David Mumford on Grothendieck and magazine “Nature”

Can one explain schemes to biologists

December 14, 2014

John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective “genius”. I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on “Existence theorems”. His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me.

So John and I agreed and wrote the obituary below. Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology. Here’s what we came up with:

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Andreas Schleicher: Seven big myths about top-performing school systems

A paper by Andreas Schleicher, , at the BBC website. The list of “seven big myths”:

  1. Disadvantaged pupils are doomed to do badly in school
  2. Immigrants lower results
  3. It’s all about money
  4. Smaller class sizes raise standards
  5. Comprehensive systems for fairness, academic selection for higher results
  6. The digital world needs new subjects and a wider curriculum
  7. Success is about being born talented

In my [AB] humble opinion,  this appears to be the case when the negations of myths are myths, too (with a possible exception of no. 7). School systems cannot, and should not, be compared without first having a close look at socio-economic, cultural, and political environments of their home countries.

OBE

No, it is not the Queen’s Honours list. It stands for Outcomes Based Education, the latest pedagogical fad. The EU has practically adopted it as its official educational project. Which probably means that to bid successfully for EU funded projects you’d stand a better chance if you insert the OBE here and there.

But I am writing this because in the Wikipedia page Outcomes-based education

I was somewhat amused to (re)read a paragraph such as this:

In a traditional education system, students are given grades and rankings compared to each other. Content and
performance expectations are based primarily on what was taught in the past to students of a given age. The goal of traditional education was to present the knowledge and skills of an older generation to the new generation of students, and to provide students with an environment in which to learn. The process paid little attention (beyond the classroom teacher) to whether or not students learn any of the material.

and guess what reference they cite at the end of this paragraph? The
Constance Kamii and Ann Dominick 1998 paper on the “harmful effects” of algorithms in Grades 1-4. They also quote this gem from the paper:

“The teaching of algorithms is based on the erroneous assumption that mathematics is a cultural heritage that must be transmitted to the next generation.”

And whole countries build their educational policies on such “findings”.

The beginning of the end?

Is this beginning of the end of the traditional model of mathematics education?

This advert for PhotoMath gone viral:  and enjoys an enthusiastic welcome.

Mathematical capabilities of PhotoMath, judging by the product website, are still relatively modest. However, if the scanning and OCR modules (“OCR” here refers to “Optical Character Recognition”, not to the well–known examination board). of PhotoMath are combined with the full version of Yuri Matiasevich‘s “Universal Math Solver“, it will solve at once any mathematical equation or inequality,  or evaluate any integral, or check convergence of any series appearing in the British school and undergraduate mathematics. Moreover, it will produce, at a level of detail that can be chosen by a user, a complete write-up of a solution, with all its cases, sub-cases, and necessary explanations (with slight Russian accent, but that can be easily fixed).

In short, smart phones can do exams better, and the system of mathematics education based on standard written examinations is dead. Perhaps, we have to wait a few years for a formal coroner’s report, but we cannot pretend that nothing has happened.

In my opinion, a system of mathematics education which focuses on deep understanding of mathematics and treats mathematics as a discipline and art of those aspects of formal reasoning which cannot be entrusted to a computer is feasible. But such alternative system cannot be set-up and developed quickly, it is expensive and raises a number of uncomfortable political issues. I can give an example of a relatively benign issue: in the new system, it is desirable to have oral examinations in place of written ones. But can you imagine all the complications that would follow?

PhotoMath gives a plenty of food for thought.

An Open Letter: To Andreas Schleicher, OECD, Paris

An Open Letter: To Andreas Schleicher, OECD, Paris

Heinz-Dieter Meyer and Katie Zahedi, and signatories – 5th May 2014

Dear Dr. Schleicher,

We write to you in your capacity as OECD’s director of the Programme of International Student Assessment (PISA). […]

Read the rest of the letter in the Global Policy journal.

To signt the open letter please go to  http://oecdpisaletter.org/.

A first meditation on being a teacher

[Reposted from multijimbo.]

I have been a teacher for many years now; in fact, we’re now rapidly approaching the point at which I’m thrice the age of my students, rather than merely twice. I teach 2 very different things. On the one hand (which I can write without violating the basic tenets of the Number Liberation Front, as I’m using ‘one’ not as a number but as an adjective, I suppose), I teach mathematics to undergraduate university students, and on the other hand, I am 1 of the instructors in the University aikido club. I’ve been thinking recently about the commonalities and differences between the 2 types of teaching.

There is an obvious difference between the 2. To practice aikido requires physical contact. Someone grabs me, or attempts a strike, and I need to do something rather quickly. (Or is it quite quickly? Having lived in 2 countries where the use of ‘quite’ and ‘rather’ is different, I am now very confused and can’t remember which is the current local usage.) The end of a good, active aikido session can be sweaty.   This physicality leads to a directness in teaching.  When I’m being thrown by a student, I have the opportunity to feel exactly what they’re doing, right and wrong, which I can then feed back to them immediately.

Mathematics, on the other hand, can be done in isolation. (And to follow a random train of memories, this brings to mind Ms Shearer, my 6th grade teacher, with whom we spent a session listening to Simon and Garfunkel’s I am a rock, I am an island, and discussing how people cannot exist in isolation, as they remain part of the cultural in which they grew up.)   Also, mathematics rarely involves physical combat.   Not never, mind you, just rarely.  In terms of teaching, though, mathematics teaching is a bit more at a distance than aikido teaching.  Part of this is that mathematics classes tend to significantly larger than the aikido classes I teach.   Also, a good, active mathematics class rarely ends in sweat.

Even so, there are for me some deep and significant similarities.   These are things that no doubt are similar to the teaching of many things, but hey, this is my meditation.  The similarity I would like to focus on here is the lead-a-horse-to-water phenomenon that is regularly, and sometimes almost brutally, brought home to me in both teaching fora.

In both aikido and mathematics, there are some basic, fundamental ideas that underlie everything that we do, and that I try to bring out and illustrate as much as I can through my teaching.  This is after all, in my mind at least, what a teacher should do.  I have spent  time studying how to do particular things, learning from my contemporaries and those who have gone before, and I can use the miracle of language to take what I’ve learned and provide my students with some short cuts, so that they can get farther along the path a bit faster than me.

In aikido, 1 of these basic, fundamental ideas is that at any moment in a technique, I should understand where my balance is and what is happening within both my own centre and my partner’s centre.  The way I like to try and embed this idea into my students’ brains is to have them go slowly through a technique, paying attention throughout.  But this requires that the student is willing to do the technique slowly, and alas not all of them are.  So I talk, I demonstrate, I cajole, but in the end, I cannot force.  Ultimately, I cannot teach anything.  All I can do is to provide guidance for my students on how they might learn and provide them with an environment within which they can learn.

In mathematics, the basic, fundamental idea on which I like to focus is that each statement, each assertion, needs to come from somewhere.  With each question, we have to start with things we know to be true and work out from there.  Part of an undergraduate mathematics education, and indeed mathematics education before university, is to provide students with a collection of facts, procedures and processes that we know to be true.  Mathematics does not come from nothing.  Mathematical facts do not spring full-grown from the head of Zeus.  Rather, mathematical facts are the product of accretion and accumulation (and this is where the sweat comes from).  We have just come to the end of the semester, and as in all previous years, I have the evidence that some of my students listened, and some didn’t.

So, what to do?  There is nothing to do besides persist.  Some students listen and some students don’t, but I have come to believe that it is these larger things, these fundamental ideas, that are by far the more important things that I teach, far beyond the individual techniques of aikido or the definitions, theorems and examples in mathematics.  And so we persist.  As Samuel Beckett once wrote, ‘Try again.  Fail again.  No matter.  Try again.  Fail again. Fail better.’

Ofsted: Low-level classroom disruption hits learning

From BBC http://www.bbc.co.uk/news/education-29342539 :

Low-level, persistent disruptive behaviour in England’s schools is affecting pupils’ learning and damaging their life chances, inspectors warn.

The report says too many school leaders, especially in secondary schools, underestimate the prevalence and negative impact of low-level disruptive behaviour and some fail to identify or tackle it at an early stage.

 

Source: Poll conducted by YouGov for Ofsted, http://www.ofsted.gov.uk/news/failure-of-leadership-tackling-poor-behaviour-costing-pupils-hour-of-learning-day

This is one of many low-level school issues that affect undergraduate mathematics teaching.  In a mathematics lecture, weaker students are more prone to “loosing the thread” than in most other courses. Also, students for whom English is not the first language,  in particular,  most from overseas are more sensitive to the signal-to-noise ratio than natives,  and, at a certain level of background noise,  their understanding of the lecture becomes seriously degraded. In my opinion,  this is one of many neglected issues of undergraduate mathematics education. I in my lectures always insist on complete silence in the audience (and usually start my first lecture with  a brief explanation of the concept of signal-to-noise ratio).

Brain works in sleep

Many mathematicians believe that that their brains continue to do mathematics during sleep. A paper

Kouider et al., Inducing Task-Relevant Responses to Speech in the Sleeping Brain, Current Biology (2014), http://dx.doi.org/10.1016/j.cub.2014.08.016

Proves that brain continues in sleep some mental activities of the day.

From the summary of the paper:

using semantic categorization and lexical decision tasks, we studied task-relevant responses triggered by spoken stimuli in the sleeping brain. Awake participants classified words as either animals or objects (experiment 1) or as either words or pseudowords (experiment 2) by pressing a button with their right or left hand, while transitioning toward sleep. The lateralized readiness potential (LRP), an electrophysiological index of response preparation, revealed that task-specific preparatory responses are preserved during sleep. These findings demonstrate that despite the absence of awareness and behavioral responsiveness, sleepers can still extract task relevant information from external stimuli and covertly prepare for appropriate motor responses.

The paper generated a huge response in mass media: BBC, New Scientist, NBC News. It is mentioned in this blog because the study of brain activity  is relevant to mathematics education. A naive question: do our students get enough sleep?