Show the Path · Sitting 1
The Path Is the Mathematics
The answer is the end of the path. The path is the mathematics. A model that gives you the answer without the path has given you a shortcut, not a proof.
- 40 min
- Practice · Reflect
- 15–18
Parent briefing · 5 minutes, before they sit
This sitting teaches the student that the path is the mathematics, not the answer. The model is excellent at answers. It is less reliable at paths. The student needs to learn to demand the path, not the answer. If the model gives an answer, the student asks: 'Show me the steps.' If the model skips a step, the student asks: 'Why is that step legal?' The model is a shortcut. The student's job is to walk the path and use the model as a checker, not a replacement.
Hard edges
- Academic integrity: a model answer submitted as a proof is plagiarism of the path. The answer is not the proof.
- Do not ban the model. The point is to use it as a path-checker, not to reject it.
If they say
- “The model's path is clearer than mine.”
- It is. That is because it has walked a million paths. Clarity is not understanding. Walk your own path, even if it is messier. The mess is where you learn. The model's clean path is a demonstration, not your learning.
- “I only need the answer for the test.”
- For the test, the answer might be enough. For the next course, the next problem, the next job, it is not. The path is reusable. The answer is disposable. Choose the reusable one.
Objective
The student can distinguish between an answer and a proof, explain why the path is the mathematics, and use a model as a path-checker, not an answer-giver.
Why the path matters
The answer to a math problem is a destination. The path is how you get there. If someone gives you the destination, you know where it is. You do not know how to get there. If you walk the path, you know both. The next problem is a different destination, but the path uses the same streets. The student who walks paths learns the streets. The student who collects answers learns nothing reusable. The model gives destinations. It is a taxi. Taxis are useful. They do not teach you to walk. Walk the path yourself. Use the taxi when you already know the streets.
Using the model as a path-checker
The model can check your path. You walk the steps. You show the model your steps. You ask: 'Is each step legal? Did I skip one?' The model reviews the path. It does not replace the walking. If the model says 'your step 3 is wrong,' you fix it. If the model says 'this is correct,' you still verify. The model is a checker, not a walker. The student who uses it this way gets the best of both: their own understanding and a tireless reviewer. The student who lets the model walk the path gets neither.
Big idea
The path is the mathematics. The answer is the destination. Walk the path. Use the model to check, not to walk.
Try this~18 min total
Walk and check
18 min- Pick a problem at the edge of your skill. Solve it on paper. Write each step.
- Ask the model: 'Here are my steps. Is each one legal? Did I skip any?'
- If the model finds a gap, fix it yourself. Do not let the model fill the gap for you.
- Compare your path to the model's path (if it gives one). Are they the same streets?
- Talk About It: what did the model catch? What did you catch that it missed?
Lesson guide
Ask after you try
After the student's path is written.
- Show the model your steps. Ask: 'Where am I wrong?' not 'What is the answer?' If it fixes the step for you, reject the fix. Ask for the reason, then fix it yourself.
- Did they walk the path themselves?
- Did they use the model as a checker, not a walker?
- Can they name what the model caught and what they caught?
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Dinner table
What path did we walk this week, and did the model check it or replace it?
Sits beside
- Science. Hypothesis before search: the hypothesis is the path. The search is the check.
- Thinking. Claim and check: the proof is the check. The claim is the door.
Integrity. Do not submit the model's path as your proof. If you walked it, it is yours. If the model walked it and you copied, it is plagiarism. The path is the proof.