Posted by: Alexandre Borovik | February 8, 2015

Soldiers and horses

From a discussion at a LinkedIn group on mathematics education:

Q: You’re the general of an army. You have many soldiers and many horses. Each soldier needs one horse. What’s the fastest, most efficient way to see if the # of soldiers = the # of horses?

A: Tell the soldiers that the war is over, and  that they can go home, and take one horse each.

There is an interesting class of mathematical problems: intentionally ambiguous, because the rules of the game (or criteria for correctness of the answer) are not set;  a solution should, first of, recover – or set –  the rules, and set in a way that makes it immediately obvious that this is the only sensible set of rules, and the only sensible answer.

I do not know what were the intentions of the person who asked this question, but the answer perfectly fits into this humorous (or comical) side of mathematics.

Breaking the rules (or a  clash between two interpretation of the rule, or finding a consistent set of rules that fit into the problem better than an inspected answer) is the essence of many situation which perceived by humans as comical.

This is the rarely discussed  “comical” aspect of mathematics.

Posted by: Alexandre Borovik | February 5, 2015

The Metaphysician’s Nightmare

I had at one time a very bad fever of which I almost died. In my fever I had a long consistent delirium. I dreamt that I was in Hell, and that Hell is a place full of all those happenings that are improbable but not impossible. The effects of this are curious. Some of the damned, when they first arrive below, imagine that they will beguile the tedium of eternity by games of cards. But they find this impossible, because, whenever a pack is shuffled, it comes out in perfect order, beginning with the Ace of Spades and ending with the King of Hearts. There is a special department of Hell for students of probability. In this department there are many typewriters and many monkeys. Every time that a monkey walks on a typewriter, it types by chance one of Shakespeare’s sonnets. There is another place of torment for physicists. In this there are kettles and fires, but when the kettles are put on the fires, the water in them freezes. There are also stuffy rooms. But experience has taught the physicists never to open a window because, when they do, all the air rushes out and leaves the room a vacuum.
— Bertrand Russell
‘The Metaphysician’s Nightmare’, Nightmares of Eminent Persons and Other Stories (1954), 38-9.

This is the last pre-publication version of my paper:

Alexandre V. Borovik, Calling a spade a spade: Mathematics in the new pattern of division of labour, arXiv:1407.1954v3 [math.HO].

Abstract:
The growing disconnection of the majority of population from mathematics is
becoming a phenomenon that is increasingly difficult to ignore. This paper
attempts to point to deeper roots of this cultural and social phenomenon. It
concentrates on mathematics education, as the most important and better
documented area of interaction of mathematics with the rest of human culture.
I argue that new patterns of division of labour have dramatically changed the
nature and role of mathematical skills needed for the labour force and
correspondingly changed the place of mathematics in popular culture and in the
mainstream education. The forces that drive these changes come from the tension
between the ever deepening specialisation of labour and ever increasing length
of specialised training required for jobs at the increasingly sharp cutting
edge of technology.
Unfortunately these deeper socio-economic origins of the current systemic
crisis of mathematics education are not clearly spelt out, neither in cultural
studies nor, even more worryingly, in the education policy discourse; at the
best, they are only euphemistically hinted at.
This paper is an attempt to describe the socio-economic landscape of
mathematics education without resorting to euphemisms.

Posted by: Alexandre Borovik | September 3, 2014

The Spellbinding Mathematical GIFs Of Dave Whyte

Have a look.

Posted by: Alexandre Borovik | August 18, 2014

Educational value of deliberate mistakes

This recent story, Confuse Students to Help Them Learn, moved me to re-publish a post from my previous blog.

The Economist (22 Sept 2007), of all journals, published a long obituary of a parrot, Alex the African Grey, who became an ex-parrot on 6 September 2007, aged 31.

Alex the African Grey

Alex the African Grey


The last time Irene Pepperberg saw Alex she said goodnight as usual. “You be good,” said Alex. “I love you.” “I love you, too.” “You’ll be in tomorrow?” “Yes, I’ll be in tomorrow”. But Alex (his name supposedly an acronym of Avial Learning Experiment) died in his cage that night, bringing to end a life spent learning complex tasks that, it had been originally thought, only primates could master.

In 1977, Dr Pepperberg bought a one-year old African Grey parrot at random from a pet shop. Then, for 30 years,

Using a training technique now employed on children with learning difficulties, in which two adults handle and discuss an object, sometimes, making deliberate mistakes, Dr Pepperberg and her collaborators at the University of Arisona began teaching Alex how to describe things, how to make his desires known and even how to ask questions.

And these are the key words which attracted my attention: making deliberate mistakes! In learning mathematics, detecting and correcting other people’s mistakes is a crucial but badly underrated component. We do not give our students a chance to analyse, criticise and correct each others’ work, and we do not reward them for detecting an error. Not surprisingly, our students’ progress is frequently less impressive than that of Alex:

By the end, said Dr Pepperberg, Alex … had a vocabulary of 150 words. He knew the names of 50 objects and could, in addition, describe their colours, shapes and the materials they were made from. He could answer questions about objects’ properties, even when he had not seen that particular combination of properties before. He could ask for things – and reject a proffered item and ask again if it was not what he wanted. He understood, and could discuss, the concepts of “bigger,” “smaller,” “same” and “different”. And he could count up to six, including the number zero.

Research publications on Alex:

Pepperberg, I.M., and Gordon, J.D. (2005). Number Comprehension by a Grey Parrot (Psittacus erithacus), Including a Zero-Like Concept. J. Comp. Psych, 2005, Vol. 119, No. 2, 197-209.

Pepperberg, I.M. (2001). Lessons from cognitive ethology: Animal models for ethological computing. Proceedings of the First Conference on Epigenetic Robotics, C. Balkenius, J. Zlatev, H. Kozima, K. Dautenhahn, & C. Breazeal, Eds., Lund University Cognitive Science Series No. 85, Lund, Sweden. 

Pepperberg, I.M., Willner, M.R., and Gravitz, L.B. (1997). Development of Piagetian object permanence in a Grey parrot (Psittacus erithacus). J. Comp. Psych. 111:63-75.

[With thanks to Jeff Burdges]

Posted by: Alexandre Borovik | August 14, 2014

Fibonacci

It appears that Fibonacci was the first person in Europe to represent a real number in a place value system: he wrote a root of the equation

x^3+2x^2+10x=20

as

1^{\circ} 26^{\prime} 7^{\prime\prime}  42^{\prime\prime\prime} .

Posted by: Alexandre Borovik | August 13, 2014

This is what I call numeracy

From Vasily Grossman‘s notebooks published in A Writer at War, pp. 161-162, an entry about the Red Army infantry fighting off Luftwaffe in Stalingrad:

The brains of the Red Army have finally turned to to the anti-tank rifle … [using] a cart wheel, fastened to a picket and rotating through 360 [degrees]. […]

Battalion Commander Captain Ilgachkin had a problem: he never could manage to hit an aircraft with a rifle. He made theoretical calculations  of the speed of bullet from an anti-tank rifle (one thousand meters per second), made a table, supplemented it with information on whether an aircraft is moving towards the firing point or away from it. Having made this table, he hit the aircraft immediately. After that, he fastened a stake in the ground, made an axle, put a wheel on it and they attached an anti-tank rifle to the spokes.

These “theoretical calculations” were made under shell fire in trenches of Stalingrad…

Posted by: Alexandre Borovik | July 20, 2014

Mathematics: The Most Misunderstood Subject

post by  Robert Lewis. Much recommended.

My new preprint:

Calling a spade a spade: Mathematics in the new pattern of division of labour.

From Introduction:

I argue that new patterns of division of labour have dramatically changed the nature and role of mathematical skills needed for the labour force and correspondingly changed the place of mathematics in popular culture and in the mainstream education. The forces that drive these changes come from the tension between the ever deepening specialisation of labour and ever increasing length of specialised training required for jobs at the increasingly sharp cutting edge of technology.

Unfortunately these deeper socio-economic origins of the current systemic crisis of mathematics education are not clearly spelt out, neither  in  cultural studies nor, even more worryingly, in the education policy discourse;  at the best, they are only euphemistically hinted at.

This paper is an attempt to describe the socio-economic landscape of mathematics education without resorting to euphemisms.

Posted by: Alexandre Borovik | July 8, 2014

Shen: Foundations of probability theory and Kolmogorov complexity

Alexander Shen, Основания теории вероятностей и колмогоровская сложность.

(Foundations of probability theory and Kolmogorov complexity).

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