(continued from Part A)
What do babies know about math?
From this point on, I'm going to be working off memory a lot... I watched a presentation about this once, very fascinating, but I don't know if I can find the data.
So basically, some researchers wanted to know if babies could distinguish between different numbers. So they took infants, less than 1 year old, into a laboratory, and asked them if they knew what the difference between 17 and 18 was.
The babies looked around and started making gurgling noises so the researchers decided they had to be a bit more clever. So instead, they showed these children two pictures with different numbers of objects, and marked which picture held the babies' attention. I'm sure it's a bit more complicated than this, but essentially they tried to find out what babies knew about numbers.
They knew the difference between 0 and 1. Between a blank screen and a screen with a dot on it, the dot is obviously more exciting.
They knew the difference between 1 and 2. Between a lonely dot and dots that weren't lonely, the babies wanted to join the more social group.
The did NOT know the difference between 2 and 3. To them, it looked basically the same - a few friendly dots. Similarly, between 3 and 4, 4 and 5, etc... a difference of 1 was meaningless after the 1~2 jump.
However, they DID know the difference between 2 and 4... between 4 and 8... between 12 and 24... But not between 12 and 18, or 40 and 50, etc.
The result: the RATIO is what the infants recognize. As long as one picture had at least twice as many dots as the other, it captivated the baby's attention.
Further study was done. This time, children of different ages were studied. They found that after a couple months, babies (or toddlers now, perhaps) could distinguish between 2 and 3. And a couple months later, between 3 and 4. Then all of a sudden there's a jump, and in an incredibly short length of time, the children recognize 4, 5, 6, 7, 8...
To people who study progress over time, a sudden jump almost always reflects a new technology. You invent the printing press, and the number of books explodes. Develop the cotton gin, the textile industry skyrockets. Here too, there is a new technology - the technology of counting.
I call counting a technology because it is NOT something humans inherently have. Maybe you've heard the news stories of tribes of people (the Pirahã are an example of this) who don't have words representing different numbers, just concepts of "smaller amount" and "larger amount." They didn't discover the technology of counting, so they couldn't distinguish between, say, 5 dots and 7 dots.
What humans do inherently have is something often called "number sense." The definition on Wikipedia says that number sense is "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Note that the process of counting is not referred to, nor are specific names of numbers - here "number" just refers to an understanding of amount. Once counting is learned, one's number sense drastically improves - reflecting the dramatic growth in a toddler's ability to recognize amounts. But you have number sense even without counting - in its most natural state, number sense is based not on counting, but on RATIOS.
As you grow, whether you learn counting and names of numbers or not, your number sense sharpens - while babies need a 2:1 ratio to be able to recognize something, most adults can recognize smaller ratios, 1.5:1 or lower, and curiously, many people who are very good at higher level math tend to also be very good at this very fundamental, basic number sense, recognizing ratios as small as 1.1:1 or lower (i.e. being able to tell the difference between 11 dots and 10 dots without counting them, just at a glance)
I found this site where you can test your basic number sense for yourself - it works because you don't have time to count the dots, so you have to rely on your more basic sense of ratio. Unfortunately it doesn't tell you your ratio, but if you keep track of the numbers of blue and yellow dots in each question and which ones you got right or wrong, you can figure out what ratio you need to consistently get it right. Unfortunately, after a bit of time of getting used to it, I got the right answer every time, so I'll need a better tester...
Actually, I'm inspired now to make a swf file which does the same thing, but actually calculates the ratio of dots you can distinguish. I'll post it to the blog when I'm done.
(Edit: I'm done.)
Ratio keeps on coming up in the way we think... for example, intuitively, we think $1 billion and $1.01 billion are extremely close. But that difference is more money than most of us will ever own in our entire lives. We don't think about the difference, which is huge - we think about the ratio, which is tiny.
So how, then, does counting, which focuses on sums and differences instead of ratios, fit in?
(continued in Part 3A)
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