How to Read a Proof (Actively)

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Most students read proofs the way they read stories: from top to bottom, passively. Mathematicians read proofs more like code or a chess game: they constantly pause, check each step, and ask 'why does this follow?'.

Active proof-reading habits:

Keep track of assumptions: what are we allowed to use right now?
At each step, ask: 'Which definition or fact is being applied?'
Try to reconstruct the next step yourself before reading it.
After finishing, summarize the key idea of the proof in 1–2 sentences.

Active reading trains your brain to think in proof mode, not just to follow text.

TL;DR — key idea

Don’t just let the proof wash over you. Pause, question each step, track assumptions, and try to predict or reconstruct steps yourself.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Describe one habit you could adopt to read proofs more actively instead of passively.
2
Think of a time you 'zoned out' reading a solution. What could you have done differently to stay engaged?

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.