How stoichiometry problems exposed the Indian Education System!!!!!

For the  past couple of days I've been at home and have been teaching my brother some of his class XI science and mathematics. So, while I juggle among molarity, molality, dimensional analysis, significant figures (?!) and the likes, he keeps barraging me with random questions - most of which I could answer thanks to my clear & conceptual understanding of my +2 level . Whether he does this out of curiosity OR whether to verify my WBJEE rank credentials - I don't know.

Anyway, at one point, he asked me a seemingly trivial question in chemistry and he was surprised at the ease of the solution. He later told me that her teacher told her that this is "high-level" and she cannot solve it. I am surprised because all that the question demanded was an understanding of the mole concept. A few other anecdotes from her and my own experience of the Indian education system lead me to a disturbing conclusion: The system is decaying, and it is decaying fast, and we are running out of time, and it won't fix itself, and they are not really bothered!

My brother has a good theory about why the senior secondary teachers suck at their job. Let me paraphrase him:

"Most of the class XI/XII syllabus is "irritating" and requires sound fundamentals to grasp the subject. Most teachers prefer to teach class X-and-below because it is "very easy" in comparison to senior classes and students hardly ask (or are encouraged to ask) "difficult" questions.

The few who actually manage to master the syllabus never return to teaching because there are greener pastures for such people. So, there is a void created and this void gets filled with sub-standard (or, as he said, "bekar") teachers."

To think of it, he makes perfect sense.

Going through his NCERT books, I realised that class XI/XII is indeed demanding (and rightly so!). To teach the thousands of these students, we need teachers of the very best caliber and it seems not forthcoming. You know that the system is rotting when your teachers advise the parents to arrange for "coaching", even for basic school syllabi; when the school lets a professional coaching institute "counsel" the students and for all their queries, there seems to only one answer - Jay E Eee. (After the counselling session, the coaching institute offered the students a discount on their JEE-coaching program.)

While Pranab Mukherjee is busy allotting thousands of crores towards education, I hope he realises that there are serious problems with the Indian education system that cannot be solved by pumping-in cash. Perhaps, we need a revamped pedagogy - one which understands the changing needs of the students while still able to deliver quality input.

The Private Engineering Colleges ( built to nurture and churn out good Technocrats!!) face a somewhat similar crunch - 60-70% shortage of faculty. And thankfully, they are appointing bekar professors (or so I'd like to believe). There was/is an attempt, though, to bring in reservations in the faculty, which would undoubtedly lead to a compromise of merit.

So, Pranab dada giving the Education System 2000-odd crores is great news but it will hardly solve any of the fundamental problems that plague this Indian Education System- and there are enough and more. (If you a new joinee to any private engineering colleges(however reputed it may be!!) - too bad I am squashing your dreams so soon :-) some of my other posts might offer solace!)

I leave you with a question that often bewilders me:

Why does India not have a network of world-class government high schools which attract and train the best minds in the country? And how come we have the world-class IITs and IIMs without a sound higher-educational foundation in the system?

Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem

Pardon me!!! For using this title for my blog post which resembles its name with the famous book " Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem" by Simon Singh, but I couldn't find a more suitable name to use. I intend to write about this great mathematician Pierre De Fermat sometime later in my blog but let me throw some light first on one of his problems which  fascinated me since my childhood. Yes you got it right!!!! It's the "Fermat's Last Theorem". I first read about this problem when I was in Junior School (Class 6 or 7 ) and from then on I was really fascinated by the intrigues and triviality of such a simple looking theorem. I actually chanced upon this theorem at an young age while I was searching the net looking for an elegant proof of the  Pythagorean Theorem ( Thanks to my curiosity!!!!! :D ) which I was taught in my junior school. Blame it on my ignorance or not but for all these years ( Blame it on those stupid  ICSE and AISSC  board exams and tuition pressure!!!) I was unaware that the proof for this theorem had come into existence, that too in 1995. Actually this was one of the reasons which initially motivated me to study Mathematics honors in my Undergrads. Sadly :(  I had to take up engineering!!!!!

 Getting back to our topic :

What is exactly the Fermat’s Last Theorem?

"No positive integers  x , y and z can satisfy the equation x^n+y^n=z^n  for all integers n greater than 2."

  Fermat’s Last Theorem is an extension of the Pythagorean Theorem. Every schoolchild knows that, for a right-angled triangle,

  x^2+y^2=z^2

where x, y and z are the lengths of the three sides of the triangle, with z the length of the longest side.

Some right-angled triangles have sides whose lengths are whole numbers. For example, a triangle whose sides have lengths 3, 4 and 5 is right-angled because

 3^2+4^2=5^2

Fermat was considering the more general equation 

x^n+y^n=z^n

where n is a positive whole number. He wondered whether positive whole numbers x, y and z could be found that satisfied this equation if n was larger than 2.

In one of his notebooks, he wrote that

I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.

believing that he actually proved the then conjecture.

Well, that was in 1637!!

However,as number theory developed, mathematicians observed that there are no positive integer triples that satisfy the equation above given larger integer exponents.Fermat left no proof of the conjecture for all n, but he did prove the special case n = 4. This reduced the problem to proving the theorem for exponents n that are prime numbers. Over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although Sophie Germain proved a special case for all primes less than 100. In the mid-19th century, Ernst Kummer proved the theorem for regular primes. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to prove the conjecture for all odd primes up to four million.

The final proof of the conjecture for all n came in the late 20th century. In 1984, Gerhard Frey suggested the approach of proving the conjecture through a proof of the Taniyama–Shimura–Weil conjecture for elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the Taniyama–Shimura–Weil conjecture to prove Fermat's Last Theorem, with the assistance of Richard Taylor.

Wiles Proof of Fermat's Last Theorem: 

Ribet's proof of the epsilon conjecture in 1986 accomplished the first half of Frey's strategy for proving Fermat's Last Theorem. Upon hearing of Ribet's proof, Andrew Wiles decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves.^[Wiles worked on that task for six years in almost complete secrecy. He based his initial approach on his area of expertise, Horizontal Iwasawa theory, but by the summer of 1991, this approach seemed inadequate to the task. In response, he exploited an Euler system recently developed by Victor Kolyvagin and Matthias Flach. Since Wiles was unfamiliar with such methods, he asked his Princeton colleague, Nick Katz, to check his reasoning over the spring semester of 1993.

By mid-1993, Wiles was sufficiently confident of his results that he presented them in three lectures delivered on June 21–23, 1993 at the Isaac Newton Institute for Mathematical Sciences. Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semi-stable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it soon became apparent that Wiles' initial proof was incorrect. A critical portion of the proof contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles' manuscript, including Katz, who alerted Wiles on 23 August 1993.

Wiles and his former student Richard Taylor spent almost a year trying to repair the proof, without success. On 19 September 1994, Wiles had a flash of insight that the proof could be saved by returning to his original Horizontal Iwasawa theory approach, which he had abandoned in favour of the Kolyvagin–Flach approach. On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.

A video featuring the story behind proving this theorem had also been made by B.B.C. which you can also check out if are quite bored up with my writings!!!

Well it's really impossible to put up Wiles full proof here.But if you really interested to see the proof( Word of Caution: Don't check out the proof if you're a novice in higher mathematics!!!) then check out the two links below and download the files. These were the original manuscripts of the proof :

http://math.stanford.edu/~lekheng/flt/wiles.pdf

http://math.stanford.edu/~lekheng/flt/taylor-wiles.pdf

There is also a power point presentation by an  IIT Kanpur Grad ( Go!! IIT!!!!) which is more simple to follow. You can also check that out:

http://www.cse.iitk.ac.in/users/manindra/presentations/OlympiadStudents.pdf

Lastly I want to say that thanks to Mr. Andrew Wiles and other such great mathematicians who had dedicated their lives behind proving this 350 year old historic problem. Guess what!! Now I have one less problem to think about ;)