Part of the Iconic Math project is to rebuild the axioms of arithmetic (i.e. group theory), based on manipulative, interactive, visceral rules. Kindergarten math uses direct manipulatives, but it is generally not known that this form of math can be structured by axioms. That is, spatial arithmetic is a formal and as rigorous as symbolic math.

This is ongoing work, 2007-2009....

The bulk of this work is a Mathematica program, the Spatial Arithmetic Calculator. Not yet done.

Here is an NSF proposal (one reviewer said that the proposal was deficient in almost every regard), a Department of Education research proposal, and a slideshow on spatial arithmetic.