A world without numbers

I have a favorite thought experiment that, for some reason, I think about a lot when I’m driving (to clarify, I’m not driving at the moment). It’s inspired by the claim that the Pirahã language, spoken by a group of people in Brazil, lacks number terms (the original paper is here). The claim is based on Pirahã speakers’ performance in two tasks. In the first, they were shown one battery and asked: how many? The researchers continued to present one at a time, continuing to ask how many there were. The responses were as expected based on previous research: the speakers all used the same term for “one,” a different term for “two,” and combinations of the “two” term and one that signifies “many” for larger quantities.

Image: http://xkcd.com/764/ Interesting post - Is "one, two, many" a myth?
Image: http://xkcd.com/764/

In experiment 2, the batteries were presented in the reverse order, so the participants first saw 10 batteries, and they were taken away one at a time. This time, the participants used the “one” term when there were as many as 6 batteries left, and they all used it when there were 3. The researchers took this as evidence that the terms that researchers believed to indicate “one” and “two” are not precise, but instead seem to be relative quantifiers. The claim is controversial, but the possibility that a language might not have any definite terms for numbers is intriguing.

Returning to my thought experiment, I often try to imagine living in a society with no ways to quantify things. If we had terms for “one,” “two,” and “many,” we could still see the difference between five apples and six, but the only way we could talk about that difference would be invoking our terms for “one” and “many.” In addition to having no words for definite quantities, we wouldn’t have numerals either. I recognize that a society without number terms would be vastly different from the modern-day American society that I know, but I like to imagine some consequences that would arise if our society suddenly lost all numbers:

We’d all have far less money. We’d have the currency that we could stash away, but no more invisible money in abstract sources like stocks and bonds. Debt would probably be a lot more manageable too.

It would be nearly impossible to be punctual. It seems natural to measure time of day by the sun, but that’s still so subjective. The pattern of the sun shifts a tiny bit every day, and we’re probably not pretty good at perceiving the sun’s exact angle in order to use it to tell time.

Life would be less competitive. In school, we wouldn’t be able to split hairs over percentage points. Many sports, like swimming or long jump, would be pointless without a precise measure of time or distance. We would have no way of knowing how many people liked our facebook posts, how many grams of fat were in the cake we just ate, or how few hours we slept last night (thank God – time for that competitive habit to die anyway).

Losing our number system would dramatically catapult our society into a much more primitive culture, and we’d lose progress in every domain of life. But at the same time, I wonder if we might see the number of people being diagnosed with ulcers and high blood pressure plummet… even without the technology to diagnose them.

P.S. An interesting post that uses the comic above as a jumping off point: Is “one, two, many” a myth

Number lines in the head?

I recently read an article by Rafael Núñez which argued that the concept of a number line is not an innate way to conceptualize numbers, although many scientists seem to have assumed that it is. In the study, the researchers had members of the Yupno tribe in New Guinea do a number line task- for example, they may be given a card with 6 dots, and were asked to place it somewhere on a number line ranging from 1 to 10 dots. Those participants who had received some formal schooling did this task almost as accurately as Westerners did, but those who had never gone to school could not seem to grasp the concept- they placed all numbers at one end or the other, seeing it more as a bi-categorical system than a continuous one. All participants did demonstrate that they had number concepts, but the concept of a number line as we know it was not universal.

Image: sparknotes.com
Image: sparknotes.com

This got me wondering: what would an elementary school teacher, who has taught the concept of a number line, think? When teaching children how to use number lines, does the teacher feel like she’s introducing a brand new concept to them, or is she pointing out an innate or intuitive tool that they already have access to? I asked my own second grade teacher (who teaches third grade now), and here’s her response:

“I had always had a number line posted in the room, again because we were told to display it since it became part of the program. If it was not there, we were told to put it up. I noticed far too many [students] were relying on the number line for far too many reasons, some making no sense at al. So, (surprise, surprise) I took it down. For the past two years, not one student asked about its absence since they had to use it in K, 1, and 2… To understand the concepts of number in grade three, they are much better able to grasp many concepts with the use of a hundreds chart since the base 10 skills are basic foundations for grade K-4 math concepts being taught.”

To me, her response suggests that the number line isn’t an innate tool, but one that’s been pushed on students, perhaps too forcefully, causing them to rely on it maybe in circumstances in which they don’t even understand why they’re using it. Even more importantly, eliminating the emphasis on a number line in the third grade classroom may even help some students, by allowing them to conceptualize numbers in other ways that may actually be more intuitive.

I also brought this up with a middle school math teacher I know, since number lines continue to be an integral part of math education. For her, they tend to show up when graphing inequalities, negative numbers, and fractions, and in the ubiquitous coordinate plane. She pointed out that many students reverse the numbers on the y-axis especially, labeling the bottom half as positive and the top half as negative. It seems, she noted, that if a number line weren’t intuitive, this wouldn’t happen. Is it possible that a horizontal number line is intuitive but not a vertical one? That would seem to undermine the concept behind it, that numbers are related to each other in a continuous stream, that the difference between 2 and 3 is the same as that between 3 and 4, regardless of the orientation.

Image: math.com
Image: math.com

To most of us, number lines seem like an obvious way to visualize the relationships of different quantities to each other, but maybe they aren’t so natural. Maybe we have to keep examining the ways of conceptualizing number that are truly natural, and maybe we’ll even want to revamp the ways we teach numbers to elementary students. I don’t know, but I’d be interested to hear from others who have maybe spent more time thinking about and teaching about number concepts than I have.