University mathematics education: two worlds

What follows is a table of distribution of UCAS Tariff Scores actually achieved by entrants into Mathematics at Cambridge and Wolverhampton in 2014 (taken from the official source, https://unistats.direct.gov.uk/Compare-Courses) :


Cambridge Wolverhampton
< 120 0% 10%
120 – 159 0% 5%
160 – 199 0% 25%
200 – 239 0% 25%
240 – 279 0% 15%
280 – 319 0% 5%
320 – 359 0% 5%
360 – 399 0% 5%
400 – 439 1% 0%
440 – 479 2% 5%
480 – 519 4% 0%
520 – 559 12% 0%
560 – 599 13% 0%
600+ 68% 0%


Let us do some aggregation:

Cambridge Wolverhampton
< 400 0% 95%
400 – 479 3% 5%
480+ 97% 0

100% of new students at Wolverhampton are within the range of the lowest 3% at Cambridge and 95% at Wolverhampton are below the Cambridge range
entirely. And this does not include STEP, compulsory at Cambridge.
We have to accept that Cambridge and Wolverhampton belong to two different nations separated by the deep socio-economic, class and caste schism. This is a “first world / third world” division. How one could measure “learning outputs” at the two universities using the same numerical scale when *inputs* are so different?



Decoupling of assessment

BBC reported on 2 May 2016 that

Thousands of parents in England plan to keep their children off school for a day next week in protest at tough new national tests, campaigners say.

Parents from the Let Our Kids Be Kids campaign said children as young as six were labelling themselves failures.

In a letter to Education Secretary Nicky Morgan, they said primary pupils were being asked to learn concepts that may be beyond their capability.

The government said the tests should not cause pupils stress.

These new tests, known as Sats, have been drawn up to assess children’s grasp of the recently introduced primary school national curriculum, which is widely considered to be harder than the previous one.

The letter from the campaign, which says it represents parents of six- and seven-year-olds across the country, says children are crying about going to school.

There is a simple solution – decoupling of assessment of schools from assessment of individual children.

As far I remember my school years back in Soviet Russia of 1960s, schools there were assessed by regular (but not frequent) “ministerial tests”. A school received, without warning, a test paper in a sealed envelope which could be open only immediately before the test; pupils’ test scripts were collected, put into an enclosed envelope, sealed and sent back. Tests were marked in the local education authority (and on some occasions even a step up in the administrative hierarchy — in the regional education authority); marked test scripts, however, were not returned to schools, and schools received only summary feedback — but no information about performance of individual students.

This policy of anonymised summary tests created a psychological environment of trust between pupils and the teacher — children knew that it was not them who were assessed, but their teacher and their school, and they tried hard to help their teacher. Good teachers could build on this trust a supportive working environment in a classroom.  Schools and teachers who performed well in such anonymised testing could be trusted to assess pupils in a formative, non-intrusive, non-intimidating way — and without individual high stakes testing.

Of course, all that are my memories from another historic epoch and from the country that no longer exists. I could be mistaken in details, but I am quite confident about the essence. In this country and in recent years, I happened to take part in a few meetings in the Department for Education, where I raised this issue. Education experts present at these meetings liked the idea but it was not followed by any discussion since it was outside of meetings’ agenda — we had to focus on the  content of the new curriculum, not assessment. I would love to see a proper public discussion of feasibility of decoupling.

I teach mathematics at a university. I think I am not alone (I heard similar concerns from my colleagues from Universities from all over the country) in feeling that many our students come to university with a deformed attitude to assessment — for example, with subconscious desire to forget everything as soon as they have sat an exam. It could happen that some of them, in their school years, suffered from overexamination but were not receiving  sufficient formative feedback. At university, such students do not know how to use teachers’ feedback. They do not know how to ask questions. Could it happen that the roots of the problem could be traced back to junior school?


Spoiler: 2016 as the sum of 3 squares, by mental arithmetic

Dave Radcliffe  @daveinstpaul   twitted:

\(2016\) is the sum of four squares. This exceptional event occurs only \(100\) times each century.

I commented:

It is a good idea to start the New Year Day by finding these four squares. This year, it is easy. And 3 squares suffice.

and added:

This year, finding the four squares can be done by mental arithmetic Honest! Try!

So, here comes a spoiler, intentionally written with minimal mathematics notation from what I first did entirely by mental arithmetic. Indeed, observe that

  •  \(2016= 2000 + 16\) and that \(16 = 4^2\);
  • \(2000\) is divisible by \(16\) because \(2000\) is \(2 \times 10^3\), hence \(2000 = 2 \times 2^3 \times 5^3 = 2 \times 8 \times 125 = 16 \times 125\);
  • hence taking out  \(16\) out of  \(2000 +16\) simplifies the problem;
  • now \(2016 =  16 \times (125 + 1)  = 16 \times 126 = 4^2 \times 126\);
  • all that remains to do is to write \(126\) as the the sum of four or less squares and then multiply each of them by \(4^2\).

Here we start trying our luck.

  • The largest square smaller that \(126\) is \(9^2 = 81\), and \(126 = 9^2 + 45\).
  • Similarly, \( 45 = 6^2 + 9 = 6^2 + 3^2\)
  • Ha! Now  \(126 = 9^2+ 6^2 +3^2\). Multiply everything by \(4^2\) and we get \(2016 = 36^2 + 24^2 + 12^2\).

So that was what I did by mental arithmetic.

However, mental arithmetic is not optimal way of solving. In calculations above I made an error and an omission which, fortunately, were not lethal, but which I noticed only now, while writing up my mental solution.

  • \(9^2 = 81\) is not the largest square smaller than \(126\); there are two others, \(10^2 = 100\) and \(11^2 = 121\), leading to decompositions \(126 = 10^2 + 5^2 +1^2\) and \(126 = 11^2 + 2^2 +1^2\), and to corresponding decompositions of \(2016\).
  • I stopped looking for square factors too early, missing \(126 = 9 \times 14\) with \(14 = 3^2 + 2^2 +1\), instantly yielding the decomposition \(2016 = 36^2 + 24^2 + 12^2\).
  • notice that we can make four non-zero square instead of three by observing that \(10^2 = 6^2 + 8^2\) and \(126 = 10^2 + 5^2 +1^2 = 8^2 + 6^2 +5^2 +1^2 \).

What is the moral of that story? It illustrates something that Tony Gardiner calls structural arithmetic, see his paper Teaching mathematics  at secondary level. This is Key Stage 3 and 4 material, and, in mathematics learning, could be  an excellent preparation to elementary algebra.  As said earlier, mental arithmetic is not optimal way of solving arithmetic problems, but structural arithmetic, with pencil and paper, is.


Stalin on Mathematics

A paper


contains a tiny, but exceptionally important piece of evidence of Stalin’s attitude to mathematics:

Текст доклада Лысенко первоначально состоял из 10 разделов и занимал 49 страниц. Сталин зачеркнул весь второй раздел доклада, который имел название “Основы буржуазной биологии ложны”, сохранив в нем только абзац с критикой физика Э.Шредингера и написав против него на полях: “ЭТО В ДРУГОМ МЕСТЕ”.37 [Примечание 37: Прописными буквами дан текст, написанный Сталиным собственноручно на первом варианте доклада Лысенко.] В этом же разделе Сталиным было подчеркнуто положение: “Любая наука — классовая” — и на полях написано: “ХА-ХА-ХА… А МАТЕМАТИКА? А ДАРВИНИЗМ?”.

So, Stalin did not believe into the class nature of mathematics. This had profound impact on the fate of mathematics in the Soviet Union. Not every direction  in science was so lucky.

Full text of paper by Esakov

Source: В.Д.Есаков. Новое о сессии ВАСХНИЛ 1948 года //
Репрссированная наука, вып.II, СПб.: Наука, 1994, с.57-75.


The only way to learn mathematics …

It is a well known dictum:

The only way to learn mathematics is to do mathematics.

I would make it a bit more sharp:

The only way to learn maths is to use it for learning more advanced maths.

This is because to learn mathematics, we have to use it, and the is no more challenging and more stimulating use of mathematics than learning the next level, more advanced

As a corollary we get The Law of Supernumerary  Learning of Mathematics:

To be able to use mathematics at a certain level you have to learn it at the next level.

The law has an obvious empirical confirmation: graduates of mathematics departments of British Universities (BSc or MMath Degree) are hired by banks and insurance companies to do office jobs that require high school level mathematical skills. When big employers really need university level mathematical skills, they advertise for people with MSc or PhD in mathematically intensive disciplines.

In respect of teaching of mathematics, The Law of Supernumerary Learning of Mathematics implies that middle- and high school mathematics teachers should have an university degree in mathematics or mathematically intensive discipline, and primary school teachers
— at least good middle school level (such as GCSE) or high school level education in mathematics.


Some motivational slogans

From a  colleague’s letter:

Here the teaching has finished and the exam period has started. I had some midterms couple of weeks ago in which a student was trying to prove \(\sqrt{2}\) is irrational. they defined a rational number \(\frac{a}{b}\) with \(b=0\).

I think at that point mathematics upped and left the country. I had another who had made some calculation mistake in the first step of a proof by induction, and ended up with \(8>9\). They dutifully then marked it as \(P(1)\) holds and continued.

I had others that started with \(8 \mid 5^{2n} -1\)  (\(n>1\) ) and put it “equal” to many things and ended up with
\dots = \frac{8}{5^{2n} -1 } = 3
At this point I think logic decided to follow maths out of the country …

What can I say?

Keep calm and carry on.

Wikipedia says about the now famous poster:

“Printing began on 23 August 1939, the day that Nazi Germany and the USSR signed the Molotov–Ribbentrop Pact, and the posters were ready to be placed up within 24 hours of the outbreak of war. Almost 2,500,000 copies of Keep Calm and Carry On were printed between 23 August 1939 and 3 September 1939 but the poster was not sanctioned for immediate public display. “

Apparently the Blitz was not judged to be desperate enough situation — I personally see a great moral lesson in that, especially for us, mathematics teachers.


A response to “Malta: new “Learning Outcomes Framework””

Thank you Alexandre for taking an interest in the curriculum being developed for the Maltese schools. (As a matter of information, this curriculum is being developed by a consortium of foreign “experts” supported by a European Social Fund grant. What is shown on the website is work-in-progress, and one hopes that the final product will be a more coherent curriculum and banalities like the one you pointed out will have been removed.)
So, let me share my answers to the same question you ask, basically why does this draft curriculum contain such a statement: I can use equivalent fractions to discuss issues of equality e.g. gender. I agree with your two responses, namely mis-use of vocabulary and the strictures imposed by an Outcomes Based (OB) curriculum. But allow me to elaborate further.
In my view, the above statement would be banal whether one uses the term “equivalent fractions” or “similar fractions” or any other notion which extrapolates from 1/2=2/4=3/6=etc to anything having to do with gender equality. The problem, in my opinion, is that some people do not realise that, in science, we expropriate a word from everyday vocabulary to use in a context which does have some similarity to the everyday use of the word, but whose meaning becomes something technical which cannot be exported back to the everyday sense of the word.  I sometimes taught classes of Arts students who felt they needed to use some mathematical jargon in their essays (a few years ago the fashionable thing to do was to drop the words “chaos” and “fractal”). One of my usual examples of how wrong this is involved the use of the word “work”, as used in science and in everyday life. Translated into the context of curricula, the analogous banal statement could be something like: I can calculate the work done by a given force moving an object through a given distance and I can use this to discuss the conditions of work in factories and industry. 
What surprises me when statements such as the one on gender equality are made is that while the ambiguity of language is appreciated outside science, in fact it can be a wonderful tool in the hands of a good writer, when transporting scientific vocabulary back into the everyday world, this variegated meaning of the same word in different contexts is sometimes forgotten. I have no explanation why this happens.
But another problem with curricula written in OB style and which could have a bearing on such wording is the necessity that the statements should be written in a way that the learning child would write them, for example, by starting the description of each outcome with “I can…” That sentences such as the one you quote about gender issues crop up is not, in itself the main problem, in my opinion. Such sentences can be edited out when reviewing the curriculum. The problem, as I see it, is that this style excludes the possibility that the curriculum contain concepts to guide the teacher but which the student would not likely be able to express. So take your improved statement of how mathematics can help understand social inequalities:
I believe in the power of mathematics and I am convinced  that comparing numbers (for example, salary)  reveals a lot about gender inequality (and other, frequently hidden,  inequalities in the world — just recall the Oaxaca Decomposition and its role in fight against discrimination of any kind).
It might be reasonable to expect a Level 5 student (aged 7-8) to express such a statement up to “gender inequality”, but hardly the rest of the statement, although the writer of the curriculum might very well want to make a reference to the Oaxaca Decomposition to give the teacher an example of a highly non-trivial use of mathematics in this context.
This OB format, I believe, betrays a fallacy about the teaching of mathematics, namely that teaching elementary mathematics to 7-year olds, say, does not involve deep knowledge of mathematics, certainly not deeper than what a 7-year old can express.
I look forward to reading other comments, especially by readers of this blog who are more familiar with OB curricula than I am.