## Study Suggests the Number-Line Concept Is Not Intuitive 404

Posted
by
samzenpus

from the learning-to-count dept.

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."*
## The Story of 1 with Terry Jones (Score:5, Interesting)

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

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 (Score:5, Funny)

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 (Score:4, Funny)

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: (Score:3)

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.

## Re: (Score:3)

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

You mean it's a relativistic metric? Wow...

## Re: (Score:3)

Have you seen those Redguard lake-women? They lob curved swords.

Curved.Swords.## Anyone who has ever taught math knows this (Score:5, Interesting)

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 (Score:4, Insightful)

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 (Score:5, Funny)

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

## Re:Anyone who has ever taught math knows this (Score:5, Insightful)

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.

## Re: (Score:3)

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.

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

## Re:Anyone who has ever taught math knows this (Score:5, Insightful)

No. The science which maps real-world phenomena onto artificial symbols and concepts is known as physics. Mathematics is only concerned with the artificial symbols and concepts, independent of whether they can be mapped to real-world entities (many things cannot).

Yes. And mathematics is about the map and its rules,

withoutcaring about the territory, or even if it corresponds to a territory at all. If you want to learn about the territory, use physics. And yes, you'll use maps (i.e. mathematics) there, too. But those maps are not arbitrary, but carefully adapted to the territory as far as we know it, and actively developed to improve how well it maps the territory.So in the map/territory picture you have:

Mathematics: The science of maps. Doesn't care about what the maps mean, or if they mean anything at all. As long as a map is consistent, it is accepted as valid map.

Physics: The science of territory. Uses maps to describe the territory. A map is considered valid only if it describes the relevant aspects of the mapped territory sufficiently well.

## Re:Anyone who has ever taught math knows this (Score:5, Funny)

Physics? Is that your name for applied maths? [xkcd.com]

## Re: (Score:3)

Similarly, human linguistics generally concerns itself only with mappings between symbolic concepts with no thought as to how those are truly internally represented nor how synthesis into external representation occurs.

No it most certainly does not. That's "semiotics". "Linguistics" is a much broader field.

## Re:Anyone who has ever taught math knows this (Score:5, Insightful)

"Logically consistent" and "able to be used to prove its own consistency" are not the same thing.

## Re: (Score:2)

## Re: (Score:3)

50% full really means 80% full

The ex-wife used to interpret the petrol gauge with a similar coversion function: E = Ehhhhnuff, (where fingers = crossed).

## Re:Anyone who has ever taught math knows this (Score:5, Funny)

In order to save time, paper, and ink, I made my number line logarithmic.

I hope you made two of them for the synergy effect.

Slide rule joke of the day:

When Noah told his menagerie "go forth and multiply", two snakes replied: "We can't, we're adders!"

Noah then built a wooden table, placed the snakes on it, and much joy and spawn ensued.

Because on a log table, even adders can multiply.

## Re:Anyone who has ever taught math knows this (Score:5, Informative)

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

notintuitive. He needed an explanation.## They have the problem ass backwards. (Score:4, Interesting)

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.

## BASIC Programming, old school (Score:2, Offtopic)

## Re: (Score:3)

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

## Re: (Score:3, Insightful)

You Forth about talking are, I think is what you're aiming for. Your sentence came across as more German than RPN.

## Counting? (Score:5, Interesting)

## Re:Counting? (Score:5, Funny)

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: (Score:2)

"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: (Score:2, Funny)

## Re: (Score:2)

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

## Re: (Score:2)

That is a seriously long wait.

## Re: (Score:3)

## Re:Counting? (Score:5, Funny)

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? (Score:4, Funny)

It was actually "Developers! Developers! Developers! Developers!"

## Re: (Score:3)

Not true. You'd end up with -1 hunters.. while irrational, is still a valid answer

No, -1 isn't irrational. If it had been the square root of two hunters, it would have been irrational.

## Re: (Score:3)

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: (Score:3)

A lot of neuroscientists reckon we can process about 4 separate things in our mind simultaneously

I think it depends on the sort of processing you need to do with those things and how long you have to do it:

http://cognitivefun.net/test/28 [cognitivefun.net]

http://cognitivefun.net/test/7 [cognitivefun.net]

http://cognitivefun.net/test/3 [cognitivefun.net]

http://cognitivefun.net/test/4 [cognitivefun.net]

http://cognitivefun.net/test/8 [cognitivefun.net]

Apparently you can train yourself to do it better, and some research claims that "dual n-back" (and n-back) training can also increase your "fluid intelligence".

FWIW I've got much better at the single "n-back" where n=2, after just a few tries over a d

## Re:Counting? (Score:5, Interesting)

## Re: (Score:3)

## Re:Counting? (Score:4, Interesting)

## That was the Peano Construction, not ZFC (Score:5, Interesting)

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

alsodefine 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.

## Re:Counting? (Score:4, Insightful)

No, not joking. There already have been studies that show different cultures have different counting systems. For example, many cultures will have only the most basic of numbers (1, 2, 3, 4, 5) and then jump into the "many" category. Another example of the non-intuitive nature of numbers? 0. That one took a while to catch on. Third example? Describe to me a forest with -10 trees or a person with -1 apple. Negative numbers were not intuitive either. Notice I am avoiding those wonderful numbers like fractions, irrational numbers (pi, e, the square root of two, etc), and complex numbers (i, the square root of -1... graph that on your number line!) - all of which are not intuitive in and of themselves. Final example? If numbers are intuitive, why does it take so long to teach our young to count? Why do so few people understand the concept of billions and trillions of dollars of debt, or the vast distances of the universe, or the very tiny number which represents the time in which million/billion/trillions of molecules collide and interact when undergoing an exothermic reaction?

No, while you have been educated and indoctrinated into a system of numbers, that does not mean it is intuitive. Or another way to think of it - take the pro basketball player who has taught his muscles how to shoot a 3-pointer... he might argue that it is intuitive, meanwhile someone like me (who couldn't make a freaking free-throw shot) would say that it is definitely not intuitive.

## Re: (Score:3)

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: (Score:2)

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)

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. (Score:4, Insightful)

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 (Score:2)

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: (Score:2)

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)

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.

## It's not just Yupno Valley - Seattle too (Score:4, Funny)

Thus you could have an axis that looked like:

1 4 7 8 14 35IMHO that sort of defeats the purpose of a line graph. I can userstand linear or log scales but a random changing scale is pointless.

## Re: (Score:3)

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.

## Re: (Score:3)

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.

## Re: (Score:2)

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: (Score:2)

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: (Score:3)

## Re: (Score:2)

## Re: (Score:2)

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.

## Re: (Score:3)

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

## Re: (Score:3)

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.

## Valleys and Language (Score:4, Insightful)

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

## Re: (Score:3)

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.

## Lingala and time (Score:3)

## Re: (Score:2)

"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

## Re: (Score:2)

Yes, that may well have been...

Source? TFA, which mentions the same re

## Re: (Score:2)

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.

## What is intuitive (Score:3)

"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: (Score:2)

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: (Score:2)

## Re: (Score:2)

## Re: (Score:3)

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.

## agriculture (Score:3)

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 ... (Score:4, Funny)

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

## Re: (Score:2)

'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.

## So what? (Score:2)

## Re: (Score:2)

## Logarithmic vs linear scale (Score:5, Interesting)

## Obviously? (Score:2)

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

## Management (Score:2)

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.

## Americans don't understand number lines either (Score:4, Informative)

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: (Score:2)

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: (Score:2)

## Counting and measurement are distinct concepts (Score:5, Insightful)

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 (Score:2)

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: (Score:3)

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

## Typing on a computer isn't "innate" (Score:3)

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? (Score:2)

## numbers are not innate (Score:2)

## follows logical from the field axioms (Score:2)

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## I have Spatial Sequence Synesthesia (Score:3)

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

## Re: (Score:3)

Well, I stand corrected then. I didn't realize AC was the final arbiter on what is or is not synesthesia. I'm sure the medical community was pleased to have been rid one more burden.

## Re: (Score:3)

Decades, months, and days of the week all have specific shapes, locations, and colors. They have always been the same as far as I can remember. Numbers you would use in calculating things have color, albeit past 10 they group in 10s. That is all the 20s are a yellow orange color, 30s purple blue, and so on. The personality of numbers is entirely about if they are prime or have prime factors or are odd. It's a simple good and bad type thing. 3 and 7 are sinister, 9 more so, 21 also. All are odd and ar

## ask your non-nerd friends (Score:4, Interesting)

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.

## Re: (Score:3)

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

## Re: (Score:3)

## Wrong questions (Score:3)

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?

## The Plural of Anectdote is not Data (Score:3)

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.

## I thought we covered this in school (Score:3)

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.

## Re: (Score:2)

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

## Re: (Score:3)

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.