Show the Path · Sitting 2
The Missing Step
A proof with a hole in it is not a proof. It is a guess with a gap.
- 28 min
- Talk About It
- 11–14
Parent briefing · 5 minutes, before they sit
A proof is a path. If a step is missing, the path has a hole. You cannot walk across a hole. The model is very good at jumping over holes. It produces the answer without showing the step that connects. The student needs to learn to look for holes. If the model says 'and then the answer is 42,' the student asks: 'What step got you from the line before to 42?' If there is no step, there is a hole. The hole is the lesson.
Hard edges
- Keep the examples simple: arithmetic with a skipped step, a logic puzzle with a jump.
- Do not shame the model for skipping. The point is the student's ability to see the gap.
If they say
- “The model probably skipped it because it's obvious.”
- Obvious to whom? If it is obvious to you, fill it. If it is not obvious to you, it is a hole. 'Obvious' is the most common word used to hide a gap. Check it.
Objective
The student can identify a missing step in a simple mathematical argument and explain why the gap breaks the path.
The hole in the path
A path has steps. If you remove a step, the path has a hole. You can see the other side, but you cannot walk there. The model is very good at jumping. It says 'the answer is 42' and skips the step that gets you there. If you cannot see the hole, you will trust the jump. If you can see it, you will ask: 'Where did 42 come from?' That question is the skill. The model's jump is not a proof. It is a leap. A proof is a walk, not a leap.
Finding the hole
To find the hole, read each step and ask: 'Does this step follow from the one before it?' If yes, walk on. If you cannot see how it follows, there is a hole. The hole might be a missing calculation, a skipped assumption, or a logical jump. Mark the hole. Ask: 'What step would fill this gap?' If you can fill it, the path holds. If you cannot, the path is broken. The model will not mark its own holes. You have to find them. That is the student's job, not the model's.
Big idea
A proof with a missing step is a guess with a gap. The model jumps. The student finds the hole.
Try this~14 min total
Find the hole
14 min- Ask the model to solve a simple problem and show its steps.
- Read each step. Ask: 'Does this follow from the one before?'
- If you find a jump, mark it. Write: HOLE.
- Ask: 'What step would fill this gap?' Try to fill it yourself.
- Talk About It: did the model skip a step? Could you fill it? Was the hole real or just hard to see?
Lesson guide
Ask after you try
After the model has shown its steps.
- Ask the model: 'Show me the step between [line A] and [line B].' If it fills the gap, the gap was real. If it says 'this is straightforward,' that is a dodge. The dodge is the hole.
- Did they find at least one hole (or confirm the path was complete)?
- Did they try to fill it themselves?
- Can they explain why a hole breaks the proof?
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Dinner table
What hole did we find in a path this week, and did we fill it?
Sits beside
- Thinking. Claim and check: the hole is the unchecked claim inside the proof.
- Science. A lab report with a missing method step is the same hole.
Integrity. If you find a hole, say so. Do not pretend the path was complete because the answer was right. A right answer with a hole is still a broken proof.