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• W174891686 abstract "Background Much of the power of mathematics comes from one heing able to both construct and work with its ideas in ways which are not dependent on physical contexts, are general and not local in nature, and are based on using symbolic forms in creative ways. Ernest [1991] suggests that mathematics done in this way is the front of (professional) mathematics, while work with specific local situations, which may serve as motivation for the fOrmal front of mathematics, is ascribed a lesser role in its social construction. In order to facilitate the learning or building up of mathematical ideas, it is inappropriate to simply present the student, even of advanced school mathematics, with formalised ideas. Hare! and Tall [1991] have suggested that such an approach leads to disjunctive generalisations which are not related to less formal schemes In these circumstances, generalisations and related understanding activities are disjoint [Piiie and Kieren, 1992a; 1992b] for many students and can lead to even university level students thinking of mathematics as a rather meaningless mechanical activity [Frid, 1992]. In the case of younger children, Hiebert and Weame [1987] and Skemp [1987], have warned that once they are taught in the manner described above and develop inappropriate, disjoint, and formal generalisations, these schemes are hard to change But, of course, as argued by Hiebert and Wearne [1989] or Hart [1988], simply mathematical ideas in a concrete setting and then later teaching formal generalisations and rules does not necessarily bring about appropriate, connected, formal understanding either The question ts, how can one provide instmction to bring about connected, formalised understanding in students? As part of the answer to this question, it is our purpose in this paper to consider the nature ofjormalising understanding in terms of our theory of mathematical understanding and to illustrate such formalising and contrast it with less formal mathematical understanding as seen in the activities of eight and twelve-year-olds functioning in what we have termed constructivist environments for mathematics [Pirie and Kieren, 1992b; Kieren 1992] We have given general accounts of our theory of the growth of mathematical understanding as a whole, dynamic, leveled but nonlinear, recursive process, and have discussed some of the features of the theory, such as folding back, in a number of places [e g Pirie and Kieren, 1989a; Pirie and Kieren, 1991; Kieren and Pirie, 1992] Figure 1 Model of a dynamical theory of the growth of mathematical understanding" @default.
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• W174891686 date "1994-01-01" @default.
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• W174891686 title "Beyond Metaphor: Formalising in Mathematical Understanding within Constructivist Environments." @default.
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