Abstract
In this chapter, we initially clarify what we mean by an emergent structure from a parallel distributed processing (PDP) point of view. Then we contrast an emergent structure from other well-known points of view of structures in cognitive science, in particular, symbol structures, theory-theory structures, and probabilistic structures. We also expound on the theory of PDP in semantic cognition in some detail and close the chapter with a discussion of the implications of the PDP theory on pattern generalization processes that matter to mathematical learning. In the closing discussion we discuss the need to modify some of the elements in the original PDP model based on cognitive factors that bear on pattern generalization processes involving school mathematical patterns. Further, we demonstrate the usefulness of a PDP network structure primarily as a thinking model that enables us to describe the complexity of students’ pattern generalization processes not in terms of transitions from, say, arithmetical to algebraic generalizations, but as parallel and graded, adaptive, and fundamentally distributed among, and dependent on, a variety of cognitive and extracognitive sources.
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Notes
- 1.
Glenberg, de Vega, and Graesser (2008) and Steels (2008) note that the meaning of symbols in cognitive science differs from, say, a Peircean view of symbols that grounds ideas from objects. Grounding refers to those processes that we and other computer-driven systems use to link mental structures onto external objects (p. 3). For Peirce, symbols as a type of signs mediate between an object and an interpretant (i.e. how the object appeals to individual learners). Cognitive science in general, however, has a more theoretical conception of symbols beyond grounding—that is, symbols convey “an arbitrary relation between symbol and referent” (ibid., p. 2). Hence, they can be “arbitrarily related to objects, abstract, or amodal (i.e. they are nonperceptual or not tied to particular sensory modes)” (ibid). For example, the word “chair” as an amodal symbol does not in any direct way resemble an actual chair with four legs.
- 2.
As Bereiter (1991) notes, older connectionist views put premium on connections and associations between individual units, say, word to word. The newer version, such as PDP, the units, “like the neurons they are modeled after—do not individually signify anything. Instead, as Rumelhart (1989), puts it, ‘all the knowledge is in the connections’” (p. 11).
- 3.
Readers who are interested in details involving the mathematical calculations that are needed to determine the weights are referred to Rogers and McClelland (2008, pp. 692–693).
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Rivera, F. (2013). A Theory of Graded Representations in Pattern Generalization. In: Teaching and Learning Patterns in School Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2712-0_4
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