Towards an Embodied, Cultural, And Material Conception of Mathematics Cognition by Luis Radford

This is an interesting counterpoint to the Lakoff & Nunez model.
A new research trend, however, offers a different approach to the understanding of human cognition. In this trend our tactile-kinesthetic bodily experience of the world and our interaction with artifacts are considered to be much more than merely auxiliary or secondary elements in our cognitive endeavours (Bautista & Roth, 2011; Borba & Villareal, 2006; Edwards, Radford, & Arzarello, 2009; Lakoff & Núñez, 2000; Sheets-Johnstone, 2009). Dwelling upon Vygotsky’s (1987-99) and Leont’ev’s (1978, 2009) work on enactivism (Maturana and Varela, 1998), in this paper I elaborate on what I have previously termed sensuous cognition (Radford, 2009a).
The author adopts, rather than an mind-body duality, “a monistic position according to which mind is a property of matter. More specifically, mind is conceptualized as a feature of living material bodies characterized by a capacity for responsive sensation.”
Ontogenetically speaking, through a tactile experience, the child can feel the weight of an orange; through a perceptual one, she can have a sense of its relative chromatic characteristics. Later, she can feel its  porous skin even if it is out of her actual tactile reach. Knowing hence appears to occur through a multi-modal sensorial experience of the world.
To sum up, instead of being something purely “mental,” reflection and its products remain, one way or another, intertwined with the environment that is been reflected and with the organism’s capacities for sensation.
In Zaporozhets’ example, preschool children develop a mathematical form of perception that allows them to distinguish between cultural categories of geometrical figures. In the course of this developmental process, the children have recourse to the material objects whose contours they cover with a finger while using numbers to count aloud. What this example shows is that our individual senses evolve intertwined not only one with the other senses, but also with the materiality of the objects in our surroundings. The materiality that shapes our senses is not, however, reduced to inert matter, but, as the example shows, matter already endowed with meaning (e.g., ‘triangularity,’ ‘quadrilarity,’ etc.).
Here’s an interesting bit, which is a bit of a sudden leap. We need to know more about this.
p. 8
At this point the relationship between variables started becoming apparent for the students. The relationship started being objectified.
The episode suggests how the target cultural knowledge is objectified as a new ideational-material psychic unity is forged. The students no longer need to see the terms of the sequence. What could only be made apparent through an intense interplay between various sensorial modalities and different signs is now contracted, subsumed and reorganized in a new complex psychic unity where no reference is made to top or bottom rows.
But how does this happen exactly – how does a thing become an idea?

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