These four chapters describe studies of using models for integer addition and subtraction. The models draw principally on the two grounding metaphors of object collection and motion along a path. A strength of all chapters is detailed analysis of how the models are and can be implemented and how they influence student's learning. Together the…

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…

The crucial role of teachers in introducing integers to children is highlighted in chapters 8-10, comprising this section. The three chapters discuss (prospective) teachers' conceptions of integer equations, of children's thinking about integer expressions, and of the role of some didactical models used in teaching integer addition and…

Maintains that findings showing numerical computation abilities in infants are considerably more robust and consistent than Wakeley, Rivera, and Langer suggest. Asserts that all the interim replication attempts have successfully replicated Wynn's original findings. Discusses possible reasons for failure to replicate in Wakeley et al. experiments.…

Presents 2 complementary techniques for promoting success in computer-based exploration environments for arithmetic instruction. Semantic constraints are explained, heuristic guidance is discussed, use of the techniques for subtraction instruction is described, and field testing with students in grades 1-3 is briefly discussed. (25 references)…

Proposes a four-category classification of semantic structures of one-step word problems involving multiplication and division: forming the n-th multiple of measures, combinatorial multiplication, composition of operators, and multiplication by formula. This classification is compatible with semantic structures of addition and subtraction word…

Asserts that findings on whether young infants look longer at incorrect addition and subtraction have been inconsistent or negative. Hypothesizes that imprecise ordinal calculating with very small numbers of objects develops in late infancy and that precise calculating develops in early childhood. (Author/KB)

What is meant by "normal science" is described, the first three steps in the "route to normal science" are outlined, and an example is given in which the trend toward research consensus is apparent. This involves current work on learning addition and subtraction. (MNS)

Several ideas for questions and settings for interviews are presented. It is noted that although subtraction is used as an example in each case, most of the material can be easily adapted to the other basic operations. Many problems are noted with standard diagnostic tests and use of them is discouraged. (MP)

Possible stages in young children's use of the number line as an aid are described. The practice of using number lines in test items is discussed and criticized. Finally, the wider role of the number line diagrams in mathematics teaching is outlined. (MNS)

One teacher's struggle with conveying a concrete realization of the subtraction algorithm to students leads to a discussion of elementary mathematics instruction in general. (MP)

Discusses the contributions in each chapter of the book, highlighting the textbook analyses for each arithmetic topic presented. (MDH)

This article argues that the reason the mathematics performance of students with mild disabilities is so limited is the mathematics. Using illustrations from subtraction, the article proposes number sense be a primary goal and that within the realm of number sense, big ideas be included among the dependent variables. (Contains references.)…

Analyzes instructional theory requirements and suggests that structural learning theories meet these requirements. Essentials of such theories are illustrated in the context of designing an instructional unit on subtraction. Structural learning theories are then described more generally and contrasted with cognitive theories. Implications for…

Provided is an examination of the methodology used to study the problems of learning addition and subtraction skills used by developmental researchers. The report has sections on categories of theory and their methodologies, which review: (1) Behaviorist, Neo-Behaviorist and Piagetian Theories; (2) the Behaviorist and Piagetian Paradigms; (3)…