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.

Pete Seeger

I met Pete Seeger once in my life, a chance meeting, a few seconds. I was sitting in the central lobby of Grand Central Station in New York City late one night.I had been to meeting in the city and was heading back out to stay with family.  It was a desolate space with echoes dancing off the walls. It is even today hard for me to imagine that this usually bustling place could be so empty.

In the distance I saw an old scraggly looking man carrying a battered guitar case plastered with stickers of all kinds. He was just buying a ticket home. At first glance, from a distance, I thought him perhaps homeless carrying that old case held together by stickers. But then I recognized that familiar iconic profile. Generally shy, I rarely approach people directly, but I could not help myself so I got up and walked over to him and mumbled the only words that I thought he might want to hear, “My children love your music.” shook hands and walked away. I don’t remember him saying anything and perhaps he just smile. We went off in different isolated directions to catch our trains.

Other than that most fortunate and for me among the most memorable events of my life, I was connected to Pete through my children who were lucky enough to attend a wonderful summer camp, Killooleet, in Vermont run by Pete’s brother John and his wife Ellie and now by their daughter Kate and her husband Dean. The camp embodied the same community values Pete sang about.

His songs, so simple and so elegant are musical patterns  we share, know, and love for they remind us of a basic tenet of our human community — essential equality. It is a bedrock principle of our educational system, and yet it is so rarely fulfilled. We do not treat our schools equitably, we do not treat our students equitably, and we only pretend to treat our teachers equitably.

Pete Seeger's Grandson's Guitar Case
Pete Seeger’s Grandson’s Guitar Case

But today, I choose not to focus on these social injustices that Pete so eloquently spoke and sang  about, but rather speak to the injustice of requiring all of our learners to achieve high levels of the same “basic” skills.  We have made the printed word and the textbook math problem, literacy and computation, the tickets to educational success for every child. We have forgotten the arts, left out creativity, and rarely given our children the chance to practice the skills they may care about. So as I remember Pete, I picture that battered, papered old guitar case, and I wonder how we can enable every one of our children to enjoy a just education that gives them such a container of their own.


If you have looked at The Pattern of Knowledge you may wonder how I was able to fill in that wide-ranging table. In that time before the Internet, when I did most of it, such research was not easy. As I describe in Elegantly Simple, I found the big pattern, the periods without a lot of difficulty once I knew what to look for. The phases, well that was another matter.

I spend many nights reading and searching, several years in part-time research, and nearly two years looking for this pattern nearly full-time. At first, I did not know what I was looking for, thinking that each period had a different pattern. But as I searched and thought further, I found all of the periods had a common pattern. I searched first in compilations of breakthroughs in subjects in appendices, art books, and history of the subject books. But I soon found them wanting. I had to go to original sources and actually read what the great inventors themselves wrote. In this endeavor I was most fortunate.

I took my undergraduate degree in physics at the University of Chicago. My physics education was first class but traditional. The rest of my education was not. Undergraduate studies, the so-called liberal arts courses at the U of C are in the College. In a tradition dating back to the Hutchins’ era, these courses in the humanities, social sciences, and even in the sciences are seminars based on using the Socratic method on original works. We rarely if ever studied pre-digested content. We had to read and understand the great thinkers themselves. We had to ask what the author meant and why.

This very non-traditional education changed my life, for it built the confidence to tackle any and every original work I cared to. I have found this a generally rare kind of confidence. I read hundreds of works and parts of works from the pre-Socratics through Einstein, Plato through Wittgenstein, Sophocles through Joyce.

Confidence is perhaps the most important quality students can build. For without confidence, we do not concentrate well. If we don’t think we can understand something, we will find it hard to keep our focus on it and we will not learn it.  Today, schools may talk of grit and sticktoitiveness, but preaching grit is not learning confidence. And while we do want students to practice working on problems until they solve them, they build that grit through building confidence.

Confidence may indeed be the most important 21st century skill. My friend George Blakeslee calls it the 5th C and the basis for the four 21st century skills, creativity, cooperation, critical thinking, and communication. It is measurable attitudinally. And true confidence, not bravado, is the mark of a student who can concentrate and who has the skills and perseverance to learn anything she or he wants to learn. Should confidence be the goal of education? Well it certainly was what enabled me to develop Elegantly Simple and all my other  inventions.. It lets me think out-of-the-box, work cooperatively, communicate effectively, and think critically. And I gained this confidence as we gain every skill, through practice.

A Poem

Robert Frost defined a poem as “a momentary stay against confusion.”

 I love this definition. My friend Larry Weinstein mentioned it in passing at lunch the other day. It describes so wonderfully why we construct unique patterns. 

The rhyme and/or rhythm in a poem helps us to remember it. It provides us with a unique, a special pattern that we associate with the poem,  and it enables us to more easily and quickly remember the next words or phrases. This is what I mean by uniqueness. Of all the possible words or sequences we could have used to describe something, the poet has chosen one or at most a very small collection of patterns overlaid on each other to grab us, hold us, and constantly remind us of his story.

For some this may be a visualization, for others a wonderful rhythm, and for still others an association, or perhaps all of those. Mending Wall has that uniqueness to me. It begins with:

Something there is that doesn’t love a wall,

And ends with:

“Good fences make good neighbors.”

It tells the story of two neighboring farmers doing their spring chore, the story of two very different people with outlooks on life we recognize, we already know their pattern. One questions, the other does not. This poem is beautiful to us. It captivates us. It tells a story that could be told in so many different ways, but it is a collection of patterns, laid one on another, special patterns of rhythm and sound,of image and action, of phrase and structure that makes this story simply one of a kind. It is like the wall, with stone carefully balanced on stone, each chosen to fit its place like this collection of words and patterns and like the unique patterns we build and use everyday.

The Simplest Patterns

Steve Wozniak is arguably among the greatest, if not the premier inventor of the second half of the 20th century. His vision of invention helps us to understand patternmaking and unique artifacts.

I found Woz’s talk fascinating. Treated as a rock star by an overflow crowd at Boston’s venerable Symphony Hall, Steve did not disappoint his audience. With history, jokes, personal anecdotes, and wonderful stories he held me and I think most of the crowd spellbound. I was most fascinated by what drove his invention.

He told us that since he was a young boy he loved mechanical things and in particular electrical circuits. And from the time he was in 6th grade he played with electronics becoming a ham radio operator at age 10. He loved to see if he could build devices with fewer and cheaper parts. With the arrival of cheap integrated circuit chips, Steve switched from the analog to the digital world. He sought to build or to rebuild circuits using fewer and less expensive chips.

This passion for simplicity and elegance made his great invention the Apple II computer an icon and the archetype for personal computers to this day. Woz talked of making chips do multiple things so that he could use fewer of them. He spoke with great pride about replacing a circuit board with 100 component with one. He loved the creative contest to see if he could make things with fewer chips than anyone else.

His passion is our human passion. For chips are fundamentally patterns, patterns etched in silicon. We want to make patterns that perform a task with as few “chips” or patterns as possible. We want to use our patterns for multiple purposes so that we don’t have to replicate them. For Woz simplicity was not only compactness, it was cheaper. Patterns are always expensive for our brains, we not only have to construct them we have to fill them and share them. If they are too complicated, if they are too hard to use, if they are too hard to remember or to share, to store or retrieve experience, then they literally cost us too much.

We are all driven as Woz was to build our artifacts to be the simplest and thus the most elegant patterns that will work to hold a portion of our experience. His genius that brought us the Apple II is our genius as we construct artifacts everyday when we cannot rest until we have made those patterns as simple and thus as elegant as possible.