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Lecture 2 - Approximate Number Sense

Listen to part of a lecture in a psychology class.

Professor: For some time now, psychologists have been aware of an ability we all share. It’s the ability to sort of judge or estimate the numbers or relative quantities of things. It’s called the Approximate Number Sense, or ANS.

ANS is a very basic, innate ability. It’s what enables you to decide at a glance whether there’re more apples than oranges on a shelf. And studies have shown that even six-month-old infants are able to use this sense to some extent. And if you think about it, you’ll realize that it’s an ability that some animals have as well.

Student: Animals have number…uh…approximate…?

Professor: Approximate Number Sense. Sure. Just think: would a bird choose to feed in a bush filled with berries or in a bush with half as many berries?

Student: Well, the bush filled with berries I guess.

Professor: And the bird certainly doesn’t count the berries. The bird uses ANS: Approximate Number Sense. And that ability is innate, it’s inborn. Now I’m not saying that all people have an equal skill or that the skill can’t be improved, but it’s present…uh…as I said…it’s present in six-month-old babies. It isn’t learned.

On the other hand, the ability to do symbolic or formal mathematics is not really what you would call universal. You need training in the symbols and in the manipulation of those symbols to work out mathematical problems. Even something as basic as counting has to be taught. Formal mathematics is not something that little children can do naturally. And it wasn’t even part of human culture until a few thousand years ago.

Well, it might be interesting to ask the question: Are these two abilities linked somehow?Are people who are good at approximating numbers also proficient in formal mathematics?So to find out, researchers created an experiment designed to test ANS in 14-year-olds. They had these teenagers sit in front of a computer screen. They then flash a series of slides in front of them. Now, these slides had varying numbers of yellow and blue dots on them. One slide might have more blue dots than yellow dots, let’s say…six yellow dots and nine blue dots. The next slide might have more yellow dots than blue dots. The slide would flash just for a fraction of a second. So you know, there was no time to count the dots. And then the subjects would press a button to indicate whether they thought there were more blue dots or yellow dots.

So the first thing that jumped out at the researchers when they looked at the result of the experiment was that between individuals, there were big differences in ANS proficiency. Some subjects were consistently able to identify which group of dots was larger even if there was a small ratio, if the numbers were almost equal, like ten to nine. Others had problems even when differences were relatively large, like twelve to eight.

Now, maybe you are asking whether some fourteen-year-olds are just faster, faster in general, not just in math. It turns out: that’s not so. We know this because the fourteen-year-olds had previously been tested in a few different areas.

For example, as eight-year-olds, they had been given a test of rapid color naming.

That’s a test to see how fast they could identify different colors. But the result didn’t show a relationship with the results of the ANS test. The ones who were great at rapidly naming colors when they were eight years old weren’t necessarily good at the ANS test when they were fourteen. And there was no relationship between ANS ability and skills like reading and word knowledge.

But among all the abilities tested over those years, there was one that correlated with the ANS results: math, symbolic math achievement. And this answered the researchers’ question. They were able to correlate learned mathematical ability with ANS.

Student: But it doesn’t really tell us which came first.

Professor: Go on, Laura.

Student: I mean, if someone’s born with good approximate number sense, um, does that cause them to be good at math?Or the other way around: If a person develops math ability, you know, and really studies formal mathematics, does ANS somehow improve?

Professor: Those are very good questions, and I don’t think they were answered in these experiments.

Student: But…wait. ANS can improve?Oh, that’s right, you said that before…even though it’s innate, it can improve. So wouldn’t it be important for teachers in grade schools to…

Professor: Teach ANS? But shouldn’t the questions Laura just posed be answered first? Before we make teaching decisions based on the idea that having a good approximate number sense helps you learn formal mathematics.