Category Archives: Mathematics

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 Bricklin & Frankstonrepresent 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.

20th Century Math

For those of you who have been following my blogs, I apologize for taking so long to get out a new one. I have been working a wonderful new project that i am not yet ready to show you, but I promise to do so very soon.

Meanwhile I had a fascinating afternoon yesterday attending a seminar on SketchUp, what they call the 3D program for everyone. Google just sold it to a company called Trimble, a construction company.This program is for designers what WordPress is for bloggers.

As I watched amazing demo after amazing demo, all I could think about was America’s K-12 math program, indeed our entire educational focus. To put it bluntly, it has nothing to do with the real world jobs of the 21st century. For here were architects, designers, engineers, interior decorators, landscape architects, and more using this program in their daily work to both design and to demonstrate. Here was an amazingly large community of people contributing their ideas and their actual work to other users of the program, developing “plug-ins” to do a variety of tasks SketchUp was not designed for.

Are we teaching our students to use technology, to work with sophisticated programs, to be part of a community of users and developers? Are they learning to create, to explore, to learn from each other? Are we preparing our children with the skills they will need for the 21st century? Are we imagining them working with tools like SketchUp? Or are we preparing them for the jobs and work of the 20th century?

Learning Math as an Experimental Science


What if math were learned as an experimental science where spreadsheets were our laboratories.

Keith Devlin wrote a fascinating biography called “The Man of Numbers” about Leonardo of Pisa. Leonardo is credited with bringing Arabic arithmetic and algebra to Europe. But his story is much more interesting and his lasting mark on not just mathematics but on education has long been overlooked. Leonardo was born in Pisa around 1170, when Pisa was the greatest trading city in Europe and had just begun building what we know of as the Leaning Tower. His father, a trader and diplomat for Pisa, brought Leonardo to Algeria when he was a boy and had him tutored in Arabic arithmetic and algebra. Leonardo became a merchant/trader traveling Africa and Eastern Asia. The mathematics of business in both Medieval Europe and the Muslim world was the Roman system using an abacus to perform calculations. But as Leonardo found, it was slow, cumbersome, and error prone. It worked for the Roman Empire which had a consistent monetary and weights and measures system. In Medieval Europe nearly every city-state had its own monetary and weights and measures system which meant that merchants had to constantly solve difficult ratio and proportion problems and equations.

So in 1200 Leonardo returned to Pisa and wrote Liber abacci The Book of Numbers, for merchants and traders to provide them with a revolutionary new arithmetic and algebra to solve their problems quickly and confidently. This book became the basis for the transformation of mathematics and business. Devlin included a picture of Leonardo’s “Table of Contents.” I was awestruck. The sequence of chapters in Liber abacci was a replica of the sequence we teach in K-12 math. The mathematics Leonardo developed for the needs of business in 1202 is the mathematics we now expect every child to master in K-12.

It is not the math business uses today!. For a business math revolution occurred in 1979 with the first personal computer spreadsheets. Spreadsheets are based on the mathematics of functions, first developed nearly 450 years after that enabled the scientific revolution and that has in the past 35 years transformed business. Leonardo’s rapid paper-based computation algorithms and equation solving is no longer necessary. Computers do that. Today business thinks in terms of input, outputs, and rules connecting them. Today business wants to ask of mathematics, What if…

Most of us have heard of Leonardo as Fibonacci, a nickname given him for unknown reasons eight centuries after he lived. His sequence was a minor diversion in his great work. Now his mathematics is obsolete, we no longer use most of it, and we no longer need our children to learn it.

I created this deck to help you envision math as an experimental science and spreadsheets as laboratories.

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To learn more go to What if Math.

The Tools of the Trade

“We would all be better off if functions were introduced in kindergarten and studied regularly thereafter. The concept of a function is one of the most important ideas in mathematics.”

Paul Sally 2008

I found out yesterday that Paul died last month. A great mathematician, a profound teacher, and a deeply caring math educator, Paul was a man of incisive intellect and wit. I had the good fortune to hear him in person at a University of Chicago seminar for alumni last June. Though blind and nearly deaf walking on two artificial legs, he was brilliant, funny, and transfixing. I went to the bookstore and found one of his latest books, Tools of the Trade, which he subtitled Introduction to Advanced Mathematics. It is a textbook like few others. Tiny, less than 200 pages long, and dense for he is speaking to advanced college students who seek to be mathematicians, it nonetheless captivated me, and it has been a very long time since I last took a serious math course. I cannot say I have worked my way through this book for it seeks to get students to “come to grips with the idea of proof in a serious way.” But I have returned to it again and again to gain its deep insights, for while he did not write this book for math educators,  we can find in it the tools of our trade.

Paul makes it clear that functions should play a fundamental role in all math education. He constructed his book with projects to provide “Inquiry Based Learning (IBL) experience(s)…” for students. And he sought to challenge their “mathematical creativity.” So as we rethink education and particularly math education we focus our attention on those things that Paul Sally thought were fundamental: functions, Problem or Inquiry Base Learning, and of course creativity and make these the foundation of mathematics education.