The book’s underlying philosophy: The files are being crafted for independent learning experiences, rather than as a teaching tool. Nonetheless, they certainly can be used to supplement teaching, and I continue to work on materials to help those who wish to teach from these files.

The goal is to provide a more integrative experience so that users do not even notice that they are learning math. Users are simply playing with numbers, either using arrow keys or by entering them in cells and seeing what happens. The images that they create will naturally present opportunities for mathematical learning albeit in a subtle way. Most of the images users encounter have a mathematical reason for being interesting, but users need not know the reason to enjoy the images. For example, even very young users will appreciate learning how to create pentagons and pentagrams from polygons with more than 5 sides. Those in grades K-2 need not know that the answers they derive (that n = 10, 15, 20, 25, …) are all multiples of five. They will see the pattern even though they cannot express it in multiplicative terms. Nonetheless, visualization of patterns such as this primes them to understand multiplication more deeply once it is formally introduced.

This book pulls together topics at various levels of mathematical sophistication. Given the highly visual nature of the material, students are able to enjoy exploring a variety of topics even when they do not fully understand the underlying mathematics. Such explorations are attuned with Sherman Stein’s suggestion that mathematical understanding is best accomplished by following what he called The Triex: Explore, Extract, Explain. Stein recommends that we encourage students to explore and gather data, seek to extract some sort of order or pattern from those explorations, and ultimately seek an explanation for what has been found. He points out that even if students are unable to attain that third step, they have been primed to appreciate the explanation provided by the instructor or another student for why the pattern exists. Such explorations encourage learning and this learning can occur outside the classroom even in the absence of complete understanding of the results.