LA 2100 · Quadrivium: The School of Arithmetic — Module 1: Number, Notation and Numeration

About this course

Arithmetic is the first of the four mathematical arts, and Nicomachus of Gerasa gave the proof rather than the assertion: abolish number and harmony perishes, the figures perish, the spheres perish. Abolish the figures and number stands entire. Boethius took that argument, named the four arts the quadrivium, and put arithmetic first because it lends to the other three and borrows from none.

The specification is James B. Thomson's New Rudiments of Arithmetic, New York, 1872 — one of the most widely used arithmetics of the American common-school era. Thomson wrote for a room with one teacher, forty pupils, and one copy of the book, in the teacher's hands rather than the pupils'. That constraint produced a textbook designed to be performed rather than read, in which every article is a question and every answer is short enough to hold.

You will need an exercise book. Thomson's pupils had slates: they worked their figures on the slate, and the master walked over and looked. So the working here is done longhand, on paper, away from the screen — columns, carries, the answer proved by a second operation. What comes into the chat is the answer. You will never be asked to type a sum in any special notation; you type numbers and sentences, and that is all this course assumes you can do. The tutor sets the layout out properly when he is demonstrating, and you copy the form into your book.

That is the axis the whole school turns on, and it is Thomson's own. Mental work uses numbers small enough to hold in your head — nothing written, the answer given straight — and its purpose is to fix the principle. Slate work uses numbers too large to hold, worked in the book, and its purpose is to fix the process. Mental always comes first on a new topic, because a pupil given the written rule first will execute it and understand nothing. And the General Rule is never stated until you have stated it.

When you get an answer wrong, you will not simply be corrected. The tutor works in four steps. He reads the wrong answer, because most faults have a signature and a sum short by exactly a hundred is a dropped carry rather than a table error. Then he asks you for one number — what was your first remainder? — which usually finds it. Then he asks you to say what you did at a particular step. And if that has not settled it, he asks you to photograph the page: take a picture of it with your phone and attach it using the paperclip icon — 📎, just to the left of the microphone at the bottom of the chat — and he can look at it.

That last is him walking across the room to look at your slate, and it is deliberately the fourth move rather than the first. A pupil who knows the master will simply look stops examining his own work — and re-examining it is frequently where you find the error yourself.

Thomson's commercial examples and obsolete coin-names have been modernised — the course works in modern decimal currency, in whatever denomination suits you. His curriculum order has not been touched.