Math is Hard

In a magazine published for college trustees, a recent short article captured the latest statistics from the ACT and SAT tests. The downward trend was notable especially in math. For example, “Forty-nine percent of the class of 2018 that took the SAT (2 million students) showed a strong chance of getting at least a C grade in a college-level math test, much lower than the 70 percent who reached the same benchmark for reading and writing, according to the Washington Post.” Math is preventing our kids from getting a college degree.

But what really broke my heart was this number: “The share of ACT test takers who showed readiness for college math fell to 40 percent–the lowest level in 14 years.” They go on “‘The negative trend in math readiness is a red flag for our country, given the growing importance of math and science skills in the increasingly tech-driven U.S. and global job market,’ said Marten Roorda, ACT chief executive officer. ‘It is vital that we turn this trend around for the next generation and make sure students are learning the math skills they need for success in college and career.’”

This is a great ad for What if Math, where math for the digital age is not hard.

Art

Black Elk Speaks: Visual Storytelling

Black Elk Speaks is one of my favorite books. I am particularly fond of its opening paragraphs. When the Web was new in the 1990’s I was exploring images and storytelling. I saw these wonderful pictures on the Eastman Kodak website and married them to Black Elk’s beautiful language. The result is one of the best usings of visual storytelling slideshows that I have ever done. I hope you feel the same way, and that it gives you ideas for your visual storytelling.

Art

Calculus for 5th Graders

“Calculus is the study of change” is a common description and all too common misunderstanding. For if calculus is the study of change then algebra becomes the “study of symbols?” I believe this confusion between algebra and calculus among math educators makes these subjects, and the choice to study them, a challenge to students. It is the reason some even advocate changing the traditional arithmetic-algebra-calculus sequence into arithmetic-algebra-statistics, believing statistics to be more important and more relevant to the everyday lives of students. I think they are wrong.

If we say instead: “Algebra is the study of change.” thenCalculus is the study of the rate of change.”

The symbol we associate with algebra, “x”, was defined by Descartes in 1637 as the symbol for a variable. Unfortunately, because today x also used to represent an unknown quantity as well as a variable, our students are confused about its meaning and the nature of algebra. For those who understand the language and concepts of mathematics, numbers represent fixed quantities, and variables like x represent changing quantities. Our students who ask, “What is x?” will likely describe math as abstract and early on make the decision that they are a member of the majority who just don’t get math.

Spreadsheets, work with discrete variables. A number or an unknown is a fixed quantity in a single cell. A variable is a row or column of numbers, a table of values. This fundamentally visual and concrete representation of variable makes the difference between an unknown and a variable vanish, and this all too often asked question, go away. Function, symbolized as f(x), often baffles students. A function can be visualized by the second column in a simple table, next to the first one, an “output” generated by a rule that transforms a variable (input) column into an output column.

We can graph the function easily on a spreadsheet. And we can make a table for its parameters so that we can change and adjust the function easily. This format gives students the ability to play with functions and to learn to visualize them and build models using them. For a mathematical model is built out of functions.

The derivative, a special class of functions, rates, ∆f(x)/∆x, could be a third column taking the differences between the function values and dividing by the variable differences. If a student makes ∆x, the interval between the input values, smaller and smaller she will be zooming in on the function and build an intuitive sense of limit and derivative. The derivative is now clearly not the study of change, it is the study of rate of change.

If rate is a concept we expect 5th and 6th graders to understand, then derivatives using discrete variables on spreadsheets is a concept they can understand as well. And they can be asked to solve real problems that ask for rates of change.

Spreadsheets enable all our students to get math, to see variables, functions, and derivatives as concrete easily calculated mathematics. And all can be easily envisioned graphically.

Try it yourself. Build a spreadsheet like this one and experiment by finding the slope of the curve at different points on the curve. Was Jim Kaput right. Can these fundamental and important ideas of algebra and even calculus be learned by 5th graders.

A Very Good Year

I feel most fortunate when I have a year I get to work on a new great idea in it. This past year has thus been one of good fortune. Some great ideas can appear huge from the start, covering wide swaths of life, and some, at first, can seem small, almost insignificant initially, but on reflection turn out to be consequential and central. This year the great idea was the latter, an idea that appeared tiny and obvious at first glance, but that grows and grows, becoming more valuable with each passing day. Some great ideas seem to be wonderful creative inventions, unique and very, very different, while others are quiet, seemingly so common that we easily miss them. This year’s great idea was again the latter. I would never have expected it to grab hold of our minds and continue to pull us in new directions. And some great ideas seem to spring whole formed, exploding like the Big Bang connecting all sorts of wonderful things together. While others, like this one, play hopscotch in our minds, jumping to and fro with little recognizable pattern to see at first, but eventually linking all sorts of ideas together.

This year’s great idea germinated late last spring as Peter and I were working on what we came to call Classic Story Problems, in particular, the familiar motion, work, and mixture problems found in middle school and high school math textbooks. These problems are painful, that is the only way to describe them, painful to learn and painful to teach. You know the type, “George leaves New York and Martha leave Washington at the same time going at different speeds, where or when will they meet?” Students fight hard to figure them out and many, I would argue most, finally succumb and memorize a formula. Then, of course, when the wording changes, even by a small amount, they are lost again.

In the process of building simple tables in Excel with time in hours in the first (input) column and distance in miles (output) connected by the rule that multiplies time by speed to get distance, we realized that we did not need to fix the input values. Originally, as I was starting to develop spreadsheet lessons, I would let Excel create my input column by putting the first and second input values into the column and then dragging the + to create the column of values. As we got more sophisticated we began to use a rule to create the column, add 1 to the previous number. Well it finally occurred to us that it would be very nice if we could enable students to easily change the input values. Why not make a parameter table with an initial value for the input and an incremental value, and build the input table using a formula and those values? This way we could easily change the start value and the increment value.

Seems such a simple change. But what power. We could enable students to easily solve story problems using a table with discrete values by letting them choose the level of accuracy they needed, developing an understanding of accuracy along with problem solving ability. We could enable them to easily change the domain of any function they choose to graph and quickly and easily control their graphs. We could enable them to zoom in on a particular aspect of a function to study it, or zoom out to picture its form. We could enable them to easily ask “What if…” in a myriad of new ways.

This tiny idea, a parameter control table that enables students to change constants as well as variables has great power. Make the increment smaller and smaller on a quadratic function and you can see that segment of the graph becoming straighter and straighter. Change the initial value and you can watch that straight line change slope as it moves around the parabola. It is almost magical. We don’t need to try to get students to understand limits or secants to picture derivatives, or find a common way to solve story problems. And we are giving them the tools to be flexible problem solving thinkers and explorers.

So as Frank Sinatra sang, “It was a very good year.”

Motion Problems

Spreadsheets and the Rule of Four

A little over 20 years ago the Harvard Calculus Consortium sought to remake the calculus curriculum. “We believe that the calculus curriculum needs to be completely re-thought,” began the text by Andrew Gleason and Deborah Hughes Hallett, both of Harvard University. They sought to get “our students to think.” In doing so they proposed “The Rule of Three.” “Our project is based on our belief that these three aspects of calculus—graphical, numerical, analytical—should all be emphasized throughout.” The Rule of Three, today often known today as The Rule of Four with the now addition of verbal, rests at the heart of math education. While the Calculus Consortium’s book may no longer own major market share, it has had a remarkable influence on all Calculus textbooks and indeed on all math textbooks in both K-12 and college. It is a widely shared belief that such multiple-linked representations must be central to 21st century pedagogy. It is clear that students learn in different ways. It is certain that they need to see mathematics from different perspectives.

Spreadsheets are Rule of Four platforms. They are function machines which naturally represent mathematics graphically,
numerically, analytically, and verbally. They show a function as a graph, as a table, as a formula, and we can describe them with text and visuals. They did not start out that way. The first spreadsheet, VisiCalc invented by Bob Frankston and Dan Bricklin was designed to be a visual calculator to automate the accountants’ worksheets. Three years after VisiCalc’s debut in 1979, Mitch Kapor added graphs and tables to create Lotus 123 which brought the IBM PC into every business. And Excel from Microsoft came out for the new Macintosh 2 years later not only simplifying the interface but adding beautiful texts and visuals to spreadsheets. Today, the mature spreadsheet technology is the standard quantitative tool for business worldwide. It is not only available on every major platform, but its format and design are the basis for displaying and interacting with quantity on the Web.

In a spreadsheet we can write a formula, use that formula to create a table of values, and use that table of values to make a wide variety of different graphs and charts. Change the formula and the table and graph changes automatically. Change the table and the graph changes automatically. Spreadsheets are dynamic and highly interactive. They even let you embed variable quantities in text to add units to quantities our dynamic values to verbal descriptions. Once a student builds a model in a spreadsheet, it is naturally a multiple-linked representation that can played with and explored. Spreadsheet models designed with functional thinking as multiple-linked representations are therefore simulations of which students can ask “What if…”

If you use Link Sheets in your classroom, if you believe that every student has a learning style, if you like to have students explore different representations, if you want to get your “students to think” then try using our What if Math spreadsheets or develop your own built on the Rule of Four.

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.

Opportunity for Creativity

I just looked at a wonderful short video by Sir Ken Robinson on creativity at https://youtu.be/63NTB7oObtw in which he describes creativity as a process that produces something original that has value.

At What if Math we seek to make learning a creative experience, a process that enables every student to produce something original that has value. Isn’t that the essence of what learning is or at least should be. Can all learning be a creative experience? Can it be problem-based with interesting problems that can appeal to a wide variety of students? Can it provide powerful tools for students to experiment with and to stretch their imaginations enabling them to produce something original and of value? We believe most, if not all, of it can!

The spreadsheet math problems and experiments in What if Math are designed to enable every student to be creative and to experience creative learning. We hope you will encourage your students to learn and use the wonderful power of spreadsheets to be a creative problem solver. And we hope you will help us create new Labs by sending us problems to develop or Labs you would like us to publish. I look forward to hearing from you.

Art

Another Sunday Ritual Soon Gone

When I was a kid, Sundays in the summer were car washing days. The stores were closed. The roads were generally quiet. And we took out the hose and the pail, filled them with water and dishwashing soap then rubbed, scrubbed, and waxed the family car…or later our own car…beautiful again. Sundays have already substantially changed and now in an article a recent Economist magazine, one of the last of the great suburban Sunday rituals will soon be going away too. For a scientist has found a pattern made by an obscure but available laser that sheds water and dirt with it. One day in the not too distant future we should be able to buy a car that not only no longer needs waxing but will never get dirty and need washing. Perhaps, about the same time, we will no longer even own cars but instead will ask for one on our cell phone, and an Uber Self-Driving car will pull up to our house to pick us up and drop us off wherever we want to go.

Crazy ideas, great changes. Yes, this is just a tiny example of the new world our students need to be preparing for. We are all linear thinkers. We think the world will continue in much the same way it has been going, with changes that take place slowly and methodically, changes that we can get used to, changes we can plan and prepare for. But change is not linear at all. It is, like the word used by the wonderful writer and biologist Stephen Jay Gould to describe evolution, “punctuated”. Sometimes change accelerates quickly and sometimes it moves at a constant speed. This has always been the nature of change and it is evermore so true today. For things that shed water don’t get dirty and don’t need cleaning — cars, windows, even clothes. And cars that don’t need drivers don’t need parking spaces on streets, driveways, or shopping centers. Our world could, and likely will, change dramatically.

How do we prepare children for work and life in this future world? What should they learn in their school years to make them ready for a lifetime punctuated with change they cannot predict? We cannot base it on the Sunday rituals of our past like counting the cash my dad brought home from his dental practice so that he could put it into the bank on Monday morning. We cannot base it on the paper and pencil calculating rituals we spent innumerable hours on, the paper algorithms that define most of the math our students practice. We cannot base it on “What is____?” habits of thought of the past, when the answer can almost always be easily Googled. We must base it on “What if…” thinking, functional thinking, the basis of science, technology, engineering, and math, the heart of business planning and quantitative reasoning, the question the future depends on.

Sunday rituals will come and go. Technology will sometimes change rapidly and sometimes slowly. But we can prepare our children for their future by making their education about “What if…”, their practice and mastery not off paper-based algorithms but off open-ended problem solving, their focus not on facts in today’s data-rich world but on thinking, their vision directed not on finding the right answer but on seeing outside the box.

The Hawthorne Effect

To make its workers more productive, the Western Electric Company, makers of phones and other parts for the Bell Telephone System, conducted one of the great scientific experiments of all time. The researchers increased the brightness of the lighting at a plant in Hawthorne, Illinois just outside of Chicago, incrementally, while measuring worker output. They alerted the workers to the experiment and they found that as the workplace got brighter the output increased. Then, in a great out-of-the-box experiment, they dimmed the lights incrementally, telling the workers the experiment was still going on. Output continue to grow!

This became known as the Hawthorne Effect.

…the novelty of being research subjects and the increased attention from such could lead to temporary increases in workers’ productivity. Wikipedia “Hawthorne effect”

I got to thinking about the Hawthorne Effect while rereading the Common Core State Standards and thinking about the testing controversy that embroils us. In their website under the Key Shifts in Mathematics menu, they shine a bright light on what they consider to be the core topics:

  • In grades K–2: Concepts, skills, and problem solving related to addition and subtraction
  • In grades 3–5: Concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions
  • In grade 6: Ratios and proportional relationships, and early algebraic expressions and equations
  • In grade 7: Ratios and proportional relationships, and arithmetic of rational numbers
  • In grade 8: Linear algebra and linear functions

And they conclude:

“This focus will help students gain strong foundations, including a solid understanding of concepts, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the classroom.”

The Hawthorne Effect caused by these brighter illuminations will no doubt improve student productivity as measured by test results. But as with the Hawthorne plant, once the experiment, and it is an experiment, is over, once the lighting no matter what the brightness level, stays constant, then productivity will return to its “normal” level. For we have not really changed either the product to make it much more integrated nor have we changed the process to make it more productive.

For the past 50 years we have put math education through a series of Hawthorne Effects. This is the reason that the NAEP scores have remained flat lined. If we truly want to substantially improve math education, as we must, then we will have to reinvent both its product and its process and not just shine brighter lights on it.

Lynn Arthur Steen

My fortune cookie today read, “If you’re happy, you’re successful.” Usually for me that is true, but not today. For during that same lunch my iPhone told me that Lynn Steen had died. I never had the good fortune to meet him in person or to even talk with him, but I loved him and learned so much from him. His words, “Mathematics is the Science of Patterns,” from an article he wrote in Science in 1987 and his other visionary works on math education have been my guiding lights. I have long wanted to write something about math education that was beautiful and compelling enough to make it worthwhile for him to read. But alas, I have waited too long. I will nonetheless not give up trying.

Lynn Steen sought a revolution in math education that would enable every child to participate in our more and more mathematical world, to see the wonder and beauty of its patterns, and to fall in love with his discipline. Today, as I remember him, I dedicate myself once again and ever more deeply to this great task he has left for us to complete.

Art

The Great American Probability Machine

Great American Probability MachineThis program started my career in digital learning. I bought my first computer, an Apple II in February 1978 on their first anniversary. I talked my wife into letting me play with a computer for doing my checkbook and turning our houselights on and off. Though I was an educator who had worked with film technology, I never imagined that this machine would send my life rocketing in a new direction. That force hit the following summer at one of New England Apple Tree monthly club meetings where we came to see the wonderful new things people were dreaming up and take them home on cassette tapes. The buzz this meeting was about the latest West Coast phenomenon. Written by a guy we never heard of, Bruce Tognazzini, it was the first program to integrate text, animation (in Apple Lores graphics), and sound to tell an interactive story. It made use of the joysticks that came with the Apple II, and the first to tell a story. I fell in love. It made me see the personal computer as a tool I could use to express my vision of curriculum, and led me to start the first of my companies to express this vision.

I met Tog a few months later and we became good friends. He was by then working at Apple as an interface designer having a major impact on Apple and Mac screen designs. He continues to help companies improve user design today. Though Tog was not an educator, indeed, when he wrote the program he was the owner of a San Francisco Sony television store who bought his first Apple II about when I did as a toy to play with on his Sony Trinitron. The Probability Machine was the first program he wrote. Did this program turn him into a brilliant designer or was he already a great designer looking for a medium? I don’t know. But he was among the first to understand how to make screens both interactive, engaging, and wonderfully simple at the same time. For example, he introduced the graphic element through a story about the building of a great machine, a massive public works project in the late 19th century, to build the understanding of probability. He tells us that the small rectangle on the bottom right of the machine is a door to let people access the machine. He took a common concept seen in many museums and made it a powerful tool that could go into the hands of students. I saw his vision as education’s vision.

As I designed educational software over the years since the Probability Machine appeared, I have often thought of Tog and his amazing 1500 lines of Woz’s Integer BASIC code, saved on and loaded from cassette tapes, and edited without a printer or any of the tools coders rely on today. I think of it now as I play with our spreadsheet version of Pascal’s Triangle that enables students to tell new stories and perhaps, just perhaps, be directed by this Spreadsheet Lab in a new life direction. There is great power in this tool when we let students use it to experiment and explore. Thank you Tog.

Art

Balance

As I watched a young woman the other day learning to ride her bike, zigzagging down the street, desperately trying to keep her balance, I thought of the Wright brothers. They owned a bicycle shop where they made and taught people to ride the then relatively new form of transportation. They, like their students, had to learn to balance these new contraptions, just as most of our children do today. They took that core skill to their work on flight. For they were not the first to try to solve the problem of mechanical flight, but they were the first to be successful at it.

That success was built on many things, on a sense of design, on an understanding of both the physics and the engineering of lift, torque, control, and material science, on observation of birds, and even on the design and manufacture of the then very new gasoline engine. But if we had to ask, “What singular idea enabled them to be the first flyers?” The answer would have to be balance. They would have taken that from their bicycle business for it is also the secret to riding a bike. The small wing sticking out in front of the main wings had a reverse curvature to balance the lift of the big wings. The wing warping mechanisms enabled them to balance the chaos of wind and air currents and use a tail as a rudder to turn. They balanced weight, size, and structure to make it possible to carry a pilot and a passenger. And they even had to balance success and failure, secrecy and publicity, business and invention.

Education today is off balance, and like the Wrights we too have to make balance not just a priority but our central driving force. The balance between the arts and the sciences, so long a key aspect of our educational system is now gone. The practical arts are no longer part of our school day. The fine arts are for all intents and purposes missing in action. And the design arts, so critical to business today are not found in our schools. The sciences which includes math, for math is the “science of patterns,” have taken over our curriculum. English Language Arts, is not artistic but scientific with word counts, difficulty formulas, and non-fiction governing that subject.

We try. We add an A to STEM and make it STEAM, but we don’t seem to have a clue about how to integrate those subjects. We talk about the importance of the need to bring the arts back, but we are so nervous about those tests that we cannot find the time in the school day to do that. And while private schools flaunt their wonderful studios, their darkrooms, their theaters, and so many of the people demanding charter schools seek to emulate the privates, the measure of a good high school remains its SAT and AP scores.

As we invent an education system for the future and not the past, we, like the Wright brothers must make balance our central design principle. Just as balance enabled the Wrights to create a new form of transportation, so too must balance, by infusing our classrooms with the creativity of the arts, enable our students to fly.

The End

Despite the many attempts to codify the creative process, it is as surprisingly individualistic as it is human. John Irving, author of iconic works like The Cider House Rules, describes his creative process as writing the conclusion before the beginning. He spends a year or more developing a story, the plot, and a set of characters in his head, writing nothing down, until he has the whole in mind. Then he writes the final paragraph. Then and only then does he start actually putting words down on paper going from beginning to end. His description of the writing process is a fascinating view into the creative act, most fascinating for me because I do it so differently.

I come up with an idea, often in the quiet mind time of the morning shower or the last thoughts as I slow my mind on the way to sleep. It is usually a blurb of an idea, often captured in a few words or a sentence or two. Never more than that. I repeat it over and over again to myself so that I will remember it, because my memory is notoriously terrible. I sometimes work on that idea for a few days, coming back to it, testing whether I still like it, adding a few connections to it, but rarely carrying it much further. I basically write that sentence of two on my brain and hold it there until I have the inclination or the time to actually develop it, compose it. For I am most fascinated not with the original idea but with what will become of it, where it will go, what it will develop into when I start keying it in for real. To me, the thrill is in discovery. It is in running the experiment and finding the result. It is in seeing the working invention. It is in printing the beautiful picture that I have tweaked and messed with. So I find writing not a painful demanding activity but a creative thrilling one that I take joy in. For I never quite know where it will go or what it will produce. I suppose it is the reason I so love asking “What if…” in our spreadsheets. What does your creative process look like? I just learned more about mine!

Mastery

The word seems so benign. Yet it has become the goto word in education. School superintendents, even the best and most advanced of them, use it all the time. “We want each of our students to reach mastery in each of the standards.” To master something is to be in command of it, to control it, to know it. It seems so logical and natural to want every student to achieve mastery that we now make it a synonym for learning, the product of schooling which we can objectively define and measure.

But if learning is to be a creative experience, how can it be mastered. Creativity by its very nature is constantly evolving, constantly pushing us to think out-of-the-box. Can we really learn to master creativity?

Apple is trying to do just that with its new headquarters in Cupertino, CA. It is seeking to make innovation a product that can be engineered, that can be structured, that can be mastered. Steve Jobs started its design, and Tim Cook and his leadership team is finishing it for Steve. It is a huge building, the size of several Pentagons, built to bring together in one place much of Apple’s creative and product development workers. It is designed to enable Apple retain its innovation leadership as well as its status as the largest company on earth. Laid out in a giant torus to bring people together to share ideas and share visions in both formal and informal opportunities, it’s goal is to master and manufacture creativity.

If Apple is right and a building designed to express and indeed enhance a corporate culture of creativity can make creativity a product, then mastery in schooling is right. If they can master creativity, build products that are inherently, creative then Apple will retain its competitive edge. If we can teach students to master concepts and thus master creativity, then a focus on mastery must assuredly be the goal of education. But if creativity is a process, a process of continuous growth and not mastery, a process of practice in which all we learn is to always ask “What if…” then creativity can never can be mastered or finished or finally learned. For it is a process of continuous growth, exploration, and experimentation, a process we become increasingly fluent in. If creativity is a process and not a product if our goal is fluency and not mastery then Apple should rethink its reliance on architecture and education should rethink its goal of mastery, for neither will produce the creative products or learning we seek for our future generations. Our goal must be creative fluency and not content mastery.

Change

“Today, it seems as if nearly everyone agrees that high school mathematics needs to change. For far too long high school mathematics has not worked for far too many students: too many students leave high school unprepared for college or a career, particularly a STEM career; too many students do not see how math is useful in their lives; too many students leave high school without an affinity for doing math; too many students leave high school without the quantitative skills necessary to make sound decisions in their personal life and in our society which is increasingly quantitative in nature.  High school mathematics has not changed substantially in my lifetime, nor has it changed substantially for most students, teachers, schools, districts, and states.  It is clearly an issue—and it is a critical issue of access, opportunity, and equity.” By Matt Larson, NCTM President
October 25, 2016

We are thrilled that NCTM agrees that the math we teach is obsolete and does not serve our children. We hope you find the great new stuff that we have coming very, very soon, does.

Art

Tradition, Tradition

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 21st 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!

The First Graph

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.

Empathy

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!