Math Intuition Builder

Math Intuition Builder

When to Use

Trigger on phrases like:

• "help me understand"
• "build intuition for"
• "what does this mean geometrically"
• "why does this work"
• "visualize this concept"
• "concrete example of"
• "what's the intuition behind"

Use before computation to establish understanding (Polya's "understand the problem" phase).

Process

Guide through Mason's specializing-generalizing cycle:

1. Restate in own words

Ask: "Can you state the problem in your own words?"

• Forces re-processing
• Catches misunderstandings early
• Verifies shared understanding

2. Try concrete examples

Ask: "What would a concrete example look like?"

• Specialize: try N=3, x=2, simple case
• Ground abstract concepts in specifics
• Use computation tools to verify examples

3. Anticipate the answer

Ask: "What form should the answer take?"

• Work backwards from expected result
• Constrain solution space
• Check dimensional consistency

4. Visualize

Ask: "Can you draw a picture or diagram?"

• Multiple representations matter
• Algebraic <-> geometric <-> numeric
• ASCII diagrams for quick sketches

5. Simplify

Ask: "What's a simpler version of this problem?"

• Reduce dimension
• Remove terms
• Special cases reveal structure

6. Find analogies

Ask: "Does this remind you of anything you've solved before?"

• Pattern matching across domains
• Transfer solution strategies

Key Questions to Ask

• What are you trying to find? (restate)
• What happens with small numbers? (specialize)
• What shape should the answer be? (anticipate)
• What does this look like visually? (represent)
• What's the simplest version? (reduce)
• What's similar? (analogize)

Output Format

Understanding Checkpoint:

Restated Problem:
[User's words, verified]

Concrete Instance:
[Specific numerical example]

Visual Interpretation:
[Description or ASCII diagram]

Simplified Version:
[Special case that's easier]

Analogous Problems:
[Similar structures from other domains]

Intuition Summary:
[1-2 sentence "aha" insight]

Ready to Proceed:
- Next step: [specific action]
- Verify by: [sanity check]

Example Interaction

User: I'm trying to understand why eigenvalues matter for stability

Claude: Let's build intuition step by step.

Q1: Restate the problem?
> I have x' = Ax and want to know if solutions grow or shrink.

Q2: Let's try a concrete example.
A = [[0, 1], [-1, 0]]

Computing eigenvalues: +/- i (purely imaginary)

What happens to solutions?
> They neither grow nor shrink?

Exactly. Solutions are circles - oscillation without growth.

Q3: Now try A = [[-1, 0], [0, -1]]
Eigenvalues: -1, -1 (negative real)

What happens?
> Shrinks to zero.

Yes. Negative real parts = decay.

Q4: Visual pattern:

  Im
   |    x unstable
   |
---+------- Re
   |
   |    x stable

Re(lambda) < 0: stable (decay)
Re(lambda) = 0: marginal (oscillate)
Re(lambda) > 0: unstable (blow up)

Q5: Analogies:
- Springs: eigenvalues = decay rates
- Populations: growth/death rates
- Circuits: time constants

Intuition Summary:
Eigenvalues are the system's "DNA" - they encode fundamental rates.
Look at real parts for stability.

Ready to Proceed:
- Next: Analyze your specific A matrix
- Verify: trace(A) < 0 and det(A) > 0 implies stability (2x2 case)

Co-activates

Suggest alongside:

• math-model-selector - if framework unclear
• math-mode - for concrete computations
• math-progress-monitor - during problem solving
• Relevant domain skill - once intuition established