Vie

Vie grew up beside a tournament field, where every summer the province came to run, wrestle, throw, and argue about who had won. Her family were the stewards — the people who, before every contest, arranged the entrants in starting order. Not the winners; the seeding. And to seed a field you did one thing, honestly, over and over: take two entrants and decide which was greater — faster, stronger, further by last year's marks.

"You never order the whole field at once," her grandfather told her, chalk in hand. "The mind won't hold thirty runners. You order them two at a time. This one or that one — set the greater ahead, then the next two. The whole field falls into order out of nothing but little comparisons." That is precisely the academy's method for ordering a set: a full ordering is built from repeated pairwise comparison, each comparison small enough for a fair mind to make.

Vie's given name was Ada, but she was called Vie from the day she settled a shouting-match between two throwers by chalking their marks side by side, drawing one line to show which reached further, and saying, "This one is greater. There is nothing to argue." She understood magnitude before she had the word: a number was a reach, and the number sitting further along the line was greater. But she also learned to read two numbers from the front — the efficient way. "Don't hunt through every digit," she teaches. "Compare the biggest place first. More digits reaches further, the way a longer stride covers more ground. Same number of digits? Compare the leading digit, then the next, and the first place where they differ decides it. The larger digit there wins, and nothing behind it can overturn the verdict." This is place-value comparison: the highest place that differs is decisive, so you compare left-to-right and stop at the first disagreement.

An academy examiner watched Vie seed a field of forty in the time it took to lace his boots. "You are ordering by repeated comparison," he said. "We teach exactly that — least to greatest — and most children find it slippery." "It is not slippery," said Vie. "It is only this one or that one, done until you run out of pairs." She surprised herself by wanting the academy, and has taught comparing and ordering for seven years.

In her classroom Vie keeps a long rope pinned with numbers, and two great wooden signs — one a mouth open wide, one a mouth closed to a point. She holds up four hundred six and four hundred sixty and asks which is greater. The children start hunting through digits; Vie stops them. "From the front. Both are four hundred — same. Then this one has no tens and that one has six tens. Six tens beats no tens. Decided at the tens; nothing after it matters." Then she brings out the sign. "Greater-than is hungry. Its wide mouth opens toward the bigger number, its point toward the smaller." She faces the open mouth at four hundred sixty. The children see the symbol is not a rule to memorise but a picture of an appetite — it eats the bigger meal, and < and > are the same mouth turned to face whichever is larger.

Then she gives them the whole field: ten numbers scattered on the floor, to be laid along the rope least to greatest. The children panic at ten-at-once — until a boy named Fen picks up two, decides which is greater, sets them in order, then two more, then two more. The rope fills; the panic drains. They have ordered ten numbers having never held more than two in their heads. "That," Vie says, "is the whole secret. A sort is not one huge act of comparing everything. It is one small comparison, repeated, until the crowd stands in line."