Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a "LT approach"--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. The present report serves to address…

Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a "LT approach"--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. The present report serves to address…

Ginsburg (1977) observed that children typically develop surprisingly powerful informal (everyday) knowledge of mathematics and that mathematical learning difficulties often arise when formal instruction does not build on this existing knowledge. By using meaningful analogies teachers can help connect new formal instruction to students' existing…

Six widely used US Grade 1 curricula do not adequately address the following three developmental prerequisites identified by a proposed learning trajectory for the meaningful learning of the subtraction-as-addition strategy (e.g., for 13-8 think "what + 8 = 13?"): (a) reverse operations (adding 8 is undone by subtracting 8); (b) common…

Addressed are four key issues regarding concrete instruction: What is concrete? What is a worthwhile concrete experience? How can concrete experiences be used effectively in early childhood mathematics instruction? Is there evidence such experiences work? I argue that concrete experiences are those that build on what is familiar to a child and can…

Addition strategies used by 36 kindergarten children were examined. Children were given written stimuli (such as "2+5" and "3+7") during two sessions taking place a week apart. Results indicated that once children came to rely on mental addition strategies, they often quickly invented more economical procedures to compute sums. Also confirmed was…

Three models have been proposed to account for the relationship between the principle of commutativity and the development of more economical addition strategies, which disregard addend order. In the first and second models, it has been proposed that either discovery or assumption of commutativity is a necessary condition for the invention of…

Use of the commutativity, addition-subtraction complement, and N+1 progression principles was studied by interviewing 54 capable pupils in grades 1-3. Commutativity was used extensively at each grade, while the addition-subtraction principle to solve subtraction varied across grades, and the N+1 pattern was seldom used. (MNS)

The effects of problem size on judgments of commutativity by 51 moderately and mildly retarded students were investigated. Results indicated that many retarded students who are given computational practice recognize the general principle that addend order does not affect the sum. (Author/DB)

Two studies pursued theory that knowledge of addition combinations facilitates learning of subtraction combinations. Study 1 involved 25 kindergartners and 15 first graders in a gifted program; study 2 involved 21 first graders in a regular program. Participants responded to two pairs of problems. Findings revealed that the complementary…

Kindergartners appeared to differ in their readiness to use a concrete counting strategy for addition. Many persisted in counting with objects. Mental counting strategies were sequenced. (MNS)

Analyzes error patterns of kindergartners' mental addition task. Eight weeks of computational practice affected the errors of unpracticed combinations on a retest. Seven of 10 children mastered previously unknown combinations involving zero by learning a relationship rather than the practice and memorization of individual facts. (Author/YP)

A training experiment with 30 mentally handicapped children tested two models of mental-arithmetic development. Children's initial responses to a mental-addition task appeared to be the product of a mechanical prescription. Errors appeared to be more errors of method than of recall. (MNS)

An experimental group (n=13) and a control group (n=15) of children with mental retardation were both shown a basic concrete counting procedure. Over six months, the experimental group was given regular opportunities to practice computing sums. Many of them invented calculational short cuts. Results suggest that children with mental retardation…

Limitations of the retrieval strategy dimension of Siegler's (1982, 1984) distributions-of-associations model of young children's estimation of sums are delineated, an alternative model is described, and findings of two studies designed to test key assumptions of the models are reported. In Study 1, kindergarten children with normal IQ and no…