Tangent lines are often first introduced to students in geometry during the study of circles. The topic may be repeatedly reintroduced to students in different contexts throughout their schooling, and often each reintroduction is accompanied by a new, nonequivalent definition of tangent lines. In calculus, tangent lines are again reintroduced to…

Conjectures are a key component of mathematical inquiry, a process in which the students raise conjectures, refute or dismiss some of them, and formulate additional ones. Taking a design-based research approach, we formulated a design principle for personal feedback in supporting the iterative process of conjecturing. We empirically explored the…

The transition from preformal and propaedeutic generalization-actions to a symbolically explicit use of the concept of variable has been a matter of significant attention in mathematics education, for example in the context of generalization processes on a preformal level and regarding the specific nature of algebraic concepts. This contribution…

Rats were trained to search for a food reward hidden under sawdust in the center of a square-shaped enclosure designed to force orientation on the basis of the overall geometry of the environment. They were then tested in a number of enclosures differing in shape and in size (rectangular-, double-side square-, and equilateral triangle-shaped…

This volume attempts to bring together a collection of reports on the Geometric Supposer, a series of computer software environments which can be a tool for exploring particulars and generalizations in geometry. The book contains the following chapters: (1) "A Personal View of the Supposer: Reflections on Particularities and Generalities in…