A new way to teach math
Mathematics: science of learning
The word "mathematics" derives from the Greek word mathesiz, meaning learning. Hence, mathematics means the "science of learning". According to Plato, "all learning is recollection", a process by which the soul remembers its eternal ideas. Mathematics was regarded as best suited to learning since it was believed that mathematics contained eternal truths which sympathetically moved the eternal soul.
Mathematics: why is it difficult to learn?
How did it happen that mathematics---once regarded as the science of learning---is today regarded as the most difficult subject to learn?
Mathematics and theology
Why is math difficult today? The answer is simple: because religious politics got mixed into mathematics. How? That is a long story.For example, in the 5th c., the philosopher Proclus argued that since the truths of mathematics were eternal, the cosmos itself was eternal. This angered the Christian priests of the time, who had invented the doctrine of apocalypse: that doomsday is round the corner (a ploy to frighten people and make them submit to the authority of the priest). The priests showed their anger by declaring Proclus a heretic, and in 529, Justinian, the Roman Christian king, banned all schools of philosophy in the Roman empire.
The full story is interesting (look out for my forthcoming book), but we will not go deeper into it, since our immediate concern is how to make mathematics easy to learn.
The point is that math was mixed with religion in the earliest Western traditions. It was for religious reasons that European priests first learnt math in the 12th c. from Arabic books on math, captured during the Crusades, and translated into Latin, starting 1125. After the failure of the Crusades, these math books were used to teach Arabic methods of argument to priests, since the priests (Aquinas and the schoolmen) now aimed to convert Arabs using reason rather than force. (Of course, Arabic math books were reinterpreted to make math consistent with Christian theology.) It was at this time that Greek authors like "Euclid" were concocted for many of these books. Ever since then, Western math has suffered from a religious hangover.
Indian math
In India, however, this religious connection was entirely absent. Mathematics developed as ganita: a means of practical calculation. There certainly was a notion of proof, but this was not the same notion of mathematical proof as developed in the West. For example, empirical techniques of mathematical proof were used in India, although this is disallowed in present-day Western math..
Because of this difference in the understanding of math, Europeans found it hard to understand Indian math when they first encountered it. Even a simple thing like the place value for numbers took some five hundred years to be accepted in Europe: starting from the recorded blunders made by Gerbert, Pope Sylvester 2 in the 10th c.
Similar difficulties attended the transmission of the calculus from India to Europe. It took over three centuries from the time of Kepler and Cavalieri, Fermat and Pascal until the formal real numbers of Dedekind and the eventual formalisation of set theory in the 20th c.
The new way
Education today is unfortunately seen as a process of learning to ape the West, a la Macaulay, so math is taught from a purely Western perspective. This way of teaching math retraces the European experience of learning math: so the historical difficulties experienced by Europeans are replayed in the math classroom.
The solution to this difficulty is to throw out the religious beliefs that have crept into mathematics, and to go back to something very close to the original understanding in which things like the calculus developed as primarily a tool for calculation. This makes math very simple, and also makes it ideally suited to the present-day technology of computation which has greatly enhanced the ability to calculate.
Some philosophical adjustments are required in the understanding of math; but this should not be a serious problem for anyone except those who are currently regarded as math experts!