Working with definitions

When I set a question asking students to prove things, the most common question I get is “How do I start?” Further investigation often reveals that the students don’t remember the definitions of the terms that appear in the question, and have not developed fluency in one relatively routine aspect of doing proofs: make sure that you know what the information you have been given means.

For those students on my first-year, first-semester module on pure maths, mathematical reasoning using definitions, proofs and examples is completely different from what they have seen at A level. My emphasis on the importance of precise definitions is seen by some as rather dry. But I am trying hard to help students to practice working with definitions, proofs and examples as much as possible for themselves.

In my first-year workshop this week, I had a question asking students to prove that if you compose two injections you get an injection, and if you compose two surjections you get a surjection. It took me a few minutes to realise that I was going to have to write the definitions of injection and surjection up at the front, because that was where most people were stuck. I’ll probably include a reminder on the worksheet next time!

The “How do I start?” issue (relating to working with definitions) is not just a first-year phenomenon: it persists through second and third year, and not just in workshops, but also in coursework. I think that I’m going to have to find some new, more exciting, ways to get this idea across, and to try to help students to get past this initial obstacle so that they can spend their time thinking about the interesting bits instead!

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One response to “Working with definitions

  1. I have written about some of what you said in the articles I mention below. One thing that might help is slogans. One slogan I use for starting with the definition is “REWRITE ACCORDING TO THE DEFINITION” which I give with examples in “Useful behaviors” http://www.abstractmath.org/MM/MMUsefulBehaviors.htm. Surely someone could come up with a better slogan, but I haven’t been able to think of one.

    You said “My emphasis on the importance of precise definitions is seen by some as rather dry.” That is what I call “dry bones” Dry bones http://www.abstractmath.org/Word%20Press/?p=39. It is important to admit that there is lots of fun in using images and metaphors to understand some piece of math, but when you have to write a proof you have to temporarily close down your rich images and go to the dry bones — make it DEAD, as someone in Bourbaki said. See also Rich and Rigorous http://abstractmath.org/MM/MMImagesMetaphors.htm#richrigorous

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