maths should evolve, with computers

Maths is not a static subject but evolves, with computers

An Exploration in the Space of Mathematics Educations (1996) by Seymour Papert
This article is "all over the place" but contains some very interesting theoretical ideas some of which can be readily implemented in practice

A thought experiment: maths does not consist of fixed elementary building blocks that must be taught first before more advanced maths can be understood. The elementary building blocks change as the available technology changes:
eg. binary arithmetic becomes more important with computers on the scene
eg. dynamic representations of say, the parabola, become possible to represent on the computer before static representations of the parabola (more realistic than when paper is the dominant medium)
eg. random is more fun and interesting in a dynamic medium than coin flipping or dice rolling in "real life"

Some principles for a different maths, in part, using computers:

• Power principle - using it before getting it
• Projects before problem solving - problems come up as part of doing projects and they are sometimes "solved" and sometimes "dissolved" (rather than teaching "problem solving" as a thing in itself)
• Media before content and Dynamics before statics - eg. it is hard to teach dynamics on the static medium of paper but much easier to teach dynamics using the computer medium - the medium of paper fights the message of dynamics
  
• Thingness principle - Object before operation, in the computer medium we can name and create icons for things that previously only existed as abstractions, eg. vertical and horizontal motion icons can be combined into composite motion. Reification is possible. (Papert prefers the down to earth word "thingness" to the academic word "reification")
  

Key ideas:
The idea that maths evolves, that it could be different from current school maths

The mathematical thing like object - eg. scratch blocks or tiles are mathematical objects, so deeply built into scratch that their mathematical nature is hardly noticeable. This paper was written before scratch was developed; scratch was developed under the influence of these ideas. Resnick was a student of Papert's.