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Arithmetic Algorithms

All standard algorithms are taught in Think Math!, but they are taught as the culmination of learning of a particular arithmetic operation, not as the introduction.

Elementary school arithmetic is often taught and learned as a set of rules applied to a memorized look-up table. To add 28 to 35, I line up the numbers a certain way, read the ones column, "look up" (in memory) 8 + 5, dissect the answer by writing one part of it on the bottom and "carrying" the other to the top, switch attention to the tens column and follow the same procedure there. To multiply a two-digit number by 5, I "look up" 5-times-the-ones-digit, write (and carry, if necessary), and so on.
The advantage of an algorithm is that it does not require thinking about the computation. Given a sufficient store of memorized facts, the algorithm (process) suffices for any computation. That is a huge help for lengthy computations that would be too taxing to perform mentally. But this advantage for performing "professional computation" (the work of bookkeepers before technology) is a disadvantage in the initial process of learning to compute, and a disadvantage to algebra, in which students must use quite consciously the properties of the operations that the algorithms of arithmetic have largely hidden from them.
In elementary school, the commutative property is often taught as a way of reducing memory load: 7 x 8 and 8 x 7 become one thing to know rather than two. But the other algebraic properties -- particularly the associative and distributive properties and the nature and utility of the inverses -- can often feel like "just more stuff to know" rather than genuinely helpful.
At some level, students already have, from a very early age, the logic that these properties encode (see, e.g., Goldenberg, Mark, and Cuoco, 2010)[1], but early arithmetic training tends neither to rely on nor to maintain that logic, and so many children "forget" it in the context of computation.