How to Get an A* in A-Level Maths

This is my first daily update thanks to a commitment to spend 1 hour per day on this project for 100 days
Part of Josie Long's 100 days challenge: http://www.hundreddays.net/

My goal here is to create a website that makes A-Level maths incredibly easy.

Immediate considerations:

How is it best to learn (and therefore teach) maths?

My opinion is that people learn best independently if they have a commitment to do well and if they have access to an expert on the complete syllabus who they have confidence in.
(I may be biased as I get a lot of my income as a private tutor)
It would be interesting to do some further research on best methods of teaching and learning.

My belief is that anybody who embarks on A-Level maths and a good attitude can at least match their GCSE grade.
This is purely anecdotal - I also believe that anyone who can answer a basic ratio question on recipes (i.e. set in a real world situation - here is a recipe for 4 cakes, how do we make 6 cakes?) has sufficient abiliity to pass a GCSE maths (A*-C) if they put in the required effort.

Out of all my many private students, I can only think of one who couldn't answer this question with reasonable prompting (he is now retaking after getting a D...)
and only one of my private students has ever failed to get a C though she could answer that question (got a D but was told by their maths teacher that they would be lucky to get an F and that there was no way they would ever get higher - one of the most satisfied customers I've ever had even though I thought she could have passed: that maths teacher should be struck off)

One problem that I consistently come across is that high school teachers in the UK are often not familiar enough with the A-Level syllabus.

Just this week I have identified
* Teacher teaching material that is not part of the course as though it is urgently important
* Teacher failing to identify consistent mistakes in working and answer (tick! tick! tick!)
* Teacher teaching incorrect material to students (teacher's own notes were wrong)
* Teacher teaching particular complicated requirements for written answers that do not gain any credit according to the exam mark scheme.
* Non-expert mathematicians teaching maths and not really able to answer intelligent questions
* Many, many teachers who fail to inspire confidence in their pupils - particularly bright, ambitious A-level students and including teachers at highly respected independent schools

I've even known cases where teachers have taught the "wrong" syllabus with entirely the wrong content and not realised - either because of an administrative error or because the syllabus had changed without the teacher's realisation.
This possible error is mitigated somewhat by a reasonable series of books covering each exam syllabus. The books are generally good and teachers following them as a guide can be sure to cover most everything - though they are lacking in some detail and particularly in links between different content areas and modules. Motivated self-studiers can potentially do well exclusively by using the books and provided exam materials, but may miss out the underlying ideas of the course and key techniques. Whether or not this provides a good bridge into post-A-level study is debatable? (There are advantages and disadvantages to learning 'by rote', 'for the exam')
With e.g. Edexcel attempting to self-publish course books (so far at least for GCSE), the quality of textbooks may take a step backwards. Their books have a sheen of authenticity but are of poor quality and not good enough for self-study.

What I want to provide is a website where students can be confident that every part of the syllabus is covered fully, in complete detail - identifying what is important, how best to answer a question, possible alternative methods, common problems to be avoided, plenty of worked examples, plenty of potential exam-style questions to try, links between content areas and referring back to previous modules.
The most important thing is to improve student confidence.
(This may be challenging at first when the website needs to build a reputation)

Potentially users can contribute by asking questions, the website is easily updatable if the syllabus changes or if alternative methods are suggested.

The idea is that students can come to be reassured that they have covered everything they need to know.

Problems

Unfortunately a website has the disadvantage (from the students' POV) that there is no immediate interaction
Can provide an on-demand tutor to help (at a cost) - online or in person?

How best to display website on the printed screen?
If the idea is to get across how best to answer an exam question, is this is best done with written examples?
Can a consistent font be used that can draw e.g. an integral sign
(LaTeX standard font is not ideal)
What about questions that require pictures e.g. circle theorems, forces diagrams

Wish to draw these in a consistent and pretty way that can easily be drawn by hand
Is a graphic designer necessary? (Hopefully not)
How do textbooks deal with this problem well/badly?

Asides:

Good name?

It seems natural to link forward to university courses - where do people use maths in their careers/lives? what uni subjects is maths A-Level necessary/useful?

Government goal to get 100,000 pupils taking maths A-level gives a decent sized market. (at the moment ~75,000 full A-level students IIRC)

Which modules to concentrate on?
- C1, C2, S1, M1
- C3, C4, S2, M2

If your probability of winning a point on your serve is say p = 0.5
And your probability of winning a point against your serve is say q = 0.5

What's the probability you'll win a match of tennis? (easier than it looks)

Then what about if you're more likely to win on your serve and less likely to win on your opponent's serve
say p = 0.6, q = 0.4. What's the probability now? (still easy enough)

And what about if you're better or worse on the pressure points - break points, set points and match points for and against you? (we can make it incrementally harder though!)

Suppose you're a reasonably good player - you usually dominate your service games, you return a good few winners, and you play reasonably consistently throughout a match with no weakness on the pressure points - maybe a weaker backhand than forehand - which area of your game do you most need to work on in order to increase your chances of winning?

If you can answer that last question intuitively, you could make a good tennis coach ;)

Just downloaded "Big Top Ten" - the point of the game is to add up to ten. Simple but addictive!

Not sure whether to be flattered that I'm getting spam despite hardly advertising the site.
I guess that's the peril of using a common (and good) Content Management System.

Unfortunately commenters will need to be confirmed as humans - for the moment I'll need to update new accounts by hand so it will take a day or two. Sorry!

Some tasks are well-suited to humans, and others are well-suited to computers.

For example - identifying and labelling images is (currently) very difficult for computers, but fairly easy for humans. Google use an online game to get humans to do the work - but it turns out that computer programs do better at the game because they are better able to cheat!

The seemingly similar task of completing a jigsaw is different... big jigsaws get increasingly difficult for humans but stay quite simple for computers. Computers can easily check which edges match quickly and easy (within reason), and simply join them up (assuming you can find unique matches). Then just check for more matches until the whole puzzle is complete. Simple!

Does anyone have a computer program that will solve a 30 piece jigsaw, say?

Can you see blue and green spirals? Don't believe your eyes. what's going on?

Washington Post have posted statistics which "leave very little room for reasonable doubt" that the Iranian election was rigged. In summary, they look at the final two digits of the vote counts and compare them with what you would expect if they were random.

The statistics applied to the problem are admirable and (as far as I can make out) follow the hypothesis test methods which can be found in the A-level Statistics modules - the article is very much worth a read. The results show significant results at 5% level, suggesting that the number of votes in the Iranian election may have been made up. I'd like to get the opportunity to check the numbers in more detail, though I have no reason to disbelieve what they've written.

However - this is not the end of the story...

Considering the weight of feeling amongst Western people that the Iranian elections are rigged, people are prepared to seek out any evidence anything which agrees with that conclusion. This is a human bias known as confirmation bias - the act of searching for information that agrees with what you believe - a very dangerous source of bias. Generally we must make a serious effort to be as neutral as possible.

So let's concede that political scientists Beber and Scacco were genuine in their investigation - that they honestly set up the test with no preconceptions, were not looking for evidence unduly, and were prepared to accept the result whichever way it landed - with no confirmation bias.
Even so, we might imagine that perhaps hundreds of similar scientiists have recently done hundreds of similar tests. For every hundred similar tests, at a 5% significance level, around 5 tests will generate a significant result incorrectly - even if the election was perfectly fair, 5% of the time we will conclude that the election was not fair.

We don't know exactly how many of these tests have been performed, that's true - but under these circumstances, we would easily be able to argue that 5% is not a strong enough significance level. If we were to make the significance level any stricter - say 1% - then we'd have to conclude that there's not enough evidence to say the election was rigged. (in some sciences, a significance level of 0.000001% is not unusual)
This is often a consideration with medical statistics and drug trials in particular. It's important that we don't allow companies to repeat hypothesis tests until they get the result that they want or do multiple parallel tests without admitting that and correcting for it.

Then we have another source of bias I'm afraid! The Washington Post are very unlikely to publish a result that said "Iranian elections might be fair" - and are much more likely to publish "Iranian elections might be rigged" (as they have here). So there's a bias which comes from the way news is reported.

I would add a final point - a hypothesis test is always better if the conditions are prepared before the results are released. If it were an international standard that elections were checked by this method, then a 5% significance level might be very reasonable. On the other hand, if that were the international standard and everyone knew it, it would be very easy to cheat the system. Oh dear!

Conclusion: Always take statistics with a pinch of salt! In trials involving humans, we must be very careful to eliminate as much bias as possible - but often it will be impossible! In the real world, we'll have to apply statistics with consideration for the situation.

In terms of the Iranian elections - I can't say whether the elections are rigged or not, but these particular statistics are not really conclusive and can't fairly be described as "strong evidence".

Hope this wasn't too ranty! Happy statistics!

Just bought a Casio fx-83ES with Natural Display.
Natural Display is the best thing about new calculators. If I type

√12 + √27

The answer displayed is simplified to

5√3

How good is that?! Of course I can press one button to get the decimal answer of 8.66 as well.

I am also very interested in the Casio fx-115. Apparently it can calculate definite integrals (very useful for checking A-level answers).

I am a little bit concerned about this - the Edexcel syllabus says "Calculators with a facility for symbolic algebra, differentiation and/or integration are not permitted" in any of the exams.

Not sure if definite integration counts as symbolic algebra as the calculator probably does it by a numeric method. The fx-115 doesn't appear to do indefinite integration which I would say was definitely crossing the line. Will try to seek clarification from Edexcel!

The 2009 US Puzzle Championships was today. Consisting of "culture fair" puzzles that are the type you find with the sudoku in the newspaper - but more bombastic versions!

You can find the puzzles here in the archive:
http://wpc.puzzles.com/

Good luck!

I'd usually argue that if you're learning how to answer A-level questions by rote (memorization rather than understanding), then you're getting yourself into trouble.

Here's an example of where learning by rote can be useful. Get a stopwatch ready.

1) Time yourself: Write down the seven times table in decimal (7,14,21,...) up to seven times thirteen
2) Time yourself: Write down the seven times table in binary (111,1110,10101,...) up to seven times thirteen

You might argue that (2) will take longer because there's more digits to write. OK then...

3) Time yourself: Write down the seven times table in base 11 (7,13,1a,...) up to seven times thirteen

Please let me know your results!

A little mathematical help

In decimal (base 10) we use 10 symbols (digits) to write numbers: 0,1,2,3,4,5,6,7,8,9.
In binary (base 2) we just use 2 symbols: 0 and 1, to produce the same effect. Instead of Units, Tens, Hundreds and Thousands we have Units, Twos, Fours and Eights.
Computers frequently use base 2 as it's easy to represent this as on and off states.

The only real reason we use base 10 is because we have 10 fingers and that's what's been selected as the standard way - there's no reason not to any other number of symbols - like in base 11 we'll use 0-9 and then 'a' to represent ten. Historically not everyone has used base 10

(Although whatever base you choose should be called "base 10" I suppose!)