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.

Svenson, O. & Hedenborg, M.‐L. 1980. Strategies for Solving Simple Subtractions as Reflected in Childrens's Verbal Reports. Scandinavian Journal of Educational Research 24,157‐172. Children, 9‐11 years old, solved subtraction problems in the form of M‐N = . . . , where 0 ≤S M ≤ 13, 0 ≤ N ≤ 13, and N ≤ M. Following the answer of a problem, the child gave a verbal report about the thought processes used to arrive at the solution. The analysis of the retrospective verbal reports started from an earlier presented model based on analyses of reaction times, and resulted in a new and more detailed process model. The verbal reports indicated that the answers were obtained either by direct retrieval from memory or in a reconstructive memory process. When problems where M=N, N=0 or N=l were excluded from the analysis, about #fr3/4> of the remaining answers were retrieved. The most frequently used reconstructive strategies were one‐unit step counting down (e.g., 7‐3: 7, 6, 5, 4), counting down greater steps than one with the number 10 as reference (e.g., 13‐7:13,10, 7), one‐unit counting up (e.g., 7‐4: 4,5, 6, 7), and solutions with reference to memory storage of decompositions of even numbers into two equal addends, ties (e.g., 9‐4; /8=4+4/, 8‐4=4, 9‐4=5). Finally, the deterministic nature of earlier models for simple arithmetic problem solving was altered in the new process model, resulting in a probabilistic description of the choices among the cognitive processes used for solving simple subtractions.

Svenson, O. & Hedenborg, M. L. 1980. Counting processes in simple addition. Scandinavian Journal of Education Research 24,93‐104. Verbal protocols and response latencies were used in this study to investigate the thought processes used by children solving simple arithmetic problems (I + J = , where 0≤I≤13 and 0< J ≤13). To specify, the verbal reports were used to classify the solutions into one of two main groups: (a) answers retrieved from long term memory and (b) answers involving active manipulation of the problem in working memory (reconstructive solutions). The response latencies in each of these groups were analyzed separately for each child. As expected, latencies for retrieved answers were shorter than for reconstructed solutions. The retrieval from long term memory of the answer to a problem with the smaller number first (e.g., 3+4) required about 0.1 sec longer time than when the numbers were in the reverse order (4 + 3). Grouping latencies according to verbal protocols made it possible to refine the analysis of the response latencies. To exemplify, counting strategies with greater units than one (e.g., 6 + 4: 6, 8, 10) were identified and parameters describing the cognitive processes listed.

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.

Svenson, O., Hedenborg, M‐L. & Lingman, L. 1976. On Children's Heuristics for Solving Simple Additions. Scand. J. educ. Res. 20, 161‐173. Children aged 9‐11 years solved all 105 additions of two addends with a sum smaller than 14. After each of 50 problems (where the addends were unequal and none of which was 1 or 0), verbal reports were given by each child about his way of handling the numerical information to arrive at the solution. The reports indicated that the answers were obtained either by direct retrieval from memory (in about one‐third of the cases) or in reconstructive memory processes, of which almost all (94%) started with the greater addend. Most answers indicating reconstructive processes were classified as one of three major types of heuristics or strategies: 1) one‐step counter strategy (57% of the reconstructive reports), 2) counter strategies with units greater than 1 counted (25%), and 3) tie reference heuristics (12%). An earlier presented cognitive model based on analyses of latencies was improved on the basis of these results.