Math and Happiness

There was an article on CNN today about a study done that relates students' confidence and happiness to their success in math.

On the surface, I was surprised by their findings that less happy students with lower confidence were better math students; conversely "confident students do worse in math". At the very least, they were contrary to my experiences, and I would have liked to see some of the actual analysis rather than the simple report of the raw data.

But I do see some possible ways to reconcile these results. From my own background, I can distinctly recall three major phases (thus far) in my mathematics education.

In the first phase - think back to your first contact with math :) - it is still a very novel thing, pretty cool and different from most other subjects you're used to, still very puzzle-like and playful. It's like learning a new language - at the beginning the new words and phrases are foreign, mysterious - you're fairly engaged.

The second phase is the downer. It involves much more rote computations and you start to get bored, frustrated, unhappy. Keeping with the learning a new language analogy, I liken this phase to learning the "grammar". Basically, this is where you pick up many of the fundamental computational tools you'll need in more advanced mathematics.

The third (final ?) phase, for me, began at the moment when I started higher mathematics and the now ingrained tools were applied in various contexts: pure mathematics, physics, engineering, computer science, etc... For example, I found myself drawn to the beauty and elegance of proofs, the synthesis of topics I've learned previously, and the amazing satisfaction of finally solving the problem. In this respect, this phase is very much like the first phase again where it becomes much more enjoyable again. In the language analogy, it's like finally being able to talk and use the language for extended periods of time - that's when the true beauty comes out.

Of course, I'm still very much in the middle of my mathematical journey and there may be more phases. Also, the phases sometimes tend to oscillate :P

So, anyway I think this is a possible confounding variable that needs to be considered when analyzing the results of the study correlating success in math and student happiness. For example, I can certainly relate to the phase two feeling of being very proficient at math from a pure computational standpoint yet not being very satisfied with the overall experience. On the flip side, proficiency with these tools, like a firm grasp of grammar in English, is essential to fully appreciate the language of mathematics. But in the end, the ability to imagine and play are just as vital for happiness in mathematics as in other fields.

-- Arkajit

Thirteen Digit ISBN

The days of the 10 digit ISBN are winding down! Starting 1/1/2007, is ISBN-13. Luckily the new ISBN system has been made backwards compatible with the old one. It seems there was always a 3-digit EAN code prefix which was always 978. Now, they're just introducing another series 979. Since the last digit in the ISBN is just a check digit, that really only gives 1 billion extra ISBN numbers. Which makes me wonder, how long can the 13 digit ISBN last before it is exhausted as well? 2100? 3000? Will we still be relying on books at that time or will all our data have become digitized and books obsolete? Or will there be some new medium by then? Just some interesting questions to ponder.

On a side note, the method of computing the check digit of the ISBN is quite interesting as well. For the 10-digit ISBN, the check digit has to be chosen such that the dot product of the vector [1..10] with the 10-digit vector representation of the ISBN (including the check digit as the first entry) should be congruent to zero modulo 11! If the check digit has to be 10, the letter X is used as the digit. Now obviously the check digit is not foolproof, but since 11 is prime, it does a pretty good job. So I wonder if for ISBN-13, the check digit will be computed modulo 13+1=14 or 13, also prime. I'm inclined to suspect that it would use modulo 13.

Anyway, this leads me to my idea for an ISBN game. So one person picks a favorite book and then finds its ISBN. He gives the other person the ISBN without the check digit and he has to go and figure out what the book is. Right now the game is fairly trivial, but you can probably add some extra steps along the way to add to the challenge. Maybe the book title can be part of a clue for a larger puzzle and maybe one other digit (besides the check digit) could be hidden as well. That at least expands the number of possible matching ISBNs - there could be other hints given to narrow down the choice. I'll have to try this sometime :)

-- Arkajit