- A levels are getting easier;
- Teachers are getting better at preparing students for the exams;
- Students work very hard and so it is insulting to them to say the exams are getting easier.

So let’s examine these arguments for A level maths, which is the subject I know about. Unlike some pundits I don’t wish to pontificate about subjects I am not familiar with. My main point is that the arguments above are *not* contradictory and it is quite possible for exams to be easier than in previous years *and* for students to find them hard.

- It is generally acknowledged in the mathematical community that A level maths exams are getting easier and it has been remarked on by a government advisor A-levels are easier says adviser. Yet this headline is misleading because it is only referring to maths and physics which shows just how difficult it is to have a rational argument about the standard of A levels.

It is interesting to see the effect this is having on university mathematics courses even in the last few years A-Levels: Gah.

However, A level mathematics and its equivalent has become steadily easier over at least the last hundred years as the subject has been ‘democratised’ and taught to a wider population. I am very much in favour of this but the cost has been the dumbing down of the syllabus as I have mentioned before. I can only repeat my question that*Is it possible to teach mathematics to a larger range of students without compromising on the level taught?* - This is true but nothing to be proud about. Teachers and schools are under extraordinary pressure to get good results. Hence they teach very much to the exams, test and retest, and have no time to explore interesting parts of mathematics which would help to motivate and put things into context.

How politicians can imply that this is good beats me. - Mathematics, as those of us who taught it know, is hard. This has been recognised (finally) by recent research, and it has been proposed that, as happens in Australia, more UCAS points are awarded for harder subjects like maths, though I can’t see that it will happen here.

Also mathematics is essentially linear so unless students are adequately prepared at one stage they will find it very difficult to proceed to the next stage. With dumbing down going right back to the early years of education most students will find each stage hard no matter what the level. The ignorance of this caused disaster in 2000 when A level maths was temporarily made tougher and so the gap from GCSE was unbridgeable for many students, leading to lots of failures, the abandonment of the subject by many and the restoration of an easier system.

Finally, with a greater percentage of the student population studying mathematics many of them are bound to find the subject hard.

It is good that Further Mathematics is becoming more popular and is to be encouraged, though it should be noticed that it too is a pale reflection of when it was last popular.

I don’t have easy answers to the problems of mathematical education but wish to see an *informed* debate on it. I just hope that Alexander Borovik is not correct when he writes

]]>

The current crisis in mathematics teaching is a dawn of a much more serious crisis of transition from the mass mathematics education of the past to a more selective and elitist education of increasingly small, in relative terms, numbers of mathematicians.

He asked 2 questions:

1.

Can the elements of all (finite) cyclic groups be arranged to give these diagonal stripes?

If you take a finite cyclic group generated by the element (we will use multiplication for the binary operation) then the natural ordering will show this pattern:

2.

Are the cyclic groups the only ones that generate these patterns?

The answer is yes, but is not quite so obvious although not difficult to prove. Suppose a (finite) group G arranged as exhibits the diagonal stripes. Use induction. Let and suppose also that for . Then we get:

It follows that so by induction assumption, and hence is the cyclic group generated by .

The level of education was highÂ including a crash course in reflection groups. One of the blackboards photographed includes the following mathematics:

If is a group then the set of automorphisms of forms a group under composition. If is abelian then .

Homework:

Find

Show that if then Â Â

But then the story goes sour. The authorities have closed the summer school – see A Blackboard under Arrest, A Blackboard under Arrest, II Â – which is such a shame. There is a petition to save the school at Save Mathematical Summer SchoolÂ you may wish to support, but also, can I ask other bloggers, mathematical or otherwise, to spread the news in the hope that it will help find some way round the problem?

]]>- InauguralÂ EditionÂ hosted by Abstract NonsenseÂ has 19 assorted links ranging from quotes exhorting people to study mathematics to algebraic topology.
- The Second Carnival Of Mathematics: The Math Geeks are Coming to Town!Â hosted by Good Math, Bad Math has 27 fascinating assorted links including the Halting problem, Using math for astronomy and Rubik’s Magic Cube.
- Carnival Of Mathematics #3Â hosted by Michi’s BlogÂ has 20 articles grouped into five ‘halves’: didactic, financial mathematics, humour, dimensions, and number theory, geometry, topology, algebra.
- Carnival of Mathematics Number 4Â hosted by EvolutionBlogÂ has 21 links from the Bernouilli process to finding the last two digits of 3
^{1000}. - Carnival of Mathematics, Ordinal 5Â hosted by Science and ReasonÂ is dedicated to the memory of Paul J. Cohen with tributes and discussion of his work. There’s at least 30Â links to other topics including tilings, Lie groups etc and surreal numbers.
- The Carnival of Mathematics Sixth EditionÂ hosted by Modulo ErrorsÂ has 19 links with many presenting problems of various difficulty.
- Carnival of Mathematics Edition #7Â hosted by nOnoscienceÂ has 28 links from Euler to German bloggers and includes a new improved number system and a calculus paradox.
- 8th carnival of mathematicsÂ hosted by The Geomblog has 20 links with “time to revisit, reflect, and ponder on things we think we already know” so has an educational section and is the first carnival to have a cartoon.
- Carnival of Mathematics IX hosted by JD2718Â is an alphabetical list of 36 links that appeal to the school teacherÂ blogger.
- The nextÂ carnival is due on 15th June at MathNotations. Carnival 11 is due to be hosted by Grey Matters on 29th June and Carnival 12 on 13th JulyÂ by Vedic Maths Forum. Do let them know if you have anything you wish to be included.

That’s over 200 links (not all mutually exclusive) – mathematical blogging is alive and well.

]]>Mathematical humorous videos are even rarer. Here are a couple. The first is very well-known and has been written about all over the net for a long time. It isÂ Finite Simple Group of Order Two by the Klein Four Group and involvesÂ very clever use of mathematical terms. The other is G103Â which is described asÂ *a (surreal) day in the life of an undergraduate on the 4-year MMath degree at the University of Warwick*. Anyone who has experience of a pure mathematics degree will recognise the accuracy of the amusing observations it makes. There’s more about the film at the G103 site.

There was also aÂ previous humorous video made at Warwick University called Maths Club. Unfortunately, it’s not available on that page or anywhere else as far as I can see. Does anyone know if it’s still available?

Links to other amusing mathematics videos are very welcome.

PS The Unapologetic MathematicianÂ links to a spoof basic mathematics tutorial produced by the BBC called Look Around You – 1 – Maths.

]]>It’s interesting that the Dr Who reference was put in the Wikipedia article half-an-hour before the programme was aired so probably an inside job. I was delighted thatÂ in the programme Dr WhoÂ asked if mathematics was so dumbed down that recreational maths wasn’t studied any more. As Russell T.Â Davies, the head writer,Â said in tonight’sÂ Dr Who Confidential, the programme reflects current concerns, so this problem has clearly reached a wider audience than I imagined was the case.

]]>

Every linear transformation of a finite-dimensional complex vector space has an eigenvalue.

Here is his proof:

Let be a non-trivial finite-dimensional complex vector space and *t*Â a linear transformation . Let be a fixed non-zero vector in and suppose that . Then the vectors are linearly dependent. Hence there exists complex numbers not all 0 such that

.

and hence

.

Now, since is algebraically closed, Â the polynomial will factorise so we get

.

where are complex numbers with . It follows that

which means that, since this is composition of functions, and , then

*either*

soÂ Â Â so that and is an eigenvector

*or*

so Â is an eigenvector

*or
…
or
* is an eigenvector,

and hence has an eigenvalue. Â Â

Discussion on this approach of not using determinants can be found at NeverEndingBooksÂ and The n-category CafÃ©.

]]>

Is math class too easy for you? Looking for a greater challenge?

You’ve come to the right place.

It has an excellent introduction to siteÂ and a forum to discuss problems. The forum has an RSS feed where students pose new problems every day. So if you’re “looking for a greater challenge” subscribe to this feed. Here is a random sample of some of the problems posed recently, which range from very easy to extremely difficult. Click on the problem number to go to the discussion on it.

- 1. We define addition in a different wayÂ to usual; an addition statement is true only if the letters in the addends is a rearrangement of the letters in the sum. For example,

10 + 6 = 16?

TEN + SIX = TENSIX = SIXTEN, but to be 16 it would need another E.

Find a “true” addition a + b = c + d.

2. Prove that . 3. Let be nonzero real numbers. Find all ordered pairs such that .

4. is a continuous complex-valued function satisfying:

i)

ii)

Find

5. If and are relatively coprime, find all possible values of .

6. Let be three angles ofÂ . Prove that .

7. For each function which is defined for all real numbers and satisfies and determine the value of .

8. Let be positive reals such that . Prove that .

9. For , we define the numbers . Find the last digit of the number .

10. The product of several distinct positive integers is divisible by . Determine the minimum value the sum of such numbers can take.

Of course there is a place for calculators and mathematical software. Checking my tax would be a pain without a calculator and graphing software for focusing in at what happens near the origin of the graph of functions like

is fascinating. Similarly, if I am marking student work I use mathematical software to check their matrix operations rather than doing them by hand. Some mathematical software is very powerful and expensive but there are free sites like QuickMath which will solve many problems. However, they should really be used to save time or offer insights *after* the techniques have been taught and understood – practice, practice and practice is often the best way to learn.

So when I was asked by ElasticLogic to review their Equation WizardÂ Â I made it clear that I would be offering an honest opinion of the program that they sent me.

Equation Wizard is a Windows only program that solves real rational equations and simplifies rational expressions (rational means ratios of polynomials). In fact it claims to solve algebraic equations but that is a misnomer as it doesn’t solve equations involving fractional powers or complex coefficients, though it will give some complexÂ roots (for some reason called *imaginary*Â roots in Help). QuickMath does this for free but Equation Wizard’s strength comes from the fact that it will show the working so the user can understand the method behind the solution.

Entry of polynomials is easy using ^ for powers or using buttons or menus or the Ctrl key and the text is previewed in mathematical form as you type, so x^2 becomes and (x^2-1)/2 becomes ; the previewer does its best to interpret ambiguous expressions such as 1/2x. It will add algebraic fractionsÂ showing theÂ working, so if you input 1/(x-1)+1/(x+1) then it is simplified to by adding using a common denominator, multiplying out and collecting the terms in the numerator to get the result, with all steps shown. It uses a similar method to solve the equation 1/(x+1)+1(x+1)=1 finding the answer to 3 decimal places (or up to 9 decimal places if required). There appears to be no limit to the degree of the polynomial equations to be solved – solving was virtually instant.

However, there are limitations. The answers given cannot give exact values so gives 1.414 and -1.414 rather than . Rational equations are solved by multiplying by the denominator but the solutions aren’t checked so gives as one of four solutions. My Norwegian students were taught to *always* check their answers, so would know what to do. On the other hand solutions can be missed so gives 0 and 0 as the two solutions, though I expect that’s just a bug. However, gives one correct answer and two wrong ones, which is rather worrying. There are language problems with being described as a quadric equation. Quadric *surfaces* are interesting and there are nice pictures here.

You can print out the calculationsÂ or save themÂ in rtf or HTML format (with the equations saved as images) but it would be nice to be able to copy the selected output to the clipboard. Having solved an equation it’s not possible to change it – you have to enter a new one, though you can copy and paste the old one or use the history button, bizarrely labelled .

Your opinions on Equation Wizard are welcome and I hope ElasticLogic will improve this early version as a result of feedback from you. Can you find other equations with missing or incorrect solutions? The software costs $29 (or Â£15.55 in real money ) and you can download a free trial version here.

]]>-
Chancellor reveals maths weakness. Gordon Brown, almost certain to be the next Prime Minister, said he wasn’t very good at maths.

*But*he also added

“I did maths at school and for one year at university but I don’t think I was ever very good at it – and some people would say it shows,” Mr Brown laughed.

I wish he hadn’t laughed (was that due to guilt?) but in his defence he has probably studied mathematics to a higher level than most politicians. It should also be noted that he was educated in Scotland where the first year of university is the equivalent to the last year of school in England and Wales (and so is more like the US system). So he is likely to have studied mathematics to A level standard.

Pupils are being discouraged from taking A-level maths as schools in England chase higher places in the league tables, scientists have claimed.

The Royal Society of Chemistry said that as maths was a difficult subject, schools feared examination failures which would threaten their standings.

Of course the DES totally miss the point when they say

The Department for Education and Skills said more pupils were studying maths.

More than what? Such is the pressure of those league tables that I can totally believe this story. I wonder sometimes if we shouldn’t rename this country Wonderland and then find an Alice who can make sense of it all.

]]>