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!

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