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:

Screen shot of Wolfram Alpha, 01 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:

Screen shot of Wolfram Alpha, 07 May 2016

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

BBC reported today 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?

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

Alexandre Borovik

Three PhD studentships at Loughborough

The Mathematics Education Centre at Loughborough University has three fully-funded PhD studentships available to start in October 2016. Each project is full time for three years.

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

Tony Gardiner Receives the 2016 Award for Excellence in Mathematics Education

Citation for the 2016 Award for Excellence in Mathematics Education to

Dr. Anthony David Gardiner

It is with great pleasure that the Award Committee hereby announces that the 2016 Award is given to Dr. Anthony D. Gardiner, currently retired from University of Birmingham, United Kingdom, in recognition of his more than forty years of sustained and multiple major contributions to enhancing the problem-solving skills of generations of mathematics students in the United Kingdom (UK) and beyond.

Gardiner’s major achievements include:

  • orchestrating teams of volunteers from many constituencies, including teachers, mathematics educators and university mathematicians, to create a portfolio of mathematics contests, leading eventually to the creation of the UK Mathematics Trust, which creates problem-solving challenges taken by well over half a million students per year;
  • creating structures that dramatically increased and broadened participation in mathematics competitions and other activities supporting UK participation in the International Mathematics Olympiad;
  • leading the UK IMO team (1990 – 95);
  • creating problem solving journals for school students (including grading thousands of solutions personally), leading eventually to the Problem Solving Journal for Secondary Students (edited by Dr. Gardiner since 2003, with a circulation over 5,000);
  • authoring 15 books on mathematical thinking and mathematical problem solving, including Understanding Infinity, Discovering Mathematics: the art of investigation, Mathematical Puzzling (all reprinted by Dover Publications), the four volume series Extension Mathematics (Oxford), and the recent Teaching mathematics at secondary level (Open Book Publishers).

In addition, Gardiner’s expertise on the problem-solving abilities of English schoolchildren, and his insights into omissions in UK mathematics education has led to his being consulted by multiple UK Ministers of State for Education, and have influenced significant changes in the UK mathematics curriculum. Gardiner has also served in multiple high level leadership positions in mathematics education both in the UK and internationally, including Council of the London Mathematical Society, and member of the Education Committee (1990s), Presidency of the (UK) Mathematical Association in 1997-98, chair of the Education Committee of the European Mathematical Society (2000-04), and Senior Vice President of the World Federation of National Mathematics Competitions (2004-08). He has addressed major teacher conferences in more than 10 countries, and he was an Invited Lecturer at the 10th International Congress of Mathematics Education in 2004. He has organized many meetings and programs to support mathematics education, teacher professional development, and to promote problem solving. He has contributed numerous articles to newspapers and magazines to communicate the goals of successful mathematics education to a broader public. Both the extent and impact of Gardiner’s efforts are remarkable. He provides an inspiring example of how a mathematician can have a positive impact on mathematics education; he is a most worthy recipient of the Texas A&M Award for Excellence in Mathematics Education.

Gardiner received his doctorate in 1973 from the University of Warwick, UK. He taught at the University of East Africa from 1968-69, University of Birmingham from 1974 to 2012. During that time he worked at the Free University of Berlin on a fellowship, and held numerous visiting positions including at the University of Bielefeld in Germany, University of Waterloo, the University of Melbourne and the University of Western Australia.


 

This  award  is  established  at the Texas  A&M  University to  recognize  works  of  lasting significance  and  impact  in advancing  mathematics  education  as  an  interdisciplinary field  that  links mathematics,  educational  studies  and  practices.  In  particular,  the award  recognizes major  contributions  to  new  knowledge  and  scholarship  as  well  as exemplary contributions  in  promoting  interdisciplinary  collaboration  in  mathematics
education.
This  is  an  annual  award  that  consists  of  a  commemorative  plaque  and  a  cash  prize ($3000).  A  recipient  will  be  selected  yearly  and  will  be  invited  to  give  a  keynote  talk, with  all  travel  expenses  covered,  at  a  workshop  dedicated  to  advancing  mathematics education.  Moreover,  subject  to  the  availability  of  the  recipient,  a  housing  allowance and  a  $5000  stipend  will  also  be  provided  to  the  recipient  to  spend  two  weeks  in residence  at  Texas  A&M  University  interacting  with  students  and  faculty  in  seminars and  informal  mentoring  sessions.

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.

Simon Jenkins: Our fixation with maths doesn’t add up

in The Guardian, Thursday 10 March 2016. A random paragraph:

There is nothing, except religion, as conservative as a school curriculum. It is drenched in archaic prejudice and vested interest. When the medieval church banned geography as an offence against the Bible, what had been the queen of the sciences never recovered. Instead Latin dominated the “grammar” curriculum into the 20th century, to the expense of all science. Today maths is the new Latin.

Read the full article. Refutation anyone?

 

A. Borovik: Sublime Symmetry: Mathematics and Art

A new paper in The De Morgan Gazette:

Form the Introduction:

This paper is a text of a talk at the opening of the Exhibition Sublime Symmetry: The Mathematics behind De Morgan’s Ceramic Designs  in the delighful Towneley Hall  Burnley, on 5 March 2016. The Exhibition is the first one in Sublime Symmetry Tour  organised by The De Morgan Foundation.

I use this opportunity to bring Sublime Symmetry Tour to the attention of the British mathematics community, and list Tour venues:

06 March to 05 June 2016 at Towneley Hall, Burnley
11 June to 04 September 2016 at Cannon Hall, Barnsley
10 September to 04 December 2016 at Torre Abbey, Torbay
10 December 2016 to 04 March 2017 at the New Walk Gallery, Leicester
12 March to 03 September 2017 at the William Morris Gallery, Walthamstow

William De Morgan, Peacock Dish. The De Morgan Foundation

William De Morgan, Peacock Dish. The De Morgan Foundation.

UK Mathematics Trust: Teacher Meetings

These one-day professional development seminars, held at various venues throughout the United Kingdom, offer maths teachers the opportunity to discover inspirational ideas for motivating pupils across the ability range. They provide a number of ways in which teachers can continue their professional development and all delegates will benefit from:

  • Developing ideas for making maths fun and engaging for students;
  • An interesting day out of the classroom with ideas for creating an engaging and motivational learning environment;
  • Receiving a delegate pack filled with resources to take back to the classroom, along with a CPD attendance certificate;
  • Meeting other mathematics professionals from their region and beyond, with the time to discuss and exchange best practice.

The dates and venues of the 2016 teacher meetings (with links to agendas/speakers) are as follows:

  • 20 May:  Edinburgh University of Edinburgh
  • 10 June: Belfast W5 Odyssey, Belfast
  • 21 June: Greenwich Greenwich Campus, University of Greenwich
  • 24 June: Cardiff Cardiff City Hall
  • 27 June: Cambridge Centre for Mathematical Studies, Cambridge
  • 1 July:    Leeds University of Leeds

Would you like to book a place? Click here for the booking form.