Author: Art Bardige

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

Why do we have to Learn the Quadratic Formula?

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 quadratic equations, and cap off their introduction to algebra with this magical formula. The intent is to show our students the power of algebra to use a sequence of symbols to find the values for x that make any second order polynomial equation true.

The Quadratic Formula is a big deal. If a student can use the Quadratic Formula to solve an equation, we deem them worthy to graduate high school and ready to learn college algebra, the algebra of functions. It involves not just the use of symbols to represent known and unknown quantities but makes use of imaginary numbers along with real numbers they learned on the numberline. It is no wonder movies and TV shows that put math symbols in blackboard background shots almost always include this iconic equation. Thus, it has long seemed obvious that this pinnacle idea should be a fundamental mathematical concept that every student should prepare for, learn, and master.

But… let me ask you a question. “When will any of our students ever use the Quadratic Formula?” Unless perhaps, they are in the 1% who finish college with a Bachelor’s Degree in mathematics or who minored in math, and now teach middle or high school, the answer has to be never! Unless, perchance, they have to solve a quadratic equation to get off of a desert island. (When they can always google it.) And if they can’t get on the Internet then, wouldn’t you suggest they treat the equation as they would any function, graph it and approximate x-intercepts? In all likelihood they would by then have forgotten the formula anyway.

If it is not necessary for our students to learn the Quadratic Formula because they will never need it, then what about the math that leads up to it? — the carefully-crafted sequence developed over centuries to prepare students to learn it? Do they have to learn to factor quadratic equations or even solve linear equations? Graph those too! We need hardly stretch our imaginations past the present to know that spreadsheets and other web tools make it unnecessary for our kids to master these processes. Understand functions, of course! But to know and be facile with the algorithms for solving equations, never!

The Quadratic Formula is a paper algorithm, designed and used to solve quadratic equations on paper. It is no longer of value in this digital age. Like other, similar paper processes we insist in teaching our students, it is obsolete. Why do students have to learn a paper-based “long division” algorithm, a process for adding fractions with unlike denominators, or even subtraction of whole numbers by regrouping? They will never need to use these paper algorithms in their work life. And even if they did, they would use them so infrequently that they would not remember them. Those myriads of homework problems that they practice to mastery, like 3 digit multiplication on paper, worthless!

So is it now time for us to jettison the obsolete and to ask, “What math should our students learn to prepare them for the digital world they will inherit?” The answer is straightforward. Students should learn to use the computational tool used in business and industry, the spreadsheet, to solve not only the traditional math problems they are taught to solve today in our schools, but the kinds of interdisciplinary, data modeling problems they will need to learn to solve to do their jobs. That means they will not need to learn to calculate on paper, they will need to compute on screens. They will need to learn to creatively solve real world problems, to analyze data, to build models, to work collaboratively, and to reason quantitatively as part of a cross disciplinary team. To know what they should learn in our schools we need only look to the real world around us and the problems people now face in the 21st century workplace.

Imagine giving students the opportunity to creatively find a new way to solve any quadratic equation, not memorizing an obsolete paper formula but using a fundamental powerful general coding tool, recursion, to build a simple spreadsheet. Imagine embedding this in a problem solving context that relates it to real world activities like throwing a ball. Imagine making sharing central to this student experience so that they can share their learning. Imagine employing these tools and this logic to deep and substantial projects and inquiry across the disciplines, and applying them to the urgent issues facing our planet. This is the future of learning. Therefore, this should be our mission — to prepare our students for their future and not our past.

Independent Learners

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 as a business experiment, forced on them by circumstances, fully expecting to return to the old form after the magical elixir in the form of a vaccine becomes available; others see it as the future, a trend, already in progress, now accelerated. Enabled by the Web, workers and companies are weighing the cost of commuting time, office space, and watercooler office chat against the disruptions typical of home offices. For some managers, the home office conjures up a supervision nightmare, with the lack of synergistic interactions suggests lower creativity. But many are finding that work from home actually increases their efficiency and effectiveness, and changes how people interact. Zoom technology is already becoming ubiquitous. It is clear that this trend will only continue and almost certainly accelerate, its current limitations in form, equity, and bandwidth will become things of the past.

The key to the success of this new business format will most certainly be workers who thrive on independence. Companies will hire those who demonstrate an independent work ethic, who can manage and complete projects on their own, who can solve problems that they meet in pursuit of such projects, and who can independently collaborate with others within their group and outside their sphere to think critically and creatively. We will all have to learn to take more responsibility for our time and for our work, to husband our energies and make efficient and effective use of our time. That being the case, we should obviously be looking to our schools to ready our children to join this new workforce with an ethic that promotes independent work and independent learning.

Independent learning requires more than telling students that an assignment is their responsibility. It is more than adding new grading heuristics to report cards. It is more than treating high school students with our expectation of college students. Independent learning requires us to rethink the kind of assignments we give students and the kinds of interactions we expect from them. These assignments will have to excite imaginations, engage concentrations, and give students the ability to choose by using real-world Web links and tools. And as teachers we must have the patience and fortitude to not “tell them what to do”, to let them fail, and to grade products and not process. This is the goal of every Exploration, to make every learner a creative independent problem solver.

Connections not Collections

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 to coins, baseball cards to cars. Museums graced big cities and barns were stopping places in many small towns. This collecting craze was more than just putting things into glass cases, it made names and naming conventions, classification, and classification systems critical intellectual activities. Collecting and collections were the central learning activity in schools.

I have in my house one of those school exemplars, a wonderful small antique 19th century cabinet with drawers filled with collections for classroom learning. It has a drawer for products of corn, another for cotton, for flax, fiber, and more. This iconic cabinet was built to grace a 19th century school classroom in Philadelphia and for us as a visual representation and definition of 19th century schooling. It continues to represent the essential nature of schooling today. We still view our students as collectors, filling their mental drawers with labelled mental things. We think of our students as museums, filling up empty cases, collecting the facts and ideas they will need as adults. We have them collect by practicing words, algorithms, science concepts and formulas, history dates and people. We plaster classroom walls with these collections, the Periodic Table, lists of formulas, images of geometric shapes, words of the day, or timelines of history. School for most of our students is the place to fill their personal mental museum, to become good collectors, and to use these collections in their work.

Our Digital Age today is no longer about making collections. It is nearly always about making connections. There is no longer a need to memorize most information when we can instantly look it up. It is indeed impossible to even collect a tiny portion of the data we receive each day. We don’t use the Web to make collections, we use it to make connections. That is what we do when we search, when we link, when we post, and when we social network. Of course, the Web is a rich treasure chest of collections, its great innovation and great power is in making connections. Good sites are full of links, well-structured spreadsheets are rich in links, and coding is in essence linking. Search engines and search engine optimization count links. Links both to physical sites and to people are central to the Web and to the Web economy.

If our schools are to prepare students for their work in their future then we must learn to make connections and not collections. They must learn to search for links and make new ones, to use links and explore them, to appreciate links and select valuable ones. They must use links to become problem solvers with “flow charts.” Our new Explorations projects are focused on connections building and using links. Rich in links and problem solving experiences using links, they help students develop the connecting skills so critical for the digital age. We share Explorations with you and invite you to explore them. They are free and require no sign in. We look forward to your links.


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 where students can discover an answer to the question: “Are there more grains of sand on earth or stars in the universe?” Sand and Stars is free, as will be the case with all future Explorations. This Explorations is designed for students to use on their own independently or in teams collaboratively, accompanied by instructional support as needed.

If you want highly interactive online content, if you want your students to work together on experiments and projects that interest them, if you want them to learn to use the power of the Web and digital tools like spreadsheets, to learn to communicate with each other, to productively search the Web, to have a STEM focus, and to become problem solvers in the digital age; we encourage you to try our first Exploration, Sand and Stars.

In this time of change, when education must now be Web-based and in the foreseeable future when the Web will play a large role, it is our dream that you and your students find Sand and Stars a valuable and thrilling learning experience, the first of many such resources for learning in the digital age.

We look forward to your thoughts and comments. More to come soon! Here is the Teacher’s Guide to get you started.

My Mentors

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 great good fortune to have had some wonderful ones. Fletcher Watson, Gerald Holton, and Jim Rutherford, in the physics project/curriculum they created, vectored me to think first about physics education and then all education as a human enterprise full of fascinating stories, real people, living concepts, and the history of ideas.

Their Harvard Project Physics program brought me to Fort Lauderdale to teach it’s beta version at Nova High School for two years, meet my wife Betty, then go to Cambridge to write a lab manual for some of its electronic equipment which turned into a stint making educational films and from there to teaching and developing a math curriculum with the same goals and objectives. It’s humanistic vision has underpinned all of my efforts to reenvision and redesign education in each of my entrepreneurial enterprises.

Today, I am working with some new mentors on a new, next generation, project to bring this same fundamental philosophy and pedagogy to learning in the digital age. I am as excited about this as I was about teaching Harvard Project Physics so long ago, and to bringing a new vision of online learning to all of our kids. Our Challenges Project follows on the work Peter, Ryan and I developed in What if Math and that is described in detail in my new book Make it Real, and the Grand Challenges work that Larry Myatt has brought to us. You can follow our progress here or send me an email if you want to join us to rethink education for our digital age.