Proofs

A question that comes up a lot in class is, “how do you get good at proofs?” (It’s usually framed as “I don’t like proofs”, but we’re not having any of that negativity here, thank you very much.)

I don’t have a silver bullet for that. I do have some advice.

Keep a notebook

In your special proofs notebook, keep track of all the tricks that seem obvious in retrospect - or that make you say “I’d never have thought of that!” When you’re stuck, flick back through it and see if anything jumps out as a possible way forward.

Do lots of proofs

I can’t explain the process of walking - when I was a kid, I just sort of kept falling over until I got passably good at it. Similar thing with proofs: try. Fall over. Try again.

The more you do, the easier it gets.

Get a feel for things

Try to get a feel for the kinds of problems that tend to use one method or another. After a while, you develop a sense of smell for it.

Look for nuggets

I find that many proofs have a “nugget” - a key flash of insight in the middle of routine work. Focus on those nuggets. Respond to them with an ‘oo!’.

Use the details

If a question gives you details - for example, that the function isn’t defined when $x=a$, or that your integer is odd, or similar, ask questions like:

  • “Why have they told me that?”
  • “What does that definition mean?” (often if helps to write out definitions explicitly)

Convince yourself, then prove it properly

Proofs are full of all sorts of rules about keeping things equivalent and what you can or can’t do. But those are for presenting proofs. When you’re trying to work out the way to a proof, it’s absolutely fine to disregard those rules and focus on convincing yourself.

You can tighten up the presentation later - but playing around is usually how you get to the nugget.


Do you have any top tips for generating proofs? I’d love to hear them in the comments.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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