> Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). This is not a "crutch", beyond the extent to wh

> Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system. Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying. Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together. 18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108 vs. 18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108 Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.