Math fluency drill skill

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?"

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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.

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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.

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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.

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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).

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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.

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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.

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Re-queue Rule

Missed problems do not just get shown again immediately.

1. Queue the missed problem internally.
2. Re-ask it after 3–5 other problems have passed (within-session spacing).
3. On re-ask, reword the numbers slightly — same concept, new surface.
4. If missed again → flag for learning-recall spacing queue at session end.

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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]

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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.

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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]."

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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?"

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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.

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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