What has the teaching of Roman numerals got in common with human embryonic development?
The story of Roman numerals in mathematics is much more than it may first appear. While it might be tempting to think of inclusion of Roman numerals in maths curricula as an important cultural and historical reference – an intermediary step on the route to an efficient counting system that helps reveal to students how easy it is to operate with decimal place value – further examination may reveal criticisms of this view. Asking whether we should include Roman numerals in a mathematics curriculum quickly gives way to a wider and much more important debate about what exactly mathematics is, and it’s a fascinating one.
Roman numerals are difficult for students to learn and work with (e.g. Selvianiresa and Jupri, 2017)1 and this is partly because they are a less sophisticated and efficient form of representing numbers than our current system (a decimal place-value structure with the digits 0-9, variously known as ‘Hindu-Arabic’, ‘Arabic’, ‘Indo-Arabic’ or ‘Latin’ numerals); see also Kathotia (2017).2 You could, if you like, try to plot the evolution of number representation in the same vein as human evolution, with a simple chronological conception, a bit like this:
hominids → homonins → homo sapiens
count marks → tallies → additive symbols → place value symbols
or, in practice, with a much more complex network diagram with much more detail. The “evolutionary” metaphor is actually much more akin to evolutionary systems than straight lines – multiple complex connections across time and space rather than a neat linear sequence. It is also true to say that we are piecing both examples together with limited and fragmentary evidence, as histo-scientists, so will likely never know the true picture – a rough roadmap using the best theory that fits the facts is the best we can do. Concentrating on human biology for a moment, If we extend the evolutionary lens outwards in time and space, we can add other species into the picture too, eventually concluding, on a very simplified basis, with the familiar but still breathtaking idea that we are “evolved” from single-celled organisms who crawled out from a primordial swamp to breathe, forage, hunt and eventually do complex biological experiments on their unwitting ancestors.
Ernst Haeckel, a zoologist, philosopher and artist of the late 19th century, made quite the splash with his extraordinary idea that in order to develop into humans, embryos in the womb go through a micro-version of this evolution (this model also covers other creatures). That is to say, the stages of evolution are replicated and accelerated for every person that has ever lived before they are born: your adorable infant was first a little fish, a froglet, a miniature lizard and a tiny bird. This is called the theory of recapitulation, after the phrase “ontogeny recapitulates phylogeny” – meaning the development of the embryo follows the same (although very compressed) development of the adult evolutionary stages.
In curriculum terms, this idea can be applied in the same way: a suggestion that students’ learning should recapitulate, or in some sense replicate, the evolution of the mathematics they are learning. “Psychological recapitulation, which transposes the biological law of recapitulation, claims that, in their intellectual development, our students naturally traverse more or less the same stages as mankind once did” (Furinghetti & Radford, 2008, p. 627).3 In other words, “The educators’ task is to make children follow the path that was followed by their fathers, passing quickly through certain stages without eliminating any of them” (Poincaré, 1899).4
The biological model however, has been generally considered untrue for many years with multiple counterexamples found to exist (e.g. Rasmussen, 1991).5 Further, in curriculum development terms, the recapitulation theory suggests not a development but a “natural” and predetermined unfolding that does not seem to fit the classroom experience of learning mathematics. Another issue is that the twists, turns and dead-ends in the history of mathematical thought may make for inefficient and confusing learning. But the most important argument against it is the question of which historical and cultural experiences should be privileged. This extract from Thom’s 1973 ICME address shows the problem:
Pedagogy must strive to recreate (according to Haeckel’s law of recapitulation—ontogenesis recapitulates phylogenesis) the fundamental experiences which, from the dawn of historical time, have given rise to mathematical entities. Of course this is not easy, for one must forget all the cultural elaborations (of which axiomatics is the last) which have been deposited on these mathematical objects, in order to restore their original freshness. One must forget culture in order to return to nature (cited in Furinghetti and Radford, 2008, p.628).3
The idea that mathematics is pure, abstract, universal, eternal and free from culture is termed an internalist view. Externalists, on the other hand, suggest that mathematical knowledge production happens within distinct cultural and social contexts, inseparable in some important ways from the mathematics itself, which in turn depends on culture and society. In terms of Roman numerals, one could suggest alternatively that they were “invented” or “discovered;” that they – or the structures they represent – are a necessary step in the unfolding of ideas about systematic mathematical representation or an arbitrary one. This is a complex question, and a particularly interesting one in light of the idea that much of mathematical history has been subject to silencing; to an intentional erasure and revisionism as part of a wider process of colonialism and imperialism.
In terms of Roman numerals, this means that including them in curricula is not something one might suggest is in itself wrong, but rather that it may be a puzzling choice to include them as the object of mathematical learning by themselves. This is not only because there are lots of other interesting and important historical number representation systems in the world (why Roman, specifically?) but also because effective mathematics teaching is about deep understanding of structure and relationships, and an unconnected focus on performative use of a single specific number system may obscure this.
How do you see ideas of cultural, historical and societal context as related to the teaching of mathematics?
References:
- Selvianiresa, D., & Jupri, A. (2017). Analysis of students’ incorrect answers on the topic of Roman numerals in the fourth-grade of primary school. Journal of Physics: Conference Series, 812, 012050.
- Kathotia, V. (2017). What is the Roman numeral for ten? Cambridge Mathematics.
- Furinghetti, F., & Radford, L. (2008). Contrasts and oblique connections between historical conceptual developments and classroom learning in mathematics. In L. English (Ed.) Handbook of international research in mathematics education (2nd ed., pp. 626–655). Routledge, Taylor and Francis.
- Poincaré, H. (1899). La logique et l’intuition dans la science mathématique et dans l’enseignement.
- Rasmussen, N. (1991). The decline of recapitulationism in early twentieth-century Biology: Disciplinary conflict and consensus on the battleground of theory. Journal of the History of Biology, 24(1), 51–89
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