Errors in subtractions: judged and observed frequencies

In a first study 10 adults, aged 24-44 years, solved all 105 subtraction problems in the form M - N = , where 0 < or = M < or = 13, 0 < or = N < or = 13 and N < or = M. Each participant solved every problem 10 times and in total there were 10 500 answers. Answers, response latencies and errors were registered. Retrospective verbal reports were also given, indicating how a solution was reached: (1) via a (conscious) reconstructive cognitive process or (2) via an (unconscious) reproductive (retrieval) process. The participants made 291 errors (2.8%) when solving the subtractions in study 1. The rate of self-correction was very high, 92%. In a second study 27 undergraduate students estimated overall error rates, including self-corrected errors for the 105 subtraction problems used in the first study. Judged and actual error rates were compared. The participants systematically underestimated error rates for error prone problems and overestimated error rates for error free problems. The participants were fairly accurate when they predicted problems that were most error prone, with a hit rate of 0.67 for the (18) problems predicted as the most error prone ones. In contrast, predictions of which problems were error free were very poor with a hit rate of only 0.20 of the problems predicted as error free really having no errors in study 1. The correlation between judged error rates and frequencies for actually made errors was 0.69 for answers belonging to reconstructive solutions. In contrast, there was no significant correlation between judged and actual error rates at all for retrieved solutions, possibly reflecting the inaccessibility to consciousness of quick retrieval processes.     

Strategies used by children when solving simple subtractions

Ten children, 9–11 years old, solved all subtraction problems in the form of M?N=…, where 0 ? M ? 13, 0 ? N ? 13 and M ? N. The solution times were analysed and used for the formulation of a process model for subtraction. The model involves memory processes on two different levels, called reproductive and reconstructive respectively. When M=N, N=1, and M=2N the answers were quickly retrieved in reproductive memory processes. The reconstructive processes were found to be analogous to one of two counting procedures, viz. counting up and counting down. In general, the counting process starts either on N (when M < 2N) and counts up to reach the answer, or on M (when M > 2N) and counts down to reach the answer. This may reflect an effort to minimize the number of steps to be counted. However, when M > 10 and N < 10 a problem is always solved in a decrementing counting process. When M=10 many subtractions are solved in a reproductive memory process and the number 10 is also important as a point of reference for solving subtractions when M > 10 and N < 10.