This story has to be apocryphal, as fractions aren't _that_ rare, especially in the U.S. with its imperial system and third of a cup measurements or quarter inches or half miles and so on.
It's not that they're rare, it's that it legitimately is an easy error to make even if you understand it to be an error. Even people who work with equations every day will occasionally make careless mistakes like this. That's why mathematicians joke about how it's important to make an even number of sign errors.
To not make this mistake, you have to be able to call to mind that the map x -> 1/x reverses the inequality sign. That's a fairly abstract thing to remember especially if you haven't taken math for years. Yes you could draw it or write down the equation, or convert to decimal... But it's enough of a cognitive barrier that it doesn't surprise me that it would impact the behavior even of people who would answer correctly on a test.
Where it does get easy is if you work with the same set of fractions every day. For example, if you work in construction in the US you can probably quickly order the fractions commonly used for measurement, e.g. 1/4, 1/8, 1/16, 3/4 etc. But 1/3 isn't one of these. Now that I think about it, they probably should have just chosen a fraction that you can find on a tape measure, like 3/8.
3/8ths is 0.375 while 1/3rd is 0.333~ so it's even bigger while still larger than 1/4th (0.250), without being that much bigger.
3/8ths is a pretty good marketing point since all the numbers are bigger and it should be intuitive, plus you can more easily see that it's also 50% bigger than 1/4th => 2/8ths. The harder sell is the 'double whatever' being equal to 3 patties of the competitor.
I do not really like the term "common sense" as it is more like common experience. It is not hard to learn what fractions are but I do not think it is something that any one is born with and there is other notation to deal with fractions that work differently.
I speak, and thus think, in both English and Japanese.
English says "1/4", or "1 over 4", or "1 quarter".
Japanese says "4 bun no 1", or the practical equivalent of saying "4 under 1" in English.
I consequently routinely say the numbers in reverse, confounding both myself and anyone around me before I realize my brain engaged in furious tentacle sex with the numbers.
> I speak, and thus think, in both English and Japanese.
The vast majority of processing is happening outside language-related areas of the brain. There's certainly leaky interfaces between areas of the brain, but if you literally thought in a language, and that distinction persisted throughout the brain, that would seem to imply that speaking 3 languages would require 3x the number of connections in the brain.
The strong Sapir-Whorf hypothesis would presumably be true if we literally thought in a language, but the strong form of the Sapir-Whorf hypothesis has been thoroughly discredited.
In other words, "thinking in a language" is an illusion.
I think this is partially correct. Inner monologue is not an illusion, and choosing a wrong linguistic construct for your audience (sometimes from another language) through temporarily forgetting to context switch does happen. However, thinking something without ever having done so in words does seem to strongly correlate with your assertion.
Tangentially, I realised in high school that I was doing almost all math operations as word transformations. I reasoned this was why even familiar procedures for which I confidently & consistently got correct results were taking substantially longer than everybody else. I was translating everything twice.
> To not make this mistake, you have to be able to call to mind that the map x -> 1/x reverses the inequality sign. That's a fairly abstract thing to remember especially if you haven't taken math for years. Yes you could draw it or write down the equation, or convert to decimal...
You absolutely don't have to remember that x |-> 1/x is order reversing, and, for most people, shouldn't—you immediately give two or three other methods (I don't understand what "write down the equation" means) that are a much better way for the average person to check this.
Yes, but I was also speaking generally about fractions since that was the context of the comment I was responding to.
For example, consider: 1/1123 < 1/1092. Is that inequality true? The fastest way to check is to compare the denominators and adjust for the way division interacts with the inequality.
You can't really draw that pie chart quickly. You could write the equation down and multiply both sides though.
For 1/3 vs 1/4 yes you could draw it quickly. Or you could fill a glass 1/3 full of water and one 1/4 and compare them. But that's a pretty special case for small enough denominators.
Forty years ago we learned fractions with chocolate bars. A trustworthy child would be chosen to walk from the primary school to the local store (about 5 minutes walk for an adult, probably about two minutes for a child who was just given money by a responsible adult to buy chocolate) and bring back some chocolate, and then kids who raise their hand and give the correct answer to fraction questions get the fraction in its physical form as chocolate. What's half of this third of the bar? A sixth, and now because I knew that I get to eat 1/6 of a bar of chocolate, whereas the kid who enthusiastically answered that it's a quarter does not because that's wrong.
One third times one fifth loses a lot of folks. As does addition and subtraction of fractions that don’t start with the same denominator, for that matter. They might figure out what to do to pass the test, but they may not get it.
Fractions are just division. When kids learn division, it's about splitting into equal groups.
Fractions are a bit different though - you're splitting a single thing into equal chunks. Hence, slices of pie.
Multiplying by 1/5 is really dividing by 5. Introduce that first. We already know how to do this. You split your 1/3 slice into 5 equal slices.
Do the same to the other 2/3 slices, count all the slices, and you have 15. Hence, 1/15.
As an aside, common core math is amazing. They gave my daughter a model for the distributive property that can be used to show how to do long multiplication.
There's a difference in type of thinking in moving from operations on numbers (basic whole number math) to operations-on-operations-on-numbers (anything with fractions).
Suddenly, you need to begin to understand the rules around operators, sequencing, and what operations are legal and illegal.
Absent that understanding, even...
1/4 x 2/5
... gets very complicated trying to reason with physical analogs.
So it's the point at which math becomes "pure" rather than strictly physically-mapped.
IDK, I did fine with them and find thinking in them natural (I think of fractional division as division, in fact, though I certainly understand the multiplication analogy); I’ve just known enough people who lost track of math at fractions that I doubt it’s coincidence.
I’m not saying I don’t get it, I’m saying others have told me that they found the explanations and instructions they were given nonsensical.
Have you tried dumping all the sockets out of a socket set and putting them back in order? Do you find it's easier to order the metric ones than the imperial ones which have a lot of different denominators on adjacent sizes? I certainly do but I'm not American so maybe it's my background limiting me.
Just to save you some time: the numbers written on the sockets indicate the size of the socket. So you don't even need to read them, just put them in order of size and you'll have them in order of number automatically.
Can confirm - While I was decent at math up to a point, fractions and long division in 4th Grade sent me down a hole that took me years to get out of...until Algebra II as a junior in HS crushed me.
I blame this on my Chemistry teacher - a class which I was also taking at the time - who spoke little English and had never taught in the United States until the year I landed in her class. I actually did reasonably well in Algebra for the first quarter or so until it all fell apart.
I re-invented what turned out to be short division (no joke! I wouldn’t learn it had a name until I was in my late 20s) because I hated long division so much, same year we started doing long division in school (4th grade? 3rd? IDK).
Fits in about the same space as the original problem unless it’s printed so small that you have to rewrite it, and way less room for transcription errors. I also find it clearer but that may just be me (fwiw I’m “bad at math”—I find it incredibly boring and basically can’t follow proof- and equation/identity-based stuff, I have to turn it all into algorithmic thinking to have a prayer of understanding it; i.e. my opinion on the superiority of short division is that of a mathematical imbecile, so, grain of salt)
> I blame this on my Chemistry teacher - a class which I was also taking at the time - who spoke little English and had never taught in the United States until the year I landed in her class.
It doesn’t help that in chemistry, 1 + 1 may be 1. Or 3. :-)
Under the “example” section, the little superscripts are what you write in by hand on the problem as you work it, at least as I did it. 9/4 in the hundreds place is 2 with 1 remainder, so write 2 up above as part of the solution and a 1 superscript next to the 5 in the problem itself (tens place), now that’s 15, divide that, 3 goes in the tens place of the solution, write the remainder (3) next to the digit in the ones place as a superscript and do it again, if you need to keep going just add a decimal point and zeroes as required.
Way faster than working long division, takes up less space, and less error prone (imo). What’s actually going on is clearer (again, imo)
I'm ok with fractions, but fractional and/or negative exponents always give me problems. I suspect it might be something to do with being taught that "multiplication is repeated addition, exponentiation is repeated multiplication". The model doesn't extend properly.
I’ve seen a later fall-off point at factoring. Feels pointless (the motivations are… distant at best), tedious as hell, lots of guessing involved. “So much for math making sense, fuck this, guess I’m out.”
It actually doesn't shock me that many people would be confused, especially if they didn't work with fractional quantities--e.g. for cooking--on a regular basis. Maybe it's a myth but it wouldn't surprise me if it weren't. And even if they've sort of internalized 1/4, 1/2, 3/4 without necessarily fully getting fractions--1/3 is something people encounter a lot less day to day.
I did teacher's college in Canada and the teacher who taught math said his biggest surprise when he moved from Europe to Canada is how terrible people were with fraction. I think he asked a barista to fill his cup to 2/3 and they couldn't do it because they didn't know what 2/3 was.
> This story has to be apocryphal, as fractions aren't _that_ rare, especially in the U.S. with its imperial system and third of a cup measurements or quarter inches or half miles and so on.
I literally had an argument with a room full of US university professors about whether or not 30% and 1/3 were the same thing.
Or perhaps all those people on here who defend US Standard measurements over metric and quote the fractions they know over decimals as an advantage are a minority?
Perhaps the average Joe would be better off with mm rather than 1/16" increments.
Based on a "focus group" discussion which are well known for selecting the brightest bunch of people who have nothing better to do than answer questions for 2 hours and get a coupon for free fries.
https://www.snopes.com/news/2022/06/17/third-pound-burger-fr...
Crazy.