Author: Art Bardige

I am a digital learning pioneer who believes that technology can play a great role in enabling every child to learn efficiently, effectively, and economically. What if Math is my latest work and the most exciting I have ever been involved with. I hope you will give it a try.

A Very Good Year

I feel most fortunate when I have a year I get to work on a new great idea in it. This past year has thus been one of good fortune. Some great ideas can appear huge from the start, covering wide swaths of life, and some, at first, can seem small, almost insignificant initially, but on reflection turn out to be consequential and central. This year the great idea was the latter, an idea that appeared tiny and obvious at first glance, but that grows and grows, becoming more valuable with each passing day. Some great ideas seem to be wonderful creative inventions, unique and very, very different, while others are quiet, seemingly so common that we easily miss them. This year’s great idea was again the latter. I would never have expected it to grab hold of our minds and continue to pull us in new directions. And some great ideas seem to spring whole formed, exploding like the Big Bang connecting all sorts of wonderful things together. While others, like this one, play hopscotch in our minds, jumping to and fro with little recognizable pattern to see at first, but eventually linking all sorts of ideas together.

This year’s great idea germinated late last spring as Peter and I were working on what we came to call Classic Story Problems, in particular, the familiar motion, work, and mixture problems found in middle school and high school math textbooks. These problems are painful, that is the only way to describe them, painful to learn and painful to teach. You know the type, “George leaves New York and Martha leave Washington at the same time going at different speeds, where or when will they meet?” Students fight hard to figure them out and many, I would argue most, finally succumb and memorize a formula. Then, of course, when the wording changes, even by a small amount, they are lost again.

In the process of building simple tables in Excel with time in hours in the first (input) column and distance in miles (output) connected by the rule that multiplies time by speed to get distance, we realized that we did not need to fix the input values. Originally, as I was starting to develop spreadsheet lessons, I would let Excel create my input column by putting the first and second input values into the column and then dragging the + to create the column of values. As we got more sophisticated we began to use a rule to create the column, add 1 to the previous number. Well it finally occurred to us that it would be very nice if we could enable students to easily change the input values. Why not make a parameter table with an initial value for the input and an incremental value, and build the input table using a formula and those values? This way we could easily change the start value and the increment value.

Seems such a simple change. But what power. We could enable students to easily solve story problems using a table with discrete values by letting them choose the level of accuracy they needed, developing an understanding of accuracy along with problem solving ability. We could enable them to easily change the domain of any function they choose to graph and quickly and easily control their graphs. We could enable them to zoom in on a particular aspect of a function to study it, or zoom out to picture its form. We could enable them to easily ask “What if…” in a myriad of new ways.

This tiny idea, a parameter control table that enables students to change constants as well as variables has great power. Make the increment smaller and smaller on a quadratic function and you can see that segment of the graph becoming straighter and straighter. Change the initial value and you can watch that straight line change slope as it moves around the parabola. It is almost magical. We don’t need to try to get students to understand limits or secants to picture derivatives, or find a common way to solve story problems. And we are giving them the tools to be flexible problem solving thinkers and explorers.

So as Frank Sinatra sang, “It was a very good year.”

Motion Problems

ATMNE 2018

Are Your Students Ready

Are you and your students ready to learn mathematics and problem solving in a digital way? You will be introduced to functional thinking, our problem solving strategy across the grade level.

Agenda — Google Sheets

Agenda — Excel  Workshop-Agenda-ATMNE-12.7.18

Baseball and Math

If you are a Bostonian by address, birth, or just a connection, you can’t help but be full of pride this morning for your baseball team. The Red Sox were amazing, keeping us up late at night and giving us so much to cheer about in a time otherwise to often dreary. And if you are a math teacher you cannot but wonder why so many of our kids find math irrelevant, uninteresting, and difficult when so much of what they heard and saw as we watched those World Series Games was math, math that helped us understand the game, math that we found fascinating, math that we understood. It is truly amazing how much math there is in our national pastime.

There are numbers and shapes everywhere from the backs of player uniforms to counting of balls, strikes, and outs, numbers for the innings and even numbers for the game, the series, the postseason, and the regular season. There are addition totals shown on the scoreboard for hits and runs along with the count for each inning. There is the shape of the diamond, measurements to the bases, the mound, and the various walls, the square bases and pentagonal plate. This is only the beginning because baseball is all about statistics. Batting averages and earn run averages, hits off right handers and hits off left handlers, even stats about which team will win the series after being down by 2 games. Pitchers throw balls that curve in left or right, go straight and rise or go straight and fall. The speed of the ball is posted for each pitch along with its position in the rectangular strike zone..

The game is full of algebra and not just arithmetic, geometry, or simple statistics. The amazing TV graphics shows the parabolic path of the ball when a player hits a homerun where we think quadratically as we wonder about height vs. distance, or time to land. It is about how long it takes for a ball to be thrown across the infield compared to how fast a runner can get to first base. It is about where on the bat the ball hit and the shape of the swing for maximum momentum transfer. It is today about building models for managing teams, salaries, salary caps, offence vs. defence, the number of starting pitchers you must acquire vs. the number of relievers and even the specialized relievers, the closers. And it is about the wealth of statistics and statistical analysis rivaling the stock market in richness and complexity.

I have only begun to scratch the surface of math and our national pastime. For as I watch and listen to the games, I hear and see math everywhere. How could it then be that our school math is so out of sync with this sport math we learn and enjoy so much? That is the question I leave you with today as I cheer Go Sox as loud as I can and think about how much mathematics helped me enjoy the game.

Here is a sophisticated baseball What if Math Lab you may want to try: Hit Streak.

ATMIN 2018

Spreadsheets Across the Curriculum

Take a Tour with us to discover an exciting new way to imagine mathematics, solve problems, learning coding fundamentals, and use spreadsheets. Workshop-Agenda-ATMIM-10.27.18

A Book or a Course?

I have long loved Maxwell’s Equations as the epitome of beauty in physics and as the source of inspiration for my teaching. But though the equations are beautiful and even familiar, very few people understand them. So, when I came across this paper by the great physicist Freeman Dyson called “Why is Maxwell’s Theory too hard to understand?” I could not resist reading it. His telling of the Maxwell Equations’ story led me in a new direction not just in thinking not about physics but about education in the digital age. It led me to ask: “What’s the difference between a book and a course today?” and to further ask, “What will they look like in the future?” Before you help me tackle those questions, I suggest you look at the story Dyson tells about Maxwell’s great work.

In the year 1865, James Clerk Maxwell published his paper “A dynamical theory of the electromagnetic field” in the Philosophical Transactions of the Royal Society. He was then thirty-four years old. We, with the advantage of hindsight, can see clearly that Maxwell’s paper was the most important event of the nineteenth century in the history of the physical sciences. If we include the biological sciences as well as the physical sciences, Maxwell’s paper was second only to Darwin’s “Origin of Species”. But the importance of Maxwell’s work was not obvious to his contemporaries. For more than twenty years, his theory of electromagnetism was largely ignored. Physicists found it hard to understand because the equations were complicated. Mathematicians found it hard to understand because Maxwell used physical language to explain it. It was regarded as an obscure speculation without much experimental evidence to support it. The physicist Michael Pupin in his autobiography “From Immigrant to Inventor” describes how he travelled from America to Europe in 1883 in search of somebody who understood Maxwell. He set out to learn the Maxwell theory like a knight in quest of the Holy Grail.

Maxwell’s Equations in the elegant form found on college student tee shirts and physics classroom posters were not the the way Maxwell wrote them down in 1865. He did not have the benefit of the power or the simplicity of vector calculus. And the idea of fields as environments was then brand new and hard to grasp. But of greater interest to me, beyond the significance and power of symbol systems which have been well known, was Dyson’s recognition that for many, maybe most new ideas, just the process of writing them down for someone to read in paper or book form is not enough. We have to be taught. We have to learn them. Dyson continues.

Pupin went first to Cambridge and enrolled as a student, hoping to learn the theory from Maxwell himself. He did not know that Maxwell had died four years earlier. After learning that Maxwell was dead, he stayed on in Cambridge and was assigned to a college tutor. But his tutor knew less about the Maxwell theory than he did, and was only interested in training him to solve mathematical tripos problems. He was amazed to discover, as he says, “how few were the physicists who had caught the meaning of the theory, even twenty years after it was stated by Maxwell in 1865”. Finally he escaped from Cambridge to Berlin and enrolled as a student with Hermann von Helmholtz. Helmholtz understood the theory and taught Pupin what he knew. Pupin returned to New York, became a professor at Columbia University, and taught the successive generations of students who subsequently spread the gospel of Maxwell all over America.

I highly recommend you read the rest of Dyson’s paper, but for now, I want to consider the question it has prompted. As books have become more interactive, as textbooks become linked to fancy interactive websites, as courses become MOOCs wrested from the tyranny of a 15 week calendar the physical classroom and the format of live teacher; we now see both methods of education in a wide variety of shapes and sizes. So, today, “What is the difference?” “Are we trying to write and publish (perhaps self-publish) a book, or are we trying to teach an online course?”

For me, these questions are not philosophical; they are real. I am in the process of putting together a book/course on the future of education. Since it is about education in this new digital age, the form and format are just as important as the ideas. So I ask your help.

As we learn from the Maxwell’s Equations story, courses help people digest and learn new ideas that simply reading them in a traditional paper or book form does not. The ideas in my vision of the future of education are radical and no doubt in need of something that looks more like a course, but certainly not a 20th century course and even less like a 19th century book. “So what does it look like, I wonder?” “What does the merger of books and courses make?”