Abstract
Although much has been written about the philosophy for children programme in the academic literature (and the press), there is very little on philosophy and the mathematics classroom, and the little there is has tended to treat mathematics within the context of this programme. By contrast, however, this chapter proposes a radically different perspective whereby the mathematics teacher is the dominant authority and directs the discourse from the front of the class. The aim of this perspective is to move the class towards a deeper understanding of mathematics through philosophy that specifically involves a cultural-historical approach. To place this perspective in context, this chapter begins with a critique of the philosophy for children programme as presented in the literature and critiques the use of philosophy as a bolt-on to the mathematics already learnt.
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Unfortunately this chapter does not discuss philosophy and undergraduate mathematics as this deserves a chapter in its own right. At this level there is not only a change in content (with an emphasis on formalism and rigour) but also a variety of teaching/learning methods that may not be so appropriate at the secondary school level. Nevertheless Pincock’s (2012) Mathematics and Scientific Representation would be most appropriate as a text for this level.
- 2.
Although there were great mathematical advances prior to the Greeks, the Greeks created deductive proof and the necessary theoretical objects to accomplish it, culminating in the axiomatic framework of the Elements. There was nothing like it beforehand and nothing like it until the nineteenth century when much of mathematics was rewritten in axiomatic form. Prior human achievements in mathematics notwithstanding, Greek deductive geometry was the most stunning advance, and there is a sense in which ‘history’ begins two and a half thousand years ago – especially since over that period to learn mathematics was to learn the Elements (to the delight or chagrin of many pupils).
This is not by any means to undervalue the educational potential in introducing the mathematics of, for example, the Babylonians (is the 360° in the angle measure of a circle now a matter of convention or was there an objective reason for adopting it? Is there a need for base-ten compared with base-60? What prompted their recipe for what is today expressed as the quadratic formula?). However, this article is primarily about engaging pupils consciously with justification and proof, the abstract theoretical objects that were created for the purpose and the impact both culturally and cognitively.
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Perhaps this whole debate concerning schoolchildren, philosophy and religious belief can be resolved if religious education and worship are thrown out of schools. Perhaps the UK should adopt the US model whereby state education excludes any form of worship and religious instruction. The school should be seen as an induction into rationality and reason to which faith and indoctrination have no place.
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From a sample of adults performing a classification task, Wason (1977) shows how even adults deny the relevance of facts or contradict themselves, and how conceptual conflict can arise when the force of a contradictory assertion is denied. This implies that even adults do not acquire the rules of logic just because they speak. The Lipman and colleagues assertion is therefore unwarranted.
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According to Vygotsky (1987), concepts are not absorbed ready-made by the child but undergo a process of development. Initially, the concept may begin as a complex; for example, the child might use ‘dog’ not as a member of ‘animal’ but as an ‘associative complex’ extended to inanimate objects with fur. A complex often relies on perceptual features. For example, children who think in complexes may successfully complete a classification problem involving geometrical shapes, so the child might appear to think in concepts – until the child attempts a borderline example, such as a trapezium looking very much like a parallelogram, in which the child classifies as a parallelogram without thinking of how a parallelogram is defined (see Vygotsky 1987). The point being the child might not even think in concepts, let alone concepts that may be deemed philosophical.
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‘Many [children] found convincing a view which could well be labelled constructivist – that infinity is not something we imagined as complete, but was a result of the fact that there was no stopping point, that infinity just kept going on. We then turned to measuring infinity and, having detoured through Cantor’s proof that the number of integers was equivalent to the number of even integers, ended on a note of indecision’ (De la Garza et al. 2000).
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This is not, however, an instructional unit on philosophy and mathematics (Jankvist’s modules approach) nor is it a course that pursues a particular philosophy of mathematics (Jankvist’s philosophy approach).
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Of course a little historical excursion can reveal how the Sophists used indivisibles and how the school allied to Plato used Eudoxus’ method of exhaustion. The two can be compared in Archimedes’s various proofs for the quadrature of the parabola (and the Palimpsest, the book where Archimedes uses indivisibles, has itself a wonderful history).
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At the secondary level the aim has more to do with developing an understanding of mathematics itself (and its place in terms of impact) than it has with understanding its history. The same with philosophy; for example, a secondary school mathematics teacher doesn’t raise Plato’s distinction between knowledge and belief because it happens to be a good thing to know, but because it illuminates the concept of proof.
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And as far as the majority of English and Welsh classrooms are concerned, proof no longer exists and perhaps because many teachers and curriculum designers can remember the negative experience of having to regurgitate proof in examinations. For some educationalists, proof means rote learning.
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Even social constructivists such as Paul Ernest love and rely on Godel’s proof, even though they don’t like proofs in general (Ernest goes so far as to regard proof as Eurocentric and its glorification racist). This and other social constructivist contradictions can be found in Rowlands et al. (2010).
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Rowlands, S. (2014). Philosophy and the Secondary School Mathematics Classroom. In: Matthews, M. (eds) International Handbook of Research in History, Philosophy and Science Teaching. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7654-8_22
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