Can preschool children invent addition algorithms?

Abstract

This research is concerned with patterns of reaction times that emerge when a child is taught a specific problem-solving procedure and then given extensive practice over many weeks. Preschool children who knew how to count but were unacquainted with arithmetic were taught a simple algorithm for solving single-digit addition problems and were then given extended practice. A subject performing this algorithm would generate reaction times proportional to the sum of the addends. The main finding was that, at the end of the extended practice phase, data of many subjects were best fitted by a different model predicting reaction times proportional to the minimum addend. This implies that these children are no longer using the algorithm they were originally taught. It is also interpreted as suggesting that they have invented a more efficient procedure.We are concerned in this article with the relationship of relatively skilled performance on tasks to the processes used during early phases of acquisition. In the typical information-processing experiments, performance is highly skilled and overlearned. The processes postulated are sufficient to account for skilled performance, but little is said regarding how such processes are acquired or how performance changes over time. There is no a priori reason, however, to assume that the processes used in skilled performance are identical to the processes used to solve the same problems in a relatively unpracticed state. Here we address the question of cognitive skill acquisition directly, by modeling the performance of individuals at different times in the course of their acquiring skill in procedures for solving simple addition problems. We consider in particular the relationship between the specific procedures taught to beginners

Why is 9+7 harder than 2+3? Strength and interference as explanations of the problem-size effect

“…In experiments with both children and adults and with addition and multiplication, it has been found that reaction times increase linearly with the magnitude of the minimum digit (Groen & Parkman, 1972). It has been found that even children as young as 4 use the min model to solve addition problems (Groen & Resnick, 1977).…”

A componential model for mental addition.

“…For example, if a subject consistently set his or her mental counter to the larger addend and then incremented the smaller, RT should be a linear function of the smaller addend, or MIN; also, the estimated regression slope for the MIN variable would reflect the time required to increment the mental counter. Groen and Parkman (1972;Groen & Resnick, 1977;Parkman & Groen, 1971) found that RT for simple addition problems was predicted better by the MIN structural variable than by any of the remaining four structural variables for both adults and children. Adults, however, had an estimated regression slope that was V2o the size of the slope parameter estimated on data from first graders; this indicates much greater proficiency by adults.…”

Des modèles de la compréhension

“…Car, tel que souligné par Vergnaud (1982), on ne pourrait ignorer la contribution possible que la symbolisation pourrait apporter au processus d'abstraction. La possibilité d'une telle contribution peut être inférée à partir d'une expérience conduite par Groen & Resnick (1977) dans laquelle ils découvrirent, qu'en présence d'objets concrets et de fiches symbolisant l'arithmétique, des enfants de quatre ans pouvaient éventuellement, dans des situations additives, développer par eux-mêmes la stratégie de compter à partir du plus grand ensemble.…”