I am not that learned in philosophy of mathematics and therefore the following thesis is quite a new word for me. I cannot classify it as belonging to any established tradition in philosophy of mathematics.
Some time ago, I came across another industrial illustration which relates specifically to school mathematics. The Cockroft Committee of Enquiry into the teaching of mathematics in schools was set up in 1978 as part of James Callghan’s Labour government’s response to widespread criticism of state education, especially from the engineering industry and the writers of the Black Papers (see Kogan, 1978; Lawton, 1980; Salter and Tapper, 1981). The Cockroft Committee was instructed to pay “… particular regard to the mathematics required in further and higher education, employment and adult life generally…” (Cockroft et al. 1982, p. ix).
As part of their response, the Committee set up three research studies. One of these studies was conducted by a team from Bath University. This study focused on the mathematical requirements of the working practices of sixteen-to-nineteen-year-olds. The researchers collected data which they categorized as ‘specific tasks incorporating mathematics ‘ (STIM) and ‘mathematics incorporated in specific tasks’ (MIST). They found that a great many young employees — a ‘vast army of people’ — did not appear to require any formal mathematics, not even counting or recording numbers. Nevertheless, they claimed that ‘all these occupations involve actions which could be described in mathematical terms’ (Bailey et al, 1981; p. 12). The researchers presented a list of these mathematical terms (MIST) as follows:
- A set, dis-joint sets.
- Mappings, one-to-one, one-to-many, many-to-one correspondences.
- Symmetry, bilateral and rotational.
- Rotation, reflection, translation and combinations of these[.] Tesselation patterns.
- Logical sequences (if … then … .). (Bailey et al, 1981; p.26)
- Tasks often have to be carried out in particular orders sometimes requiring simple decisions, but which would not often be verbalised. For example, a creeler in a carpet factory: ‘Is the spool empty? Yes! Replace with another of the same colour’. Awareness of the conseuences of not following the prescribed order may be important. (ibid; pp.25-6)The terminology used there is very much out of the ‘modern mathematics’ tradition of the era. The tasks (STIM) to which these MIST items corespond are:
- (Articles are sorted into separate collections for packing or on an accept/reject basis.
- Articles are moved into particular orientations involving moving sideways, turning over or round.
- Articles, such as wine glasses[,] are checked for uniformity of shape.
- Packed articles form regular patterns.
- Assembly tasks can involve matching parts, such as connecting wires to correct terminals.The researchers are not claiming that creelers need to study formal logic in order to be abe to replace a carpet spool when it’s empty rather than when it’s full. On the contrary, they are clear that such tasks are successfully carried out in the abscence of mathematical knowledge. Yet in making this claim, they are […] establishing a division between the mathematical-intellectual and the manual and costituting the former as generative of commentaries upon the latter. It is as if the mathematician casts a knowing gaze upon the non-mathematical world and describes it in mathematical terms. I want to claim that the myth is that the resulting descriptions and commentaries are about that which they appear to describe, that mathematics can refer to something other than itself. I shall refer to this myth as the myth of reference.
[P. C. Dowling, The Sociology of Mathematics Education: Mathematical Myths/Pedagogic Texts. Routledge, 1997. ISBN-10: 0750707917. ISBN-13: 978-0750707916.]