This whole thing about understanding disturbs me. I don't understand understanding because, like creativity, it is a vague, nonspecific term and difficult to measure. If you can't measure something, then what is it? A teacher said, "Yesterday when I was going over the problem (-12ab) + 4ab ... The kids said, 'Can you just work the problem?' They don't want to understand math; they just want the procedure and move on." (Comment: Combining like terms is a simple task. How does the teacher define understanding? I don't know. Did the teacher link combining like terms to prior knowledge? Kids are novices, not experts.)
Well, that's what Issac Newton did with his calculations (a.k.a calculus). He moved on! He used the procedures even though he didn't understand why they worked. Why? The calculus just worked! It agreed with experimental observations. That was good enough, but it took another 200 years "to hammer out the formal details," explain Klein and Bauman. (Limits)
Note: For novices, the act of performing or applying procedures implies some level of "understanding" that is difficult to quantify or explain. It is a functional understanding. Novice: At first, I "understand" something when I perform or apply it. For example, 3 x 4 = 4 + 4 + 4 or 12 is useful at first, but 345 x 876 needs "something more sophisticated" than repeated addition, says Ian Stewart, mathematician. "Mathematics builds new ideas on old ones." Stewart (Letters to a Young Mathematician, 2006, also points out the practice does not cause talent, but it can improve performance.
The next time a teacher says they stopped teaching long division because kids won't understand it, Think, Newton. Students can learn to do it and apply it, even if they don't completely understand it. How do you think students learn arithmetic, algebra, trig, and calculus? (With perfect understanding? Right?) The idea of division is easy and can be taught in 1st grade as intuitive division. The understanding of the standard algorithm takes repeated practice and time. To do the standard algorithm for multiplication, you should memorize basic number facts, such as 3 x 7 = 21 or 8 x 7 = 56, no later than the 3rd grade. (Half in the 2nd grade.) In short, children will not learn much algebra when they don't know basic arithmetic. My Teach Kids Algebra program fused algebra ideas to traditional arithmetic. Hence, the importance of memorizing math facts that support basic arithmetic (e.g., standard algorithms) was emphasized.
We are told that not understanding math stuff will result in poor algebra scores. It's mostly nonsense. Being able to perform arithmetic or algebra on paper indicates that the student has acquired some understanding, which is a functional understanding. I did not understand arithmetic, algebra, precalc/trig, or calc as well as I do today at age 78, yet, in college, I could apply the procedures correctly in chemistry and physics. Understanding grows slowly and is intertwined with the mechanics of procedures such as the standard algorithms and other operations such as "taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power," etc. You will not understand addition unless you can do sums efficiently, in your head, or on paper by using standard algorithms based on place value. And you cannot do division of fractions without finding reciprocals, a mini functional procedure.
Note: "Symbolic reasoning and calculations with symbols are central in algebra." (California 1997 Algebra Content Standards) When symbolic manipulations are marginalized, it's "pretend" math, not algebra.
✍️ Marxist radicals have taken over education at the local, state, and federal levels. The only way to close gaps, they say, is to lower content standards and eliminate excellence. No child gets ahead. The radicals have redefined achievement as privilege, says Thomas Sowell. These are terrible ideas! "Equalizing down, by lowering those at the top" is a crazy idea, a "fallacy of fairness," not equity, explains Sowell in his book of essays and his latest book below.