One of the four C’s, perhaps for many the most important 21^st 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 19^th 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.

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 4^th 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

The Genius Behind Accounting Shortcut? It Wasn’t Einstein

The Rule of 72 is a nifty shortcut for estimating investment returns; first published mention was in 15th century

Great article in last weeks Wall Street Journal on the Rule of 72 by By Jo Craven McGinty.

Learn more about the Rule of 72 in our fun Lab.

 

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.

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 8^th 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!