A more natural proof of Taylor’s Theorem?

If you (like me) have trouble remembering “the trick” for proving Taylor’s Theorem (with the Lagrange form of the remainder), have a look at Timothy Gowers’s blog post on this, where you will find what is, perhaps, a more natural proof than the usual one.

Timothy Gowers’s blog

I have just emailed the following to my first-year students.

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Professor Sir Timothy Gowers, FRS, Fields Medal winner, Cambridge, is an incredibly successful research mathematician who, nevertheless, really cares about teaching mathematics to first-year undergraduates. 

Although the syllabus they are following in Cambridge is a bit different to the one in Nottingham, there is enough in common that you should find what he says helpful.

See http://gowers.wordpress.com/2014/01/11/introduction-to-cambridge-ia-analysis-i-2014/ for his introduction to first-year analysis in Cambridge.

See http://gowers.wordpress.com/2014/02/03/how-to-work-out-proofs-in-analysis-i/ for some discussion of proofs. I like his immediate illustrative example of how understanding aids memory, which I quote here:

“Suppose I were to ask you to memorize the sequence 5432187654321. Would you have to learn a string of 13 symbols? No, because after studying the sequence you would see that it is just counting down from 5 and then counting down from 8. What you want is for your memory of a proof to be like that too: you just keep doing the obvious thing except that from time to time the next step isn’t obvious, so you need to remember it. Even then, the better you can understand why the non-obvious step was in fact sensible, the easier it will be to memorize it, and as you get more experienced you may find that steps that previously seemed clever and nonobvious start to seem like the natural thing to do.”

Exam solutions and feedback

It is the policy here in Nottingham to provide approximately three years’ worth of past exam papers, solutions and feedback (comments, common errors, pointers for improvement) for each of our modules.

With such a policy in place, it is important to make sure that students are aware (assuming this is the case) that studying three years’ worth of exam questions and solutions is no substitute for actually understanding the material taught in the module. Although I said this a few times to my first year Pure Maths students in Autumn 2012-13, it appears that I wasn’t convincing enough. I made a more significant effort this year at the start of the module to explain what students needed to do if they actually wanted to do well in the module (rather than just scraping a pass). I hope that this had some effect: I even got the students to vote on whether they now knew what they needed to do in order to do well in the module, and (perhaps unsurprisingly!) they almost unanimously said yes.

Having said all that, it is of course worth trying past questions and comparing your attempts with model answers, if they are available. Reading the feedback on common errors etc. is probably at least equally important. However, web logs reveal that far more students read the solutions than read the separate examination feedback document. Knowing this, I made sure to include some of the most important feedback directly in the model answers document when I wrote the solutions and feedback last year.

The exam is coming up in about a week’s time, so I will soon see whether there has been any improvement from last year to this!

2013 in review

The message below was automatically generated by WordPress.com for me.

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The WordPress.com stats helper monkeys prepared a 2013 annual report for this blog.

Here’s an excerpt:

The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 25,000 times in 2013. If it were a concert at Sydney Opera House, it would take about 9 sold-out performances for that many people to see it.

Click here to see the complete report.

When is 4 exceptional?

What makes the number 4 interesting?

Obviously there will be many answers to this. Here are a couple of special things about 4 which have turned up in my first year module recently (and in one case this was unplanned!)

  • 4 is the only square number which is exactly one more than  a prime number
  • 4 is the only positive integer n such that n is not prime, and yet (n-1)! is not divisible by n. (Here we follow the convention that 0!=1.)

Do you have any other favourite exceptional properties of 4?

(OT) Review of Elaine Feinstein’s memoirs

I see that there is a very nice review of my mother’s memoirs in the Daily Telegraph:

http://www.telegraph.co.uk/culture/books/10483123/It-Goes-with-the-Territory-Memoirs-of-a-Poet-by-Elaine-Feinstein-review.html

 

100,000 views

I see my blog has just passed the 100,000 views mark. 

(Rather more people watch my YouTube videos!)