“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… Continue

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… Continue

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… Continue

Art Bardige

9 Reasons for Using Spreadsheets in Schools

July 22, 2021

Spreadsheets are equity platforms available to all students at no cost. They can give every student a fresh start in math. Spreadsheets from Microsoft, Google,… Continue

Art Bardige

A Book or a Course?

September 13, 2018

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… Continue

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.… Continue

Art Bardige

A Very Good Year

March 8, 2019

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… Continue

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… Continue

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.… Continue

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… Continue

Art Bardige

Baseball and Math

October 29, 2018

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… Continue

Art Bardige

Black Hole

April 11, 2019

This picture made the front page of the New York Times this morning. It is not very often that a science experiment makes the headlines… Continue

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… Continue

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… Continue

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… Continue

Art Bardige

Connections not Collections

June 2, 2020

Museums were one of the great inventions of the 19th century. People loved collecting, collecting all sorts, and often funded buildings to display those collections… Continue

Art Bardige


June 11, 2018

The words curious and curiosity do not appear in the Mathematics Common Core Standards document, yet they are arguably the most important words in mathematics… Continue

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… Continue

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… Continue

Art Bardige


April 7, 2020

Welcome to the first of our new Explorations. Joining the work of What if Math and Education Resources Consortium (ERC), Explorations provide students with a… Continue

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… Continue

Art Bardige

Functional Thinking

July 5, 2018

We call our problem-solving process, functional thinking. When we apply functional thinking to digital age problem solving, we find a few fundamental models give us… Continue

Art Bardige

Functional Thinking

April 11, 2017

We call our problem solving process, functional thinking. When we apply functional thinking to problem solving in the digital age, we find that a few… Continue

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… Continue

Art Bardige

Independent Learners

June 4, 2020

Perhaps the most profound and lasting effect of the Covid Virus pandemic on our economy will be in the change in the way people work.… Continue

Art Bardige

Is the Textbook Dead?

July 24, 2018

It caught my eye, this headline/story posted on EdWeek recently. Seems there was a panel at a conference that was supposed to debate what they… Continue

Art Bardige

Labs Posts

July 15, 2021


Ryan McQuade

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… Continue

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… Continue

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… Continue

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… Continue

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… Continue

Art Bardige

Math as a Laboratory Science

April 17, 2018

Math is not only the last letter in STEM or STEAM, it is the only one that we do not picture as experimental. We don’t… Continue

Art Bardige

Math is Hard

April 10, 2019

In a magazine published for college trustees, a recent short article captured the latest statistics from the ACT and SAT tests. The downward trend was… Continue

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.… Continue

Art Bardige

My Favorite Teacher

February 26, 2020

Walt Hunter was a quiet man, slight and balding with a Great Plains accent. You might mark him as a teacher, but likely not the… Continue

Art Bardige

My Mentors

March 27, 2020

Recently, in a history of physics magazine, I came across this picture and a short story on Harvard Project Physics, and I thought about the… Continue

Art Bardige

My New Book

March 25, 2020

Our education model is broken. Despite the economic promise of and documented need for a bachelor’s degree: graduation rates are stagnant, achievement gaps are widening,… Continue

Art Bardige

Opportunity for Creativity

June 3, 2015

I just looked at a wonderful short video by Sir Ken Robinson on creativity at https://youtu.be/63NTB7oObtw in which he describes creativity as a process that… Continue

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… Continue

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… Continue

Art Bardige

Personalizing Learning

June 14, 2018

Envisioning technology that reinvents our schools not automates them should, I believe, be our goal and our dream for personalizing learning. Continue

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… Continue

Art Bardige

Revolutionary Math

March 1, 2018

Cape Cod in the winter is one of those marvelous places filled with interesting shops and people waiting in the quiet winter time for the… Continue

Art Bardige

Rows and Columns

May 24, 2018

This picture from a recent blog post sends shivers down my spine. It is our picture of a “modern” classroom with the desks lined up… Continue

Art Bardige

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… Continue

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… Continue

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… Continue

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.… Continue

Art Bardige


August 6, 2022

We are pleased to be featured in Continue

Art Bardige

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… Continue

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… Continue

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… Continue

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… Continue

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… Continue

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… Continue

Art Bardige

The Hardest Question

June 28, 2018

What is the hardest question a teacher has to answer? As teachers, especially math teachers, we face this most painful question all too often, rarely… Continue

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… Continue

Art Bardige

The Los Alamos Primer

June 5, 2018

Or how to build an atomic bomb. One of the best curriculum ideas I ever had was to use this book as the text for… Continue

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… Continue

Art Bardige

The Math Guys

January 24, 2020

Our good friend Larry Myatt, one of the great thinkers and leaders on the future of education, recently sent out a New Years greeting that… Continue

Art Bardige

The Problem with MOOCs

July 9, 2018

When MOOCs were the rage in higher education, I asked my friend David Kaiser, a physicist and professor of the history of science at MIT,… Continue

Art Bardige

The Science of Patterns

October 23, 2015

As Lynn Steen said, mathematics is the study of patterns, the “science of patterns”. We focus and want students to focus on patterns, on seeing… Continue

Art Bardige

The Summer Challenge Problem of the Week

May 1, 2016

“How do you keep students engaged in math while they are having fun?” We think we have come up with the perfect solution for teachers… Continue

Art Bardige

The Tour

November 20, 2017

Take this tour of functional thinking applied to the key concepts of mathematics. Visualize and experience the power of the spreadsheet to unify and simplify… Continue

Ryan McQuade

This is Why I Love Graphs!

March 2, 2018

This graph appeared on one of my favorite websites – Statista. Given the “breaking news” of the day, that the President wants to impose new… Continue

Art Bardige

Touching the Sun

August 13, 2018

The Parker Solar Probe was launched yesterday to study the sun. Sixty years ago, Eugene Parker launched my scientific career. A young physics research scientist… Continue

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… Continue

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… Continue

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… Continue

Art Bardige

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… Continue

Art Bardige

What if?

September 11, 2015

My favorite question is, “Why?” (And my favorite answer is, “Because.”) But not far behind is the question, “What if?” Read about us on my… Continue

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… Continue

Art Bardige

Why do we have to Learn the Quadratic Formula?

November 4, 2020

Mastering the quadratic formula has long been the culmination of the high school algebra courses, the capstone of “Algebra I and Algebra 2”. We endeavor… Continue

Art Bardige