Arrows as a primary representation

Arrows first, because theyre easy to reason with

Velocity, acceleration and displacement are of course vector quantities, and therefore both magnitude and direction are essential to their nature. 'Simplifying' such quantities by restricting them to a single dimension does entangle teachers in verbiage, as evidenced by this series of quotations from experienced teachers: 'signed quantities, but not like power', 'the deceleration is negative', 'these are one-dimensional vectors, but not just like numbers on the number line', 'acceleration happens when the speed is not in same direction', 'directed speed'. All of these, which cause significant cognitive loading for teachers and children, could be sidestepped, I think, by using vectors in their natural introductory environment of two dimensions. This may be one place where the affordances of the computer as a (two-dimensional!) representation machine can kickstart a reformulation of our customs and rituals. Adding vectors is now easy: 'just place the arrows tip to tail'. With computational assistance, the arrows become objects-to-reason with, tools for thought. Later, perhaps much later, one can burrow down into components and algebraic representations. But in introductory teaching, we might try and get the physics established with the most approachable representations that we share with mathematics: just arrows. Arrows such as those which appear on sketch maps, on GPS displays and in diagrams always show velocity from a point of view.