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…

How best to promote fluency with basic sums and differences is still not entirely clear. Some advocate a direct approach--using drill to foster memorization of basic facts by rote. Others recommend an indirect approach that first involves learning reasoning strategies. The purpose of the present study was to evaluate the efficacy of 2…

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…

Subtraction combinations are particularly challenging for children to learn (Kraner, 1980; Smith, 1921; see Cowan, 2003, for a review). This study examines whether the group of children receiving the "experimental subtraction-as-addition" training outperform the "control" group, which received training on a different reasoning…

Little research has focused on an informal understanding of subtractive negation (e.g., 3 - 3 = 0) and subtractive identity (e.g., 3 - 0 = 3). Previous research indicates that preschoolers may have a fragile (i.e., unreliable or localized) understanding of the addition-subtraction inverse principle (e.g., 2 + 1 - 1 = 2). Recognition of a small…

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…

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 informal subtraction strategies that children develop are discussed in detail, with the role of the counting-down strategy described. Problems and remedies for difficulties caused by counting backward and the double count are also presented. (MNS)

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…

A model of subtraction development and the computing difficulties and research issues suggested by the model are outlined. Demands of simultaneous processes, difficulties with informal subtraction, and the impact on the counting-up procedure are discussed. (MNS)