Teaching the Conceptual Revolutions in Geometry

Abstract

Mathematics begins in human experience thousands of years ago as empirical and intuitive experiences. It takes the deliberate naming of concepts to help crystallize and secure those observations and intuitions as abstract concepts, and to begin separating the concept of number from specific instances of objects. It takes the creation of compact symbols to enable efficient calculation and to begin raising a consciousness of this activity we call mathematics. And it takes the sustained development and discussion of mathematical conventions and practices to create entire domains of mathematical thought, such as we find in geometry. The major innovations and conceptual reformulations are few in number, but these represent perhaps the greatest challenges to learners. Historically significant transformative events have their counterpart in the cognitive growth of the individual. This article examines the interplay between these big ideas in cultural history and the deliberate processes of cognitive change that are their counterpart in the educational process.

Problems with Fallibilism as a Philosophy of Mathematics Education

“…Although Platonism is a highly respected position in philosophy (to which we agree with educationalists that Platonism is somewhat ontologically indefensible, although there is no space to enter into the debate), Plato's Forms may have been a heuristic to emphasise the distinction between abstract concepts with concepts that are contextually bound. Here, historically, was the formation of theoretical objects (such as the geometric straight-line) distinct from the concrete exemplars that once served to represent what is to count as a straight line, such as the stretched rope (see Carson and Rowlands 2007). The Forms emphasised a distinction between the two that could have easily collapsed (and still could!).…”

Bharath Sriraman and Simon Goodchild (eds): Relatively and Philosophically Earnest: Festschrift in Honor of Paul Ernest’s 65th Birthday, A volume in The Montana Mathematics Enthusiast

“…but would also situate the learner in the society that she belongs-the justification problem would then evaporate. The teacher that can teach deductive geometry by situating the formalism culturally and historically, discussing what the terms of discourse mean in that context and raising philosophical awareness in the process, can be achieved with any age group from 11 years of age upwards and at almost any level of development (see Carson and Rowlands 2007;Rowlands 2010. Of course, the teaching would have to adapt to the level of development).…”

Strategies for Affecting the Necessary Course of Cognitive Growth as an Integral Part of Curricular and Instructional Planning

“…The following is a very brief description of that exercise (taken from Rowlands, 2006. For a theoretical underpinning, see Carson and Rowlands, 2005). Following a narrative treatment of Thales' visit to Egypt, four stakes are pounded into the ground with two intersecting ropes forming an X.…”