Tony Gardiner: “The Man Who Knew Infinity”

The film The Man Who Knew Infinity  goes on UK general release from 8th April.

It is a compressed, and beautifully dramatised version of the theme treated more fully in Robert Kanigel’s double biography of the same name – which treats Ramanujan alongside a partial portrait of G.H.Hardy.
Mathematicians can be remarkably unforgiving about attempts to present mathematics to a general audience.  And Ramanujan’s story could so easily be cheapened – with awkward aspects being trivialised, in order to pander to current prejudices.  The Good News is that, not only has this been avoided, but the film manages to incorporate much of the detail and spirit of what we know, while using its dramatic freedom to confront important issues that are often either treated too tritely, or passed over in silence.  The project may have taken 10 years in the making, but the result has been worth it.
As someone who does not usually watch movies, I simply encourage everyone to see it
(perhaps several times), to encourage others to see it, and to use it to discuss the issues which it raises.
A film is not meant to be a reflection of reality.  This film would seem to be a fairly faithful representation of what we know in those areas where fidelity matters. In other respects it  exercises flexibility.  In contrast to Ramanujan, Dev Patel is slim and beautifully formed; yet he manages to capture an essential seriousness and devotion which is entirely plausible.  His wife is portrayed as older and I suspect much more beautiful than the real Janaki; yet her portrayal of profound simplicity is moving in a way that seems entirely appropriate (whether or not it is documented).
In his review for the February issue of the Notices of the AMS
George Andrews suggested that the film will help students appreciate the importance of “proofs”.  In fact, the struggle between proof and intuition, between Hardy and Ramanujan, is not so cleanly resolved, and there is a danger that the film may leave many strengthened in their belief in mathematical invention as “magical intuition”.  So the film should be used to actively encourage a deeper discussion of the relative importance of proof, and what is too often simply labelled “intuition” (as if it were not susceptible to, any further explanation).
Here is a chance to grapple with the often neglected interplay between
   (a) technical, or formal, training in universal methods – whereby my individual “mental
universe” is disciplined to fit with yours (or with some imaginary “Platonic ideal”),
   (b) our individual, idiosyncratic way of thinking about these shared objects and processes – whereby my thoughts avoid being mechanical replicas of everyone else’s, and so provide scope for originality.
Without the second, we are little better than machines.  And without the first, we are almost bound to go astray.
Almost all students need a significant dose of (a) before their (b)-type thoughts can become fruitful.  But some individuals’ (b)-type thoughts flourish – mostly unerringly – with relatively little (a)-type formalism. One thinks of Euler, or Schubert, or 19th century Italian algebraic geometers, or Feynman, or Thurston, or … .  The problem is then how to check the resulting claimed insights, to embed them within mathematics as a whole, and to make the methods available to the rest of us.  By neglecting such delicate matters we leave a vacuum that is too easily filled by half-truths.
Tony Gardiner

Response to Simon Jenkins

I have read his paper with mixed feelings:

Charge the maths lobby with the uselessness of its subject and the answer is a mix of chauvinism and vacuity. Maths must be taught if we are to beat the Chinese (at maths) (Only those arguments that can be linked to immediate pragmatism are regarded as worth voicing!). Or it falls back on primitivism, that maths “trains the mind”. So does learning the Qur’an and reciting Latin verbs. (So what? I would adore an education system that offers the opportunity of learning such things, provided that it is not compulsory. When I was 15 years old I was annoyed by the idea that I – as a child of the 20th century- had to miss the opportunity of learning Latin, so I took private Latin lessons. I was lucky enough that I was in the German highschool such that the wife of one of our teachers could teach me Latin. Later I did the same for Ancient Greek, too.)

Meanwhile, the curriculum systematically denies pupils what might be of real use to them and society. There is no “need” for more mathematicians. The nation needs, and therefore pays most for, more executives, accountants, salesmen, designers and creative thinkers. (Who has the priviledge to decide what the society needs? After all, those who have this priviledge are able to create these needs in the first place. So, it is a tautology.)

At the very least, today’s pupils should go into the world with a knowledge of their history and geography, their environment, the working of their bodies, the upbringing of children, law, money, the economy and civil rights.

This is in addition to self-confidence, emotional intelligence and the culture of the English imagination. (As if these attributes can be acquired in a way that is isolated from learning mathematics!) All are crowded out by a political obsession with maths.

The reason is depressingly clear. Maths is merely an easy subject to measure, nationally and internationally. It thus facilitates the bureaucratic craving for targetry and control. (With this part I agree. In fact, this is closedly connected with my above comment on “determining the needs”. Quantitative measurements and statistics are important to give the decisions an objective aura and disguise their unavoidably ideological nature. For this purpose, one has to make sure to raise statistics-literate generations, which is not what mathematics education means to me.)

Altogether the article has brought to my mind the verses from “Murder in the Cathedral” (T.S. Eliott):

The last temptation is the greatest treason:

To do the right deed for the wrong reason.

Higher Education and Human Capital

A few weeks ago I attended a UCU conference in London on the future of UK higher education which had one or two interesting speakers.  The most interesting for me was Andrew McGettigan, an academic who studies the economics and ideology of government education policy in depth.  He explained that because the current system of financing universities is unsustainable (70% of the student loan book will never be repaid) government ideologues are planning to introduce differential support to universities  for educating students in different subjects, which will depend on the expected  “added value” to the student of their degree, as measured  by increase in expected lifetime earnings.  Quite how this would be determined without the projections being hopelessly out of date is not apparent.  The idea would be to encourage the production by universities of “valuable” citizens who will be in a position to repay the cost of their education, since it is expected that universities will naturally try to maximise their income.  So on this model we can expect a big expansion of law, medicine, and accountancy departments and the virtual disappearance of nursing, history, and the arts.  I suspect pure mathematics might not do too well either.
I found a very interesting paper of McGettigan in which he explains the background to this part of his talk in detail.
See also the remarkable table at Fig. 12, frame 19, of the slides from McGettigan’s UCU talk which details the expected lifetime financial benefit of a first degree in various subjects. I reproduce the table here:
Individual net Lifetime benefit of undegraduate degrees

Individual net Lifetime benefit of undegraduate degrees

When did you stop?

I heard a famous French illustrator on the radio this morning and one of the thing he said strongly resonated with me. There were several versions of his background circulating in the press, publisher blurbs, web pages. In some of them he was an alumni of a famous Art School and in some of them he never had any formal training in drawing or painting.  When asked by the interviewer about it, he simply said he did not go to an art school. He remarked he was often asked about his training, for instance: “When did you start drawing ?” He usually turns that around:

“You see, most children start expressing themselves through drawings, from a very early age, at home, in kindergarten. I did that, too, that’s nothing remarkable. Usually during primary school, they don’t do it anymore. I just didn’t stop. I never stopped. I kept drawing every day, every kind of things, and it happened partly because my parents did not block me or frown upon this activity. And I still do it. So when people ask me that kind of question, I ask them back: when did you stop drawing?”

Hearing him, I recalled my own frequent feeling of powerlessness when I try do draw something or see the kind of work I would like to produce myself. I told a friend who was listening with me to the radio program: “I think that’s what I did with mathematics. I started early playing with numbers, object combinations, dots, lines, a compass, gridded paper, I never stopped, and I never asked for permission.”

That’s probably what I should have done with drawing. Thinking about my terrible mandatory middle-school art hours may give me an insight into what people experience in the ordinary math classes — and what disgusts them.

To all those people feeling bad about mathematics,
we could ask : when did you stop doing “it”?

I write “it”, because many activities that are profoundly mathematical are not recognized as such by teachers and family of young children, while art is more commonly seen as a continuum. Parents are more open to their child expressing themselves in pre-art activities, to the point it becomes a nuisance to everyone else.

If I follow the analogy with early childhood drawings, it suggests that when helping people who have a failed or non-existent relation with mathematics, most approaches start with an excessive level of sophistication, preconception and structuration. We take for granted cognitive processes, standard viewpoints, rhetorics and expectations most mathematicians have acquired unknowingly from many clues. We are the ones who “got it”. We expect to bring people to reconciliation and insight within a few hours of structured exposure, we do not help them practice some accessible, spontaneous, “proto”-mathematics that could be formative, nor do we really prepare and aim for life-long practice, enjoyment and learning.

I hope I could “restart” drawing, as if I was 3 years old, discovering the fun of playing with color pens and sheets of paper.

There are many entry points for mathematics, many of them we have yet to find.

There are several ways to relate to mathematics, many ways to excel at it. This is not so widely known. Alexander Borovik, in several of his books, describes people warming to mathematics very early or others in their late adolescence or young adulthood (entering the University). I am rather of the first kind. Furthermore, I tend to quickly identify and entertain ancient connections between what I am studying and doing now and what I felt and longed for when I was in my school years, even as a very young child. Part of it is probably a self-serving fabrication: I take pleasure into the sense of cognitive continuity it offers. Genealogy conforts me and provides valuable analogies and insights.

But another part is linked to the fact that academic published mathematics has always been to me an extensive, wonderful and bewildering area of mathematics, not the whole.  Before I was initiated to the global mathematical culture, I accumulated a store of pre-mathematical facts, experiences, tastes, concerns, implicit problems and naïve research programs that are still nagging me today. The corresponding perspective in art is very common: art is not restricted to what you can see in museums or what is labelled or publicized as such. You grow a sense of aesthetics, you look at some art and you see something that you always wanted to see or feel that something is missing. You know that art can be found almost anywhere, with various degree of sophistication, and that many starting points exist, many of them we have yet to find. I wish it were a more widespread opinion about mathematics too, especially among mathematicians.

Olivier Gérard

The Politics of Math Education

NYT Opinion Page, 3 December 2915, by

A quote:

The new math was widely praised at first as a model bipartisan reform effort. It was developed in the 1950s as part of the “Cold War of the classrooms,” and the resulting textbooks were most widely disseminated in the 1960s, with liberals and academic elites promoting it as a central component of education for the modern world. The United States Chamber of Commerce and political conservatives also praised federal support of curriculum reforms like the new math, in part because these reforms were led by mathematicians, not so-called progressive educators.

By the 1970s, however, conservative critics claimed the reforms had replaced rigorous mathematics with useless abstractions, a curriculum of “frills,” in the words of Congressman John M. Ashbrook, Republican of Ohio. States quickly beat a retreat from new math in the mid-1970s and though the material never totally disappeared from the curriculum, by the end of the decade the label “new math” had become toxic to many publishers and districts.

Though critics of the new math often used reports of declining test scores to justify their stance, studies routinely showed mixed test score trends. What had really changed were attitudes toward elite knowledge, as well as levels of trust in federal initiatives that reached into traditionally local domains. That is, the politics had changed.

Whereas many conservatives in 1958 felt that the sensible thing to do was to put elite academic mathematicians in charge of the school curriculum, by 1978 the conservative thing to do was to restore the math curriculum to local control and emphasize tradition — to go “back to basics.” This was a claim both about who controlled intellectual training and about what forms of mental discipline should be promoted. The idea that the complex problems students would face required training in the flexible, creative mathematics of elite practitioners was replaced by claims that modern students needed grounding in memorization, militaristic discipline and rapid recall of arithmetic facts.

The fate of the new math suggests that much of today’s debate about the Common Core’s mathematics reforms may be misplaced. Both proponents and critics of the Common Core’s promise to promote “adaptive reasoning” alongside “procedural fluency” are engaged in this long tradition of disagreements about the math curriculum. These controversies are unlikely to be resolved, because there’s not one right approach to how we should train students to think.

We need to get away from the idea that math education is only a matter of selecting the right textbook and finding good teachers (though of course those remain very important). The new math’s reception was fundamentally shaped by Americans’ trust in federal initiatives and elite experts, their demands for local control and their beliefs about the skills citizens needed to face the problems of the modern world. Today these same political concerns will ultimately determine the future of the Common Core.

As long as learning math counts as learning to think, the fortunes of any math curriculum will almost certainly be closely tied to claims about what constitutes rigorous thought — and who gets to decide. [Emphasis is by AB]

Christopher J. Phillips teaches history at Carnegie Mellon University and is the author of “The New Math: A Political History.”

Jo Johnson on graduate employment

Jo Johnson, Minister for Universities and Science, said in his recent speech
Higher education: fulfilling our potential:

We have all been reminded of the scale of the challenge by a recent CIPD survey suggesting that almost 60% of graduates are in non-graduate jobs.

While it may overstate matters — official statistics show that in fact only 20% of recent graduates did not find a graduate level job within 3 years of leaving college — it is clear that universities must do more to demonstrate they add real and lasting value for all students.

In my humble opinion, there are essentially two ways to improve the percentage of graduates finding graduate level jobs:

(a) Increase the number of  vacant graduate positions available, and

(b) decrease the number of graduates.

All other solutions are log-linear combinations of these two. The only option under control of universities is (b). Is this what Jo Johnson wants from the universities?

Added 11 September 2015: A detailed analysis of Jo Johnson’s speech is given by Martin Paul Eve in his post at THE blog, TEF, REF, QR, deregulation: thoughts on Jo Johnson’s HE talk.

The Secret Question (Are We Actually Good at Math?)

Posted on the AMS Blogs on September 1, 2015 by Ben Braun

“How many of you feel, deep down in your most private thoughts, that you aren’t actually any good at math? That by some miracle, you’ve managed to fake your way to this point, but you’re always at least a little worried that your secret will be revealed? That you’ll be found out?”

Over half of my students’ hands went into the air in response to my question, some shooting up decisively from eagerness, others hesitantly, gingerly, eyes glancing around to check the responses of their peers before fully extending their reach.  Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren’t raised looked around in surprised confusion at the general response.

The use of the term ‘Expected Frequency’

The June 2015 GCSE Subject Level Conditions and Requirements for Mathematics includes (P3)

“relate relative expected frequencies to theoretical probability, using appropriate language and the 0 – 1 probability scale”

and this leads to questions like

“If you rolled a die 600 times, how many sixes would you expect to get’.

which is taken from the CIMT MEP Pupil’s textbook on probability, and is given the answer

‘You would expect to get a 6 in 1/6 of the cases, so 100 sixes’.

This seems a confusing and misleading term. What exactly is an ‘expected frequency?’ The obvious meaning is the frequency that you expect. But we are trying to support the concept of a random variable, with ideas that a random variable is unpredictable in terms of value, that values do not form patterns or sequences, and can only be forecast and predicted in some general ways.

If you roll a die 600 times, I do not expect any value for the number of sixes. That is the most significant aspect of a random variable.

The implied sub-text is that

Expected frequency = probability X number of trials

So that, for example, if we toss a fair coin 100 times, what is the expected frequency of heads? Well, 50. So does that mean we expect to get 50 heads? This is a Bernoulli trial, and the probability of getting precisely 50 heads in 100 tosses is about 0.08. So we would need to say to a pupil

‘The expected frequency is 50; but it is unlikely that you would get 50 heads’

which hardly makes sense.

The probability of 51 is about .078, and 52 is .074. So, of course, 50 is the most likely frequency.

The phrase ‘most likely frequency’ is straight-forward, makes sense, and says what it means, unlike ‘expected frequency’.

Please can we stop using the phrase ‘expected frequency’?

The Great Mystery of Malta’s Learning Outcomes Framework

Important update below: it is no longer a mystery.

Malta’s new Learning Outcomes Framework is an important case study of the European Union’s approaches to implementation of its education policies in member countries. For that reason the Framework deserves a close attention.

An attempt to study the official website

immediately leads to a question:

Who had actually developed the Framework?

According to Wikipedia, population of Malta is about 445,000. When compared with the City of Manchester (about 514,000), it becomes clear that development of the Framework is a job beyond capabilities of a small nation.

So, external consultants were hired, some institutions or companies from English speaking parts of Europe. Taking into consideration traditional cultural connections, this part of Europe is likely to be the UK.

Added 24 August 2015:  Indeed I could not locate contractor’s names using advanced Google search on,but  serendipitously discovered their logos in the document Joint Venture Presentation dated 28  Jan 2015:

IoE_24Aug15Outlook Coop is a company on Malta specialising in project management with expertise in EU funded projects. 

East Cost Education Ltd is a small private company based in Northumbira with specialism, judging by their website,  concentrated mostly in vocational education and training. In recent years, they worked on Malta on several projects in vocational training.

Institute of Education, London, is

the world’s leading centre for education and applied social science.

Outcome Based Education

In 1990, South Africa regarded Outcome Based Education (OBE) as its preferential educational paradigm, and designed Curriculum 2005. The South African Department of Education was very influenced by William Spady — an American proponent of OBE, who visited South Africa as a consultant on the issue. The National Qualification Framework went into effect in 1997 with great expectations, but these expectations were not met. It became evident even to the most vocal OBE-proponents that the educational approach gave inculcate skills not conducive to pursue any university education in mathematics and science. Since then, the curriculum underwent several corrections, and now is at stage of Curriculum Schooling 2025. Meanwhile, William Spady distanced himself from the South African version of OBE, describing it as a professional embarrassment:

“So now, with a decade of confusion about OBE behind us, I would encourage my South African colleagues to stop referring to OBE in any form. It never existed in 1997, and has only faded farther from the scene since. The real issue facing the country is to mobilize behind educational practice that is sound and makes a significant difference in the lives of ALL South African learners. Empty labels and flowery rhetoric are no longer needed; but principled thinking and constructive action are.”

Educational experts may argue whether it was Outcome Based Education, or some kind of Education Based on Outcomes. These experts may further argue on the terminology, but the fact remains it was supposed to be transformational OBE. A close look at their mathematics curriculum reveals that it is not so different from the proposed new Learning Outcomes Framework (LOF) for school mathematics in Malta, and in some aspects is even better. What is however completely identical in both is the educational utopia of outcomes coming from nowhere.

Essential mathematical skills are not just about a computational answer, for it is not the answer that is of the greatest importance to school children’s mathematical development. Rather it is children’s ability to apprehend mathematics as a conceptual system. Many education systems are emphasising on this, here is an excerpt from the Secondary Mathematics Syllabuses in Singapore:

“Although students should become competent in the various mathematical skills, over-emphasising procedural skills without understanding the underlying mathematical principles should be avoided… Students should develop and explore the mathematics ideas in depth, and see that mathematics is an integrated whole, not merely isolated pieces of knowledge.”

Unfortunately, in Malta’s case the design falls far short of such goals. Here is an example from level 5:

(COGNITIVE LEARNING) 16. I understand that multiplication is repeated addition.

Accordingly, a factor can only be added to itself a counting number of times. In Singapore’s Primary Mathematics Syllabus, multiplication and division are conceptualised gradually, and still on that level are introduced area and various square units. In contrast, square units are not present in Malta’s LOF for school mathematics. In fact, the proposed LOF is teeming with conceptual deficiencies. For instance, there is some kind of misconception between “equation” and “function”. Equations were never related to unknown variables, while functions are assumed to be somehow equations between the variables “x” and “y“. Use of radian measurement is not present, but learners are supposed to “plot graphs of trigonometric functions”.

Perhaps, Malta can learn from Singapore’s remarkable success since independence and the policies underlying its achievements in mathematical education.