I’m working in the math department library today and have gotten distracted by a collection of historical math textbooks that’s just gone on the shelves.

From *College Mathematics: A First Course* (1940), by W. W. Elliott and E. Roy C. Miles:

The authors believe that college students who take only one year of mathematics should acquire a knowledge of the essentials of several of the traditional subjects. From teaching experience, however, they are convinced that a better understanding is gained if these subjects are presented in the traditional order. Students who take only one year of college mathematics are usually primarily interested in the natural sciences or in business administration.

The book covers algebra, trigonometry, Cartesian geometry, and calculus. The definition of the derivative as a limit is given, but the epsilon-delta definition of limit is not. Startling to think that science majors came to college never having taken algebra or analytic geometry.

Further back in time we get *Milne’s Progressive Arithmetic*, from 1906. This copy was used by Maggie Rappel, of Reedsville, WI, and is dated January 15th, 1908. Someone — Maggie or a later owner — wrote in the flyleaf, “Look on page 133.”

On the top of p .133 is written

Auh! Shut up your gab you big lobster, you c?

You got me, Maggie!

I can’t tell what grades this book is intended for, but certainly a wide range; it starts with addition of single digits and ends with reduction of fractions to lowest terms. What’s interesting is that the book doesn’t really fit our stereotype that math instruction in olden times was pure drill with no attention paid to conceptual instruction and explanation. Here’s a problem from early in the book:

How many ones are 3 ones and 4 ones? Write the sum of the ones under the ones. How many tens are 6 tens and 2 tens? Write the sum of the tens under the tens. How do you read 8 tens and 7 ones? What, then, is the sum of 24 and 63? Tell what you did to find the sum.

From the introduction:

Yet the book is not merely a book of exercises. Each new concept is carefully presented by questions designed to bring to the understanding of the pupil the ideas he should grasp, and then his knowledge is applied. The formal statement of principles and definitions is, however, reserved for a later stage of the pupil’s progress.

Would these sentiments be so out of place in a contemporary “discovery” curriculum?

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They would only be considered out of place in modern schools because they set a certain bar for students—and expect them to meet it. Not every child can succeed at every subject, but I sometimes feel like we sell most of our kids short. Thanks for the insightful post.

Occasionally I have the opportunity to browse older mathematics texts, mainly at the graduate level, and I have the impression that authors in those days were far more generous in attempting to convey intuition.