## "Why Johnny Can't Add" by Michael Cothran

My father was an aerospace engineer. When I was growing up, I never really knew what he did, since his job involved mostly top-secret projects. In fact, I never once visited his office, which required a high-security clearance even to enter. But over the years, I did manage to piece together a few facts about what he did.

One day, when I was rummaging around in a supply closet, I came across a paperback book titled

That explained his frequent visits to Vandenburg Air Force Base, from which test missile launches were frequently made, their vapor trails visible from our house in the south bay of Los Angeles County. And it explained why he could frequently be found in the garage at night, firing different materials to test their melting point.

And that explained why he said with such authority when we sat on our living-room couch watching CNN as the Israelis launched Patriot missiles to shoot down the Iraqi SCUD rockets during the first Iraq War, “Oh, that’s nothing. That technology is at least twenty years old. You should see what we have now.” I later found out, through various conversations with my father and others, that he had headed the project to put the first three spy satellites into space. After being released from their cargo bay, their wings folded out into a 150-foot wingspan that had to be positioned within the accuracy of the width of a nickel. “We got it right every time,” he said. Had I known all this at the time, I would have been even more comforted when, as a college student having just read an article on the nation’s missile defense system and the various strategies the Air Force employed to make sure we could launch our ICBMs in the face of a massive Soviet missile attack, my dad, to whom I was reading it while he was shaving, finally lowered his razor, looked at me, and said, “Don’t worry about it. You have no idea what we have,” and calmly finished shaving.

One night, when he headed payload processing for the Space Shuttle, my dad came home somewhat aggravated. As it turned out, there was a problem getting the cargo out of the Shuttle cargo bay if a launch had to be aborted. The cargo was inserted in the cargo bay with the Shuttle standing on its end, vertically, on the launch pad. But how to get the cargo out so it could be transported back to the U.S.?

For three nights we heard hammering and drilling in the garage. After the third night, he came in holding a board with various hydraulics and pulleys and other things I would never be able to identify. “I figured it out,” he said.

He was on at least two space shuttle launch teams, and was at Cape Canaveral in 1986 when the Space Shuttle

In his later years, before he had lost the ability to speak, we were sitting, watching a television news report on the Iranian nuclear centrifuges, which are still a problematic issue because that is the only way to produce weapons-grade plutonium. “They must have somehow gotten the report I wrote on that,” he said. “What do you mean?” I asked. “Well, you know, I wrote the design specifications for the nuclear centrifuge.”

No, I didn’t know that.

But the curious thing about all this was the kind of math education he received. My father grew up poor in the hills of South Carolina. When he was five years old, he convinced his mother to let him go to school. So early one morning, he turned from the window of their house, from which he could see the one-room schoolhouse down the road, and said, “Ma, the smoke’s a-rollin’!” They had stoked up the wood stove, which meant school was about to start.

He ran down the hill and snuck in the back door, sitting in a seat in the back row, near the water barrel.

When the teacher asked him who he was, he wouldn’t tell her.

Being an education writer, I was naturally interested in how they taught math there. “We just learned how to figure,” he said enigmatically.

“What kind of curriculum did they use?” I asked.

“We didn’t have any. Couldn’t afford books. They just taught us arithmetic.” The teacher didn’t need a book because she knew the system of arithmetic and she taught it to students using practices that would make a modern progressive educator cringe.

When my dad graduated from high school, he applied at Clemson University, which was just a couple miles down the road. He walked into the admissions office one day during a break from laying cement with a job crew. They told him to come back when he had a shirt and shoes and they would take his application.

He graduated with a bachelor’s degree in ceramic engineering, having taken every math course the university offered. And the interesting thing is that he never acquired another degree. Everything he did, from missiles to satellites to nuclear centrifuges, he did with a bachelor’s degree―and an education in arithmetic that today’s education experts would call primitive.

This was not uncommon. In fact, if you talk to people who were educated in the old rural, one-room schoolhouses, it is not uncommon to be treated to a litany of the successes of those who graduated from them.

I told my dad once about the kind of math they were doing in the early 1990s in schools in Kentucky where I lived. “That sounds like what they were doing in California in the 1950s,” he said. “In fact, if you give me a report from one of my engineers and told me he was educated in California, I’ll tell you how old he is.” These engineers, he said, would use computers to compute complex calculations, but would make basic mistakes that no good mathematician should make.

It was the newest version of the New Math (invented in the 1950s and implemented mostly in the 60s), an approach to math that has characterized every education reform since.

The central tenet of the New Math was that, rather than teaching the “how” of math, we should teach the “why.” Rather than spending all that time memorizing the fact that 2 + 2 = 4, it would get children to understand why 2 + 2 = 4. The purveyors of the New Math were impatient with the traditional curriculum’s reliance on memorization and drill, and believed math education could be improved by putting more emphasis on understanding as opposed to mastery.

In his famous 1973 book

[T]he students respond at once: “9 and 1 are 10 and 1 more is 11.”

“Wrong,” the teacher exclaims.

“The correct answer is that by the definition of 2, 9 + 2 = 9 + (1 + 1). But because the associative law of addition holds, 9 + (1 + 1) = (9 + 1) + 1.

Now 9 + 1 is 10 by the definition of 10 and 10 + 1 is 11 by the definition of 11.”

In the Outcomes-Based Education of the early 1990s, the New Math returned with a vengeance, this time armed with the “math essay,” another way of facilitating an “understanding” of mathematics.

Today, textbook publishers, who are still stuck in the 60s, have interpreted the new Common Core Math Standards through their 1960s lens and now emphasize “multiple strategies” for learning basic arithmetical operations. This is what was behind the now infamous math quiz posted on the website Reddit, in which a student was marked down for saying that 5 + 5 + 5 = 15 rather than saying that 3 + 3 + 3 + 3 + 3 = 15. The quiz question had to do with one of many new math “strategies” now being employed by teachers and publishers, in this case the “Repeated Addition” strategy for multiplication.

This is one of the most controversial aspects of modern math pedagogy: the complicated way in which it proposes to teach simple arithmetic. This is not an accident and is, in fact, very intentional. The idea behind this “multiple strategies” approach is to intentionally complicate the process of learning arithmetic in order to force the student to think about the procedures he is learning. The more strategies employed to teach a child to solve simple arithmetic problems, goes the reasoning, the better off he will be.

This is exactly the opposite approach to that taken by traditional educators. If we were to haul the teacher from my father’s one-room schoolhouse into a modern math classroom, her first impression would be that the education world had been turned upside down. She would wonder why we had completely overcomplicated a simple procedure. And if we were to tell her that the reason we were doing it was to get children to understand arithmetic better, she would get out the dunce cap and put us on a stool in the corner.

What she would know was that this is precisely

Unlike the modern math educator, she would know that arithmetic is a tool—that is, a means, not an end. Arithmetic is not something you learn—it is something you should have learned. It has no value other than as a tool for learning other things, namely more advanced, conceptual math.

She would know, unlike many modern math educators, that the point is not to understand arithmetic, but to use arithmetic in order to understand. You learn arithmetic, not in order to think about arithmetic, but in order

“We should do all we can,” said Kline, “to make the elementary operations so habitual that students do not have to think about them any more than one thinks when he ties his shoelaces.”

No one believes that a carpenter is made better able to build a house by contemplating the complex process by which his tools are made. No one believes that you can become a better writer by learning more about how the alphabet system was developed.

To think too much about the things that are supposed to help you think can be positively detrimental. Kline tells the story of a centipede who was walking along and met a toad, who remarked to the centipede, “Isn’t it wonderful? You have one hundred feet and yet you know when to use each one,” at which point, “the centipede began to think about which foot to use next and was unable to move.”

“It is a profoundly erroneous truism,” said the great mathematician Alfred North Whitehead, “that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”

The math education of my father’s day produced a whole generation of great scientists and engineers. They are the ones who sent men to the moon and who pioneered the computer revolution. They followed the great German scientists of the nineteenth century whose classical education enabled them to conduct the relativity and quantum revolutions in physics, and the development of genetics in biology.

I asked my father one time whether he took calculus in high school. I was curious because so many of today’s schools push advanced mathematical subjects younger and younger in the curriculum. “I never even heard of calculus until after I graduated from high school,” he said.

Poor man. If only he’d had some of the advantages of today’s students, maybe he could have achieved something in life.

Originally published in

One day, when I was rummaging around in a supply closet, I came across a paperback book titled

*The Nike Zeus Propulsion System*. At the bottom was my father’s name. When I brought it in and asked him about it, he took it from me and said, nicely but firmly, “You’re not supposed to have that.” The*Nike Zeus*was the nation’s first anti-ballistic missile (a missile that shoots down other missiles). He had apparently—I later learned—designed it.That explained his frequent visits to Vandenburg Air Force Base, from which test missile launches were frequently made, their vapor trails visible from our house in the south bay of Los Angeles County. And it explained why he could frequently be found in the garage at night, firing different materials to test their melting point.

And that explained why he said with such authority when we sat on our living-room couch watching CNN as the Israelis launched Patriot missiles to shoot down the Iraqi SCUD rockets during the first Iraq War, “Oh, that’s nothing. That technology is at least twenty years old. You should see what we have now.” I later found out, through various conversations with my father and others, that he had headed the project to put the first three spy satellites into space. After being released from their cargo bay, their wings folded out into a 150-foot wingspan that had to be positioned within the accuracy of the width of a nickel. “We got it right every time,” he said. Had I known all this at the time, I would have been even more comforted when, as a college student having just read an article on the nation’s missile defense system and the various strategies the Air Force employed to make sure we could launch our ICBMs in the face of a massive Soviet missile attack, my dad, to whom I was reading it while he was shaving, finally lowered his razor, looked at me, and said, “Don’t worry about it. You have no idea what we have,” and calmly finished shaving.

One night, when he headed payload processing for the Space Shuttle, my dad came home somewhat aggravated. As it turned out, there was a problem getting the cargo out of the Shuttle cargo bay if a launch had to be aborted. The cargo was inserted in the cargo bay with the Shuttle standing on its end, vertically, on the launch pad. But how to get the cargo out so it could be transported back to the U.S.?

For three nights we heard hammering and drilling in the garage. After the third night, he came in holding a board with various hydraulics and pulleys and other things I would never be able to identify. “I figured it out,” he said.

He was on at least two space shuttle launch teams, and was at Cape Canaveral in 1986 when the Space Shuttle

*Challenger*broke apart 73 seconds into its launch, killing seven astronauts.In his later years, before he had lost the ability to speak, we were sitting, watching a television news report on the Iranian nuclear centrifuges, which are still a problematic issue because that is the only way to produce weapons-grade plutonium. “They must have somehow gotten the report I wrote on that,” he said. “What do you mean?” I asked. “Well, you know, I wrote the design specifications for the nuclear centrifuge.”

No, I didn’t know that.

But the curious thing about all this was the kind of math education he received. My father grew up poor in the hills of South Carolina. When he was five years old, he convinced his mother to let him go to school. So early one morning, he turned from the window of their house, from which he could see the one-room schoolhouse down the road, and said, “Ma, the smoke’s a-rollin’!” They had stoked up the wood stove, which meant school was about to start.

He ran down the hill and snuck in the back door, sitting in a seat in the back row, near the water barrel.

When the teacher asked him who he was, he wouldn’t tell her.

Being an education writer, I was naturally interested in how they taught math there. “We just learned how to figure,” he said enigmatically.

“What kind of curriculum did they use?” I asked.

“We didn’t have any. Couldn’t afford books. They just taught us arithmetic.” The teacher didn’t need a book because she knew the system of arithmetic and she taught it to students using practices that would make a modern progressive educator cringe.

**They memorized their addition and subtraction facts and their multiplication tables, and mastered long division through repeated drill and practice. I asked whether he thought that was boring. “We didn’t know enough to be bored,” he said. In fact, this is how most everyone was taught arithmetic in that time.**

That schoolhouse was abandoned by the time I first saw it when I was little. Presumably all the children had been sent to large county schools, where they knew better.

When my dad graduated from high school, he applied at Clemson University, which was just a couple miles down the road. He walked into the admissions office one day during a break from laying cement with a job crew. They told him to come back when he had a shirt and shoes and they would take his application.

He graduated with a bachelor’s degree in ceramic engineering, having taken every math course the university offered. And the interesting thing is that he never acquired another degree. Everything he did, from missiles to satellites to nuclear centrifuges, he did with a bachelor’s degree―and an education in arithmetic that today’s education experts would call primitive.

This was not uncommon. In fact, if you talk to people who were educated in the old rural, one-room schoolhouses, it is not uncommon to be treated to a litany of the successes of those who graduated from them.

I told my dad once about the kind of math they were doing in the early 1990s in schools in Kentucky where I lived. “That sounds like what they were doing in California in the 1950s,” he said. “In fact, if you give me a report from one of my engineers and told me he was educated in California, I’ll tell you how old he is.” These engineers, he said, would use computers to compute complex calculations, but would make basic mistakes that no good mathematician should make.

It was the newest version of the New Math (invented in the 1950s and implemented mostly in the 60s), an approach to math that has characterized every education reform since.

The central tenet of the New Math was that, rather than teaching the “how” of math, we should teach the “why.” Rather than spending all that time memorizing the fact that 2 + 2 = 4, it would get children to understand why 2 + 2 = 4. The purveyors of the New Math were impatient with the traditional curriculum’s reliance on memorization and drill, and believed math education could be improved by putting more emphasis on understanding as opposed to mastery.

In his famous 1973 book

*Why Johnny Can’t Add*, mathematician Morris Kline gave an example of how this worked out in the classroom of the late 1960s. The teacher asks, “Why is 9 + 2 = 11?”[T]he students respond at once: “9 and 1 are 10 and 1 more is 11.”

“Wrong,” the teacher exclaims.

“The correct answer is that by the definition of 2, 9 + 2 = 9 + (1 + 1). But because the associative law of addition holds, 9 + (1 + 1) = (9 + 1) + 1.

Now 9 + 1 is 10 by the definition of 10 and 10 + 1 is 11 by the definition of 11.”

In the Outcomes-Based Education of the early 1990s, the New Math returned with a vengeance, this time armed with the “math essay,” another way of facilitating an “understanding” of mathematics.

Today, textbook publishers, who are still stuck in the 60s, have interpreted the new Common Core Math Standards through their 1960s lens and now emphasize “multiple strategies” for learning basic arithmetical operations. This is what was behind the now infamous math quiz posted on the website Reddit, in which a student was marked down for saying that 5 + 5 + 5 = 15 rather than saying that 3 + 3 + 3 + 3 + 3 = 15. The quiz question had to do with one of many new math “strategies” now being employed by teachers and publishers, in this case the “Repeated Addition” strategy for multiplication.

This is one of the most controversial aspects of modern math pedagogy: the complicated way in which it proposes to teach simple arithmetic. This is not an accident and is, in fact, very intentional. The idea behind this “multiple strategies” approach is to intentionally complicate the process of learning arithmetic in order to force the student to think about the procedures he is learning. The more strategies employed to teach a child to solve simple arithmetic problems, goes the reasoning, the better off he will be.

This is exactly the opposite approach to that taken by traditional educators. If we were to haul the teacher from my father’s one-room schoolhouse into a modern math classroom, her first impression would be that the education world had been turned upside down. She would wonder why we had completely overcomplicated a simple procedure. And if we were to tell her that the reason we were doing it was to get children to understand arithmetic better, she would get out the dunce cap and put us on a stool in the corner.

What she would know was that this is precisely

*not*the purpose of teaching arithmetic. She would understand implicitly that there are things you learn in order that you may think about them and things you learn so that you do not have to think about them, and that arithmetic is an example of the latter.Unlike the modern math educator, she would know that arithmetic is a tool—that is, a means, not an end. Arithmetic is not something you learn—it is something you should have learned. It has no value other than as a tool for learning other things, namely more advanced, conceptual math.

She would know, unlike many modern math educators, that the point is not to understand arithmetic, but to use arithmetic in order to understand. You learn arithmetic, not in order to think about arithmetic, but in order

*not*to think about it.“We should do all we can,” said Kline, “to make the elementary operations so habitual that students do not have to think about them any more than one thinks when he ties his shoelaces.”

No one believes that a carpenter is made better able to build a house by contemplating the complex process by which his tools are made. No one believes that you can become a better writer by learning more about how the alphabet system was developed.

To think too much about the things that are supposed to help you think can be positively detrimental. Kline tells the story of a centipede who was walking along and met a toad, who remarked to the centipede, “Isn’t it wonderful? You have one hundred feet and yet you know when to use each one,” at which point, “the centipede began to think about which foot to use next and was unable to move.”

“It is a profoundly erroneous truism,” said the great mathematician Alfred North Whitehead, “that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”

The math education of my father’s day produced a whole generation of great scientists and engineers. They are the ones who sent men to the moon and who pioneered the computer revolution. They followed the great German scientists of the nineteenth century whose classical education enabled them to conduct the relativity and quantum revolutions in physics, and the development of genetics in biology.

I asked my father one time whether he took calculus in high school. I was curious because so many of today’s schools push advanced mathematical subjects younger and younger in the curriculum. “I never even heard of calculus until after I graduated from high school,” he said.

Poor man. If only he’d had some of the advantages of today’s students, maybe he could have achieved something in life.

Originally published in

*The Classical Teacher*Spring 2016 editionMartin Cothran, the author of Memoria Press’ Traditional Logic, Material Logic and Classical Rhetoric programs, is an instructor of Latin, Logic, Rhetoric, and Classical Studies at Highlands Latin School. Martin holds a B.A. in philosophy and economics from the University of California at Santa Barbara and an M.A. in Christian Apologetics from the Simon Greenleaf School (now a part of Trinity University). He currently serves as senior policy analyst with The Family Foundation of Kentucky. His articles has have appeared in the Cincinnati Enquirer, the Louisville Courier-Journal, and various other newspapers, as well as on radio and television. He has also served as a registered agent (or “lobbyist”) at the Kentucky State Capitol for over 12 years and has served on various state committees that oversee education policy, where he continues to be an influential voice on education policy issues. Martin is the managing editor of “The Classical Teacher” magazine, which also serves as Memoria Press’ product catalog. He and his family live in Danville, KY.