Mathematics

Dear Student, 

As you are about to discover, the transition from school mathematics to senior school-level mathematics is a difficult one. Not because the mathematics gets harder. Those students who have successfully made the transition will likely agree that senior school math feels in many ways easier. What causes the problem for many—as I mentioned above—is only the change in emphasis. In school, mathematics is understood as a collection of techniques for measuring, counting, and accounting. The focus is primarily on mastering procedures to solve various kinds of problems. At senior school and beyond, the focus is on learning to think in a different, specific and powerful way—to think like a mathematician!

Many think mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than performing calculations, manipulating symbols, and putting numbers into formulas. Mathematics is the science of patterns . Mathematician identifies and analyzes abstract patterns—numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. Mathematics is about reasoning, finding counterexamples, and arguing logically. Mathematics is a way of thinking. It is an activity that is both extremely creative and challenging.

Mathematical thinking includes logical and analytic thinking as well as quantitative reasoning, all crucial abilities.  For life in the twenty-first century, everyone benefits from being able to think mathematically to some extent.  Today, good analytic thinking skills are essential for anyone who wants to take full advantage of the opportunities for self-growth and advancement that contemporary democratic societies offer. After senior school you may or may not pursue  mathematics or some math-dependent subject but you will certainly benefit from learning to think in this very powerful way. 

An explosion in the way mathematics is used in society mirrors the explosion in mathematics itself. Today we encounter uses of mathematics in every corner of our lives. Graphs, charts, and statistical data appear on television and in newspapers. The results of opinion polls are reported along with their margins of error. Lending institutions advertise variously computed interest rates for loans. We listen to music composed and performed with the aid of computers, and we watch the fantastically detailed pictures of imaginary worlds that computers draw. Computers also do a host of ordinary tasks. They scan bar codes on purchases, keep track of inventories, make travel reservations, and fill out income tax forms. The citizen's need to perform simple calculations may have decreased, but there has been a dramatic increase in the need to interpret, evaluate, and understand quantitative information presented in a variety of contexts.

Although some people do not need or use highly technical mathematics in their daily jobs, many others do. The complexity of daily life requires that we all be able to reason with numbers. Any car or home buyer ought to understand how interest rates work even though a computer may be doing the calculation. Anyone building a house or redecorating a room should be able to make and read a scale drawing. Newspaper readers and television viewers should be able to draw correct inferences from data on social problems such as pollution, crime, drugs, and disease. Buyers of insurance and purchasers of stock need to know something about the calculation of risk. All adults should be both able and disposed to use mathematics to make practical decisions, to understand public policy issues, to do their job better, to enhance their leisure time, and to understand their culture.6 Mathematics and the ways it is used are changing

CBSE Curriculum

NCERT Textbook   https://ncert.nic.in/textbook.php?kemh1=0-14

The History

Mathematics can be said to have begun with the invention of numbers and arithmetic, which is believed to have occurred around ten thousand years ago. Over the ensuing centuries, the ancient Egyptians and Babylonians expanded the subject to include geometry and trigonometry. In those civilizations, mathematics was largely utilitarian, and very much of a “cookbook” variety - "Do such and such to a number or a geometric figure, and you will get the answer". The period from around 500 BCE to 300 CE was the era of Greek mathematics. They treated numbers geometrically, as measurements of length, and they discovered that there were lengths to which their numbers did not correspond (essentially the discovery of irrational numbers). They introduced the idea that the precisely stated assertions of mathematics could be logically proved by formal arguments. This innovation marked the birth of the theorem, now the bedrock of mathematics. This formal approach by the Greeks culminated in the publication of Euclid’s Elements, reputedly the most widely circulated book of all time after the Bible.

School mathematics is based on all these developments listed above, together with just two further advances, both from the seventeenth century: calculus and probability theory. Virtually nothing from the last three hundred years has found its way into the classroom. 

The Birth of New Math

Up until 150 years ago, although mathematicians had long ago expanded the realm of objects they studied beyond numbers, they still regarded mathematics as primarily about calculation. But during the nineteenth century, as mathematicians tackled problems of ever greater complexity, they began to discover that these earlier intuitions about mathematics were sometimes inadequate to guide their work. Counter-intuitive (and occasionally paradoxical) results made them realize that some of the methods they had developed to solve important, real-world problems had consequences they could not explain. The increasing complexity in mathematics led mathematicians in the nineteenth century to shift (broaden, if you prefer) the focus from computational skills to the underlying, foundational, conceptual thinking ability.  Mathematicians turned the methods of mathematics inwards, and used them to examine the subject itself.

This revolution—for that is what it amounted to—completely changed the way mathematicians thought of their subject. Yet, for the rest of the world, the shift may as well have not occurred. Now, 150 years later, the changes in society that were facilitated in part by that more complex mathematics, have made that focal shift important not just for professional mathematicians but for everyone who learns math with a view to using it in the world. If you, find yourself reeling after your first encounter with this “new math, ” you can lay the blame at the feet of the mathematicians like Lejeune Dirichlet, Richard Dedekind, Bernhard Riemann, and all the others who ushered in the new approach 

Explosion of Modern Mathematics

The explosion of mathematical activity that has taken place over the past hundred years or so in particular has been dramatic. At the start of the twentieth century, mathematics could reasonably be regarded as consisting of about twelve distinct subjects: arithmetic, geometry, calculus, and several more. Today, the number of distinct categories is somewhere between sixty and seventy, depending how you count them. Some subjects, like algebra or topology, have split into various subfields; others, such as complexity theory or dynamical systems theory, are completely new areas of study. 

Mathematics Education in 21st century

 There are so many different mathematical techniques, with new ones being developed all the time, that it is impossible to cover them all in school and college education. By the time the student enters the workforce, many of the specific techniques learned in those college-years are likely to be no longer as important, while new ones are all the rage. 

"Give a man a fish, he’ll eat for a day. Teach a man to fish, he’ll eat for a lifetime.” The educational focus has to be on learning how to learn.