Khan Academy and Calculus

This semester I am teaching Calculus II.  My class is small and intimate with a dozen students, half of whom were in my Calculus I course last semester.  The biggest challenge I have as an instructor of Calculus is the variety of knowledge students enter the course with.  Some students have taken Calculus in High School, while other students may not have seen the prerequisite material before.  As an instructor, how can I make Calculus refreshing and enjoyable for all students in the course?



Recently, I came across a great resource to supplement my Calculus lectures when I was searching the internet for ways to enhance tomorrow's lecture on L'Hospital's Rule:  Khan Academy.  Salman Khan, not to be confused with the Bollywood actor, has posted 1800+ self-narrated, self-produced online tutorials on various topics in math, science and humanities.  The collection of Calculus topics spans an introduction of limits to Green's theorem.  He quit his job as a hedge fund manager to fulfill his mission to provide a world-class education to anyone, anywhere.  Amazing!

Khan Academy's free mini-leactures are a wonderful resource for students who need to brush up on their Algrebra and Pre-Calculus, or those who would like to revise concepts discussed during lectures.  I highly recommend Khan's conversational approach to communicating the ideas of Calculus in a simple, understandable way.   See it for yourself at http://www.khanacademy.org/ and maybe you can find ways to contribute to this fantastic endeavor in making Mathematics and other subjects accessible to all.   

Adding and Counting by Ken Ono

Last Friday, I attended an Emory Public Lecture titled "Adding and Counting" by Ken Ono.  During this well attended lecture, Professor Ono beautifully revealed the new theory he and his colleagues have discovered:  that partition functions behave like fractals and a closed formula to count the number of partitions for any number n.  The results confirm an observation made by Ramanujan in 1919.



The partition function p(n) counts the number of ways a number n can be partitioned.  For instance, over addition, n=4 may be partition in five distinct ways:  1+1+1+1=1+1+2=1+3=2+2=4.  Thus, p(4)=5.  For a better idea of what it means to "behave like fractals", watch the video above and notice the same structure appears repeatedly upon zooming in.  A formula for p(n) has intrigued mathematicians for centuries.

Professor Ono joined the faculty of the Mathematics Department at Emory University last semester and I have had the pleasure of taking a Number Theory course with him last semester.  For more information on this new theory, please see the Emory eScienceCommons and AMS press release.  Congrats to Professor Ono and his team on this remarkable advancement of Mathematics!

Back to School

With 4-6 inches of snow and only 6 plows, icy Atlanta transformed to Hoth-lanta last week, resulting in a 5-day extended winter break for University students.  Although there was a delay in the start of the semester, there was no delay in discussing interesting topics in my Combinatorics course this week. 

A Steiner system with parameters l, m, n such that l<m<n is an n-element set S with an m-element subset of S (called blocks) such that any l-element subset of S is contained in exactly one block.  To simplify, let l=2 and m=3 to form a Steiner triple system S(2,3,n).  For what n does a Steiner triple system exist?
 

This question inspired Reverend Thomas Kirkman and he solved a special case of this problem in 1847.  His solution is widely known as the Kirkman Schoolgirl Problem.  So in the spirit of a fresh, new semester, here is the riddle for you:  15 girls walk to school each day in five groups of three.  How many days does it take for each pair of girls to walk together exactly once?  Note this is a Steiner triple system S(2,3,n) where n is the number of days.  Hint:  See picture.