# Study Suggests the Number-Line Concept Is Not Intuitive

#### samzenpus posted more than 2 years ago | from the learning-to-count dept.

404
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."*

## Did they test males or females? (1, Funny)

## Anonymous Coward | more than 2 years ago | (#39802793)

Because I know most males know the number of their line, or at least what they think it is.

## Re:Did they test males or females? (-1)

## Anonymous Coward | more than 2 years ago | (#39802993)

Whoi is the blackest of them all?!

Oh, it's you, dear Slashdot reader

Welfare queens are gifted breeders!

Seeing that doesn't take a pH.D, They proudly wave their E.B.T.!

Niggers, coons, and jigaboos,

On what side of the fence are you?!

## Re:Did they test males or females? (0)

## mlow82 (889294) | more than 2 years ago | (#39803295)

## Re:Did they test males or females? (0)

## BenJCarter (902199) | more than 2 years ago | (#39803349)

Because I know most males know the number of their line, or at least what they think it is.

So true. This behavior has deep roots in human nature. To have phallus [wikipedia.org] (at it's most NSFW PG rating), is to exaggerate the number line.

## The Story of 1 with Terry Jones (4, Interesting)

## StarWreck (695075) | more than 2 years ago | (#39802795)

## Re:The Story of 1 with Terry Jones (5, Funny)

## Anonymous Coward | more than 2 years ago | (#39802829)

I thought the concept of "ruler" started with King Arthur, after a watery tart lobbed a scimitar at him.

## Re:The Story of 1 with Terry Jones (5, Funny)

## Anonymous Coward | more than 2 years ago | (#39802965)

Bah. Farcical aquatic ceremonies are no basis for a system of measurement.

Use of the number line is derived from a mandate of the masses. Everyone knows that.

## Re:The Story of 1 with Terry Jones (3, Funny)

## ArsonSmith (13997) | more than 2 years ago | (#39803099)

I need to know the "watery tart lobbing scimitars" to miles conversion. The other day they was an asteroid the size of a strange woman distributing swords that burned up over California.

## Re:The Story of 1 with Terry Jones (2)

## Black Parrot (19622) | more than 2 years ago | (#39803273)

I need to know the "watery tart lobbing scimitars" to miles conversion.

Unfortunately there's not a constant conversion, since the number of watery tarts lobbing scimitars per mile varies with geographical location.

## Anyone who has ever taught math knows this (5, Interesting)

## Anonymous Coward | more than 2 years ago | (#39802811)

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.

## Re:Anyone who has ever taught math knows this (4, Insightful)

## slippyblade (962288) | more than 2 years ago | (#39803231)

If your 10 year old doesn't ALREADY understand the number line, you have failed. Hell, if your 6 year old doesn't understand it, you've failed.

## Re:Anyone who has ever taught math knows this (4, Funny)

## Brian Feldman (350) | more than 2 years ago | (#39803309)

-1 Completely misunderstanding the point of the article and comment.

## Re:Anyone who has ever taught math knows this (1)

## phantomfive (622387) | more than 2 years ago | (#39803357)

## Re:Anyone who has ever taught math knows this (1)

## jaymemaurice (2024752) | more than 2 years ago | (#39803437)

To be fair though, they should all get basic concepts like the battery meter on a smartphone or the gas gauge on your car... or a glass being about a 1/3 full

## BASIC Programming, old school (1, Offtopic)

## OakDragon (885217) | more than 2 years ago | (#39802825)

## Re:BASIC Programming, old school (1)

## Anonymous Coward | more than 2 years ago | (#39802921)

That would be line number not number line.

## Re:BASIC Programming, old school (2)

## Boronx (228853) | more than 2 years ago | (#39803433)

You must use one of those languages weird that puts the modifier before the modified.

## Counting? (5, Interesting)

## deodiaus2 (980169) | more than 2 years ago | (#39802851)

## Re:Counting? (5, Funny)

## pthisis (27352) | more than 2 years ago | (#39802865)

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.I should hope not: if there are four hunters initially, then one by one four hunters leave, there are no hunters left to shoot the monkey. And if there are 3 or fewer hunters initially than the scenario's impossible.

## Re:Counting? (1)

## dcollins (135727) | more than 2 years ago | (#39802917)

"And if there are 3 or fewer hunters initially than the scenario's impossible."

Not if, after all 4 leave, at least 2 go back.

## Re:Counting? (1, Funny)

## Mannfred (2543170) | more than 2 years ago | (#39803165)

## Re:Counting? (0)

## TheInternetGuy (2006682) | more than 2 years ago | (#39803277)

Once you go silverback, there's no going back.

Oh... I don't know, there are some pretty good permanent and semi permanent hair coloring agents out there.

## Re:Counting? (1, Funny)

## Brucelet (1857158) | more than 2 years ago | (#39802933)

## Re:Counting? (1)

## Ambvai (1106941) | more than 2 years ago | (#39802955)

Not if there're male and female hunters in the group...

## Re:Counting? (1)

## mooingyak (720677) | more than 2 years ago | (#39803225)

That is a seriously long wait.

## Re:Counting? (2)

## ChrisMaple (607946) | more than 2 years ago | (#39803005)

## Re:Counting? (4, Funny)

## initialE (758110) | more than 2 years ago | (#39803347)

There was this bald monkey coming out, screaming in his monkey language: "There... Are... Four... Hunters!"

And then, he died. Apparently a bad day to wear his red shirt.

## Re:Counting? (2)

## sg_oneill (159032) | more than 2 years ago | (#39802925)

Thats not necessarily even counting on the monkeys behalf. A lot of neuroscientists reckon we can process about 4 separate things in our mind simultaneously , and then use a variety of clever tricks to work around it (Ie counting!) and if that stretches across species. So conciably the monkeys are just at their limit of how many dudes they can track at once, rather than an inability to count beyond 4.

## Re:Counting? (5, Interesting)

## blankinthefill (665181) | more than 2 years ago | (#39803071)

## Re:Counting? (2)

## Vellmont (569020) | more than 2 years ago | (#39803217)

I wonder how far this goes! Is the notion of the counting numbers innate?Counting exact numbers is not innate. There are some cultures that don't have words for an exact number beyond 3. That doesn't mean they don't understand quantities, just that they can't name a specific amount. It'd be like if somone showed you a thousand of something, and 1100 of something. You'd know the 1100 was more, but you wouldn't be certain by exactly how much more.

## Re:Counting? (1)

## mveloso (325617) | more than 2 years ago | (#39803239)

There are studies that show the "natural" conception of numbers is one, two, many, a lot.

http://numberwarrior.wordpress.com/2010/07/30/is-one-two-many-a-myth/ [wordpress.com]

You can search the real literature if you'd like; that was one of the first hits I found. Once you can teach your kids to count beyond 5, they've already beaten most of humanity.

## Time? (0)

## overshoot (39700) | more than 2 years ago | (#39802853)

## Re:Time? (-1)

## Anonymous Coward | more than 2 years ago | (#39803137)

Start by studying the Hopi.

Best way to study something is from the inside.

That's how I studied yo mama's vagina.

## Re:Time? (0)

## Black Parrot (19622) | more than 2 years ago | (#39803293)

Start by studying the Hopi.

Best way to study something is from the inside.

That's how I studied yo mama's vagina.

You're siblings?

## Vertically, it is. (5, Insightful)

## pushing-robot (1037830) | more than 2 years ago | (#39802863)

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.

## Re:Vertically, it is. (3, Insightful)

## b4dc0d3r (1268512) | more than 2 years ago | (#39802889)

I'm inclined not to believe your oversimplification. I remember elementary school math, with whole chapters devoted to teaching the number line. Concepts such as greater/less, constant distance, visual estimation, and numberless comparisons are, or were, part of what gets taught in a school setting.

If you don't have the concept of a number line already, is it really that intuitive to stack 1 cup on top of another and consider it a measurement rather than an amount? Stacking things and coming up with a ruler based on that stacking seem like they are fairly distinct concepts, that one won't lead to the other.

## Ordered sets (1)

## overshoot (39700) | more than 2 years ago | (#39802913)

Show me a culture that doesn't have the concept of ordered sets -- which is all that a "number line" is.

And, no, I don't mean the fancy mathematical formalism. I mean things like narratives, directions from A to B, etc.

## Re:Ordered sets (1)

## dcollins (135727) | more than 2 years ago | (#39803029)

Those things can be ordered in time without being mapped to space.

(Especially if you don't have written language yet.)

## Re:Ordered sets (5, Informative)

## Anonymous Coward | more than 2 years ago | (#39803049)

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.

## Re:Ordered sets (0)

## Anonymous Coward | more than 2 years ago | (#39803253)

Well, with directions you're setting pretty broad goal posts. I'm not aware of a human culture that doesn't use geographical communication like that. Other primates, however, may lack this ability and so learn routes only by being directly shown. If that has been demonstrated, it would suggest that human cultures either learned route-language at some point in their collective past, and that no non-route-using cultures remain or that our primate ancestors learned it, and no non-route-using human cultures existed.

As for other ordered sets, all do seem to be learned behaviour, since we have apparent examples of their absence. Counting is at least done in several different ways, with the Walpiri in Australia apparently using a one-two-many system (and also a fascinating, and very different way of tracing ancestry).

And time? There are claims that the Piriha don't have the tenses to tell stories about the past, although I think that research is disputed, while other tribes have many more tenses than English (up to eight in each direction I think) making our temporal naratives rather vague by comparison. I'm pretty sure I remember reading about one that blended spatial and temporal separation into the same tenses as well (so that very far away, and in the remote past, would use the same tense).

So while some use of sequencing may be inate, it seems clear that just about everything we do with sequencing isn't inate, which means - to me at any rate - that claiming sequencing is inate is rather meaningless.

## Re:Ordered sets (1)

## Brian Feldman (350) | more than 2 years ago | (#39803343)

You would probably quite enjoy Noam Chomsky's latest work, The Science of Language. In it, he claims nothing is innate except the concept of Merge. Basically, it is only set theory and construction/deconstruction based upon that. Counting numbers is not innate; it is consequential of a certain kind of indoctrination. All humans can potentially do it, but it is not something inborn. Likewise, all humans can learn a spoken/written/signed language, but it is not inborn.

## Re:Ordered sets (1)

## Black Parrot (19622) | more than 2 years ago | (#39803311)

Show me a culture that doesn't have the concept of ordered sets -- which is all that a "number line" is.

No, the number line has a metric in addition to an ordering.

There's a sort of hierarchy of these things, but I never can remember the terminology.

## Re:Vertically, it is. (1)

## The Wild Norseman (1404891) | more than 2 years ago | (#39802975)

I'm inclinednot to believe your oversimplification.Hey, we're discussing algebra, not geometry!

## Re:Vertically, it is. (1)

## Khyber (864651) | more than 2 years ago | (#39803365)

"with whole chapters devoted to teaching the number line"

Never had that in my classes. No problems understanding a number line, here.

## Re:Vertically, it is. (2)

## gargleblast (683147) | more than 2 years ago | (#39802931)

## Re:Vertically, it is. (1)

## Black Parrot (19622) | more than 2 years ago | (#39803321)

Intuitive? Watch a toddler try and fill a cup from a jug sometime.

A toddler trying to fill a bathtub from a jug gives just about the same result.

## English, Mofo. (1)

## Anonymous Coward | more than 2 years ago | (#39802871)

Do you speak (or write) it? Intuitive and instinctual are different words.

## A-ha! (0)

## Anonymous Coward | more than 2 years ago | (#39802875)

> the familiar concept of time may be cultural as well.

So *that's* why in some cultures I can eat a fish before I've caught it.

## Re:A-ha! (1)

## PPH (736903) | more than 2 years ago | (#39803003)

Well, here in the USA, you certainly can eat it before you've paid for it.

## Valleys and Language (4, Insightful)

## IntentionalStance (1197099) | more than 2 years ago | (#39802877)

## Re:Valleys and Language (5, Informative)

## JoshuaZ (1134087) | more than 2 years ago | (#39803175)

## Re:Valleys and Language (1)

## IntentionalStance (1197099) | more than 2 years ago | (#39803271)

## Re:Valleys and Language (1)

## Garble Snarky (715674) | more than 2 years ago | (#39803181)

"In their time study with the Yupno, now in press at the journal Cognition, Nunez and colleagues find that the Yupno don't use their bodies as reference points for time – but rather their valley's slope and terrain. Analysis of their gestures suggests they co-locate the present with themselves, as do all previously studied groups. (Picture for a moment how you probably point down at the ground when you talk about "now.") But, regardless of which way they are facing at the moment, the Yupno point uphill when talking about the future and downhill when talking about the past."

"In earlier research, Nunez found that the Aymara of the Andes seem to do the reverse, placing the past in front and the future behind."

The second example seemed odd to me, but makes a lot of sense with your analogy.

## Re:Valleys and Language (1)

## IntentionalStance (1197099) | more than 2 years ago | (#39803257)

I speak some Thai and it is really difficult for English speakers to grasp

Imagine - no word for yes or no. Verbs don't change their form for person or tense.

In English we only really have 'it' as the third person singular. In French they have 'il' (masculine it) and 'elle' (Feminine it). In Thai they have literally different hundreds of pronouns for stuff like 'things with handles', 'long thin things', 'containers', 'things with limbs that are not people'

Language is so much more diverse than you would imagine if you don't study it.

Interestingly, this does not make the Sapir–Whorf hypothesis (http://en.wikipedia.org/wiki/Linguistic_relativity) true.

The structure of the language has little, if any, relationship to the deeper mental understanding of the 'way things work'

## Re:Valleys and Language (1)

## Brian Feldman (350) | more than 2 years ago | (#39803409)

Even in math, we don't necessarily limit ourselves to Euclidean space. For every X/Y/Z, there's also a theta/phi/rho.

## Re:Valleys and Language (1)

## QuasiSteve (2042606) | more than 2 years ago | (#39803191)

Yes, that may well have been...

Source? TFA, which mentions the same researcher was involved with that finding.

## Re:Valleys and Language (0)

## Anonymous Coward | more than 2 years ago | (#39803317)

I hope that "educators" in the US will not use it as an excuse to eliminate "legacy" mathematics from the curriculum. Well, there is not much left to be eliminated, but still, unlike those New Guinea folks we managed to go some distance in the past 5000 years.

On the other hand, if I were living on a tropical island I probably wouldn't bother myself with mathematics either.

## Re:Valleys and Language (1)

## Black Parrot (19622) | more than 2 years ago | (#39803337)

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.

I don't think left & right are very intuitive. For most of my life I had to stop, close my eyes, imagine the plane of symmetry of my body, and ask myself which side of the plane something was on.

Of course, that may have just been a cognitive disorder, rather than in indication that the distinction is unintuitive. Either way, I finally outgrew it.

## Re:Valleys and Language (1)

## Black Parrot (19622) | more than 2 years ago | (#39803375)

I should have mentioned in the same post:

I used to work in an environment where we assembled handrails, which came in mirror-image pairs based on where the bolt holes were located. Some people could glance at a scrambled pile of them and immediately pull out a symmetrical pair, but others would have to fish them out of the pile and put them side-by-side in order to determine whether two were the same or mirror images. And it didn't seem to have anything to do with how long they had been doing it.

That doesn't seem like a task that depends on "smart", but rather, whether you can visualize an object and change its orientation in your mind. Perhaps it's learned, but it's not obvious why some people would learn it and others wouldn't - especially since the people who worked in the task described above seemed to either have it or not-have it, without regard to experience. So maybe it's easier to learn as a child, or maybe it's just an innate difference between people.

Also oddly, IMO, is that I was one of the best at it, despite what I said about my difficulties with left-vs-right in the parent post.

## Re:Valleys and Language (0)

## Anonymous Coward | more than 2 years ago | (#39803431)

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

There are probably more variations than you think. In Korean, the word "ap" means front, and "dwi" means back. "10 years

dwi" will mean "10 years later". "10 yearsap" will mean... "10 years into the future".So, front or back, you go into the future. Sometimes I think my language is totally messed up.

## who proof reads these ? (1)

## wulfmans (794904) | more than 2 years ago | (#39802891)

## Re:who proof reads these ? (1)

## sub67 (979309) | more than 2 years ago | (#39802949)

One could never be wrong.

## What is intuitive (2)

## AK Marc (707885) | more than 2 years ago | (#39802895)

"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.

## Re:What is intuitive (1)

## smileygladhands (1909508) | more than 2 years ago | (#39802905)

## Re:What is intuitive (1)

## _merlin (160982) | more than 2 years ago | (#39803013)

It's sensible to keep your tank low - vehicles are more efficient if they aren't hauling extra fuel weight. Aircraft operators have this down to a fine art.

## Re:What is intuitive (1)

## Kinky Bass Junk (880011) | more than 2 years ago | (#39803223)

## Re:What is intuitive (1)

## Anonymous Coward | more than 2 years ago | (#39803335)

Wrong its better to never let the tank become regularly low as it is better to have any crap that's in there pass through as lots of small chunks rather than all at once just as the tank runs empty.

## Re:What is intuitive (1)

## Kinky Bass Junk (880011) | more than 2 years ago | (#39803339)

## Re:What is intuitive (2)

## Black Parrot (19622) | more than 2 years ago | (#39803383)

It's sensible to keep your tank low - vehicles are more efficient if they aren't hauling extra fuel weight. Aircraft operators have this down to a fine art.

I prefer to let it run reasonably low (but not so low as to risk getting stranded), then fill it all the way up.

Because that means less stops at the pump.

## Seems Obvious (1)

## bazald (886779) | more than 2 years ago | (#39802899)

I have taught a number of people to code 2D and 3D games. Both 2D and 3D involve a lot of coordinate axis transformations that are almost universally non-intuitive at first. This is true in 2D despite there being a

direct correlationbetween the data being mapped from the model/simulation to the screen (2D to 2D).This study finds it is non-intuitive to go from an

abstractnumber or count to a line segment? Sure. What I'd like to see is the "sources of evidence [which] suggest that humans naturally associate numbers with space". They would surprise me.## agriculture (2)

## chichilalescu (1647065) | more than 2 years ago | (#39802909)

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.

## The number line does not work for me ... (3, Funny)

## Skapare (16644) | more than 2 years ago | (#39802915)

... because I use complex numbers for everything, you insensitive clod. Don't you have any feelings for the one dimensionally-challenged?

## Re:The number line does not work for me ... (1)

## goodgod43 (1993368) | more than 2 years ago | (#39803161)

FTFY

1+i dimensionally challenged.

## Re:The number line does not work for me ... (1)

## ThorGod (456163) | more than 2 years ago | (#39803249)

'eh The complex numbers are (one) logical extension of the real number system (aka 'number line'). Can't have a complex plane without two real number lines.

## Re:The number line does not work for me ... (1)

## Black Parrot (19622) | more than 2 years ago | (#39803393)

'eh The complex numbers are (one) logical extension of the real number system (aka 'number line'). Can't have a complex plane without two real number lines.

Or (for the anal retentive among us) one real number line and one imaginary number line.

## So what? (2)

## Trapick (1163389) | more than 2 years ago | (#39802927)

## Re:So what? (1)

## mcavic (2007672) | more than 2 years ago | (#39803015)

## Logarithmic vs linear scale (5, Interesting)

## tukang (1209392) | more than 2 years ago | (#39802969)

## Re:Logarithmic vs linear scale (0)

## Anonymous Coward | more than 2 years ago | (#39803203)

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.

Not to pick on you or anything, but you made an error. 2 to 3 is a 50% increase.

## Obviously? (1)

## GiMP (10923) | more than 2 years ago | (#39802971)

I imagine that a thickness gauge (which is what is *really* intuitive in the measuring-cup example) or a color-gauge would be more intuitive. The critical point here is that thicker is "more" and thinner is "less". Even with colors you can have "more red" or "less red". Numbers are a higher-form thought process. When dealing with a line system, your general intention is to gauge this same "more or less" comparions, but is abstracted through numbers which is based on a complex thought process of reading and comprehension.

## Useless (1)

## Anonymous Coward | more than 2 years ago | (#39802987)

Can they measure how much time of my life was wasted reading this stupid article? Thanks for the information about nothing...

"Mathematics... is largely taught dogmatically, as objective fact, black and white, right/wrong," Nunez said. "But our work shows that there are meaningful human ideas in math, ingenious solutions and designs that have been mediated by writing and notational devices... Perhaps we should think about bringing the human saga to the subject – instead of continuing to treat it romantically, as the 'universal language' it's not. " ummm ok? someone didn't do well in algebra.

whether anyone gets it or no one gets it, math is inherent and intrinsic in numbers.. 1 + 1 is 2 and 3+3 is 6, regardless of what words and symbols we use to describe it or what we know about it or don't.

Obviously people don't automatically know it by design. Math, like any language, evolves and develops over time. Similar to researchers in other fields, mathematicians research and make discoveries about math and share their results, and those results spread into the collective human body of knowledge. It started out as nothing, then something really basic, and developed over... well, as long as there has been humanity. Math is a pure language because its rules exist on their own, by nature, not by human convention. Applications of the laws of math and numbers, such are rulers and gauges, are human convention - of course they aren't universally known abstracts. Nor is the concept of measuring. I'd bet tho, that given the right experiment, they would find that the people of Papua New Guinea do indeed have notions of amount, and measure.

What next? Are they going to report that chemistry isn't universal because the people of Papua New Guinea don't have any concept of it?

## Management (1)

## PPH (736903) | more than 2 years ago | (#39803023)

In earlier research, Nunez found that the Aymara of the Andes seem to do the reverse, placing the past in front and the future behind.

I've worked for a number of PHBs who seemed content with the future sneaking up behind them and smacking them in the back of the skull.

## Learned vs. Innate Intuition (1)

## Anonymous Coward | more than 2 years ago | (#39803027)

I'm tired of all the BS splitting hairs over whether something is "intuitive" or not.

Intuitive just means: given your current knowledge, does the thing make immediate sense? If so, it's intuitive.

Very few things are innate. You learn the rest, even if you don't realize that you learned it somewhere. Maybe you saw someone else do it, or maybe you deduced it from other things you already knew. But once you learned the concept, if you can always apply it without thinking, then it's intuitive.

Think of it like mathematical axioms/proofs. You start with a few axioms. As you learn, you gain some new theorems that are so elegant to use that you end up using them as frequently as axioms. These special theorems are what we call "intuitive" facts; they allow us to quickly and "intuitively" deduce things that would be difficult to prove all the way from axioms.

p.s. Intuition is also based on guesswork. For example, suppose you guessed the right theorem(s) to complete the proof; if you guessed right the first time (or 'quickly enough'), then you'll describe the proof as intuitive. If you had to try N different things and finally get an "aha" to use unexpected Theorem XYZ, then it's not intuitive.

## Re:Learned vs. Innate Intuition (0)

## Anonymous Coward | more than 2 years ago | (#39803063)

Argh. I forgot to mention: The hypothetical theorem proven by non-intuitive theorem XYZ in the above "p.s." will often become an "intuitive" fact, if it appears frequently in subsequent proofs. Also: You may not even remember how to prove it, but you will remember that it's true!

## That's not challenging the mainstream (0)

## Intropy (2009018) | more than 2 years ago | (#39803037)

The number line represents the continuum, that is the real numbers. The debates about the foundations of mathematics are old and ongoing, but most hold that the natural numbers {1, 2, 3, ...} are intuitive and axiomatic and prove the rest formally. Heck, Kroneckers well-known quote, "God made the natural numbers; all else is the work of man" is saying exactly that.

## Re:That's not challenging the mainstream (2)

## mevets (322601) | more than 2 years ago | (#39803121)

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.

## Americans don't understand number lines either (3, Informative)

## phantomfive (622387) | more than 2 years ago | (#39803079)

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.

## Re:Americans don't understand number lines either (1)

## Your.Master (1088569) | more than 2 years ago | (#39803259)

Are you sure? Is there a study?

That just doesn't seem obviously true (or false) to me. It's somewhat justifiable on a logarithmic axis too.

## Re:Americans don't understand number lines either (1)

## phantomfive (622387) | more than 2 years ago | (#39803279)

## Counting and measurement are distinct concepts (4, Insightful)

## FoolishOwl (1698506) | more than 2 years ago | (#39803105)

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.

## I find them unintuitive (1)

## Fished (574624) | more than 2 years ago | (#39803219)

Oddly enough, I was telling my girlfriend just tonight that I'm not very visual, and tend to approach concepts best through symbols (numbers, words, etc.) I've always found graphical representations of math more-or-less useless (although they are cool sometimes) and prefer my math without the diagrams. She told me that I'm deeply weird. :)

## Re:I find them unintuitive (1)

## Black Parrot (19622) | more than 2 years ago | (#39803423)

Oddly enough, I was telling my girlfriend just tonight that I'm not very visual, and tend to approach concepts best through symbols (numbers, words, etc.) I've always found graphical representations of math more-or-less useless (although they are cool sometimes) and prefer my math without the diagrams. She told me that I'm deeply weird. :)

Many educators think people have different learning modalities (hearing, seeing, touching, etc.), and suggest combining all of them when teaching, so that all the learners can benefit from what works best for them.

OTOH, some people think the whole idea is a crock. I don't have any opinion, though your anecdote seems to support the idea.

## Typing on a computer isn't "innate" (2)

## GodfatherofSoul (174979) | more than 2 years ago | (#39803243)

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.

## Numberwang? (1)

## x0 (32926) | more than 2 years ago | (#39803265)

## Asking wrong question (0)

## Anonymous Coward | more than 2 years ago | (#39803323)

Wrong question: "Here is 1, and here is 10. Where is the number 7

Yunpo: so, I have to classify, and determine, to which of the two numbers, number 7 is nearer?

California: so, I have to put 10 consecutively-equidistant points on the line, and count up to the 7-th point.

No, go ask Yunpo the right question: "Put 10 consecutive equidistant points, and count up to the 7-th point."

## Old news - again. (1)

## DrCJM (827451) | more than 2 years ago | (#39803327)

http://alexbellos.com/wp-content/uploads/2010/04/maths.pdf

Log scale = intuitive (ratios - there's twice as many of those as these)

Numberline != intuitive (counting, ordering etc.)

## numbers are not innate (1)

## amoeba1911 (978485) | more than 2 years ago | (#39803359)

## follows logical from the field axioms (1)

## ThorGod (456163) | more than 2 years ago | (#39803363)

Personally, studying

unintuitive concepts via the language of mathematics interests me. That's how mathematics allows you to expand the list of things that you find intuitive. First, only the abstract language of mathematics describes some logical object. The logical object itself may or may not be 'intuitive' from the outset. Eventually, after studying a logical concept via math for a time, I can eventually gain some intuition concerning the object. I've done this with the real and complex number systems (separately), partial and ordinary differential equations, vector/inner product spaces, mathematical knots, and etc.PS If you're looking for a way to study calculus or the real numbers in a "more intuitive sense", I suggest you look up the hyperreal number system.

## Re:follows logical from the field axioms (1)

## ThorGod (456163) | more than 2 years ago | (#39803367)

## Education system corrupted, continuum hypothesis (0)

## Anonymous Coward | more than 2 years ago | (#39803395)

Kids are taught calculus in most schools without informing about the existence of continuum hypothesis :-(