I wonder if we can compare what happens over the long therm with people who intuitively understand math vs. people who learn to understand it with the help of metaphors such as above?
Are such un-abstractions helpful for teaching basic math knowledge? Advanced math skills? Both?
I want think these things are great for a start, to reach that first epiphany which clicks a concept in place, but I'm thinking they should be applied selectively and discarded as soon as possible.
I base this pretty much entirely on my own frustrating experience being "taught" with methods that focused entirely on and metaphors and procedures for "how" of things, not the "why" which I believe is essential to any real understanding.
In my experience teaching science and math, I've found this sort of analogy to be of limitted utility. It is often possible to get a whole class on board with the story and understanding the "rules" and able to come to the right conclusions. They're thrilled. Then you try to bring up the more abstract version and connect it and you get blank stares. Some kids make the connection quickly, but they tend to be the ones that wouldn't have needed the story version in the first place.
I have found that people who understand a topic well often mistake frameworks of understanding that they come up with after they understand for good ways to teach the concept to people who don't yet understand. It's like people think, "Oh I get it! Here's the big picture!" And they explain it that way. But the big picture they came up with was the result, not the cause, of their understanding.
I certainly learn abstractions better when I have a solid concept to work from. I understand it is a crutch/bridge/scaffold for a true understanding, but such a thing really helps me learn anything. This has been true for me of maths, various programming concepts, human language grammar rules, and on.
I really really dig the power abstractions and abstract thought provide, but for whatever reason, I can't grasp a lot of abstracts without first learning how to apply them to a specific case and working out from there. As I have gotten older, I have been able to see quicker how to apply the abstractions from the initial case study, and regularly can apply abstraction in interesting ways - I am particularly good at seeing where various design patterns can be used on some software we are architecting, and doing the mental refactorings on the fly. Yet when I learn about a new design pattern, I still need to see it in a working piece of code, and preferably a step-by-step transformation to grasp it.
Are such un-abstractions helpful for teaching basic math knowledge? Advanced math skills? Both?