Sep 10 2010
How’s Your Number Line?
As you might imagine, I am fascinated by how the mind works. It is, arguably, the most important thing to each of us – our own brain. We are our brains. It is our universal tool, the one tool to rule them all. And so understanding the strengths and weaknesses, the quirks and foibles of the human brain in general, and our own brains in particular, should be of paramount interest.
One aspect of brain function that is very interesting is how we model abstract concepts in our brain, like numbers. We often assign physical properties to abstract concepts, perhaps as a handle so that we can better think about them. For example, with numbers we tend to picture them spaced out along a line, with linear distance being proportional to number value. We put numbers into a physical space. An extreme example of this in number synesthesia – some people actually feel that numbers have a shape, color, or texture. In some cases they use these properties of numbers to help them do calculations in their heads.
A recent psychological study looked at the number line of children. Young children just learning numbers tend to have a logarithmic number line, with low numbers spaced out and higher number increasingly squished together. But as children mature their number line tends to become more linear, with spacing being proportional to value. The recent study asked children to place various numbers on a line where they belong, to see what their mental number lines look like. They then gave them a mental task that required remembering numbers. What they found was that the more linear the child’s number line, the better they performed on number memory tasks.
This result suggests that our mental map of numbers is important to our ability to think about numbers, and perhaps manipulate them in our minds. It also shows that this ability is variable. That much is not a surprise – the ability to perform mental calculations varies widely among people. There are those who are mathematically challenged and those who are gifted. While education certainly plays a huge role in our mathematical abilities, raw talent is clearly also important – like with music and many other mental abilities. Perhaps facility with numbers is closely tied to our ability to think of numbers as having a physical property, as this study suggests.
An interesting follow up question is – can this ability be taught, or is it a consequence of genetically determined hard-wiring. Perhaps this research may eventually point the way to teaching strategies that encourage children to develop their number lines and other mental maps as a way of enhancing their ability to deal with numbers.
This is an appealing idea – using our knowledge of how the brain works to craft optimal teaching programs. However – if history is any guide, preliminary knowledge about brain function can lead to teaching fads that are not themselves evidence-based, and common sense approaches can get displaced by fanciful and ultimately pseudoscientific teaching programs. Like any other area of science, translating it into specific applications needs to be done carefully and itself based upon evidence. It is not a trivial thing to extrapolate from basic science to real work application.
16 Responses to “How’s Your Number Line?”
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My first thought was, how do we know we are testing the “mental number line” and not just the child’s understanding of fractions?
I see the study uses some young children who wouldn’t have learned about fractions yet, but certainly some of the elementary children have. So then we’re saying children who are good at math are good at math.
I thought that studies showed that raw talent had little to do with musical ability – that it was hours of practice that made the great musicians.
@eean I like that question. Dr. Novella, could you address it? Are the number line and fractions the same mental process, at different scales?
I’m not normally speculative, but the specific questions I have are a result of the inherent difficulty of controlling studies with humans. Is this difference a result of people going over how to count to ten with children when they are young? I know that I have seen this time and time again with children? Were the children separated socioeconomically? That might have controlled for this effect somewhat. It would also help control the effect of toys that teach pattern recognition, memorization, and rudimentary understanding of numbers.
I would like to point out, that in my experience (anecdotal, I know) in number ranges I use very often, I do much better memorizing amounts in those ranges. The less I deal with a range, the harder it is to remember. The way I get around this is by using techniques taught early on in math, but obviously later than what this age group would have been exposed to. These include scientific notation and metric measurements. Instead of remembering 13; 1,300; 1,300,000; 15,000; 12 I would remember 12, 13, 15K, 13 hundred, 1.3 Million. I wonder if this is a skill that is related to the same process being studied.
@Sara This may be ignorant on my part, as I have not looked into that debate in some time. I thought one hypothesis was that being naturally prone to music assisted in motivating the practice needed. Practice is all you need to be proficient, but there are tendencies that tend to propel people to great heights. This may have been disputed or incorrect, but it’s floating around in my head. If it is incorrect, someone knock it out of there!
Tendencies that tend to be redundantly redundant… Proof reading, it’s important kids!
My mental number line is logarithmic, which I think makes more sense than a linear spacing. Also, I didn’t really decide that it should be logarithmic; I suspect it developed that way due to the fact that we are taught numbers in powers of ten. First, we learn to count to ten. Then we learn the “teens” and to count to one hundred. Then we learn to keep adding powers of ten.
Interestingly (at least to me), my logarithmic mental number line is three-dimensional, which I suspect also reflects the order in which I learned numbers. The numbers 0 through 10 are linear on a short, horizontal line going left to right. The teens branch 90 degrees from the end of that line, upward and slightly away. (That part of the scale is not purely logarithmic, but I think it reflects my learning the teens before learning to count to 100.) From 20, my line turns back to the left 90 degrees and slants upwards to 100, with a reduced scale. From 100, it makes another 90-degree turn away and slightly upward to 1000, again with a reduced scale. From 1000 to 10,000, the line doesn’t change much in angle, just in scale. From 10,000 to 100,000, it turns 90 degrees left and slightly upward. From 100,000, it turns back 90 degrees away and slightly upward. From a million on, it’s purely logarithmic on a slightly curved line pointing upward and away.
I don’t have a separate line for negative numbers, other than the one shooting off to the left of zero forever.
I’ve wondered about this mental number line image before, and I’ve asked some people about theirs. So far, I haven’t discovered anyone with a similar model to mine. By the way, I’m an engineer, and I’ve always been pretty good with math.
In “What Do You Care What Other People Think?”, Richard Feynman looked at how people count. He and most of his friends could keep a steady mental count going while they were touch-typing or reading, but not when talking. They all said they counted by mentally “saying the numbers to themselves”, and couldn’t talk to themselves and others at the same time.
But one friend could keep count while talking, but couldn’t while typing or reading. When he counted, he said he mentally saw the numbers flow by as if on a stock ticker.
Are concepts like numbers and mathematics represented in similar parts of the brain in fMRI studies, or are there different “types” of mathematical thinkers – some more visual than others? And if there are different types, I wonder if that influences the results of the above number line study.
I thought of that anecdote Rickk. I don’t have a number line, numbers are just numbers to me. I recall them verbally, I remember phone numbers as groups of words, or with stories as mnemonics , but definitely no visual picture.
I work with a graphic designer who can’t spell, and has real trouble with any abstract idea he can’t put a picture to. But describe something to him, and he can imagine it, improve on the picture in his imagination, and then draw it for you. Of course whether these different views are innate, or just that we learnt things in a different way or order is going to be difficult to establish concretely, but the difference is so stark that one feels intuitively it must be innate (but I don’t place much credence on that).
The one that I find hard to comprehend at the moment is the accuracy of the top chess players. I’m an okay chess player, but I marvel at how consistently accurate the very best are. Almost of if they are made of different stuff, I assume they must be thinking of these things in a quite different way to myself. Perhaps like driving or many motor skills, they have relegated more of the process to some part of the brain that processes it without demanding as much conscious attention.
I didn’t go back and read the paper, but from what’s described here and in the link provided, I have another interpretation:
Did anyone consider improvement of children’s spatial skills in general, rather than in as they relate to numbers themselves? I distinctly remember when I was a kid and had to write out a number line, the higher numbers would get more squished together because I’d misjudge how much room I had to write the numbers out when I started with the lower numbers. I’d do the same thing if asked to write the alphabet on one line.
Also, a side note, my boyfriend has mild synesthesia, where days of the week and numbers have color. (also, interestingly, the days of the week are apparently arranged on a color wheel in his mind, rather than in typical calendar format) As far as I know, though, he’s no better at the task where you pick out a 2 in a field of 5s all printed in black ink because you see the written numbers as different colors… not that I’ve ever tested it (hmm, fun game for tonight…) Anyway, I’ve started to wonder if synesthesia this mild is really synesthesia at all. To me, 7 is orange; Tuesday is blue. And I know why. We had magnets on the refrigerator when I was growing up – all different colors, 0-9… 7 WAS orange. And in elementary school, we were taught to put our homework in color-coded folders based on the day the assignment was due… Tuesday WAS blue. Where and how do you draw the line between true synesthesia and learned association?
I tend to think in numbers as blocks rather than linearly, kind of like tetris blocks. So 63 and 37 fit together to make 100. Anyone else do this? I always put it down to playing too much tetris as a child
Jim, my number line is quite similar to yours, though I don’t ususally deal with big numbers so it only goes up to the hundred thousands.
Steve this reminds me of a comment that was made on one of your skeptic’s podcasts about visualizing time. Evan or one of your brothers said they visualize the year as a clock, and described quite a complicated visualization. The rest of you didn’t do this. I visualize a clock for the year, with summer at the bottom, winter at the top, spring at the left and fall to the right (my clock runs counter-clockwise, go figure). Whenever I hear a date I visualize it on that clock and I ‘see’ all the other things that happened on or near that date, past years are stacked one on top of the other. I have a really good memory for past events and when they occurred in relation to each other.
Correlation does not equate to causation. Especially one this weak.
It could be that an innate or precocious grasp of linear measure & logarithmic measure may well influence the way one winds up visualising the “number line”.
I remember learning the left-to-right “number line” at infant school, *well after* I had formed my concept of both linear & logarithmic extent. And the “number line” that I was taught seemed to shackle my relative quantitative thoughts, rather than free them.
Until then, I had thought of numbers as multi-dimensional and continuous, rather than integral & linear, irrespective of the scaling.
Perhaps I am an exception. I chose to do a math degree after all!
“But one friend [of Feynman's] could keep count while talking, but couldn’t while typing or reading. When he counted, he said he mentally saw the numbers flow by as if on a stock ticker.”
Every week day I exercise while I watch the ABC news and the 7:30 Report. I used to count the repetitions but found the mention of a number by someone on one of these programs would annoyingly interrupt my count. When I heard about Feynman’s experiment, I changed to visualising the number count in my head instead of counting.
Anyone interested in this topic should read Number Sense by Stanislas Dehaene, which covers this topic in great depth and was published back in 1998. Not only is the development of number sense in humans described in detail but also for animals, with some amazing facts. It’s not a dry technical book, but very readable for the general popular science reader (apart from the fact that the 1998 paperback version used a rather small font). I found it particularly fascinating to read about aspects of development where the generally accepted “fact” was wrong for a number of years, until someone came along with the right way to do the experiment.
My number lines include a separate axis for imaginary numbers. It is because I am one of those engineers who really likes and actually uses Euler’s Formula.
Though seriously, as a kid I saw numbers as chunks. I could see four and five items as a chunk and realize how big they were. I had no problem when the numbers got bigger, and I found logarithmic scales a good way to look at things (and it really helped during the age of the sliderule, and having to deal with sound decibels).
My line is not logarithmic by any means. Things are smaller when looked at from a distance, and the line is actually more like a fuzzy tape winding through a pale yellow and pinkish space, think of the Gloria Dei rose. Upon a closer look, the numbers appear to have their colours and textures. Also, timeline is similar tape in space to me, just winding slightly differently and months and years have different colours. Synaesthesia, I have it.
Also, I suck at math in general. I have excellent spatial imagination and I can sort of reverse engineer things from how I ‘see’ them. If I count, I just go bit by bit, adding or chopping off pieces of something to get them in a block to get an actual number. Arithmetics and me are almost mutually exclusive.
Funny thing, this imagination (or what the hell it may be).
I read the article, but not the paper — yet. My thoughts agree with the other commenters. The mental number line correlates with memory for numbers, but I’m not sure the study demonstrates that it affects it. I wonder if the cause for both is simple familiarity. (Familiarity affects number line. Familiarity affects memory.)
The scale they describe in detail was 1-20. I wonder if the, seemingly, logarithmic kids were more familiar with lower number names and kind of bunched up the higher number names because they were less familiar with them. Kind of like, “That’s one of those higher numbers up at the top of the scale.”
The linear children could have been the ones who were taught to count to twenty as opposed to ten or twelve as their first introduction to numbers. (I’ve classified our children’s early number toys, flash cards, etc. by which marker they count to 10, 12, 20)
Another aspect is that the article mentions “western culture”. I would guess from the location of the study, it was conducted in English. I don’t think it’s too surprising that you would spread the lower numbers and bunch the higher numbers considering the number names in English. Note — they weren’t SHOWN numbers and asked to place them on the line, they heard them.
Our numbers have very discrete sounding names from 0-12. Numbers 10, 11, and 12 are a bit odd in that they continue the pattern of unique names, but their numerals start the repeating pattern. Thirteen and fifteen are odd in that they aren’t “three-teen” and “five-teen” as you might expect based on fourteen and sixteen through nineteen.
I would guess that those who haven’t been exposed to them as much would kind of see the “blank-teens” as kind of grouped together at the high end of the 1-20 scale (especially 16-19).
So I would say that English number names for 0-20 decrease in uniqueness as their numerical values increase and this could lead to grouping on the number scale.
This would decrease as the child became more familiar with the number names.
The article also says they found this for the 1-1000 scale with older children. If the finding is as pronounced as the 1-20 scale, I may be off here with respect to the number names.
And as far as how memory comes into play — if, for the test, they asked the children to recall the names of the characters in the story as opposed to the numbers, they would probably find that the children would remember the names with which they were more familiar.
So, kind of like what the other commenter said, the more familiar a child is with number names, the more familiar they are with number names?