# Peter Gates: Teaching Mathematics for Social Justice: Meaningful Projects for the Secondary Mathematics Classroom

I’d like to draw your attention to a new book: ‘Teaching Mathematics for Social Justice: Meaningful Projects for the Secondary Mathematics Classroom’. The aim of the book is to share teaching resources and ideas generated from the TMSJ Research Project (a participatory action research project). The book was published by the Association of Teachers of Mathematics in April 2016.
The book is:

* Aimed at teachers of mathematics who are interested in addressing issues of social justice in their classrooms.
* Based on the premise that conventional approaches to teaching maths do not adequately address the needs of all learners or the needs of society as a whole.
* Suitable for students in Key Stages 3 and 4, those studying the new ‘core mathematics’ curriculum and for those on post-compulsory numeracy courses.
* Written in a style that allows teachers to use the ideas in a flexible, creative and non-prescriptive way.
The book contains:

* Seven projects addressing issues of social justice in the mathematics classroom;
* Twenty task sheets designed to be photocopied for students;
* Teachers’ notes offering ideas for supporting and developing classroom practice;
* Six accessible research articles exploring the theories underlying the teaching ideas.

Further details of the book can be found on:
http://maths-socialjustice.weebly.com/teaching-mathematics-for-social-justice-book.html

and on the ATM website:
https://www.atm.org.uk/shop/teaching-maths-for-social-justice-book-and-pdf/act099pk

Dr Peter Gates

# Graham Brown-Martin: Disrupt Assessment

An important post by Graham Brown-Martin. A quote:

The notion that the assessment tail wags the dog of learning seems so illogical and yet it drives the entire process of educating our children as they get processed through the conveyor belt of the school system.

Work hard, get good grades, go to university, get a good job. Why do we continue to collude in this illusion when even a degree from the best university doesn’t guarantee wellbeing and employment for life?

“This is clearly the wrong way around and yet this reality is beyond humour and sets a ticking time bomb for present and future generations in what will be their imperative to reimagine society to solve the challenges of their generation.”

# Mathematics in the news this week

France DGSE: Spy service sets school code-breaking challenge

France’s external intelligence service, the DGSE, has sponsored a school competition to find the nation’s most talented young code-breakers.

It is the first time the DGSE has got involved in such a project in schools.

The first round drew in 18,000 pupils, and just 38 competed in the final on Wednesday, won by a Parisian team.

“The main message is mathematics is not about numbers and figures,” [Mark] Saul said. “It’s about figuring things out. Whenever you’re figuring something out, you’re doing something mathematical.”

Rebecca Hanson has opened her agency Authentic Maths to help Primary School Teachers in the UK offering solutions to the difficulties being experienced with the implementation of the Government’s changes to the primary mathematics curriculum.

UK follows Russia’s example to set up specialist sixth form maths colleges:

A key figure in the establishment of specialist maths institutions in the UK was Baroness (Alison) Wolf, a professor at King’s College London. She knew about Russian maths skills because of her work in universities, where maths departments often attract a fair few Russian academics.

Initially, the idea in the UK was for universities to set up a nationwide network of specialist maths schools. However, only King’s College London and Exeter have taken the plunge.

# Why undegraduate students should not use online matrix calculators

Since 1 April 2011 I from time to time was trying to convince Wolfram Alpha to fix a bug in the way they computed eigenvectors, see my post of 28 April 2012. It survived until May 2016:

As you can see, Wolfram Alpha was thinking that the zero vector is eigenvector. On 5 May 2016 this bug was finally fixed:

But there is still one glitch which can send an undergraduate student on a wrong path. The use of round brackets as delimeters for both matrices and vectors suggests that the vector $$(1,0)$$ is treated as a $$1 \times 2$$ matrix, that is a row vector. This determines which way it can be multiplied by a $$2 \times 2$$ matrix: on the right, that way:
$(1,0) \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right)$
and not that way
$\left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right)(1,0),$
the latter is simply not defined. Therefore the correct answer is not
$\mathbf{v}_1 = (1,0)$
but
$\mathbf{u} = (0,1) \quad\mbox{ or }\quad \mathbf{w} = (1,0)^T = \left(\begin{array}{c} 1 \\ 0\end{array}\right),$
depending on convention used for vectors: row vectors or column vectors. Indeed if
$A = \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right),$
then
$\mathbf{v}_1A = (1,0)\left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) = (1,2) \ne 1\cdot \mathbf{v}_1,$
while
$A\mathbf{w} = \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) \left(\begin{array}{c} 1 \\ 0\end{array}\right) = \left(\begin{array}{c} 1 \\ 0\end{array}\right) = 1\cdot \mathbf{w}$
and
$\mathbf{u} A = (0,1) \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) = (0,1) = 1\cdot \textbf{u}.$
The bug is likely to sit somewhere in the module which converts matrices and vectors from their internal representation within the computational engine into the format for graphics output. It should be very easy to fix. It is not an issue of computer programming, it is just lack of attention to basic principle of exposition of mathematics and didactics of mathematics education.

# Alexandre Borovik: Decoupling of Assessment

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?

Disclaimer. The views expressed do not necessarily represent the position of my employer or any other person, organisation, or institution.

Alexandre Borovik

# 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”),
and
(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:

# 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.

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.”