Clearly, many children in the United States have difficulty learning place-value skills and concepts, and the teaching of these fundamental competencies needs to improve. Fuson correctly concludes that her research (Fuson, 1986, 1988; Fuson & Briars, 1990) indicates that substantial improvement over traditional textbookbased instruction is possible. However, existing research does not clearly address the issues of how place-value instruction should be changed and when it should be introduced into the curriculum (Baroody, in press).

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- Author or Editor: Arthur J. Baroody x

### Arthur J. Baroody

This study was undertaken to reevaluate whether preschoolers understand that counting in different orders yields the same number before they learn tag-reassigning skills.

### Arthur J. Baroody

A model of subtraction development and the computing difficulties and research issues suggested by the model are outlined. It is argued that, in order to mentally compute the differences for written, symbolic problems such as 5 − 3 = □, children first use a counting-down procedure because counting down is consistent with their informal concept of subtraction as “take away” and represents a logical extension of their mental procedure for (*N* − 1) problems. Counting down requires an ability to count backward *while* keeping track of the number of backward steps. The demands of the simultaneous processes help to explain the difficulty of subtraction relative to addition, the difficulties some children have with informal subtraction, and why—as larger problems are introduced—children tend to supplement counting down with a counting-up procedure.

### Arthur J. Baroody

Many children with learning handicaps have difficulty learning to write numerals. To be effective, remedial instruction must redress the real problem. Otherwise, students (and teacher) may suffer unnecessary frustration. What, then, is the source of numeral-writing difficulties? And what can be done to teach or remedy this skill effectively?

### Arthur J. Baroody

The thesis is advanced that children do not learn and store basic number combinations as so many separate entities or bonds (as hundreds of specific numerical associations) but as a system of rules, procedures, and principles as well as arbitrary associations. In this view, “mastering the basic facts” largely involves discovering, labeling, and internalizing relationships--processes encouraged by teaching thinking strategies. Moreover, internalized rules, procedures, and principles may become routinized and may help to account for the efficient production of number combinations in adults. Given an infinitely large arithmetic system, the use of such automatic reconstructive processes would make sense--would be cognitively economical. Accessibility, which has been advanced to account for anomalous retrieval time results, could be affected by input from semantic and procedural knowledge.

### Arthur J. Baroody

This article is an elaboration on the theoretical point made by Paul Cobb (1985) in “A Reaction to Three Early Number Papers” that appeared in the March issue of this journal. Cobb notes a discrepancy between a model of subtraction that posits a developmental progression from counting down to a choice between counting down and counting up (Baroody, 1984b) and longitudinal data (Carpenter & Moser, 1984) that indicate that counting up develops at least as early as counting down. He presents longitudinal data from four case studies that corroborate the model (and, I should add, the earlier cross-sectional data and position of Woods, Resnick, & Groen, 1975). To reconcile these results with those of Carpenter and Moser, Cobb argues that a child's use of an informal arithmetic procedure may not be an accurate reflection of his or her conceptual sophistication.

### Arthur J. Baroody

Evidence of basic counting principles has been found in retarded children (Gelman, 1982), including those who are moderately handicapped (Baroody & Snyder, 1983). However, Gelman found no evidence of a stable-order or a cardinality principle in mentally handicapped children with a mental age (MA) of less than 4½ years. The current study examined retarded children in the same MA category to (a) evaluate the hypothesis of a critical MA for learning basic counting principles and (b) further examine how an understanding of counting develops.

### Arthur J. Baroody

Over 9 months, structured clinical interviews with 17 kindergartners were used to study (a) the learning of a concrete counting strategy for addition, (b) the transition from concrete to mental counting strategies, and (c) the role of the commutativity principle in developing more economical counting strategies. Kindergartners appear to differ in their readiness to use a concrete strategy. Many children persisted in counting all with objects. The most common sequence of mental counting strategies was counting all starting with the first addend, counting all starting with the larger addend, and then counting on from the larger addend. A knowledge of commutativity does not appear to be necessary to invent counting strategies that disregard addend order.

### Arthur J. Baroody

School mathematic is my terious, foreign, and threatening to many students. For too many children, school mathematics is something that happens to them rather than something that they make happen. Some children are so overwhelmed that they are intellectually and emotionally paralyzed by school mathematics. How can we make school mathematics sensible, familiar, and enjoyable to children—especially those with learning difficulties? Ginburg (1982) suggest that we should relate formal mathematical instruction to a child's informal knowledge and skills, which are often based on counting. This principle is applicable to children across the whole range of abilities, from kindergarten through eighth grade, and to a range of topics in school mathematics. Such an approach may help children feel more in control of their work and, as a result, feel better about themselves and school mathematics.

### Arthur J. Baroody

Even when they are presented with basic addition combinations (1 + 1, 2 + 1, up to 8 + 9, 9 + 9) and basic subtraction combinations (1 − 1, 2 − 1, up to 17 − 9, 18 − 9), primary-age children tend to reJy on informal counting strategies to obtain sums or differences (Ginsburg 1982; Resnick and Ford 1981; Steffe, Thompson, and Richards 1982). Indeed, without instruction, children invent increasingly sophisticated counting strategies for adding and subtracting (Baroody 1984a; Carpenter and Moser 1982; Groen and Resnick 1977). Moreover, informal arithmetic provides a developmental basis for more advanced strategies, such as reasoning out or recalling combinations of numbers (e.g., Brownell [1935]). As a result of their informal experience, children discover numerical relationships that can later be ued to deduce sums and differences, for example, 5 + 4 = 9 because 4 + 4 = 8 and 5 is 1 more than 4, or 6 − 4 = 2 because 2 added to 4 makes 6 (Baroody, Berent, and Packman 1982; Baroody. Ginsburg, and Waxman 1983; Rathmell 1978; Suydam and Weaver 1975). Only gradually are these relationships internalized, enabling children to generate basic addition and subtraction combinations automaticall y (Baroody 1983b; Brownell 1935). Thus, despite the emphasis placed in school on memorizing the basic addition and subtraction facts, primary-age children typically rely on their informal counting procedures to do basic arithmetic because, initially, these are more meaningful.