Is Math History Important? -- Talk at NIAS 27 Jul 2016

Pre-University Mathematics: Looking Back to Look Forward

India has a long mathematical tradition spanning over two millennia. A substantial portion of what is regarded as standard school level and part of undergraduate level mathematics has a history and tradition in India. The characteristics of a systematically developed body of knowledge – namely the existence of specific terminology, temporal continuity, geographic span, cross-referenced and consolidated literature, and a body of solved real-life and book problems, examples, formulas, algorithms, illustrations and so on – are all available in this tradition. Yet, this tradition seems to have had no impact upon mathematics education in India, which is a matter of both curiosity and concern.

Epistemological issues have been cited as the reason for the marginalization of Ganita from the taught material. Yet, a close look at the curricular content of modern mathematics textbooks suggests, first, that this objection is inconsequential– i.e. that high-school level mathematics, in the way it is increasingly taught, makes few if any appeals to the requirements of formalism, and therefore, can be supplanted or supplemented with Ganita mathematics without much philosophical consequence; and second, that, as machines increasingly take over repetitive tasks in problem-solving, the focus of math pedagogy worldwide is on problem-framing or modeling abilities, where it is advantageous for Indian students, to dip generously into their own authentic traditions of enquiry.

An examination of curricular objectives, textbooks and examination patterns of popular curricula indicates certain systemic trends in Pre-Univ Math, showing a preference among students and faculty for modularized content, more critical thinking components, less formal theory and more intuitive and practical problem-solving skills. Theorem-proof is far less emphasized in mainstream curricula at the high-school level, than it used to be.

We examine below some examples from Ganita, which map onto current high-school syllabi. It is postulated that awareness of and facility with an indigenous tradition will enhance confidence among math students at the high-school level and provoke independent thought, questioning and innovation.

The scope of Ganita has been discussed in ancient and modern literature. It includes both the act of computation and the science underlying computation. The scope has significant overlaps with modern high-school math, for example in areas such as: Quadratic Equations, Sequences and Series, Limits, Calculus, Trigonometry, Combinatorics, etc.

Quadratic equations find repeated mention in Ganita literature. The technical term used is: Madhyamāharana. The earliest recorded quadratic equation problem is attributed to Aryabhata (c.500 CE) and is in the nature of a moneylending problem.

A Principal P is lent out for a month and a return of x unit is obtained on it. The earned income of x units is further lent out for a period of t months at the same rate. The total earnings on the Principal is desired to be A units. What is the rate of interest x to be charged?

The problem is modelled as the following quadratic equation:

The solution is:

This can be described as a problem of . Brahmagupta (Brahmasphutasiddhanta, c. 658 CE) gives a more generalized version of the above problem, in which P is lent out for  months, yielding a phala of x, which is then lent out for a further period of  months, gaining a total yield of A. The equation for interest is

which can be solved in the same manner, leading to an even more successful moneylending operation! Students can create look-up tables of interest rates, similar to those used by insurance agents, using these formulae. This unique problem is not seen in math textbooks.

The Ganitashastra Sangraha of Mahavira (c.850 CE), poses problems in Arithmetic Progressions involving multiple incremental series, and their sums and characteristics. The use of quadratic equations to solve for ‘n’ is also demonstrated.

Mahavira also gives a recursive search procedure for finding the common ratio of a Geometric Progression, given the first term and sum to n terms, entailing the scaling down and stepping down of the series. The procedure will draw the student to think about an efficient algorithm for the search. For example the student may observe how a ‘gradient descent’ method can be used in conjunction with Mahavira’s algorithm for the common ratio.

Limits are the precursor to the discussion of calculus and a vital step in higher mathematics. Yet it is one of the most challenging topics for teachers and students alike. We examine how limits are evaluated or understood in Ganita. Making use of the example

to show that as we sum more terms on the LHS, the difference on the RHS becomes smaller and smaller but not zero, Nilakantha Somayaji (Aryabhatiya-Bhashyam c.1500 CE) demonstrates the formula for the sum of an infinite geometric series. In this approach, we discard very small quantities, depending upon the practical context. The terms of the infinite series give the instantaneous position at any chosen instant. This is the foundation of differential calculus.

Note that the formal way of summing geometric series, involves subtracting infinity from infinity, leading to difficult or unanswerable questions, for example in the second step of the following derivation:

The derivation and application of the Infinite series and Calculus is a fascinating story of Yukti, or clever, ingenious methods, using results from geometry and summations of power series. In his commentary Aryabhata’s observations on the ratio of circumference to diameter, Nilakantha says:

“The unit which measures the circumference without a remainder, will leave a remainder when used for measuring the diameter [and vice versa].  … We can achieve smallness of the remainder, but never remainderlessness.”

Students, to their great loss, are never introduced to this idea in their curriculum. Thus, their understanding of  is ambiguous. And this ambiguity carries on to infinite series and calculus.

Scholars have answered the question, “What makes Math hard to learn?” by pointing not to the content of math, as much as to its history and philosophy. Fields’ Medallist Manjul Bhargava has urged students of math to “go to the original sources” and has cited his own experience of reading “Gauss and Dirichlet, and Pingala and Hemachandra”.  It is very useful to understand how various concepts arose and in what context. Insights in those writings, if recovered, could stimulate the mind and engender creativity.

Thus, the study of Ganita literature, in the context of modern-day high school math education, and in conjunction with it, could, if properly undertaken, hold the key to improving the quality of math education and outcomes in line with modern-day requirements, in India.