Instrumentality, relationality, and terrible memory.

I've always had a problem with pure instrumental understanding. I imagine part of that is because I have terrible memory, but also because I always wanted to know why. I was curious and loved the feeling of finding things out. I understand some people just want to know the easiest way to solve something, a "just give me the formula" kind of approach. Shortcuts, mnemonics, and memorizing formulas can all help with efficiency, but for me that never really stuck. Having a really bad memory could make me remember the formulas wrong. Memorizing the whole formula and mistaking a plus with a minus sign could sound like a small mistake, but that could also be as big as mistaking the gravity going down with it going up. I've seen a lot of sign mistakes on memorized formulas or on calculations but never found someone who thought gravity was pushing things up. That just shows how fragile the instrumental understanding can be, with small mistakes being huge disconnects with reality. Not that focusing on relational understanding makes calculations fail proof, but at least, if you know what the big picture is, you can look at the result and say "this doesn't seem right", and double-check your work. If you just follow a procedure with no understanding of the relations between that and the world around you, the result is just a number, and a wrong number and a right number are both just numbers.

I agree with most of what was said, including the part where the author mentions the possible reasons for instrumental understanding. But I think when the author mentioned the difficulty of assessing relational understanding, the biggest mistake was trying to do that with traditional math questions. "Solve this" and "calculate this" questions will only check the execution, not necessarily the thinking. Maybe assessments should just be different. Make some math assessments be more like humanities studies, asking students things like "what would be the next step?", or "why would we do this?", or "when would we use this?" instead of just asking for final answers. Math has number problems and word problems, but those are usually just answered with numbers. Maybe math also deserves word problems with word answers.