All posts by Art Bardige

I am a math educator and entrepreneur. I have just published a book on the evolution and theory of knowledge called Elegantly Simple.

Learning Math as an Experimental Science


What if math were learned as an experimental science where spreadsheets were our laboratories.

Keith Devlin wrote a fascinating biography called “The Man of Numbers” about Leonardo of Pisa. Leonardo is credited with bringing Arabic arithmetic and algebra to Europe. But his story is much more interesting and his lasting mark on not just mathematics but on education has long been overlooked. Leonardo was born in Pisa around 1170, when Pisa was the greatest trading city in Europe and had just begun building what we know of as the Leaning Tower. His father, a trader and diplomat for Pisa, brought Leonardo to Algeria when he was a boy and had him tutored in Arabic arithmetic and algebra. Leonardo became a merchant/trader traveling Africa and Eastern Asia. The mathematics of business in both Medieval Europe and the Muslim world was the Roman system using an abacus to perform calculations. But as Leonardo found, it was slow, cumbersome, and error prone. It worked for the Roman Empire which had a consistent monetary and weights and measures system. In Medieval Europe nearly every city-state had its own monetary and weights and measures system which meant that merchants had to constantly solve difficult ratio and proportion problems and equations.

So in 1200 Leonardo returned to Pisa and wrote Liber abacci The Book of Numbers, for merchants and traders to provide them with a revolutionary new arithmetic and algebra to solve their problems quickly and confidently. This book became the basis for the transformation of mathematics and business. Devlin included a picture of Leonardo’s “Table of Contents.” I was awestruck. The sequence of chapters in Liber abacci was a replica of the sequence we teach in K-12 math. The mathematics Leonardo developed for the needs of business in 1202 is the mathematics we now expect every child to master in K-12.

It is not the math business uses today!. For a business math revolution occurred in 1979 with the first personal computer spreadsheets. Spreadsheets are based on the mathematics of functions, first developed nearly 450 years after that enabled the scientific revolution and that has in the past 35 years transformed business. Leonardo’s rapid paper-based computation algorithms and equation solving is no longer necessary. Computers do that. Today business thinks in terms of input, outputs, and rules connecting them. Today business wants to ask of mathematics, What if…

Most of us have heard of Leonardo as Fibonacci, a nickname given him for unknown reasons eight centuries after he lived. His sequence was a minor diversion in his great work. Now his mathematics is obsolete, we no longer use most of it, and we no longer need our children to learn it.

I created this deck to help you envision math as an experimental science and spreadsheets as laboratories.

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To learn more go to What if Math.

The Tools of the Trade

“We would all be better off if functions were introduced in kindergarten and studied regularly thereafter. The concept of a function is one of the most important ideas in mathematics.”

Paul Sally 2008

I found out yesterday that Paul died last month. A great mathematician, a profound teacher, and a deeply caring math educator, Paul was a man of incisive intellect and wit. I had the good fortune to hear him in person at a University of Chicago seminar for alumni last June. Though blind and nearly deaf walking on two artificial legs, he was brilliant, funny, and transfixing. I went to the bookstore and found one of his latest books, Tools of the Trade, which he subtitled Introduction to Advanced Mathematics. It is a textbook like few others. Tiny, less than 200 pages long, and dense for he is speaking to advanced college students who seek to be mathematicians, it nonetheless captivated me, and it has been a very long time since I last took a serious math course. I cannot say I have worked my way through this book for it seeks to get students to “come to grips with the idea of proof in a serious way.” But I have returned to it again and again to gain its deep insights, for while he did not write this book for math educators,  we can find in it the tools of our trade.

Paul makes it clear that functions should play a fundamental role in all math education. He constructed his book with projects to provide “Inquiry Based Learning (IBL) experience(s)…” for students. And he sought to challenge their “mathematical creativity.” So as we rethink education and particularly math education we focus our attention on those things that Paul Sally thought were fundamental: functions, Problem or Inquiry Base Learning, and of course creativity and make these the foundation of mathematics education.

No Room for Creativity

I love roadtrips. I love to drive and see new things at ground level. And I find that every roadtrip I take has a profound effect on my thinking. This one my wife Betty and I took over the Christmas, New Years holiday this year was no exception. We drove to Florida stopping along the Atlantic Coast. On the return trip we went inland where our first stop was a small north Florida town, White Springs, known for its hot springs and Stephen Foster’s Swanee River. This small town with one restaurant, a Dollar store, and four gas stations because it is on US 41 supplying a constant stream of trucks, and one B&B where we stayed. The town has clearly seen better days though it is trying to rise again. Though Stephen Foster is highlighted in the town with a huge and beautiful park and visitor center, he never actually was there and found the name Suwannee while looking for a southern river to write a song about.

The B&B is a small, quite and very unpretentious house. Judith calls it a boarding house bed and breakfast. Curious about this work in progress with painted windows decorating its porch and a book or two by her on the coffee table, at breakfast I asked about her past. She told me she had taught in Phoenix for some 25 years but retired four years ago. She taught mainly Hispanic and Native American third graders. She said she retired because she no longer enjoyed teaching. She loved the kids and it was obvious to me that she was a great teacher. “Teaching,” she said, “was no longer creative.” The curriculum and the lessons were dictated. The principal would come into her classroom to make sure she was following the lesson plans defined by the district. She was sure here students were learning less and certainly enjoying it less. She could not be creative nor could her students.

With a long trip to Chattanooga ahead of us we could not stay and talk, but I have heard this story so many times in the last couple of years I was sure I could tell it. What creativity there was in education has been driven out, the arts are nearly gone; the opportunity for teachers to develop their own lessons, all but gone; and the chances for kids to let their imaginations roam, well when are they tested on that. Judith’s story may be all too familiar, but for me it was profound and personal, very personal. For we share a dream, and I hereby dedicate this website and blog to that dream, the dream of productive imaginations, the dream of schools rich in creative experiences, the dream of children loving learning because they are creating it.