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.

Sir Cumference and the First Round Table by Cindy Neuschwander

I spent the first week of the New Year with my niece in New York City.  At age one, she has brilliantly mastered the art of counting from one to ten.  I tried to take it a step further by teaching her the even numbers.  When I said, "one," she replied "two."  To "three," she replied "four," and so on.  I was thrilled until I reintroduced the odd numbers and somehow her number system omitted "three."  I'm happy to report that this issue has since been fixed and she can now count to 15.  With these types of counting skills, perhaps she will be a Combinatorialist like her aunt!  And what would life without three be anyway?

I then strayed from counting and taught her some geometric shapes.  What's a better way than with Sir Cumference and the First Round Table by Cindy Neuschwander?  In this clever children's book, Sir Cumference is challenged to construct a table to accommodate King Arthur and his 11 other Knights to meet to discuss the invasion of the Circumscribers.  After trying a rectangle, square, parallelogram and octagon shaped table, with the help of his wife Lady Di of Ameter, his young son Radius and the carpenter Geo of Metry, it becomes clear to Sir Cumference that a circle shaped table is most suitable for the King’s long awaited discussion for peace.

There are several books in the Sir Cumference series.  Looking forward to the next one!

Happy New Year!

2011 is prime number since the only numbers divisible by 2011 are one and itself.  So 2011 will be a prime year!   My husband and I have come up with 10 goals for ourselves to ensure our precious time this year is used wisely.  Of course, weekly blog posts made my list! 

When was the last prime year?   When is the next prime year?  Were these years prime for you?  Just because a year may be prime, what would you like to achieve to make it prime?  Reach for the stars!!  A Happy New Year to you and yours!