I think at least some of what has gone wrong in Math education is that the linguists have infected the teaching of Math with a whole lot of over-descriptive buzz words. One that I recall from my sisters years at primary (elementary) school was the "commutative law of addition". She was 3 years younger than me, and has never really dealt with Math well. I don't think it helps when kids have to learn lots of wordy rules, instead of just getting in and tackling the numbers.
Mathematics is a language in it's own right : you don't need to overload it with extras to make it comprehensible.
What I've read in other comments about Common Core Math seems to simply be a different way of breaking down the numbers into easily handled bits.
The way I was taught was with simple sums at first: 2 + 3 = 5; 7 - 4 = 3. But our Math books had squares, not just lines, so we were taught to structure the sums to give numbers a proper place to simplify the operations we carried out on them:
3 5 +
The significance of the additional columns to the left was that they were 10 times the immediate neighbour to it's right.
So, a large subtraction operated by adding 10 (in this case) to the number in the "units column", and 1 to the number at the bottom of the "tens column".
Same value (10 units / 1 ten), different number to express it.
3 9 -
So, descriptively it operated as "2 minus 9 won't go, add 10, 12 minus 9 is 3, 1 (to 'put the 10 back') plus 3 is 4, 8 minus 4 is 4".
It's an array, with a handy sub-array, to facilitate operations that rely on the relationship of 1 and 10 and 100 (etc.) each in it's proper place.
The operation described in the Common Core examples is linear, they're "climbing a ladder, a step at a time" using addition to find the value between the two numbers. It teaches a linear operation that is more easily described in words, but is less structured in mathematical terms.