“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!


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 power. The Web is open and free. And the two person desks are arranged in cloverleaf pattern with 4 tables extending from a central pivot. Most of the folks are studying or working on their own, some are working together across from each other. The café keeps the food and drink flowing, and often free food sits atop the long professional magazine counter to provide tempting refreshment, nourishment, and up-to-date research.

But, like nearly every other college and university space, even this one, devoted as it is to the future of education, is a cloister in the fullest sense of the word. Cloister comes for the Latin for enclosure, a place separated from the real world, a place where devotees poured over manuscripts, copying and recopying documents, listening to lectures, and joining discussions, intellectual and otherwise. It was and remains a world devoted to nuance, and like all schools much of the talk is about what is demanded of us, what did he or she mean by that, what will we be evaluated on. So, like students everywhere, these Harvard education students plug miniature speakers into their ears, consult their latest messages on iPhones, and turn to laptops to read, trying to concentrate on understanding, absorbing, and finding meaning in ideas that for the most part are inherently ambiguous and often irrelevant.

Like the cloisters of old, today’s students, like all previous students, are for-all-intents-and-purposes monks, listening to a master speak, then working for the most part on their own, in great buildings where silence is golden and talk is in whispers. When they complete this great institution, they will walk in crisp lines wearing medieval costumes and receive pieces of paper printed in elegant old fonts that say they have behaved according to the ancient rules and rituals that govern schooling.

As was true in the medieval religious cloisters, change in our learning institutions is astonishingly slow. Few new ideas penetrate the walls segregating them from the external world. Rather, our schools like the monasteries of old, seek to perfect their mission and processes. Like the cloisters they are modeled after, our schools are profoundly regulated systems with clocks and bells controlling movement and calendars defining activities. To meet ever growing demands, they have become overly optimized, leading to exhaustion, criticism, and failure. Their mission no longer relates to the modern digital world, and yet they remain steadfast in form and substance, recycling old ideas again and again.

Surprisingly, the solution is at their fingertips. At the Ed School library, most of the students have their phone sitting on the desk right next to their computer. They consult it often. They are looking for the latest news, for the latest posts, for information. They are looking at it to take a break and to get educated, to answer a message or to check on their plans. Yet, as ubiquitous as these small connections to the Web and the outside world are in graduate schools and in elementary schools, they are rarely, if ever, viewed as learning tools in our classrooms. The cell phone in their hands connecting them to the outside world and the computer on their table giving them tools to work with that information are so clearly objects of continuous learning, and yet they are not used for school learning. How could they be, when our schools remain cloisters? Our task is to break down those cloister walls, to open our schools to the real world, and to enable our students to use technology for learning as they are already trying to do.

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.”  https://en.wikipedia.org/wiki/Hermann_Minkowski

Collaboration is Cheating?

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 if their paper looks like another. In English or social studies, if you are caught copying something or someone without attribution then you are plagiarizing and treated as if you have committed a crime, cheating. And if you are doing a project with a group of students, be sure your work and effort are your own not the work of others or you are cheating. We are training students from the earliest school age to work individually, to “do your own work,” to not cheat. We are still teaching our students to be rugged individualists, independent, self-motivated, and self-reliant. We are teaching 19th century skills.

Today, collaboration is one of the four C’s skills because it is seen, in survey after survey of business, to be critical to digital age problem solving. Creative problem solving is considered a group activity today, and business would no more consider isolating individuals in the workplace than taking them off the Internet. Offices and universities are designed to breakdown silos, to have courtyard and corridors, like this design of the new Google headquarters in London, where people can constantly meet, share ideas, and engage in group problem solving. The best employees are considered to be the ones who work well in teams, who are good collaborators.

Yet, we educators act as if collaboration is either a skill we are born with or magically gain when we require it. Despite the importance given to teamwork and collaboration in sports, we still do not consider it a skill we should learn in school, a skill we should practice in school, a skill that is no different from reading or numbersense. Learning to collaborate in school as a central mission certainly requires us to rethink education from the ground up. But even if we are not ready to take on that big task, we can start by making our classrooms meeting places where silos are not just torn down between subjects but torn down between students, where students are supported and encouraged to learn to collaborate.

“Just try it on!”

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, highest on my list, is when Flor, who has just started as a housekeeper at the Clasky home, comes back to her own house after witnessing the mother embarrassing her daughter over her weight by buying her clothes a size too small. Flor, who had never learned English, asks, no she demands, that her daughter teach her how to say, “Just try it on!” They repeat it in synchrony over and over again. Flor arrives at the Clasky house before dawn to let out the new clothes, then she wakes Bernice up and holding up the clothes, she speaks English for the first time. “Just try it on!” “Just try it on!” she demands again and again until Bernice finally succumbs and the smile returns to her face.

I think of that poignant scene when I contemplate students and teachers in math classes today, for in conversations with teachers, parents, administrators, and yes students, I hear story after story about classrooms filled with kids who have given up on their ability to learn math, are just plain bored, or who see no reason to learn the math they are being taught. Teachers, who are themselves bored with a curriculum so heavily structured and predetermined it leaves no opportunity for creativity or even fun, try to pretend they are not. It is clear that teachers, students, parents, and administrators are dealing with a subject that no longer fits, a subject which has shrunken beyond recognition, a subject no longer relevant, no longer meaningful. Though, it claims to be conceptual, to enable students to learn to think and to solve problems, in reality it is mechanical in an age where machines have taken over most of those functions. It is too small for our digital age students and teachers who are wanting more, more relevance, more creativity, more fun, more learning. And it is too small for our classrooms where teachers want more discretion and more opportunities to engage.

In What if Math we have done more than just let out the old curriculum. We have started from scratch to build a digital age curriculum. Did we get it right? Do your students enjoy it more, learn more, feel better about their math ability? Are you having more fun? There is only one way to know, so I ask you, “What have you got to lose?”, and as Flor coaxed Bernice, “Just try it on!”, “Just try it on!”

The Democratization of Knowledge

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 course of human thought. For in this first decade of the second millennium we will be most affected by the democratization* of knowledge. Before this decade, knowledge was rare, a scarce and valuable commodity to be bought, sold, and transferred in schools, workplaces, and books. It was owned by the few and sold to the many. Our schools sold knowledge to students in the form of teacher presentations and textbook descriptions. Our workplaces sold it as apprenticeships, training, and professional development. And our encyclopedias, sold these compilations of all knowledge door-to-door, and libraries were handsomely constructed to give all the opportunity to acquire it in printed form.

During the first decade of our new century, knowledge ceased to be owned by the elite, ceased to be rare, ceased to be precious. It is now owned by all, ubiquitous, and, for all intents-and-purposes, free. Wikipedia, a tiny but highly significant reflection of this revolution, began in 2001. Its rate of growth, as we can see from this fascinating graph, inflated around mid-decade and is even now slowing down as it has filled up with much of the past knowledge. And Wikipedia is but one of billions of posts and sites that today freely contribute knowledge in every sphere of human endeavor. For knowledge continues to expand at, what to most of us seems like an accelerating pace. Knowledge for the first time in human history, and for as far as we can see in the future, is now democratically produced and distributed.

From an educator’s perspective, this revolution profoundly changes our job. We can no longer be paid to carefully dole out the precious knowledge, mainly in text form that we acquired from our sweat and treasure, for it is now freely available on the Web. As painful as this may be to those of us who are or have taught, we will no longer be valued for what we know and show, but instead for how we help our students find what they want or need to know.

Applying this to problem solving demands we change our role from telling to asking, from answering to encouraging, from giving to wondering. Our job is now to encourage our students to use the Web to find things out, to learn how to solve a problem, to search for conceptual understanding, to explore possibilities, and perhaps most of important of all, to choose problems of interest to them to practice and build their skills. While students likely know Khan Academy and YouTube videos, most still need to be invited and yes encouraged to use all the tools available on the Internet to solve problems (in our case to learn to use spreadsheets to solve problems). In the digital age, they and their collaborators need to learn to aggressively find and use the digital opportunities to work on and solve problems. They need to learn to separate good from bad, valuable from worthless, helpful from wasteful. They need to be encouraged to find things out for themselves, to try again, to try harder, to keep trying. And they need to be told to get help from their peers and from experts if they need it. They may, of course, at times need some direction, need us to bend their path a bit, but we will have to remember it is their path and not ours!

*I borrow this term, as it applies to education, from my dear friend Jim Kaput.

The Challenge of New

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 6 weeks on a tramp steamer across the whole of the Pacific Ocean to Seattle to meet the father he had not known, who brought him and his mother to Chicago to settle in a new land. This immense voyage in space and in time must have been, disorienting, yet that is far too weak a word for the changes he had to confront. He left a town shattered by war, likely without electricity, telephones, or motor driven transport, an agrarian country where farming, as it had been done for centuries, was the main industry and where industrialization had virtually no impact. Transported into a new world by steam railroad and ship to an industrialized world with electric lights, indoor plumbing, central heating, electric streetcars, telephones, and an abundance of food; he would soon see know air travel, radio, refrigeration, skyscrapers, and of course automobiles. During his all too short lifetime of 45 more years, he would travel on a jet plane, use the modern air powered tools of dentistry to save people from the pain of tooth decay, watch television, buy a house in the suburbs, and travel comfortably throughout much of the United States.

To my lasting regret, I never thought to ask him how he navigated such profound changes, changes in technology, in community, in governance, and in economics. He was just 20 years-old when the Great Depression started. He was 33 years-old when, as a brand-new father, he went into the army. And he was 43 years-old when he left the dirty crowded big city to join the suburban green migration. I often think about the profound and yes unprecedented changes he lived through as I consider what we are going through today. Like him, I am, and actually all of us are, experiencing changes we could never have even imagined in our best science fiction, changes in the nature of work, in the technology we use and the capabilities and comforts it brings, in our political systems, our communications, our personal interactions, in who we marry and what we spend our time on. I wonder often how he felt about those last generation changes, how disorienting they were to him and to others, and how that concerned him. I wonder, as I talk with family, friends, and associates, how our disorienting profound and unprecedented changes compare to his. I wonder if the first Americans, who invaded virgin land, faced with unseen animals, unfamiliar plants, and unknown geological challenges. I wonder if we are, after all, so very different, and if as humans we are most fortunate to have the gift of a flexible mind to enable us to change, to meet the challenges of the new.

My father was by most standards an incredibly successful human being. He was happily married, had three boys who have all had their own families and made useful contributions to their communities. Despite coming here with no English and likely little formal educational experience he learned, becoming a dentist, a community leader and in his later years started a second career as a teacher building a revolutionary new program to educate dental assistants. He not only served his new country and his various communities, he played a key role in their growth and transformations. I can only hope his grandchildren and great grandchildren should be so successful.

He managed to do all of this during a time of change in every area of his life. I have, therefore, hope that in our time of amazing change, when our world, our jobs, our technologies, our communities are so profoundly transforming, we take comfort that our generation is not alone. Our forefathers too have, from time to time, succeeded in navigating such amazing periods. For, I am sure my father would agree, these times of tumult are also times of great energy, these times that look so scary are also times of promise, these times when we cannot see the future are ones we can instinctually navigate. I picture my dad riding on a wobbly rail car across the great stretches of Russia comforting his very frightened mother and believing in a better future. This was his gift to me, and it is the gift I pass on to my children, grandchildren, and all my students. It is why I am so deeply sure that technology not only shakes up our world, it gives us the power to reinvent it. It will give us the power to reimagine education and enable all our children to thrive in their new world.

Real Feedback vs Artificial Feedback

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 Davidson designed and built this first in a lineage of educational products.

Today, most of our digital learning programs pretend to be more sophisticated. They deal with multistep problems and less rote practice. Their feedback is less shoot ‘em up, and their interactions are no longer pure gamey response. Yet they remain wedded to the same kind of “artificial” feedback where the computer knows the answer and students guess it. Students may look like they are doing authentic problem solving, but they expect their computers to check each step they take, and with the same artificial feedback keep them on track to solve the problem. The computer now knows not only the answer, the product, it knows a particular process as well, and it tries to ensure that students get the problem right by following that process.

Unfortunately, in the real world outside of schools, we seldom get artificial feedback to help us solve problems. We must learn to use real feedback to keep us on the right track, to help us solve real problems, especially those requiring creative solutions. We need to learn to build real feedback into spreadsheets to catch selection and rule mistakes, so we learn to build in redundant calculations as checks. We make graphs to give us feedback, to see whether all our numbers fit the pattern. Though spreadsheets may seem intelligent, we have to constantly find ways of assuring that we have not made mistakes in the models we build, so if we are good spreadsheet developers we us headmath to check whether our algorithms make sense.

Students need to be learning to build feedback skills into their problem-solving processes and strategies. Artificial intelligence may be an interesting arena today, but our kids need to learn to create and use real feedback and not rely on any program’s artificial feedback if they are to become good problem solvers.

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 whatifmath.org 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 Bit

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 few of us would today recognize. Claude Shannon grew up in Minnesota when radio was becoming the means of communication to all, broad cast. It was the age when sound was added to movies, when phonographs and records storing sound became a must in every home, when the first facsimile machines were used to transmit photographs and text, and when everyone could take their own pictures with the Kodak Brownie camera.

Each of these transforming inventions used a different analog means of storing or transmitting data. Analog data is continuous; on a graph it is a line, sometimes smooth, sometimes jagged. All of these inventions had to deal with the problem of noisy data and of separating the noise from the data. This was the problem Claude chose to work on. Before him the common way of dealing with noisy data was to turn up the volume. If the radio static was bad, make it louder. If the picture was muddy, increase its contrast. If the telephone call was hard to understand, yell.

To solve this problem of noisy data both in storage and in transmission, Claude came up with a truly brilliant, surprising, and original idea. Think about all data as digital. Think about it as being broken down into discrete bits, a collection of just 1’s and 0’s. No longer would data be stored or transmitted as a wave like the grooves in a phonograph record, a continuous quantity. In Claude’s new world it would be like atoms, discrete, separate, objects. Bits, the word he chose, came from binary digits; where his “atoms” took two and only two forms. It was transmitted in bits, stored in bits, and processed in the same bits. He then figured out how to find corrupted noisy data, how to minimize it, and how to replace it. When he died at the turn of this century, his vision for data was just becoming an overwhelming reality. Because today, we have the bandwidth, the storage, and the processing power to handle all data digitally, and the processes that make noise no longer a problem we concern ourselves with.

Isn’t it time our schools deal with its noise problem by becoming digital and focusing on discrete data? Today’s “analog” continuous variable algebra makes the concept of variable abstract and difficult for many students to understand. It requires students to learn a complex set of special cases to solve abstract equations. It turns algebra into collection of mechanical processes focused on cases that are easy to solve. What if we were to follow Claude Shannon’s lead and treat variables as discrete, digital quantities? Spreadsheets make this easy. Variables become concrete, easy to understand, iterate, build into functions, and use those functions to build models. They give us the means to focus on real, messy, interesting data to solve fascinating problems.

Try this new way of thinking for yourself. Go to our Tour to see apply the digital world to algebra. Try it with your students. Tell us what you think.


Revolutionary Math

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 on the Cape and then take a lovely drive to empty beaches, delicious lobster rolls, and of course a bookstore or two. It was on that last stop, just before the bridge, that I came upon a hidden treasure, a math textbook from 1788. The author Nicolas Pike entitled it, A New and Complete System of Arithmetic: Composed for the Use of the Citizens of the United States. Pike, proud of his brand new nation so recently created, says that it needed a book to educate its newly minted citizens in mathematics.

It is however the opinion of not a few, who are conspicuous for their knowledge in the mathematics, that the books, now in use among us, are generally deficient in illustration and application of the rules; of the truth of which, the general complaint among schoolmasters is a strong confirmation….as the United States are now an independent nation, it was judged that a system might be calculated more suitable to our meridian, than those heretofore published.
Pikes Arithmetic, Nicholas Pike, 1788, Preface.

The book follows, for the most part, the sequence and topics laid out by Leonardo of Pisa in Liber abbaci. Though it lacks pictures, it is full of contemporary problems, problems faced by farmers, shopkeepers, traders, surveyors, sailors, and even militia. It is full of such real-world problems and full of tables to help the users to calculate the answers to those problems. It was a reference book as well as a textbook. It was designed for this new country, “suitable to our meridian” including decimal currency.

Over the past 230 we have desiccated this work, taking out its focus on problem solving in the real world, both in the problems given and in the tools for solving them. I love the thought that in What if Math we are returning to Nicholas Pike’s 1788 vision, to focus school learning on the kinds of problems students will need to solve and giving them training in the tools and skills they will need to use. Yes, this was a revolutionary vision then and it is a revolutionary vision today. But it is a vision for a nation whose promise has been: to enable all of its citizens to thrive. I would love to have been able to take Nicholas Pike on our Tour. I think he would have liked it.


This is Why I Love Graphs!

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 is a perfect example of the power of visualization, of graphs, to tell a story, and the reason we consider them fundamental to our What if Math Labs. Take the Tour to see more.


Math as a Laboratory Science

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 without messing with paint, clay, or paper. Yet, we easily imagine learning math without experimenting. In fact, it is rare that students ever do a math experiment or think about math that does not have a “right” answer.

I learned to experiment from one of my great teachers, Walt Hunter. I even had the great good fortune to also being his chemistry lab assistant my senior year in high school. That I did not fall in love with chemistry was not his fault; I had just loved physics since I was 7 years old. But I did fall in love with experimentation, and like Walt I gained a deep belief that learning to experiment should be an essential aspect of every student’s education. I brought that belief to my physics classes replacing teacher demonstration with student experimentation. I took it to my Jr. High math classes, where I made my students worksheets that let them play with numbers and mathematical patterns. I carried it to my focus on manipulatives as a math coordinator, and I bring it to What if Math.

Using spreadsheets as basic learning tools for math has many advantages, but I think the most important one is that it turns math into a laboratory science. It enables students to experiment, to build and iterate models, to test those models, and to apply them to real-world data, complex rich data. It lets them ask and answer what if… questions. And it turns them into explorers who love to use math and who gain Walt’s experimental habits of mind, the thrill of discovery. It is this, I now know, that Lynn Steen saw when he described mathematics as the “Science of Patterns,” for math does belong to STEM/STEAM after all. So, when you plan your math classes, imagine your chemistry teacher, and the twice weekly labs where you learned to act like a scientist, to explore, to discover, to ask, “What if…”


*Portrait of Antoine-Laurent Lavoisier and his wife by Jacques-Louis David, ca. 1788, Wikipedia

Rows and Columns

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 of the head of the teacher talking and writing on the board. Principals should ban this arrangement, ban rows and columns classrooms.

How is it student centered? How does it enable students to work together on projects or problems? How can it make our modern schools interesting places where students want to be? It looks like the world of business did in the 1920’s, not today. It is a constant reminder that schools are places where you sit at attention and focus on the quietly listening, not active, not collaborating, not communicating with other students. It is a static place not a dynamic place, a place where the teacher is the focus and not the student, a place you cannot wait to leave and not do not want to go. The moment you walk into such a classroom you know you are in the past and not preparing for the future, you feel you are in a problem-making room and not a problem-solving one; a room where students are not valued and where creative thinking out-of-the-box is not encouraged.

It is a room I do not want to be in, why would I want our kids to be in either.


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.



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 curious, to question, to experiment, to wonder, to imagine what we will do or become. Isn’t this what it means to reason quantitatively?

Curiosity is certainly the most human of traits. It is the reason we explore, the reason we invent, the reason we question, the reason we love mysteries and games. We are naturally curious. And curiosity is at the heart of mathematics as well. We wonder if particular numbers form a pattern, we want to know whether all triangles that fit in a semi-circle are right, whether all right triangles have two sides that when squared are equal to the square of the longest side. We wonder if our business will make money or our budget will enable us to buy that new computer. We are curious big and small, is the universe infinite or finite, is the chance of winning the lottery worth the price of the ticket, is global warming really that bad.

At What if Math we are all about curiosity. How many of the products in a 12 by 12 times table are odd numbers? If we choose any whole number and divide it by 2 if it is even and multiply it by 3 and add 1 of it is odd will repeating that pattern always make a sequence, 4, 2, 1…? What if we make a table using 1 simple rule, add the two cells in the row directly above it? What does the graph of a quadratic equation do if I change the b term? Was Ted Williams or Joe DiMaggio a greater hitter in the 1941 season? Was Napoleon right about using the Great Pyramid to build a wall around France or Moore right about the exponential rate of growth of microprocessors? What does absolute value do to the graphs of polynomials? Is the rate of change of C02 increasing or decreasing? Should I lease or buy that new car?

If curiosity is so central to our lives, then I am curious to know why it doesn’t appear in our math standards or in all too many of our math classrooms. I am curious to know whether students can care about what they are learning in their math classes if they are not curious about the problems we give them or the concepts they are supposed to learn. I am curious to know whether learning math can be fun for kids if they are not curious about it. I am curious to know…


Personalizing Learning

The Bill & Melinda Gates (Gates) Foundation and the Chan Zuckerberg Initiative (CZI) have recently initiated a joint effort to envision breakthroughs in education. While their dream is laudable and no doubt exciting, their Request for Information, it seems to me, looks backward rather than forward. Fortunately, it gives us another opportunity to talk about defining a new vision for problem solving in the digital age.

The RFI focuses on personalized learning and imagines:

There are promising field-developed approaches emerging that help teachers to address individual student needs by mirroring the same personalized approaches used by the best 1:1 tutors—especially when combined with expert, well-supported teachers in an active learning environment. Highly personalized learning experiences and tools have the potential to analyze student responses to understand barriers to student learning, provide immediate feedback, and apply immediate and effective remediation to students when needed.

Though they call for “active learning environment(s)” their view of technology and personalized learning looks to me more like an automated factory than a school of the future. An intelligent machine like an intelligent tutor, “analyze(s) student responses” then provides “effective feedback” and finally applies “remediation”. This description sounds as if we are to think of the student as a passive recipient of the machine’s intelligence, the machine learns and passes this learning onto the student. It is an all too common vision of the future of education, an automated version of 19th century educational perfection, Mark Hopkins on one end of a log and the student on the other. How does it help students become 21st century problem solvers? How does it enable them to think critically and creatively? How does it cause students to be active engaged learners who work in collaboration with other students on projects and problems in which they and not their machines determine whether they have made and learned something of value? How does it give students choice, the real power of personalization?

Technology, in every other walk of our lives, enables us to do old things more easily and competently; relieves the burden of tasks we no longer need perform; and makes possible things we have not before imagined. Technology, if it is to transform education, must enable students to more easily do old things like multiplying 3-digit numbers or solving equations. It should get rid of unneeded things like all the paper-based algorithms and practice students will never find useful. And it should enable students to work with the most powerful math concepts to find patterns and solve real world problems we never dreamed they could tackle before.

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


The Hardest Question

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 a bleating call for relevance. For students, it is one more reason to ignore or at best inattend to a teacher’s presentation. For teachers, it reaches into the deepest sense of who we are and what our job is. Our inability to answer it directly and cogently can feel like a failure in the traditional analog classroom where our primary role is threefold, present concepts and skills, develop examples as prototypes, and motivate students to attend and learn. Our inability to give meaningful answers to the hardest question destroys their faith and our ability to motivate.

I always found the effort to motivate learning to be the most taxing part of my job as a teacher. Learning is work, pure and simple, and to get students to learn we have to get them to work, motivate them by placing what they have to learn into context, making it part of an interesting story, providing a strong rationale, finding a reason for students to care, or as the last resort bribing students with grades.

In our digital age the Web is the resource center, a library for concepts and skills. Whether the result of a Google search, a YouTube video, or Facebook connection, students today can find information, concepts, and even demonstrations of the skills they are to learn. The traditional roles we long assigned to teachers as presenter and prototyper are now generally obsolete. (Technology has even replaced most of the traditional teacher evaluation role with computerized high-stakes tests.) The motivator role remains, which brings me back to the hardest question teachers face all the time.

“Why do I have to learn this?”

Difficult, if not impossible to answer in the traditional analog classroom, it can be even more painful in a digital one that follows the traditional form, because in this time of accelerating change, amazing tools, the Web as an infinite library, and cell phones as communicators, our usual answers are all too often irrelevant!, Therefore, in the digital age, the student who has to answer the hard question. The student has to find his or her own meaning, we cannot give it to them. And to do that the student obviously has to have interesting assignments and choice.

The only way I know to give students interesting choices is to make our assignments project-based! So when you next get that hard question, think digital and think Project-Based-Learning.

The Problem with MOOCs

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 better to do a MOOC or two bringing his wonderful style of teaching and presentation of important physical ideas to more people. But he was not at all interested, and as far as I can tell several years later has not done any.

“Why” I asked. “Because you can’t change them.” he replied. As he explained, one of the most wonderful aspects of teaching a course year after year for a great teacher is the opportunity, indeed the necessity, to change and adapt the courses in general and the presentations in particular. His reaction brought back a vivid memory of my first couple of years of teaching high school physics. I usually carefully prepared my lectures which were the standard fare for most of my classes. Occasionally too busy, too tired, or too lazy to develop a new one, I would grab my lecture notes from the previous year which I thought pretty good. The class usually started all right, but I soon got into trouble. The coherence was gone, the presentation no longer seemed to make sense to me. I don’t know if my students realized that I was stumbling, they were too busy taking notes, but I did. So, I would stop lecturing, told my class what I had done, apologized, would come back the next day with a fresh lecture and gave them time to work on their assignments. One of the things that makes teaching such a great job is the year-to-year, day-to-day, and even student to student opportunity for improvement, for growth, for learning. This has not been true of curriculum.

MOOCs like textbooks are expensive to produce. They are linear, moving from topic to topic in a standard form, a continuous line of lesson following lesson. They are thus difficult, often impossible, to update or change. Once created, except for minor revisions they are for all practical purposes, fixed. Yet, the world is constantly changing, and even more importantly students are constantly changing. A fixed curriculum or presentation cannot work. It will no longer work to expect textbooks to have a 7 year lifespan. Nor will MOOCs, made once and used again and again, work either. The analog continuous linear sequence of lessons that represent a course is no longer functional in the digital world.

The digital world is a discrete world. It needs education to be flexible, easy to change, constantly renewing, and growing. The metaphor for the analog age and the MOOC is the book, done once and then published. The metaphor for digital age educational content is the newspaper, renewed and reimagined everyday. One is fixed, unchanging, the other constantly refreshed. One is designed to be the same for all students, the other can be different to suite the needs and interests of every individual student. One is the education of the past, the other is the education of the future.