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.
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.