Counting Strategies Research Articles

Critiques: Cognitive Development and Children's Solutions to Verbal Arithmetic Problems: A Critique

One primary reason for writing this critique of the report by Hiebert, Carpenter, and Moser (1982) is to emphasize the intricate relationship between theory and data. First, we question the classical empiricist assumption that observation is free from the prejudices of theory. We then consider the role that explanatory theoretical constructs played in the study. Finally, we discuss briefly a claim that number conservation is not a readiness variable for acquiring counting strategies.

Cognitive Development and Children's Solutions to Verbal Arithmetic Problems: A Critique

One primary reason for writing this critique of the report by Hiebert, Carpenter, and Moser (1982) is to emphasize the intricate relationship between theory and data. First, we question the classical empiricist assumption that observation is free from the prejudices of theory. We then consider the role that explanatory theoretical constructs played in the study. Finally, we discuss briefly a claim that number conservation is not a readiness variable for acquiring counting strategies. Hiebert, Carpenter, and Moser make the following distinction between

Defining software science counting strategies

Although the Software Science metrics originally proposed by Halstead are appealing, calculation of the metrics depends on the existence of well-defined counting strategies. The strategies require precise definitions of operators and operands. It is important that the strategies employed be described in research papers. Furthermore, the presentation of helpful examples of the application of the strategies is recommended. Good descriptions do not imply correct strategies, but they do ensure that the strategies can be understood, tested, and evaluated. Appendices to this paper provide the description of a Pascal counting strategy and an example of applying the strategy.

The Calculator for Concept Formation: A Clinical Status Study

Kindergarten and first-grade children (N=30) used successive punches of a handheld calculator as a means for counting. Each child was presented 16 tasks in two individually videotaped interviews. Data concerning three questions were obtained: (a) Can children maintain a one-to-one correspondence between successive punches of a handheld calculator count and (i) an oral count, (ii) a manipulation of a set of objects, and (iii) a second calculator count?, (b) How do children account for an experimenter induced discrepancy in each of these correspondences?, and (c) With a calculator can children model counting strategies known to be used to process basic addition and subtraction facts? Data suggest an affirmative answer to each question. The authors conclude that it may be possible to facilitate a child's acquisition of addition and subtraction concepts by using the calculator to augment counting behaviors.

Counting Processes in Simple Addition1

Abstract: Svenson, O. & Hedenborg, M. L. 1980. Counting processes in simple addition. Scandinavian Journal of Education Research 24,93‐104. Verbal protocols and response latencies were used in this study to investigate the thought processes used by children solving simple arithmetic problems (I + J = , where 0≤I≤13 and 0< J ≤13). To specify, the verbal reports were used to classify the solutions into one of two main groups: (a) answers retrieved from long term memory and (b) answers involving active manipulation of the problem in working memory (reconstructive solutions). The response latencies in each of these groups were analyzed separately for each child. As expected, latencies for retrieved answers were shorter than for reconstructed solutions. The retrieval from long term memory of the answer to a problem with the smaller number first (e.g., 3+4) required about 0.1 sec longer time than when the numbers were in the reverse order (4 + 3). Grouping latencies according to verbal protocols made it possible to refine the analysis of the response latencies. To exemplify, counting strategies with greater units than one (e.g., 6 + 4: 6, 8, 10) were identified and parameters describing the cognitive processes listed.

Parameter Estimation of a Distributed Model of Ferrokinetics: Effects of Data Errors on Parameter Estimate Accuracy

A distributed model of ferrokinetics is herein illustra ted. Model parameters were identified fitting experimental dataobtained by measuring the radioactivity in the plasma and in red blood cells after an intravenous injection of transferrin-bound 59Fe. The results obtained in a normal subject and some anaemic patients are presented. Moreover, this paper aims at studying the design of the experimental conditions so that the experiment is maximally informative. The problem of the choice of sample si zes, sampling instants and counting strategy is discussed taking into account the constraints on the allowable experimental condi tions. These experiment design variables proved to have a significant bearing upon the information provided by an experiment.

Role of selected encoding strategies on short-term retention of kinesthetic information.

30 volunteer subjects were required to use either a counting encoding strategy or an imagery encoding strategy to facilitate recall of information from short-term storage. Subjects were also required to use two reproducation cues in combination with two encoding strategies. Analysis indicated no differences between the accuracy at recall of subjects using imagery or counting strategies of encoding, nor was the predicted interaction between encoding strategies and reproduction cue supported.

Developmental Relations between Notational Counting and Number Conservation

The 2 studies of this report sought to determine the developmental relationship between the child's use of counting as a notational symbol system to extract, compare, and reproduce numerical information and the development of number conservation. In study 1, children between 4 and 6 years of age were administered notational-counting and number-conservation tasks. Analysis of children's profiles across tasks indicated that children develop quantitative counting strategies (but do not necessarily count accurately) before they develop number-conservation concepts. In study 2, the generality of this sequence was tested. A population of 7- to 9-year-old "learning-disabled" children who reportedly were developing atypical counting skills were administered notational-counting and number-conservation tasks. All of these children who conserved number also used quantitative counting strategies, although some of these children frequently counted arrays inaccurately. The significance of these findings is discussed with respect to existing models of counting/number-conservation relations, and an alternative formulation is suggested based upon the new findings.

Left-right sequencing in unschooled children: a function of learning or maturation.

The relative influence of experience and maturation on the development of left-right directionality was investigated. The subjects were 15 3- and 9 4-yr.-old black and Puerto Rican children who had failed to use a left-right sequence in counting arrays of checkers. Within each age level approximately equal numbers of girls and boys were assigned to experimental and control groups (ns = 12). Experimental children were trained to count using the left entry and the rightwards directional movement. Trained girls and boys used significantly more left-right and alternative linear counting strategies on a posttest given three days following training than did untrained children. Trained girls, but not boys, retained the left-right strategy in the second posttest 2 wk. later. The findings suggest that both learning and maturation may influence directionality in perceptual exploration.

The Relation between Two Strategies of Cardinal Number: Correspondence and Counting

RUSSAc, R. J. The Relation between Two Strategies of Cardinal Number: Correspondence and Counting. CHILD DEVELOPMENT, 1978, 49, 728-735. 60 kindergarten, first-grade, and secondgrade children were required to construct numerically equivalent collections and to determine asymmetric-transitive relations among nonequivalent collections, using two strategies of cardinal number: and counting. A novel correspondence task was also introduced and compared with a standard conservation of number procedure. Equivalency by collinear was significantly more difficult than number conservation. This was seen to indicate that the ability to conserve number is but an intermediate step in the development of quantitative correspondence. Children were able to apply a counting strategy to comparisons of small equivalent and nonequivalent collections prior to making the same comparisons by correspondence. There was little indication that children in this study understood the principle of transitive inference in either the or counting conditions.

Statistical evaluation and measuring strategy for extremely small line shifts

For a measuring situation limited by counting statistics, but where the level of precision is such that possible systematic errors are a major concern, it is proposed to determine the position of a spectral line from a measured line segment by applying a bias correction to the centre of gravity of the segment. This procedure is statistically highly efficient and not sensitive to small errors in assumptions about the line shape. The counting strategy for an instrument that takes data point by point is also considered. It is shown that an optimum (“two-point”) strategy exists; a scan of the central part of the line is 68% efficient by this standard.

A Developmental Analysis of Notational Counting

SAXE, GEOFFREY B. A Developmental Analysis of Notational Counting. CmLD DEVELOPMENT, 1977, 48, 1512-1520. The 2 studies of this report trace the child's acquisition of counting as a means to extract, compare, and reproduce number from arrays of objects. In study 1, 3-, 4-, and 7-year-olds' use of counting to compare and reproduce arrays numerically was analyzed with respect to 2 features: (1) counting accuracy, and (2) counting strategy. In study 2, 9 of the 3-year-olds who participated in study 1 were retested 12 and 18 months following their initial assessments. As predicted, in both studies there was an age-related improvement in accuracy which was interrelated with a progression from the use of prequantitative to the use of quantitative counting strategies to compare and reproduce number.

Counting strategies and semantic analysis as applied to class inclusion

This study examined strategic and semantic aspects of the answers given by preschool children to class inclusion problems. The Piagetian logical model for class inclusion was contrasted with an alternative, problem processing model in three experiments. A major component of the alternative model is an enumeration strategy which is advantageous for learning reliable counting skills. The counting strategy was found to explain the inclusion errors of young children better than did the logic of the task. It was also found that young children understand the semantics of inclusion but are unable to coordinate their semantic knowledge with their counting strategy. Methodologically, one of the experiments suggested a fruitful extension of task analysis ( Simon, 1969) to experimental design.

Study of the Precision of X-ray Stress Analysis

Software is described for complete computer control of residual stress measurements. One program (that incorporates either the two tilt method, the sins| procedure, or the Cohen-Marion technique) has been developed for use with either a normal detector or a position sensitive detector. The operator inputs the desired error in stress and various instrumental parameters that determine systematic errors. The counting strategy to obtain the total error is then determined by the software.Employing this automated system, an investigation of a parabolic fit to the top of a diffraction profile indicates that a three point fit is satisfactory only for sharp profiles.

Research perspectives concerning asbestos minerals and their effects on biological systems.

The interaction of asbestos minerals with biological materials is under intensive investigation throughout the world today. Physical scientists should be responsible for the selection of fibers, and their varietal types, for biological studies; they should characterize these experimental materials physically and chemically to the level of sophistication which currently exists in their fields; they should develop definitive assay methods to monitor changes in such characteristics after biological residence; they should actively participate in the formulation of theories or mechanisms of particle interaction in biological systems.Physical scientists should begin the characterization of minerals in the environment and the determination of ambient fiber levels in air and potable water supplies. Such characterization of the environment requires standardized instrumentation and preparation techniques. Acquisition of data in numbers large enough to achieve statistical significance requires the development of automated counting strategies. Instrumentation and software have yet to be developed. The training of physical scientists in environmental areas is lagging behind current national needs and must be accelerated.

Research Perspectives concerning Asbestos Minerals and Their Effects on Biological Systems

The interaction of asbestos minerals with biological materials is under intensive investigation throughout the world today. Physical scientists should be responsible for the selection of fibers, and their varietal types, for biological studies; they should characterize these experimental materials physically and chemically to the level of sophistication which currently exists in their fields; they should develop definitive assay methods to monitor changes in such characteristics after biological residence; they should actively participate in the formulation of theories or mechanisms of particle interaction in biological systems.Physical scientists should begin the characterization of minerals in the environment and the determination of ambient fiber levels in air and potable water supplies. Such characterization of the environment requires standardized instrumentation and preparation techniques. Acquisition of data in numbers large enough to achieve statistical significance requires the development of automated counting strategies. Instrumentation and software have yet to be developed. The training of physical scientists in environmental areas is lagging behind current national needs and must be accelerated.

A sequential approach to the counting of plankton organisms

A counting strategy is described by means of which it is possible to decide, after carrying out a few counts, whether a sampled plankton population can be considered homogeneously distributed. The time saving of the proposed strategy lies in the fact that if the underlying population is homogeneously distributed only one count, i.e., the one of the mixture of aliquots taken from all individual samples collected, would be necessary to characterize the population sufficiently. The count of this mixture may be considered as a ‘mean value’ equivalent to the mean obtained from the counts of all individual samples. With some experience in predicting where homogeneously and where heterogeneously distributed populations are sampled, it is possible to mix large numbers of samples and thus save a great amount of effort in the analysis of plankton populations. The samples may be collected during a survey in any way desired, e.g., in normal hydrographie casts, with pumps, with the air bubble lift etc. A numerical example is given in the Appendix illustrating the method.

Anmerkungen zur besetzung von zellen unter verschiedenen modellen

It is to be examined, which counting strategies are optimal for determining, if a given cell-occupation-pattern is one to one. In this the counting of cells is regarded as the time-consuming element. The question is, whether the counting of full cells or the counting of empty cells is more favorable. The essential parameter, which enters the problem besides the number of cells, M, and of elements, N, is the chosen probability-model controlling the occupancy process. The models examined here (polynomial-model, Bose-Einstein-model) show a significant superiority of counting the full cells. Independently of any model this superiority also holds for the case M > 2N. Under the given premises, the result can be applied to checking-processes in documentation, where it is to be determined, if a given mapping is one to one.