Synopsis: It’s no secret: American students are way down the international list when it comes to math scores. Why is this? What makes math so hard for U.S. kids to learn, and how can we help make math easier? Our two guests have answers to both of those questions along with some innovative ideas on how to help kids – and adults – become better at math.
Host: Marty Peterson. Guests: Bob Sun, inventor of The 24 Game and First in Math; Jason Wilkes, author of the book, Burn Math Class: And reinvent mathematics for yourself.
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Teaching Math Better
Marty Peterson: Bring up the subject of mathematics to any group of people and you’re sure to get some groans, some eye rolling and more than a few admissions that math was hard, they weren’t good at it and they’re glad they no longer have to work algebraic equations or find the area of a circle. Why is this? Is math inherently difficult? Are some people just cursed with a non-mathematical brain? Our two guests think that math shouldn’t be dreaded, but enjoyed. And they have some ideas about why so many Americans – from first grade to adulthood – do poorly at math in school and end up hating it. First, Bob Sun, inventor of the 24 Game and the First in Math program. He was born in China and came to the U.S. when he was a child without knowing any English. He says he was teased in school, but those same bullies ran to him to for help in math. He studied hard and eventually became an electrical engineer, but decided he’d use his skills to help people do better at math. He approaches the numbers like he approached learning English all of those years ago…with practice. And that’s what he says is the key to kids – and adults – learning math.
Bob Sun: I realized that, you know, a lot of people have trouble with math because they haven’t really practiced the skill. So then the next question became, “Well, why don’t kids like to practice math? They practice sports, they practice music.” And the number one reason is we don’t have a built-in feedback mechanism for doing mathematics. In the realm of sports and music our physical senses serve that function. So, as an example, if you’re learning to ride a bike and you lose balance, you don’t have to turn to anybody else to tell you that. You’re going to know right away. Playing baseball, swing the bat miss the ball, your eyes and ears will tell you, and you figure out what kind of adjustments you’ve got to make so you’re engaged and ready for the next pitch. But when you shift over to the mental realm of doing math, our physical senses are useless and, you know, we don’t have a built-in feedback mechanism to we’ve essentially been asking people to get good at a skill with a blindfold on.
Peterson: Imagine someone learning to make free throws in basketball without being able to see the basket! They’d lose interest and refuse to practice. To remedy the situation of feedback and practice, Sun devised the 24 Game way back in 1988.
Sun: That was kind of my low-tech way of solving this feedback issue. The game is quite simple. You have cards that are printed with four numbers on it. You try to be the first player to turn those four numbers into the number 24. Use any of the four operations. Now, I said to the children, “I’m going to give you the answer right away. The answer is always 24,” because that’s the target they’re always going for. So 50% of the feedback issue is taken off the table because you don’t have to worry about what the answer is. How to get to 24, however, they had to still solve that, but by turning it into a game, running tournaments, it brought the children together and the children served the function of giving each other feedback and that was my low-tech way of solving this.
Peterson: The game became quite popular, and Sun says they staged tournaments all over the world with kids and adults. When technology became common in homes and schools, he launched the First in Math program online. He says that to date, players have solved about 18-billion math problems correctly on the site because it’s fun.
Sun: Children love to practice mathematics and of course when they practice they get good and then they lose that fear.
Peterson: It’s not just drilling player with math problems on the site. Sun says that they’ve designed it like a video game, giving kids more challenging problems as they go along, keeping kids – and adults – interested and engaged…
Sun: The engagement is really what makes video games so powerful in motivating children. We have hundreds of entry points and no matter what skill level of math you are, you’re going to find a game you can succeed at. I found that the quickest way to engage a child is give him something they can do. Because the attitude is, “Oh, I can do this.” You immediately have their attention. The problem is that if you do it too easily, they quickly get bored. So engagement is really a continuous process. I give you something you can do, I confirm that you can do it very quickly – in a matter of seconds – and then immediately I start pushing you in a very gradual way so that you’re constantly operating right at the edge of your skill, because that’s where all the active learning occurs. And the fact that you’re constantly mastering new skills, new challenges, that’s really the engaging power.
Peterson: Our next guest was a C student in math, and was glad when he got out of high school and didn’t have to take the subject anymore. When he happened on a book on calculus, though, he was fascinated by it. Jason Wilkes went on to study math in college, and is now a grad student in evolutionary psychology at the University of California-Santa Barbara. He’s also author of the book, Burn Math Class — And reinvent mathematics for yourself. It’s about his experience teaching himself calculus, but without mastering algebra, trigonometry or even fractions first. He says he found that learning calculus backwards made much more sense and was far easier than the more traditional method.
Jason Wilkes: There’s really no reason to study some of the content that they teach you before calculus until after you understand calculus itself. Things like natural logarithms are only interesting insofar as they relate to this idea of the derivative of mysterious number “e” they’re always talking about. And trigonometry, in some sense, is not a solvable problem. The problem of sort of breaking up a tilted thing into horizontal and vertical bits without calculus, that’s not really something that you can fully understand – at least not easily. Just for sort of historical reasons I ended up learning the subject in reverse and a lot of people since then have been saying that that’s arguably a better way to learn the beginning parts of the subject.
Peterson: He shows in the book that many of the conventions of math, such as “x” being the symbol for what’s unknown, can trip students up. He’s all for using symbols that mean something to the learner if it makes more sense.
Wilkes: And the purpose of that is sort of to break the magical properties that people sometimes accidentally assign to mathematical symbols. Even if everybody knows explicitly that we can abbreviate things however we want – we can call things whatever we want – they sort of don’t realize that in practice. And so I think its helpful to not get tied to any one particular system of notation and to say if the reason why we use “x” as the symbol for an unknown thing has to do with some long, historical process involving a mistranslation from Arabic, maybe we should realize that and just, you know, use the first letter of whatever we’re abbreviating to stand for the idea in question.
Peterson: Wilkes wants readers to understand that they can learn for themselves how many math operations and concepts work by “reinventing” them all by themselves. Take the simplest geometry calculation you learned in school – finding the area of a rectangle. Wilkes says that this concept can be rooted in our everyday experiences without any math to start.
Wilkes: So you can sort of write down English sentences that describe your everyday, non-mathematical concept of area. And then by abbreviating those sentences more and more you eventually end up with sort of a series of mathematical sentences that say, you know, “Whatever area is, it should be have this way.” Technically, in doing this you’re writing down something called a “functional equation.” But ignore that term. That’s sort of jargon. But basically you’re writing down these English sentences that you then abbreviate into mathematical sentences. If you sort of do that enough, you’ll eventually end up reinventing the concept that you wanted to understand in the first place. So with area, even if we don’t know anything about particular numbers we want to assign to particular areas, it makes sense that if you sort of keep the width of a rectangle-y thing the same then double the length then the area should double whatever we mean by area. And if you do that and, you know, a few more steps that turns out to force the area of a rectangle to be the normal definition we were all taught in mathematics courses.
Peterson: It sounds tricky, but in his book he shows how simple it really is. This is the part of math that Wilkes says is missing from schools until you get into the upper levels of the subject in college. That’s where students are taught to question math to create new concepts.
Wilkes: Mathematicians will talk about “non-standard models of the natural numbers,” which are sort of mathematical systems that behave in most ways like the common system of whole numbers, 0,1,2…and so on, that sort of satisfy all the sentences that define the system but that don’t behave like the intuitive number system we think of when we talk about numbers. Or where you redefine what “equals” means, or generalize what “equals” means, then you get some related but not entirely identical notion, like an equivalence relation. Nothing just “is” in mathematics. Basically everything in mathematics is true because of some assumption or something we agreed upon earlier in the game. If there’s any field where there’s always an answer to the “why” question, it’s mathematics.
Peterson: Bob Sun says that learning how to add, subtract, multiply and divide numbers is important for a good grounding in math. But, like Jason Wilkes says, there’s more to it than that. Understanding the connection between mathematical concepts, and how they work in nature and our lives that are important as well.
Sun: I don’t have to know what the meaning of nine is, what the meaning of six is. I only have to understand how nines can relate to sixes, or relate to threes, or relate to 27s. So math is all about relationships. And in some ways math should be much easier to learn because we’re spared two, three years of struggling to learn the vocabulary. It’s the fascination of how numbers can relate to each other, the patterns that occur and how they reflect what’s going on in nature. That’s what’s fascinating, and if you don’t present that math to children in that way, and try to drill it like a language, they’re going to get turned off and not understand how the world works, because understanding how things relate is a critical part.
Peterson: You can find out how math can be better understood by children and adults by checking on Bob Sun’s websites: the number 24 Game.com, and at FirstinMath.com. To read up on how Jason Wilkes learned calculus by himself, and how you can learn math in a more conceptual, intuitive way, pick up his book, Burn Math Class available in stores, online and at BasicBooks.com. For more information about all of our guests, log onto our site at viewpointsonline.net. You can find archives of past programs there and on iTunes and Stitcher. Our show is written and produced by Pat Reuter. Our production directors are Sean Waldron and Reed Pence. I’m Marty Peterson.