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!

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.

This picture was first published in 1638! It is from Galileo’s great work Two New Sciences, that he smuggled out of his home imprisonment in Florence, when he was 72 years old and effectively blind. Though famous for his telescope and the first images of the surface of the moon, he had not before published his seminal work on motion. It is easy today to gloss over his extraordinary achievements, even being called the “father of science,” and to make him a caricature battling for the Copernican theory. But this view of the renegade, the persistent critic who fathered three illegitimate children, fails to recognize his profound contribution to humanity. Galileo invented the experiment. Before him people “observed” nature. They developed instruments that simulated phenomena like the motion of the planets. But they did not “experiment.” They did not ask “What if…?

An experiment is a process in which we can change not only inputs, but the rules connecting inputs to outputs as well. Galileo’s classic experiment, the motion of objects sliding down inclined planes, enabled him to dilute gravity, to slow down the motion of a falling body, so that he could measure the distance traveled in each time unit. To do this: he built the first accurate way of measuring short periods of time, he constructed an inclined plane so he could ask “What if I change its slope?” He developed the concept of repeated trials to measure, re-measure, and measure again so fundamental to experimental science.

In the fourth chapter, which he called the fourth day, in Two New Sciences (the first new science being the science of proportion) on the science of motion, he showed how projectile motion, the motion of objects shot out of cannon, thrown, or dropped, can be envisioned as the composition of two motions horizontal and vertical. The horizontal motion of a projectile is constant, it goes the same distance in every unit of time. The vertical motion of a projectile, like the motion of any falling body is accelerated, the distance it travels increases as the square of the time. Added together the motions to produce the path of the projectile and that path is a parabola. We would call the graph that Galileo drew a distance/distance graph (both axes are distance measures). At each point in time, we move across and down. This compounding of motions like the compounding of functions in the Parametric Equations Lab enables us to put together two separate functions linked by a parameter (a parametric variable) to model motion.

As you experiment with the Parametric Equations Lab, imagine you are Galileo, experimenting with inclined planes, dreaming of dropping balls from the Leaning Tower, and explaining why, if Copernicus is right that the earth rotates, we don’t feel ourselves moving. And as you experiment replicate has graph, the first ever drawn and imagine how he would have used it to explain the motion of projectiles.

As part of the process of designing and developing new Labs, I visit math content sites all the time to help me think about the kinds of questions to ask and the way to explain or represent a concept. I am constantly struck by how talkative these sites are. As teachers, words are our currency, and with few exceptions they are the main way we have always communicated skills and ideas. We come from a very long “stand and deliver” tradition. We seek to replicate Socrates talking to his disciples. When that oral tradition was turned into a printed one, teachers used words, even more words, to communicate ideas. Surely, in some cases, we draw pictures, knowing a picture is worth a thousand words, particularly in mathematics. But we rarely let images stand alone, but embed them in a sea of words, for words have remained our currency and our tradition for 2500 years.

Today, as we turn into the digital age using screens instead of dead trees, we continue to find it so very difficult to get past our tradition. We make videos, draw and animate images on screens, but still we fill them with words. Even when we create content with dynamic, interactive images, we still embed them in a sea of words to either be read or listened to. We, it seems, cannot leave our long tradition of making words our learning currency. Even when our visionaries preach teaching in the tradition of the great Socrates by asking questions, having conversations, seeking roots of concepts, we continue to apply his words based pedagogy to build 21^st century skills.

We have yet to learn the lessons of this new digital medium, the lessons of PowerPoint slide shows, Twitter and Facebook, emails and especially messaging. We have not applied the “less is more” use of words to digital learning. It is not easy to make lessons with just few words that do not have to tell, show, or direct. It is not easy to ask simple questions that suggest. It is not easy to picture concepts in visual representations as tables or graphs or animations. It is not easy to change tradition. But just as in Fiddler, we must!