]]>He writes a lot of Ikea proofs — all the parts are there,

but the reader has to assemble them, and sometimes there are a few

extra parts at the end.

The first essay sets out the theoretical foundations of mathematics (and therefore of mathematics education), and boy is it a trip. Mathematics is divided into four areas: arithmetic, geometry, analysis, and mechanics. There is a long discussion of the importance of understanding the distinction between postulates and axioms, and another about the distinction between deductions and proofs. We get a list of The Three Functions of Mathematics: apparently, starting with a collection of “postulates, the truth of which is no concern of mathematics”, they are to perform deductions via the use of logical axioms, “to ascertain whether the set of postulates is consistent”, and finally to demonstrate that the chosen postulates are irredundant. (This description sounds comical to a modern mathematician, but Gödel’s work was two decades in the future when Carson was writing.)

The absurd high-level description is followed by quite reasonable practical discussions: for example, Carson argues that using technical jargon can be challenging for students and that demanding extreme rigor while proving obvious things is alienating, and he suggests that instructors should instead use rigorous methods in the exploration of new ideas. The second essay contains a good discussion of the value of relatability in word problems. The fifth essay includes makes a good case for the importance of introducing fractions via concrete examples before abstracting, and for developing their properties conceptually before developing mechanical rules for manipulating them.

Some other things that caught my eye: the fifth essay recommends teaching (e.g.) base 8 to help students appreciate the significance of base 10 (fifty years before Lehrer), and he advocates introducing students to elementary combinatorics (though he does not call it by that name) as part of arithmetic. There’s a discussion of whether students should learn to use ready reckoners (he’s for it: “Those who deprecate this suggestion might as well deprecate the use of sewing-machines, and their introduction into girls’ schools”) and of infinite series (he takes the view that they are not useful, and thinks people who want to keep them in the curriculum should not pretend otherwise). The discussion of arithmetic wraps up with the suggestion that “the meaning of the value of money and its variations in time and place as matters which should be considered by teachers of arithmetic; the fallacy of the thirty-shilling wage would find no wide acceptance if education were all that it should be” and I would love it if someone could explain to me what exactly this is a reference to.

Anyhow, in summary: as a curiosity, certainly worth the $4 I paid for it.

]]>I do not know if you have had any experience of suburban literary societies, but the one that flourished under the eye of Mrs Willoughby Smethurst at Wood Hills was rather more so than the average.

P.G. Wodehouse, “The Clicking of Cuthbert”

He’s a good author to have on the bedside table to read while drifting off. The amount of recycling that goes on is extreme: for example, “The Code of the Woosters” and “Stiff Upper Lip, Jeeves” have precisely the same characters playing out precisely the same story (the first is probably funnier and also has fewer blatantly racist bits). Perhaps when you write 90 books + hundreds of other pieces that’s going to happen, but in his case it seems like overkill.

]]>I received several pieces of pie-making equipment/guidance for my birthday, and am starting to put them to use. First, a simple apple tart:

Second, a pear and anise pie with a lattice crust:

The recipe for the latter came from *The Four and Twenty Blackbirds Pie Book* by Emily and Melissa Elsen, which is a collection of “uncommon” recipes. They were both tasty.

Are there four segments from which it is possible to construct a quadrilateral, but from which it is not possible to construct a trapezoid?

The answer, I suppose, is “it depends whether you think a parallelogram is a trapezoid”: if the opposite pairs of edges are equal in length, you’ll always have a parallelogram. But if there are a pair of opposite unequal sides, you can always do something: if we have edges *a*, *b*, *c*, *d* with *a* > *c*, then we can draw the figure thusly

with and . (These formulas allow *x* and *y* to be negative, in case *b* and *d* differ quite a lot, pushing the top edge off to one side or the other. Hopefully I have not missed any subtleties about different drawings!)

When the edge lengths are obliging, one can think in terms of hinges:

Since the right vertex of the blue edge begins below the left vertex but ends above it, at some point they must be level, i.e., the blue edge must be parallel to the black edge.

]]>Country | G | S | B | Population (thousands) | Medals per 1 M people |

San Marino | 0 | 1 | 2 | 34 | 87.04 |

Bermuda | 1 | 0 | 0 | 72 | 13.87 |

Grenada | 0 | 0 | 1 | 113 | 8.81 |

Bahamas | 2 | 0 | 0 | 353 | 5.67 |

New Zealand | 7 | 6 | 7 | 4,991 | 4.01 |

Jamaica | 4 | 1 | 4 | 2,816 | 3.19 |

Slovenia | 3 | 1 | 1 | 2,102 | 2.38 |

Fiji | 1 | 0 | 1 | 939 | 2.13 |

Netherlands | 10 | 12 | 14 | 17,337 | 2.08 |

Hungary | 6 | 7 | 7 | 9,728 | 2.06 |

Because I’m friends with a lot of people with PhDs in the sciences, this led to some methodological nit-picking. One possible alternative measure is medals / GDP, where there is a much higher correlation coefficient (0.79 versus 0.42); here are the top ten in that measure.

Country | G | S | B | GDP (B $ PPP) | Medals/GDP |

San Marino | 0 | 1 | 2 | 2.008 | 149 |

Grenada | 0 | 0 | 1 | 1.908 | 52 |

Jamaica | 4 | 1 | 4 | 28.78 | 31 |

Bermuda | 1 | 0 | 0 | 5.288 | 19 |

Fiji | 1 | 0 | 1 | 12.178 | 16 |

Georgia | 2 | 5 | 1 | 55.78 | 14.3 |

Bahamas | 2 | 0 | 0 | 14.45 | 13.8 |

Cuba | 7 | 3 | 5 | 137.0 | 11 |

Mongolia | 0 | 1 | 3 | 39.72 | 10 |

Armenia | 0 | 2 | 2 | 40.38 | 9.9 |

Obviously someone at the *Washington Post* was following me avidly, because Chuck Culpepper wrote a whole article about the last-place finisher among nations that won a medal: With 1.3 billion people and 35 medals ever, India remains an Olympic mystery. But I feel like the WaPo article really missed a crucial angle, namely, that all the large South Asian nations are chronic under-performers at the Olympics: Pakistan has only ever won 10 Olympic medals, and the most recent was in 1992; Sri Lanka has only ever won 2, and Bangladesh and Nepal have never won any. (India was not quite last among medal-winning nations on a medals / GDP basis this year: they just squeaked past Saudi Arabia.)

]]>One day, at the New York World’s Fair in 1964, I entered the Hall of Free Enterprise to escape the rain. Inside, prominently displayed, was an ant colony bearing the sign: “Twenty million years of evolutionary stagnation. Why? Because the ant colony is a socialist, totalitarian system.”

Gould, “Sizing up human intelligence”