In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the "Elements" of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in…

Students are faced with many transitions in their middle school mathematics classes. To build knowledge, skills, and confidence in the key areas of algebra and geometry, students often need to practice using numbers and polygons in a variety of contexts. Teachers also want students to explore ideas from probability and statistics. Teachers know…

In some secondary mathematics curricula, there is a topic called Stereometry that deals with investigating the position and finding the intersection, angle, and distance of lines and planes defined within a prism or pyramid. Coordinate system is not used. The metric tasks are solved using Pythagoras' theorem, trigonometric functions, and sine and…

Transformations are a central organizing idea in geometry. They are included in most geometry curricula and are likely to appear with even greater emphasis in the future, given the central role they play in the "Common Core State Standards" for K-12 mathematics. One of the attractions of geometry is the ability to draw and construct the…

Trends in International Mathematics and Science Study raised the concern that the United States (US) mathematics teachers place less emphasis on topics and teach a variety of content strands (Goldsmith, Mark & Kantrov, 2000; Kerachsky, 2008). The overall pattern in the US mathematics classrooms today is that primary and upper elementary…

This book, photographically reproduced from its original 1942 edition, is an extended essay on one of the three problems of the ancients. The first chapter reduces the problem of trisecting an angle to the solution of a cubic equation, shows that straightedge and compasses constructions can only give lengths of a certain form, and then proves that…