Thinking logarithmically

An article in NewScientist reports about dyscalculia; a mathematics disorder where people have a selective deficit between a set of numbers and the numerical symbol that represents it.
Interesting stuff about "exact numbers" and "approximate number sense":

"This concept of "exact number" is known to be unique to humans, but there is long-standing disagreement about where it comes from. One school of thought argues that at least some elements of it are innate, and that babies are born with an exact-number "module" in their brain. Others say exact number is learned and that it builds upon an innate and evolutionarily ancient number system which we share with many other species. This "approximate number sense" (ANS) is what you use when you look at two heavily laden apple trees and, without actually counting the apples, make a judgement as to which has more..."

One, two, lots

Amazonian hunter-gatherers called the Mundurucú only have words for numbers up to 5. Does this affect the way they think about mathematical problems? Experts who think that the human concept of exact number is innate would predict not. However, Stan Dehaene of the Collège de France in Paris is among a growing number who believe that exact number is learned and therefore affected by our culture. He decided to test this idea with the Mundurucú.

Working with his colleague in the field, Pierre Pica, and others, Dehaene has found that the Mundurucú can add and subtract with numbers under 5, and do approximate magnitude comparisons as successfully as a control group. But last year the team discovered a big cultural difference. They asked volunteers to look at a horizontal line on a computer screen that had one dot at the far left and 10 dots to the right. They were then presented with a series of quantities between 1 and 10, in different sensory modalities - a picture of dots, say, or a series of audible tones - and asked to point to the place on the line where they thought that quantity belonged.

English-speakers will typically place 5 about halfway between 1 and 10. But the Mundurucú put 3 in the middle, and 5 nearer to 10 (Science, vol 320, p 1217). Dehaene reckons this is because they think in terms of ratios - logarithmically - rather than in terms of a number line. By the Mundurucú way of thinking, 10 is only twice as big as 5, but 5 is five times as big as 1, so 5 is judged to be closer to 10 than to 1.

The team conclude that "the concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education". With only limited tools for counting, the Mundurucú fall back on the default mode of thinking about number, the so-called "approximate number system" (ANS). This is logarithmic, says Dehaene. When it comes to negotiating the natural world - sizing up an enemy troop or a food haul - ratios or percentages are what count. "I don't know of any survival situation where you need to know the difference between 37 and 38," he says. "What you need to know is 37 plus-or-minus 20 per cent."

It is just so intriguing; the idea of thinking logarithmically".