From The Guardian:

One of our first patients, just 17 years old, was brought to us in a wheelchair,” says Professor Christer Lindquist,  a pioneer in the use of a brain surgery technique for people with Obsessive Compulsive Disorder   (OCD),  known as Gamma Knife. “This boy would set himself maths problems, which he had to solve before he could eat. His OCD had become so severe, and the maths problems he set himself so complex, that he couldn’t solve them any more, so he couldn’t eat.

(With thanks to muriel)

As for the limits, we were told to imagine an animal getting closer to point on the plane but not really arriving at it.  This is a very poor discription and gave me many years of unnecessary pain while dealing with limits.  The things is this animal might actually decompose itself into many different animals who have nothing to do with the first one and whoever acts more dramatically close to a certain point will guide the limit close to that point.

Meaning is one of the recent terms which have gained great currency in mathematics education. It is generally used as a correlate of individuals’ intention and considered a central element in contemporary accounts of knowledge formation. One  important question that arises in this context is the following: if, in one way or another, knowledge rests on the intrinsically subjective intentions and deeds of the individual, how can the objectivity of conceptual mathematical entities be guaranteed? In the first part of this paper, both Peirce’s and Husserl’s theories of meaning are discussed in light of the aforementioned question. I examine their attempts to reconcile the subjective dimension of knowing with the alleged transcendental nature of mathematical objects. I argue that transcendentalism, either in Peirce’s or Husserl’s theory of meaning, leads to an irresolvable tension between subject and object. In the final part of the article, I sketch a notion of meaning and conceptual objects based on a semiotic-cultural approach to cognition and knowledge which gives up transcendentalism and instead conveys the notion of contextual objectivity.

I found your blog via the entry on women and mathematics.

I know that this is not exactly what you are searching for with your demand for stories, but I have to comment on the “intrinsic competitiveness” of mathematics.

In mathematics olympiads and in professional mathematics, competitiveness is ritualized and contained by ethical rules. In my olympiad training session, you were expected to explain things to newer participants even though they might well be better than you at the competition. In olympiads and real life, I have found that this kind of ethics almost always corresponded to mathematical capability.

So here is my anecdote on competitiveness:

In primary school (age 6-10), I had about half the running speed of any other child. On the other hand, I understood everything in maths immediately and was very quick with mental calculations. In sports, we had weekly running competitions of groups by four, where my group never had a chance at all.  In maths, we had weekly speed calculating competitions where each pair of students were posed a question and the quicker one would remain standing until one student was left and got a little prize. I won every time and was only allowed to participate every other week.

The lesson that good performance in math is bad for your social standing with your peers could not have been more evident.

More anecdotes on learning:

Age: 3
Gender: female
languages involved: German
Brother aged 5

My brother and I had learned (presumably from our parents) how counting goes on and on without an end. We understood the construction but we were left with some doubt that you could *really* count to high numbers, so we decided to count up to a million by dividing the work and doing it in the obligatory nap time in kindergarten in our heads. After a couple of days, we had to admit that it took too long, so we debated whether it was ok to count in steps of thousands or ten-thousands, now that we had counted to thousand many times. We ended up being convinced that it is possible to count to a million but slightly unhappy that we could not *really* do so ourselves.

Age: 14
Gender: female
languages involved: German

I participated in a mathematical competition where an inequality had to be proven by induction. I adhered strictly to a ”one-line-format” where you start with the left-hand-side of the desired inequality, apply induction hypothesis etc until you arrive at the right-hand-side.

When I afterwards saw someone at some step adding the desired right-hand-side and manipulating the unproven inequality, I understood the advantage immediately and felt stupid for not considering this possibility, but I could never have asked a question about this because I was not aware of any restriction in my thinking. My teachers also could not notice any problems because my proofs were correct.

SUNDAY MAGAZINE | October 30, 2009 Motherlode: Math’s Too Hard for a Parent’s Help By Lisa Belkin Many parents would rather talk to their kids about sex and drugs than math and science.

(with thanks to muriel)

Apparently, by a woman mathematician:

http://ideafoundlings.blogspot.com/

http://www.logicomix.com/en/

(with thanks to Jean-Michel Kantor).

I have always been rather partial to plane geometry; probably because it was the only branch of mathematics that was ever taught to me in such a way that I could understand it. For though I have no belief in the power of education to turn public school boys into Newtons (it being quite obvious that, whatever opportunity may be offered, it is only those rare beings desirous of learning and pssessing a certain amount of native ability who ever do learn anything), yet I must insist, in my own defence, that the system of mathematics instruction of which, at Eton, I was the unfortunate victim, was calculated not only to turn my desire to learn into stubborn passive resistance, but also to stifle whatever rudimentary aptitude in this direction I might have possessed. But let that pass. Suffice to say that, in spite of my education and my congenital ineptitude, plane geometry has always charmed me by its simplicity and elegance, its elimination of detail and the individual case, its insistence on generalities.

(From Aldous Huxley’s essay “Views of Holland”, page 98 of Thomas R. Cook’s Essays in modern thought on Google books (http://books.google.ca/books?id=k1yrBlylOigC). With thanks to Dan MacKinnon.

He had not become a mathematician, but geometry had imprinted on Huxley as an aesthetic paradigm obvious from the description of the Dutch landscape that immediately follows his childhood recollections:

My love for plane geometry prepared me to feel a special affection to Holland. For the Dutch landscape has all the qualities that make geometry so delightful. A tour of Holland is a tour trough the first books of Euclid. Over a country that is the ideal plane surface of the geometry books, the roads and the canals trace out the shortest distances between point and point.

[...] I may be free to admire the farmhouse on the opposite bank of the canal on our right. How perfectly it fits into the geometrical scheme! On a cube, cut down to about a third of its height, is placed a tall pyramid. This is the house. A plantation of trees, set in a quincunx formation, surrounds it; the limits of its rectangular garden are drawn, in water on the green plain, and beyond these neat ditches extend the interminable flat fields. There are no outhouses, no barns, no farm-yard with untidy stacks. The hay is stored under the huge pyramidal roof, and in the truncated cube below live, on the one side the farmer and his family, on the other side (during winter  only; for during the rest of the year they sleep in the fields) his black and white Cuyp cows. Every farmhouse in North Holland conforms to this type, which is traditional, and so perfectly fitted to the landscape that it it would have been impossible to devise anything more suitable. An English farm with its ranges of straggling buildings, its untidy yard, full of animals, its haystacks and pigeon-cotes, would be horribly out of place here. In the English landscape, which is all accidents, variety, detail and particular cases, it is perfect. But here, in this generalised and Euclidean North Holland, it would be a blot and a discord.Geometry calls for geometry; with a sense of the aesthetic proprieties which one cannot too highly admire, the Dutch have responded to the appeal of the landscape and have dotted the plane surface of their country with cubes and pyramids.

Delightful landscape! I know of no country that it is more mentally exhilarating to travel in. No wonder Descartes preferred the Dutch to any other scene. It is the rationalist’s paradise. One feels as one flies along in the teeth of one’s own forty-mile-an-hour wind like a Cartesian Encyclopaedist—flushed with mental intoxication, convinced that Euclid is absolute reality, that God is a mathematician, that the universe is a simple affair that can be explained in terms of physics and mechanics …

I was studying at the FeMeSha 18 in Moscow around ‘73. I recall being comfortable with the definition of derivative as a limit. On the other hand, the alternative definition that the instructor provided caused me no end of anxiety. Namely, he said the derivative is a number such that

As you correctly point out, it takes a considerable amount of mathematical training to formulate precisely what the problem was. The problem was that the definition says absolutely nothing about how one could find such a ““, or how to go about SIMULTANEOUSLY (in what sequence?) finding and ““. In retrospect, what I must have been bothered by is the non-constructive nature of this definition.

Actually I am currently writing a text on constructivism, and it could be that even after all these years I would still be unable to identify the source of the anxiety were it not for the fact of having understood constructivism better recently.