Why We Learn Math Lessons That Date Back 500 Years

Jul 23, 2016
Originally published on August 16, 2016 6:37 pm

Open up a classic mathematics textbook written by a Welshman, Robert Recorde, and flip past the preface and the table of contents.

There, you'll see the bold statement that math is "contemptible and vile."

Recorde didn't believe that himself. Quite the opposite. But, writing in 1543, he had to acknowledge that this was a prevailing view at the time — that math was profane. Not worth knowing.

That's something many school children today might agree with, but why such a disdain back in the 16th century? And not from kids, but from lots of people.

Houman Harouni, who teaches at Harvard University, says he knows the answer. Back then, mathematics was associated with banking and trade and so "was shunned among the upper classes and the educated classes in Europe."

Recorde's math textbook was far from unique.

"Almost all of these books start with an apology," says Harouni.

While you won't find an apology in math books these days, you will find a curriculum — word problems and all — not that different from the lessons that filled textbooks 500 years ago.

Those old textbooks use "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


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.


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.

This story was updated on Aug. 16, 2016.

Copyright 2018 NPR. To see more, visit http://www.npr.org/.


As kids head back to school, they are likely to be full of questions. One of them might be, why do we have to study this, especially when it comes to math? The NPR Ed team's Gabrielle Emanuel went in search of the answer.

UNIDENTIFIED CHILDREN: (Singing) Ones, tens, ones, tens, hundreds.

GABRIELLE EMANUEL, BYLINE: On the second floor of the Baldwin School in Cambridge, Mass., Amy Moylan's first graders are getting in the mood for math.

UNIDENTIFIED CHILDREN: (Singing) Four, five, six, seven, eight, then comes nine.

EMANUEL: Next, these kids gather around a small table, each student with a little dry erase board.


SKYLER: Ten plus five - that equals 15.

EMANUEL: Skyler gets it, but when I was in school, I was constantly answering math questions with that famous question, why? So I asked these kids.

OK, guys, I want to know why we study math.

SAM: So we can become experts at math.

ASHKAN: Because so I can be very quick, and then I can go play.

UNIDENTIFIED CHILD: You learn so you don't have a tiny, tiny, eensy-weensy brain.

EMANUEL: That's basically what I was told. It's good for you. It makes you smarter. But that answer wasn't satisfying to Houman Harouni at Harvard, where he lectures at the Graduate School of Education. For Harouni, it wasn't so much why we teach math but why we teach this math.

HOUMAN HAROUNI: Why these topics? Why in this order? Why in this way?

EMANUEL: He found the answer in history.

HAROUNI: You can go back all the way to Babylonia, ancient Sumer, ancient Egypt.

EMANUEL: Looking at how they teach math, Harouni found three main approaches. The first one is one we all know.

HAROUNI: Four plus five equals...

EMANUEL: Nine - addition followed by subtraction, then multiplication and so on - the primary goal...

HAROUNI: Calculating and predicting something.

EMANUEL: This, Harouni says, is the dominant form of math, and it has been for centuries. It goes all the way back to merchants in Florence in the 1300s.

HAROUNI: They have a curriculum that is so similar to the curriculum that we have right now that it might as well have been written by the good folks who wrote the Common Core.

EMANUEL: But there's one difference. It was associated with banking and seen as un-Christian.

HAROUNI: A vile, corrupt and corrupting science.

EMANUEL: Nevertheless, all those accountants, traders and merchants had to know this stuff, so they created schools to teach their kids. But Harouni says there are actually two other equally valid ways to teach math. It doesn't always have to be four plus five. In the second model, nine isn't the answer. Instead, it's the question.

HAROUNI: Nine equals question mark.

EMANUEL: This is the philosophical approach.

HAROUNI: To leave the student to come up with all kinds of answers. And the more math you know, the more answers you can come up with.

EMANUEL: Students might answer 3 squared or the square root of 81. The emphasis wouldn't be on the outcome of an equation but on the relationship between numbers. So for a long time in Europe, this is how math was taught to the elites - numbers as a concept, as an idea. So there's the economic approach and the philosophical approach. And that third one - it's the stuff taught in apprenticeships.

HAROUNI: People who are building things, artisans.

EMANUEL: Artisanal math - here it's more about measuring than counting. Tools and materials are key. Like, if you're told to divide a plank into thirds, you can figure that out with a piece of string. And if there are numbers, it's not always as simple as four plus five.

HAROUNI: Four of what, five of what - four cats plus five chickens.

EMANUEL: If you have 4 inches and 5 feet, the answer certainly is not nine. So there are lots of ways to teach math, and 7-year-old Ashkan in Amy Moylan's class can tell us which approach won.

ASHKAN: The nine plus four equals 13.

EMANUEL: The economic model. Houman Harouni says whether you like it or not, we have to study this stuff. Why? Well, we live in a world where money matters, and our math curriculum can prove it. Gabrielle Emanuel, NPR News. Transcript provided by NPR, Copyright NPR.