The link below will take you to a short movie (QuickTime Required). You’ll see a series of seven easy addition problems, which will flash by at the rate of one every two seconds. Your job is to solve the problems as quickly as possible (ideally, you should say the answers out loud).
Apart from the possibility that you might have a better memory for some math facts than others, were any of the problems easier for you?
If you’re like most people, you probably responded faster when a problem was repeated, as was the case with 3 + 5. And since the order of operations doesn’t matter in addition, you might have also had an easier time with 7 + 2 since you had seen 2 + 7 earlier. You might have even thought you saw 7 + 2 twice — as both Nora and I did when we tried this demo (yes, it happened to me even though I’m the one who made the demo!).
If seeing “2 + 7” earlier helps you solve “7 + 2” later, then maybe some other mathematical operations can be facilitated in similar ways. In fact, there has been a good deal of research on multiplication and division problems, with a bit of a startling result: While seeing “9 × 4” helps you solve “4 × 9” later, seeing “36 ÷ 9” doesn’t seem to help with “36 ÷ 4”. Aren’t these all equivalent sorts of problems?
Timothy Rickard and others have come up with a model that explains the difference in multiplication and division problems: The multiplication problems have identical elements — the same numbers, just in a different order. The division problems actually involve different numbers; only when the answer is known are the numbers the same. But Jamie Campbell has found that division sometimes is speeded in cases like “36 ÷ 9” and “36 ÷ 4”. Campbell argues that because of the close relationship between these problems and 9 × 4, we do have some advantage solving “36 ÷ 9” when we’ve seen “36 ÷ 4”. But when it comes to addition and subtraction, the rules change somewhat.
While 36 only has a few common factors (2, 4, 6, 9, 18), there are many different numbers that can add up to, say, 12 — every number lower than 12 is a possibility. So practicing “12 – 5” isn’t much help when we try to solve “12 – 7” later on. So Campbell and two other researchers decided to present 48 volunteers with addition and subtraction problems to see whether their model of memory for math facts held up.
The researchers presented every possible addition problem involving digits between 2 and 9 (that’s 28 different problems when “ties” such as 5 + 5 are removed) and the corresponding subtraction problems. They carefully monitored when associated addition and subtraction problems (like 2 + 3 and 5 – 2) appeared (sometimes soon after one another [4-6 problems away], and sometimes separated by as many as 22 problems). The speed of response was carefully tracked.
So, does seeing the same problem again make it easier to solve? What about seeing a similar problem? This graph charts the results:
For both addition and subtraction, seeing the identical problem a second time resulted in significant time savings. For both addition and subtraction, seeing the same problem with only the order of the elements changed (2 + 3 and 3 + 2, or 5 – 3 and 5 -2) also resulted in a significant time savings. But when the problem type changed (2 + 3 and 5 – 2, or 5 – 3 and 2 + 3), there was not time savings.
Unlike in multiplication and division, seeing the associated subtraction problem doesn’t help with an addition problem, and vice versa. And while seeing a previous problem with a different order does help in both addition and subtraction, for subtraction the savings for seeing a problem in a different order is just a half to a third of the advantage for seeing the identical problem.
Campbell’s team argues that this demonstrates that memory for addition and subtraction is managed differently than multiplication and division.
Campbell, J.I., Fuchs-Lacelle, S., Phenix, T.L. (2006). Identical elements model of arithmetic memory: Extension to addition and subtraction. Memory & Cognition, 34(3), 633-647.