On the relation of abstract and concrete in scientists’ graph interpretations: A case study

Does mathematical learning occur in going from concrete to abstract or in going from abstract to concrete?

“…the way in which thought appropriates the concrete, reproduces it as the concrete in the mind" (Marx, 1973, p. 101). This statement is consistent with our experience described in the opening part of the companion text (Roth & Hwang, 2006), where vectors first were merely concrete images and words on a page in a high school mathematics book (abstract), and subsequently were concrete, almost palpable entities in the hands and minds of a mature physicist and statistician. Marx's original framing therefore is opposite to the recently published interpretation some mathematics educators offered (Hershkowitz et al, 2001).…”

“…Second, interpretation has to be understood as a continual movement of consciousness that goes from subject to object and from object to subject. This aspect stands out in our case study featuring Eddie's interpretation of one graph (Roth & Hwang, 2006).…”

“…Rightly so, this position, which constitutes the foundation of Piaget's work and (radical) constructivism, was recognized as constituting "subjective idealism" (Hegel, 1975). The Kantian approach remains insufficient to theorize processes of scientific activity such as interpreting graphs, the one our research participant Eddie conducted (Roth & Hwang, 2006). Because graphs express something about the experienced world and therefore are of the order of empirical concepts rather than pure ideas, we need to begin by articulating the relationship of the subject of consciousness (knowing) and its object of consciousness (knowing).…”

“…However, those scientists who succeeded articulated a lot of concrete detail and used their lived experiences during their interpretations. To better understand the relation between the abstract (abstraction) and concrete (concretion), we constructed a detailed case study of one scientist (Eddie) interpreting a graph featuring the distributions of three types of plants; the graph is used in ecology courses in support of a theory of adaptation (see Roth & Hwang, 2006).…”

Keeping the local local: Recalibrating the status of science and traditional ecological knowledge (TEK) in education

“…We cannot say that scientific knowledge applies in this or that particular context, but rather, enormous situated and situating work has to be done to specify local contingencies in such a way that anything like scientific knowledge can be brought to bear; and it frequently turns out that it does not bear at all on the situation unless the situation itself is changed to fit the contingencies of scientific knowledge. This is a universal rather than a particular characteristic of scientific knowledge, given the many similar findings in social studies of science in various disciplines such as ecology and physics (Bowen, Roth, & McGinn, 1999;Roth, 2004), marine science (Roth & Hwang, 2006), rocket science (Collins & Pinch, 1998), medicine (Epstein, 1996;Lambert & Rose, 1996), econometry (Evans, 1997), agricultural science (Shepherd, 2006), and ergonomics (Hyysalo, 2006). This feature of scientific knowledge can be seen as a particular of the situated nature of human cognition shown in studies on navigation (Hutchins, 1983), weaving (Childs & Greenfield, 1980), coal mining (Billett, 1993), dairy workers (Scribner, 1986), midwifery (Jordan, 1989), and tailoring (Lave, 1990), which causes people to encounter problems when changing local contexts in such a way that decontextualized knowledge can be brought to bear within them (Lave, 1988;Miller & Gildea, 1987).…”