6.5.5 Multiplicative operators
· A study by LeFevre, Bisanz, Daley, Buffone, Greenham, and Sadesky
gave subjects single-digit
multiplication problems to solve and then asked them what procedure they had used to solve the
problems. In 80% of trials subjects reported retrieving the answer from memory; other common
solution techniques included: 6.4% deriving it (e.g., 6×7 6×6 + 1), 4.5% number series (e.g.,
3×5 5, 10, 15), and 3.8% repeated addition (e.g., 2×4 2 + 2).
· A study by Campbell
measured the time taken for subjects to multiply numbers between two and
nine, and to divide a number (that gave an exact result) by a number between two and nine. The results
showed a number of performance parallels (i.e., plots of their response times and error rates, shown in
, had a similar shape) between the two operations, although division was more difficult.
These parallels suggest that similar internal processes are used to perform both operations.
· A study by LeFevre and Morris
supported the idea that multiplication and division are stored in
separate mental representations and that sometimes the solution to difficult division problems was
recast as a multiplication problem (e.g., 56/8 as 8×? = 56).
Adults (who spent five to six years as children learning their times tables) can recall answers to single digit
multiplication questions in under a second, with an error rate of 7.6%.
The types of errors made can be
put into the following categories
· Operand errors. One of the digits in an operand being multiplied is replaced by another digit. For
example, 8×8 = 40 is an operand error because it shares an operand, 8, with 5×8 = 40.
· Close operand errors. This is a subclass of operand errors, with the replaced digit being within ±2 of
the actual digit (e.g., 5×4 = 24). This behavior is referred to as the operand distance effect.
· Frequent product errors. The result of the multiplication is given as one of the numbers 12, 16, 18, 24
or 36. These five numbers frequently occur as the result of multiplying operands between 2 and 9.
· Table errors. Here the result is a value that is an answer to multiplying two unrelated digits, than those
actually given (e.g., 4×5 = 12).
· Operation errors. Here the multiplication operation is replaced by an additional operation (e.g.,
4×5 = 9).
· Non-table errors. Here the result is a value that is not an answer to multiplying any two single digits;
for instance, 4×3 = 13, where 13 is not the product of any pair of integers.
Operand family value
· · ·
Figure 1143.1: Mean percentage of errors in simple multiplication (e.g., 3×7) and division (e.g., 81/9) problems as a
function of the operand value (see paper for a discussion of the effect of the relative position of the minimum/maximum
operand values). Adapted from Campbell.
January 29, 2008