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…

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…

A 9-month training experiment was conducted to evaluate the efficacy of highly and minimally guided discovery interventions targeting the add-1 rule (the sum of a number and one is the next number on the mental number list) and doubles relations (e.g., an everyday example of the double 5 + 5 is five fingers on the left hand and five fingers on the…

The primary purpose of the study was to determine if computer-based training programs promoted fluent and flexible use of reasoning strategies to solve addition problems using different tasks. Specifically, does participation in strategy training result in the fluent application of the target strategy on a traditional mental arithmetic task? Does…

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…

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…

Shows how to relate formal instruction to students' informal knowledge. Indicates that the approach (which is applicable across topics, ability, and age) may help children feel more in control of their work and feel better about themselves and mathematics. (JN)

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)

Four three-year-old girls were studied in seven sessions during a three-month period for their performance on rote and object counting tasks. Two exhibited behavior consistent with underlying stable-order and uniqueness principles. (MNS)

Eleven retarded children with mental ages (MA) less than 4.5 years were examined to evaluate whether there is a critical MA for learning basic counting principles and how an understanding of counting develops. Results are discussed in terms of individual needs. (MNS)

Use of manipulatives is neither a sufficient nor a necessary condition for meaningful learning. Provides some incidents in support of the argument. Lists 14 references. (YP)

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)

Mastering the basic number combinations involves discovering, labeling, and internalizing relationships, not merely drill-based memorization. Counting procedures and thinking strategies are components, and it may be that using stored procedures, rules, or principles to quickly construct combinations is cognitively more economical than relying…

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)