The Encuvati was the quintessential Tamil multiplication table book, used in precolonial South Indian schools. Nowadays they are available as palm leaf manuscripts, collected from different geographical locations of the Tamil-speaking region of South India and stored in various manuscript libraries there as well as in other collections inside and outside the country. But why is something as innocuous as a multiplication table book important for us? Their contents look nothing like modern tables, with lines and columns, but the numbers are arranged the same way. Their importance emerges from the simple fact that countless children participated in their making from about the seventeenth century, if not earlier.
These children, usually male, went to the local village schools run by a single, usually male teacher, who taught the elementary arts of reading, writing, and counting on the veranda of someone’s house. The Encuvatis were products of these schools. Along with other primers such as the Ponnilakkam and the Nellilakkam, they are available to us as records of pedagogic practices in the learning of arithmetic to achieve proficiency in the computation of numbers, weights, and measures. Learning in these schools was grounded in the cultivation of the arts of memory, like in many cultures of the world, where, to recollect with prudence and efficiency constituted education, not just learning to read and write. Not that reading and writing were not accomplished, but they were merely aids to train recollective memory; not ends in themselves.[1]
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Why were the numbers in the Encuvati arranged as if in rows and columns?[2] In these primers, numbers came together, always representing themselves in relationship to each other, as series resulting out of arithmetic operations like addition and multiplication. It is as if their use and place was anticipated in advance, in another context of computation, like say a simple arithmetic problem when you need to know the product of two numbers, or of two units of measures of volume or weight. These tables anticipated such situations in all their possibilities, making them an assemblage in anticipation. They created an arrangement of recall, an apparatus which inscribed numbers in set orders to facilitate computational practice. At the same time, they provided a rationale for numbers as entities that could constantly transform.
Such exercises in recollective memory building did not begin with a book, not with a manual or handbook that you could refer back to, turning its pages, knowing where to look for that particular instance when two numbers or quantities or units could come together to transform themselves into another number. Instead, exercising of the mind proceeded in a direction to organize, assemble, arrange numbers in a manner that the logic of their transformation remained like a shadow (if 2 is added to 4, then 6 or such like; but if x, then y), but its primary function was to accumulate possibilities of transformations. We could read these practices as assemblages in anticipation of the potential transformation of numbers and units, not as mechanical tools to ease the act of computation. In early modern Tamil country, the act of manipulating quantities was trained in a deliberate manner to equip the mind to solve not problems in which numbers stood alone but instead real-world problems such as counting objects, measuring land, and weighing its produce, that is, calculating productive resources and commodities.
Such anticipation of the use and function of numbers as both material and cognitive entities required practice. The elementary veranda schools had to deliver on a social necessity and ensure that such practices succeeded when it mattered in the real world of transactions, in the daily lives of the people. When such institutional modes became the medium, then orders for numbers were imagined, dexterous enough such that the acts of recall assigned them a concrete function. At this instance, these institutions brought together the abstract and the concrete.
This was ensured by the child rehearsing the mind to recall through the bodily acts of loud recital and writing. The Tamil people nourished these acts of practicing the tongue and the hand together. The children pronounced the name of the number as they wrote it down, as everyone in the class repeatedly heard the sound of the number’s name and became familiar with its written shape. Together, these acts constituted the mental act of inscribing possibilities and ensuring recall when numbers had to operate as quantities of grains, the dimensions of some land, or even as the curves of a beautiful sculpture. Such moments of recall and ordering of numbers through the simultaneous acts of reciting and writing made manuals in the mind. The physical artifact that you see in the image in the form of a palm leaf manuscript is a product of that practice.
But with the onset of the British colonial rule, this was not accepted as learning at all. The logic of the relationship between numbers that had remained like a shadow, only to become apparent at the instance of transformation in a context of real world computation, had to come to the forefront. Cognition and the material practice of counting now had to prioritize numbers in their transformative relationship as operations that stood independently of any immediate function. Numbers had to make sense by themselves. Children needed to learn addition, subtraction, multiplication, and division and then apply these operations as rules, as nonnegotiable truths. Of course, logics such as “if x, then y” remained, but their purpose changed. If the children could add, that is, compute in the abstract, then they could calculate. The mental manual of earlier generations now had to be printed and followed or obeyed in its particular order. The objective was first to cultivate a skill through the book and then to use the book when the situation demanded. The Encuvatis were cut up, altered, and printed as textbooks in schools during the nineteenth century in the Tamil-speaking regions of South India.[3]
In this situation, two historically significant changes occurred.[4] The purpose of learning was not to be necessarily related to functionality. In other words, it was sealed from within. The acts of memorization of number facts had to be generative, not in anticipation of real world transactions, but also as entities coming together to serve a function for themselves. Exercising of the mind had to be reoriented to depend on the new physical device of a printed book of tables with neatly drawn lines. The printed books of tables became necessary not only as a device to verify computational correctness but to begin the process of computation in the first place, as a procedure- and rule-driven world of mathematical pedagogy was ushered in.
Could we then ask if procedure- and rule-driven regimes in history required manuals?
Such a question could be asked not merely as a comparative aid but in order to understand which regimes of skill making required what kinds of oriented cognition and knowledge organization?[5] The act and necessity of swift referencing, as a guide to knowing, acquiring, and accumulating informative facts, could then be placed alongside Encuvati like registers, products of skills in the making. Does that mean that the the transition of the Encuvati from palm leaves to the printed book was embedded in a history of shifting regimes of skill, cognition, and material practices? Such a realignment was also steeped in the logic of establishing private property in land which began in the early nineteenth-century South Indian society under the rule of the English East India company, which was no longer a mere trading firm. But that is another story!
D. Senthil Babu is an associated researcher at the French Institute of Pondicherry.
- For a detailed discussion of these schools and arithmetic teaching in them, see D. Senthil Babu, “Memory and Mathematics in the Tamil Tiṇṇai Schools of South India in the Eighteenth and Nineteenth Centuries,” International Journal for the History of Mathematics Education 2, no. 1 (2007): 15–37. ↩
- For the significance of tables in the history of mathematics, see Martin Campbell Kelly et.al., eds., The History of Mathematical Tables: From Sumer to Spreadsheets, (New York: Oxford University Press, 2003). ↩
- For a brief overview of these changes, see D. Senthil Babu, D., “Indigenous Traditions and the Colonial Encounter: A Historical Perspective on Mathematics Education in India,” in Mathematics Education in India: Status and Outlook, ed. R. Ramanujam, and K. Subramaniam (Mumbai: Homi Bhaba Centre for Science Education and Tata Institute of Fundamental Research, 2012), 37–62. ↩
- For a discussion on changes to these school and their pedagogies, see Jana Tschurenev, “Diffusing Useful Knowledge: The Monitorial System of Education in Madras, London and Bengal, 1789–1840,” Pedagogica Historica 44, no. 3 (2008): 245–64. For the changing historical context during this period, see Bhavani Raman, Document Raj: Writing and Scribes in Early Colonial South India (Chicago, IL: University Press of Chicago, 2012). ↩
- For an important discussion on material practices and the cognition of numbers in the history of mathematics, see Peter Damerow, “The Material Culture of Calculation: A Theoretical Framework for a Historical Epistemology of the Concept of Number,” in Mathematisation and Demathematisation: Social, Philosophical and Educational Ramifications, ed. Eva Jablonka and Uwe Gellert (Rotterdam: Sense Publishers, 2007), 19–56. ↩