Category: 6 What If…

Small Changes

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 classrooms a little over a half century ago. School desks were unscrewed from the floor. That seemingly small change, which on its surface seemed to be just about furniture, precipitated a major reduction in class size and a revolution in expectations of good teaching. Desks bolted to the floor, locking students in straight rows facing a teacher in the front of the classroom, optimized the use of space.

My 5th grade Chicago classroom with fixed desks held 51 students in 6 rows with 8 desks per row and three portables. It also defined Miss O’Hearn’s teaching style. My 6th grade suburban classroom with moving desks had 25 students. Desks could be rearranged, students could interact with each other, learning in groups was enabled, and teachers could give students individual attention toward student-centered learning. Small changes can have great effects even in education.

We have the opportunity to make such a small, seemingly inconsequential change that could profoundly transform our schools by allowing students to use the internet on their Common Core Math tests.

We need only change the wording in the test’s directions to allow and not prevent student use of a computer/tablet/smart phone. The tests are designed to be given online already. They give the students digital tools to use to solve some of the problems. What if we simply extended that existing open technology requirement to every question and enable students to use most any available program or website? What if they could use Google search to solve an arithmetic problem, or open Excel, Sheets, Numbers, Wolfram Alpha, Khan Academy, Wikipedia or any website they wanted to find an answer? What if, as the PARCC initials stand for, we are serious about the tests assessing “college and career readiness?” A realistic 21st century college or career problem would quite naturally expect the solver to have internet access. College tests are generally open book and every online course must, by its very nature, allow internet access. So why not really prepare our students for college and career?

The consequences of such a minor change in the assessment directions would be far reaching and revolutionary. Teachers would stop teaching the algorithms and stop giving students arithmetic and algebra algorithm worksheets. Why teach long division if the tests don’t require it? Why spend all of that classroom and homework time on operations on fractions if students won’t be tested on it? Why teach students to factor equations using paper and pencil algorithms if they can get the answer online? This mechanical symbol manipulation that today makes up the bulk of student practice time would simply vanish. Creative experiences using technology to solve math problems would naturally replace it, for those will be the “basic skills” required by the tests. Spreadsheets and other quantitative technologies would replace pencil and paper. Mathematics would become more interesting to students for they would no longer need to ask, “Why am I learning this stuff for when I can solve this problem on my old phone or calculator?” Math classrooms could be filled with creative “What if…” experiences.

Not only would there be more time for authentic problem solving in math, but there would be more time for the other STEM subjects, and more time for the arts, for physical education, for history, for the manual arts, for project and performance oriented activities. So many of us dream of an educational system that is rich and creative, but we are overwhelmed by a system seemingly sluggish to innovate, overwhelming in complexity, and demanding in tradition that it seems to make substantial change all but impossible. Yet there are times and circumstances when small, seemingly inconsequential acts can have monumental impacts. Allowing students to use the Web when they take their Common Core math assessments could well be as revolutionary for students today as unscrewing the desks were in the 1950’s.

The Magic Wand

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 or need it to be. We all know it lacks the essential creativity our children will need for 21st century work and life. We all know it fails even the least stringent tests for those who need it most. We hear the same refrain again and again. More money — even though we have tripled our per pupil expenditures over the past half century without any significant improvement in performance. Better teachers and teacher training – even as we have taken away the opportunity to be imaginative and entrepreneurial which brings the best and brightest into a discipline. More demanding – even though we have not a shred of evidence that our children are responding by caring more and working harder. Science – even if what we are measuring will not be a useful life and work skill. To make matters even worse, education in America seems so resistant to change, so overwhelmingly complex, so replete with attempts to transform it, no wonder my question stumps most people.

No doubt great teachers are transformative but until we make their jobs transformative we will just have to hope research into cloning makes a huge breakthrough, because we will never get enough great teachers, never ever. No doubt more money is necessary in many places, but we have cloaked our schools in so many narrow demands that we starve the essential. No doubt we need to demand more of our children but more of what, for we have asked them to work harder and not work smarter even though school today should be about smarter and not harder. And no doubt we require more learning science and more learning research, but when we study the same old we get the same old, it does not take any scientific research to find out that barely 1/5th of our students really master the required content.

We know what we need. Sir Ken Robinson’s TED Talk has been watched more times than any other. Not just any other on education, but any other TED Talk, now approaching 30 million times. Ken talks about creativity in education. We are starving for it. So how do we get it to happen?

What could a magic wand do?

If I had such a magic wand I would use it to make a minor change in the instructions of our new Common Core Tests. The tests are designed to be given on internet-connected computers. I would change the instructions to enable every student full educational access to the Internet at any time during the tests. The Web is certainly full of strange beasts and of course we should protect our students from accessing those, but everything else, sites like YouTube that are today blocked by many school systems should be accessible, spreadsheets, Google searches, Wikipedia, sites designed to help students take tests, yes even help from a friend. Anything of educational value.

When our students leave our classrooms and go to work they will live in the real world where they will have such full Web access. Don’t we want them to learn those things that will enable them to use that access to solve problems and to learn to do things? Don’t we want them to use this incredible new tool that is defining 21st century work and life to its fullest advantage? Don’t they need their schools to prepare them for the kinds of tasks they will perform which almost certainly will involve the Web?

The Common Core tests are redefining education. If we give our students full educational Web access when they take these tests, they will be able to take charge of their learning, and they, not the tests, will redefine education. Oh, for sure the tests will have to change and our classrooms and teachers will most certainly change, because there will be no reason to teach our children the long division algorithm any more then we should go back to teaching them the square root algorithm. They will not get “What is ____?” questions whose answer could be easily found on the internet. They will have to get “What if…” questions that will challenge their creativity, demand knowledge, and engage their insight. The tests will have to be designed to represent the real world and not an artificial one that has produced profound and fatal flaws as described by Steve Rasmussen.

Such a minor change in the instructions will change our schools in dramatic ways and open a floodgate to creativity for both our teachers and our children. Magic wands are simple things, but they have the capability to make wonderful happenings. Opening the internet door to our tests and our schools would profoundly change education.


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 Learning process where Stanford’s D School tells students to empathize “you observe, engage, watch and listen.” In short, you begin the creative problem-solving process by looking at problems in human terms, from the standpoint of the people who have the problem they need to solve. In the Functional Thinking problem solving process that mirrors Design Learning, we ask students to visualize the problem to make a problem real, see it in context, and picture the kind of answer they will be looking for. We believe that students who visualize a problem will naturally empathize with it.

There is an even greater role that empathy must play in our schools. For if we want our students to care about solving the problems we assign to them, then we must develop and assign those problems empathetically. An empathy-based curriculum may seek to find those few amazing problems that nearly every student empathizes with, or problems that have such a potent human emotion attached to them that students engage immediately. Those great projects are worthy targets. But there is another way.

Imagine instead a future STEM curriculum made up of thousands of creative problem-solving Labs so that students can, in large measure, choose those they want to work on, those they find interesting, those they have observed, those that engage them, those they already have established empathy with. If our goal is no longer the mere acquisition of knowledge, the development of personal libraries of information or techniques, because such libraries are available to all on the Web, then we can focus on practicing creative problem-solving, the skills they will need for the digital age. And they can build these skills because they have also developed the empathy to truly understand how to solve problems in the digital age. Those of us who create these Labs must thus hold empathy as our core vision and first step.

“Algebra before Acne”

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 that should be taught from the very beginning. He believed that the great abstractions which make mathematics so powerful and so beautiful could and should be taught from the very beginning. He would not have been happy with today’s Standards though he helped write the original National Council of Teachers of Mathematics (NCTM) Standards. He would not have been happy to see the Common Core push variables down only a grade or two from their traditional place in the math curriculum. He would believe that we continue to silo variables and to make them difficult, very difficult, for so many kids. He would not understand why we do not apply technology to represent x.

Unlike Jim, most of us continue to think of variable as an abstract idea which Piaget decided required students to be in the Formal operational stage (the acne stage). No doubt, we have heard students ask the painful question, “What is x?” when they do get to “real” algebra in 8th grade, for which we likely has no succinct answer. And of course, math historians make the excuse that is took 800 years for the “unknown” of al Khwarizmi to become the variable of Descartes. So, we let it go, think Kaput a dreamer who would oversimplify this abstract idea to present it to even young kids. But we would be wrong. We would miss his genius and the real point. And what is more, we would miss a great opportunity to give all of our students interesting problems to solve.

We are so wedded, in the standard math curriculum, to dealing with and thinking about variables as continuous quantities that we do not recognized the concrete power and utter simplicity of dealing with variables as discrete quantities. Students have no problem with discrete quantities; after all arithmetic is all about discrete quantity. I did not recognize this profound intuition until a decade after Jim’s untimely death, when I started working on learning math using spreadsheets. Spreadsheets, born of the digital world, are a natural medium for dealing with discrete quantity. Variables are represented by tables of values, generally by a column or perhaps a row of discrete numbers. To operate on the variable is to operate on each number in turn. Functions are discrete and usually link one column to another. Indeed, in the application of math today in both STEM and business, spreadsheets are the primary quantitative vehicle, and discrete variables are the standard quantities. Spreadsheets are digital tools and as such are built to handle discrete variables and functions.

If we ask students to build a table of values from 1 to 10 on a spreadsheet, and label that column x, then “What is x?” It is simply the name of that column! It is a variable because it can take on different values, any of those values. And if we ask them to make a second column that adds 2 to the variable x, they will have no difficulty doing that, creating a function of x, labeling the second column f(x), a machine that adds 2 to every value of x. First graders can do this. We can teach algebra from the very beginning if we use discrete variables. Spreadsheets make it easy, and you can do it at any level. Jim Kaput was right, we can and we should teach algebra before acne, way before acne. Try it!

Minkowski’s Connections

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 process, physics as well. I had been interested in Hermann Minkowski’s contribution to the theory of relativity from the time I wrote my master’s thesis on the teaching of special relativity to high school students over 50 years ago. Both the human story and the physics story are fascinating.

Minkowski had been Einstein’s mathematics professor at ETH Zurich also known as the Polytechnic. Neither one, teacher or student, thought much of the other’s gifts. Soon after, Einstein went off to work in the Swiss Patent office and Minkowski to teach in Göttingen. Einstein published his paper on “The Electrodynamics of Moving Bodies” in the summer of 1905 and despite its publication in a respected journal, the paper was far from an instant hit. Three years later, Minkowski gave a talk on a new formulation of Einstein’s work. He linked time with distance to envision a four-dimensional world out of which Special Relativity naturally flowed. Einstein was not impressed, he thought Minkowski’s work a mathematical trick that did little to improve an understanding of the physics. It was not until later when he incorporated Minkowski’s ideas into General Relativity that he came to appreciate their profound importance. Sadly, by that time Minkowski had died of appendicitis at age 44 just 4 months after his presentation.

He began that talk with what I consider some of the most beautiful and powerful prose ever used to describe science:

“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

He then lays out a formula to take measure of this new space-time. In the Newtonian world, distance is measured using the “sum of the squares”. In two dimensions, the square of the distance (s) from one point to another, envisioned as the diagonal of the right triangle, is the sum of the squares of the two sides. s2 = x2 + y2. To find the distance we take the square root of the sum of the squares. In three-dimensional space s2 = x2 + y2 + z2. Minkowski combining time and distance defines four-dimensional space as s2 = x2 + y2 + z2 – c2t2, where c is the speed of light and ct is just the distance light would travel in an interval of time. This looks like a simple extension of the Pythagorean Theorem with the clever way of turning time into a distance by measuring it with light making it equivalent to the other dimensions. But there is one surprising element in Minkowski’s equation, the minus sign. Why does he subtract distance-time, shouldn’t we be adding it? This subtraction puzzled me greatly. I had tried to find out the answer, asking physicists and looking for it on the Internet, but no luck until I ready Muller’s book.

For he explained that Minkowski thought about the fourth-dimension using an idea physicists and mathematicians were already well versed in, imaginary numbers. These oddly named quantities, which seem so esoteric to most students are surprisingly valuable ideas. If the t dimension is imaginary then mathematicians and physicists have well-defined powerful ways of dealing with it and thus with four-dimensions. We learn in elementary school that the square root of negative one is an imaginary number that we write with an i, and thus the square of an imaginary number is -1. So that’s where the equation comes from, a different way to think of the 4th dimension. We are still summing the squares, but since the time dimension in this fourth dimension is an imaginary number, its square is negative. This small connection not only solved my long-held puzzle, it enabled me to understand and make some amazing new connections. I will leave you to discover more about this one and perhaps to find new ones of your own. It is connections like these that drive our creativity and enable us to build our abstractions. As you solve problems in this digital age, look for such puzzling ideas to make such new and wondrous connections.

David Hilbert, widely acknowledged as the greatest mathematician of the 20th century, who very nearly beat Einstein to the fundamental equation of General Relativity, wrote this for Minkowski’s obituary:

“Since my student years Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us.”

Learning to Swim

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 required quarter of swimming classes. Now, unfortunately, I am a slow learner, particularly on the physical side, and I failed my final swim exam that first quarter. I had to take the course again the next quarter.

I went into my first class in the old Bartlett Gymnasium pool that
cold winter day and dutifully lined up at the edge of the pool like my swim coach told me to do, the edge of the pool at the deep end And I patiently waited instructions. None came. He, instead, moved in back of me and pushed me in. “Sink or swim,” he said. I started to swim. I had been taught the strokes, I had been taught to float, I had been taught to kick and to breathe before. Now I just needed to be pushed off into the deep end where I had to use those skills, conquer my fears, and swim. I am still here today, so though I am not a world class swimmer, I do know how to survive in the water and even enjoy swimming.

I think about my swimming lesson when I talk to students, teachers, and others who fear spreadsheets. I want them to get a couple of simple skills under their belt and just jump into the deep end. No practice dogpaddling in the shallow end of the pool, no kicking exercise holding on to the edge of the pool, no holding your breadth and swimming underwater across the pool are going to turn you into a swimmer. If you want to learn to swim, practice breathing in the water, kicking, and using your arms in a crawl stroke then jump into the deep end of the pool, jump in and swim.

The same holds true for spreadsheets, arguably the most important and most feared business technology we have. Practice using a rule on an input cell to put its output into another cell thus creating a function. Use relative and absolute addressing to copy those functions and build models combining them. And start with a parameter table to enable you to ask, “What if…” of those models.

Go to and click on Tour to see how simple models can enable you to build and do math across the curriculum. So, I say to teachers and students, “Jump in! The water is fine. You won’t drown. And I promise you will learn how to swim in the spreadsheet ocean and come to really enjoy learning math with spreadsheets in this Happy New Year.”


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 and judges teachers on things they cannot control, on a lack of money, on cell phones. The list is endless, personal, and the results exhausting.

If we seek not blame but instead deep cause, we will see that much of the pain teachers are now rightly feeling is due to the new digital technology, technology that has had a positive affect on most other aspects of our lives. Digital technology in the form of cell phones not only distracts students, it invades teachers’ lives, for they feel the need to answer students queries 24/7. Email which has become a primary form of communication opens the door to parent-teacher and student-teacher dialog again extending the school day and adding burdensome demands. Powerful computers now enable standardized testers to analyze data and grade teachers on student progress. Shared syllabi on common instructional platforms rigidly sequence and control teacher lessons removing any opportunity for creativity and innovation. The scope and sequence that used to weight down teacher desks collecting dust in the bottom right hand drawer are now online controlling the day.

Word processors, while making it easier for teachers to read written work, also make it easier for students to write more and to demand that teachers immediately read, respond to, and grade it. PowerPoint presentations of content are not as easily erased as chalkboards, saving class time but demanding more preparation time. And like doctors today, teachers too, feel the need to be up on the latest info available on the Web. Last year’s lecture notes just won’t do any more.

New technologies can be insidious. While as teachers we may worry about big tech issues like flipped classrooms, online assignments and tests, personalization, and the need to ensure our students have equitable tech treatment; we must also prepare for the future of blended classrooms and online courses. Digital technology has made our lives harder, much harder.

Technology is always like that. It starts out by making us work harder. It requires us to follow a learning curve. It demands we learn new ways to do old things that do not make them easier or save us time. And it is invasive, causing us to add new problems like student security to all the old problems. This is where most teachers at both K-12 and college find themselves, fighting digital technology instead of enjoying its benefits. For powerful new technologies, technologies that change our lives, require us to not just adapt our old forms, methods, and content but to rethink them. It requires us to learn to fully use technology and integrate it with what we are doing. And it requires us to imagine our role in a new way.

What can you do to make technology work for you and for your students? What can you do to make your job easier instead of harder? What can you do to prepare your students for a world you were not prepared for?

The Los Alamos Primer

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 newly arrived scientists and technicians on the fundamentals of the Manhattan Project they were involved with. It is a fascinating book, released from its top secret status only in 1992. It is a fascinating overview of the most important concepts in physics in 1942 that were both fundamental and essential to building the bomb.

I suggested it to my son Brenan who was assigned an intro high school physics course at a private high school without any time to prepare or text or materials to use. He planned to combine some of its simple lessons in physics with lessons in history and morality. For just as the physicists at Los Alamos questioned its ultimate usage, he sought to engage students not only in physics concepts and real-world problems but in the responsibility that physicists and all of us have in use of the ideas we develop or support. It was a brilliant and creative way for him to begin a physics course and engage his students in this great subject. Unfortunately, his classes started on September 10, 2001. Needless to say, he ended up pivoting in a different direction.

But it is still well worth thinking about the lesson and ask, “How do we connect our physics classes with the world our students are living in?” “What do our students have to know to prepare themselves for their future?” “Would you rather learn fundamental physics by dealing with the atomic bomb problem or by learning the definition of time, distance, and velocity?” And when we think about engaging students during those critical first six weeks of a physics course gets me wondering the same about our math courses. What kinds of projects can we develop into our math classrooms to bring them alive, to insure our students crave learning math and do not sit back asking what is perhaps the ugliest of all questions, “Why do I have to learn this?”

When I think about project-based-learning I picture The Los Alamos Primer, Leo Szilard’s The Voice of the Dolphins, and Brenan’s moral questions.