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.

“Wild Enchantment”

One of my most beloved lines comes from Boris Pasternak’s brilliant novel, Dr. Zhivago which I read nearly 50 years ago. Pasternak describes Lara as seeking “to call each thing by its right name.”

“Lara walked along the tracks following a path worn by pilgrims and then turned into the fields. Here she stopped and, closing her eyes, took a deep breath of the flower-scented air of the broad expanse around her. It was dearer to her than her kin, better than a lover, wiser than a book. For a moment she rediscovered the purpose of her life. She was here on earth to grasp the meaning of its wild enchantment and to call each thing by its right name, or, if this were not within her power, to give birth out of love for life to successors who would do it in her place.”

Doctor Zhivago, Boris Pasternak 1957

That phrase has been my lighthouse as I have made my path in life. It seems to me it is the essence of poetry, for does not every poet, and both Pasternak and Zhivago were poets, seek to call each thing by its right name. Now that certainly does not necessarily mean a single word or a definition, it must mean much more than that. For a right name could be metaphor, it could be a phrase, it could be a description. Is this not want poetry is all about?

And is this not what science and mathematics are all about as well? For like poetry they require us to learn a new language, to use metaphor and definition. I am not much of a poetry reader. I think it is because it takes effort to read poetry. We cannot just scan it. We have to read it, often aloud. We have to think it and imagine it and wonder it. And this is just what we must do with science and math. We can’t just scan those equations those symbolic presentations. We have to think them, say them, imagine them, and of course wonder them.

So, just as poetry is the language of life so too are the symbolic sentences we see in our science and math books. Like poetry we have to read them slowly, to ingest them, to think about them. We have to analyze them and rewrite them in our own “right names.” And does this not tell us a great deal about the importance of poetry in our schools and in our lives. For it can help us learn to seek right names in our world of science and art, and “to grasp the meaning of its wild enchantment.”

Patterns and Printing

It was smaller than I had expected it to be. And it was in two colors which I definitely had not imagined. Encased in its Plexiglas shield, it was still magnificent. I kept staring and staring at this open page of a Gutenberg Bible. This one printed on paper was one of three at the J. Pierpont Morgan Library in New York City. It was the first time I had ever seen one. Here was the first printed book, the first product of its kind. But try as we can to imagine what Gutenberg had to invent, had to go through. We describe him as creating moveable type, but he had to invent the font, learn to make molds and to find the right alloy to pour into them, then perfect that process for thousands of tiny objects. He took a wine press and converted it to print large pages, get high quality paper, develop a new kind of ink, print page after page hanging them to dry and then turning them over to print the other side, cut them apart and bound them into a great book. What an amazing task that took him nearly five years. He printed over 100 copies about a third on vellum of which 48 remain and of those the Morgan has three.  We glibly today talk about the origin of the printing press as among the greatest inventions, but we do not even try to imagine Gutenberg’s incredible invention and process.

I thought back to one I was more familiar with, for it was not too long ago that our early flat screen TV’s and monitors came out. The standard pixel count was quickly set for both at a 640 pixels per row and 480 rows, under half a million separate picture elements. We had to be tolerant of those early screens with up to five “dead” pixels in them which usually shown black in an otherwise lighted screen. The picture was not nearly of the quality of the Sony Trinitron but the set was a not lighter and thinner. This error level, about 1 per 100,000, was considered very good and non-returnable. We have come a long way in a dozen or so years. Today we are buying retina or 4K or high definition displays with more than 10 times as many pixels in the same screen sizes and no dead pixels in those screen. The screen manufacturers have perfected a remarkable printing process.

It led me to think again about Gutenberg and his Bible. For he must also have faced a similar error rate problem. He must have thrown an incredible number of pages away, burning them to warm his great warehouse where hundreds of pages at a time were hung out to dry. His Bible, nearly 1250 pages long with a little over 70 copies on paper meant he would have printed over 100,000 pages, 50,000 double-sided pages, and with two pages for each sheet, 25,000. He would have run one or more proof sheets for each page to check the type setting and the spelling against a hand written Bible. Then he would ink the type and press the sheets page after page. He began by printing each page twice to add the red rubrics to them, but he later found that doing rubrics by hand was less work.

Each line of type would have about 70 characters in Latin, separated into two columns by 42 lines per page would mean in the order of 3,000 characters per page that would have to be checked and individually changed, by over 1,000 pages, means over 3 million characters (effectively pixels). What error rate would he have tolerated?

All of this thinking and calculation leads me back to patterns and patternmaking. For whether a Gutenberg Bible or a computer screen, the technology of printing enables us to replicate things with amazingly few errors. LCD/LED displays have made portable computers and big screen TV’s cheap, plentiful, and beautiful in an amazingly short time. Gutenberg’s printing press made books cheap, plentiful, and beautiful and the too did this in a surprisingly short time. For in both cases they involve the replication of simple patterns.

Spreadsheets and the Rule of Four

A little over 20 years ago the Harvard Calculus Consortium sought to remake the calculus curriculum. “We believe that the calculus curriculum needs to be completely re-thought,” began the text by Andrew Gleason and Deborah Hughes Hallett, both of Harvard University. They sought to get “our students to think.” In doing so they proposed “The Rule of Three.” “Our project is based on our belief that these three aspects of calculus—graphical, numerical, analytical—should all be emphasized throughout.” The Rule of Three, today often known today as The Rule of Four with the now addition of verbal, rests at the heart of math education. While the Calculus Consortium’s book may no longer own major market share, it has had a remarkable influence on all Calculus textbooks and indeed on all math textbooks in both K-12 and college. It is a widely shared belief that such multiple-linked representations must be central to 21st century pedagogy. It is clear that students learn in different ways. It is certain that they need to see mathematics from different perspectives.

Spreadsheets are Rule of Four platforms. They are function machines which naturally Bricklin & Frankstonrepresent mathematics graphically, numerically, analytically, and verbally. They show a function as a graph, as a table, as a formula, and we can describe them with text and visuals. They did not start out that way. The first spreadsheet, VisiCalc invented by Bob Frankston and Dan Bricklin was designed to be a visual calculator to automate the accountants’ worksheets. Three years after VisiCalc’s debut in 1979, Mitch Kapor added graphs and tables to create Lotus 123 which brought the IBM PC into every business. And Excel from Microsoft came out for the new Macintosh 2 years later not only simplifying the interface but adding beautiful texts and visuals to spreadsheets. Today, the mature spreadsheet technology is the standard quantitative tool for business worldwide. It is not only available on every major platform, but its format and design are the basis for displaying and interacting with quantity on the Web.

In a spreadsheet we can write a formula, use that formula to create a table of values, and use that table of values to make a wide variety of different graphs and charts. Change the formula and the table and graph changes automatically. Change the table and the graph changes automatically. Spreadsheets are dynamic and highly interactive. They even let you embed variable quantities in text to add units to quantities our dynamic values to verbal descriptions. Once a student builds a model in a spreadsheet, it is naturally a multiple-linked representation that can played with and explored. Spreadsheet models designed with functional thinking as multiple-linked representations are therefore simulations of which students can ask “What if…”

If you use Link Sheets in your classroom, if you believe that every student has a learning style, if you like to have students explore different representations, if you want to get your “students to think” then try using our What if Math spreadsheets or develop your own built on the Rule of Four.

Small Changes

Small changes, seemingly inconsequential acts, can have momentous repercussions. Dead birds set off the environmental movement. An assassin’s bullet protesting an exhausted empire started a world war that brought down the ruling monarchies of Europe. A tax on tea turned into a revolution. Such a small change occurred in America’s classrooms a little over a half century ago. School desks were unscrewed from the floor. That seemingly small change, which on its surface seemed to be just about furniture, precipitated a major reduction in class size and a revolution in expectations of good teaching. Desks bolted to the floor, locking students in straight rows facing a teacher in the front of the classroom, optimized the use of space. My 5th grade Chicago classroom with fixed desks held 51 students in 6 rows with 8 desks per row and three portables. It also defined Miss O’Hearn’s teaching style. My 6th grade suburban classroom with moving desks had 25 students. Desks could be rearranged, students could interact with each other, learning in groups was enabled, and teachers could give students individual attention toward student-centered learning. Small changes can have great effects even in education.

We have the opportunity to make such a small, seemingly inconsequential change that could profoundly transform our schools by allowing students to use the internet on their Common Core Math tests.

We need only change the wording in the test’s directions to allow and not prevent student use of a computer/tablet/smart phone. The tests are designed to be given online already. They give the students digital tools to use to solve some of the problems. What if we simply extended that existing open technology requirement to every question and enable students to use most any available program or website? What if they could use Google search to solve an arithmetic problem, or open Excel, Sheets, Numbers, Wolfram Alpha, Khan Academy, Wikipedia or any website they wanted to find an answer? What if, as the PARCC initials stand for, we are serious about the tests assessing “college and career readiness?” A realistic 21st century college or career problem would quite naturally expect the solver to have internet access. College tests are generally open book and every online course must, by its very nature, allow internet access. So why not really prepare our students for college and career?

The consequences of such a minor change in the assessment directions would be far reaching and revolutionary. Teachers would stop teaching the algorithms and stop giving students arithmetic and algebra algorithm worksheets. Why teach long division if the tests don’t require it? Why spend all of that classroom and homework time on operations on fractions if students won’t be tested on it? Why teach students to factor equations using paper and pencil algorithms if they can get the answer online? This mechanical symbol manipulation that today makes up the bulk of student practice time would simply vanish. Creative experiences using technology to solve math problems would naturally replace it, for those will be the “basic skills” required by the tests. Spreadsheets and other quantitative technologies would replace pencil and paper. Mathematics would become more interesting to students for they would no longer need to ask, “Why am I learning this stuff for when I can solve this problem on my old phone or calculator?” Math classrooms could be filled with creative “What if…” experiences.

Not only would there be more time for authentic problem solving in math, but there would be more time for the other STEM subjects, and more time for the arts, for physical education, for history, for the manual arts, for project and performance oriented activities. So many of us dream of an educational system that is rich and creative, but we are overwhelmed by a system seemingly sluggish to innovate, overwhelming in complexity, and demanding in tradition that it seems to make substantial change all but impossible. Yet there are times and circumstances when small, seemingly inconsequential acts can have monumental impacts. Allowing students to use the Web when they take their Common Core math assessments could well be as revolutionary for students today as unscrewing the desks were in the 1950’s.

Discovery vs. Invention

He repeated it over and over again and each time I cringed. I wanted to shout “No, not pattern recognition, pattern construction.” We were watching the new Cosmos series hosted by Neil deGrasse Tyson. In this week’s episode tracing the evolution of life on earth, Tyson kept talking about pattern recognition as the source of human survival and the reason our species flourished. This vision, though certainly broadly held and nearly canonical as a foundation of science, needs to be questioned. To put it succinctly: Are we pattern recognizers or pattern makers? Do we discover or do we invent? Now while this distinction at first glance may seem to be just rhetoric and definition, it has profound and broad consequences. For it is the distinction that has driven the schism between the sciences and the arts.

The sciences think we are discovering the “real” nature of things. These patterns are in nature for us to discover. The arts deal with patterns too, of course, but clearly arts patterns are constructed. We need only think of the patterns in today’s abstract art which artists clearly construct and other than the general “rules” of art these patterns came out of the minds of the artists. As long as one side thinks they are a voyage of discovery and the other side thinks they are in a workshop inventing, how will we ever see unity between them.
In my book Elegantly Simple I argue through a variety of examples, particularly Maxwell’s Equations, that the patterns of science are invented. Two very, very different visions of nature in Newton’s Mechanics and Maxwell’s Electrodynamics coexisted for more than 50 years as true representations of nature. These patterns were as different as we can imagine. The same is true for General Relativity and Quantum Theory. Oh, we can argue that each of these theories covered different aspects of nature, but do we really expect nature to be schizophrenic and work in entirely different ways depending upon what part we are looking at?

I argued that we construct patterns in science, unique patterns for storing and sharing nature’s information, patterns that can be used to predict new information. We invent these patterns, and these patterns enable us to discover new experience. Maxwell’s Equations were such inventions. They not only defined all of the possible shapes of electromagnetic fields, they showed that changing fields would produce electromagnetic waves which acted exactly light visible light. Since this light we see is just a small part of the possible electromagnetic wavelengths, the Equations predicted that there would be other forms of these waves that we do not see. Heinrich Hertz actually discovered non-visible electromagnetic waves some 20 years after Maxwell published his invention. All patterns are inventions, human inventions. They enable us to discover new experience, and thus broaden our view of nature.

So scientists like artists are patternmakers who value creativity and invention. We invent new patterns. We evaluate patterns esthetically. And we test new patterns by their capability to discover new experience.

Intellectual Flexibility

Yesterday Eric Schmidt on MSNBC’s Morning Joe was asked what Google was looking for in the people they hire. Without hesitation Eric said “intellectual flexibility.” I thought this was an original and useful way of talking about the ability to “think out of the box” to be imaginative, to be creative.

We know what we are looking for in 21st century workers, but every time we talk about our schools producing such workers we talk first about the “basic skills.” Those skills we deem foundational, the skills we think everything rests on literacy and computation have reached canonical status. We do not question their validity, we do not question their importance, we take them as givens.

Today, they absorb all of the oxygen in our classrooms, they are the focus of the school day, and they are supposed to be practiced in every curricular activity as well as in students’ out-of-classroom time. Whatever we may think of their importance and necessity, if we believe that Eric Schmidt and so many other leading business and economic thinkers are right in looking for employees who can think out of the box, then we must ask the central question: “Does learning these basic skills lead to intellectual flexibility?”

I leave you with that question. For consider that Asia countries, who have been much more successful in basic skill development than we seem to have been, are seeking ways to help their students learn to be as creative as American workers!

20th Century Math

For those of you who have been following my blogs, I apologize for taking so long to get out a new one. I have been working a wonderful new project that i am not yet ready to show you, but I promise to do so very soon.

Meanwhile I had a fascinating afternoon yesterday attending a seminar on SketchUp, what they call the 3D program for everyone. Google just sold it to a company called Trimble, a construction company.This program is for designers what WordPress is for bloggers.

As I watched amazing demo after amazing demo, all I could think about was America’s K-12 math program, indeed our entire educational focus. To put it bluntly, it has nothing to do with the real world jobs of the 21st century. For here were architects, designers, engineers, interior decorators, landscape architects, and more using this program in their daily work to both design and to demonstrate. Here was an amazingly large community of people contributing their ideas and their actual work to other users of the program, developing “plug-ins” to do a variety of tasks SketchUp was not designed for.

Are we teaching our students to use technology, to work with sophisticated programs, to be part of a community of users and developers? Are they learning to create, to explore, to learn from each other? Are we preparing our children with the skills they will need for the 21st century? Are we imagining them working with tools like SketchUp? Or are we preparing them for the jobs and work of the 20th century?