math-progress-monitor skill

Math Progress Monitor

When to Use

Trigger on phrases like:

• "am I on the right track"
• "is this approach working"
• "I'm stuck"
• "should I try something else"
• "verify my progress"
• "check my reasoning"
• "is this getting too complicated"

Use mid-work to assess whether to continue, pivot, or decompose (Schoenfeld's metacognitive control).

Process

Run a structured progress assessment:

1. Inventory attempts

Ask: "What have you tried so far?"

• List each approach
• Order by when attempted
• Note time spent

2. Extract learnings

Ask: "What did each attempt tell you?"

• Even failures provide information
• What was ruled out?
• What patterns emerged?

3. Complexity check

Ask: "Is complexity growing faster than expected?"

• Warning signs:
• More terms than you started with
• New variables appearing
• Calculation getting messier
• Normal: complexity stays flat or decreases

4. Spot-check verification

Ask: "Can you verify any intermediate results?"

• Run numerical examples
• Check limiting cases
• Dimensional analysis

5. Decomposition check

Ask: "Is there a simpler sub-problem to solve first?"

• Identify dependencies
• Find stepping stones
• Reduce scope

6. Sunk cost assessment

Ask: "What would abandoning this approach cost?"

• Time already spent (irrelevant)
• Insights that transfer (relevant)
• Probability of success if continue

Key Questions to Ask

1. What have you tried? (inventory)
2. What did you learn? (extract)
3. Is it getting simpler or harder? (complexity)
4. Can you check a piece? (verify)
5. What's the smallest step? (decompose)
6. Should you pivot? (decide)

Output Format

Progress Assessment:

Status: [On track / Hitting resistance / Should pivot]

Attempts Summary:
1. [Approach] -> [Outcome/Learning]
2. [Approach] -> [Outcome/Learning]

Complexity Check:
- Expected: [what should this look like]
- Actual: [what it actually looks like]
- Verdict: [Normal / Warning / Abandon]

Verification Points:
- [Intermediate result to check]
- [Numerical test case]

Recommendation:
[Continue / Modify approach / Switch strategy / Decompose / Get more info]

If Pivoting:
- Alternative: [specific new approach]
- Why it might work: [reasoning]

Example Interaction

User: I've been trying to evaluate this integral for an hour

Claude: Let's do a progress check.

Q1: What have you tried?
> u-substitution with u = sin(x), then integration by parts twice.

Q2: What did each tell you?
> Substitution made it longer. Parts created a cycle.

Q3: Complexity now vs start?
> Much worse. Started with 3 terms, now have 8.

Complexity Check:
- Expected: Should simplify or stay similar
- Actual: 3 terms -> 8 terms after 2 techniques
- Verdict: WARNING - likely wrong approach

Let me verify an intermediate step...

[Runs sympy on original integral]

This integral has closed form using:
- Beta function (special function)
- Or contour integration

Recommendation: PIVOT

Your algebraic approaches won't simplify this pattern.

Alternative Approach:
1. Recognize: integral of sin^m(x) * cos^n(x)
2. Use Beta function identity
3. Or convert to complex exponentials

Related Skills:
- complex-analysis/contour-integrals
- numerical-methods/numerical-integration

Would you like the Beta function approach, or get a numerical answer first to verify?

Co-activates

When user is stuck, suggest:

• math-intuition-builder - reset understanding
• math-model-selector - if framework was wrong
• math-mode - for verification computations
• Relevant domain skill - for alternative approach