Everyone knows that you don’t build a house on a weak foundation, you don’t have a five-year-old practice shooting on a ten-foot basketball hoop, and you don’t learn to drive in Manhattan. The same “crawl before you walk before you run” approach pretty much applies to everything.
This includes education. If a student reading at a fourth-grade level is handed The Great Gatsby, she will get next to nothing out of it. Slogging through paragraph after paragraph of indecipherable prose could also result in her hating to read. On the flip side, a student reading at an eleventh-grade level doesn’t want to read a second-grade-level book. Teaching the right level, making sure students are challenged but not drowned, is obvious and by no means brilliant. You can’t go from fractions to calculus in a year, not if you actually want to understand it. As writer Jerry Jesness pointed out in his article “Stand and Deliver Revisited,” which appeared in the July 2002 edition of Reason magazine, the Hollywood treatment of education in Stand and Deliver painted an overly optimistic portrait of what’s possible in a short amount of time. In reality, Jaime Escalante needed a system in place over several years to establish fundamentals, and only then would he build calculus on top of them. It took him a decade to make this happen. The shiny finish didn’t come until well after the foundation was solid.
When I moved to Abaarso, I had no idea what it meant to live in a war-torn country, to be recovering from the complete collapse of all systems and all institutions. When our students got to campus, even the very best were like partially built homes for which there had been no architect. The foundations were almost nonexistent, yet there were a couple of doors and windows precariously hanging in place. The house had a few stories, but the second floor couldn’t bear any weight. One of our greatest challenges the first few years was figuring out what was and what wasn’t sturdy. Eventually, we’d conclude that none of it was; the house needed rebuilding from the base.
In the third year, some of our top students are taking the SSAT—Secondary School Admission Test—an exam that many American boarding schools use to judge applicants. The students have been practicing with a math teacher but found out that calculators weren’t allowed on the test. I heard they were panicking, so I went to talk to them.
“Look,” I tell them, “you don’t even need a calculator for this stuff. Any arithmetic can easily be done in your head. Let’s do #4 in the practice book. That’ll show you.”
I open up to a geometry problem. A circle is inscribed in a square, meaning that the edge of the circle hits all four sides of the square, with the circle completely inside the square. There are four areas of the square that the circle does not also enclose. These are shaded. The problem asks to calculate the shaded areas. You are told that the circle has a radius of 1.
The SSAT exam board no doubt viewed this as a good test of how students can apply geometry. As geometry does not require a lot of foundation before high school, our students are pretty good at it. They quickly get to the heart of the problem. The shaded areas that are inside the square but not in the circle will equal the area of the square minus the area of the circle. They know a circle’s area is Pi times the radius squared, so they square 1, which is still 1, and multiply it by 3.14. The answer is 3.14. They then do the next step, which is to see that 2 radii equal a side of the square. That means the square has sides that are 2, which means that the area of the square is 4.
For many who have not done geometry in some time, that last paragraph may sound dizzying. You may even have skipped it when you saw that math was involved. Don’t feel bad. Solving this SSAT problem takes a reasonable understanding of geometry, and our students have that. We’ve done a nice job building the geometry floor of the house. That’s when I see the foundation below it collapse.
“So which answer is it, a, b, c, or d?” I ask. I am able to eyeball it in a second.
“I don’t know. How am I supposed to do that without a calculator?” one of our top students responds.
“What do you mean? It is 4 minus 3.14. The answer is clear.”
It takes them twenty seconds to hand calculate that answer, one that should be automatic, and students taking exams like the SSAT are so pressed for time that losing twenty seconds means not getting to a later problem. Their mental math skills and understanding of number bonds, such as that 14 will need a 6 to get to 20, which will need 80 to get to 100, are nonexistent. This is a major issue, but even this isn’t the real problem.
“It is .86. But the answers are in fractions.”
This is true. The SSAT has given the answer choices in fractions, so the .86 needs to be converted. However, the SSAT has basically done it for you assuming you have any understanding of what a fraction is. The four choices are 1/6, 6/7, and then two more choices that are both greater than 1. You don’t need to calculate anything. The answer has to be 6/7. This is my whole point. No calculators are needed or even helpful in the problem.
They can’t answer it. Instead they start one by one converting the answer choices from fractions to decimals using long division with lengthy hand calculations. They can’t rule out any, not 1/6 because it is clearly less than .50, or the two fractions that are clearly greater than 1. On the actual exam, taking three minutes on this problem would be testing suicide. Their scores would be crushed by how few problems they’d get to, and the worst part about that is that it isn’t the SSAT or some “test bias” that’s to blame. The test had picked up a legitimate gaping hole in our students’ fundamentals.
After this session, I try the problem on Deqa, who hadn’t been there. She is arguably the top student in the school. Deqa is no better at solving it. Like the others, she cannot see how a fraction relates to a decimal other than by calculating, which means she cannot see it at all. This further means that she won’t be able to catch any mistake in calculating. What’s worse, without such understanding, none of the students can actually apply math. They are just going through mechanical operations and don’t know when to use them or why. The pretty geometry level has been built without any beams to support it.
Fractions are an offshoot of division, which is to say that 6/7 is six units divided seven ways. Might they not even understand division?
“Deqa,” I say, “if we have 90 kilos of rice and we use 12 kilos a day, how many days can we eat rice before we need to get more?” I know she can get 90 divided by 12, but this problem checks that she knows what division is and can apply it. She doesn’t.
Digging through our library, I find fourth-grade math textbooks. One has an assessment in the back, which I have every student take. The ninth graders have now been at Abaarso for half a year. They score 50 percent. The tenth graders are at 66 percent and the eleventh graders, 75 percent. On a fourth-grade test, I’ll call that a disaster.
From then on the “Orange Book” becomes famous at Abaarso. It is a Macmillan/McGraw-Hill fourth-grade textbook with an orange cover. Every student goes through it, even those already in advanced grades. You can’t understand fractions without understanding division; you can’t understand division without multiplication, and multiplication is just a series of additions. So future students will start by visualizing and applying addition.
This lesson in starting at the foundation, one that Jaime Escalante understood and practiced, holds true across all subjects. Yes, you can learn some chemistry before you understand basic science and critical thinking, but you won’t learn it right. And, like the fourth-grade reader picking up Gatsby, it will be a frustrating and off-putting experience.
Let’s skip ahead to a beginner English class at Abaarso, when we had a much better understanding of what we were doing. The book of choice? Green Eggs and Ham by Dr. Seuss. For those who don’t remember, Sam is asking the other character if he likes green eggs and ham. The other character says no, but Sam won’t give up. Instead, he keeps asking him if he’d like them if they were this or that, giving slight variations. “Would you like them in a house? Would you like them with a mouse?” Invariably, the character says no. “I do not like them in a house. I do not like them with a mouse. I do not like them here or there. I do not like them anywhere.” This goes on and on. Nine times Sam tries, and nine times he is rejected.
Our teacher Natalie is patiently working with the class, challenging them to really think about the book in a way you would with something far more advanced. She’s asking them who the characters are and what the conflict is, and she even has them predict what will happen next. The students are new to Abaarso and have never been asked to read a story this way. It is understandably challenging, as is the English, which is why Natalie is starting with a simple story using simple words.
“So what do you think will happen next?” she asks, Sam having been rejected several times already, plenty of book left to be read, and Sam now asking, “Would you eat them in a box? Would you eat them with a fox?”
Much of the room is sure of their answer. “This time he will try them,” they say.
Natalie, hiding her perplexity, patiently asks them to explain why. They come up with a few explanations, but the truth is that this, too, is a very new question for them. It will take lessons like these for them to hone such a skill.
Then she turns the page.
Not in a box.
Not with a fox …
I would not eat green eggs and ham.
I do not like them, Sam-I-am.
Shock and disappointment fill the classroom. The start of some understanding, too.