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… Read more

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.… Read more

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… Read more

Art Bardige

Explorations

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… Read more

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… Read more

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,… Read more

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… Read more

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… Read more

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… Read more

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… Read more

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… Read more

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… Read more

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… Read more

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… Read more

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… Read more

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,… Read more

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… Read more

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. Read more

Art Bardige

Curiosity

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… Read more

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… Read more

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… Read more

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… Read more

Art Bardige

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… Read more

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… Read more

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… Read more

Art Bardige

Exhausted

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… 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… 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… 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… 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… Read more

Art Bardige

“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… 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… 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.… Read more

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… Read more

Art Bardige

Cloisters

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… 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… Read more

Art Bardige

Empathy

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… 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… Read more

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… 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… 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… Read more

Art Bardige

Change

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… 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.… Read more

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… Read more

Art Bardige

Mastery

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… 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… 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… 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… 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.… Read more

Art Bardige

Balance

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… 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… Read more

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… 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.… 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… 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… 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… 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… 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… 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 https://youtu.be/63NTB7oObtw in which he describes creativity as a process that… 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… 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… 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… 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… 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… 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… 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… 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… 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… Read more

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 to prepare our students for it by teaching them arithmetic to do its operations, and algebra to solve linear equations, graph linear functions, factor special… Read more

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. Companies large and small have moved much of their workforce from office to home for the duration of this epidemic. Though some view this move… Read more

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 for others to enjoy and admire. The museum craze ran from animals to furniture, stamps to tools, swords to seeds, fossils to bottle caps, rocks… Read more

Explorations

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 comprehensive learning experience with fascinating questions, interesting places to go, and powerful tools to use to solve problems. Our first Exploration is Sand and Stars… Read more

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 effect that this project, and these three people in particular, have had on my view of education. We all have mentors and I have the… Read more

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, and costs are bankrupting our kids. While Digital-technology has transformed work, our schools retain their 19th century form and function defined by medieval paper-technology. The… Read more

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 dynamic personality with a certain brashness that you would think of as a “favorite” teacher. He did not convince me to change my major from… Read more

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 featured What if Math. We are partnering with Larry on a number of exciting projects in 2020 that we will keep you informed of. Meanwhile,… Read more

NCTM Nashville October 4, 2019

June 17, 2019

Exploring Mathematical Concepts with Dynamically Linked Multiple Representations Using Spreadsheets Fri, 10/4: 11:00 AM – 12:00 PM 1448 Session 60 Minutes Music City Center Room: 207 AB Description How can we build multiple representations into teaching and learning activities? This can be a challenging task, but technology provides the means… Read more

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 in a great newspaper, particularly above the fold. It is a thrilling discovery. “The First Image of a Black Hole.” The concept of black holes,… Read more

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 notable especially in math. For example, “Forty-nine percent of the class of 2018 that took the SAT (2 million students) showed a strong chance of… Read more

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 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… Read more

ATMNE 2018

October 31, 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 Read more

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 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… Read more

ATMIN 2018

October 24, 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 Read more

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 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… Read more

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 at the University of Chicago, Parker volunteered to be a mentor to encourage science-promise high school students to develop their own science projects. I had… Read more

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 obviously thought would be an attention grabbing, contentious, and controversial topic. Their conclusion: NO! All I can say is: “You have got to be kidding!”… Read more

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, when he was going to do a MOOC. Dave has won teaching awards at MIT and writes brilliant books on the history of physics. Who… Read more

Getting Started

July 6, 2018

Our goal is to enable every student to use mathematics to become a better creative problem solver. In this digital age, awash in data and technology tools for building and using models to analyze and solve problems, we believe a new approach to problem solving is required. Our approach, based… Read more

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 the tools to creatively solve quantitative problems. Think of functions as LEGOs, add columns using new rules, use outputs as new inputs, combine simple functions… Read more

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 do we have a good answer to it, and even more rarely does our answer enlighten students. The question is less a query and more… Read more

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. Read more

Curiosity

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 education. If there is any single habit of mind or critical skill I want our students to learn, don’t you agree, it is to be… Read more

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 an intro to physics course. It was written in 1942/3 by Robert Serber who had been tasked by Robert Oppenheimer to teach a course to… Read more

MassMate Symposium

May 29, 2018

2018 MassMATE Symposium Massachusetts Mathematics Association of Teacher Educators May 30, 2018 Stonehill College Leading Change with Digital Problem Solving Workshop Agenda MassMate 5.30.18. Read more

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 as they have been for 200 years in rows and columns, students looking at the backs of the heads of other students and the back… Read more

Magic Number (Infinite Series)

May 7, 2018

What if you made a series of unit fractions with the odd numbers as their denominator, added the first two together, subtracted the next one, added the next one and so on. And finally multiplied that number by 4? What number would you get? Read more

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 imagine students learning science without doing experiments. We don’t imagine them learning technology without writing code, or learning engineering without building models, or learning art… Read more

Framingham What If Workshop

April 12, 2018

Excel Version Google Version Read more

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 tariffs on steel imports, it is fascinating to see from this graph the countries most affected, certainly not the ones we might have thought. It… Read more

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 soon to come crowds. It was on one of those pretend spring is here days in February that we went to visit a dear friend… Read more

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

Exhausted

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

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

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

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

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

National Numeracy Network Conference

November 27, 2017

November 17-19, 2017, Barnard College, NY. Scientific Reasoning and QR in the Digital Age Introducing our new Tour of mathematics and problem solving in the digital age. Read more

The Tour

November 20, 2017

Functional Thinking, the problem solving process we follow in our Labs, enables students to see and learn a powerful common form and format across much of K-12 mathematics. Take this Tour of a collection of typical Labs to visualize and experience what mathematics of the Digital Age could look like… Read more

ATMNE Conference 2017, Nov 2, 2017

October 24, 2017

New Marlboro, MA, Nov 2, 2017, Session #11 10:45 – 11:45 Sudbury Grades 6 – 12 Peter Mili & Art Bardige Problem Solving in the Digital Age Known as Word problems, Story Problems, or Application Problems, making sense of and solving these problems is challenge for many students and teachers.… Read more

Inverse Variation

October 11, 2017

The variables in most of the functions we are used to working with vary directly, as one goes up the other goes up. What do functions that vary inversely, when one variable goes up the other goes down, look and act like? Read more

“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

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

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

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

Cloisters

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

“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

Empathy

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

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

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 fundamental models give us the tools to creatively solve quantitative problems. Think of functions as LEGOs, add columns using new rules, use outputs as new… Read more

GDP

April 6, 2017

The GDP or Gross Domestic Product of a country is one measure of its wealth. What can this data tell us about the U.S. economy? Are we getting wealthier? Are each of us really wealthier after inflation? What else does this data tell you? Read more

Parametric Equations

April 5, 2017

Parametric equations are powerful tools to model projectile motions and to graph things that are not functions like circle or ellipses. The x and y coordinates are defined as two separate functions with a common independent variable often labelled “t”. Read more

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

Solar System

April 4, 2017

When I was young I loved to play with planetary data, to explore their patterns, to learn more about astronomy, and to deal with large numbers. Spreadsheets make it much easier to study the solar system and to find relationships between the planets that are fascinating and unexpected. In the… Read more

Coin Problems

April 3, 2017

Suppose Briley has 10 coins in quarters and dimes and has a total of $1.45. How many of each coin does she have? Read more

Spreadsheets 101

January 29, 2017

Though powerful enough to run a business, spreadsheets are simple enough for everyone to quickly and easily learn to use. Here we take you through the things you have to know about spreadsheets and functional thinking. Read more

Work Problems

October 28, 2016

Suppose Tom can paint the entire fence in twelve hours, and Huck takes eight hours. How long would it take the two of them together to paint the fence? Read more

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

Massachusetts STEM Summit 2016

October 26, 2016

Join us on Tuesday November 1 at the DCU in Worcester at 2:55 when we talk about What if Math and why it is so important to make Learning Math as a Creative Experience. Read more

Change

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

What is x?

October 20, 2016

What is “x”? Or how do we represent variables and functions on spreadsheets? Read more

Solving Equations Digitally

October 20, 2016

This Lab introduces a method for solving or estimating the solution to an equation digitally that can be applied to many types of equations. This Functional Thinking approach reduces the need to remember a variety of rules and procedures. It is 1 of 3 Labs on this topic. Read more

Napoleon’s Pyramid

October 19, 2016

When Napoleon conquered Egypt in 1798 he went to the Great Pyramid of Giza. While his men climbed to its peak, he figured out that “There is stone enough to build a wall 3 meters high and 1/3 meter thick around the whole of France.” Was Napoleon right? How and… Read more

ATMNE 2016

October 4, 2016

Friday October 21 Radisson Hotel, Manchester NH Room: Dartmouth Join us at ATMNE 2016 at 10:30 to talk about: Learning Math as a Creative Experience with Spreadsheets Read more

Mixture Problems

September 9, 2016

How many gallons of a 70% alcohol solution must be added to 30 gallons of a 10% alcohol solution in order to produce a new mixture, a 20% alcohol solution? Read more

Lissajous Figures

September 9, 2016

We often see Lissajous figures in old sci-fi movies because they are so cool. As you play with them I think you will find them as fascinating as I do. They created those figures by graphing points as a function of a third variable (the parameter). We can do the… Read more

Lease or Buy

September 9, 2016

If you are in the market for a new car, you often have a decision to make. Do you want to buy the car or lease it. I have, over the years, developed a simple model to help me decide. I call it my rule of 6. Read more

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

Inverse of Linear Functions

August 30, 2016

What does a linear function look like when we interchange the inputs and outputs, that is make the x-axis the y-axis and vice versa. Read more

Introducing Subtraction

August 30, 2016

What if you built a rule that would enable you to count backwards? How would it be the opposite of addition? What would happen if it goes past zero? Read more

Graphing

August 30, 2016

Graphs as we know them were first invented by Galileo. They are powerful images of functions. I will introduce you to graphs by letting you graph different functions represented by a table. Fill in each table visualize its graph. Read more

Exponential Functions

August 25, 2016

What if you created a function where the exponent is a variable? As you might expect, this would be called an exponential function. When you hear someone talk about “growing exponentially,” they are talking about an exponential function. Exponential functions have some very interesting patterns. 42.3857326-71.1266351 Read more

Equivalent Fractions

August 25, 2016

Fractions are ratios that’s why we call them rational numbers (ratio numbers). If you think about fractions as ratios, how does this help you to understand them? Read more

Motion Problems

August 25, 2016

George is in New York and Martha is in Washington. They leave at the same time and follow the same road to meet each other on the way. The distance between New York and Washington is 229 miles. George has a fast horse and averages 16 miles/hr. Martha has a… Read more

Birthday

August 25, 2016

In a class of 23 students, the chances are fifty-fifty that two of them will have the same birthday. Now that may sound impossible since there are 365 different possibilities, but we can use probability to see that it is true. Read more

Air Pollution

August 24, 2016

This chart recently came out from the American Lung Association about air pollution in the United States. How would you present this data to Congress to get them to deal with this serious problem? Read more

Absolute Value Polynomials

August 24, 2016

What would you guess the absolute value of a polynomial function would look like? Try some here. I think you will be very surprised. Can you explain what you see? Read more

Absolute Value Functions

August 24, 2016

The absolute value (ABS) function is the positive value of a number or quantity. Its graph has a very peculiar shape, a V. Since absolute values are always positive we can think of them as distances on a numberline. Read more

Powers of Ten

August 18, 2016

Picturing exponential growth, powers of 10, can be hard for any of us to imagine. The spreadsheet has the flexibility to enable us to explore the powers of 10 and to get a visual image of them. We can see the difference in shape between odd and even powers, and get a sense… Read more

Make a Hundreds Table

August 13, 2016

This is the graduation exercise for the basic use of spreadsheets. We combine rules and addressing to have students build their own hundreds table in the fewest steps. There are many ways of doing this and students can be as creative and exploratory as they want. Nor should they feel… Read more

Build a Times Table

August 13, 2016

Students are tasked to build a times table in just two steps. They have to learn to use absolute as well as relative addressing to do, and the Lab takes them through using them. We encourage students to work with just a row or a column rather than with the… Read more

Gas Money

August 11, 2016

If you commute to work or school, you understand how expensive driving can be. Choose a car in this interactive spreadsheet as it breaks down the cost of driving, using MPG, distance traveled, and cost of gas. Read more

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

Moore’s Law

August 4, 2016

It was one of the most amazing visions of the future ever made. In 1965 Gordon Moore, one of the founders of Intel, proposed a law governing the future of computing. He originally proposed that the number of components on a chip would double every year. Later he revised that… Read more

Rows and Columns

July 28, 2016

We use the hundreds table to introduce rows and columns and focus students on seeing the patterns in these tables. Again and again we go back to making rules and using rules to ask and answer questions. For example, what rule would you make to fill in a column on… Read more

Addressing

July 28, 2016

We have been using cell addresses informally until now, but now we can be more formal and explicit. Different spreadsheets have different types of address bars, but all use the same format, letters for columns and numbers for rows with letters first and numbers second. We introduce this on the… Read more

Probability: Flipping a Coin

July 28, 2016

That probability is multiplicative is not an easy concept for many of us. Using the spreadsheet with our ability to make tables and to cut and paste can make this important concept much more transparent. We look forward to your thoughts about what we have done. Read more

Number Lines

July 27, 2016

Number Lines introduce students to functional thinking and the use of formulas in spreadsheets. For younger students we call these formulas “rules” and ask students to build a variety of number lines using rules. For example they can build a whole number line by creating a rule that adds 1… Read more

Introducing Spreadsheets

July 26, 2016

In introducing Spreadsheets we want you to learn to build a numberline by using a rule (a formula). We begin with a simple rule that you can copy and paste into the entire numberline. Then we want to add an input from another cell into the rule to give you a chance to change… Read more

Fibonacci’s Sequence

July 21, 2016

Fibonacci, the nickname given the great medieval mathematician Leonardo of Pisa, is connected in most of our minds with the Fibonacci sequence. Spreadsheets make wonderful tools for creating such sequences. This one is amazingly simple. Just select a cell, any cell and write a formula in that cell that adds… Read more

Mastery

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

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

Triangular Numbers

July 14, 2016

1, 3, 6, 10… are called the triangular numbers because they can be stacked up to form a triangle. They are very interesting numbers, and they form a very interesting pattern when graphed. Can you guess the next triangular number? Can you guess the shape of the graph of the… Read more

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

String Diagrams

July 7, 2016

The usual way to make string diagrams using rubber bands or yarn on a board with nails does not allow much exploration. Mary Boole meant them as exercises in visualization. Building these diagrams using spreadsheets not only shows their versatility and capability for artistic expression, it helps students get used… Read more

Enigma Machine

June 30, 2016

Spreadsheets are great for creating secret codes and for breaking them. During World War II the German military used a machine they called Enigma to send coded messages. In a box about the size of a typewriter, wheels with letters on them were spun around to encode or to decode… Read more

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

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

Balance

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

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

Odd Times

June 6, 2016

How many of the products in a 12 by 12 times table are odd numbers? This is a question we rarely ask in paper-based math classrooms, yet it is an important and a very interesting question. We ask students to explore it, learning to Show and Hide rows and columns… Read more

Drawing Triangles

May 26, 2016

Though spreadsheets lets you put geometric shapes on the screen, those shapes are not connected with the cells and cannot be changed by using different values. We wanted to make geometric shapes that we could control and change by changing parameters. This is our first try. You can learn to… Read more

2016 MTA Summer Conference Aug 1 & 2

May 24, 2016

Learning Math as a Creative Experience with Spreadsheets UMASS Amherst   MTA Summer Conference Guide   OUR WORKSHOP: What if your students had the opportunity to learn mathematics in new ways, observe patterns, experience connections between varieties of representations and generate their own solutions to interesting problems? In this 2-session… Read more

Sudoku Challenge

May 18, 2016

Have you ever played Sudoku? It is fun and challenging. You have to find the numbers from 1 to 9 in each cell so that that all of the numbers appear only once in every row, column, and grid square. Ryan added a sweet wrinkle to the traditional Sudoku game,… Read more

Massachusetts Mathematical Association of Teacher Educators 5.25.16

May 16, 2016

Peter and Art are speaking at the: MassMATE 2016 Conference May 25, 2016 Bridgewater State University Read more

Why Math Education Doesn’t Add Up

May 16, 2016

Steve Strogatz talking on WGBH radio. Read more

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 and parents, the What if Math Summer Challenge. Choose a Lab from the Explore menu and mail it to your students. They learn problem solving,… Read more

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

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

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

ATMIM March 19, 2016

March 15, 2016

Spreadsheet Resources for the Learning and Doing of Mathematics The spreadsheet is a tool that presents opportunities for learners of all ages and abilities to engage in learning and doing mathematics in a novel way. A selection of spreadsheet labs that challenge learners to think and problem solve creatively will… Read more

Projectile Motion

February 17, 2016

Let’s do a little target practice with this spreadsheet projectile simulator, which will map out the flight path of an arrow shooting toward a target. Read more

Subtraction Tables

February 12, 2016

What would a subtraction table look like? How would its pattern be different from an addition or multiplication table? Is subtraction commutative? Read more

Hindu Algebra Problem

February 8, 2016

This problem is typical of the earliest algebra problems that likely came out of India. It is interesting historically, and it is the kind of problem students are still taught to solve today. We can do it very differently using spreadsheets. Read more

How Many Times

November 23, 2015

How many of the numbers from 1 to 100 are in the times table? All, Most, Less than half? I think you will find in this exploration of the relationship between the multiplication table products and the whole numbers as fascinating as I have. It is one of my favorite… Read more

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 them and on building them. There is great power in patternmaking, it is the most human of all activities. And there is great power in… Read more

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

Syracuse Problem

October 11, 2015

I built a Lab for you to play with the Syracuse Problem and to learn to use spreadsheets to play with like problems in fun and interesting ways. The Syracuse Problem is a simple one. Pick a number, any whole number. If it is even divide it by 2, if… Read more

Prime Numbers

October 11, 2015

The prime numbers are among the most fascinating objects in all of mathematics. While we can generate them, we do not know or understand their pattern. Yet, they have some fascinating patterns that we can easily see like the twin primes. We found on the Web a Conditional Formatting formula… Read more

Powerball

October 11, 2015

You just won the Lottery worth $600,000,000. You have a choice between taking it as a lump sum of $376.9 million or in yearly payments of $20,000,000/year for 30 years. Which should you choose? You will build a simulator of your payouts and you can decide yourself which plan is… Read more

Phone

October 11, 2015

Use spreadsheets to compare popular phone carriers and the plans they offer. Read more

Personal Budget

October 11, 2015

How much money do you spend? How much money can you save? We all need to know these things, but you can’t know them until you build and track a budget. Here is your template for a personal budget to track your income and your expenses and figure out what… Read more

Number Series

October 11, 2015

Spreadsheets make it easy for us to explore patterns in the whole numbers. This Lab does that and helps you learn the basics of spreadsheets like cell addressing, copy and pasting, and making rules. It is designed for every learner including young students. Read more

Number Patterns

October 11, 2015

Use numberlines and spreadsheet rules to explore the amazing patterns we find in our whole numbers. Did you know that you can get the odd numbers by subtracting the square numbers? I won’t give away any other secrets, but I know you will find in these patterns some wonders. And… Read more

Margin vs Markup

October 11, 2015

How should you figure out your profit? If you have an item that you are selling, should you price it at 25% over its cost to you, or should you price it so that your margin is 25%. Many people think that their profit will be the same either way.… Read more

Interest

October 11, 2015

Which form of interest, simple or compound, is the fairest? If you were buying a house or a car which would you rather have, which would you consider fair? Or if you had your money in a bank account earning interest, which would you consider the fairest way to calculate… Read more

Exploring Triangles

October 11, 2015

Create, manipulate, and explore triangles in this live interactive spreadsheet. On the surface, a triangle is being drawn on a graph, but the real magic lies in the formulas used to make this spreadsheet work. Read more

Hit Streak

October 11, 2015

In 1941, Joe DiMaggio got a hit in 56 straight games. This record has never been beaten. Some say it is the greatest record in all of sports. You can develop a simulation of DiMaggio’s streak to see why it is considered so difficult to beat. And you can decide… Read more

Decimal Subtraction

October 11, 2015

Decimal subtraction is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Addition and Decimal Multiplication and Decimal Division.… Read more

Decimal Multiplication

October 11, 2015

Decimal Multiplication is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Addition and Decimal Subtraction and Decimal Division.… Read more

Decimal Division

October 11, 2015

Decimal Division is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Addition and Decimal Subtraction and Decimal Multiplication.… Read more

Decimal Addition

October 11, 2015

Decimal addition is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Subtraction and Decimal Multiplication and Decimal Division.… Read more

Credit Cards

October 11, 2015

How much are your credit cards costing you? If you are like most of us, they are very expensive because we have credit card debt. This Lab will help you to see what your credit cards are costing you and help you to keep track and pay off your credit… Read more

Coffee Money

October 11, 2015

Are you a coffee drinker? Small daily expenses like coffee can quickly add up. By using spreadsheets to organize cost, you can monitor how much you’re really spending. Read more

Battleship

October 11, 2015

Spreadsheets can be a great place for you to build your own games. Ryan has built one of his early favorites, Battleship, where you learn graphing as you try to sink your opponent’s battleships. Making your own games can be great fun. Try it. Read more

Adding Machine

October 11, 2015

Multiplication is often thought of as repeated addition. By building a times table by using a repeated addition rule you will have a fun practice of the multiplication facts, and you will learn to build rules in spreadsheets. Rules let you program spreadsheets to do remarkable things like make the… Read more

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 friend Patrick Vennebush’s great website. Read more

Spreadsheets

September 11, 2015

This year we are celebrating the 35th anniversary of spreadsheets. The first showing of VisiCalc was October 17, 1979. NPR did a terrific podcast about spreadsheets we suggest you listen to. We think this quote symbolizes our spreadsheet/functional thinking vision. KESTENBAUM: Spreadsheets have left us in a different world, though.… Read more

ATMNE Annual Conference 10.30.15

August 27, 2015

  ATMNE Annual Conference, October 29 and 30, 2015 in Portland, Maine  Spreadsheet Math:  A Powerful Tool for the Learning and Practice of Mathematics    Friday October 30, 2:15 – 3:15. Come and learn about the power of spreadsheets as tools to encourage creativity and challenge for students of all ages… Read more

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

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

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 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

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

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

Spreadsheet Math: A Powerful Tool for the Practice of Mathematics

April 8, 2015

NCTM Annual Conference April 15-18   We were thrilled to have had over 250 people come to our session and with their response and questions. As we promised, here is the PDF and PowerPoint that we presented. We look forward to implementing the suggestions and to seeing the results as you… Read more

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

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

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

Lemonade Stand

January 20, 2015

Manage a business. The Lemonade Stand has been a business simulation developed in many different versions. We believe this is the first time that it has been created as a spreadsheet simulation. If you are a young entrepreneur who wants to learn how a business works and how you can use spreadsheets… Read more

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

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

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

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

Making Fractions

August 22, 2014

Spreadsheet math generally focuses on ratio and proportion to develop the concept of fraction. But fractions is such a big problem in today’s curriculum that it seemed only right to use the power of spreadsheets to help students and teachers to gain some fraction-sense. This is a very simple spreadsheet… Read more

String Challenge

August 22, 2014

Strings need not begin and end on axes that are at right angles to each other which we call Cartesian. It is quite interesting that Descartes himself did not use axes at right angles. We consider this a challenge because students have to figure out how to move both the… Read more

Similar Triangles

August 22, 2014

Scatterplot graphs enable us to build shapes using spreadsheets and to practice transformational geometry. They are surprisingly flexible tools. And since they depend upon a table of value and that table can have both fixed numbers and rules, we can not only build shapes but change them and watch the… Read more

Multiplying Integers

August 22, 2014

We have made a big deal of the times table and of other tables.Now we extend the times table to negative numbers and thus to all 4 quadrants of the real number space. We hope to build student intuition about this space and to gain a spatial sense of graphing… Read more

Systems of Equations

August 21, 2014

Solving systems of equations sometimes called simultaneous equations with graphs is simply a matter of finding out where they intersect. One of the most valuable things students can learn is to be able to visualize linear equations and systems of equations so that they can tell the quadrant where the… Read more

Composition of Functions

August 21, 2014

One of the most powerful aspects of the mathematics of functions is our ability to treat them as abstract quantities (essentially numbers) and then combine them with standard operations. But with functions we can go further and develop a new operation we call composition or taking a function of a… Read more

Inverse of a Function

August 21, 2014

Spreadsheets make it very easy to switch axes and add graphs. They enable students to play with what may have been difficult and abstract concepts like the inverse of a function. You may want to approach the inverse of a function by challenging students to fill in a table of… Read more

Rule of 72

August 21, 2014

The rule of 72 is an old banker’s rule of thumb to find out how long it will take to double your money at different interest rates. Financial literacy has become an increasingly important topic for K-12 education and we believe spreadsheets and headmath or mental estimation should be central… Read more

Peter’s Taxi

August 21, 2014

There are a wide variety of financial literacy problems. This is the kind of problem that appears on many tests. It does not ask you directly to find the cost of a taxi ride which includes both a fixed and variable amount. See if you can use this Lab to… Read more

Place Value: Thousands

August 20, 2014

We extend the place value generator to 100’s of thousands to show you how the pattern of 1’s, 10’s, 100’s, continues to 1,000’s, 10,000’s, 100,000’s. Enter numbers and watch the expanded and compact forms of place value change. Pay special attention to using text units and take a look at… Read more

Place Value: Decimals

August 20, 2014

We take our place value generator to decimals to help students see the simplicity of the place value pattern going right as well as left. Read more

The Chessboard

August 14, 2014

We take that great old problem of the inventor of chess and the ruler of India and use it to see how powers of 2 grow in size. We start out with a chessboard and look at doubling each successive number. Then we seek a method of representing this doubling… Read more

Sine Function

August 14, 2014

Spreadsheets are not limited to algebraic functions, they can also display trigonometric functions. We are modeling the sine function, but you can try any of the trig functions by going to the Formulas menu and choosing it. We have the 3 most significant parameters of the sine function, to control… Read more

Polynomial Functions

August 14, 2014

Polynomial functions are not limited to the highest term and while that term is most important in determining the shape of its graph, additional terms play a role. Try out additional terms to see how they affect the shapes of the graph. Focus on the patterns! Read more

Power Functions

August 14, 2014

Adding an exponent, sometimes referred to as a ‘power’, to the input variable of a linear function that passes through the origin creates a power function. Changing the parameters of these functions reveal some important and interesting patterns. Read more

Quadratic Functions

August 14, 2014

What does each of the coefficients do? How does it change the graph of the parabola. What does a do, what does c do and a question still rarely asked, what does changing b do to change the graph. To see what b does more clearly we have you add… Read more

Solving Equations

August 14, 2014

Typical algebra courses start with equations and solving equations and then move to graphing and functions. We start with functions and use them to solve equations. We treat an equation as the equality of two functions, graph each one and then look at their intersection. This is a powerful way… Read more

Linear Functions

August 14, 2014

Linear functions are the most important family of functions. They pervade our everyday lives and our work. Their graph is a line, and their general form is f(x)=mx+b where m is the slope of the function and b is the y-intercept, the value where the line crosses the y-axis. This… Read more

Rate of Growth

August 14, 2014

We look at world population over the past 60+ years and ask whether the earth’s population is growing faster or slower today. Is it out of control and something we should all worry about or are we getting it under control? This is another problem directly related to climate change… Read more

CO2 Growth

August 14, 2014

Spreadsheets offer us a nearly unlimited ability to develop and learn from case studies using real world data. We will focus mainly on climate change which is an area rich in possibilities and of great interest to students. In this case study we look at the production of carbon dioxide… Read more

Sierpinski Fractals

August 14, 2014

Fractals are a new 21st century mathematics. They are patterns that repeat themselves at various scales. This one is based on the odd numbers in Pascal’s triangle. We learn to create it easily by using Conditional Formatting which enables us to color cells or text based on a quantitative relationship.… Read more

Normal Distribution

August 14, 2014

Most museums with math exhibits have a Pascal’s triangle made up of pegs with balls falling down between them and bouncing off of them. One of the things they want to show is probability and the Normal or Bell curve produced by these balls as they fall down most of… Read more

Pascal’s Triangle

August 14, 2014

Another famous pattern, Pascal’s triangle, is easy to construct and explore on spreadsheets. Create a formula for any cell that adds the two cells in a row (horizontal) above it. This pattern is like Fibonacci’s in that both are the addition of two cells, but Pascal’s is spatially different and… Read more

Parentheses and Pi

August 14, 2014

Parentheses are very important in spreadsheets because like all programming, spreadsheet formulas have to be very specific. A big formula, especially one like Viete’s approximation of pi, likely will require us to think both in parentheses and in creating formulas that naturally build a series. This one is quite interesting… Read more

Pennies to Heaven

August 14, 2014

Pennies to Heaven is a Fermi Problem, basically a “headmath” experiment. Fermi Problems, originally developed by Enrico Fermi, one of the greatest experimental and theoretical physicists of the 20th century, are real-world estimation problems. So we ask, “If we had a stack of pennies as tall as the Empire State… Read more

Decimals and Percents

August 14, 2014

Ratios can be written in a wide variety of different way: as fractions, as decimals, and as percents.,with a colon, with a slash, as a fraction and even as a baseball batting average. Here we compare a decimal ratio and a percent by building decimal and percent tables in the… Read more

Common Denominators

August 14, 2014

We can use these proportions to compare two ratios with different denominators by finding a denominator that their proportions have in common. Thus the common denominator of 2/3 and 3/4 is 12. We then can use the common denominator to add/subtract and divide common ratios (fractions). This approach to division… Read more

Ratio and Proportion

August 14, 2014

We think about ratio tables in terms of motion. Move up 2 and over 1, or move up 1 and over 2. In this way we build proportional patterns. By coloring the cells we land on like knights in a chess table, we can see the proportions of different ratios.… Read more

Division and Ratio

August 14, 2014

We can make a division table just like we made a multiplication table. Division is surprisingly our most important operation in terms of most of the problems we solve in our daily lives. Division produces numbers we call fractions or rationals and functions we call ratios. With spreadsheets we concentrate… Read more

Square Numbers

August 13, 2014

The square numbers form an interesting pattern on the times (multiplication) table. They run along a diagonal from 1 to the top right of the table separating the table into two halves. This is the first step in looking at patterns in the multiplication table. Students build a new square… Read more

Magic Rectangle

August 13, 2014

Multiplication tables have some wonderful and quite surprising patterns. This is one of them. Draw any rectangle in a multiplication table and you will find that the products of opposite corners are equal. For example a rectangle around a full 12 by 12 table will be 1144 and 1212. Try… Read more

Shapes

August 13, 2014

Shapes introduces student to changing the colors in cells and to changing the shapes of cells by dragging the column or row separators in the address axes. Students can use spreadsheets as drawing tools and can create some wonderful pictures with them. Spreadsheets can thus be tools for visualizing mathematics… Read more

Products as Areas

August 13, 2014

Using the times table, students can see that products are always rectangles, and that they represent the area of that rectangle. They should explore the times table by playing with these rectangles whose sides are the factor of the products. Read more

Place Value

August 13, 2014

Our number system inherited from India and from the Medieval Arab world enables us to use just 10 symbols to write any number we can imagine. Students learn in this spreadsheet to enter numbers, to compare compact and expanded forms of those symbols and to add units to any number.… Read more

More Number Lines

August 13, 2014

We use rules to build new numberlines. For example we can start in the middle and go both forward or backward using adding and subtracting rules. You can even generate and experiment with negative numbers by subtracting below 0. As you build numberlines on spreadsheets you are building them in your mind. And by thinking of numberlines in terms of rules you… Read more

Lights Out

August 13, 2014

This is one of those math puzzles that come up in contests but which turn out to be quite interesting mathematically. Imagine a long hallway with lights in the ceiling, all on and each controlled by its own chain. A long line of people (as many as there are lights)… Read more

Hundreds Table Patterns

August 13, 2014

We introduce students to more complex patterns and rules in hundreds tables. In particular, we have students look at diagonal patterns and develop rules to fill them. This spreadsheet hundreds table practice is designed to build numbersense, the primary building block for a strong math education. We encourage students and… Read more

Factor Table

August 13, 2014

Spreadsheets always automatically perform the operations you ask them to do. But sometimes we want to see the process. We can make spreadsheets show us that in several ways. Here we show the factors and have students build a times table showing the factors by using a special formula called… Read more

Factor Pairs

August 13, 2014

Multiplying creates products, factoring separates a product into the numbers that make it up. We thus start with the table and then look at the axes to find the factor pairs that make the product. Once again we focus on the patterns in the times table so that you can… Read more

Distributivity

August 13, 2014

The distributive property turns out to be central to a surprising variety of important mathematics. One of the most valuable is to use it to break products into two pieces to make them easier to compute. Thus 56+510 is easier to solve in your head than 5*16. Here again we… Read more

Counting By

August 13, 2014

Counting-By introduces multiplication. Counting-by or skip-counting is, we believe, the best way to help students build their multiplication facts, and though they will live in an age of ubiquitous spreadsheets and calculators, they still need to have mastered their multiplication facts to do any interesting math in their heads. Here… Read more

Commutativity

August 13, 2014

The symmetry of the multiplication table around the square numbers diagonal we call commutativity or the commutative property. It means that in a 12 by 12 multiplication table we need only learn 72 or so facts and not 144. It also means that the square numbers are not the only… Read more

Build a House

August 13, 2014

Spreadsheets with their natural grid make a great, though not entirely flexible, platform for architectural design and for working with shapes. Build a house introduces students to using spreadsheets to create floor plans and to measure area. Most spreadsheets have amazing graphic flexibility. You may want to encourage students to… Read more

Associativity

August 13, 2014

Parentheses are not only important in paper math, they are critical in spreadsheets. To make sure that terms are handled properly by spreadsheets, we have to be sure we use parentheses to write our formulas so that there is no ambiguity. Associativity gives us the principle behind this grouping, for… Read more

Addition Table

August 13, 2014

From here on students can go in most any order they choose. They can start with addition or with multiplication. They should imagine themselves on an elevator able to go up or down anytime they want, to find interesting floors to explore. NOTE: The addition table, like the other tables,… Read more

Addition Patterns

August 13, 2014

Now that students can build addition tables, they should look for the patterns in them. We introduce them to a variety of things they might want to look for. We also suggest that some students may want to try to make a subtraction table. We have another Lab for making… Read more

Adding

August 13, 2014

Adding to 10 is but on example of a large number of spreadsheets that could be developed to practice addition. We give students two number lines (number bars) to 10, and have them copy a number from each to build a pattern of all the sums to 10. We suggest… Read more