Children who understand that subtraction is an inversion of addition would be able to quickly and correctly answer “a” if asked to solve the problem, a + b – b.

How much of that understanding of inversion in addition and subtraction transfers to multiplication and division? asks a recent study in *Cognitive Development*.

Not much, find the researchers. Based on this study of 163 students from grades 6, 7 and 8, only a few students used the inversion shortcut when solving multiplication and division problems in the form of d x e ÷ e.

Even by grade 8, children are not using the inversion shortcut in multiplication and division problems as frequently as they do addition and subtraction problems, the researchers report.

### Use of shortcut

Only 16.4% of students chose the shortcut method while 55.2% did the left-to-right computation and 27.4 began doing the computations and realized they could use the inversion shortcut to solve the problem.

“Piaget contended that despite formal instruction on arithmetic operations, children are slow to understand the inverse relationship between addition and subtraction and particularly the inverse relationship between multiplication and division,” write the two researchers from the University of Regina in Saskatchewan.

The study highlights the need for more emphasis on children’s conceptual understanding, they write. It also underscores the complexity in the development of children’s conceptual knowledge

Previous research has found that both adults and children are less likely to use the inversion shortcut on multiplication and division inversion problems than on addition and subtraction inversion problems.

That earlier research also found that the inversion shortcut on one type of problem did not predict use of the shortcut on the other type of problem, the authors report.

The computer-based test in the study comprised 10 multiplication and division inversion problems and 6 standard problems in the form of d x e ÷ f.

Half of the problems in the computer-based test used small numbers under 25 and the others used numbers higher than 25.

### Liberal standard in study

To take into account that individuals may understand the inversion concept but choose not to apply it, researchers had students verbally report their solution procedure after solving each problem on the computer. Two solution approaches to the problem 5 x 4 ÷ 4, for example, were shown on the screen, one illustrating left-to-right computation and the shortcut approach.

The experimenter used the following language in getting a response on whether or not the student approved of each approach: “This girl/boy I spoke to said that when s/he solved this problem s/he first multiplied 5 times 4 and figured out the answer was 20 and then divided 20 by 4 and figured out that the answer was 5. Is that a good way to solve the problem?” Similar language was used to elicit a response on the shortcut approach.

The researchers report that 97.5% of participants approved of the left-to-right strategy as a good way to solve inversion problems and 82.8% approved of the inversion shortcut. Although no grade effects were found for actual use of the shortcut method, researchers say, preference for the inversion shortcut was 56.3%, 39.0% and 58.9% for students in grades 6, 7 and 8, respectively.

Participants gave a number of justifications for why they did or did not approve of the inversion shortcut strategy. Examples of why inversion was a good strategy included: “That way you have an even better chance of getting it right because the answer is right there” and “It’s just easy. They just cancel each other out, so there’s just one answer left, and that’s your answer.”

Examples of why inversion was not a good strategy included: “If you just knock out the numbers you never know if it’s right” and “It’s confusing. I like to do it without thinking, so I do it the same way every time.” Some students thought that the shortcut was a form of cheating and that the only way to solve the problems was to use left-to-right computation. Most students cited accuracy, speed, or general efficiency as the reason they thought the inversion strategy was a good problem-solving procedure.

Of the few participants who did not think it was a good strategy, most stated that the strategy was too confusing. Those students who were more fluent in mathematical operations, based on a fluency task that was also administered to students during the study, were more likely to use the inversion shortcut, researchers report.

### Why so few students used shortcut

There are several reasons why so few students used the inversion shortcut in the multiplication and division problems, the researchers write. When computation is well-learned, it may be harder for students to shift their attention away from fast, automatic processes and focus instead on the pattern in the problems. Another possibility is that the addition-subtraction inversion and multiplication-division inversion are 2 separate schemas and not the same schema.

“If there are two separate schemas, the pattern of development for each may develop independently,” the researchers write.

*“Children’s Understanding of the Inverse Relation between Multiplication and Division,” by Katherine Robinson and Adam Dube, Cognitive Development, Volume 24, 2009, pps. 310-321.*