I first discovered and was intrigued by the R. L. Moore approach to teaching mathematics when reading the article “The Moore Method: What Discovery Learning Is and How It works” in FOCUS (August/September 1999). Moore’s approach to “discovery learning” was developed from 1920 to 1969 at the University of Texas, Austin, and has since been known as “Texas school” or the “Moore Method.” This approach of axioms, questions, and proofs is designed to challenge students while leading them to points of discovery. The idea is to teach mathematical thinking, not manipulation. H. S. Wall worked along with Moore and other colleagues in transmitting this style to 139 Ph.D. students, many of whom became prolific researchers and teachers.
Professor H. S. Wall (1902–1971) developed this book over those years of working with students at the University of Texas. Applying the Moore Method, his aim was to lead students to develop their mathematical abilities and intuition. Wall himself called this book “a sketchbook in which readers try their hands at mathematical discovery.” That is a fair and accurate assessment. What it lacks in depth it makes up for in breadth. Over less than two hundred pages the reader travels from elementary number theory to simple graphs, from integrals and surfaces to linear spaces of simple graphs.
Requiring little formal mathematical knowledge from the reader, this book is an excellent if breathless tour of a wide swath of basic mathematics. It can also work as an adjunct to more traditional study, whether in a classroom or out. Obviously, these techniques have a proven history in the classroom, but the material on any given specific subject is given too brief a treatment here to support more than a lecture or two. As part of the MAA Classroom Resource Materials series, the book is intended for just that: supplementary classroom material for students with an unusual approach for presenting mathematical ideas.
I would be interested in seeing those sections at work in situ. I would think an undergraduate, having been exposed to logarithms, sine, and cosine, would feel there is something suspicious about being presented the functions’ descriptions axiomatically while pointedly referring to them only as L, S, and C.
I found it easy, however, to give myself over to this unadorned presentation of axioms and examples followed by probing questions and a few meaty theorems to prove. This approach lends freshness to these topics and invokes a spirit of playful experimentation wherein learning and understanding can become a happy byproduct. There are many places where I felt this way, such the measure-theoretic approach to defining the integral. It invokes a feeling that this is a natural and practical, not abstract, direction to go in.
While this work is admittedly a compendium of sketches, I still feel it could have been improved for this updated edition by bringing in a bit more coherence. While there is an index of graphs and a glossary, there is not a subject index. This addition would be helpful for independent readers, particularly since instances where topics are introduced well before they are defined have been left in, such as with inner sums.
The “More About…” chapters and the one on Mechanical Systems could really use more structure to become a cohesive departure point for the dedicated student. As they are, they seem an intriguing if cluttered collection of unfinished ideas. Some topics are so slightly covered, such as the single paragraph given over to vectors, that the book might have been better off without them, especially if this made space for expanding the remaining material for depth while staying within the spirit of the Moore Method.