The roles of estimation and the commutativity principle in the development of third graders' mental multiplication.

In this training experiment, pretesting identified 36 third graders (mean age = 8 years 5 months, SD = 4 months) with negligible mastery of multiplication combinations involving factors from 3 to 9. Participants were randomly assigned to 2 groups, which practiced different subsets of combinations, and were then retested. The results were inconsistent with R. S. Siegler's (1988) proposal that item-specific computational practice is necessary to promote changes in error patterns and combination mastery. The results were consistent with the hypotheses that children devise increasingly flexible and accurate estimation strategies and use relational knowledge such as the commutativity principle to master combinations. Because even fast mental-arithmetic errors and correct responses may be due to strategies other than retrieval, researchers need to craft ways of distinguishing between retrieval and nonretrieval strategies.     

Fostering Taiwanese preschoolers’ understanding of the addition–subtraction inverse principle

The present research involved gauging preschoolers’ learning potential for a key arithmetic concept, the addition–subtraction inverse principle (e.g., 2 + 1 − 1 = 2). Sixty 4- and 5-year-old Taiwanese children from two public preschools serving low- and middle-income families participated in the training experiment. Half were randomly assigned to an experimental group; half, to a control condition. Participants were tested for an understanding of inversion before and after intervention. One-third of the 5 year olds from both groups performed at the marginally or reliably successful levels before the intervention, and three quarters of them did so in the posttest. Only one of the 4 year olds was marginally successful before the intervention and 4 year olds in the experimental group somewhat benefited from the intervention. Significant social class effect were evident.     

A commentary on Chen and Campbell (2017): Is there a clear case for addition fact recall?

Psychonomic Bulletin & Review - In their review of Uittenhove, Thevenot and Barrouillet (Cognition, 146, 289–303, 2016), Chen and Campbell (Psychonomic Bulletin & Review, 2017...     

Fostering First Graders’ Fluency With Basic Subtraction and Larger Addition Combinations Via Computer-Assisted Instruction

Achieving fluency with basic subtraction and add-with-8 or -9 combinations is difficult for primary grade children. A 9-month training experiment entailed evaluating the efficacy of software designed to promote such fluency via guided learning of reasoning strategies. Seventy-five eligible first graders were randomly assigned to one of three conditions: guided subtraction (e.g., If 5 + 3 = 8, then 8 – 3 is 5), guided use-a-10 (e.g., If 10 + 7 = 17, then 9 + 7 is 16), or an unguided-practice condition for 30-minute sessions twice a week for 12 weeks. An ANCOVA revealed that at the delayed posttest, the guided-subtraction group outperformed both comparison groups on unpracticed subtraction combinations. Analyses of gains in slow but appropriate use of reasoning and decreases in inefficient strategy use indicated that both types of guided training promoted the learning of a targeted reasoning strategy.     

Assessing Young Children's Number Magnitude Representation: A Comparison Between Novel and Conventional Tasks

Previously, researchers have relied on asking young children to plot a given number on a 0-to-10 number line to assess their mental representation of numbers 1 to 9. However, such a (“conventional”) number-to-position (N-P) task may underestimate the accuracy of young children's magnitude estimates and misrepresent the nature of their number representation. The purpose of this study was to compare young children's performance on the conventional N-P task and a “modified” N-P task that is more consistent with a discrete-quantity view of number and with measures of theoretically related mathematical competencies. Participants (n = 45), ranging in age from 4;0 to 6;0, were administered both versions of the N-P task twice during 4 sessions in 1 of 2 randomly assigned and counterbalanced orders. Between and within conditions, children were significantly more accurate on the modified version than on the conventional task. The results indicate that the conventional task, in particular, may be confusing and that several simple modifications can make it more understandable for young children. However, when performance on theoretically related number tasks is taken into account, both the conventional and the modified N-P tasks appeared to underestimate competence.     

A reexamination of mental arithmetic models and data: A reply to Ashcraft

In a review of the chronometric literature, M. Ashcraft (1982, Developmental Review, 2, 213–236) concludes that adults store each basic arithmetic fact in a table-like retrieval network. In my commentary (1983, Developmental Review, 3, 225–230), I argued that procedural knowledge (stored rules, principles, or heuristics) might be a cognitively more economical basis for generating many number combinations. In this paper, I draw an analogy between this alternative model of number fact representation and how computers efficiently reconstruct arithmetic combinations, note that the research findings do not clearly support any one model of mental arithmetic, and attempt to address Ashcraft's (1983, Developmental Review, 3, 231–235) criticisms of my model.