Rebecca Zook - Math Tutoring Online

I’ve been thinking a lot about the math learning discussed by Malcolm Gladwell in Outliers. In one part, a student takes twenty-two minutes to solve a single math problem. In another, a KIPP student takes twenty minutes to solve a math problem on the board with the help of his classmates .

Obviously, one way to master material is to have more time: at KIPP, ninety minutes of math class per day. Or in my own tutoring, a luxurious hour or more to discuss whatever the student wants to go over without any pressure or grades.

Slowing down and diving deep is awesome if you have time. But what do you do when you don’t have time?

When I was in eighth grade and routinely cried myself to sleep over my math homework, if someone had suggested to me that I spend twenty minutes on a single problem until I got it, I probably would have just cried harder. I, like many other students before and after me, had way too many problems to finish.

More time is not always an option.

However, as a student, I would have been a lot more open to the idea of slowing down and exploring if I only had to do it for a few problems. If I, or my teacher, had given myself permission and said, “Why don’t you just try to solve one of these problems, and take as much time as you need,” I would have been more willing to try diving deep.

I’m not talking about dumbing things down or making students less responsible. My philosophy has two parts. If you give a student a page of twenty math problems they don’t think they can do, they’ll feel pressured to do them all so at least they can show you they tried, but they probably only have time to attempt to do them poorly.

But if you give a student one to three difficult math problems instead of twenty, there’s a much better chance that the student will actually solve the problems. Doing it correctly, once, is more effective than doing it incorrectly or incompletely twenty times. And once they’ve untangled the process correctly, they’ll be in a better position to replicate that process later.

Also, reducing the amount of material can be used as a temporary measure to get a particular student through a rough patch and help them overcome a block.

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In his book Outliers, Malcolm Gladwell spends a whole wonderful chapter discussing cultural attitudes towards learning math, and he wraps up by profiling the Bronx Knowledge is Power Program Academy (also known as “KIPP”).

With high expectations and extra-long school hours (among other things), KIPP takes students from poorest of neighborhoods and gives them a chance to pull themselves out of poverty. Founder David Levin observes that when students leave KIPP, “they rock in math.”

So how do they do it? For one, all students do ninety minutes of math every day. Eighth grade math teacher Frank Corcoran explains:

I find that the problem with math education is the sink-or-swim approach. Everything is rapid fire, and the kids who get it first are the ones who are rewarded. … It seems counterintuitive but we do things at a slower pace and as a result we get through a lot more. There’s a lot more retention, better understanding of the material.

Wow! I totally agree! Corcoran’s astute observations that math classes today have a sink-or-swim approach really resonated with me. I don’t think this approach is acceptable, because it leaves so many students behind. I used to be one of them.

When I revisted this quote, I loved hearing how having more time to go over the material helped both the students and the teacher relax, and how going over it more slowly actually helped them cover more material. That has totally been my experience in my tutoring sessions with students.

A sink-or-swim approach also perpetuates the myth that one is either a “math person” or “not a math person,” because it doesn’t give students a chance to fill in the missing pieces in their prerequisite knowledge, really internalize the material, or explore how they learn best.

Moving slower also helps students who otherwise would think of themselves as “not math people” to grow their math abilities through persistent effort, and creates a world richer for having more mathematicians in it!

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I was so excited to discover that in Malcolm Gladwell, his recent book, Outliers, presents a bunch of new research on learning math!

There’s so much good stuff in there that I can’t even begin to tell you all about it. But one thing that struck me in particular was Gladwell’s discussion of the cultural differences between Asian and Western attitudes towards learning math. (You can read an excerpt from the chapter here.)

To start, language differences give Asians a linguistic advantage. In Asian languages, numbers are more transparent. For example, when an English speaker has to do mental math, they need to translate words into numbers first. Before we add “forty-three” to something else, we have to break it down into “four tens and a three.” By comparison, in Chinese the word for “forty-three” is already broken down: “four-tens-three.”

Similarly, we say “three-fifths” to describe a fraction in English. But the Chinese for the same number literally translates as, “out of five parts, take three”: the definition of how a fraction works is built in. These linguistic differences make calculation easier in Asian languages. And because it’s easier to figure out what things mean just from the words, there’s an attitude that it’s normal to be able to figure math out.

This creates what Gladwell calls a “virtuous circle”: because the names for numbers are a little bit easier to understand, arithmetic is a little bit easier to do, which means that maybe students like math a little bit more, which means that maybe they take more math classes and ultimately achieve more in math. In contrast, Western children, by third and fourth grade, start to feel that “math doesn’t seem to make sense; it’s linguistic structure is clumsy; and its basic rules seem arbitrary and complicated.” And the trouble begins…

When I mentioned this to my friend, the Future Doctor Jones, she said, “We’re stuck with this language! What are we supposed to do with it?” Her question is valid—if I tell my tutoring students to say “two-tens-seven” for 27, will they just get beat up on the playground for talking crazy numbers?

So recently I was working with a fifth grader on fractions, and I casually mentioned that in Chinese, they say fractions like, “out of four parts, take one,” instead of “one-fourth.”

I was totally surprised when later in the lesson, this same student spontaneously started saying fractions “the Chinese way.” “Out of seven parts, take four!” “Out of two parts, take one!” When I slipped up and said, “Out of two parts take five,” she corrected me immediately, which meant she completely understood the concept.

Most importantly, she didn’t want to stop doing fractions. She was begging for more!

I’m grateful to my student for spontaneously showing me how we, as English-speakers, can adopt a “Chinese” way of thinking about numbers.

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I’ve been so impressed and intrigued with Malcolm Gladwell’s observations on the relationship between persistence and success in math. So, at an appropriate moment, I eagerly told one of my students about the woman Gladwell describes in Outliers, who kept trying to understand the slope of a vertical line until she finally got it after quite some time.

“How long do you think she kept working on that problem?” I asked my student.

“I don’t know,” my student said. “Maybe three hours?”

Then it hit me. I want all of my students to cultivate persistence, and some of my students definitely need to work harder. But maybe the woman I was telling my student about wasn’t exceptional because she kept trying. Maybe she’s exceptional because she kept trying and she finally got it.

What if students are already persistent and diligent and still not able to understand the material? Is telling them to persist and try harder really the answer?

When I was in middle school, I was an extremely diligent Latin student. I would dutifully copy out the text we were translating, look up every single word in the dictionary and in declension and conjugation charts, and list my English translation under the Latin word. Then I would randomly try to string the words together into a complete sentence.

For whatever reason, my Latin teacher adored me and repeatedly praised my thorough preparation in front of the class. But wasn’t it completely obvious that I had absolutely no idea what was going on?

Even though she was a world-reknowned Latin scholar who cracked jokes in fluent Latin with her friends at the Vatican, she didn’t seem to notice (or care) that I had no idea how the Latin words worked together to create meaning.

Maybe I wasn’t paying attention. Maybe I didn’t understand her explanations. Or maybe she never actually explained it. Even though I was totally clueless, I got straight As in Latin for three years. But my effort was not enough. I never understood Latin.

It happened to me in other classes too. When I was an Algebra 2 student, I’d work on a math problem until I got totally stuck, and I’d approach the teacher’s desk for help. “You need to try harder to figure it out for yourself,” he’d tell me dismissively, and then send me back to my desk.

Now, some of my students confide in me that when they ask their teacher a question, the teacher responds, “If you had been paying attention when I explained it, you wouldn’t need to ask that question. So shut up and pay attention!” But the student was paying attention, and still didn’t understand!

It’s clear that “trying harder” and “paying more attention” aren’t going to fix anything if the effort is misguided, or if what you’re paying attention to doesn’t make sense in the first place. So why are students chastised to work harder and pay more attention as if those are the only variables in the equation that can be changed? I’ve found that frequently the missing link isn’t more effort or focus, but a better explanation or an alternative version of the procedure.

Students (and teachers) can actually change many variables in the equation. They can get a book that works better for them, ask someone else for help in hopes of getting a better explanation, watch an instructional video, or even switch to a different instructor entirely.

Maybe teachers hesitate to encouage students to explore these alternatives because it might undermine their authority. But it’s to the detriment of many hard-working and attentive students who struggle in silence, mistakenly believing that if they just try harder or pay more attention they’ll finally get it—or fearing that if they don’t, they must be incapable.

I’ve really been enjoying Malcolm Gladwell‘s excellent book, Outliers. There’s so much good stuff in this book about the relationship between learning math and: language, cultural attitudes, and agriculture (?!!) that I can’t even describe it all here–you should really just read the whole thing!

One juicy niblet in particular from the book really struck me:

A few years ago, Alan Schoenfeld, a math professor at Berkeley, made a videotape of a woman named Renee as she was trying to solve a math problem. [… ] Twenty-two minutes pass from the moment Renee begins playing with the computer program to the moment she says, “Ahhhh. That means something now.” That’s a long time. [The researcher Schoenfeld remarked,] “If I put the average eighth grader in the same position as Renee, I’m guessing that after the first few attempts, they would have said, ‘I don’t get it. I need you to explain it.’” Schoenfeld once asked a group of high school students how long they would work on a homework question before they concluded that it was too hard for them to solve. Their answers ranged from thirty seconds to five minutes, with the average answer two minutes.

But Renee took twenty-two minutes! Gladwell goes on to explain:

We sometimes think of being good at mathematics as an innate ability. You either have “it” or you don’t. But to Schoenfeld, it’s not so much ability as attitude. You master mathematics if you’re willing to try. That’s what Schoenfeld attempts to teach his students. Success is a function of persitence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds.

Gladwell doesn’t try to explain what made Renee so exceptional. But it definitely made me wonder what I can do to help my students cultivate these qualities in themselves.