Children’s Logical and Mathematical Cognition: Progress in Cognitive Development Research
Abstract
For some time now, the study of cognitive development has been far and away the most active discipline within developmental psychology. Although there would be much disagreement as to the exact proportion of papers published in developmen tal journals that could be considered cognitive, 50% seems like a conservative estimate. Hence, a series of scholarly books to be devoted to work in cognitive development is especially appropriate at this time. The Springer Series in Cognitive Development contains two basic types of books, namely, edited collections of original chapters by several authors, and original volumes written by one author or a small group of authors. The flagship for the Springer Series will be a serial publication of the "advances" type, carrying the subtitle Progress in Cognitive Development Research. Each volume in the Progress sequence will be strongly thematic, in that it will be limited to some well-defined domain of cognitive-developmental research (e. g. , logical and mathematical de velopment, semantic development). All Progress volumes will be edited collec tions. Editors of such collections, upon consultation with the Series Editor, may elect to have their books published either as contributions to the Progress sequence or as separate volumes. All books written by one author or a small group of authors will be published as separate volumes within the series. A fairly broad definition of cognitive development is being used in the selection of books for this series.
Chapters (6)
As cognitively mature adults we know that a quantity remains constant across a transformation as long as there is no addition or subtraction of the specific quantity in question. That is, we appear to be aware of an identity rule: In the absence of addition or subtraction quantity (amount) is maintained. Furthermore, we appear to know that this rule is more than just one of many available cues for judging quantity. To the extent that the possibility of addition or subtraction can be monitored during a transformation, we know that the identity rule should take precedence over any other potential cue for judging the presence or absence of a change in quantity.
In this chapter we describe children’s acquisition and elaboration of the sequence of counting words from its beginnings around age two up to its general extension to the base ten system notions beyond one hundred (around age eight). This development occurs, in our view, in two distinct, though overlapping, phases: an initial acquisition phase of learning the conventional sequence of number words and an elaboration phase, during which this sequence is decomposed into separate words and relations upon these pieces and words are established. During acquisition, the sequence begins to be used for counting objects. Near the end of the elaborative phase, the words in the sequence themselves become items which are counted for arithmetic and relational purposes.
Interest in children’s concepts of chance and probability has been prompted by several questions. Assuming that the development of a concept of chance and probability is influenced by experience, what are the conditions that bring it about? What are its precursors? Is it acquired all at once, or is it acquired gradually over a relatively long period of time? At what age is its development complete? Does every mature adult have a similarly functioning concept of chance, or are there individual differences? If so, how are they to be explained? To what extent is a concept of chance a result of formal instruction in school? What kinds of training are likely to improve upon immature or deficient concepts of chance or probability? When making probability judgments, is there one optimum strategy that can be said to be correct in each type of situation, or is there a variety of strategies more or less adequate or appropriate? To what extent is performance in a probability setting controlled by the reinforcing consequences of previous outcomes? What is the relationship between chance and probability concepts, on the one hand, and the development of linguistic ability to articulate them, on the other? In what ways are various probability tasks alike, and how do they differ? What makes some tasks seem harder than others? What is the relationship between the development of concepts of chance or probability and cognitive development in general?
In this chapter I shall be concerned with the development of various quantity concepts and some of the factors which influence their development. Two aspects of early quantity concepts will be examined: (a) linguistic factors and (b) perceptual factors. The relationship between children’s linguistic skills and their quantitative concepts will be considered in an attempt to separate cognitive processes from linguistic abilities. I shall also examine the perceptual and nonquantitative factors that influence the development of these concepts in order to understand the growth of number as a conceptual dimension.
Psychologists concerned with the development of cognition have largely studied age-related changes in Western middle class children. Although some classic findings have been produced by this approach, it has its limitations. By studying development in only one society, we are blind to the way in which culture may influence cognitive development. It is only through the analysis of development in different cultural contexts that some perspective on the links between culture and cognitive development can be achieved. In this chapter, research concerned with the numerical concepts of a remote and recently contacted group in Papua New Guinea, the Oksapmin, is discussed. The Oksapmin people are just emerging from Stone Age conditions and hence present a radical contrast to the West in their patterns of social life as well as in their practices involving number concepts.
One of the principal routes to generality in any science is the formulation and testing of mathematical models of the events that one studies. Mathematical models offer investigators a number of technical advantages in the treatment and reporting of data, with the most obvious ones being elegance, precision, and predictive power. In addition, however, the vigorous application of mathematical models to well- defined data spaces often produces more than mere technical advantages. Models that have especially simple and comprehensible forms may precipitate advances in theoretical understanding by focusing our attention on abstract communalities between seemingly disparate phenomena. The classic example of this effect in psychology is the remarkable degree of theoretical unification that was achieved in Bush and Mosteller’s (1955) application of linear difference equations to conditioning paradigms. More recently, the impetus for many hypotheses about the mechanics of adult memory has come from simple stochastic models (see Greeno, 1974; Greeno, James, DaPolito, & Poison, 1978). The distinction between short-term and long- term storage, for example, was motivated in large measure by the application of finite Markov chains to paired-associate data (e.g., Atkinson & Crothers, 1964; Greeno, 1967).
... At the same time and shortly thereafter, three-state Markov processes were successfully applied to paired-associate learning (Kintsch, 1963), free recall (Kintsch & Morris, 1965;Waugh & Smith, 1962), conditioning of the nictating membrane in rabbits (Theios & Brelsford, 1966a), and reversal transfer in rats (Theois, 1965). Lately, such models have been shown to be in close agreement with data from standard verbal-transfer paradigms (Greeno, James, & DaPolito, 1971;Pagel, 1973) and data from concept-learning experiments with children (Brainerd, 1979a(Brainerd, , 1979b(Brainerd, , 1982. Without putting too fine a point on it, then, there does seem to be presumptive evidence that data from learning paradigms of the sort we are considering are roughly three-state Markovian. ...
... These points can be illustrated with an example of nonidentifiability from the concept-development literature (Brainerd, 1978(Brainerd, , 1981(Brainerd, , 1982. A question of considerable theoretical interest in this literature is whether concepts that children acquire during their spontaneous cognitive development emerge in immutable sequences. ...
... We have seen, for example, that Theios's (1963;Theios & Brelsford, 1966b) theories provide such restrictions for avoidance conditioning. Likewise, Theios and Brelsford's (1966a) theory of Pavlovian conditioning contains the restriction b = 0, c = d, and g = h, and Brainerd's (1982) theory of children's concept learning contains the restriction 1-/•=!-e = g and c = 0. When a testable identifying restriction for a paradigm is either available from a quantitative theory or can be deduced from the analysis of a qualitative theory, it is the log-ical place to begin the process of identifying Equation 1. ...
Presents a review of the statistical machinery used to apply 3-state Markov processes to data. Material is organized around 3 general issues: parameter estimation, goodness of fit, and hypothesis testing. Parameter identifiability, 3 "parameterizations" of the 3-state model, and practical issues concerned with the selection of optimization programs are considered. Statistical tests that assess the parsimoniousness of the 3-state model and its comprehensiveness are examined. Techniques for obtaining identifying restrictions are considered, likelihood-ratio tests are developed for evaluating exact and inexact numerical hypotheses about parameters. Major developments are illustrated with the data of 3 experiments concerned with age changes in the effects of pictures and words on cued recall data. (57 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
... Naturalmente, tal dualidad, ha incidido en el estudio del origen de los errores algorítmicos. Como hemos señalado, numerosas investigaciones advierten de la existencia de una relación entre la adquisición conceptual de las estructuras numéricas y la ejecución procesal de los algoritmos (Richards y Briars 1982; Fuson, 1992; Fuson y Briars, 1990; Hiebert y Carpenter, 1992; Hiebert Carpenter, Fennema, Fuson, Wearne, Murray, Oliver y Human, 1997; Stein, Grover, y Henningsen, 1996). Teniendo en cuenta las aportaciones de las investigaciones precedentes,en el presente artículo, como objetivo general tratamos de analizar el error en la sustracción y el tipo de relación que se establece entre el conocimiento conceptual y el procesal en la generación del mismo.Todo ello, en el contexto específico de nuestro sistema educativo. ...
... Para avanzar un paso más en el análisis del conocimiento conceptual que poseían los niños/as, evaluamos los porcentajes de respuestas correctas que manifestaban en cada ítem, en relación a los objetivos establecidos para cada uno de ellos (Tabla III). El porcentaje de aciertos en los ítems 1 y 2 que evaluaban la lectura y escritura de las cantidades, con un 92, 9% en el primero y 91,2 % en el segundo; aunque es un porcentaje elevado, permite concluir que los niños leen y escriben cifras de manera mecánica y memorística (Fuson et al., 1982). Idea que podemos observar si relacionamos los resultados porcentuales de estos ítems con el resultado del ítem número 5 (21,5%),que evaluaba la relación entre los números y el control de la serie numérica. ...
... A modo de conclusiones finales, podemos indicar que a la luz de los resultados obtenidos hemos de formular las siguientes consideraciones.En primer lugar, que los resultados globales obtenidos en la prueba número 1, ponen en evidencia que, tres de los principios necesarios para la adquisición del algoritmo , como son: la composición aditiva de las cantidades, los valores convencionales de la notación decimal y la capacidad de establecer relaciones entre los números en una serie, no son comprendidos significativamente. Concluimos que, los conceptos relacionados con las unidades de recuento, no son significativamente aprehendidos.Este resultado,evidencia los porcentajes que obtuvimos en los ítems número 1, 2 y 5; que obligan a pensar que la mayor parte de los niños/as aprende la serie numérica de memoria sin comprender las estructuras conceptuales que la gobiernan, hecho que ya ha sido informado por autores como Fuson et al. (1982), Fuson, (1988). Como consecuencia, coincidiendo con estos autores, observamos la presencia de la percepción cognitiva del número compuesto por varios dígitos, como una estructura conceptual única,que no les permite valorar el lugar posicional que ocupa cada dígito dentro de la cantidad, y por tanto, no posibilita la representación mental de las distintas agrupaciones de 10, 100, 1000…, que forman parte del basamento conceptual de la resta. ...
The primary aim of this study was to investigate the implied algorithmic processes in the
learning of the subtraction, both in the conceptual and procedural planes, and to establish the
relation between both components in the origin of the errors. For that reason, this study is
classified as a scientific analysis on the nature and sources of error during the algorithmic
processes in the learning of the subtraction.The analysis and final conclusions presented here
have been drawn from an experimental cross-sectional test in which 357 students (belonging
to the last five years of Primary Education and to four different teaching institutions) have
participated.They were all evaluated on conceptual and procedural knowledge in relation to the
algorithm of the subtraction.This article offers the results obtained, but also shows a comparative
analysis, with results from other investigations in other contexts,completed by authors known
internationally and part of the literature that approaches the subject. The results provided
demonstrate that, in the teaching-learning process, the conceptual and procedural types of
components, specifically applied to the algorithm, are determining factors in the generation of
errors. The analysis of the results points at the necessity for a meaningful education that
teaches basic concepts and prioritises the development of the numerical sense, in contrast to
an algorithm learning process, based on the mechanical and memory-drawn succession of
rules or procedures whose nature is not understood.
Computational thinking (CT) is increasingly incorporated into curricular planning across various educational levels in numerous countries. Presently, CT is being integrated into preschool and primary education. To effectively implement CT at the classroom level, the design and study of techniques and tasks are crucial. This research empirically evaluates a didactic sequence using programmable educational robots for problem-solving challenges rooted in mathematical concepts. The study consists of two sets of activities: computational localisation of elements on a regular grid, where students program robots to navigate, and problem-solving tasks involving sum calculations using distinct pre-operational strategies. The study sample is a class of 16 students at the preschool level. The results indicate an increasing complexity in the success of the designed sequence, with the ’counting all’ strategy demonstrating higher efficacy. These promising findings highlight the potential for further research, aiming to establish a strong foundation for early educational levels through the integration of CT via programmable robots and engaging problem-solving challenges.
In modern cognitive theories, memorizing has been conceptualized as a process of storing information and remembering as a process of retrieving information. These memory processes have commonly been studied in standard list-learning paradigm tasks such as free recall, paired-associate learning, cued recall, serial learning, and recognition memory. When several seconds of distracting activity are inserted between consecutive study and test trials, these tasks can be considered to be long-term memory tasks (Brainerd, in press). Since the early 1950s, the simple mathematical concept of finite Markov chains has been used to account for the data of adults in these long-term memory tasks (Estes, 1962; Feller, 1950; Kemeny & Snell, 1960). Many fruitful hypotheses about the mechanisms of memory in adults have been generated by using these finite Markov chains (Greeno, 1970, 1974; Greeno, James, DaPolito, & Polson, 1978; Levine & Burke, 1972; Norman, 1972).
Occasionally, the proscriptions of a theory have more influence on the behavior of scientists than do its positive predictions (see Chapter 2 for additional comments about this phenomenon). A situation of this sort occurred some two decades ago in connection with what Piagetian theory anticipates about the laboratory learning of its stage-related concepts. At that time it was widely supposed that the theory predicts no learning effects. Although this interpretation was not entirely accurate, it was correct in spirit, and it spawned a large number of learning experiments on Piagetian concepts, especially the concrete-operational concepts of middle childhood. What began as some modest attempts to assess the trainability of conservation ultimately blossomed into a substantial literature containing multiple experiments on concepts such as perspective taking (e.g., Cox, 1977; Iannotti, 1978), sedation (e.g., Bingham-Newman & Hooper, 1974; Coxford, 1964), identity (e.g., Hamel & Riksen, 1973; Litrownik, Franzini, Livingston, & Harvey, 1978), proportionality (e.g., Brainerd, 1971; Brainerd & Allen, 1971b), isolation of variables (e.g., Case, 1974; Siegler, Liebert, & Liebert, 1973), ordinal and cardinal number (e.g., Brainerd, 1973, 1974b), subjective moral reasoning (e.g., Arbuthnot, 1975; Jensen & Larm, 1970; also Chapter 3 in this volume) and many others. In fact, I would venture to say that there is no concept that figures prominently in Piaget’s summary writings on his stages (e.g., Piaget, 1970; Piaget & Inhelder, 1969) that has not been subjected to training in several experiments.
As everyone knows, the distinction between processes that lead to the formation of traces, commonly called storage, and processes that permit access to such traces, commonly called retrieval, is fundamental to modern theories of memory. Students of the memory literature usually credit Melton (1963) with being the first to focus attention on the storage-retrieval distinction. Melton remarked, “What, then, are the principal issues in a theory of memory? These are about either the storage or the retrieval of traces” (1963, p. 4). Although this observation appeared more than two decades ago, it was only during the past decade that concepts of storage and retrieval replaced older associative ideas as cornerstones of memory theories.
It often happens that a field of inquiry comes to be so dominated by a theoretical tradition that new developments are inclined to be smothered. By the end of the Renaissance, for example, the authority of Aristotle had become so pervasive that, as Bertrand Russell remarked in his history of philosophy, “Ever since the beginning of the seventeenth century, almost every serious intellectual advance has had to begin with an attack on some Aristotlian doctrine.” Much the same could be said of the effect of Newtonian mechanics on physics by the end of the 19th century, or the effect of psychoanalysis on abnormal psychology by the 1940s, or closer to home, the effect of Piagetian theory on the contemporary science of cognitive development. The enormous influence that Piaget’s ideas achieved during the preceding two decades, together with the theory’s almost universal scope, mean that, for the near term at least, any significant reorientation of our assumptions about cognitive development is destined to conflict with some Piagetian tenet or another.
Numerical abilities are important for autonomy in daily life. Except specific deficits commonly named dyscalculia, some disorders affecting others components, like language or motor activity, could impede numerical development. Many numerical activities require language (quantification, building collections, solving problems) and motor activity plays as well an important role in learning to count, which is the root of arithmetic (Camos and al, 1998). The goal of this study is to evaluate the TEDI-MATH test ability to distinguish different patterns of basic skills affected construction in mathematic and particularly numerical activities, through the case children benefiting speech rehabilitation because of language disorder, and children benefiting psychomotor rehabilitation because of motor or spatial perception disorder.
In a recent observation of 1848 Chinese children of 0‐6 years old, it was found that the ability of Chinese children in counting, counting backward, repeating digits, repeating digits reversed etc. is much higher than that given by the Stanford‐Binet Scale #opSBS#cp and other data. Some of the observed data are quoted here. This observation confirmed the obvious fact that for children of different cultural background, different development test scales should be used in order to obtain correct measure of child development.
Cross-cultural applications of developmental theories have raised a number of is sues related to the developmental status of children in Third World countries. This paper analyses the outcomes of research carried out with children in the Indian setting using some major theories of cognitive development. It is argued that group differences in test performance are most likely to be interpreted as "deficits" if one fails to analyse the context in which children are brought up. On the other hand, differences in testperformance can be meaningfully understood and interpreted by adopting an ecocultural perspective.
This article reviews the evidence from the many studies of young children which together trace the sequence of development from counting to the beginnings of formal arithmetic. The author is senior psychologist at Cheeverstown House, Dublin. He proposes that knowledge of the normal progression in this area is invaluable in meeting special educational needs.
For about ten years, new research data have led us to enlarge the Piagetian model of the learning of the concept of number.
Research investigations of counting have especially emphasized the importance of some abilities related to the construction
of the concept of number. We think these researches should more greatly influence the practice of assessment. But valid tests
should be available for psychologists and teachers to appraise the child development of the mastery of counting. For this
purpose, we developed a set of tasks on the base of a critical analysis of Gelman and Gallistel counting principles. In a
first step of our research, we used these tasks in a large sample of children from the French-speaking community of Belgium.
This sample was composed of pupils from the third year of the nursery school (N=439) and from the first year of the primary
school (N=103). Some of our results are quite different from those of previous investigations. We observe that many 5-year-old
children are far from mastering and to coordinating the counting principles. A significant percent of pupils in the first
year of the primary school are in the same situation. The implications of these difficulties for the beginning of the learning
of arithmetic are emphasized.
This study examines both oral and object counting, and the ability to read numerals in a group of 51 children with a moderate degree of mental handicap. The findings indicate deficiencies in basic counting competencies across a wide spread of ability. There is also evidence for a distinction between oral and object counting on a cognitive skills basis. The findings are discussed within a remediation context.
ResearchGate has not been able to resolve any references for this publication.