习算纲目

Third of September, 2036, seven-forty in the morning, the Aksu station. The first new goods came down two days ago; five or six three-wheelers, plastic sheets weighted with bricks; a new turn of black tape on the chair's armrest. Eli in the chair. Half a step behind him, a young man — twenty-two, half a head taller, a pale grey shell jacket, a phone hanging at his chest. He Yuan: Liu's nephew, grew up in Aksu, studied logistics in Urumqi, a year out of college without the right job. Liu to the station master: let him stand behind Eli for a season. Eli had written: I'm nineteen; I don't know how to teach anyone. I replied: early September I'll come and look. From the wall: Eli grading steadily, far steadier than last September. He Yuan photographing every load — the goods, the scale, the grade Eli wrote — then looking down at the phone. Twenty-odd photos in an hour. Eli never once turned round. At lunch: "I don't know what he's learning. He photographs everything and looks at the phone. I want to tell him to stop and don't know what to tell him to do instead. Grandpa Luo said one sentence a day, eight days, eight sentences. I tried to do the same and couldn't think of eight in three days. His eight came from twenty-six years in the chair. I've sat one season."

Yang Hui came in the afternoon, when the sun had moved over the scale: fifty or so, thin, a Song scholar's gown and soft cap, a small bundle of equal bamboo sticks tied with cord in his hand, a thin book under his arm. He watched the two young men — one sitting, one photographing — then looked at the notebook in my hand, a single column: "Teaching a person starts with ordering the days."

He untied the sticks on his knee: counting rods — we reckoned with these, not with an abacus. Two across, three down. I wrote a few arithmetic books. In one there's a short piece, an outline for learning to reckon: a course for beginners — recite the table until it's familiar; then addition and subtraction until familiar; then multiplication; then division. I wrote roughly how many days each stage takes. So the learner moves on after that many days? No. The days are so the teacher has a sense of it. Moving on depends not on how many days have passed but on whether it's familiar. I wrote: advance when it's familiar; if it isn't, you may not skip. And: every day, go over what came before once more. And, and I picked up the pen: an outline sets the order, not the dates; you advance on familiarity, not because the time has come.

Eli, hearing it through me: "Grandpa Luo's eight had an order too." The eight still at the back of his notebook. "I used to think they were eight sentences. They were eight stages. One a day because I learned fast — he saw I was ready and said the next. If I'd been slow, the third might have taken three days." Yang, at the other end of the step, arranging his rods, smiled slightly. — M.

Memo to file. Aksu, 3–20 September 2036. The successor's first apprentice. A curriculum gated by competence, and a lineage that closed a circle.

1. The problem. Eli (19), in his second season in the chair, has a 22-year-old standing behind him who photographs every load and studies the phone. Eli tried to imitate Luo's "one sentence a day" and couldn't produce eight sentences: "His came from twenty-six years; I've sat one season."

2. The precedent. A Song mathematician's short study outline for beginners: table, then addition and subtraction, then multiplication, then division; a rough span of days for each; advance only when familiar; review earlier stages daily. The days orient the teacher; readiness gates the learner.

3. Eli's reading of his own apprenticeship. Luo's eight sentences were not sentences; they were stages, released one per day because Eli was quick. Slower, and the third might have taken three days.

4. Eli's outline (one card, three revisions). — Stage 1: stand and watch; no photos, no notes, no questions. Gate: without looking, say where I'll touch the next load first. — Stage 2: record my grades. Gate: 90%+ of a day's lines match my book. — Stage 3: record mine and yours, two columns. Gate: most of a day's lines agree. — Stage 4: each evening pull the lines where we differ; next day check in the store who was right. Gate: once, yours is better than mine. — Pass: once yours beats mine, I let you sit. — Daily: before grading, pinch the station's ruler bags one and five; if wrong, no grading that day. Daily: review the previous stage. — Added last: phone photos only after close of work; photograph, then check against the book. The mathematician's comment: his own outline gated on "familiar" but never said how to tell. Eli's writes what "familiar" looks like at each stage — so the teacher can change and the learner still knows where he is. Dates are for the teacher; descriptions of competence are for the learner.

5. Day one of stage one. "No photos?" — "No." — "I could look at them slowly at home." — "At home you'd be looking slowly at the goods in a photo. Here you're looking at my hands." Phone pocketed; no photos all day. At close: where do I touch the next load first? "The tail." — "No, the head." — "Yesterday you touched the tail first." — "Yesterday's load had bricks on the head. Today's has none." He wrote his first line in the notebook.

6. The circle. Luo came at noon — not to stand, to bring Liu a basket of tomatoes from his yard. He asked the young man's name, then his grandfather's. He Yuan is the grandson of Master He — who, in the winter of 1992, standing by this scale, said "who will look after these twelve rules." Liu: her sister married Master He's second son; the old man died early; the boy never knew him. Luo stood beside him — beside, not behind — for about a minute, said nothing, and left: "Master He's sentence has gone round and come back to his own family. Don't tell the boy. Let him stand for himself."

6a. He found out himself. On day ten He Yuan came early: "Last night I told my father I'm standing at the station. He said your grandfather stood by the scale at that station his whole life. I didn't know." Eli said nothing of Luo or the twelve rules — only "Then stand well." He stood straighter that day than any other. Liu photocopied Eli's card and pinned it on the wall behind the scale, very straight. Told of Master He that night, Eli: "Then my card was written for Master He's grandson. It feels heavier." — "It was always this heavy. You only learned it today." Liu: fifteen years here and the first time she has seen a nineteen-year-old write a course for a twenty-two-year-old — clearer than the station master could. 6b. Progress, by the card. Stage 1 passed on day four (four of five loads called right: ticked 7 September); stage 2's nine-in-ten on day nine; on 20 September, stage 3: 22 of 30 lines agreeing. On the 19th, one line where he said three and Eli two; next morning in the store Eli was right. "The card says once, not this time." — "Then there's a next time." When can I sit? Eli pointed at the card: "It's written." And Eli discovered something about himself through being asked: he touches the tail of loads from Wensu first and the head of all others — Wensu families load their best at the back. "I didn't know I did it. He asked, and then I knew." 6c. The year's travel. One trip fixed (10 March); seven called: Shanghai, Shanshan, Hami, Turpan twice, Aksu twice, Kuqa. The further I step back, the more I'm called — each time to look, not to do. Eli's card goes to Shao Min for Gu and Little Xu — credited: Eli, Aksu, nineteen. The mathematician, watching me pick up a rod: "Wrong — units upright, tens across. You keep accounts; you don't reckon."

7. Craft and curriculum. Luo's eight sentences: one man's twenty-six years, not learnable by others. Eli's seven stages: Luo's eight taken apart, each with a written picture of readiness, usable by others. A craft passes to one person; an outline can pass to several. Master He, winter 1992 → Luo, September 2035 → Eli, September 2036 → Master He's grandson, behind Eli. Forty-four years round.

8. Next. The printer, 20 September: "Mr Ma, the fourth edition of The Two Ends — proofs are ready. Please come and check them."

"An outline sets the order, not the dates; you advance on familiarity, not because the time has come."
A thirteenth-century arithmetician from Hangzhou included in one of his books a short study plan for beginners, often cited as one of the earliest written curricula in Chinese mathematics. Learn the multiplication table first, then addition and subtraction, then multiplication and division; each stage has an expected span of days; review the earlier stages daily; do not advance until the current one is secure.
The document encodes two distinctions that training programmes still blur. The first is between sequence and schedule. The order of stages is fixed; the time each takes is an estimate for the teacher, not a promise to the learner. The second is between time served and competence reached. A stage ends when the skill is there, not when the calendar says so.
The young grader in this chapter adds the piece the old outline lacked. For each stage he writes down what readiness looks like, in observable terms: say where I will touch the next load before I do; match my recorded grades on nine lines in ten; once, be right where I was wrong. With that, the plan no longer depends on the teacher's private judgement. Another teacher could use it, and the learner can see for himself where he stands.
Such an outline is also how a craft outlives one person. A master's eight remarks, released one a day to a quick apprentice, are an art. The same remarks, unpacked into stages with written gates, are something several people can inherit.
Dates are for the teacher; descriptions of competence are for the learner.