In the Oscar-winning film A Beautiful Mind, the brilliant mathematician John Nash solved math problems by seeing them. He’d visualize it, and patterns and possible solutions would pop up right before his eyes. In this Science Update, you’ll learn why the rest of us may think about numbers in a surprisingly similar way.
Why it’s normal to space out in math class. I'm Bob Hirshon and this is Science Update.
Many top mathematicians say they think about numbers as points in space, not symbols on a page. And according to psychologist Marco Zorzi of the University of Padua in Italy, even you might think that way sometimes.
He and his colleagues studied patients with an unusual brain disorder. They disregard everything in the left half of their field of vision, even though there’s nothing wrong with their eyesight.
So these patients may completely neglect things on the left side – perhaps they would shave only the right half of the face, eat only the food on the right side of the plate and so on.
Dr. Zorzi’s team found that these patients also had some problems with math. They could add and subtract just fine, but they couldn’t identify the mid-point in a range of numbers -- for example, between two and eight.
So instead of telling us the midpoint, which in this example would be five, they would say six or even seven.
That suggests that we all may rely on a kind of mental map to understand relationships between numbers -- whether you’re solving an algebraic proof, or comparing cereal prices at the grocery store. For the American Association for the Advancement of Science, I'm Bob Hirshon.
Making Sense of the Research
People with brain damage often can tell us a lot about how the brain normally works. By looking at what happens when one particular kind of thought, reasoning, or perception is impaired, scientists can figure out how important that kind of thinking is to the way we normally function.
In this case, they’re working with patients who have a condition called visual neglect: they ignore whatever’s in the left side of their field of vision. There’s nothing wrong with their eyesight; it’s a problem in perceiving space.
Now imagine that you’re looking at a number line: say, the numbers between one and eight. Most likely, you’d picture them in order from 1 to 8, left to right. And if someone asked you to find the mid-point of this set of numbers, you might look right in the middle of the line to find the correct answer (5).
But what if you neglected the left half of the line? In other words, suppose you ignored the first couple of numbers? You might see 4, 5, 6, 7, and 8, but not 1, 2, and 3. You’d end up looking for the mid-point of these numbers instead of the whole range. And you’d end up picking a number that was higher than the real mid-point, like 6 or 7.
That’s what seems to be happening in these patients, even though they weren’t asked to picture a number line. And that suggests that in some way, they were imagining the numbers as points in space, like you might see on a graph in math class. If these patients were doing it, we probably do it too, probably without even thinking about it.
Interestingly, these patients have no problems with basic addition, subtraction, and multiplication. Zorzi says that these kinds of calculations take place in a different part of the brain than the one that was damaged. So it seems that math requires a combination of different skills that are centered in different parts of the brain.
Now try and answer these questions:
- What is visual neglect?
- Why were patients with visual neglect chosen for Zorzi’s study?
- What does this suggest about the relationship between spatial thinking and math skills?
- Which of these do you think a patient with visual neglect could do correctly? Give reasons for your answers:
a) Balance a checkbook
b) Compare prices of different televisions in a store
c) Make change for a $20 bill
d) Solve algebra problems
e) Figure the tip in a restaurant
f) Estimate the size of a room
- Great mathematicians often talk about “visualizing” math problems and “seeing” numbers as points in space. Have you ever used techniques like these consciously? Give examples.
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Untangling the Mathematics of Knots has lots of information and activities relating to knots, which can be understood both mathematically and spatially.
The Abacus: The Art of Counting with Beads has information about the history and use of the abacus, an early visual calculation