Abstract
We illustrate, theoretically and empirically, the dialectical relationship between learning and development. We do so in the context of children’s understanding of multiplication concepts. We briefly summarize the Theory of Constructive Operators and examine an important correspondence between the theory’s stages of growth in mental-attentional (M-) capacity and Case’s staircase model of development. We present task analyses of concepts underlying various kinds of multiplication word problems. These analyses lead to predictions about the mental (M-) demand of understanding multiplication concepts. There is a predicted trade-off between a child’s M-capacity and a task’s M-demand, such that a child should not be able to pass a task if its M-demand exceeds the child’s M-capacity. We test this prediction in a study in which children received training in the target multiplication concepts. Results show that children benefit from brief training only when their M-capacity is at least equal to the M-demand of the concept in question. To conclude, we briefly relate our theory to the developing brain.
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Notes
- 1.
Notice that in addition to the complexity of conservation of substance, conservation of weight requires an understanding of the balance scale’s functioning; and conservation of volume requires an understanding that “heavy” solids sink into water and that concurrently the water rises forced by the object as it sinks.
- 2.
The words in square brackets were omitted by error from the printed version of the article.
- 3.
Footnote in the original 1987 paper: “Dynamic/creative syntheses are the little understood processes that enable the organism to produce truly-novel, behavioral or mental, performances that were never learned, are not computationally implied by the combinatorial possibilities of the repertoire, nor are, properly speaking, innate or prewired.”
- 4.
According to Edelman (1987) this lateral inhibition, i.e., activity by local inhibitory neurons that sharpen and segregate dynamic responses of the more highly activated neurons against other competing activated neurons, takes place in layer 4 of the neocortex. Edelman (1987, pp. 163–173) offers a model of how this happens.
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Acknowledgments
Preparation of this chapter was facilitated by an operating grant (#410-2001-1077) to the first two authors from the Social Sciences and Humanities Research Council of Canada (SSHRC). The reported research was funded by a SSHRC operating grant (#410-2006-2325) and a York University Faculty of Arts Research Grant. We are grateful to the following for assistance with data collection: S. Bauer, S. Cunning, D. Fisher, N. Im-Bolter, L. Hakim, and E. Polak.
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Pascual-Leone, J., Johnson, J., Agostino, A. (2010). Mental Attention, Multiplicative Structures, and the Causal Problems of Cognitive Development. In: Ferrari, M., Vuletic, L. (eds) The Developmental Relations among Mind, Brain and Education. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3666-7_3
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