This notebook presents tables for addition and subtraction that have a better use of the place value system. The method is already used in Holland for addition in levels but this notebook extends for addition in differences and subtraction in levels and differences. The method is only intended for an intermediate stage in teaching before addition and subtraction are mentally fully automated. For example 99 + 21 can be added per digit position. To keep digits in the range [0, 9] we remove underflow or overflow. We work from right to left, since the numbers come from India and Arabia. Then we get 99 + 21 -> {9, 9} + {2, 1} = {11, 10} = {11, 10} + {1, -10} = {12, 0} = {0, 12, 0} + {1, -10, 0} = {1, 2, 0} -> 120. Compared to existing methods: (1) This method does not change the original sum. (2) The workflow is into a single direction. (3) Allowing positions to have values outside the [0 , 9] range focuses attention upon the place value. (4) There is a unity of approach to both addition and subtraction. The method fits within the US Common Core when we tell kids that a step with {1, -10} represents the subtraction 1 * 10 - 10 * 1 = 0. It would be more fundamental to adapt the curriculum for negative numbers though. Routines to create such tables for addition and subtraction are available in a package. Obviously pupils in elementary school must master the method by hand, but the package allows for examples and checking.

This is a Mathematica notebook that includes a package in an initialisation cell. See the PDF at https://zenodo.org/record/1241350