Ilan Samson may be the kindest, most patient man you’ll ever hate talking to. He’ll make you do math. Worse yet, he’ll make you defend your answer.
Samson invented a tool, called the QAMA calculator (QAMA means how much in Hebrew, but it’s also an acronym for quick approximate mental arithmetic). It functions a lot like traditional calculators, with one major difference: You have to guess an answer before the calculator solves the problem.
If your guess is reasonably close – you’ve got more leeway for “reasonable” as problems become more complex – QAMA tells you the actual answer. Guess too far off, and you’ll have to try again.
This press release from QAMA’s launch gives a helpful example: “An ‘easy’ calculation such as 5 multiplied by 6, would have zero tolerance for any incorrect answer. Calculating 23 to the power 2.1 (answer: 723.81) would ‘tolerate’ an estimate of 550.”
Samson’s from Israel, and went to a university in England. He’s a physicist by training, but lives largely off the royalties from other inventions. He’s got a cabinet full of them.
In 2004, he published a book, “Demathtifying.” One of its central lessons: Many people are afraid of math, not because we’re inherently bad at it, but because we’re told to follow instructions without understanding why.
Samson said that about six years ago, the vice chancellor at UCSD heard about his work, and invited him to San Diego, where he’s now a visiting scholar and inventor in-residence.
He set out to make people rethink how they learn math long before 43 states adopted new Common Core state standards. But now that Common Core is in full-swing in San Diego, he’s hoping more schools will see the benefit of having a tool like QAMA in their classrooms.
The calculator is already in students’ hands at High Tech High and the Preuss School at UCSD, among other schools.
Currently, the standards mean changes to just two subjects – language arts and math – though they’ll most likely change for other subjects, too. Math students will focus on fewer topics at each grade level, but take them to deeper levels.
The goal is to help students connect math concepts, and build on them, instead of learning lessons in fragments. Ideally, students will be able to explain how they arrived at an answer, which is why parents might notice – and sometimes reject – an increased focus on word problems.
But if we want American students to become more competitive internationally, it may be time to rethink how we teach. Elizabeth Green, editor of Chalkbeat, a nonprofit news site covering education, took a detailed look at why Americans stink at math.
Green determined that Americans can learn a lot from how math is taught in other countries. One or her points fit with Samson’s take: American students spend too much time memorizing arbitrary rituals. This approach is divorced from true learning, and undervalues the process of finding the answer.
I recently visited Samson at the home he rents on the bluffs of La Jolla. Even before I sat down, Samson put his QAMA calculator in my hand and jumped into a three-hour math lesson.
Our conversation has been edited for length and clarity.
“I’m sure you remember from school how you were terrorized into learning quadratic equations, yes? Have you ever used that after you left school? Have you ever even seen a quadratic equation since you left school?” Samson said.
Um, what’s a quadratic equation, again?
My point exactly. Just instructing students is of little use, if they cannot connect that knowledge to other concepts.
Most instruction follows a predictable pattern: The teacher writes a problem, then brings students to the answer. But this doesn’t bring the students out of the woods in terms of how the answer was found.
So part of the solution is getting students to ask questions they would naturally ask themselves.
Students calculate to shoulder height, not involving their heads, because they do it all with calculators.
Students start using calculators from age 11 on. That’s bad in itself, but it’s prototypical: They get used to arriving at results without understanding how.
The solution isn’t removing calculators, because calculators are everywhere. The solution is modifying the tool, so it works to students’ benefit.
Let’s back up a step and talk about your calculator. How close does your guess have to be before it gives you the actual answer?
This is a question that takes five seconds to ask, but took me about 15 years to answer.
The guess needs to be reasonable before QAMA gives you the answer. But what does “reasonable” mean?
Reasonable has to change as the problem gets more complex, and there are infinite possibilities.
If the problem is simple, like 6 times 5, you have to guess correctly before QAMA confirms it. But as problems get more complicated, you get more leeway. This is making sure you’re following the right process.
If you take a student who learned to multiply by memorizing times-tables, and ask him to multiple 6 times 7, he says 42. What happens if you say no? What does he do when you ask him to prove it?
What do you think of the way Americans learn math?
Whether it’s England or Israel, or America, the problem is the same: The result is valued more than the process.
Some countries just slog harder. And the countries that get waived around as successes are the ones that work harder.
Let’s look at this problem: (Samson holds up a flash card showing the fraction a/b over by c/d.). On the board, a teacher writes the problem as a stack of vertical letters. Then she tells students to simplify it into one fraction.
But why? Is there a height limit? Is the fraction some sort of truck that’s going to go under a bridge?
Young children are constantly asking why. It’s their favorite question. You can’t get anything past them that doesn’t make sense.
The most common response from adults is: “Because I tell you to.” But this is an abuse of the word “why,” because the adult isn’t giving a real answer to the question.
By the time children are in high school, they stop asking why. We can get them to do almost anything we want. But we haven’t stopped to think about whether that’s a good thing.
Let me turn the question around on you, then. Why is it important that we learn math if we already have calculators to do the work?
Yes, this goes to the question of why we should do it at all.
If you ask a student a question, he plugs it into a calculator and a second later he has an answer. But if you cover up the calculator, a moment later he’s already forgotten it.
What if the answer to the question determined how much he’d pay for an iPod? The student would have no idea.
We use math every day, but we have no concept of quantity. We miss out on money. We don’t understand public policies, or what they will mean for our pockets. If a nurse does math wrong, and administers too much medication, somebody gets hurts, or worse. If an engineer makes a mistake, a bridge collapses. People die.
The point is: When we’re actually using math, we might not call it math, because it’s divorced from the rituals.
There is no ‘how much’ without understanding how. That means knowing how to get to an answer.
Why do we find ourselves in this situation?
Because it is so much simpler to test performance than to test understanding. Consider the time it takes teachers to administer a multiple-choice test, versus an oral exam for each student.
You know the best way to learn something? Teach it. It would be wonderful if students needed to teach somebody else a concept as part of their examination.
How are we going to teach educators to learn the material?
The first step is to get people to understand why we are doing it in the first place. Are we using it to arrive at results, or are we doing it to teach accurate reasoning?
The object is to get students to understand the methods, so if they encounter that situation in the world, they can solve it using only the contents of their skulls. That’s what teaching should be.