by Dan Heath
Understanding Subtraction
What makes something “concrete”? If you can examine something with your senses, it’s concrete. A V8 engine is concrete. “Highperformance” is abstract. Most of the time, concreteness boils down to specific people doing specific things. In the “Unexpected” chapter, we talked about Nordstrom’s world-class customer service. “World-class customer service” is abstract. A Nordie ironing a customer’s shirt is concrete.
Concrete language helps people, especially novices, understand new concepts. Abstraction is the luxury of the expert. If you’ve got to teach an idea to a room full of people, and you aren’t certain what they know, concreteness is the only safe language.
To see this, we can start by studying math classrooms in Asia. We know, from the news over the years, that East Asian children outperform American children in, well, just about everything (except the consumption of fatty foods). This is especially evident in math. The math skills of Americans fall behind those of Asians early—the gap is apparent in the first grade, and it widens throughout elementary school.
What are Asian schools doing differently? Our stereotype is that these schools operate with almost robotic efficiency: Hours are long and discipline is strict. We think of East Asian students as being less “creative” somehow; we like to think they outperform our students through rote mechanics and memorization. The truth, it turns out, is almost exactly the opposite.
In 1993, a group of researchers studied ten schools in Japan, ten in Taiwan, and twenty in the United States. In each school, they picked two different math teachers to observe, and they observed four lessons with each teacher. The researchers found that all the teachers used rote recall quite a bit; it was standard procedure in at least half the lessons observed in every country. But other techniques varied greatly among the three countries.
For instance, consider this question by a Japanese teacher: “You had 100 yen but then you bought a notebook for 70 yen. How much money do you still have?” Or this question, posed by a teacher in Taiwan: “Originally there are three kids playing ball. Two more came later, and then one more joined them. How many are playing now?” As she talked, she drew stick figures on the board and wrote down the equation 3 + 2 + 1.
Notice that these teachers are explaining abstract mathematical concepts by emphasizing things that are concrete and familiar—buying school supplies and playing ball. Their explanations take advantage of preexisting schemas, a tactic we explored in the “Simple” chapter. Teachers take an existing schema—the dynamics of a six-person ball game—and overlay a new layer of abstraction.
The researchers called this style of questioning Computing in Context. It is pretty much the opposite of “rote recall.” And, contrary to our stereotypes, it occurred about twice as much in Asia as it did in the United States (61 percent of lessons versus 31 percent).
In another case, a Japanese teacher placed on a desk 5 rows of 10 tiles each. Then she took away 3 rows of 10 tiles. She asked a student how many tiles were left, and he gave the correct answer: 20. The teacher then asked the students how they knew that this was a subtraction problem. This teacher provided her students with a visual image of subtraction. Students could build an abstract concept—“subtraction”—on a concrete foundation: 30 tiles being yanked away from an original set of 50. The researchers coded questions like this one as Conceptual Knowledge questions. This type of question was asked in 37 percent of lessons in Japan, 20 percent in Taiwan, but only 2 percent in the United States.
Using concreteness as a foundation for abstraction is not just good for mathematical instruction; it is a basic principle of understanding. Novices crave concreteness. Have you ever read an academic paper or a technical article or even a memo and found yourself so flummoxed by the fancy abstract language that you were crying out for an example?
Or maybe you’ve experienced the frustration of cooking from a recipe that was too abstract: “Cook until the mixture reaches a hearty consistency.” Huh? Just tell me how many minutes to stir! Show me a picture of what it looks like! After we’ve cooked the dish a few times, then the phrase “hearty consistency” might start to make sense. We build a sensory image of what that phrase represents. But the first time it’s as meaningless as 3 + 2 + 1 would be to a three-year-old.
This is how concreteness helps us understand—it helps us construct higher, more abstract insights on the building blocks of our existing knowledge and perceptions. Abstraction demands some concrete foundation. Trying to teach an abstract principle without concrete foundations is like trying to start a house by building a roof in the air.
Concrete Is Memorable
Concrete ideas are easier to remember. Take individual words, for instance. Experiments in human memory have shown that people are better at remembering concrete, easily visualized nouns (“bicycle” or “avocado”) than abstract ones (“justice” or “personality”).
Naturally sticky ideas are stuffed full of concrete words and images—think of the Kentucky Fried Rat or the Kidney Heist’s ice-filled bathtub. The Kidney Heist legend would have been far less sticky if the man had woken up and found that someone had absconded with his self-esteem.
Yale researcher Eric Havelock studies tales that have been passed down by word of mouth, such as the Iliad and the Odyssey. He notes that these tales are characterized by lots of concrete actions, with few abstractions. Why? The ancient Greeks certainly had no problem with abstraction—this was the society that produced Plato and Aristotle, after all. Havelock believes that the stories evolved away from abstraction over time. When they were passed along from generation to generation, the more memorable concrete details survived and the abstractions evaporated.
Let’s skip to the modern world and another timeless and beautiful domain of expression: accounting. Put yourself in the shoes of an accounting professor who has to introduce accounting principles to college students. To a new student, accounting can seem bewilderingly abstract—the income statement, the balance sheet, T-accounts, accounts receivable, treasury stock. No people or sensory data in sight.
As the teacher, how do you make accounting concepts vivid? Two professors from Georgia State University, Carol Springer and Faye Borthick, decided to try something radically different. In the fall of 2000, Springer and Borthick taught a semester of accounting using, as a centerpiece, a semester-long case study. The case study followed a new business launched by two imaginary college sophomores, Kris and Sandy, at LeGrande State University.
Kris and Sandy had an idea for a new product called Safe Night Out (SNO), a device targeted at parents with teenagers who were old enough to drive. Installed in the teenager’s car, the device would record the route and speed of the car. For the first time, parents could confirm whether their car was being driven responsibly.
At this point you, as a student in introductory accounting, become part of the story. Kris and Sandy are your friends, and they’ve heard that you’re taking an accounting class. They need your help. They ask, Is our business idea feasible? How many units would we have to sell in order to pay for our tuition? You are given guidance on how to track down the costs of the relevant materials (GPS receivers, storage hardware) and partnerships (how much it would cost to sell it on eBay).
The semester-long Kris and Sandy soap opera revealed the role that accounting plays in business life. Every accounting course defines the distinction between fixed and variable costs, but in the soap opera this distinction wasn’t so much defined as discovered. Kris and Sandy have to pay some costs no matter what, such as the programming expense for developing the product. Those are fixed costs. Other costs are incurred only when products are made or sold—the cost of the materials or eBay’s commission, for example. Those are variable costs. If your friends are pouring their tuition money into a start-up business, those distinctions matter.
The case study is an example of learning in context, similar to the teachers in the Asian math classrooms. But in the math classrooms a student mig
ht encounter 300 different examples over the course of a semester. In the accounting class, students had one example that was sufficiently rich to encompass a semester’s worth of material.
As the semester progresses, you witness, from your hot seat as Kris and Sandy’s accountant, the evolution of their business. A local court approaches Kris and Sandy wanting to use the SNO device for its parolees, but it wants to lease the device rather than buy it. How should Kris and Sandy respond? Later, the business begins to grow rapidly, but suddenly Kris and Sandy make a panicked call to you, having bounced a check. They’ve been selling more units than ever, yet there’s no cash in the bank. How is that possible? (This problem is faced by many startup businesses, and it introduces the difference between profitability and cash flow.) The answer becomes clear to you only after you’ve worked through a month of payment slips and eBay receipts.
So, did the students learn better? At first it was hard to say. The changes to the course made it hard to compare final exams directly with those of previous years. Some students seemed more enthusiastic about the new course, but others groused because the case study demanded a lot of time. Over time, however, the benefits of the concrete case study became increasingly obvious. After experiencing the case study, students with high GPAs were more likely to major in accounting. The concreteness actually made the most capable students want to become accountants.
But the case study also had positive effects for regular students. In the next accounting course—taken an average of two years later—the first section of the course built heavily on the concepts that students were supposed to have learned in introductory accounting. Students who had worked through the case study scored noticeably higher on this first exam. In fact, the difference in scores was particularly dramatic for students with a C average overall. Generally speaking, they scored twelve points higher. And remember, this is two years after the case study ended. Concreteness sticks.
The Velcro Theory of Memory
What is it about concreteness that makes ideas stick? The answer lies with the nature of our memories.
Many of us have a sense that remembering something is a bit like putting it in storage. To remember a story is to file it away in our cerebral filing cabinets. There’s nothing wrong with that analogy. But the surprising thing is that there may be completely different filing cabinets for different kinds of memories.
You can actually test this idea for yourself. The following set of sentences will ask you to remember various ideas. Spend five or ten seconds lingering on each one—don’t rush through them. As you move from one sentence to another, you’ll notice that it feels different to remember different kinds of things.
Remember the capital of Kansas.
Remember the first line of “Hey Jude” (or some other song that you know well).
Remember the Mona Lisa.
Remember the house where you spent most of your childhood.
Remember the definition of “truth.”
Remember the definition of “watermelon.”
David Rubin, a cognitive psychologist at Duke University, uses this exercise to illustrate the nature of memory. Each command to remember seems to trigger a different mental activity. Remembering the capital of Kansas is an abstract exercise, unless you happen to live in Topeka. By contrast, when you think about “Hey Jude,” you may hear Paul McCartney’s voice and piano playing. (If the phrase “Hey Jude” drew a blank, please exchange this book for a Beatles album. You’ll be happier.)
No doubt the Mona Lisa memory conjured a visual image of that famously enigmatic smile. Remembering your childhood home might have evoked a host of memories—smells, sounds, sights. You might even have felt yourself running through your home, or remembering where your parents used to sit.
The definition of “truth” may have been a bit harder to summon—you certainly have a sense of what “truth” means, but you probably had no preformulated definition to pluck out of memory, as with the Mona Lisa. You might have had to create a definition on the fly that seemed to fit with your sense of what “truth” means.
The definition of “watermelon” might also have involved some mental gyrations. The word “watermelon” immediately evoked sense memories—the striped green rind and red fruit, the sweet smell and taste, the heft of a whole watermelon. Then you might have felt your gears switch as you tried to encapsulate these sense memories into a definition.
Memory, then, is not like a single filing cabinet. It is more like Velcro. If you look at the two sides of Velcro material, you’ll see that one is covered with thousands of tiny hooks and the other is covered with thousands of tiny loops. When you press the two sides together, a huge number of hooks get snagged inside the loops, and that’s what causes Velcro to seal.
Your brain hosts a truly staggering number of loops. The more hooks an idea has, the better it will cling to memory. Your childhood home has a gazillion hooks in your brain. A new credit card number has one, if it’s lucky.
Great teachers have a knack for multiplying the hooks in a particular idea. A teacher from Iowa named Jane Elliott once designed a message so powerful—tapping into so many different aspects of emotion and memory—that, twenty years later, her students still remember it vividly.
Brown Eyes, Blue Eyes
Martin Luther King, Jr., was assassinated on April 4, 1968. The next day, Jane Elliott, an elementary-school teacher in Iowa, found herself trying to explain his death to her classroom of third-graders. In the all-white town of Riceville, Iowa, students were familiar with King but could not understand who would want him dead, or why.
Elliott said, “I knew it was time to deal with this in a concrete way, because we’d talked about discrimination since the first day of school. But the shooting of Martin Luther King, one of our ‘Heroes of the Month’ two months earlier, couldn’t be explained to little third-graders in Riceville, Iowa.”
She came to class the next day with a plan: She aimed to make prejudice tangible to her students. At the start of class, she divided the students into two groups: brown-eyed kids and blue-eyed kids. She then made a shocking announcement: Brown-eyed kids were superior to blue-eyed kids—“They’re the better people in this room.” The groups were separated: Blue-eyed kids were forced to sit at the back of the classroom. Brown-eyed kids were told that they were smarter. They were given extra time at recess. The blue-eyed kids had to wear special collars, so that everyone would know their eye color from a distance. The two groups were not allowed to mix at recess.
Elliott was shocked at how quickly the class was transformed. “I watched those kids turn into nasty, vicious, discriminating third-graders … it was ghastly,” she said. “Friendships seemed to dissolve instantly, as brown-eyed kids taunted their blue-eyed former friends. One brown-eyed student asked Elliott how she could be the teacher “if you’ve got dem blue eyes.”
At the start of class the following day, Elliott walked in and announced that she had been wrong. It was actually the brown-eyed children who were inferior. This reversal of fortune was embraced instantly. A shout of glee went up from the blue-eyed kids as they ran to place their collars on their lesser, brown-eyed counterparts.
On the day when they were in the inferior group, students described themselves as sad, bad, stupid, and mean. “When we were down,” one boy said, his voice cracking, “it felt like everything bad was happening to us.” When they were on top, the students felt happy, good, and smart.
Even their performance on academic tasks changed. One of the reading exercises was a phonics card pack that the kids were supposed to go through as quickly as possible. The first day, when the blue-eyed kids were on the bottom, it took them 5.5 minutes. On the second day, when they were on top, it took 2.5 minutes. “Why couldn’t you go this fast yesterday?” Elliott asked. One blue-eyed girl said, “We had those collars on….” Another student chimed in, “We couldn’t stop thinking about those collars.”
Elliott’s simulation made prejudice concrete—brutal
ly concrete. It also had an enduring impact on the students’ lives. Studies conducted ten and twenty years later showed that Elliott’s students were significantly less prejudiced than their peers who had not been through the exercise.
Students still remember the simulation vividly. A fifteen-year reunion of Elliott’s students broadcast on the PBS series Frontline revealed how deeply it had moved them. Ray Hansen, remembering the way his understanding changed from one day to the next, said, “It was one of the most profound learning experiences I’ve ever had.” Sue Ginder Rolland said, “Prejudice has to be worked out young or it will be with you all your life. Sometimes I catch myself [discriminating], stop myself, think back to the third grade, and remember what it was like to be put down.”
Jane Elliott put hooks into the idea of prejudice. It would have been easy for her to treat the idea of prejudice the way other classroom ideas are treated—as an important but abstract bit of knowledge, like the capital of Kansas or the definition of “truth.” She could have treated prejudice as something to be learned, like the story of a World War II battle. Instead, Elliott turned prejudice into an experience. Think of the “hooks” involved: The sight of a friend suddenly sneering at you. The feel of a collar around your neck. The despair at feeling inferior. The shock you get when you look at your own eyes in the mirror. This experience put so many hooks into the students’ memories that, decades later, it could not be forgotten.
The Path to Abstraction:
The Blueprint and the Machine
Jane Elliott’s simulation of prejudice is compelling evidence of the power of concreteness. But if concreteness is so powerful, why do we slip so easily into abstraction?