Math is "contemptible and vile."
That's not from a disgruntled student. It's from a textbook.
The author, 16th century mathematician Robert Recorde, nestled the line just after his preface, table of contents and a biblical quote citing God's command to measure and number all things.
Recorde didn't believe in math's awfulness — quite the opposite. He was simply reflecting popular opinion on his way to a spirited defense of math. Why?
Mathematics was associated with banking and trade and so "was shunned among the upper classes and the educated classes in Europe," explains Houman Harouni of Harvard University.
Recorde's math textbook — published in 1543 — is far from unique.
"Almost all of these books start with an apology," says Harouni.
An apology. An apologia. A quote from the Bible trying to defend the subject. While you won't find such a thing these days, the word problems that followed those apologies are nearly identical to what fill today's math textbooks.
These old textbooks outline "a curriculum that is so similar to the curriculum we have right now it might as well have been written by the good folks who wrote the Common Core," says Harouni.
Why do we study math?
Harouni didn't always spend his days studying antique math curricula and century-old textbooks. He used to be an elementary school teacher in Cambridge Public Schools.
It was there that his math students started asking that all-too-familiar question: Why?
Instead of offering the easy answer — Because math is good for you! — Harouni promised them a real answer. And that required that he do some homework of his own. A lot of it.
His research led him to Harvard, where he now lectures at the Graduate School of Education, and to the American Academy of Arts and Sciences, where he is writing a book on math education.
But it isn't just why we teach math that fascinates Harouni. He is particularly interested in why we teach math the way we do: "Why these topics? Why in this order? Why in this way?"
He says history offers the best answer.
Harouni has studied texts dating to ancient Babylonia, ancient Sumer and ancient Egypt, and, he says, he has found three main ways of teaching math, each associated with a different economic group.
Money math
5+4=9
For people all over the world, this first approach triggers memories of elementary school. Students study addition, then subtraction, followed by multiplication and so on.
There are word problems: You have four apples, and your mother gives you five more. How many apples do you have?
Here, the primary goal is calculation and prediction.
This is the dominant approach to teaching math today, and it has been for centuries.
To understand why, Harouni returns to Recorde's textbook. Immediately after acknowledging math's bad reputation, Recorde — who invented the equals sign — defended math's importance.
"It is the ground of all men's affairs," Recorde wrote. "No communication without it can be long continued, no bargaining without it can be duly ended, nor no business that man hath justly completed."
No matter how corrupting math may be, Recorde argued, accountants, administrators, traders and merchants needed to know it. And so, merchants created little schools where their children could learn math.
In Europe, these schools were called reckoning schools. There was one fee to learn addition, another to learn division by a two-digit number, another for fractions.
Reckoning schools were first seen in Italy in the 1300s, but soon merchants from other places wanted their children to learn too. As a result, reckoning schools started springing up elsewhere, following trade routes.
But Harouni says that this money math — critical for commerce and administration — is not the only way to teach math. It's not always 5+4=9. Indeed, he says there are two other equally valid approaches.
Philosophical math
Here, 9 is not the answer. It's the question.
9=?
Students could answer "3 squared" or "the square root of 81."
"The more math you know, the more answers you can come up with," says Harouni.
In this approach, the emphasis is not on the outcome of an equation. It's about revealing patterns and discovering the relationship between numbers. Harouni says this philosophical approach can be seen in how Plato taught and thought about math.
For a long time in Europe, this was the math for elites. And it wasn't for children.
Grammar schools were about languages, as the name implies. The sons — and occasionally daughters — of lawyers, town officials and pastors spent their time studying Latin and Greek.
Until the late 1500s, students wouldn't set foot in a math class until they had made it to university. And then it was about numbers as concepts and ideas.
Artisanal math
Although math wasn't in grammar schools, it was in shipyards and wood shops. Carpenters, masons and other craftsmen learned math in their apprenticeships.
Rulers and compasses replaced blackboard and textbook. It was more about measuring than counting.
A master might ask his apprentice to divide a plank of wood into thirds. This, Harouni points out, could be done simply with a piece of string. Indeed, in artisanal math, tools, instruments and materials are key.
So are units. Harouni says no artisan would ask "5+4=?" because it would leave everyone bewildered. "Four of what? Five of what? Four cats plus five chickens?"
After all, 4 feet plus 5 inches most certainly does not equal 9.
In artisanal math there isn't a set curriculum. The task at hand determines what the apprentice learns.
The mathematical winner
Reckoning schools, universities and shipyards each had their own style and system. But with the rise of capitalism and the mercantile economy, a clear favorite emerged. The economic approach to teaching math started to spread.
By the late 1600s, merchants were gaining power, and they were tired of paying for two types of education: grammar schools for reading and reckoning schools for math. Plus, the elites had become more acquainted with commerce.
Gradually, money math became less taboo and eventually sneaked into elementary school curricula, where it remains today.
Now, when kids ask Harouni why they have to study math, he knows the answer: Like it or not, we live in a world where money matters. And our math curricula can prove it.
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