There are no laboratories where formulae of maths are realized or derived. The laboratory of maths is your pen and paper. Everything you need can be proved in black and white. The best part of mathematics is that nothing has been invented in laboratories; each theory has been driven by logic and argument. And most certainly mathematics is the language of Science (Physics, Chemistry etc.)
In primary school the teachers taught us as if we knew stuff already. As a AIEEE / IIT JEE coach interacting with class 11 and class 12 students even I used to believe that the stuff might have been taught earlier, but infact it was not. I remembered that once I asked my class teacher why a0 = 1, he was not able to explain it or he just said the explanation is not in the scope. Later on I realized that
a0 = ab – b = ab / ab = 1
“Why is a0 = 1?”
Many students think that mathematics problems can be solved by killer concept shortcuts. But actually shortcuts are developed only by practice. Different students have different shortcut for the same type of problem depending upon the level of their practice.
“shortcuts are developed only by practice”
Many students and parents have one common question, How is 11th class and 12th class mathematics different from earlier maths? Although there is a big gap in the syllabus, but more than that there is an underlying difference in the way one should approach / study maths in the higher classes. Students have to learn to change their techniques. Till 10th our technique is that of memorizing algorithms and solving problems with manipulative techniques that work in certain well-defined situations. In class 11 and class 12 mathematics much amount of emphasis I put is on what the student would call Theory — the precise statement of definitions and theorems and the logical processes by which those theorems are established.
This type of learning techniques I believe should have been developed in lower classes. But unfortunately in those classes students don’t have the aptitude to raise such queries in class and apply the logic.
Even in class 11, and as an example – many think that trigonometry is boring, is full of formulas. Contrary to that trigonometry is much interesting if one knows how each formula has been derived. Derivation of the formula helps to understand when to apply it. If you try to understand how trigonometry shifts from acute angled triangle to obtuse angled triangle, what the range-domain of trigonometric functions are, and by plotting their graphs that gives a solid footing in the subject.
“Derivation of the formula helps to understand when to apply it”
I have noticed students usually do not pay attention to the fundamentals, like how a function varies, its graphs etc. For example in class 11, there are graphs of many functions like f(x) = x2, f(x) = 1/x, but how many of us even take notice of that. I tell students how to read the graphs. Graphs tells us many things about the function. Infact it is the horoscope of the function. I remember when I usually start my 11th class session, I ask students some elementary inequalities, like if – 2 < x < 3 then what are the values of x2 ? The replies are 4 < x2 < 9. Students would square it directly! But if we take help of real number line and understand the graph of f(x) = x2, we find that 0 ≤ x2 < 9. Also when I ask if –2 < x < 3, then what are the values of 1/x ? Students generally reverse the sign of inequality and get – 1/2 > 1/x > 1/3, then they realize that something has gone wrong as –1/2 cannot be more than 1/3. Then they correct it wrongly i.e. –1/2 < 1/x < 1/3. If we understand the graph of reciprocal function y = 1/x, say from the NCERT book, we find that for –2 < x < 3, 1/x ∈ (– ∞, –1/2) ∪ (1/3, ∞). These simple concepts help a lot in finding domain, range of functions in calculus and solving many inequalities.
“Graphs tells us many things about the function. Infact it is the horoscope of the function.”
The point is – more we apply logic in mathematics easier it becomes. You must have noticed that whenever mathematics starts becoming easier, one or two brain hacker questions just blow away your interest. Now if you work harder, and develop that creativity (comes with practice), the brain hackers turn to normal questions automatically.
“The point is more we apply logic in mathematics more it becomes easier.”
I always advice students whatever you hear or learn, try to feel it (isn’t it funny). You must have your own opinion/logic for a particular theory or problem which will help you feel confident in your approach.
Albert Einstein had said correctly – “ Pure mathematics is, in its way, the poetry of logical ideas”