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Study Suggests the Number-Line Concept Is Not Intuitive
An anonymous reader writes "The Yupno people of New Guinea have provided clues to the origins of the number-line concept, and suggest that the familiar concept of time may be cultural as well. From the article: 'Tape measures. Rulers. Graphs. The gas gauge in your car, and the icon on your favorite digital device showing battery power. The number line and its cousins – notations that map numbers onto space and often represent magnitude – are everywhere. Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don't stop to wonder: Is it 'natural'? Is it cultural? Now, challenging a mainstream scholarly position that the number-line concept is innate, a study suggests it is learned."
I just watched a documentary about this on Netflix, called The Story of 1, starring Terry Jones of Monty Python fame.I think it mentioned the ruler wasn't invented until sometime in ancient egypt.
... and in the DRM, bind them.
Try getting a bunch of 10-year-olds to understand the number line concept and you will find out in approximately 3 seconds that it is not innate.
I wonder how far this goes! Is the notion of the counting numbers innate? I have heard that monkeys cannot count beyond 4. The way that people figured this out is that if five hunters go into a forest as a group, split up and hide. Then one by one, four hunters leave one at a time. The fifth hunter stays in hiding, the monkeys come out of hunting, and the hunter shoots a monkey. This does not happen when there are less than five hunters initially.
Any measuring cup will tell you a number line can be very intuitive. Stacking objects, filling a container; many everyday tasks are perfect physical examples of a number line.
Rulers are another example, though perhaps a bit less physical or intuitive.
How can I believe you when you tell me what I don't want to hear?
I don't have the reference to hand but I recall there is a South American tribe which don't have words for left and right as most languages do. There words are equivalent to "Up Valley" and "Down Valley" Similarly, if I recall correctly, there's a Native American language that uses before and behind as an analog for time but the other way around to most languages. Their analogy is that you know the past and you can see what it in front of you so forward = the past. You can't see behind you and you don't know the future so behind = the future
Figuring out what isn't intuitive isn't useful, unless we also know what is. Pie graphs for gas gauges, showing the shrinkage of the tank fractionally? Or a circle in a circle shrinking within the "full" one?
"Also, we document that precise number concepts can exist independently of linear or other metric-driven spatial representations."
But TFA doesn't mention any of them, or what we could change a gas gauge to to be intuitive.
Perhaps one day they can figure out why my mother compulsively fills up once the gauge goes under 1/2, but my sister runs cars to empty on a regular basis, usually filling up only after the "e" is lit, sometimes long after.
Learn to love Alaska
Once a significant percentage of the population becomes interested in measuring pieces of land for various purposes, people will start associating numbers to lines.
Because the amount of food is proportional to the surface of your land, and then... I personally feel it's quite natural, in this context, to associate numbers to geometrical constructs.
new sig
... because I use complex numbers for everything, you insensitive clod. Don't you have any feelings for the one dimensionally-challenged?
now we need to go OSS in diesel cars
What does it matter if it's intuitive? English (and any other language, though possibly not language in the abstract) is learned, and it works just fine.
The same subject has been covered in "Here's looking to Euclid". It describes tests done on an Amazon tribe to see how they visually interpret numbers. Unlike most modern adults who visualize number spaced linearly, they visualized them spaced logarithmically. Their reasoning was that the intervals between numbers start (relatively) large and become smaller as the numbers get larger. i.e. from 1 to 2 it's a 100% increase but from 2 to 3 it's only a 33% increase and so on.
If you read the article, you'll see that the subjects of the study do understand order, but that they lack the intuition of another property of the number line that you are so accustomed to that you're not aware of it. When asked to place numbers from 1 to 10 in order, control subjects (from the US) produce an arrangement like this:
1...2...3...4...5...6...7...8...9...10
The people of the Yupno Valley tend to do something more like this:
1.2.3.4...................5.6.7.8.9.10
A number line has more than order; it also has equal spacing. That idea seems not to be innate.
In the original task, people are shown a line and are asked to place numbers onto the line according to their size, with "1" going on the left endpoint and "10" (or sometimes "100" or "1000") going on the right endpoint.
Go to a class of college students in america, ask them to mark 10, 1 million, and 1 billion on a line, and 99% of them will draw 1 million closer to 1 billion. Usually a lot closer.
I read the article, and it wasn't clear to me what these people have discovered. Maybe I'll have to read the actual study. Or maybe anthropologists are better at understanding primitive cultures than their own.
"First they came for the slanderers and i said nothing."
I don't know why this result is surprising. I thought it was generally understand that counting (there are 10 sheep) and measurement (this fence is 10 feet long) were distinct concepts. The point of the number line is to establish a relationship between the two concepts.
Come to think of it, it should be obvious that a number line relates two distinct concepts, just from the form they usually take. A number line, with its regularly spaced markings perpendicular to the main line, has a form similar to that of a line graph, which shows a relationship between two distinct variables.
Fixed measurements, such as a number line or the 'natural numbers' offer a poor model of reality. Comparing apples to apples; few are equal. Some are bigger, more bruised, less ripe, more bitter.
Hardly anything could be more alien than Euclidean space - we live on a mottled sphere. Straight lines are very much the exception.
While convenient, 'intuitive' or 'natural' are hardly the best way to describe abstract shortcuts.
Neither is reading. Human beings evolved to see "in the round" and not in focused linear scans. When we were children, both my sister and I went through periods when we were just learning to write where we wrote everything "exactly" backwards, like a mirror image. And, it wasn't all the time. We both outgrew it very quickly, but I'm sure it's been studied by some -ologist out there.
I swear to God...I swear to God! That is NOT how you treat your human!
You must use one of those languages weird that puts the modifier before the modified.
Play Command HQ online
as well as number form and personification. Numbers - depending on if they are simply numbers or dates - have a specific "geography", color, and personality.
46 & 2
I once took a course in "Math philosophy" (a simple introduction course, with e.g. Gödel numbers, introduction to infinity, and things like that), and at the end of that course we were asked to write about something. I decided to ask friends about how they viewed numbers. To my surprise, everyone had pretty much their own unique way. I think I asked about 10 people. Some viewed numbers as colors ("the number 2 is of course blue" or something along that line), some viewed the numbers as on a traditional line, one guy thought of the numbers as being in a circle and you took one out as you wanted to use it and then had to put it back. Not everyone included the number zero (or negative numbers) in their explanation. My self, I see the natural numbers on a line, but the line has "angles" at the numbers 10 and 20. Perhaps this is because in my native language, the spoken words for 10..19 are not constructed in the same simple manner as 30..39, 40..49, and so on.
In Lingala (Kingshasa area in Congo), they only have one word which both means "yesterday" and "tomorrow". Basically things happen today or they happen not-today. This kind of makes sense in a climate that has no cold and hot season, and where it is useless (or even a very bad idea) to do typical northern stuff like plan way ahead, conserve food or make warm clothes. Most pre-Columbus south american indians saw time as a strictly circular thing, with everything always comming back.
10 ?"Hello World" life was simple then
It also contains an error: Peano defined 2 as { {}, {{}} } = {0,1}. 3 is 2 U {2} = { 2, 1, 0 }. Larger numbers are defined inductively as (n+1) := n U {n}.
You can tell it was supposed to be the Peano construction (and not something else) because the GP defined zero as the empty set and 2 as {0,1}. The error was to also define 2 as {{{}}}, which is clearly not equivalent to {0,1} (since the former set has cardinality 1 and the latter has cardinality 2).
This is an incredibly common mistake even for math undergrads and good evidence that set theory really isn't very intuitive. There's a reason New Math failed.
Well, numbers are abstract. I'm not sure how a number line representation, which can take real shape would be an intuitive extension of an artificial concept. It isn't. Actually, it's the other way around, I would think. The number lines help us understand numbers and it's numbers that aren't intuitive.
I'm not sure if they've fixed it yet, but the defaults for line charts in MS Excel were insanely set to have equal spacing between data points on one axis no matter what values they have.
Thus you could have an axis that looked like:
1 4 7 8 14 35
IMHO that sort of defeats the purpose of a line graph. I can userstand linear or log scales but a random changing scale is pointless.
You Forth about talking are, I think is what you're aiming for. Your sentence came across as more German than RPN.
My little brother was having problems with vector math. So, I threw together a vector visualiser in my game engine, and illustrated basic vector primitives, and operations. Within 15 minutes of moving them around on the screen and seeing the values and vectors change he understood normalising, and dot and cross products, as well as trigonomic primitives like sine and cosine, and tangent. I showed him how dot products are used to cull faces in games, and in lighting equations, and how cross products make homing missiles work. I even showed him operations involving a unit-Quaternion, and while he didn't completely grasp the mathematics behind it, he understood how to work the numbers and what he could use them for.
He told me that he learned more about geometry and numbers in an hour at the computer than he had in his entire schooling of 10 years...
Of course, when he gets to higher dimensions, this may prove more difficult. The point being: Humans are tool using creatures. Math that's taught for the sake of learning without any direct application holds no inherent value for us. "When will I ever use this in real life," crosses EVERYONE's mind at some point. We need a better answer than silence. Also: Watching an animation is far more informative than reading a book -- We have the technology. Kids LOVE games. This isn't rocket science people... It's quite obvious what needs to happen.
In the summer between 6th and 7th grade I independently invented trigonometry while manually mapping line slopes to angles trying to make a space ship game in BASIC (with LINETO, MOVETO) -- The sin, cos, atan2, etc. functions didn't have good descriptions of what they were used for -- They were tools that I hadn't yet learned to use, so I created my own. Trigonometry (or 'length/slope/angle' ratios as I called it) was obvious to me as a 11 year old, simply because I could map the relationships between sets of numbers on my computer in real time...
IMOH, we shouldn't be teaching math without also teaching a bit of simple computer programming, or at least using SOME animated application to utilise the new tools with. In this day and age who wouldn't benefit from being able to tell their computers how to automate simple tasks?
We don't stop to wonder: Is it 'natural'? Is it cultural?
'Cultural' is natural for us humans, so it is a daft question. A better question would be to ask whether this is something we are most likely to have learned through our early experience - and how. And I think the answer is likely to be that we learn the idea of "moreness" being a continuous thing from observing varying amounts of things - water in a glass etc, or the length of a piece of string; these concepts are clearly learned as and when you learn the words to describe them - ie. it is 'cultural'.
But many - maybe most - animals have the ability to gauge the relative size of things, and some, like the corvids - even seem able to count. Thus that would count as a 'natural' ability, I suppose.
The case with the Yupno seems to be that measurements aren't needed in their culture; one can muse over where that need arises from - it could be a result of trade, perhaps?
I haven't read the book, but what about subitizing, i.e. the ability to "perceive" a small number of items? If a three-week old baby can subitize up to three objects, I'd say that's an inborn ability.
True confidence comes not from realising you are as good as your peers, but that your peers are as bad as you are.
I read the article pointed to in the summary (which is a summary of the scholarly article). The study authors seem to have confused the idea that finding a single population that behaves this way (not arranging piles of oranges linearly along a line according to the number of oranges in a pile) with determining true innate human behavior. Find another dozen isolated groups, and then maybe. Find groups that have been only recently isolated and it will be more impressive.
Put my fist through my alarm clock with its ding-dong death inside my ear. - The Blackjacks.
Of course it's learned. We teach it in school, every year, from somewhere around second grade right on up through college. Obviously it's learned.
Is that supposed to have some kind of significance? I don't see it. Virtually everything we know is learned. Arithmetic is learned. Color is learned. Language is learned. Food preferences are learned, including even the ability to tell the difference between food and non-food. The notion that a stove burner is hot and you don't want to put your hand on it is learned.
Cut that out, or I will ship you to Norilsk in a box.
If you grew up with the metric system you might not realize that common measurements used to be based on supposedly common items, so you had measurements dealing with what a man could hold with his arms around it, and the length of the King's erect cock or whatever. It's a natural advance to go from measuring things in terms of a fingertip to so many fingertip-units. I imagine it would have started with measuring distance, but it could as easily have been someone figuring it out by volume, this container holds so many of that container. Or this stick rolls over x times when it passes down the side of this object.
"You're right," Fisheye says. "I should have set it on 'whip' or 'chop.'"