ABSTRACT
Students begin math in school with manipulatives and pictures. At some later stage, conversation and prompting in math switches to statements such as, “One-half divided by one-third and two times three.” In making this transition from the visible to the abstract, it works well to pair/translate the more abstract wordings with more visual, or concrete, wordings, such as, “How many one-thirds are contained in one-half?” Or, “Two groups of three/three groups of two.” It is worth inquiring into how much of math inadequacy is related to the “cognitive bends” incurred when the language of math isn’t more gradually brought from the concrete depths to the abstract surface.
