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With your example of the quadratic equation, what should have been taught (and was taught to me in Algebra I) is how to derive the quadratic formula by completing the square. Not only do you have a better understanding of the quadratic equation but you can derive it at any time. This is what all math students should do. Same with trig formulas, learn how they are derived.
If you think of the quadratic equation as some magic formula, without realizing its roots, you will struggle.
I find the OP has an interesting point, perhaps as just one of several possible techniques for aiding memory. I went to a school with traditional methods of learning the times tables by chanting them out loud. While I am good at mental arithmetic I am poor at anything like calculus, algebra or trigonometry.
I have always found I remember things better if I write them down, but I have also found that I am very good at remembering 10 digit phone numbers. I assume this is because I was in the cellular phone business for 21 years and so practiced on a daily basis. On the other hand I am not good at remembering names or faces or facts like names of capital cities.
Everyone has different levels of ability in different subjects and kids develop at different rates at different points in their schooling. My son, for example, was average in math and in fact most subjects until he got to high school, when either his development or better teaching caused a sudden improvement. He eventually earned a Master's degree in Pure Math, specializing in set theory and topology. Pure math is basically a matter of proving or disproving a proposition, so it involves concepts and logic rather than numbers- there is no "answer" in the form of a number.
I am sure different subjects require different teaching methods- I have always thought history one of the most poorly taught subjects in that it is seen as a series of wars, and kings and queens without the social and economic events in the background being considered.
I can't do the exercise in the OP even today. My brain simply does not work that way. That's why I did so well in geometry - it was required that we drew out everything and worked through the proofs step by step. It was like a light turned on. Suddenly, math made sense and was incredibly easy, even as the rest of the class struggled to grasp basic concepts.
Everyone learns in different ways.
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