by Sian Beilock
Discrete entities, such as handshakes or dogs, occur only in whole units, unlike water or the height of a tree, which can be measured in fractions. Although students may not realize it at first, by engaging in the simple Math Dance handshake, they are doing discrete math, called combinatorics, the area of mathematics that deals with counting combinations of things. Experiencing the physical element helps students understand the abstractions of math, specifically what it means for something to be a distinct entity.
Understanding combinations of things and how to test all the possible permutations helps students grapple with difficult math concepts they will encounter from elementary school through college. Take the following algebra problems a middle schooler might see:
John has two shirts and three pairs of pants. How many possible outfits does he have?
Answer: 2 × 3 = 6 possible outfits (as long as John isn’t a nudist and always wears a shirt and pants)
Sally has a 6-CD player in her car and 100 CDs. How many unique ways can she load the player?
Answer: There are 100 ways to choose the first CD, 99 ways to choose the second, 98 ways to choose the third, 97 ways to choose the fourth, 96 ways to choose the fifth, and 95 ways to choose the sixth. So 100 × 99 × 98 × 97 × 96 × 95 = 858,277,728,000 (as long as Sally always loads 6 CDs at a time)
Students who are able to physically experience the concept of discrete are better equipped to link these equations to their context, even drawing out the different possible combinations as a way to determine whether the algebra equation they derived is correct. Just like Glenberg’s third graders counting out the fish to give to each of the animals in the math story problem, understanding the concept of discrete, and that a finite number of possible combinations exists, helps ground the meaning of abstract algebra in something concrete.
In another Math Dance exercise, each student starts by creating a movement. Then students pair off and toss a coin 10 times. Heads represents one partner’s movement, tails the other partner’s. Before they actually toss the coin, students predict how many times they will each have to enact their movement. Before the exercise, most students assume that they’ll be doing each movement roughly an equal number of times. But they soon discover this isn’t the case and that the 50 percent probability of hitting heads versus tails doesn’t actually come true, at least until you have done several thousand iterations. Kids learn that getting close to 50 percent gets more likely the more tosses they do—a key component of the concept of probability.
Perhaps most surprising about Math Dance is that the movement itself matters a lot. Doing the dance along with the coin tosses is an important part of Schaffer and Stern’s lesson on probability because, when we move, we often remember concepts and ideas better than when we stand still.
Dancers have long recognized the power of the body as an aid to memory. When ballet dancers learn new choreography, they physically act out the movement sequences to commit the steps to memory. When asked to recall what they learned, dancers tend to recall the dance movements in chunks, based on which body positions flow together. They use their body as a mnemonic device to help them organize their steps, which makes the steps easier to recall. In the same way, performing movements tied to math concepts helps students “choreograph” the math, helping them understand how different concepts fit together, which makes each concept easier to load into memory.
Physical performers besides dancers understand the link between body and mind. Athletes from figure skaters, to gymnasts, to Olympic-caliber divers know every inch of their body and also know that the amazing tricks they perform are based on principles of math and physics. Take the British diver Tom Daley. After wowing the international diving scene at the 2010 Commonwealth Games in Delhi with two Gold Medal performances, as well as his boyish good looks and charm, he was expected to repeat his wins at the 2012 London Olympics. The problem was that Tom was only sixteen and still growing. “I’m five foot eight and if you get taller than five foot 10 then there could be some problems,” he told a BBC interviewer after his winning performance in India. “If you’re too tall you start spinning too slowly so you can’t fit in all the rotations in time before you hit the water. You just have to cross your fingers and hope you don’t grow too tall.”14
By the time the 2012 Olympic Games rolled around, Tom had grown over an inch, to 5 foot 91/2. Thankfully, with a spectacular last dive that ensured his place on the podium, Tom walked away with a Bronze Medal in London and the love of his hometown crowd. David Beckham texted him congratulations, and Prime Minister David Cameron went to see him.15 But it wasn’t an easy road to victory. Tom had to learn several new dives in the years leading up to London to ensure that, despite his growing frame, he would be able to do multiple rotations so that his dives would be graded at a high level of difficulty. There is no doubt that his coaches’ and his own understanding of physics was crucial in coming up with his new repertoire of dives.
A grasp of physics helps athletes understand how best to move and rotate their body, but how we move can also aid how we think about math and science in the first place.
* * *
Susan Fischer bounces around at the front of her introductory physics class at DePaul University in Chicago, desperately trying to get her students interested in the topic of the day: moment of inertia. But she’s not having much success. It is fall in Chicago, which means that lots and lots of ice and snow are just around the corner, and Chicagoans make it a point to cherish every last sunny day. Students are oscillating between attending to the lecture and looking out the two large picture windows in the left-hand wall of the lecture hall, where the warm sunlight is streaming in. From my vantage point in the back row, I can also see that there is a fair amount of email checking and Internet surfing going on. The girl sitting directly in front of me is even buying a pair of shoes from Zappos.com. Until, that is, Fischer puts up a pop quiz on her PowerPoint. All of a sudden, everyone looks up—panicked. Even the shoe buying stops.
Here’s the question Fischer put on the screen:
A solid Disc and a Ring, both of equal mass and diameter, are held at the top of a wooden ramp. When released, both will begin to roll down the ramp without slipping under the influence of gravitational force. If the Disc and Ring are released at exactly the same time, which of the following statements are true?
A. The Disc will arrive at the bottom of the ramp first.
B. The Ring will arrive at the bottom of the ramp first.
C. The Disc and the Ring will arrive at precisely the same time.
Except for the sound of students rustling through their bags and purses to find their clickers (hand-held devices that allow instructors to test students on the fly), there is total silence in the room. But when Fischer announces that the clickers won’t be needed, there is an audible and collective sigh of relief. She tells her students that they are going to use their body to figure out the answer. Teaching assistants appear in the aisles, handing out plastic rulers and black binder clips to each student. I am given a ruler and a binder clip too. Fischer tells us to hold the ruler at one end between our thumb and index finger and to feel how easy it is to make the ruler bounce up and down. She then directs us to attach the binder clip to the opposite end of the ruler. “Now do the same thing,” she says. All of a sudden it becomes much harder to make the ruler bounce. Fischer then demonstrates that as you move the binder clip closer to where your thumb and index finger are positioned, the ruler becomes easier and easier to bounce up and down. You can actually feel the difference, and, when students are eventually told to vote for their answer to the pop quiz question, an overwhelming majority of the class gets the problem right. (The answer is A, by the way.)
Fischer says students never understood the Disc and Ring problem before she added the interactive element. That’s why high school physics classes sometimes go on field trips to amusement parks: experiencing moment of inertia firsthand while riding upside down on a roller coaster, screaming
, gives physical meaning to an otherwise abstract concept.
Like mass, moment of inertia is a property of an object. But unlike mass, which an object has regardless of how you look at it or manipulate it, moment of inertia depends on how far the mass of the object is distributed away from its axis or point of rotation. The closer the mass is to the point of rotation, the smaller the moment of inertia, and the easier it is to move. That’s why the ruler with the binder clip attached is easier to bounce up and down when most of the mass (the binder clip) is close to the axis of rotation (in this case, a student’s thumb and forefinger). It’s also why the Ring will arrive at the bottom of the wooden ramp after the Disc. As long as the Disc and Ring are of equal mass, the Ring must have a greater moment of inertia, in effect, making it harder to get rolling, so it arrives at the bottom after the Disc.
Fischer thinks that teaching students to feel the properties of moment of inertia firsthand helps to engage the motor areas of their brain that are used to registering mass and rotation in daily life. After all, our motor system evolved to help us deal with rotating objects and to aid us in manipulating tools of different masses. Getting our brain’s action centers involved in contemplating physics concepts based on movement is the best way to learn.
Fischer didn’t stumble accidentally on the power of experiencing. You wouldn’t know it from her fairly tall frame, but she was once a competitive diver—a sport, like gymnastics, in which being taller is a disadvantage, as Olympian Tom Daley noted. In diving, the degree of difficulty and thus high scores depend on doing lots of rotations in the air. The taller you are, the greater your moment of inertia, so the slower you spin and the fewer rotations you can do. In other words, if you are too tall, you spin too slowly and can’t get in all the rotations you need before you hit the water. That’s why figure skaters spin so quickly when they are all tucked in. When you bring your hands in toward your body, you are shrinking the moment of inertia, so you spin faster; when the hands come up and out, you slow down. In diving, Fischer was the Ring and all her smaller competitors were the Disc.
Fischer has her students become figure skaters themselves by sitting in a spinning chair with their feet off the ground. If you hold a book in each hand and extend your arms, then pull your arms in toward your body, the spinning chair speeds up. She is convinced that, when students feel this change in moment of inertia, when they engage their body in understanding the concept, they do better on tests of this concept. She told me about this idea a few years ago, when we were first introduced at a gathering of women scientists in Chicago. I was intrigued by her theory and the idea that our bodily experiences can affect our thinking, so I offered to help her test her hunches (with the help of one of my graduate students, Carly Kontra).
We have found that becoming part of a physical system enhances learning. We’ve had students move their arms while on a spinning chair, engage in the binder clip experiment, and move an axle with a bicycle wheel spinning on it from vertical to horizontal and back, thus changing the direction that the wheel is spinning in the world. Indeed compared to just watching a demonstration in class or simply reading about the physics of mechanics in a textbook, bodily experience leads to marked learning gains on homework assignments, quizzes, and tests that can be seen even weeks later.16
Why? Using fMRI to peer inside the brains of students who have actively engaged with physics concepts like moment of inertia, angular momentum, and torque, Fischer and my research group have found that the motor cortex, the chunk of brain tissue involved in planning and initiating movement, is activated. After physically experiencing these concepts, students later activate their motor cortex when they just think about, say, angular momentum when taking a quiz on the topic. It’s as if their motor system is replaying their previous experiences, helping them reason about what they can’t actually see and feel in the moment. The more the motor cortex is involved, the better students do on related test questions about the physics of mechanics.
In short, getting the body involved helps the mind learn.
CHAPTER 4
Don’t Just Stand There
HOW MOVING SPARKS CREATIVITY
Moved to Insight
Google’s corporate headquarters (otherwise known as the Googleplex) is set on twenty-six acres in Mountain View, California. There are four main buildings, each of which houses an eclectic mix of computer scientists, engineers, and managers. While it might seem most straightforward to group Google employees by what they do—engineers in one building and managers in another—that isn’t how the company works. Instead the space is designed to foster an atmosphere of interaction. People with different jobs are mixed together across the Google campus, and structures such as an indoor tree house and a volleyball court encourage employees to get up and move.
Google says the whole point of their campus’s design is to encourage interactions between different teams of Googlers, to spark conversations that might not normally take place. But the interactive atmosphere also encourages movement. We have already seen how movement can help kids learn and adults remember. Movement also helps us solve problems and can even increase our productivity because thinking involves moving the body as well as the mind.
To get a better sense of how movement can enhance problem solving, consider the following scenario:
You are a doctor and have realized that your patient has an inoperable stomach tumor. There are certain lasers that can destroy this tumor if their intensity is great enough. That’s the good news. The bad news is that, at the intensity the lasers need to be to destroy the tumor, they will also destroy the healthy tissue that surrounds the tumor. The tumor is malignant, so if you don’t operate on it, the patient is going to die. How can you destroy the tumor without damaging the healthy tissue through which the lasers must travel? Is there a type of procedure you can do that will obliterate the tumor while at the same time making sure that the healthy tissue that lives around the tumor isn’t damaged?
If you conclude that your patient is screwed, you are in good company. This problem proves to be a difficult one to crack. In fact only around 10 percent of undergraduate students get this problem correct when asked to solve it the first time around.1 But there is a pretty simple way to increase success, and—as you may have guessed—it involves the body. People who are given a computer diagram of the problem (a circle depicting the tumor inside surrounded by a thick layer of healthy tissue outside) and are asked to contemplate possible solutions, but who are also asked to simultaneously keep track of a tiny dot bouncing around the screen, are much more likely to come up with the right answer—as long as the tiny dot is moving back and forth through the healthy tissue to the tumor at various points around the tissue’s perimeter.2
In case you haven’t figured it out yet, the solution to the problem is to position a number of separate lasers around the patient, each directed at the stomach tumor. If each laser delivers a small amount of the radiation, you end up with enough accumulated radiation to destroy the tumor while at the same time saving the healthy tissue surrounding it.
By moving the body (in this case the eyes) in a way that mimics the solution, people have thoughts about the problem that they wouldn’t have otherwise had. Students think the dot task is designed to distract them, that it will make it harder to solve the tumor problem. But when the eyes draw a path that shows many different lasers converging on the tumor from many different areas, the bouncing dot actually leads to increased success.
Moving the body can alter the mind by unconsciously putting ideas in our head before we are able to consciously contemplate them on our own. People use their body all the time when problem solving, without even knowing it. In the case of the tumor problem, researchers found that we often unconsciously work our way through the scenario, testing out possible solutions with our eye movements. Most interesting, before students realize they have come up with the correct answer, you can actually see the solution in their eye movements.3
What acc
ounts for this direct link between body and mind? We draw on our concrete physical experiences to construct our reality. For example, the tactile sensation of warmth causes us to think about social closeness, and making a fist leads us to feel more assertive. Getting a person to move lowers his threshold for experiencing thoughts that share something in common with the movement. That’s why moving your eyes in a way that mimics the solution to the tumor problem heightens the probability that you will find that solution.
Sometimes the best way to crack a problem is to get moving. This is advice that dancers have followed for years. They constantly use movement to create new ideas. When dancers try to develop a new movement, their body is their medium, similar to the way an artist might use paint or a violinist uses the sounds of her violin to create. Just as choosing a different instrument or shifting from paint to pencil changes the art form, so too does changing the body. Make a body rigid, and the style and form of the dance are altered. The mechanics of the body are front and center in creativity. The thinking process is extended over the body. In other words, many performers literally think with their body.4
More evidence that our actions influence our thoughts—and specifically our ability to be creative—comes from research on metaphor. We use metaphors constantly, whether it is thinking “outside the box” or “putting two and two together” or first considering a problem “on the one hand, then on the other.”
But here’s where things get really interesting: our creative ability is enhanced when we literally act out creative metaphors. When people are asked, say, to come up with a word to add to measure, worm, and video to form three new compound words, most find the task difficult. (The answer is tape: tape measure, tapeworm, and videotape.) Finding the answer involves searching our mind for some broad connection among the words, a creative way to fit them together. It’s not easy. However, when people literally embody the “thinking outside the box” metaphor, they become more creative and better able to solve these types of compound-word puzzles.
