Using concreteness as a foundation for abstraction is not just good for mathematical instruction; it is a basic principle of understanding. Novices crave concreteness. Have you ever read an academic paper or a technical article or even a memo and found yourself so flummoxed by the fancy abstract language that you were crying out for an example?
Or maybe youâve experienced the frustration of cooking from a recipe that was too abstract: âCook until the mixture reaches a hearty consistency.â Huh? Just tell me how many minutes to stir! Show me a picture of what it looks like! After weâve cooked the dish a few times, then the phrase âhearty consistencyâ might start to make sense. We build a sensory image of what that phrase represents. But the first time itâs as meaningless as 3 + 2 + 1 would be to a three-year-old.
This is how concreteness helps us understandâit helps us construct higher, more abstract insights on the building blocks of our existing knowledge and perceptions. Abstraction demands some concrete foundation. Trying to teach an abstract principle without concrete foundations is like trying to start a house by building a roof in the air.