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

Abstract

This paper explores the public awareness that there presently exists a crisis in mathematics education and a "dumbingdown" of the curriculum, examines the nature of this crisis and argues that there has been a lowering of cultural, pedagogical and cognitive expectations with respect to most learners. The notion of cognitive development in mathematics education is re-examined and a model of how the concepts of learners can be transformed in the very process of engaging with the conceptual revolutions that defined geometry is proposed. The importance of cultivating a met a-narrative in support of metacognition and the development of cognitive growth are stressed.

A Pilot Study of a Cultural-Historical Approach to Teaching Geometry

“…Indeed, experience has shown that 'low ability' secondary students are quite capable of abstract reasoning regarding proof. A theoretical justification of this can be found in Carson and Rowlands (2007a).…”

“…Hershkowitz et al 2001) and Carson and Rowlands (2007a) have criticised Piaget's notion of reflective abstraction and used a bridge metaphor to show the ascent from the abstract to the concrete. Unfortunately there is no room to summarise the discussion; needless to say that this project involves a teacher initiated activity that purposefully leads students to consciously undergo what is meant by abstraction.…”

An investigation into the efficacy of the Algebra Project

“…He referenced studies which indicated that many students have only a cursory understanding of math concepts, and that they often lack the deeper understanding necessary to transfer their learning to out-of-school settings. Additionally, empirical evidence from the United States presented by White-Clark et al (2008) pointed to low percentages of students who are proficient in basic math nationwide; with anecdotal evidence from Carson & Rowlands (2007) suggesting that a general sense o f "resistance and ennui" are increasingly found in certain students as a result of the traditional approaches used to teach to mathematics.…”

“…However, while there is evidence to suggest that the implementation of a constructivist-based approach can lead to improved learning results, it also comes with some additional challenges. Several researchers have noted that that a lack of time can influence the success of constructivist-based activities in mathematics classrooms (Andrew, 2007;Carson & Rowlands, 2007;Gadanidis, 1994;Kamii, 1990;Morrison et al, 2008;White-Clark et al, 2008). Other documented challenges include: a) the need for additional teacher pre-service and in-service training, b) the low tolerance of some students for trial-and-error learning, c) the difficulty in monitoring and providing feedback during instruction, and d) the fact that some of the concepts under investigation took extensive time to discover and develop (Andrew 2007;Carson & Rowlands, 2007;Jones & Southern, 2003).…”

“…Several researchers have noted that that a lack of time can influence the success of constructivist-based activities in mathematics classrooms (Andrew, 2007;Carson & Rowlands, 2007;Gadanidis, 1994;Kamii, 1990;Morrison et al, 2008;White-Clark et al, 2008). Other documented challenges include: a) the need for additional teacher pre-service and in-service training, b) the low tolerance of some students for trial-and-error learning, c) the difficulty in monitoring and providing feedback during instruction, and d) the fact that some of the concepts under investigation took extensive time to discover and develop (Andrew 2007;Carson & Rowlands, 2007;Jones & Southern, 2003). Finally, at least one set of researchers documented the concern that not all students have the necessary academic readiness to engage in some of the interactive activities, which can limit the effectiveness of the learning situations and activities if not monitored by the instructor (Carson & Rowlands, 2007).…”