Objects to think with
Also: objects-to-think-with, transitional objects
Tangible or computational objects that carry a powerful idea and let you reason with it directly.
Papert opens Mindstorms with the gears of his childhood. Before he understood multiplication tables, he understood gear ratios — he could feel them, turn them in his mind, and later he found that the same mental machinery let him grasp equations. The gears were an object to think with: a concrete thing that carried an abstract idea.
The Logo turtle was Papert's deliberate attempt to manufacture such objects on demand. A child commanding the turtle can reason about geometry body-syntonically — by imagining themselves as the turtle, walking and turning. The abstract becomes physical, and the physical becomes thinkable.
The phrase has since spread well beyond education. Any good notation, model, or diagram is an object to think with. Read Objects to Think With.