How to study and understand Mathematics?

Everyone has some notion of what mathematics is. A layperson who is not much exposed to math would say math is just numbers, shapes, and some simple arithmetic. To a serious student of math, math encompasses much more than just numbers and geometrical shapes, of course.

To answer the question of how to study and understand math, one must first examine the preliminary issue of whether math is to be studied as a school, academic subject, or that it is to be studied as a recreational, non-academic undertaking. For the former, the student has to be systematic in approach, committed and disciplined in pursuit of studying math. For the recreational, non-academician math can still be studied and understood albeit without taking too much pains since the purpose and objectives are different.

Still, whether one is studying math at the High School and college level, or whether one is interested in math purely because of its intrinsic attractions, one must first realize that math is a 'pure science' in the sense that it is a system of coherent constructs in which mathematical statements of various degrees of complexity interplay to form a logical and consistent structure. So the various branches of math - algebra, geometry, trigonometry, etc - all contain fundamental statements ( definitions, lemmas, postulates, theorems and laws etc ) which are inter-related and inter-connected forming an independent, internally consistent system.

I would like to suggest the following approaches and strategies for our readers who are embarking on the study of math:-

1. Do some background reading on the history of mathematics.

It is instructive to learn a little about the beginnings of the various branches of mathematics; you would learn how Descartes constructed his system of co-ordinates, thus inventing Cartesian or Co-ordinate Geometry; how and why the system of numerical symbols came into being, from ancient Greco-Roman to Indian and Arabic symbolization; system of simultaneous equations in two or more unknowns, as enunciated by ancient Chinese mathematicians; how the fundamental concepts of Calculus were laid down independently by Newton and Leibniz.

From a glean of the above, you would learn to appreciate how some mathematical ideas were born and how they led to a more elaborate and coherent system, constituting a particular corpus of math knowledge. In this way, you learn to better understand and appreciate the workings of mathematics.

2. When studying a serious math topic, make sure to grasp the basic postulates and fundamental theorems, laws etc

Just as one can't really learn to run before one learns to walk, it is absolutely essential that upon embarking the serious study of a math topic, say trigonometry, one must spend some time and effort mastering the fundamental definitions and related concepts. So, in trigonometry, one needs to understand the notions of tangent, sine, and cosine of an angle before one can venture into the more complex trigonometric identities and equations. Similarly, in the study of differential calculus, one must understand thoroughly the idea of the differential coefficient or derivative ( or dy/dx symbolically ) and what it signifies, before one can hope to go into the various rules of finding the derivatives of other algebraic and trigonometric functions.

3. Work out as many of the math problems as possible in the course of your math study

Math is one academic subject for which you need to solve and work out as many problems as possible: this is the only way to consolidate your understanding of the math topic under study. When studying algebra, after grasping the fundamental rules and perusing the many examples (from textbooks, or from regular classes in schools, colleges ) one must make a concerted effort to try out the problems set in homework or assignments. This point cannot be over-emphasized: you do not know whether you have grasped or understood topic being studied if you do not know how to solve the associated problems. There is no other way, I would like to re-iterate.

4. Read books not just on the academic aspects of math; read books on recreational math as well

This is an advice not just meant for the person who is interested in math as a recreational subject, but also for the serious student of math. There are many books, written by well-known mathematicians and non-mathematicians ( e.g. Martin Gardner ) on how math can be used to create aesthetically appealing geometrical patterns and also how certain math knowledge can be utilized to solve real-life problems in an interesting manner. Mathematical games, puzzles and quizzes are to be found in many of such books. Even if you are a non-mathematician, you would find yourself entertained by the novel use of mathematical knowledge.

I hope in putting down the above suggestions for the study and understanding of math, I have at the same time succeeded in encouraging more people to look at mathematics from a new perspective, not just regarding math as a dull and difficult academic subject that is devoid of practical significance and utility.