Learning from the Calculus of Education

The Calculus of education that we have tried to evolve in our study has several lessons for us. Some of the lessons are known to us. But there is no harm in underlining them with new emphasis. One of the important lessons, is that we can draw is that education is a journey. It takes student, teacher and the system of education on a journey. ‘The journey’ metaphor, is fascinating as well as dynamic and process oriented. It is, therefore, marked by a continuously changing ecosystem of education where everything and everyone is changing. All the three primary coordinates (student, teacher and system of education) which undergo change and dynamically influence each other. But above all, the learning student is the primary agent of learning. We put the onus of responsibility mainly on the student because there is difference between teaching and learning. Teaching may not always produce learning. Today we seem to be plagued by this issue. We seem to have more of teaching and less of learning. Hence, the challenge is to convert our classrooms into learning spaces rather than domesticating or silencing spaces of the taught. When we have student-centric education, we will produce a learning community rather than a ‘silently’ taught crowd.

The mathematically modeling of the process of education in calculus invites us to simplify the teaching-learning dynamics of the classroom. What we have arrived here close to what Picasso ironically said when he stated that ‘art is a lie that make us realize the truth’. Our stupid indulgence into the calculus of education takes us not just to the simplification of the entire process of education by converting into derivatives approaching infinity, but here as we are collecting our learnings from it, we have the challenge of summation or integration of the derivatives. It would require us to square (area) the curve of the education journey of an individual student. It means it make us face the truth of our education system. Newton and Leibniz had already linked derivatives to integrals and hence, integration not just tells us about the state of education at any particular interval of time but it also can help us predict how the curve might unfold in the future. This is what we are trying to do when we are trying to milk the learnings from the calculus of education. Calculus has provided us an algorithm to analyze as well as synthesize the curves and, therefore, by implication the curve of education.

The learning function is related to the teaching and learning dynamics as well as operational education system. To improve the learning, the dedication and dexterity of the students as well as the dedication and dexterity of the teachers and the manner in which both the students and the teachers draw from the operation the education system to create a learning environment. Student-centric learning needs fluidity in the relations of the three coordinates of education system. It has to treat the process of education dynamically as Newton thought of area under the curve as a liquid while he was doing his calculus. Newton dynamized the xy plane of Descartes by inserting time and motion into it. Thus, drawing our inspiration from Newton, we have to treat the ecosystem of education in a fluid manner and not in a rigid manner. This does not mean everything is treated in a fluid manner. There are things that remain constant while others keep changing or become variable. This environment of fluidity or elasticity with a pinch of constancy opens the education system to creativity and inventive thrust. Students are encouraged to think freely and become pioneers, young thinkers and scientists. This means the education has to have sufficient freedom for the students to choose what they like to pursue and should have teachers who have sufficient expertise and interest in the subjects of their specialization as well as the processes and operational systems of the educational ecosystem have to be sufficiently efficient to bring about the desired effects in the students, teachers and in the teaching-learning dynamics.

Our study reveals that each student starts from a different starting point although the education system makes us think everyone starts at the same starting point. Hence, the teachers have the challenge to accompany the journey of each student. Each student is on the move. His/her graph of education may go up or done . It seldom remains steady. What is important is that the teachers have to continuously accompany this dynamic journey of an individual student. Thus, while the teacher deals with the class, he/she cannot forget the individual student. The journey of a student is like a continues changing areas under a curve on an xy plane. We can imagine that x is constantly sliding to the right on the x-axis. When the student is on a progressive march and his/her progressive curve where the length of the curve on the y-axis is y, we can say the student’s progresses manifested when x is moving on the right on the x-axis. This means the progress or regress of the student is a f (x). Here x is not a fixated number but is in a continues change depending on the progress of the student. Here we have to understand that f (x) depends on progress or regress that the student is making continuously which is given by y (x) {infinitesimal point on the curve} which is the curve of his/her progress which then accompanies it corresponding to the changing (x).

May be we can imagine it like a problem that challenges us to calculate the area of curve where x is constantly sliding. This means the area is continuously growing. Perhaps, we can imagine a magical paint roller moving sideways. With the sliding of x , the area under the curve is changing. As the paint roller moves on the right, it paints the region under the curve. In order to paint the curve neatly, the paint roller magically instantly shrinks and stretches in the vertical direction exactly as needed to reach the curve on the top and the x-axis on the bottom without ever needing to cross the boundaries. We then, can ask at what rate does the painted area expand as x moves to the right? If we answer this question, we may get an insight in the progressive journey of the student. The answer takes us to what happens in the next infinitesimal interval of time. The roller rolls a infinitesimal distance dx, keeping the length at the curve y in the vertical direction almost constant as there is no time ( infinitesimal) for it to change its length during the infinitesimally brief roll of the paint roller. What it really does is that it paints an infinitesimally small rectangle of a height y and width dx. The area of the infinitesimally small rectangle is dA = ydx . Taking dx on the left side of the equation by dx, then we get the rate at which the area accumulates or grows. Therefore, dA/dx = y. Thus, we are enabled to quantify the growth of an individual student given his/her curve of journey of education and respond to him and her accordingly.

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Hypocrisy is the tribute that vice pays to virtue.

- Fr Victor Ferrao