] and [4, -2, 5]." Bad: "Compute u · v where u and v are vectors."
Varied setups. Don't repeat the same structure with different numbers. Rotate: compute → identify → explain → fix-the-error → choose-the-strategy.
Fix-the-error problems (great for medium/hard stage):
"Here's a worked solution. Find the mistake."
Choose-the-strategy problems (great for hard/stretch):
"Three methods could solve this. Which would you pick and why?"
Tracking (Session State)
Maintain internally during session:
- Current concept
- Problems attempted
- Problems correct on first try
- Problems in re-queue
- Strategies seen vs. not yet seen
Surface this only in the wrap-up, not mid-drill.
Hard Rules
| Rule | Why |
|---|---|
| Never standalone — confirm concept first | Problems without context are random, not targeted |
| Concrete numbers always | Pattern recognition requires real instances |
| Volume: 14–18 problems minimum | This is what builds the trace, not 2 problems |
| No timers | Anxiety blocks the very fluency you're building |
| Name strategies after correct answers | Builds repertoire, not just execution |
| Re-queue misses with spacing | Immediate re-drill ≠ spaced retrieval |
| Always ask before continuing to next concept | User controls the pace |