Skills not covered by manipulatives

Note that the list below excludes manipulatives that could be made to demonstrate visual proofs, as those would bloat the list. Further, the vast majority of such manipulatives have only a single use, and so it's better to demonstrate them using inexpensive paper cutouts, etc.

• I have yet to see a single use of manipulatives for teaching arithmetic with square roots.
• I have yet to see a single use of manipulatives for teaching anything involving imaginary numbers.
• The use of algebra manipulatives is limited to polynomials with terms of degree 3 or less. A term with degree 4 or higher would require 4-dimensional space. We could project higher-dimensional spaces onto 3-dimensional space via computer program, but it might be the case that thinking about cuboids in higher-dimensional space is more difficult than doing algebra as usual.