Some Readings for a Rainy Weekend
We teach mathematics...something that has been done for a long time. At the same time, we often complain at how slow things change...either in the mathematics we teach or how it is taught. But, do you really know how mathematics was taught in the pastwhat were the issues confronted by teachers (e.g. no WASL), the pedagogical aides (e.g. no computing technologies), and required qualifications (e.g. No "highlyqualified").
To gain some sense of the past, you might want to peruse some of the following textsdigitized and offered free to readers. Most were written in the late 1800s and early 1900s, giving a good flavor of mathematics education at that timeboth in Great Britain and the U.S. Note: Of the "download" formats available, I prefer the Flip Book for ease of reading.
 On the Study and Difficulties of Mathematics (1910), by Augustus De Morgan (of DeMorgan's Law fame). Though the book is interesting (and quaint), I suggest the sections on fractions, algebraic notation, the negative sign, the study of algebra, geometrical reasoning, and proportions. You'll find little has changed...except no mention of things like algebraic thinking, functions, modeling, or transformations!
 The Logic and Utility of Mathematics With the Best Methods of Instruction Explained and Illustrated (1850), by Charles Davies, prolific author of secondary math textbooks in the late 1800s. The Preface's second paragraph states how mathematics should be taught (i.e. gaining comprehension of principals and laws before application). As the first part (Book I) is a standard treatment of logic, I suggest skipping to Book II, focusing on Chapter 1 (Language of Mathematics), Chapter III (teaching geometry), and Chapter IV (teaching algebra, analysis, and calculus). Read Book III only if you enjoy "goal" discussions, as Davies elaborates on the utility of mathematics as a "means of intellectual training and curiosity," a "means of gaining knowledge," and "furnishing those rules of Art which make knowledge practically effective." Sounds lofty when compared to our modern goals.
 Essays on mathematical education (1913), by George Carson, a leader in early English mathematics education. He claims that "mathematics can be saved...only through an improvement in our methods of teaching and in our selection of material." Sound familiar? Though all of the chapters are of interest, I highly recommend the two on "Some Unrealized Possibilities of Mathematics Education" and "The Educational Value of Geometry." The latter is important as at this time, geometric proof was taught (i.e. justified) because it transferred and made studentasadultinsociety a better logical reasoner.
 Graphical Methods (1912), by Carl Runge (of RungeKutta algorithm fame). Runge's focus is the use of graphs to calculate and represent....a book ahead of its time! It includes some interesting problems as well, as Runge believes that the reader must strive to both understand the underlying ideas and master their use. Without enough of the latter, he suggests "You might as well try to learn piano playing only by attending concerts...."
 A History of Elementary Mathematics, with Hints on Methods of Teaching (1917), by Florian Cajori (premier mathematics historian). Its all here...a brief history of mathematics (through trigonometry)...in a rich pedagogical context. Complete with an index so you can focus on specific terms, content areas, or mathematicians.
There are more texts like these...but the above should keep you busy through a rainy weekend. Happy reading!
