There are really two kinds of mathematics we do every day. I like to call one headmath and the other handmath, one is the mental arithmetic and problem solving we all have to do and the other is the math on paper or more likely today, if you are not a student, on computers. Though we generally think of them as being the same, they are, in fact, very different beasts and this confusion is the source of great misunderstandings in our nation.

Many people today complain that students are not learning math because they cannot make change in stores, not estimate financial transactions, not understand a business proposal. They think that our students are not being taught math well and are too wedded to technology and fail to learn the basics. It has become an all too common refrain that our students need to return to the paper and pencil and practice their skills. As I talk to both the business and STEM communities I hear their pain but I do not agree with their cause. For I believe that while this problem is quite real, its cause is not a lack of paper practice but our failure to differentiate between headmath and handmath.

For the real problem is not in handmath, but rather in headmath, in solving math problems mentally, in guessing an answer, estimating a number, finding tricks and shortcuts to approximating it, and to mentally modeling problems so that you can decide if an opportunity is worth pursuing. The clerk who cannot make change is not writing the problem down on paper and solving it, he or she is trying to figure it out in his or her head. The worker who is asked in a meeting, “What do you think?” is tasked to come up with a headmath best guess. The buyer in the store trying to figure out the best deal is rarely using paper or calculator but trying to compute in her head.

Head math is stuff we have to do all the time, whether it is figuring a tip or planning a trip; yet today with our near total focus on standardized tests, most of what we do in school is handmath because we do not test headmath. Oh sure, we ask students on tests to do estimation problems, but they are just handmath in disguise. How many teachers ask students to guess an answer or start their class with oral drills? How many parents play mental math games with their kids? We don’t do these things anymore because all math has become handmath. And you get good at what you practice!

Recognizing the importance of headmath means making sure kids know their multiplication facts, that they learn the tricks of doing mental arithmetic and practice them, like using approximations. 95 * 63 is about 100 * 60 or 6,000. It means focusing on order of magnitude. In the days of slide rules, learning order of magnitude was crucial, today with calculators we rarely attend to it. So what is 20% of 10,000? Recognizing the significance of headmath also means that we need a lot less of some of the handmath things we are led to believe are basic. For while knowing the multiplication facts for headmath is critical, knowing how to rapidly and accurately perform the multiplication algorithm on paper is obsolete. Fractions are very useful for headmath but we hardly ever use them for handmath in business. So why do we waste time teaching student to solve hairy fraction problems on paper. In the 21^st century spreadsheet world we need students who can use spreadsheets and headmath to judge results!

Three years ago I read a wonderful book by Keith Devlin called The Man of Numbers. It told the story of Leonardo of Pisa who was the first to convert Arabic arithmetic and algebra for European use. Devlin told Leonardo’s story and he described the process by which Leonardo’s book Liber abbaci (The Book of Calculation) became the basis for both the teaching of students and the development of European mathematics. Leonardo viewed his book and his task as providing a new means for merchants and businessmen to do the calculation they needed, replacing the slow, cumbersome, and error prone Roman Numerals they were then using. Devlin included the table of contents of Liber abbaci.

When I took a close look at that TOC, I saw the K-12 math curriculum we require every student today to master. I was awestruck. The math we teach our students today is the math Leonardo defined to meet the needs of medieval traders and bankers. It was not “basic.” It was not fundamental. It was the math needed and designed for 13^th century business. It is obsolete! Business today does not use most of it and has no need for most of it. Do our children need to learn “long division”?

Since the invention of the personal computer spreadsheet in 1979, business has focused on the math of functions and not of solving equations with machines calculating we no longer require pencil and paper algorithms to do arithmetic. I have been a math educator for over 40 years and not only did it finally dawn on me that much of the math we teach is no longer necessary, but that the math we should be teaching, the math our students will need to learn and use, is the math of spreadsheets and not the math of Leonardo.

I started working with some wonderful friends to think about what this reinvention of mathematics education would look like. Peter Mili a truly great math teacher, Larry Reeves my longtime collaborator, George Blakeslee my educational mentor, Steve Bayle my technology mentor and one of the spreadsheet pioneers, my sons Brenan and Arran, and wonderful other friends who helped us think through and grow these ideas.

We developed our first version of What if Math two years ago as case studies for problem-based-learning. They were difficult to develop and difficult to use. Last fall I started making some spreadsheets for my friend Megan Peterson to try in her 2^nd grade classrooms, and in January my friend, Craig Kelley, after seeing those primitive lessons, challenged me during one of our many breakfasts and lunches talking about the future of education. “I get the need, but if you want me to believe in a spreadsheet-based curriculum you need to show me what a 2^nd grade curriculum would look like.”

I took up Craig’s challenge and over the past year months we have been developing these lessons for, by, and on spreadsheets. We have more than 60, spread across the curriculum to serve as models for many more. We have another 40 in our pipeline and more in our imaginations. This summer and winter break, Ryan McQuade, still an art student at Lesley University, has made them beautiful and developed a great website to make them accessible. We give them to you at no cost to begin to build the math curriculum of the future. Our dream is to make learning mathematics a creative, challenging, and collaborative experience for every student.

I look forward to your experiences and your thoughts and hope your students get the same thrill in learning that we have experienced.

Art