Every thinking educator has grappled with the basic questions that have dogged the profession from earliest times: What is the best way to teach, or to learn? What subjects should children learn? How responsible are children? How much of a say should they have in what they do? How should schools be run in a … Read more
Though calculus represents a new sophistication in the understanding of sequences, that idea, like so many others, had already been familiar to the Greeks. In the fifth century b.c., in fact, the Greek philosopher Zeno employed a curious sequence to formulate a paradox that is still debated among college philosophy students today, especially after a … Read more
State governments tend to ignore arguments about the possible bad effects of lotteries. That’s because, for the most part, they know enough about mathematical expectation to arrange that for each ticket purchased, the expected winnings—the total prize money divided by the number of tickets sold—is less than the cost of the ticket. This generally leaves … Read more
A few years ago Canadian lottery officials learned the importance of careful counting the hard way when they decided to give back some unclaimed prize money that had accumulated. They purchased 500 automobiles as bonus prizes and programmed a computer to determine the winners by randomly selecting 500 numbers from their list of 2.4 million … Read more
In the two-daughter problem, an additional question is usually asked: What are the chances, given that one of the children is a girl, that both children will be girls? One might reason this way: since it is given that one of the children is a girl, there is only one child left to look at. … Read more
Unfortunately, it is hard to achieve quantitative dexterity when you’re juggling VIIIs and XIVs. In the end, though Roman law had a certain legal rationality and coherence, it fell short of mathematical validity. In Roman law, for example, two half proofs constituted a complete proof. That might sound reasonable to a mind unaccustomed to quantitative … Read more
Which is greater: the number of six-letter English words having n as their fifth letter or the number of six-letter English words ending in ing? Most people choose the group of words ending in ing. Why? Because words ending in ing are easier to think of than generic six-letter words having n as their fifth … Read more
Humans were aware of the cycles that govern our lives long before we understood them. The most obvious cycle is the day/night cycle. Objects floating in space, or orbiting other objects under the law of gravity, have a natural tendency to spin on their own axis. There are exceptions, but our planet is not one … Read more
The human body abounds with what, in one sense; we could call imperfections but, in another sense, should be seen as inescapable compromises resulting from our long ancestral history of descent from other kinds of animal. Imperfections are inevitable when ‘back to the drawing board’ is not an option — when improvements can be achieved … Read more
Just as ants and their subterranean fellow-travelers lose their wings underground, so numerous different kinds of animals that live in the depths of dark caves where there is no light have reduced or lost their eyes, and are, as Darwin himself noted, more or less completely blind. The word ‘troglobite’ has been coined for an … Read more
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