Students will always need help with problems and exercises that are set.* When a student asks me for help with a particular problem, then I have a go-to question for them: “What have you done so far?”. I learned this from Bill Bruce when I was his Research Assistant many years ago. It’s a great question because it stops me launching in to an explanation only to discover that the student already knows that part of the solution — my students are generally quite polite (lucky me!) and tend not to interrupt me in full flow. It forces them to think rather than rely on me. Quite often the reply is “Nothing. I don’t even know where to start”. Even that is useful information and once oriented to what they do or don’t know I can begin follow up questions “Do you know how to do X?”, “Can you break Y it into parts?”, or “What do we know about Z?”. * If they don’t, then the questions need to be a little bit...

## The five methods of proof...

posted by Kevin Houston

In essence there are five methods of proof: Direct: A string of implications , or some straightforward calculation. Induction: true and true for all means that is true for all . Contradiction. Show a statement is true by assuming it is false and showing that this implies something that isn’t true. Contrapositive: To show implies we show false implies false. Cases: Divide the statement into separate cases. For example, , and . All proofs are constructed from these basic building blocks. In my first year course a key early aim is to make sure students know this. To achieve this I mark up some proofs during lectures noting the methods used (and where the assumptions are used, etc). They are then set the exercise of finding which methods of proof are used in a different example. The exercise has a number of benefits. One, it tests the students on the methods. Two, it exposes the students to the structure of proofs. (An added benefit is that they are more likely to remember the proof when they know the structure.) Three, having explicitly seen the breaking down of proofs into pieces they are more likely to use that in the creation of their own proofs. (For the latter I have found that, generally, a prompt is needed, for example “how could we break it down?”) As an example consider the following. To prove a statement of the form “ if and only if ” students often try . That’s a great proof and it would be wonderful if it worked all the time but, unfortunately, generally it doesn’t and that causes many students to give up. However, if they have observed* repeatedly that can been divided into the cases and and those cases can be...

## D’Arcy Thompson radio programme...

posted by Kevin Houston

D’Arcy Thompson has been of interest to me since I was a teenager. There was recently a programme on BBC World Service. Have a listen on the BBC iPlayer if it’s available in your area: Science Stories: D’Arcy...

## It’s coming…...

posted by Kevin Houston

Posts have been a bit scarce round here recently. Something’s coming soon…

## Best of xkcd: Educational...

posted by Kevin Houston

The web comic xkcd is well-known for its humour. I’ve also got a soft spot for the educational infographics, such as the following. (Not all the information should be...

## Summer Hiatus…

posted by Kevin Houston

My unscheduled summer hiatus from blogging arose from a family emergency that meant I was in stuck in Brazil for about 5 weeks. Not a bad place to be stuck but under the circumstances it was difficult to find time to finish my planned summer posts. For example, I had planned to post about the Fields Medals and as it happened lost the opportunity to report from Brazil about a Brazilian winner. The short story is that, to me, the Brazilians seemed mostly proud but surprised that a Brazilian was even in the running. The hiatus also meant that the deadline for my next textbook wooshed by. Ok, in reality I was never going to make it but I’m even further behind than expected. Anyhow, service will resume when I get back on track after all the time lost. For instance, I’ll let you know what the book is...