Number words in young children's conceptual and procedural knowledge of addition, subtraction and inversion

Three studies addressed children's arithmetic. First, 50 3- to 5-year-olds judged physical demonstrations of addition, subtraction and inversion, with and without number words. Second, 20 3- to 4-year-olds made equivalence judgments of additions and subtractions. Third, 60 4- to 6-year-olds solved addition, subtraction and inversion problems that varied according to the inclusion of concrete referents and number words. The results indicate that number words play a different role in conceptual and procedural development. Children have strong addition and subtraction concepts before they can translate the physical effects of these operations into number words. However, using number words does not detract from their calculation procedures. Moreover, consistent with iterative relations between conceptual and procedural development, the results suggest that inversion acquisition depends on children's calculation procedures and that inversion understanding influences these procedures.     

Concept-procedure interactions in children's addition and subtraction

A 3-week problem-solving practice phase was used to investigate concept-procedure interactions in children’s addition and subtraction. A total of 72 7- and 8-year-olds completed a pretest and posttest in which their accuracy and procedures on randomly ordered problems were recorded along with their reports of using concept-based relations in problem solving and their conceptual explanations. The results revealed that conceptual sequencing of practice problems enhances children’s ability to extend their procedural learning to new unpracticed problems. They also showed that well-structured procedural practice leads to improvement in children’s ability to verbalize key concepts. Moreover, children’s conceptual advances were predicted by their initial procedural skills. These results support an iterative account of the development of basic concepts and key skills in children’s addition and subtraction.     

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.     

Children's patterns of reasoning about reading and addition concepts

Children's reasoning was examined within two educational contexts (word reading and addition) so as to understand the factors that contribute to relational reasoning in the two domains. Sixty‐seven 5‐ to 7‐year‐olds were given a series of related words to read or single‐digit addition items to solve (interspersed with unrelated items). The frequency, accuracy, and response times of children's self‐reports on the conceptually related items provided a measure of relational reasoning, while performance on the unrelated addition and reading items provided a measure of procedural skill. The results indicated that the children's ability to use conceptual relations to solve both reading and addition problems enhanced speed and accuracy levels, increased with age, and was related to procedural skill. However, regression analyses revealed that domain‐specific competencies can best explain the use of conceptual relations in both reading and addition. Moreover, a cluster analysis revealed that children differ according to the academic domain in which they first apply conceptual relations and these differences are related to individual variation in their procedural skills within these particular domains. These results highlight the developmental significance of relational reasoning in the context of reading and addition and underscore the importance of concept‐procedure links in explaining children's literacy and arithmetical development.     

Patterns of knowledge in children's addition

Patterns of conceptual and procedural knowledge of addition were examined in 5- to 8-year-olds (N = 80). Conceptual knowledge was measured by assessing children's responses to problems in which addends were reordered or decomposed and recombined. Problems were presented using abstract symbols, numbers, and physical objects. Children were more successful in noticing that addends had been reordered rather than decomposed and in noticing the decomposition of addends presented with objects rather than with symbols. Distinct profiles of procedural competence were derived from an analysis of children's problem-solving accuracy and strategies. Profiles were associated with patterns of conceptual knowledge and with age, although age and conceptual knowledge were not related. Findings highlight the usefulness of identifying profiles of procedural and conceptual knowledge for understanding the development of children's knowledge of addition.