I happen to hail from a country India. Here the school education system in recent years is getting totally ruined due to an examination called IIT-JEE, most of the Indians know this fact. This kind of situation is particularly prevalent in towns and cities. The thing which disturbs me sometimes is the title I have put here. I have also studied in an IIT coaching school so I know what it is. But when I was a student there, the practices were different. My friends and I used to go library read books because we used to have the freedom and time. But now a day’s the situation is entirely different, the students are experiencing a lot of unnecessary burden due to tests, assignments, exercises, home-works so that no student has time to go to the library and read books even. Now coming to Mathematics (my favorite subject due to its elegance) I find myself very unhappy at its situation now a days due to these practices. I’ll explain how. Mathematics in schools is just taught as something as in which one has to prove something or solve a problem to get a solution (either in terms of numerals or variables). I don’t think that’s what mathematics is. Problem solving is only a part of mathematics. A student has to be taught logic which is the basic essence of mathematics because mathematical theorems are derived only by the application of rigorous logic on a set of axioms. It is sad that high school math textbooks give less or no emphasis on mathematical logic. Math teachers also play a major role in learning math in the sense that the kind of training they received to teach math, the teaching methods they follow in the classroom, their interest attitude towards math etc. Mathematics is always incomplete without that rigorous logic so the teachers should inculcate the habit of looking things in a rigorous way to the students in math.
Now-a-days teaching calculus to high school students has become a fashion (fancy thing). I don’t think that high school students have that maturity to understand the elegance and importance of the theorems in the calculus. I don’t think they have the necessity to find the limit of a function, to solve differential equations, either integrate or differentiate a function etc., At the most, they are asked to solve a quadratic equation and for analyzing the properties of a quadratic equation. I don’t think one has to learn integration or differentiation to do this job. This doesn’t stop at this point (due to stiff competitions among schools for students) even vector algebra has been introduced. I think these ridiculous things have to stop at one point or the other. According to me, a high school student must solve good geometry and algebra problems. After solving them one should develop the attitude to look into the solution for improving the present solution to a more elegant solution, try to connect the problems by observing common things between the present solved and unsolved to the past problems draw some parallel lines etc., this will improve the observation skills and intuition. If this is just a brief description of the situation of math in the schools, just think about the other science subjects like physics and chemistry they are in much worse situation (may be). Only the degree of worseness changes from school to school but that worseness is present everywhere. More over what’s condition of languages god only should know.
One of the reason which caused these ridiculous activities is that physics people also started teaching some Senior High school concepts for which they needed the mathematical concepts like Vector Algebra and Calculus etc., to high school students. Here I am writing a few examples to show those physics people, that you can actually avoid calculus and produce more elegant solutions using the basic math theorems.
Ex.1 We will consider a simple kinematics problem. If the position of a particle (x) varies with time (t) as . Find the acceleration.
Sol: This a high school problem which just requires a simple observation. For this problem if a high school student starts applying calculus on this, then one will end up with a solution like this. Here’s the solution.
Differentiating on both sides and rearranging the terms,
Squaring on both sides,
Again differentiating on both sides,
One can take pains like that or can simply get the solution like this.
Squaring and rearranging terms on both sides,
Compare the coefficients of equation with ,
We can easily see that
Now just think which method is better.Okay If you are not satisfied with this example lets take another example.
Ex.2 Now I will take a classic example I am sure most of you have seen this. In a given circuit with resistor R and non ideal battery of emf E with internal resistance r, then prove that if that if the power dissipation through R is maximum when R=r.
Sol: I am not writing the solution using calculus cause it looks tedious to me. I’ll try to write some what simple solution here it goes.
The Power dissipated through the resistor R,
This expression can be rewritten as follows ,
Now carefully see the part containing R and r, for the power P to be maximum this part should be minimum. We know that if the product of two variables is constant then their sum would be minimum if the variables are equal (another form of saying AM-GM inequality). Here the product of and is a constant , so the sum would be minimum if
Isn’t this elegant? Its almost like a three line solution. I don’t understand why people want to use calculus for this problem. One can also use their knowledge of quadratic equations to solve this question. It doesn’t matter how much subject you learned but how perfectly you learnt it.
What I want to tell finally is that Calculus is intended for the analysis of complex functions in some particular situations where the you can’t use basic theorems anymore to analyze its behavior. And no high school students are required to deal with such functions. So its better not to use it unnecessarily without any purpose, otherwise if you use it will just make things tedious and complex.
P.S. Thanks to my friend for her valuable suggestions and corrections in the draft I have written.