The Bit

February 21, 2018

The key to the digital age is also the key to learning algebra. Despite what many of us may believe, our digital age did not began with the microprocessor, or the personal computer, or even the iPhone; it began with a single amazingly simple idea by a quiet man who… Read more

Art Bardige


February 12, 2018

Teaching done right has always been a hard job, but it is now substantially harder. Talk to any teacher and they will tell you that they are overwhelmed. Blame it on kids more distracted, on parents more demanding, on the misery of an over reliance on testing that saps creativity… Read more

Art Bardige

Learning to Swim

January 25, 2018

The University of Chicago is not known for its athletics, so when I entered it as a first-year student I was very surprised that I had to take and pass a swimming test. Despite my parent’s best efforts, I had never learned to swim, and thus had to take a… Read more

Art Bardige

Real Feedback vs Artificial Feedback

January 8, 2018

Math Blaster was the biggest hit educational product in the 1980’s, the first decade of the personal computer age. Flying saucer like objects would vaporize before your eyes when you solved a simple math question. It thus gave you immediate feedback and like pinball, it kept score. My friend Jan… Read more

Art Bardige

The Challenge of New

December 21, 2017

One hundred years ago my father at age 9 entered America. He had traveled from his birthplace in a town in what is now Ukraine across the vast expanse of Siberia on the Trans-Siberian railroad to its eastern extreme at Vladivostok, from there by ship to Kyoto, Japan and then… Read more

Art Bardige

The Democratization of Knowledge

December 13, 2017

On this 10th anniversary of the iPhone it is worth remembering that this invention, as world-changing as it was, will not be deemed the most important one of that decade. It will, in the long thrust of history, take second place to an event that truly and profoundly changes the… Read more

Art Bardige

A Tour of Spreadsheet Mathematics

November 20, 2017

Take a tour to visualize and experience what mathematics of the Digital Age could look like to your students. Open in Excel Open in Google Sheets Tweet Read more

Ryan McQuade

“Just try it on!”

July 14, 2017

Spanglish is one of those movies that grows on you. A coming to America story filled with themes that move us: a dedicated and resourceful woman, a dysfunctional but caring family, a highly successful artist, and of course love. It has many scenes that touch us deeply. One of those,… Read more

Art Bardige

Collaboration is Cheating?

June 29, 2017

One of the four C’s, perhaps for many the most important 21st century skill, is considered in our schools, cheating. Students caught talking to each other during exams are either yelled at or disciplined for cheating. Homework is supposed to be an individual activity and students are punished for cheating… Read more

Art Bardige

Minkowski’s Connections

June 23, 2017

I still feel it months later, the thrill and awe I knew from finding an answer to a question I have long been troubled by. I was reading a delightful book on physics by Richard Muller called, Now, in which mixing physics and history, he explained time and in that… Read more

Art Bardige

The Rule of 72

June 19, 2017

The Genius Behind Accounting Shortcut? It Wasn’t Einstein The Rule of 72 is a nifty shortcut for estimating investment returns; first published mention was in 15th century Great article in last weeks Wall Street Journal on the Rule of 72 by By Jo Craven McGinty. Learn more about the Rule… Read more

Art Bardige


June 15, 2017

I like to hang out in the Harvard Graduate School of Education library. It has a good vibe, is usually full of students focused on my favorite topic, and is set up to enable technology as you well might expect. Every student has their own laptop. Lots of tables have… Read more

Art Bardige

“Algebra before Acne”

June 14, 2017

As I was again reading the Common Core Standards, I was struck by their introduction of variables in grade 6. Jim, I could not help but think of you, my old dear friend, and your wonderful command, “Algebra before acne.” Kaput envisioned algebra and algebraic reasoning as fundamental mathematical ideas… Read more

Art Bardige


June 6, 2017

Empathy is an odd idea to discuss in math or even in STEM/STEAM education. It is usually thought of as an issue in psychology or sociology, perhaps in the humanities, a topic for English or history classes to consider in school. Yet, it is the first step in the Design… Read more

Art Bardige

What if Math 2.0

May 31, 2017

Over the past year, we have been working to combine our spreadsheet math lessons (downloaded more than 20,000 times) into digital age problem solving Courses that promote readiness for school transitions and prepare students for STEM careers. We developed a new model for creatively solving digital age problems that we… Read more

Art Bardige

The First Graph

April 5, 2017

This picture was first published in 1638! It is from Galileo’s great work Two New Sciences, that he smuggled out of his home imprisonment in Florence, when he was 72 years old and effectively blind. Though famous for his telescope and the first images of the surface of the moon,… Read more

Art Bardige

Tradition, Tradition

October 28, 2016

As part of the process of designing and developing new Labs, I visit math content sites all the time to help me think about the kinds of questions to ask and the way to explain or represent a concept. I am constantly struck by how talkative these sites are. As… Read more

Art Bardige


October 26, 2016

“Today, it seems as if nearly everyone agrees that high school mathematics needs to change. For far too long high school mathematics has not worked for far too many students: too many students leave high school unprepared for college or a career, particularly a STEM career; too many students do… Read more

Art Bardige

Back- to-School – add 10%

September 8, 2016

It was forty years ago this September that I started my career as a high school mathematics teacher, a career that spanned 36+ continuous years. I continued some teaching and other classroom work over the last few years, but this will be the first September I do not find myself… Read more

Peter Mili

What Algebra?

August 10, 2016

Each summer, as schools get ready for a new school year, the question returns, “Should we be teaching algebra to our children?” it seems to have been started by Andrew Hacker who has tried to argue and continues to argue that algebra is both difficult and an unnecessary burden for… Read more

Art Bardige


July 20, 2016

The word seems so benign. Yet it has become the goto word in education. School superintendents, even the best and most advanced of them, use it all the time. “We want each of our students to reach mastery in each of the standards.” To master something is to be in… Read more

Art Bardige

The End

July 18, 2016

Despite the many attempts to codify the creative process, it is as surprisingly individualistic as it is human. John Irving, author of iconic works like The Cider House Rules, describes his creative process as writing the conclusion before the beginning. He spends a year or more developing a story, the… Read more

Art Bardige

Personalized Learning

July 7, 2016

These two words have caught the imagination of educators and parents. They were designed to be the frames for talking about the value of digital learning. They were to replace the bland “individualized learning.” They were meant to symbolize a focus on the student, student-centered, and on that great American… Read more

Art Bardige

Function Machines

June 29, 2016

I do not know who, when, or where this iconic mathematical representation was developed. It is, however, one of the most powerful and ubiquitous of all mathematical images, and I think the most important. It is taught to 2nd graders and used by STEAM professionals. It is called a function… Read more

Art Bardige

Stand and Deliver

June 20, 2016

It was an appropriate title for the movie about Jaime Escalante and it is an appropriate title for the role that teachers continue to play. We all too frequently see our role in both K-12 and in college as an actor standing and delivering. As problematic as that vision may… Read more

Art Bardige


June 16, 2016

As I watched a young woman the other day learning to ride her bike, zigzagging down the street, desperately trying to keep her balance, I thought of the Wright brothers. They owned a bicycle shop where they made and taught people to ride the then relatively new form of transportation.… Read more

Art Bardige

Over the Rainbow

June 13, 2016

Over the Rainbow by Harold Arlen and E. Y. Harburg is considered the greatest song of the century and the greatest song in a movie of all time, and it is my very favorite song. I never tire of listening to it. E. Y. ”Yip” Harburg wrote not only all… Read more

Art Bardige

A Maker of Patterns

April 25, 2016

G.H. Hardy, one of the greatest mathematicians of the 20th century wrote this: A mathematician, like a painter or a poet, is a maker of patterns. The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a… Read more

Art Bardige

Learning Math as a Creative Experience

April 20, 2016

As mathematics takes an increasing role in work and life, creativity must become central to its learning, because: 1) creativity and creative problem solving are essential 21st century skills, 2) creativity drives engagement and enjoyment, and 3) creativity builds understanding. 1. 21st Century Skills: In survey after survey, business and… Read more

Art Bardige

What if…

March 16, 2016

“Rather than ask why our students fail to measure up, this film asks us to reconsider the greater purpose of education. What if our education system valued personal growth over test scores? Put inquiry over mimicry? Encouraged passion over rankings? What if we decided that the higher aim of school… Read more

Art Bardige

The Great American Probability Machine

October 22, 2015

This program started my career in digital learning. I bought my first computer, an Apple II in February 1978 on their first anniversary. I talked my wife into letting me play with a computer for doing my checkbook and turning our houselights on and off. Though I was an educator… Read more

Art Bardige

Lynn Steen

June 24, 2015

My fortune cookie today read, “If you’re happy, you’re successful.” Usually for me that is true, but not today. For during that same lunch my iPhone told me that Lynn Steen had died. I never had the good fortune to meet him in person or to even talk with him,… Read more

Art Bardige

The Hawthorne Effect

June 16, 2015

To make its workers more productive, the Western Electric Company, makers of phones and other parts for the Bell Telephone System, conducted one of the great scientific experiments of all time. The researchers increased the brightness of the lighting at a plant in Hawthorne, Illinois just outside of Chicago, incrementally,… Read more

Art Bardige

Opportunity for Creativity

June 3, 2015

I just looked at a wonderful short video by Sir Ken Robinson on creativity at in which he describes creativity as a process that produces something original that has value. At What if Math we seek to make learning a creative experience, a process that enables every student to… Read more

Art Bardige

Another Sunday Ritual Soon Gone

June 1, 2015

When I was a kid, Sundays in the summer were car washing days. The stores were closed. The roads were generally quiet. And we took out the hose and the pail, filled them with water and dishwashing soap then rubbed, scrubbed, and waxed the family car…or later our own car…beautiful… Read more

Art Bardige

Tradition, Tradition

May 20, 2015

Today, I attended an ancient ceremony. It is called “Hooding”. An elaborate and beautiful hood is given to students who have completed their scholarship and are ready to receive a doctoral degree. The Hooding Ceremony at Lesley University today with its rich pageant and sweet music took me back to… Read more

Art Bardige

The Magic Wand

April 8, 2015

What if I could give you a magic wand to wave over our educational system and make it fulfill our dreams for our children? What would you have it do? I find this question stumps most people. We all know education in America is far from what we either want… Read more

Art Bardige

209 to 7

February 10, 2015

If a mathematician were asked what these two numbers had in common, she might wonder if they were both primes. They are not. A gambler might consider them lucky numbers because one of the prime factors of 209 is 11 (as you could find out by asking google) and 11… Read more

Art Bardige

Headmath vs. Handmath

February 9, 2015

There are really two kinds of mathematics we do every day. I like to call one headmath and the other handmath, one is the mental arithmetic and problem solving we all have to do and the other is the math on paper or more likely today, if you are not… Read more

Art Bardige

Welcome to What if Math

January 19, 2015

Three years ago I read a wonderful book by Keith Devlin called The Man of Numbers. It told the story of Leonardo of Pisa who was the first to convert Arabic arithmetic and algebra for European use. Devlin told Leonardo’s story and he described the process by which Leonardo’s book Liber abbaci (The… Read more

Art Bardige

Learning as a Creative Experience

January 1, 2015

We are in a time of dramatic, some would say, revolutionary change in education, “challenging” as Sir Ken Robinson says, “what we take for granted.” His How Schools Kill Creativity, the most watched Ted Talk of all time, shows we hunger for learning as a creative experience. Yet we continue to… Read more

Art Bardige

Spreadsheets and the Rule of Four

October 29, 2014

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… Read more

Art Bardige

Small Changes

October 7, 2014

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… Read more

Art Bardige