Mathematics is dominated by strings of tokens. Although plenty of math symbols are containers –– like the square root sign √... , matrix brackets [...], and the integration sign ∫...dx –– computation is universally string processing.

In kindergarten, kids learn to count and to add using manipulable concrete objects -- like Cuisenaire rods and balance beams. I got interested in creating manipulable objects that taught the concepts of elementary algebra.

The first piece comments on a manipulative logic tool that unfortunately works for only the simplest logic problems.

A learning tool must address both notation and manipulation, here both the writing of logic and the proof of logical tautologies. Block logic and spatial algebra use boundary math to provide a solution.

Bill Winn and I wrote an NSF proposal about a concrete approach to algebra. Bill was interested in teaching math in virtual environments, and the spatial algebra was sort of a concrete-but-not-really approach. I've included comments on the review feedback to highlight the NSF perspective. The NSF pieces have some redundancy.