Consonance and Conflict

Okay, I lied. I'll get to that framing thing soon. And, of course, I have to come back to the skills-based thing also.

But allow me in the meantime to post a nice quote from Uri Leron, which is not altogether unrelated to the framing issue. And it certainly relates to the cinemathematics present in mathematics education today:

According to the algebraic image of functions, an operation is acting on an object. The agent performing the operation takes an object and does something to it. For example, a child playing with a toy may move it, squeeze it, or color it. The object before the action is the input and the object after the action is the output. The operation is thus transforming the input into the output. The proposed origin of the algebraic image of functions is the child's experience of acting on objects in the physical world. . . . Inherent to this image is the experience that an operation changes its input—after all, that's why we engage it in the first place: you move something to change its place, squeeze it to change its shape, color it to change its look.

But this is not what happens in modern mathematics or in functional programming. In the modern formalism of functions, nothing really changes! The function is a "mapping between two fixed sets" or even, in its most extreme form, a set of ordered pairs. As is the universal trend in modern mathematics, an algebraic formalism has been adopted that completely suppresses the images of process, time, and change.

REFERENCE: Leron, Uri. Mathematical Thinking and Human Nature: Consonance and Conflict. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004. (3) 217-224.