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By Courant R.

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For the diagram itself "is an icon or schematic image embodying the meaning of a general predicate" (ibid. 238), and the relation it represents - the general predicate - is a rational one, an intrinsically general formal relation. It is "not merely one of those relations which we know by experience, but know not how to comprehend, but one of those relations which anybody who reasons at all must have an inward acquaintance with" (ibid. 316). For instance, it would be impossible, Peirce says, to represent in a diagram the mere relation of killer to killed, because it is something that is not intelligible.

Where the construction and concatenation of the sequence of semiotic actions deployed is not automatic, that is has not been practised on similar tasks until it has become routinised for this particular student, it is appropriate to call it creative. It corresponds to non-routine problem solving and involves the student or person in constructing and combining in novel ways (new to herself, at least) different signs and procedures. Carrying out tasks individually or in groups may be the most common higher level activity in speaking/writing in school mathematics.

Et al. Eds (1956). Taxonomy of Educational Objectives 1, Cognitive Domain. New York: David McKay. Chomsky, N. (1965). Aspects of the Theory of Syntax. Cambridge, Massachusetts: MIT Press. Davis, C. (1974). MateriaUst philosophy of mathematics. In R. S. Cohen, J. Stachel & M. W. , For Dirk Struik. Dordrecht: Reidel. Dubinsky E (1988). On Helping Students Construct The Concept of Quantification. In A. ) Proceedings ofPME 12. Veszprem, Hungary, Vol. 1, 255 - 262 Ernest, P. (1991). The Philosophy of Mathematics Education.

### Advanced methods in applied mathematics, lecture course by Courant R.

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