I came across his name recently while searching my house for old letters, pictures, and documents. He had sent me a letter, while I was then a new teacher at our old high school, inviting me to come to Missouri to see him at Meramec Community College where he had become a dean of instruction. I never went. Instead I headed to Fort Lauderdale to stay in high school teaching and to my great good fortune to meet the woman who I would spend most of my life with.

So as our digital world calls us to do, I looked him up and found an article he published in 1971 called Self-Directed Learning at Meramec Community College. It was a short report on a program he had initiated. It was written in his plain style and tells an amazing story, for this program he developed some 50 years ago reverberates in our time. He argues for Universal Higher Education and for a system of education under the heading Modern Challenge in which:

<em>“…college departments need to develop more learning modes, leading to individualized achievement of well defined objectives. Thus, the student can, with guidance, select objectives which are appropriate to his needs and learning modes which fit his learning style.”

He calls for “A New Kind of College” </em>

<em>“a college based on a new philosophy of student learning and achievement.” “…colleges will be able to make significant gains in the efficient and effective utilization of available talents, spaces and facilities. Students would be freed to pursue learning individually via the most appropriate pathway…colleges would become learning centers…”</em>

You can see why I now think him my favorite teacher. Teach as he taught. I learned so much from him including the vision of learning as a laboratory science. I wonder what he would have imagined if he had the technology we have today. I wish I could tell him that his student is continuing to carry on his mission.

]]>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

]]>Art

]]>If we say instead: “Algebra is the study of *change.” *then* “*Calculus 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 6^{th} 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 5 ^{th} graders.**

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.”

]]>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*.

My 5^{th} 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 6^{th} 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 21^{st} 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.

]]>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 21^{st} 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 1/5^{th} 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 21^{st} 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.

]]>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

]]>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.