> ½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.
On the contrary, continuing to conceptualize fractional multiplication exactly as your "1/2 X 1/2" as addition example served me well. My wife—with a higher measured IQ than mine, FWIW—evidently can't see it that way. She basically stopped learning math when multiplication of fractions was introduced (it's interesting that one can continue to earn decent math grades, regardless) as all the results felt arbitrary and magical to her, and still do in middle-age. This is actually a very common point for students to stop following WTF is going on in their math classes and never really get back on track (yes, all the way down in, what, 2nd or 3rd grade) from what I've seen; factoring is another, later on.
"1/2, added 1/2 times" is precisely how I think of it. If I can't just pattern-match or rule-follow my way to a solution (because I've forgotten the rules, say) that's still my line of reasoning to figure out what to do to get the solution.
I do exactly the same sort of thing to come back to my senses if I get lost or forget exactly what is happening in fractional division. "How many times does 1/2 fit in 1/4? 1/2. How many times does 1/4 fit in 1/2? 2." Even if I have to manipulate some things to figure out the result with uglier fractions, that's absolutely how I think about what I'm doing as I do it, and it keeps me focused on the ultimate purpose of the calculation. I imagine that's also wrong, according to the author.
In short I can vouch that yes, that exact thing was very helpful to this particular person, including for multiplication of fractions. I truly don't know how else I might have understood those problems, to avoid joining the ranks of the mathematically-lost as early as lower elementary school.
On the contrary, continuing to conceptualize fractional multiplication exactly as your "1/2 X 1/2" as addition example served me well. My wife—with a higher measured IQ than mine, FWIW—evidently can't see it that way. She basically stopped learning math when multiplication of fractions was introduced (it's interesting that one can continue to earn decent math grades, regardless) as all the results felt arbitrary and magical to her, and still do in middle-age. This is actually a very common point for students to stop following WTF is going on in their math classes and never really get back on track (yes, all the way down in, what, 2nd or 3rd grade) from what I've seen; factoring is another, later on.
"1/2, added 1/2 times" is precisely how I think of it. If I can't just pattern-match or rule-follow my way to a solution (because I've forgotten the rules, say) that's still my line of reasoning to figure out what to do to get the solution.
I do exactly the same sort of thing to come back to my senses if I get lost or forget exactly what is happening in fractional division. "How many times does 1/2 fit in 1/4? 1/2. How many times does 1/4 fit in 1/2? 2." Even if I have to manipulate some things to figure out the result with uglier fractions, that's absolutely how I think about what I'm doing as I do it, and it keeps me focused on the ultimate purpose of the calculation. I imagine that's also wrong, according to the author.
In short I can vouch that yes, that exact thing was very helpful to this particular person, including for multiplication of fractions. I truly don't know how else I might have understood those problems, to avoid joining the ranks of the mathematically-lost as early as lower elementary school.