Algebra vs. Calculus

While interacting with some students of 12th grade (possibly some undergraduates also) in Orkut Math Community, I found something about which I feel very sad. This has something to do with their approach of problem solving. What I found out was that their brains have been so trained to a formula based or rule based approach. They don't understand the concepts and their meaning. The whole procedure of solving problems has been mechanized. After a certain amount of thinking I found the root cause of this behavior.

Well, it has something to do with the difference between Algebra and Calculus. A more respectable name for Calculus is Analysis but this term is normally not known to students in 12th grade, so I will stick to Calculus only. Right from the school level most of the mathematics is concerned with algebra. And it involves manipulating expressions consisting of some variables and numbers with operations like +, -, *, / and =. Using these manipulations a lot of standard formulae like (a + b)^2 = a^2 + 2ab + b^2 are derived and then these are memorized.

What the student understands is that for a given problem he has to use some such formula and the answer will be arrived at easily. So the learning acquired is mostly mechanical. You can program a computer to do such stuff by giving all standard rules and formulae to it. Some bright students do understand the meaning of these algebraic operations also. But somehow they are kept away from the notions of inequalities. Thus we don't have much of symbols like <> in algebra. The order relations are simply ignored. Whatever inequalities the student learns (like AM > GM) is more like an exercise in algebraical manipulations, rather than understanding what happens in those order relations.

Another thing entirely missing out entirely from the school curriculum is the concept of an irrational number. Now this is somewhat difficult to grasp at high school level, but definitely possible in 12th grade. This also involves an appreciation of the order relations <, >. The student having trained algebraically has hell lot of problem understanding the concept of irrationals. This is primarily because irrationals cannot be derived out of rationals by means of algebraical operations of +, -, * and /. So at the end of 10th grade the students has the knowledge of few irrationals consisting of radicals like sqrt(2), 3 + sqrt(5) and a few isolated numbers like pi.

The concepts of irrationals numbers as a whole class is entirely alien to them. Some smart students do understand the class of irrationals as non terminating and non repeating decimals. But they have no idea about what such decimals mean. Again this is because of a lack of emphasis on order relations like <>.

And that brings us to calculus. Calculus primarily deals with the properties of real numbers (rationals and irrationals together) and most of these properties have to do with order relations. And the student is left wondering about these messy concepts of zero and infinity. Somehow he reaches upto derivatives and integrals without having any notions of limits and continuity and he becomes happy. There are some standard cookbook rules of differentiation and integration which resemble algebraical formulae and the students again gets into this mechanical learning mode. Ask him a problem related to mean value theorems and he is lost. Ask him the meaning of an integral he is lost.

The sole problem in calculus is the lack of understanding of concepts like real numbers, and order relations. Rather than teaching the students these concepts in 11th grade, they are taught infinite series like binomial, logarithmic and exponential series. These series and the functions involved in them cannot be appreciated without developing calculus first. Students are again kept wondering about these. Add to that the plethora of books by S L Loney, Hall & Knight who don't teach a grain of concepts and load the student with a pile of mechanical problems.

By the time he is ready to learn calculus at 12th grade, it is ensured that he will not understand a single line except those mechanical rules of differentiation and integration. The reason for not providing conceptual framework for calculus is that these concepts are difficult to grasp at that level. This is not the actual reason though, it is the reason provided by book authors and curriculum designers while the real reason is that they don't know how to present these concepts.

As an example to illustrate my point, I just need to tell that these concepts are not presented even at the undergraduate level or post graduate level in a standard textbook. Thus Walter Rudin, supposedly the God of Analysis Books, says in preface to Principles of Mathematical Analysis, "Experience has convinced me that it is pedagogically unsound to start off with the construction of real numbers from the rational ones". How the hell does Rudin think it is unsound? It is because of such a pedagogical attitude from school level that the student will never appreciate these concepts. On the contrary I think that any student, who can understand the tough concepts of metric spaces, point set topology and to top it all the Lebesgue measure, can definitely and much more easily understand the concept of a real number. Its a child's play for someone with the ability to grasp Lebesgue Measure. Rudin explains all these tough concepts with lot of proofs, examples and problems while he dedicates only 3-4 pages (that also in appendix) to the theory of reals with no proper examples.

It is teachers/authors like Rudin who have made that whole thing pedagogically unsound. It is not that the student cannot understand these concepts, but in reality it is Rudin who does not know how to teach these concepts to a student. Given proper training to students at 11th or 12th grade about the importance of order relation and introducing them to theory of real numbers will definitely get rid of this pedagogically unsound situation.

There are indeed very few books of Calculus which present the theory of reals and most of them present the theory in a set theoretic notation without any examples and problems. The way student is taught about rational numbers in class 6 or 7 with lot of examples of +, -, *, / of rationals, that way he is never taught the theory of reals. I believe that learning a theory of reals at 12th grade takes at least as much time and material as it takes to learn rationals in 6th or 7th grade. That means roughly 30 pages of textbook and 3-4 weeks of learning time for solved examples and problems.

The teachers/authors want to keep the algebraical attitude as long as possible and never let the student have an appreciation of the wonderful world of calculus with its deepest concepts. And when they do teach the stuff, it is made boring by a ton of set theoretic concepts like metric spaces, topological spaces, and measure theory. All that is presented in the fashion of abstract algebra. But there is a difference. In algebra the students know the basic algebra of integers, rationals and polynomials before learning the Abstract Algebra concepts of groups, rings and fields. But in calculus the students don't know the real number system and thus limit/continuity/derivative integral concepts in reals and are taught Abstract Analysis consisting of metric spaces, point set topology, topological spaces and measure theory. Looks like a step fatherly approach towards calculus.

It is because of these reasons that students never appreciate calculus and I sometime feel like laughing when I hear about these notions from a student of 12th grade. Most of them live in some dreamland alien from the real world of calculus concepts.

Those of you who had the patience to read this much of the post, for them I have a good news. Its never late in learning and there is a book (and it is the only one) which teaches calculus the right way beginning with a chapter on real numbers with almost 30 pages. That is the masterpiece created by G. H. Hardy and is titled "A Course of Pure Mathematics". This book is suitable for students of 11th grade but it needs good language skills to understand and appreciate the book. Go and get it from your college library or get it from Amazon if you have $50 to spend. The value of the book is more than the value of all books written by Rudin/Royden/Apostol and other Analysis gurus.

Being Smart

A month ago, I was traveling to my native place Bokaro by Rajdhani Express. Rajdhani Express is the best train provided by Indian Railways in terms of speed and services. So the fare is also the highest here. So I was expecting that the passengers will be more or less decent enough compared to those traveling in some other class of trains. But this is what I actually experienced.

There was a guy, about the same age as myself, in our compartment who had returned from USA and was going to his native place Ranchi. He offered the attendant a dew dollars (not much for a guy returning from US) as some kind of advance tip, so that he would take proper care of this US returned gentleman. And it turned out to be exactly like that. The attendant used to keep coming to our compartment every now and then asking "koi dikkat to nahi hai saahab?" (are you having any problem sir?). He made sure that we were given all the services on time.

But the best was the food service. Normally the amount of food provided to a person in Rajdhani is fixed. If you need extra you have to pay apart from the fare. But since our attendant was given a tip in dollars, he made sure that everyone in our compartment (actually we were only 3 - 4 people) ate to the fullest of our satisfaction. He even asked if anybody needed any extra icecreams.

Then that US returned fellow started stating in a tone of moral authority that if one pays to people from lower strata of the society, they tend to remember it and pay you back in some way, but the fellows from the upper strata will just grab your money. Hence we should always keep such people (like waiters, rickshaw pullers, auto drivers, and attendants in train) happy by giving a few bucks. And that would improve the quality of their services by an unimaginable extent.

The guy was right, or let's say, was smart, and he demonstrated his formula effectively. However I was wondering whether what he did was moral? Was that the really right thing to do in those circumstances? First of all, the attendant was serving us with extra food which was not his. The money spent on to get that food and other services comes from the Indian Railways. So in a way the attendant got paid for something which was not his by right. Secondly, this led to the discrimination among the passengers who had paid the same fare to Indian Railways. How did they feel about all this?

Just to make a comparison, suppose you go to a shop and you give tip to the person who is making a bill of the items purchased. (Its a big shop, so imagine that billing is being done by an employee and not by the owner of the shop.) And you get the goods at a very low price. Isn't that equivalent to stealing? And here our US returned Indian hero was proclaiming that this is the moral thing to do, whereas in fact he was acting like a thief.

By the way I did not participate in all this. No extra food and any stuff like that. Another passenger who was sitting beside me was a fellow who did MTech from IIT Kanpur. He was offering some advice about waste management for the Railways. He suggested that the compartment should be provided with small windows through which this garbage could be thrown from the train while running, so as to litter the area nearby the track with all sorts of garbage including plastic, food, aluminium foils etc.. I objected immediately that because of such an attitude the country is dirty. The waste should be disposed in proper way by throwing them in dustbin which are collected by the municipality. The attendant also said that all this Railways waste is dumped in dustbins at various stopping stations.

I was glad to know that this guy was just an MTech and not a BTech from IIT, thereby saving the cream of the country from a bad name.

Now both of these guys were perfectly good natured in all other respects. "Nice to have friends like that" kind of people. The US guy showed us a comedy movie on his laptop and kept us entertaining all throughout the journey. So I must say that I had a great time with them, perhaps one of my most entertaining journeys.

I am really perplexed by such phenomena. There are criminals, thieves and other bad elements in our society, but they have at least some sense of shame and guilt. That's why they keep hiding themselves and their actions from the general public. But here we find people who are well educated and having lots of money who behave like a thief and somehow proclaim that they are moral or smart.

Being smart is not about fooling people, but about doing things a better way (efficient, economical etc.) compared to others. When you bribe someone you are not actually being smart, but you are committing a legal crime.

And from my very limited experience with girls, I find that these are the kind of smart men who are preferred by them for their mate over the more principled ones. The reasoning is that they have the kind of survival talent which the current world requires. Thus according to Darwinism they are making the right choice. I wonder what these girls would teach to their kids when they have one. Probably they would create an even smarter kid. God bless them all!!