In this follow-up to his popular Science Secrets, Alberto A. Martínez discusses various popular myths from the history of mathematics: that Pythagoras proved the hypotenuse theorem, that Archimedes figured out how to test the purity of a gold crown while he was in a bathtub, that the Golden Ratio is in nature and ancient architecture, that the young Galois created group theory the night before the pistol duel that killed him, and more. Some stories are partly true, others are entirely false, but all show the power of invention in history. Pythagoras emerges as a symbol of the urge to conjecture and “fill in the gaps” of history. He has been credited with fundamental discoveries in mathematics and the sciences, yet there is nearly no evidence that he really contributed anything to such fields at all. This book asks: how does history change when we subtract the many small exaggerations and interpolations that writers have added for over two thousand years?

The Cult of Pythagoras is also about invention in a positive sense. Most people view mathematical breakthroughs as “discoveries” rather than invention or creativity, believing that mathematics describes a realm of eternal ideas. But mathematicians have disagreed about what is possible and impossible, about what counts as a proof, and even about the results of certain operations. Was there ever invention in the history of concepts such as zero, negative numbers, imaginary numbers, quaternions, infinity, and infinitesimals?

Martínez inspects a wealth of primary sources, in several languages, over a span of many centuries. By exploring disagreements and ambiguities in the history of the elements of mathematics, The Cult of Pythagoras dispels myths that obscure the actual origins of mathematical concepts. Martínez argues that an accurate history that analyzes myths reveals neglected aspects of mathematics that can encourage creativity in students and mathematicians.

To achieve quality education in American schools, we need a better understanding of the way classroom instruction works. Susan S. Stodolsky addresses this need with her pioneering analysis of the interrelations between forms of instruction, levels of student involvement, and subject matter. Her intensive observation of fifth-grade math and social studies classes reveals that subject matter, a variable overlooked in recent research, has a profound effect on instructional practice.

Stodolsky presents a challenge to educational research. She shows that classroom activities are coherent actions shaped by the instructional context—especially what is taught. Stodolsky contradicts the received view of both teaching and learning as uniform and consistent. Individual teachers arrange instruction very differently, depending on what they are teaching, and students respond to instruction very differently, depending on the structure and demands of the lesson.

The instructional forms used in math classes, a "basic" subject, and social studies classes, an "enrichment" subject, differ even when the same teacher conducts both classes. Social studies classes show more diversity in activities, while math classes are very similar to one another. Greater variety is found in social studies within a given teacher's class and when different teachers' classes are compared. Nevertheless, in the classrooms Stodolsky studied, the range of instructional arrangements is very constricted.

Challenging the "back to basics" movement, Stodolsky's study indicates that, regardless of subject matter, students are more responsive to instruction that requires a higher degree of intellectual complexity and performance, to learning situations that involve them in interaction with their peers, and to active modes of learning. Stodolsky also argues that students develop ideas about how to learn a school subject, such as math, by participating in particular activities tied to instruction in the subject. These conceptions about learning are unplanned but enduring and significant consequences of schooling.

The Subject Matters has important implications for instructional practice and the training, education, and supervision of teachers. Here is a new way of understanding the dynamics of teaching and learning that will transform how we think about schools and how we study them.