Abstract

In this article I sketch three key concepts of a cultural-historical theory of mathematics teaching and learning—the theory of objectification. The concepts are: knowledge, knowing and learning. The philosophical underpinning of the theory revolves around the work of Georg W. F. Hegel and its further development in the philosophical works of K. Marx and the dialectic tradition (including Vygotsky and Leont’ev). Knowledge, I argue, is movement. More specifically, knowledge is a historically and culturally codified fluid form of thinking and doing. Knowledge is pure possibility and can only acquire reality through activity—the activity that mediates knowledge and knowing. The inherent mediated nature of knowing requires learning, which I theorize as social, sensuous and material processes of objectification. The ideas are illustrated through a detailed classroom example with 9–1 0-year-old students.

Keywords: objectification; knowledge, knowing, learning, consciousness.

In this article I sketch three key concepts of a cultural-historical theory of mathematics teaching and learning—the theory of objectification. The concepts are: knowledge, knowing and learning. The theory rests on the fundamental idea that learning is both about knowing and becoming. It considers the goal of mathematics education as a dynamic political, societal, historical, and cultural endeavour aiming at the dialectical creation of reflexive and ethical subjects who critically position themselves in historically and culturally constituted and always evolving mathematical discourses and practices. The philosophical underpinning of the theory revolves around the work of the German philosopher Georg Wilhelm Friedrich Hegel (1 977, 2009) and its further development in the philosophical works of Karl Marx (1 973, 1 998) and the dialectic tradition—Ilyenkov (1 997), Mikhailov (1 980), and Vygotsky (1 987-1 999), among others.

Most of the article is devoted to the concept of learning. I start, however, with a discussion about the concepts of knowledge and knowing. Although a discussion about knowledge and knowing may seem esoteric and even futile, I claim that if mathematics education theories want to provide suitable accounts of learning they need to clarify what they believe constitutes knowledge and knowing in the first place. Learning is, indeed, always about something (e.g., learning about probabilities, about geometric properties of figures, etc.). As a result, we cannot understand learning if we do not provide a satisfactory explanation of what learning is about. The next section starts with a discussion of knowledge as construction, followed by a discussion of knowledge as it is understood in the theory of objectification.

Knowledge

Knowledge as Construction

It is now common in mathematics education discourse to talk about knowledge as something that you make or something that you construct. The fundamental metaphor behind this idea is that knowledge is somehow similar to the concrete objects of the world. You construct, build or assemble knowledge, as you construct, build or assemble chairs. This idea of knowledge as construction is relatively recent. It emerged slowly in the course of the 1 6th and 1 7th centuries, when manufacturing and the commercial production of things became the main form of human production in Europe. Hanna Arendt summarizes this conception of knowledge as follows: a “I ‘know’ a thing whenever I understand how it has come into being.” (Arendt, 1 958, p. 585) It is within the general 1 6th and 1 7th centuries’ outlook of a manufactured world that knowledge is first conceived of as a form of manufacture as well. A limpid exposition of this view appeared at the end of the 1 8th century in Kant’s Critique of Pure Reason. In this monumental book whose influence has not ceased to affect us Kant presents mathematics as the most achieved way of knowing and tells us that “Mathematics alone (…) derives its knowledge not from concepts but from the construction of them” (Kant, 2003, p. 590 [A 734/ B 762] ). This conception of knowledge as construction was featured by Piaget in his genetic epistemology and was widely adopted in mathematics education where an emphasis was put on the personal dimension of knowledge
construction: You and only you construct your own knowledge. For, in this view, knowledge is not something that I can construct and pass on to you; what you know is what results from your own experience.

As many scholars have pointed out, such a view of knowledge is problematic on several counts. For instance, it leaves little room to account for the important role of others and material culture in the way we come to know, leading to a simplified view of cognition, interaction, intersubjectivity and the ethical dimension. It removes the crucial role of social institutions and the values and tensions they convey, and it dehistoricizes knowledge (see, e.g., Campbell, 2002; Lerman, 1 996; Otte, 1 998; Roth, 2011 ; Valero, 2004; Zevenbergen, 1 996). As we shall see in the next subsection, there are other ways in which to consider knowledge and the students’ relationship to it.

Artículo Completo

REDIMAT- Journal ofResearch in Mathematics Education Vol. 2 No. 1
February 2013 pp. 7-44.

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