Thursday, July 13, 2006

Recommendations for Math Prodigies

This blog originated from a visit to a friend's place. His son, who is not yet 6, is very interested in math (he was writing out the Fibonacci series on a sheet of paper when I saw him). I asked him to sum up the numbers from 1 to 100 (like Gauss), and to my surprise, he gave an answer in a couple of minutes --- he had inferred the right idea on summing an arithmetic progression, though he made a slight error in the calculation, but it was still incredibly impressive. We then asked him to name two primes that sum to 19, he immediately replied 2 and 17. Then we asked him for another pair, he asked me if I was sure there was another, saying that there were no others as 2 is the only even prime.

So it was clear that this kid is unusually gifted in mathematics. I asked my colleagues how we should nurture his interest further, without pressuring him in anyway, but just pique his curiosity about numbers further (and he is very curious indeed!). This is a summary of their responses. I will keep adding more ideas, please send them to vineetgupta AT gmail DOT com.

Interesting books to read: In many cases these books are too difficult to read for a 6 year old, so the parents would end up reading the books and then talking to the child.

Recreations in Number Theory - Beiler
The Lore of Large Numbers - Davis
Martin Gardner's Mathematical puzzle books. Aha! Insight.
Adventures in Mathematics
Mathematician's Delight - W W Sawyer
Sideways Arithmetic from Wayside School - Sacher
What is Mathematics? - Courant and Robbins. For older kids.
Smullyan's logic puzzle books : What is the name of this book?, Alice in Puzzleland, The Lady or the Tiger (inspired by Frank Stockton's beautiful story), The Riddle of Scheherazade, Forever Undecided.
Asimov's Realm of Algebra
Weeks' Shape of Space
The Number Devil
How to solve it - Polya
Flatland - Abbott
John Allen Paulos' books.
Proofs from the Book
The man who counted.
The Penguin Dictionary of Curious and Interesting Geometry

Software, tools, websites:
POV-ray an open source ray tracing engine for geometric intuition, coding etc. (programming).
Happy Cube
Soma cube(you can make soma cubes at home)
Wire puzzles

Math Programs:
Math summer camp Ross at Ohio state
PROMYS at Boston U after a few years.
EPGY Stanford.
Find a local Math Circle (Ask Tom Davis ).
The Gelfand Program for talented and inquisitive students.
Arrange a visit MSRI Berkeley and chat with mathematicians.
Math circles at Stanford and Berkeley

Math areas to focus: (These are not usually taught in schools).
Number Theory
Graph theory
Set Theory and set theoretic definitions of numbers, arithmeic etc.
Abstract Algebra
Mathematical puzzles like Sudoku (solve and later build).
CS ideas like binary search, sorting algos, dfs and bfs, probability computations, optimal algorithms for games like Mastermind)

Interesting questions to ask and encourage discovery:
Find Euler's tour in a graph
Ratio of circumference to diameter
Find rules for divisibility in different bases ( e.g. for 3 and 5 in base 16)
Limits and infinite series 1 + 1/2 + 1/4 + ....
Convergence of sequences
Show countability of rationals and uncountability of reals to demonstrate different kinds of infinity
Show that there are infinitely many primes.
Show that sqrt(2) is irrational.
Given the numbers 1 through n, choose n/2 + 1 numbers. Show that this set must contain two relatively-prime numbers.
Show that the sum of odd numbers from 1 to 2n+1 equals (n+1)^2.

Other ideas:
Don't focus only on math - let him be a kid - play soccer, build sandcastles etc.
Focus on things not taught in school.
Do not hold him back!
Talk to teachers and find out about techniques used for kids so far above norm. Do not force him to sit through standard math classes as he already knows this stuff.
Find peers who share his affinity to numbers for interaction.
Chess clubs for peers.
Leave lots of fun books lying around, especially a math and science encyclopedia.
Do other fun activities that apply math: build a robot, perform experiments with chemicals and electricity, build a radio, program a computer game, build a tree house etc.
Let him invest for his education.
Find a smart caring adult to talk to him and generally goof off about math.


Blogger Sudheer Marisetti said...

Excellent compiled list.I don't know if my kids will go through these books or concepts but I am going to go through them myself. It would be interesting to know in each of the cases what made the kid a prodigy - is it genetics or environment or something else.

1:17 AM  
Blogger Kiran said...

Vineet, tanks for a very interesting blog, and links to good math books.

5:52 AM  

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