The Eighth Principle

  Look for and express regularity in repeated reasoning.

     This principle reads:

 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. 

The 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. 

This principle relates so closely to the last where students are looking for patterns and structure to base their reasoning and explain their mathematical thinking. The investigative or inquiry approach to math leads students to discover patterns and structure in mathematical concepts. 

One example from Thinkmath is that "children can recognize that adding 9 can be simplified by treating it as adding 10 and subtracting 1and that this 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." (Thinkmath)

"Through investigation and discovery, students can use repeated addition to begin their exploration of linear growth and repeated multiplication to extend to geometric growth. Drawing on their observations of geometric properties, students make and test conjectures, develop generalizations, and write formulas."  (Discovering Math) The more children can relate repeated patterns the more connections they will build and the more meaning they will bring to their math.