Mathematical Practice Standard #8

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal…. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. —CCSS

The standard in elementary school

A central idea here is that mathematics is open to drawing general results (or at least good conjectures) from trying examples, looking for regularity, and describing the pattern both in what you have done and in the results.

A multiplication fact-practice exercise might, for example, ask children to choose three numbers in a row (e.g., 5, 6, and 7) and compare the middle number times itself to the product of the two outer numbers. In this example, the two products are 36 and 35. After they do this for several triplets of numbers, they are likely to conjecture a pattern that allows them to multiply 29 × 31 mentally because they expect it to be one less than 30 × 30, which they can do easily in their heads. Seeing that regularity is typically easy for fourth graders; expressing it clearly is much harder. Initial attempts are generally inarticulate until students are given the idea of naming the numbers. A simple non-algebraic “naming” scheme was used above to describe the pattern: the numbers were named “middle” and “outer” and that was sufficient. A slightly more sophisticated scheme would distinguish the outer numbers as something like “middle plus 1” and “middle minus 1.” Then children can state (middle – 1) × (middle + 1) = middle² – 1. The step from this statement to standard algebra is just a matter of adopting algebraic conventions: naming numbers with a single letter like m instead of a whole word like “middle,” and omitting the × sign.

The recognition that adding 9 can be simplified by treating it as adding 10 and subtracting 1 can be a discovery rather than a taught strategy. In one activity—there are obviously many other ways of doing this—children start, e.g., with 28 and respond as the teacher repeat only the words “ten more” (38), “ten more” (48), “ten more” (58), and so on. They may even be counting, initially, to verify that they are actually adding 10, but they soon hear the pattern in their responses (because no other explanatory or instruction words are interfering) and express that discovery from their repeated reasoning by saying the 68, 78, 88 almost without even the request for “ten more.” When, at some point, the teacher changes and asks for “9 more,” even young students often see it as “almost ten more” and make the correction spontaneously. Describing the discovery then becomes a case of “expressing regularity” that was found through “repeated reasoning.” Young students then find it very exciting to add 99 the same way, first by repeating the experience of getting used to a simple computation, adding 100, and then by coming up with their own adjustment to add 99.