To announce, as Euclid did… that ‘A line is a breadth-less length’ is to pack an awful lot of presumption into [an] incomprehensible statement – incomprehensible because human imagination cannot conceive of a line that has no width… Now the student is left with a dilemma, either to accept on faith something that violates all prior experience and common sense, or to simply refuse to proceed further… We routinely treat this and other cognitive crises in a cavalier and dismissive manner, expecting learning to violate their own internal warning system and go along with what we say, simply based on teacher authority… [I]ntellectual confidence should be build up on the basis of evidence and reason, not on a system of authoritative fiat, wherever possible. (Rowlands & Carson, 2022, p. 11)

It’s rare to come across a genuinely groundbreaking idea in curriculum design, but I think education researchers Stuart Rowlands and Robert Carson are onto something revolutionary. Their latest article (reproduced in full below, with permission) fleshes out their ideas for structuring a high school maths curriculum around philosophically-rich inflection points in the history of mathematical discovery. The authors offer tantalising and authentic examples of narrative stimuli that can be used in maths classrooms to convey the electrifying energy that would have been felt in particular historical and socio-cultural contexts when geometric proofs were discovered for the first time. It’s no surprise that the 11- and 12-year-old students who experienced this new curriculum were riveted and and intellectually excited by it, or that they responded so thoughtfully to its provocations.

The article ‘Philosophy and history as an epic narrative in secondary school mathematics’ (Mathematics in School, vol. 51 no. 4) details a novel program for curricular reform which promises to enliven and deepen the learning of high school mathematics. More than that, it provides a needed corrective to the mistaken notion that philosophy in maths classrooms can be nothing more than a ‘bolt-on’ to conventional instruction. Rowlands and Carson show that philosophy can in fact be integral: situating students as active players in a dramatic recapitulation of the history of ideas, and reconnecting academic subject matter with the broader intellectual culture – subject matter that in the modern era has become unfortunately siloed and dissociated from wider concerns.

In contemporary schooling, the authors observe, studying maths is almost invariably reduced to an “other-worldly pursuit operating in its own existential universe… so that students are left to wonder how any of this is ‘relevant’, or ‘meaningful’, or even useful” (p. 9). The authors are intent on restoring this lost meaning by giving students ‘a seat at the table’ as they survey the stepwise accretion of geometrical understanding. Their instruments of choice are vivid and relatable (if somewhat speculative) phenomenological accounts of puzzlement and insight, localised in place and time. Rowlands and Carson argue that the intrinsic beauty and fascination of the discipline of mathematics are best elucidated through narrative accounts that reveal an evolution “from its naturalistic origins through cultural-historical processes of development” (p. 11).

Thales and [the] Egyptian priests… are examining a configuration of four wooden stakes, pounded into the ground, with two ropes stretched tightly between opposing stakes to form an elongated ‘X’… [T]he older Egyptian priest has pointed out that opposite angles are equal. And Thales has agreed, saying ‘Yes, that is obvious… But, how would you prove it?’ The Egyptians are silent for a moment, then burst out in laughter, and finally the older fellow asks, ‘Why on earth would you need to prove something when it is so obvious?’…

In the classroom… students came up with several very good questions, including ‘Right, why would you want to prove it if you already know the answer?’ and ‘Is it really obvious? I am not sure’. And ‘Yeah, that is a good question. How could you prove it? … Thales puzzled deeply over the question of what would constitute an acceptable form of demonstration…

With this line of inquiry … students [enter] into a conceptual problem space, the ‘point of inflection’ in the course of mathematical history from which the concept and practice of ‘formal proofs’, based on logic rather than measurement or calculation, is born. In a subsequent story, Thales will pass on his interest in geometry to Pythagoras, who actually did develop the beginnings of formal proof, and achieved the transfiguration of geometric figures from concrete objects acted upon by empirical measurement, to theoretical objects acted upon by means of pure reason. It is in that transcendence that a ‘line’ sheds its width, and the conversation in geometry becomes one of logic rather than measurement. (pp. 11–12)

Rowlands and Carson argue that even where students succeed in correctly applying rule-of-thumb procedures to solve mathematical problems correctly, teachers systematically fail to convey a deep understanding of the central concepts at play and the relationships among them. The well-crafted and compelling examples of narrative history in Rowlands and Carson’s article reveal how their proposed approach to teaching maths as an ‘epic narrative’ not only contextualises the difficulties that students face as they struggle to understand new mathematical concepts, but also highlights the cultural significance of the concepts in question. The potency of this approach lies in animating the birth of modern scientific rationality: “the ability to sequentially and logically from premises to conclusions, and to make those thought processes visible to inspection and open to critique” (p. 12). Teaching maths through epic narrative therefore provides what may well be “the single best opportunity most students will ever have in understanding the uniqueness of scientific culture as a human achievement… Without this human and cultural context, mathematics is too often experienced by learners as an unending series of facts and procedures to be memorised, for no apparent reason” (p. 13).

It is late. The sun has gone down, calming the intense heat of the Egyptian desert. Thales looks up into the night sky and sees three stars. His mind connects the three stars with invisible lines. For just an instant he believes that he can see those lines too, just as clearly as he sees the stars. ‘Those three stars have formed a triangle,’ he says to himself. He blinks. The image fades… ‘The stars are real’, [he says to himself], ‘for the evidence of them persists. But the lines I saw … they were only ideas.’ But what about ideas? Aren’t they real too? (p. 11)

I was honoured to have the opportunity to comment on an earlier draft of this article, and I’m grateful to the authors for generously endorsing my paper, Unveiling and Packaging: A Model of Presenting Philosophy in Schools, in their endnote.

You can read Rowlands & Carson’s innovative article in full below. Please share it with the maths teachers and curriculum designers in your networks. The proper citation is as follows: Rowlands, S. & Carson, R. N. (2022), Philosophy and history as an epic narrative in secondary school mathematics. Mathematics in School 51(4), pp. 9–13.

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The Philosophy Club works with students and teachers to develop a culture of critical and creative thinking through collaborative inquiry and dialogue.