Nothing is perfect. Or is it? Some mathematicians out there might like to try to prove that statement wrong. Our beautiful subject is one of the few places where we have managed to prove that things are perfectly true … or at least they are perfectly true if the basis on which those things are claimed is itself true.
We only need to turn to Gödel’s incompleteness theorems to realise that there are limitations to arithmetic systems, but ultimately these are so specialised and ‘remote’ from the kind of mathematics we usually do that we don’t need to worry. What matters to all mathematical development is the idea that we need to prove that connections, ideas, theories and so on are true wherever we possibly can. It is this idea of proof that makes the subject the most perfect of all.
Where do I start?
Before teaching anything about proof, it is a really good idea to discuss why it matters with the students. You would want to bring out the following:
• You can use formulae, results and facts with confidence if you know they have a proof
• You don’t run the risk of finding that your work is based on a false claim if you have made sure that everything you have done is based on something that has been proven to be true
• It brings students of mathematics the closest they will ever get to being perfectly correct about something, allowing them to get a feel for watertight, logical argument
• It allows students to enter a whole new world of connections and dependencies, much of which is very appealing to newcomers, bringing as it does much of the ‘magic’ of mathematics
Then it is a very good idea to discuss the ingredients of a proof:
• Which definitions do you need? Have you got them precisely correct?
• What ‘facts’ are you going to use and how do you know they are correct? Have they been proven?
• Do you have a clear statement of the theory you are trying to prove (the ‘theorem’)?
• Finally, the method itself; the clear synthesis of all of the above into a coherent, logical whole. The proof needs to cover all possible cases, with no assumptions made.
Not yet!
I would never advise a teacher to leap into an example of a great proof quite yet. You need to give some examples where things can go wrong if you don’t provide a proof. Please click below to see some nice examples.
Further ideas
The choices you make for next steps will depend on the abilities of your students and the stage of their development. However, at IGCSE level there are some very good examples that will make the point and structure of a properly executed proof clear:
• A general piece of research into ‘proof by contradiction’ will be fruitful, especially if the students pick up the proof that there are infinitely many primes
• Circle theorem proofs use a nice combination of approaches and should be considered, some of which also use contradiction
• Angle problems are good examples of direct proof and underline the need to write out each step carefully and to state theorems accurately. You can always argue, once again, that the students are merely setting out the ingredients of a proof before actually following the process.
• Later in the course, the use of completing the square to derive the quadratic formula by direct proof always works well. It also provides wonderful challenge with algebraic fractions and understanding square roots.
• Differentiation from first principles takes students to the very edge of what they can do at this level and is always warmly encouraged. With differentiation, to merely present the rules seems mystifying. Indeed, this is also true with the quadratic formula and circle theorems. AND Pythagoras.
Above all when you are teaching children about proof, you must never let any detail go unchallenged. All such discussion will help your students to become more precise and efficient mathematicians as well as more creative problem solvers.
Have you enjoyed reading this blog post? Watch the below video to gain more useful tips from Nick Hamshaw and explore how our new resources support your teaching strategies.
Nick Hamshaw is an author of the first edition of the current Cambridge IGCSE™ Mathematics Core & Extended coursebook. Since graduating from Oxford in 1999, Nick has taught mathematics throughout his career in a mixture of state and independent schools in the UK. Although now a senior leader, Nick continues to teach mathematics regularly.
