Portrait of a Mathematician, I

I am moving to my Selected Passages From Correspondence With Friends blog a collection of portraits of mathematicians which was spread on some my old blogs. But this is a new entry:

Musa Diplomatcia (Karin Kosina), by Fernando Mircala

Karen Cosina is a diplomat and an information security expert. The latter makes her a mathematician.


Physical intuition in the imaginary world

Paging through a wonderful book “An imaginary tale: The story of \(\sqrt{-1}\)” by Paul J. Nahin (strongly recommended!), I discovered this episode of history.

On 18 October 1740 Euler wrote to John Bernoulli that the solution to differential equation of a harmonic oscillator

\(y”+y=0\),  \(y(0)=2\), \(y'(0)=0\)

can be written in two ways:

\(y(x) = 2 \cos x\)


\(y(x) = e^{ix} + e^{-ix}.\)

He concluded from that

\(2\cos x =e^{ix} + e^{-ix}.\)

which was first step to his famous formula.

Obviously, Euler was using the uniqueness of a solution with given initial values. I bet his belief in the uniqueness was rooted in physical intuition. For him, expansion of mathematical language did not change his vision of the world.

Perhaps, 2oth century physicists weret thinking that an “imaginary” solution corresponds to something in the real world, something that was not discovered yet.


Sexism in action

[I repost my old post from now defunct blog. I wonder how much has changed over the years that passed?]
I had a conversation with a colleague from a Mathematics Department in a decent British University. Her Department adopted a remarkable policy in respect of pastoral care of students: all undergraduate students are assigned as personal tutees to 6 or 8 members of staff (of 32). My colleague, as a result, has about 60 personal tutees.

Of 32 members of academic staff, 3 are women. As the reader perhaps already expects, all of them got tutees. Apparently, pastoral care is considered to be women’s natural duty.

This shameful episode is a manifestation of a general principle that a “care” component, and, more generally, a “person-to-person interaction” component of work, so prominent in teaching, is systematically undervalued – and underpaid.

I quote from Paula England, Emerging theories of carework, Annu. Rev. Sociol. 2005. 31: 381–99 ; doi: 10.1146/annurev.soc.31.041304.122317 :

In more recent work, England and colleagues (2002) operationalized care work as those occupations providing a service to people that helps develop their capabilities. The main categories of jobs termed care work were child care, all levels of teaching (from preschool through university professors), and health care workers of all types (nurses aides, nurses, doctors, physical and psychological therapists). Controlling for skill demands, educational requirements, industry, and sex composition, we found a net penalty of 5%–10% for working in an occupation involving care.

One of the reason why “care component” is penalised because it is considered a more feminine function. The conclusion is striking:

We (male teachers) suffer from a typically anti-female form of discrimination.

Perhaps, this can help to convince even the worst male chauvinist pig that we all have to fight for gender equality in the workplace.


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.