Solving Math Problems Approximately: A Developmental Perspective

Abstract

Although solving arithmetic problems approximately is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems approximately, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults’ ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the approximated calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an approximate number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner.

Approximation processes in arithmetic in old adulthood

“…The young adults in the present study and in past research [ 14 , 15 ] tended to use the approximated calculation strategy more often for close reference numbers and the sense of magnitude strategy for the far reference numbers. This pattern was viewed as evidence for an adaptive strategy choice process as it involves using the more time-consuming and attention demanding approximated calculation strategy when it is needed and when the sense of magnitude strategy cannot guarantee a correct response.…”

“…The stimuli set for the estimation task was composed of 40 2x2 multiplication problems taken from Ganor-Stern [ 15 ]. The exact answers were in the range of 768–8178.…”

“…Ganor-Stern [ 15 ] has looked at the development of performance in this task from childhood to adulthood. With age, there is an improvement in speed and in accuracy.…”

Cognitive predictors of children's development in mathematics achievement: A latent growth modeling approach

“…Until quite recently most research in this field focused primarily on understanding children's computational estimation skills from 10 years of age and above (Dowker, ; Ganor‐Stern, ; LeFevre, Greenham, & Waheed, ; Lemaire, Lecacheur, & Farioli, ). Actually, there are large individual differences even in adults’ computational estimation explained by the large variability in the types of strategies that they employ (Dowker, ).…”

“…Actually, there are large individual differences even in adults’ computational estimation explained by the large variability in the types of strategies that they employ (Dowker, ). Estimation strategies become more sophisticated with development, for example, Ganor‐Stern () suggests that at 10 years of age the most common strategy employed is the “sense of magnitude”, that is, an intuitive sense of magnitude without any calculation, whereas adults use the “approximate calculation strategy”, that is, rounding either one or two operands, calculating the result, and comparing that to the reference number. But, at 10 years of age estimation has already been partially established.…”