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Tall, David – For the Learning of Mathematics, 2011

This paper introduces the notion of "crystalline concept" as a focal idea in long-term mathematical thinking, bringing together the geometric development of Van Hiele, process-object encapsulation, and formal axiomatic systems. Each of these is a strand in the framework of "three worlds of mathematics" with its own special characteristics, but all…

Descriptors: Geometric Concepts, Mathematics Instruction, Mathematical Models, Mathematical Concepts

Tall, David – For the Learning of Mathematics, 2004

In this commentary on Matthew Inglis' "Three Worlds and the Imaginary Sphere" (see EJ1106688), David Tall develops the theme that the building of theories is not an easy process. A theory in progress is a particularly delicate creation. Theories do not appear fully formed. There is a period of exploration and incubation that precedes the…

Descriptors: Theories, Mathematics, Mathematical Concepts, Perception

Peer reviewed

Pinto, Marcia; Tall, David – For the Learning of Mathematics, 2002

Reports on a case study of a student who constructs formalism not from processes of quantification, but from his own visuospatial imagery. Rather than construct new objects from cognitive processes, the student reflects on mental objects already in mind and refines them to build an interpretation of the formal theory. This example leads to…

Descriptors: Case Studies, Communication (Thought Transfer), Higher Education, Mathematics

Peer reviewed

Tall, David – For the Learning of Mathematics, 1989

Discusses using the computer to promote versatile learning of higher order concepts in algebra and calculus. Generic organizers, generic difficulties, and differences between mathematical and cognitive approaches are considered. (YP)

Descriptors: Algebra, Calculus, Computer Uses in Education, Computers

Peer reviewed

Harel, Guershon; Tall, David – For the Learning of Mathematics, 1991

The terms generalization and abstraction are used with various shades of meaning by mathematicians and mathematics educators. Introduced is the idea of "generic abstraction" that gives the student an operative sense of a mathematical concept and provides a passage point in the process toward formal abstraction. (MDH)

Descriptors: Cognitive Processes, Concept Formation, Generalization, Learning Theories