May
1^{st} update – the BBC are reporting that the
RSC has admitted receiving two thousand entries to their maths test...

The RSC had made a very
clumsy comparison between maths in the UK and China. British school students
often study calculus, matrices, probability and statistics, all largely absent from
the standard Chinese curriculum, instead of lots of traditional geometry.
Chinese students are coming to Britain in large numbers at both the A level and
university stage, and one reason for this may be that our own approach is much
better suited to modern science and technology. We should not therefore damage
our international “brand” with uninformed comparisons. Neither should we
denigrate our own students and teachers – the number of submissions received by
the RSC says it all. It would be of interest to all of us for them to state how
many correct answers they got from British 6^{th} formers.

Media and Chemistry make bad maths smell….

BBC news is carrying a story based on the Royal Society of Chemistry posting a geometry
question from China. So yet again we have the media finding an excuse to have a
go at British education.

I have taught
students from many countries and the lesson of this is that at the end of high
school students will have received mathematical training with very different
emphases in different countries, and that the emphasis in any country will vary
with time.

Here in the UK school
mathematics, many years ago, consisted of a huge amount of geometry and
algebra. Later the emphasis was changed to include a great deal of calculus,
especially at A-level, and for advanced students, some calculus was introduced
at O-level in “Additional Maths syllabuses”. Mechanics used to feature strongly
as well. These days many students spend more time on statistics, probability
and data interpretation, and less on mechanics and a LOT less on geometry.

The RSC have put up a
story saying “Chinese maths level embarrasses English system”. While there are
perhaps many ways we can work on improving UK maths education, this is a cheap
and uninformed shot at UK maths teaching, and those involved in UK maths
teaching deserve better. The RSC have posted a geometry question. The Chinese
love geometry and teach it a lot. This is indeed to their credit.

But chemists need to
learn that people with a half-decent grasp of mathematics understand that the
first rule of comparison is to do so on a like-with-like basis.

I recall admitting and then teaching
some very bright and well-taught Chinese students while a tutor in Oxford. At
school these students had done lots of geometry, algebra and all kinds of
puzzle-solving. But I found myself having to teach *introductory calculus* to some of them. Some had just not done it. Similarly,
in the United States, many good students might not learn calculus until they
get to college, unless they were in a certain type of programme (Advanced
Placement or some equivalent).

So I could set a test
for university entrants in China (or the US) which many British 6^{th}
form maths students could do, based on some calculus, which could potentially
make a similarly unbalanced media story in the Chinese papers. This would not
imply Chinese education is bad (it is not), just that the media can distort
things in the opposite way if they choose to do so.

There are real problems in UK
quantitative science education. What we really need to do in the UK is indeed
to excite more students about maths, the sciences, and engineering;
improve the number of good graduates going into teaching, and expand university
quantitative science resourcing, preferably by shutting down mumbo-jumbo
courses in “homeopathy”
and other wastes of time. But this story is just nonsense. Logically it is
almost as bad as setting British kids who have done some French a Russian
literature test from Russian schools and claiming that this is evidence for a
crisis in language teaching.

**Some
Tentative Information on Chinese School Mathematics**

Here is some material I found on Chinese High School
Mathematics (see the internet links below – I would appreciate independent
confirmation). Note the emphasis on geometry, and the last line which is a list
of things UK students *do* do, and which,
interestingly, is a list of topics kinda vital for experimental data analysis
and quantum mechanics, which tend to have a little bit to do with experimental
and theoretical chemistry.

“Chinese mathematics
education stresses the theory of functions, trigonometry

and solid geometry.

[Background] The Chinese mathematics curriculum for
high schools includes

four courses:

1. Sets and functions, including all elementary
functions--exponential,

logarithmic, trigonometric.

2. Solid geometry.

3. Analytical geometry.

4. Algebra: equations, combinatorics.

**No probability, statistics, data analysis,
calculus, matrix theory, etc.”**

So that rules out both the Schroedinger and Heisenberg
approaches to quantum theory I guess.

I have received also some
e-mails on this topic! So now I have to do the thing of issuing a
"clarification" that is longer than my initial comment. The following
should cover most of the e-mails I have had.

My comments
above should be taken ONLY as a criticism of the intellectual dishonesty
displayed by the RSC, and of the media's initial unqualified reporting of it,
in picking an example heavily biased in favour of Chinese students and against
UK students, in order to draw an inference. More generally any type of
anecdotal evidence is deeply questionable. But it would be entirely wrong for
anyone (especially HM Gov.) to cite me in any claim that there are no concerns
about the level of preparedness of UK science and maths students at university
entry. I taught undergraduates in Oxford between 1990 and 2005 and, as a result
of my experience of first year students, have very real concerns on this front,
which are no doubt shared by the RSC, in respect of UK university entrants. My own
experience (annoyingly anecdotal!) tells me that UK maths students were less
well prepared in 2005 than in 1990, for rigorous university study of
mathematics.

In this
respect, the phrase “this story is nonsense” I used above is a little too
easily extrapolated beyond what I intended. But this situation, although it is
what I have experienced, does give a very blinkered view.

That very
same block of experience also revealed that some Chinese students starting the
same course were very badly prepared in calculus and also probability, despite
having previously performed very well on an entrance test. They were wizards at
geometry and algebra. Hence my concerns about the particular approach of the
RSC. This debate needs to be intellectually honest. When it came to my own
personal teaching experience, the times I recall having to do
"catch-up" tutorials were with Chinese students who had not done UK A
levels, some students from North America and some International Baccalaureate
students.

That
anecdotal list brings me to another point. (For the record, there is no implied
criticism of the IB/US/Canada/China etc.) A second issue is the need for there
to be some national consensus about the level of specialization we as a nation
expect from our 6th form students. While I grew increasingly frustrated at the
level of mathematical preparation, I also saw many students who, rather than
the "Double Maths, Physics and Chemistry" I did years ago, who had
mixed maths (often single subject) with languages and literary A levels. I had
wanted to take French A level while at school but it was not possible to do
that with double maths in a school system that at that time was more rigid on
arts-science divisions. No doubt university teachers of literature are in a
similar state of despair at the state of things.

However, we
cannot rationally tell 6th formers on the one hand that they need to take a
broader set of options and at the same time moan at or about them for not being
so well prepared in any one speciality. At the same time HMGov must recognize
that it has to fund 4+ year degree programmes if it is going to push towards an
American model of secondary school preparation but not compromise the standard
of university graduates. The first year might then indeed, for UK
traditionalists, be "remedial maths". But I remain of the view that
it is very wrong to blame teachers and students for the current situation,
where we tell school kids to broaden what they study, moan about them when they
arrive at university, and then don't give them a full four years at university
to get to an uncompromised degree. This blame for that mess lies firmly at the
door of HMGov and nowhere else. Gordon Brown needs to be told very firmly to
cough up the cash.

When we
consider the very particular question of the meaning of a particular grade,
e.g. in A level maths, the debate on "dumbing down", "grade
inflation" so beloved of the media is really a separate matter and not one
on which I have proper evidence. Others who do should comment. But I will say,
again, that the comparisons need to be like-with-like wherever possible.
Another cheap trick is to dig out an old O level/A level maths paper and
confound current students with it. At A level you just have to find a question
on solving cubic and quartic equations and watch the chaos. (I am personally so
annoyed that this topic is lost from school maths that I made a point of
emphasizing it in my book "Complex Analysis with Mathematica".) Sadly
the requisite balancing act for the benefit of the media would require a time
machine, but the “thought experiment” of giving a paper on statistics to a
1930s A level maths student is a fun one. [A school mathematics teacher has
since e-mailed me to suggest that he would have foundered on a modern statistics
S2 paper, having done his 6^{th} form work around *1973*, and that many of his colleagues are now learning new material to
teach discrete mathematics.] Similarly we need to look at the big picture in
time. First I think it would help A level grade assessments and comparisons if
we just allocated a fixed percentage to the A, B, C etc. slots. Second, let us
not forget that undergraduates in maths and physics often take courses in
general relativity and quantum theory, today, that were at the forefront of research
in the first half of the 20th century. A student of mathematical finance
now at the first year graduate level might well look at how to improve on
Markowitz's 1990 Nobel prize-winning theory of portfolio optimization, which is
something of day to day interest for all kinds of pension funds and hedge
funds. I do have to ask myself how much it REALLY matters if four or so years
previously, these people now working close to front-line research did not do
Further Maths at A level on the way, or if they did, did not fully solve all
linear second order ordinary differential equations the way we used to, when
perhaps what matters more is that a devoted school teacher inspired them to
study the subject at university and beyond. I do not expect everyone to
agree with me on this and already have a mental list of people who would shoot
me for it – in one case literally.

Some PC
Waffle:

At the risk
of wandering off into philosophy and being branded a "PC
Educationalist" type there is really something else we need to debate. I
will introduce it by revealing an slightly contradictory aspect of my own
views. I find myself amusing myself in shops by picking up several things and
then presenting the person at the checkout with the exact change. It rarely
fails to get a "look" when I present them with the results of a
calculation done in my head when they have waited for the till to add it up. I
stand there wondering "Why can't they do this themselves". But then
again I recall the look of disgust I got from Dr (now Prof.) Douglas Gough in
the Institute of Astronomy in Cambridge, when during an interview do to
PhD in Astrophysics, he asked me how far away the nearest pulsars were and I
did not have a clue. But I was sitting there wondering why he thought I should
know something I can always look up in a book in a moment. We do live in
an age where knowledge is both massive and changing, and indeed exploding in
quantity, and we do have to confront the fact that our students might just be
better prepared for the 21st, 22nd centuries by, in addition, being educated
about how to find things out and how to use technology to solve problems,
rather than JUST knowing some hard stuff. In reality a balance is needed
between knowing, knowing how to find out, problem solving, and (with a
heavy heart), knowing how to get the computer to do it. We should be teaching
ALL of this properly (and preferably not as an uncritical dumbo
"look it up on Wikipedia" exercise, nor as a pointless "let's
make some pretty presentation in Power Point and add some gratuitous animation
and pointless 3D graph" exercise ). Teaching all of these skills properly
might take some time out from traditional "hard stuff", (although
there is perhaps some time to be found more effectively by first cutting
mumbo-jumbo, and hobbies disguised as subjects, from a crowded curriculum), but
if we want to stay competitive with China, India and other emerging powers this
is the way we have to go. I do not believe that dragging up biased examples
along the lines of the RSC will ever get us there, nor does it add to the
quality of the debate. I also remain appalled at the supermarket checkout.

**Internet Links**

Some of the following
internet links might help people understand why the RSC approach is flawed. I
have just started putting this together and would like very much to receive
other links to make this a more informed discussion.

1.
This on-line discussion
on a maths forum at a US university is probably worth checking out. It reflects
on US-China differences. You should see the discussion of the Chinese high
school syllabus at the bottom. While I would prefer to have some more detailed
and independent confirmation, the final comment is striking, and is probably a
good list of things you need to know to do the analysis of experimental data
and to do any kind of quantum theory (aka theoretical chemistry).

“No probability, statistics, data analysis, calculus, matrix theory,
etc.”

2.
This article written by a
Chinese researcher raises other issues.

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