A Student's Skill-Level Should be Private

A student's skill level should be a private matter, between him and the teacher, and students who are behind should be able to work comfortably, without embarrassment. "They know they should know more. They know they should not be working on tens and ones when their friends are doing division and fractions and all that, and there's no shame in working on it with the computer." Actually, the same principle applies to kids who are off-the-charts advanced: if they just want to relax and do high-...

 Mathematical Cue Words

Addition: add, plus, sum, total, altogether, increased by, grew, gained, total of, combined, more than (as in, “3 more than 7 is 10”), put together, in all Subtraction: minus, take away, diff erence, less than, from, remove, subtract, gives away, sells, loses, fewer than, decreased by, diff erence between Multiplication: product, times, doubled (tripled, etc.), some problems give information about one and ask for total amounts (also, when dealing with multiplication of fractions, of us...

 We Teach Kids Mathematics in the Wrong Order

The familiar, hierarchical sequence of math instruction starts with counting, followed by addition and subtraction, then multiplication and division. The computational set expands to include bigger and bigger numbers, and at some point, fractions enter the picture, too. Then in early adolescence, students are introduced to patterns of numbers and letters, in the entirely new subject of algebra. A minority of students then wend their way through geometry, trigonometry and, finally, calculus, w...

 All Human Acheivement is the Result of Networking

Human achievement is entirely a networking phenomenon. It is by putting brains together through the division of labor—through trade and specialization—that human society stumbled upon a way to raise the living standards, carrying capacity, technological virtuosity, and knowledge base of the species. We can see this in all sorts of phenomena: the correlation between technology and connected population size in Pacific islands; the collapse of technology in people who became isolated, like n...

 The Circle of the Brain cannot be Squared

A Circle round divided in four partsHath been great Study 'mongst the men of Arts;Since Archimed's or Euclid's time, each BrainHath on a Line been stretched, yet all in Vain;And every Thought hath been a Figure set,Doubts Cyphers were, Hopes as Triangles met;There was Division and Subtraction made,And Lines drawn out, and Points exactly laid,But none hath yet by Demonstration foundThe way, by which to Square a Circle round:For while the Brain is round, no Square will be,While Thoughts divi...