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…

While preschool-aged children display some skills with quantitative relations, later learning of related fraction concepts is difficult for many students. We present two studies that investigate young children's tendency of Spontaneous Focusing On quantitative Relations (SFOR), which may help explain individual differences in the development…

External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one…

Predictions of the Identical Elements (IE) model of arithmetic fact representation (Rickard, 2005; Rickard & Bourne, 1996) about transfer between arithmetic facts were tested in primary school children. The aim of the study was to test whether the IE model, constructed to explain adult performance, also applies to children. The IE model predicts…

This study builds on two lines of research that have so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. We investigated the development with age of both kinds of erroneous solution methods. We gave a test containing missing-value…

This study examines presentations of the distributive property (DP) in two widely used U.S. elementary text series and one main Chinese text series along three dimensions: problem contexts, typical problem types within each problem context, and variability in using the DP. In general, the two U.S. texts were found to resemble each other but to…

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students' understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The…

Looking away from an interlocutor's face during demanding cognitive activity can help adults and children answer challenging mental arithmetic and verbal-reasoning questions (Glenberg, Schroeder, & Robertson, 1998; Phelps, Doherty-Sneddon, & Warnock, 2006). While such "gaze aversion" (GA) is used far less by 5-year-old school children, its use…

The aim of the study was to analyze the development of children's adaptive expertise in computing sums and differences up to 100. We defined the adaptive nature of children's strategy choices on the basis of problem (addition, subtraction), achievement, and strategy performance (accuracy, speed). Sixty-nine 2nd graders of high, above-average, or…

A schema based view of addition development is compared with Siegler's latest strategy-choice model, which includes an addition goal sketch (a basic understanding of "the goals and causal relations" of addition; Siegler & Crowley, 1994, p. 196). This metacognitive component in the latter model is presumed to develop as a child practices a basic…

Simple arithmetic word problems are often featured in elementary school education. One type of problem, "compare with unknown reference set," ranks among the most difficult to solve. Differences in item difficulty for compare problems with unknown reference set are observed depending on the direction of the relational statement (more than vs. less…

We examined the relation between self-explaining and encoding among kindergartners. For 5 days, children (n = 27) took turns solving addition problems with an adult expert who always used an advanced addition strategy. During the game, children explained the expert's answers (Explain-Expert), explained their own answers (Explain-Novice), or did…

Previous research (e.g., De Bock, 2002) has shown that--due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education--many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of…

This study investigated the developmental paths of Japanese Grade 1 students' understanding of quantities through the examination of their addition solution methods over the school year period. The individual students exhibited a wide range of experiences with and knowledge of addition from the beginning to the end of the school year. Students…

This study explored first graders' fraction learning when the teacher elicited and built on children's informal knowledge of fractions during discussions of equal-sharing tasks; the study focused on key classroom interactions resulting in cognitive change. Findings suggest that thinking about fractions is influenced by the structure of the child's…