Mental math is manipulating numbers in the mind to turn unfamiliar problems into familiar ones. For the number facts, it is based on certain rules and mathematical principles like the following for M and N any numbers:

• “Zero” rules, that N + 0 = N, N – 0 = N, N – N = 0, and N x 0 = 0
• “One” rules, that N x 1 = N, N ÷ 1 = N, and (N + 1) – N = 1
• Commutative Properties of Addition and Multiplication, that M + N = N + M and M x N = N x M, respectively
• “Nine” rule, that 9+N = 10+(N–1), for example, 9+8 = 10+(8–1) = 10+7 = 17

Other rules are the “Make a 10” rule and the rules that stem from knowing the doubles, that 1 + 1 = 2, 2 + 2 = 4, 3 + 3 = 6, ..., 9 + 9 = 18:

• “Make a 10” rule where, say, 7+4 is thought of as 7+(3+1) = (7+3)+1 = 10+1 = 11
• “Double Plus One” rule, for example, 5+6 = 5+(5+1) = (5+5)+1 = 10+1 = 11
• “Double Less One” rule, for example, 5+4 = 5+(5–1) = (5+5)–1 = 10–1 = 9

A bonus in teaching mental math is that it shows that mathematics is understandable.

In general, research supports mental math over drill and practice in the areas of learning, retention, and transfer. Unfortunately, mental math is limited in its coverage of the number facts. The rules for mental math that are reasonably easy to teach, like the ones listed above, yield only some—not all—of the 390 number facts. This deficiency is noted passionately by Ashcraft (1985), although in defense of drill and practice for teaching the number facts:

“What are the other rules? What rule, for instance, yields the answer 13 to the problem 8+5? What rule or procedure generates the 56 for 8x7? The 7 for 4+3? The 36 for 9x4? To suggest that ’some subtraction combinations may be efficiently reconstructed from their addition counterparts (e.g., 10–7 is 3 because 7+3 is 10)’ is completely evasive—what rule generates 7+3 = 10? This is not a minor, nagging incompleteness, ...—it is an absolutely central flaw.” —Mark Ashcraft, “Is it farfetched that some of us remember our arithmetic facts?” Journal for Research in Mathematics Education 16 (2): 99-105, 1985.