Ronnie Brown: from Esquisse d’un Programme by A Grothendieck

I just came across again the following (English translation):
 The demands of university teaching, addressed to students (including
those said to be “advanced”) with a modest (and frequently less than mod-
est) mathematical baggage, led me to a Draconian renewal of the themes
of reflection I proposed to my students, and gradually to myself as well.
It seemed important to me to start from an intuitive baggage common to
everyone,  independent of any technical language used to express it,  and
anterior to any such language
– it turned out that the geometric and topo-
logical intuition of shapes, particularly two-dimensional shapes, formed such
a common ground.
(my emphasis)
It seems to me a good idea, and expressed with AG’s usual mastery of language.
Ronnie

Chinese maths textbooks to be translated for UK schools

The Guardian, 20 March 2017. Some quotes:

British students may soon study mathematics with Chinese textbooks after a “historic” deal between HarperCollins and a Shanghai publishing house in which books will be translated for use in UK schools.

 

HarperCollins signs ‘historic’ deal with Shanghai publishers amid hopes it will boost British students’ performance.

 

The textbook deal is part of wider cooperation between the UK and China, and the government hopes to boost British students’ performance in maths, Hughes added.

Most likely, an attempt to introduce Chinese maths textbooks in English schools will lay bare the basic fact still not accepted by policymakers. Quoting the article,

Primary school maths teachers in Shanghai are specialists, who will have spent five years at university studying primary maths teaching. They teach only maths, for perhaps two hours a day, and the rest of the day is spent debriefing, refining and improving lessons. English primary teachers, in contrast, are generalists, teaching all subjects, all of the time.

See the whole article here.

What Students Like

A new paper  in The De Morgan Gazette:

A. Borovik, What Students Like, The De Morgan Gazette 9 no.~1 (2017), 1–6. bit.ly/2ie2WLz

Abstract: I analyse students’ assessment of tutorial classes supplementing my lecture course and share some observations on what students like in mathematics tutorials. I hope my observations couldbe useful to my university colleagues around the world. However, this is not a proper sociologicalstudy (in particular, no statistics is used), just expression of my personal opinion.

Call for Nominations for the 2017 ICMI Felix Klein and Hans Freudenthal Awards

From IMCI Newsletter November 2016:

[All nominations must be sent by e-mail to the Chair of the Committee (annasd >>at<< edu.haifa.ac.il, sfard >>at<< netvision.net.il) no later than 15 April 2017.]

Since 2003, the International Commission on Mathematical Instruction (ICMI) awards biannually two awards to recognise outstanding accomplishments in mathematics education research: the Felix Klein Medal and the Hans Freudenthal Medal.

The Felix Klein medal is awarded for life-time achievement in mathematics education research. This award is aimed at acknowledging excellent senior scholars who have made a field-defining contribution over their professional life. Past candidates have been influential and have had an impact both at the national level within their own countries and at the international level. We have valued in the past those candidates who not only have made substantial research contributions, but also have introduced new issues, ideas, perspectives, and critical reflections. Additional considerations have included leadership roles, mentoring, and peer recognition, as well as the actual or potential relationship between the research done and improvement of mathematics education at large, through connections between research and practice.

The Hans Freudenthal medal is aimed at acknowledging the outstanding contributions of an individual’s theoretically robust and highly coherent research programme. It honours a scholar who has initiated a new research programme and has brought it to maturation over the past 10 years. The research programme is one that has had an impact on our community. Freudenthal awardees should also be researchers whose work is ongoing and who can be expected to continue contributing to the field. In brief, the criteria for this award are depth, novelty, sustainability, and impact of the research programme.
For further information about the awards and for the names of past awardees (seven Freudenthal Medals and seven Klein Medals, to date), see http://www.mathunion.org/icmi/activities/awards/the-klein-and-freudenthal-medals/

Continue reading

Misha Gavrilovich: Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups

Misha Gavrilovich’s paper Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups, available from

is a follow-up to his paper in The De Morgan Gazette,

M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette, 5 no. 4 (2014), 23-32

The paper raises important questions about optimal approaches to exposition of elementary group theory: quite a number of group-theoretic concepts (for example, solvable, nilpotent group, p-group and prime-to-p group, abelian, perfect, subnormal subgroup, injective and surjective homomorphism) can be expressed as diagram chasing in the category theoretic language.

Mathematics in the news this week

The week of 30 May 2016

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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.