Problem Solving in the Digital Age

An Audacious Goal for American Education

Art Bardige

I was reading it for fun, a biography of an obscure 13th century mathematician, Leonardo of Pisa, when I made the discovery that profoundly changed my life’s direction. And I believe it shows us all the path to the future of digital learning where technology changes not just how we learn but what we learn.


The Past

In the year 1200, Leonardo, a 30-some-year-old trader, returned to his native Pisa from the Arab world, to bring the power of the Arabic arithmetic and algebra he learned as a boy to solve medieval business problems. Math used by medieval merchants, in both the European and Islamic world, was Roman, and the calculation technology was an abacus. Good enough for the Empire with a common currency and weights & measures, it handicapped medieval merchants facing ratio rich problems. Leonardo’s book, Liber abbaci (The Book of Calculation), applied math that had been until then academic to business problems.

The biography included an image of Liber abbaci’s table of contents. It left me breathless. It was a near replica of today’s K-12 math curriculum. The math our kids are required to master in the 21st century is, I then realized, neither basic or fundamental. Designed by Leonardo of Pisa for medieval business, from concepts developed by Arab scholars; it used paper, introduced to Europe a century earlier, as its technology for algorithmic calculation.

Today, business no longer calculates on paper. It uses digital technology, mainly spreadsheets, with digital instead of paper algorithms, discrete instead of continuous variables, and functions instead of equations, it asks “What if…” instead of “What is___?”

The Present

What if we extracted paper algorithms, no longer useful in this age of smartphones and spreadsheets, from the curriculum; we could free up the thousand learning hours a student spends practicing

3-digit multiplication, long division, operations on fractions, etc. What if we took out the algebra of solving equations? That would free up another math learning year at least. Along with time, many students would save the pain of trying to learn Leonardo’s math and teachers would gain valuable flexibility by eliminating:

• Scope & Sequence based on the complexity of paper algorithms like long division.
• Troublesome concepts like operations on fractions, irrelevant in a digital age
• Ratio conversion because these silos are unnecessary on spreadsheets
• Algebra focused on x’s as continuous variables – no longer needed for today’s discrete data
• Utter failure to include or apply powerful tools of our age for solving interesting data-rich relevant problems, one reason math is the “most boring subject .”

Though, I have spent most of my working life focused on using technology to change the way kids learn, th

is revelation led me to understand that technology changes not just the way schools transfer knowledge, but the knowledge they transfer. What is the value of putting jet engines on planes designed for propellers, using steam to drive the oars of a Greek galley, or teaching Leonardo’s math with digital age tools?

Freed from paper algorithmic practice, students can focus on authentic problem solving. Freed by the power of spreadsheets, the support, and the rich data available on the Web; they can engage in solving fascinating problems. Freed from the tyranny and constraints of Leonardo’s staircase determined by the difficulty of the algorithms, STEM subjects and problems can be integrated, fascinating, and motivating.

The Future

Over the past 4 years a small group of us envisioned and developed a range of such digital age “Labs” at as exemplars where students can practice integrated STEM learning as:

• Scientists using spreadsheets as laboratories to build and experiment with models.
• Coders thinking functionally.
• Engineers using design thinking for problem-solving.
• Mathematicians applying their most powerful tool, functions, to real world problem solving.

But it was not enough for us to substitute problem solving for paper algorithmic manipulation, not enough to claim we were moving up Ben Bloom’s taxonomy if we did not make the concepts easier to understand and the problem solving easier for all students to do.

Fortunately, the history of invention shows time and again that new technology increases effectiveness as well as efficiency. It makes it possible for less highly well-trained people to perform jobs at the same level or above those who had become very highly skilled. It is utterly amazing that most of the population can today safely drive 80 MPH on busy interstates, a speed previously limited to race car drivers, because our cars and our highways reduce the specialized skills needed to be competent. Could technology have a similar impact on learning? Could the ability to learn math depend not so much on native “intelligence”, teacher quality, or student interest? Could technology actually help students gain understanding? So, we turned to a question I liked to ask people who would complain to me about not getting math “When did you start having trouble with math?”

When it got abstract,” they all answered. For some, math got abstract early when fractions became non-intuitive because multiplication shrinks them and division grows them. For classic problem solvers it was when they got to two-step problems. For others, it was algebra when their teacher could not answer the, “What is x?” question to explain variables. Still others got to calculus but tripped over limits. In every case, abstract meant a failure of intuition. Can spreadsheets make math less abstract; make it easier to concretely visualize fractions, story problems, variables, and infinities?

Spreadsheets are function machines. A function is itself a “machine” with an input, a unique output, and a rule connecting them. A spreadsheet cell can contain an input value or an output value which is defined by a rule. Functions are considered the most important concept in mathematics because they enable us to represent cause and effect, they are the building blocks that we join to create models that can replicate natural processes and solve real-world problems. We built What if Labs to enable students working with spreadsheets to use these ideas concretely and intuitively. A variable is a column (or row) of numbers. A function has an input in one column and output created by a rule, usually in an adjacent column. Discrete variables, typical of the digital age, are concrete, simply a column of numbers.

Functional Thinking

We developed a standard problem-solving methodology to give students the ability and the common language to imagine, collaborate, and solve a very wide range of STEM and business problems. Functional Thinking starts the problem-solving process with a Parameter-Table to build and then flexibly control a Function-Table containing inputs and outputs, that can then be linked to a Graph to visually represent and communicate the function.

These 3 elements of functional thinking make solving legacy school problems much easier and provide the essential structure for all students to solve new relevant, real-world problems rich in data.

Take the Tour at to see for yourself how this technological simplification can enable all students to see math as a stepping stone and not a stumble stone to becoming effective, creative problem solvers who ask, “What if…” This should not come as a surprise, for technology simplifies complexity enabling us to perform tasks with less effort and easily handle complex machines. This simplification of math and STEM problem-solving through thinking functionally gives us confidence that most students can learn to effectively apply math at a level only the few did before. And it gives us confidence that technology can enable us to rethink both teaching and learning across all of schooling.

Just-in-time technology that revolutionized manufacturing gives us another example. It can make most lectures obsolete. Today, we see just-in-time instruction used in flipped classrooms, MOOC’s, Mazur’s Peer Instruction , and Khan Academy’s conceptual presentations. Web technologies including video, search, open educational resources, and communication make it possible for students to get the support they need, or questions answered, just in time.

Though lectures and textbooks might have been efficient for condensing knowledge in a sparse information analog world, they are less effective for managing the new information and concepts flooding our full spectrum of students in a digital world. Today, just-in-time self-instruction using digital technology makes it possible to abandon most paper-based technologies and enable all students to efficiently find relevant information on their own. Without lectures students would have more lab/studio/discussion time, without textbooks more real-world problem-solving time, more opportunities for choice, more time for developing 21st century skills, more time for learning important but ignored subjects like financial reasoning and design, and more time to explore and enjoy the arts.

Our Challenge

The great problem of our age, Robert Putnam believes, is the ever-widening gap between the middle-class families who achieve a bachelor’s degree, who thrive, and the families who do not, who struggle against further decline. Captured by his famous scissors graph, this is our great challenge.

Today, just 40% of our kids achieve a bachelor’s degree, a number frozen for several decades. Ever-increasing demands placed on the 21st century workforce, by automation that competes for repetitive jobs, higher order skills and continuous learning now required, and unmanageable costs, have caught students and American education in a vice grip, squeezed by need, affordability, and capability.

The American college system exploded after the Civil War because industry and agriculture needed managers. The bachelor’s degree became the symbol of management skills. It remains so today, a sign to employers that a person has developed the 21st century “4C’s” skills—the creativity, critical thinking, collaboration, and communication—ability to solve problems. This capacity is rapidly becoming essential for all our kids, whether they will manage a business, a practice, a project, or a family. For our next generation to thrive and perfect our union, I believe we must therefore challenge ourselves to:

Double college graduation rates at half the cost.

Can digital technology enable our schools to meet this audacious challenge? Can it do for education what it does for all other products and services, substantially increase effectiveness, efficiency, and relevance?

Effectiveness: Digital age technology makes students more effective learners by giving them just-in-time feedback as they solve exercises, learn from responses by other students, naturally iterate when they code, build models, explore parameters, ask, “What if…”, or even just search Google for answers. But beyond instant feedback, beyond the uses we commonly find today, digital age technology, as we have seen already with the mathematical concept of function, enables and empowers students to use powerful conceptual tools for patternmaking, for working with their real experience, and for bringing understanding.

Efficiency: The student debt crisis, the mounting financial and enrollment struggles colleges face, as well as the increasing disparity between rich and poor school districts are symptoms of the bigger problem. Education just plain costs too much.

Students using digital learning won’t just save the time wasted on learning obsolete content or practicing skills no longer needed, they will become more efficient and economical learners by practicing digital age problem solving skills. When students use technology to collaborate to solve problems, they take the burden off teachers to provide explanations. When students learn to be critical thinkers who can search for and judge their own answers, they relieve teachers of grading each step. When students have real choices, teachers no longer need to spend exhausting hours being cheerleaders. And when students learn general principles by solving problems rather than listening to lectures, they center the burden of performance on themselves. Student-centered learning enables more efficient learning and increased student-teacher ratios.

Relevance: As long as students view school as an irrelevant distraction asking, “Why do I have to learn this?” they will be inattentive, even disruptive, and fail to learn. When schools view digital technology as providing essential learning tools focused on 21st century skills and design thinking, when they reinvent basic content, the essential curriculum, to represent digital age knowledge, then and only then, will all our kids find school relevant. Just as What if Math views math education in the larger context of digital age STEM problem solving, so too can ELA become visually focused digital age communication and collaboration.

Open the Web

As the first step into the digital age of learning, we should open the Web to all students all the time to provide the resources and relationships they need to learn and to work in the digital age.

Opening the Web in schools at all times, including during high stakes tests, will necessarily change what and how we measure, which will change the curriculum, which will change what teachers do, which will change what students do. Opening the Web tears down the wall separating school and life. Opening the Web to students is the first step in meeting our audacious challenge.