Show the Path · Sitting 3
The Model Is a Hostile Reviewer
The best use of a model in mathematics is as a hostile reviewer: it attacks your path and finds the weak steps. You fix them. The path gets stronger.
- 40 min
- Practice · Reflect
- 15–18
Parent briefing · 5 minutes, before they sit
This sitting teaches the student to use the model as a hostile reviewer. Not a friendly helper. A hostile reviewer. The student walks the path. Then asks the model: 'Attack my proof. Find the weakest step. Tell me where it breaks.' The model finds the weak points. The student fixes them. This is the professional workflow: you write, a reviewer attacks, you fix. The model is a tireless reviewer. The student is the owner of the path. The model does not fix. It finds. The student fixes.
Hard edges
- Academic integrity: the model reviews. The student writes. The final proof is the student's, with the model cited as a reviewer if it changed a step.
- Do not let the model rewrite the proof. It finds weaknesses. It does not rewrite. The student rewrites.
If they say
- “The model said my proof was good.”
- Ask it to be hostile. 'Find the weakest step.' A model that says 'good' is being friendly, not useful. A model that finds the weak step is being a reviewer. Demand the review, not the compliment.
- “I'll just let the model fix it.”
- The model finds the weakness. You fix it. If the model fixes it, the fix is the model's, not yours. The proof is only yours if you wrote every fix. The model is the reviewer. You are the author.
Objective
The student can use a model as a hostile reviewer of their mathematical work, ask it to find the weakest step, and fix the weaknesses independently.
The hostile review
In professional mathematics, proofs are reviewed. The reviewer does not rewrite the proof. They attack it. They find the weak step, the unjustified claim, the missing assumption. The author fixes it. The proof gets stronger. The model can be that reviewer. You walk the path. You show the model each step. You ask: 'Attack this. Where is the weakest step? Where would a skeptic push?' The model finds the weak points. You fix them. The model does not rewrite. You rewrite. The model is the reviewer. You are the author. That is the right relationship.
Why hostile, not friendly
A friendly model says 'your proof looks good.' That is useless. A hostile model says 'step 3 assumes something you did not prove. Step 5 jumps without justification.' That is useful. The student who asks for a friendly review gets nothing. The student who asks for a hostile review gets the weak points. Ask the model to be hostile: 'Find the weakest step. Tell me where a skeptic would push.' The model is better at being hostile than at being right. Use that. The hostile review is the value. The friendly review is flattery.
Big idea
Use the model as a hostile reviewer. It attacks. You fix. The path gets stronger. The model does not rewrite. You do.
Try this~22 min total
Hostile review
22 min- Walk a proof on a problem at your level. Write each step.
- Show the model. Ask: 'Attack my proof. Find the weakest step. Where would a skeptic push?'
- Write each weakness the model finds. Do not let it rewrite. Ask for the weakness, not the fix.
- Fix each weakness yourself. Rewrite the proof.
- Show the model the revision. Ask: 'Is the weakness fixed? Where is the next weakness?'
- Talk About It: how many weaknesses did the hostile review find? Did the proof get stronger?
Lesson guide
Ask after you try
After the hostile review.
- Ask the model: 'Be hostile. Find the weakest step in my proof.' If it says 'your proof is well-structured,' ask again: 'No. Attack it.' If it finds a weakness, fix it yourself. If it cannot find one, ask a different question: 'What assumption am I making that I did not state?'
- Did they ask for a hostile review, not a friendly one?
- Did they fix the weaknesses themselves?
- Did the proof get stronger after the revision?
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Dinner table
What proof did we strengthen this week with a hostile review?
Sits beside
- Science. Hostile method review: the same workflow, applied to experimental design.
- English. The hostile editor in You Write It is the same role: the model attacks, you fix.
Integrity. Cite the model as a reviewer if its hostile review changed your proof. The review is the model's. The fix is yours. The proof is yours. The citation is honest.