Exploding Dots: Thinking Outside the Box in Math

James Tanton, a mathematician and educator, visited ASIJ on February 26th. During his visit, James taught High School students and teachers his theories on how to perform mathematical processes. High School junior, Siddharth Ray, shares the experience with us:

When I entered my first period math classroom, I could immediately feel the energy given by James Tanton to the rest of the room. As it was a first period class, I never imagined that he would be able to sustain this type of energy for the rest of the period.

Mr. Tanton started by talking about how addition is taught and how it should be done: from left to right (the same way that we do our reading) instead of right to left. He went on to talk about subtraction, multiplication and division using his “exploding dot method,” which used a left-to-right approach as opposed to the conventional ways of doing arithmetic. It was interesting, but given that we were calculus students who had just started exploring series and sequences, we thought it was something trivial.

He later showed how his method could be used to find solutions to polynomial division, which was something that seemed slightly more relevant to the class. Slowly but surely, he went through all levels of mathematics using his “exploding dots method,” and he ended by showing how dividing 1 by (1-x) (using his exploding dots method) would result in 1+x+x^2+… one of the most common series that exists. Mr. Tanton also showed us that 1/(1-x-x^2) would result in coefficients derived from the Fibonacci sequence. At this point, we had gone all the way from basic arithmetic to calculus-level mathematics, all using his unique “exploding dots method.” It was a pretty long journey through a 75-minute class!

What James Tanton taught me was more than just how to add, subtract, multiply and divide with his cool “exploding dot” method; he taught me the importance of thinking outside of the box. From the beginning of his lecture, he challenged the norm and made us question why we were doing what we did. “Why do we do addition, subtraction and multiplication from right to left when humans read from left to right? Let’s do these operations from left to right.” He showed us how to do well-known operations in a unique and easy way that helped tie in more complex ideas. Simple arithmetic, different bases (base-2, base-6, etc.), polynomial operations and Taylor series were all brought together using his “exploding dots and boxes.”

Finding these connections, as Tanton showed, can help with endeavors into more difficult levels of mathematics. A link between something as simple as division and something as excruciatingly difficult as Taylor series help make the complicated bits more bearable. To top all of this off, Mr. Tanton showed how cool math could be. Connecting basic division with calculus material? Never in my wildest could I have imagined such a strong connection between those two seemingly distant concepts. Thanks to Mr. Tanton, math just became 100 times cooler. (Siddharth Ray, grade 11)

You can visit James Tanton’s website here: Thinking Mathematics!