Anecdotally, it seems like working with fractions is where a lot of people fall off the math-train and never get back on.
… yes, that early.
… yes, that early.
And, to be fair, they're the first math concepts that aren't intuitive.
Which IMHO leads to some people not studying, then feeling lost, then rationalizing their lack of effort as "I'm bad at math" or "Math is hard."
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.
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. :-)
[edit] short division:
https://en.m.wikipedia.org/wiki/Short_division
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)
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.
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.
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
So it's the point at which math becomes "pure" rather than strictly physically-mapped.
Actually I think we do very little addition of fractions later in math. But it is a concept that confuses the multiplication or division.