Children's Addition Research Articles

Bruneian children's addition and subtraction methods

A research methodology, which incorporated written tests and individual interviews, was adopted to study the methods used by Bruneian children in solving addition and subtraction problems structured around ten. The analysis of pupils' written and verbal responses to questions suggest that they have, through the use of their native language, assimilated the primary requirements necessary in solving problems structured around ten.

Associativity and Understanding of the Operation of Addition in Children with Learning Differences

This study sample consisted of children from two learning categories, learning disabled (LD, n = 27, x̄ = 9.08) and not identified as learning disabled (NLD, n = 42, x̄ = 7.46), who were individually tested on three different mathematics tasks. The modified nonverbal task and the associativity of length task investigated the quality of students' structures of organizing activity by noting the complexity of grouping relationships abstracted among and between object sets (i.e., composite unit structures). Additionally measured in these two tasks was response accuracy. The flashcard task measured accuracy in response to the same number problems as in the modified nonverbal task as well as strategy type used. However, the strategies scored on the flashcard task were indicative of explicitly taught procedures regardless of children's structures of organizing activity. Significantly more NLD children abstracted composite unit structures suggestive of operational logic on the modified nonverbal and associativity of length tasks, although there were no significant differences in the rate of success on the modified nonverbal task. On the flashcard task, there were no significant differences between the two groups on strategy type used, although the LD children achieved greater success. These results suggest that although children state correct answers on the flashcard and modified nonverbal tasks, they may be reflecting on the tasks using thought structures that are not yet operational.

Simple Mental Addition in Children with and without Mild Mental Retardation

The speeded performance on simple mental addition problems of 6- and 7-year-old children with and without mild mental retardation is modeled from a person perspective and an item perspective. On the person side, it was found that a single cognitive dimension spanned the performance differences between the two ability groups. However, a discontinuity, or “jump,” was observed in the performance of the normal ability group on the easier items. On the item side, the addition problems were almost perfectly ordered in difficulty according to their problem size. Differences in difficulty were explained by factors related to the difficulty of executing nonretrieval strategies. All findings were interpreted within the framework of Siegler's (e.g., R. S. Siegler & C. Shipley, 1995) model of children's strategy choices in arithmetic. Models from item response theory were used to test the hypotheses.

SCADS: A Model of Children's Strategy Choices and Strategy Discoveries

Preschoolers show surprising competence in choosing adaptively among alternative strategies and in discovering new approaches. The SCADS computer simulation illustrates how simple processes can generate this impressive competence. The model's behavior parallels data on children's addition in at least eight ways: It uses diverse strategies over prolonged periods of time, makes adaptive choices among strategies, discovers the same strategies as children, discovers strategies in the same sequence as children, makes discoveries without trial and error, makes discoveries without having experienced failure, narrowly generalizes new approaches, and generalizes more broadly following challenging problems. SCADS thus indicates plausible sources of young children's surprising competence at strategy choice and strategy discovery.

The role of conceptual understanding in children's addition problem solving.

The study examined the relationship between children's conceptual understanding and addition problem-solving procedures. Forty-eight 6- to 8-year-olds solved addition problems and, in a 2nd task, were prompted to judge whether a puppet could use the arithmetic properties of one problem to solve the next problem. Relational properties between consecutive problems were manipulated to reflect aspects of additive composition, commutativity, and associativity principles. Conceptual understanding was assessed by the ability to spontaneously use such relational properties in problem solving (Task 1) and to recognize and explain them when prompted (Task 2). Results revealed that conceptual understanding was related to using order-indifferent, decomposition, and retrieval strategies and speed and accuracy in solving unrelated problems. The importance of conceptual understanding for addition development is discussed.

The role of conceptual understanding in children's addition problem solving.

The study examined the relationship between children's conceptual understanding and addition problem-solving procedures. Forty-eight 6- to 8-year-olds solved addition problems and, in a 2nd task, were prompted to judge whether a puppet could use the arithmetic properties of one problem to solve the next problem. Relational properties between consecutive problems were manipulated to reflect aspects of additive composition, commutativity, and associativity principles. Conceptual understanding was assessed by the ability to spontaneously use such relational properties in problem solving (Task 1) and to recognize and explain them when prompted (Task 2). Results revealed that conceptual understanding was related to using order-indifferent, decomposition, and retrieval strategies and speed and accuracy in solving unrelated problems. The importance of conceptual understanding for addition development is discussed.

Training in Place-Value Concepts Improves Children's Addition Skills

The present study examined whether children's variation in arithmetic performance was related to differences in their understanding of place-value. Training in place-value concepts was provided to a group of Chinese children who were poor in arithmetic. Their performance before and after the training was compared to that of the children in two control groups. The results showed that there were reliable connections between place-value understanding and addition and subtraction skills. Furthermore, training in place-value concepts was found to be effective in enhancing the children's place-value understanding and addition skills. Implications for instructions in arithmetic were discussed.

Young Children's Addition Estimates

Two hundred and fifteen children aged between 5 and 9 years were asked to estimate the answers to addition sums. Their competence at addition was first assessed, and they were accordingly divided into five groups. Children of each level were given a set of estimation problems involving sums a little too difficult for them to calculate (termed their “base correspondence”). Of the 215 children, 108 were then given sets of estimation problems corresponding to levels higher than their own. In the base correspondence, children of higher levels tended to produce more reasonable estimates than did children of lower levels. As difficulty increased beyond the base correspondence, the reasonableness of the estimates declined. The existence and nature of a zone of partial knowledge and understanding are discussed.

Measuring individual differences in children's addition strategy choices

A test of individual differences in addition strategy choices was devised and its psychometric properties were analyzed. The test was used to examine how individuals' accuracy of retrieval was related to their use of backup strategies (e.g., counting on fingers) and the accuracy of their backup strategies. Strategy use and errors of 105 6-year-olds was recorded on all 100 single-digit addition problems. Children were randomly assigned to the “exploratory” or the “validation” subsample. In the exploratory subsample, discriminant analyses reduced the 100 items to 2 sets of 20 items: each test discriminated good students, not-so-good students, and perfectionists. In the validation subsample, the 2 forms' reliabilities exceeded .90 on all variables, classified 74% of the children similarly, and showed similar relations to other variables. Not-so-good students had lower achievement scores than good students or perfectionists in math computations but not math applications. Children's frequency and accuracy of backup strategy use and backup strategy accuracy functioned independently to reduce their retrieval errors.

Children of alcoholics. Predictors for psychopathology and addiction.

Children of alcoholics have a higher risk of psychopathology and alcoholism. Therefore, in 1993 the Amsterdam Institute for Addiction Research initiated a study on vulnerability markers and risk factors in children of alcoholics, aimed at identifying predictors for the development of psychopathology and addiction in children of alcoholics. This article provides a summary of the background, rationales and aims of the study. With more specific and sensitive biological vulnerability markers that indicate risk status, more effective preventive interventions might become available. The biochemical part of the study aims at answering the question whether adenylate cyclase is a vulnerability marker for alcoholism. The psychophysiological part is directed at event-related potentials during task performance to clarify the nature of the brain and cognitive functions that may underlie or relate to vulnerability to alcoholism. The third part, the psychological component, aims at possible psychological mechanisms of enhanced risk of addition in children of alcoholics as well as the relationship with childhood psychopathology.

Acquisition of children's addition strategies: A model of impasse-free, knowledge-level learning

When children learn to add, they count on their fingers, beginning with the simple SUM strategy and gradually developing the more sophisticated and efficient MIN strategy. The shift from SUM to MIN provides an ideal domain for the study of naturally occurring discovery processes in cognitive skill acquisition. The SUM-to-MIN transition poses a number of challenges for machine-learning systems that would model the phenomenon. First, in addition to the SUM and MIN strategies, Siegler and Jenkins (1989) found that children exhibit two transitional strategies, but not a strategy proposed by an earlier model. Second, they found that children do not invent the MIN strategy in response to impasses, or gaps in their knowledge. Rather, MIN develops spontaneously and gradually replaces earlier strategies. Third, intricate structural differences between the SUM and MIN strategies make it difficult, if not impossible, for standard, symbol-level machine-learning algorithms to model the transition. We present a computer model, called GIPS, that meets these challenges. GIPS combines a relatively simple algorithm for problem solving with a probabilistic learning algorithm that performs symbol-level and knowledge-level learning, both in the presence and absence of impasses. In addition, GIPS makes psychologically plausible demands on local processing and memory. Most importantly, the system successfully models the shift from SUM to MIN, as well as the two transitional strategies found by Siegler and Jenkins.

Acquisition of Children's Addition Strategies: A Model of Impasse-Free, Knowledge-Level Learning

When children learn to add, they count on their fingers, beginning with the simple SUM strategy and gradually developing the more sophisticated and efficient MIN strategy. The shift from SUM to MIN provides an ideal domain for the study of naturally occurring discovery processes in cognitive skill acquisition. The SUM-to-MIN transition poses a number of challenges for machine-learning systems that would model the phenomenon. First, in addition to the SUM and MIN strategies, Siegler and Jenkins (1989) found that children exhibit two transitional strategies, but not a strategy proposed by an earlier model. Second, they found that children do not invent the MIN strategy in response to impasses, or gaps in their knowledge. Rather, MIN develops spontaneously and gradually replaces earlier strategies. Third, intricate structural differences between the SUM and MIN strategies make it difficult, if not impossible, for standard, symbol-level machine-learning algorithms to model the transition. We present a computer model, called GIPS, that meets these challenges. GIPS combines a relatively simple algorithm for problem solving with a probabilistic learning algorithm that performs symbol-level and knowledge-level learning, both in the presence and absence of impasses. In addition, GIPS makes psychologically plausible demands on local processing and memory. Most importantly, the system successfully models the shift from SUM to MIN, as well as the two transitional strategies found by Siegler and Jenkins.

Development and Evaluation of a Presurgical Preparation Program

Three presurgical preparation programs were developed and evaluated in an Australian hospital utilizing an additive component design. The component basic to all three preparation programs was modeling. This technique was compared with the additional components of teaching child coping skills and parent coping skills via videotape. Subjects were 28 children between the ages of 4 and 13 years who were scheduled for elective surgery. Anxiety of both the children and parents was assessed by self-report and behavior observation. Results indicated that there was no further anxiety reduction by the addition of child and parent coping skills. Results are discussed in terms of the viability of teaching coping skills via videotape particularly to parents. Methodological difficulties associated with research in this area are examined.

Starting to add worse: Effects of learning to multiply on children's addition

A major stumbling block in acquiring a new skill can be integrating it with old but related knowledge. Learning multiplication is a case in point, because it involves integrating new relations with previously acquired arithmetic knowledge (in particular, addition). Two studies explored developmental changes in the relations between single-digit addition and multiplication. In the first study, third-graders, fifth-graders, and adults performed simple addition or multiplication in mixed-and blocked-operations formats. Substantial interfering effects from related knowledge were found at all age levels, but were more pronounced for younger subjects. Thus in the early stages of learning multiplication, one consequence of learning a new operation is interference in performance of an earlier, related, but less recently studied skill. Consideration of error patterns supported the view that the problem of integrating operations is a prominent one even in the early stages of mastering multiplication. Patterns of errors were generally consistent across all age groups, and all groups were much more likely to give a correct multiplication response to an addition problem than the reverse. A second, langitudinal study confirmed this finding, showing evidence for impaired performance of addition over time within individual children (second-, third-, and fourth-graders) tested on simple addition and multiplication over a 5-month period. Analysis of reaction times for addition indicated that second- graders in advanced math classes and third-graders in regular math classes tended to slow down over the year in responses to addition problems. Fourth- graders, on the other hand, tended to increase their speed of addition over the course of the year. Multiplication showed a different pattern during this period, with no evidence for slowing among children who were able to perform this task. Disruption of previously learned knowledge in the course of acquiring new skills provides evidence that new knowledge and old knowledge are being integrated. This kind of non-monotonic development may provide an empirical method for determining the functional limits of a domain of knowledge.

Children in the student family

Abstract Although marriage and matrimony do constitute life values, it is essential that they be multiplied with the addition of children. Essentially, the formation of the family is fully realized only with the birth of a child.

The perils of averaging data over strategies: An example from children's addition.

The perils of averaging data over strategies: An example from children's addition.

The perils of averaging data over strategies: An example from children's addition.

The perils of averaging data over strategies: An example from children's addition.

The Case of Felicia: A Young Child's Strategies for Reducing Memory Demand During Mental Addition

A semi-structured clinical interview technique was used in an intensive case study of a preschool child's addition ability. The interviews uncovered a novel mental addition strategy: counting-all starting with the larger addend. This strategy minimizes the cognitively demanding double count required by mental addition strategies. Moreover, another invented strategy, a counting-like variation, was employed with problems involving a double-digit, addend to reduce mental effort.

Review: Looking at the Pluses and Minuses

The book is a collection of articles, mainly research studies, edited after presentation and discussion at a seminar in Wisconsin in 1979. In an introductory chapter, Romberg states the aim: “To formulate precise models that describe children's addition and subtraction skills and how these skills change over time” (p. 4). He points to a growing consensus on the acute problems to be studied and the appropriate methodology for studying them.

Herbert Read on Education, Art, and Individual Liberty

Outside of occasional references to a child's addition of a personal touch to a drawing or painting or the imprint of each child's originality, there is little sustained attention in the art education literature to the nature and source of individuality in child art. Studies of child art have predominantly focused on the common, uniform developmental tendencies.1 An exception to this state of affairs is the work of Herbert Read. Knighted man of letters, Read was widely published as an art critic, poet, and author on politics and education. His Education through Art, revised in 1956 and reprinted several times since its first publication in 1943, is perhaps the most definitive, theoretically comprehensive statement to date on the development of art abilities. Its influence on European art education parallels that of Lowenfeld's Creative and Mental Growth (1947) in America. While the Lowenfeld text can be managed handily by the overburdened teacher, Education through Art is, by contrast, uncompromising in its scholarship and intricacy of argument. The writing is philosophically blunt and politically iconoclastic with many complex, technically demanding passages.