Forgot your password?
typodupeerror
Math Science

Study Suggests the Number-Line Concept Is Not Intuitive 404

Posted by samzenpus
from the learning-to-count dept.
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."
This discussion has been archived. No new comments can be posted.

Study Suggests the Number-Line Concept Is Not Intuitive

Comments Filter:
  • by StarWreck (695075) on Wednesday April 25, 2012 @10:59PM (#39802795) Homepage Journal
    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.
  • by Anonymous Coward on Wednesday April 25, 2012 @11:02PM (#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.

    • by slippyblade (962288) on Thursday April 26, 2012 @12:17AM (#39803231) Homepage

      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.

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

        • by MojoRilla (591502) on Thursday April 26, 2012 @01:31AM (#39803595)
          I don't get your comment. I teach math to six year olds once a week. They "get" the number line, in that they use it as a useful tool for calculation, and can understand how numbers equate to divisions on the paper. Is it innate? Probably not. Is it something that many six year olds in the US culture have? From my experience, yes.

          Where the article veers into the absurd is the suggestion that we should consider "bringing the human saga" into teaching math, and that math isn't objective fact, or black and white. Math is freaking math. There is right and wrong, black and white.
          • by tehcyder (746570)

            I teach math to six year olds once a week. They "get" the number line, in that they use it as a useful tool for calculation, and can understand how numbers equate to divisions on the paper. Is it innate? Probably not. Is it something that many six year olds in the US culture have? From my experience, yes.

            When my kids started school, they had to be taught how to use number lines, number grids for multiplication, how to divide by 2 and so on, just as much as they had to be taught how to read. None of it is innate, as far as I can see.

      • Your ten year old probably doesn't understand the number line. Sure, he can put a few numbers on a line, but ask him to put a million, and a thousand on the line. Try it yourself, you may be surprised.
      • by ShanghaiBill (739463) on Thursday April 26, 2012 @01:48AM (#39803651)

        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.

        Then I guess I failed. My seven year old son is at the top of his 2nd grade class in math. Be he was doing the number line exercise in Khan Academy [khanacademy.org] about two weeks ago, and he needed some help. Once I explained the concept, and gave him a few examples, he "got it", and was able to do the exercises. But it was not intuitive. He needed an explanation.

  • Did anyone else think about older versions of interpreted BASIC first?
  • Counting? (Score:5, Interesting)

    by deodiaus2 (980169) on Wednesday April 25, 2012 @11:07PM (#39802851)
    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.
    • by pthisis (27352) on Wednesday April 25, 2012 @11:10PM (#39802865) Homepage Journal

      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.

    • by sg_oneill (159032)

      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? (Score:5, Interesting)

      by blankinthefill (665181) <blachanc@gmaiEULERl.com minus math_god> on Wednesday April 25, 2012 @11:46PM (#39803071) Journal
      Numbers are not an intuitive concept. As I've learned more and more math, I've had numerous discussions about this topic. The conclusions that tend to be reached are that sets are intuitive. A set is very intuitive, it's just a bunch of objects that are grouped together. You may not THINK of these things as sets, but that's what they are. You have a pile of apples, or a herd of sheep, or a group of hunters. Those are all sets of objects (or some philosophers would argue that there's a difference between the set and the group of physical objects, but I don't think that this ruins the intuition here). You can also label those things however you want, or not label them at all. Very intuitive. But numbers are when intuition starts to get messed up. A number can be disassociated from a concrete set, and that can make it hard to deal with, if you're not used to it. What is 1? What does it mean? What does it even mean to talk about 1 sheep, if it's completely hypothetical? There's no concrete sheep there, so what does it MEAN to be talking about 1 sheep? It's not even like you're talking about a sheep that's going to be born, or that belongs to your neighbors. This sheep is basically just imaginary. That's really a huge jump in cognition, especially when you start to consider other crazy things about numbers, like what's the biggest number, and what's a negative number, and what if you can't divide your numbers evenly. Anyways, nothing scholarly to back this up, just my experience in mathematics :)
      • Hmm, but if sets _are_ intuitive then it follows that numbers are too based on set theory [wikipedia.org]. {} = 0 {{}} = 1 2 = {{{}}} = {0,1} ...
        • Re:Counting? (Score:4, Interesting)

          by blankinthefill (665181) <blachanc@gmaiEULERl.com minus math_god> on Thursday April 26, 2012 @01:20AM (#39803545) Journal
          The problem with this argument is that it assumes that set THEORY is intuitive, which I do not agree with. While a SET is an intuitive concept, the ZF axioms of set theory and what they imply are NOT intuitive. There may be basic operations that are more intuitive, like the union of two sets or the intersection of two sets, but that intuition is almost entirely tied to the physical manifestation of the set. As soon as you introduce the formal idea of a set, especially as an abstract construct, I believe that, just like what I said about numbers, you remove a large amount of the basic intuition behind them. While a lot of the things that happen here seem intuitive to us, I feel like that is almost solely due to the fact that we are introduced to this abstraction at such an early age, and we deal with it so much, that we internalize it. Without that exposure, I'm not so sure the abstractions of sets and numbers is totally intuitive.
          • by TheEmperorOfSlashdot (1830272) on Thursday April 26, 2012 @02:47AM (#39803885)
            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.
    • by Vellmont (569020)

      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.

    • by mveloso (325617)

      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.

  • Vertically, it is. (Score:5, Insightful)

    by pushing-robot (1037830) on Wednesday April 25, 2012 @11:09PM (#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.

    • by b4dc0d3r (1268512) on Wednesday April 25, 2012 @11:15PM (#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.

      • 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.

        • by dcollins (135727)

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

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

        • Re:Ordered sets (Score:5, Informative)

          by Anonymous Coward on Wednesday April 25, 2012 @11:41PM (#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.

          • by dbIII (701233) on Thursday April 26, 2012 @04:17AM (#39804225)
            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.
            • 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.

              That's what happens when you take the programmer who worked on Windows progress bars and tell him to use his talents on Excel graphs.

            • by jonnythan (79727)

              The line charts use the x-values as labels only. The scatter plots interpret the x-values as quantities. That's why both exist in Excel.

        • 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.

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

    • Do you intuitively know what a continent is? If you said yes, post a reply then check out What are Continents? [youtube.com] - then post another reply to that.

      As to the measuring cup example: if a number line is so intuitive to a measuring cups, why are so many sets of unmarked 1/4 cup, 1/3 cup, 1/2 cup, and 1 cup measuring cups sold? After all, shouldn't anyone just need a 1 cup measuring cup? For that matter, why need tablespoons and teaspoons? After all, a tablespoon is merely 1/16 of a cup and a teaspoon is 1/48

      • by retchdog (1319261)

        the single-measure cups are for scooping the right amount of dry material directly out of a bag, especially flour. in fact, that's what they are called: dry measuring cups.

        not only is it much more convenient (have you ever tried to pour flour?), but flour volumes in recipes are based on it being loosely-packed, which is easier if you just scoop it.

  • by IntentionalStance (1197099) on Wednesday April 25, 2012 @11:12PM (#39802877)
    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
    • by JoshuaZ (1134087) on Thursday April 26, 2012 @12:06AM (#39803175) Homepage
      The Piraha are in South America and they have a language that is lacking many words considered normal in other cultures. http://en.wikipedia.org/wiki/Pirah%C3%A3_language [wikipedia.org]. They give directions primarily in terms of the relation to the river (towards or away from the river or up or down the river) which may be what you are thinking of. There's a highly readable book about the tribe and their language- "Don't Sleep, There Are Snakes" by Daniel Everett, a linguist who spent decades with them. However, there's some degree of question by other scholars about how accurate Everett's description of their language was, and research is ongoing.
      • by Animats (122034)

        Hawaiian has a radial notion of location: makai, towards the sea, mauka, away from the sea. Rotational direction is expressed as toward one of a few key shore points.

      • 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.
    • FTA:
      "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
    • 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

      Yes, that may well have been...

      the Aymara of the Andes [seem to do the reverse, placing the past in front and the future behind]

      Source? TFA, which mentions the same re

    • 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.

  • by AK Marc (707885) on Wednesday April 25, 2012 @11:16PM (#39802895)
    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.
  • by chichilalescu (1647065) on Wednesday April 25, 2012 @11:21PM (#39802909) Homepage Journal

    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.

  • by Skapare (16644) on Wednesday April 25, 2012 @11:21PM (#39802915) Homepage

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

    • by ThorGod (456163)

      '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.

  • 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.
    • by mcavic (2007672)
      Right. The concept is intuitive, but putting it into words takes education, just like everything else.
  • by tukang (1209392) on Wednesday April 25, 2012 @11:30PM (#39802969)
    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.
  • 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

  • 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.

  • by phantomfive (622387) on Wednesday April 25, 2012 @11:47PM (#39803079) Journal

    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.

  • by FoolishOwl (1698506) on Wednesday April 25, 2012 @11:53PM (#39803105) Journal

    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.

  • 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. :)

    • 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

  • by GodfatherofSoul (174979) on Thursday April 26, 2012 @12:19AM (#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.

  • by x0 (32926)
    Seriously, I can't be the only one who read the title and thought: Numberwang! m
  • Logical or not, the number line is equivalent to a finite list of axioms (field axioms, look 'em up, maybe with some stuff I forget atm). When we accept the truth of those axioms, all at once, then we begin studying 'the number line'.

    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
  • by pigwiggle (882643) on Thursday April 26, 2012 @01:23AM (#39803561) Homepage

    as well as number form and personification. Numbers - depending on if they are simply numbers or dates - have a specific "geography", color, and personality.

  • by gavare (15141) on Thursday April 26, 2012 @01:28AM (#39803579)

    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.

    • by ledow (319597)

      And how many thought in binary? Although I don't count every day in binary (the indoctrination into the decimal system is almost impossible to avoid in the Western world), I often catch myself finding binary patterns and thinking about things in a binary way (and if someone asks me to remember a number, the best way is to try to calculate its binary expression - the calculation and the resulting string fix into my memory a lot easier). Hell, when I run out of fingers counting in decimal, it's easier for m

    • Fascinating. When I was a kid numbers used to talk and fight with each other. Some numbers were good and some were bad. Not sure that's a very useful way to think of numbers because I am horrid at arithmetic.
  • by jandersen (462034) on Thursday April 26, 2012 @05:42AM (#39804577)

    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?

  • by pz (113803) on Thursday April 26, 2012 @07:25AM (#39804989) Journal

    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.

  • 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.

"Even if you're on the right track, you'll get run over if you just sit there." -- Will Rogers

Working...