Studies in Reflecting Abstraction

Jean Piaget, Robert L. Campell

• Publisher: Psychology Press
• Year: 2014
• eBook ISBN: 9781317762737
• Topic: Psychology
• Subtopic: Developmental Psychology
• Index: Psychology

This translation of the French Recherches sur l'abstraction reflechissante (1977), make available in English Piaget's only treatise on reflecting abstraction - a process he came to attribute considerable importance to in his later thinking and which he believed to be responsible for many of the advances that take place in human development, especially our understanding of mathematics.
Rich with empirical research on reflecting abstraction at work in the thinking of 4 to 12 year olds, the studies in this volume examine its role in many contexts of cognitive development such as: reasoning about mathematics; forming analogies; putting objects in order by size and comparing the resulting series; and navigating through a wire maze. His theoretical discussions explore the relationships between reflecting abstraction and other central processes in his later theory, such as generalization, becoming conscious, and equilibration, as the differentiation of possibilities and their integration into necessities. These discussions indicate which aspects of his later theorizing were settled and which require further thought and investigation.
Studies in Reflecting Abstraction will be of interest to developmental and cognitive psychologists, educationalists, philosophers and anyone who seeks to understand human knowledge and its development.

PART ONE

Logico-arithmetical or algebraic abstraction

[p. 31]

In 1950 the author of these lines was already insisting^1,2 on the need to distinguish between empirical abstraction and reflecting abstraction. Reflecting abstraction proceeds from the actions or operations of the knowing subject and transfers to a higher plane what has been taken from a lower level of activity; it leads to differentiations that necessarily imply new, generalizing compositions at the higher level. Though this hypothesis seemed obvious to us, none of the many programs of study that we have carried out at our Center for Genetic Epistemology has previously been devoted to the problems of abstraction or to the relationships between its empirical and reflecting forms. The present work aims to fill that gap.³

The kind of abstraction that ranges over physical objects or the material aspects of one's own action (such as movements, pushes, and the like) we will call “empirical.” Let us note right away that this type of abstraction, even in its most elementary forms, cannot be a pure “read-off” of data from the environment. To abstract any property whatsoever from an object, such as its weight or its color, the knowing subject must already be using instruments of assimilation (meanings and acts of putting into relation) that depend on sensorimotor or conceptual schemes. And such schemes are constructed in advance by the subject, not furnished by the object.

However, these schemes are only instrumentally necessary for empirical abstraction [p. 31]. Empirical abstraction does not range over the schemes themselves; it aims only at the data that remain external to them. The facts are the content; the knowing subject's schemes merely embody the forms that make it possible to grasp that content.

By contrast, “reflecting” abstraction ranges over those very forms and over all of the subject's cognitive activities (schemes or coordinations of actions, operations, cognitive structures, etc.). Reflecting abstraction separates out certain characteristics of those cognitive activities and uses them for other ends (new adaptations, new problems, etc.). It is “reflecting” in two complementary senses. First, it transposes onto a higher plane what it borrows from the lower level (for instance, in conceptualizing an action). We will call this transfer or projection a réfléchissement.⁴ Second, it must therefore reconstruct on the new level B what was taken from the previous level A, or establish a relationship between the elements extracted from A and those already situated in B. This reorganization that is forced by the projection will be called a reflection in the strict sense.⁵

Reflecting abstraction, with its two components projection and reflection, can be observed at every major stage of development. During the sensorimotor substages⁶ (as will be seen in Chapter 18, “The rotation of a bar around a pivot”) toddlers can solve problems by borrowing certain coordinations from previously constructed cognitive structures and reorganizing them in light of new data. In these cases, we have no idea what the knowing subject becomes conscious of. By contrast, at the higher levels, when reflection is a product of thinking as this is normally understood, it becomes necessary to make a further distinction. There is reflecting abstraction as a constructive process, and there is its retroactive thematization,⁷ which then becomes a reflection on the reflection. In these cases, we will speak of “reflected abstraction”⁸ or reflective thought.⁹

It is useful to add a final distinction. At those developmental levels that are representational but preoperational,¹⁰ as well as at the level of concrete operations,¹¹ it sometimes happens that the knowing subject cannot carry out some constructions (which later on will become purely deductive) without relying constantly on their observable results (cf., using the abacus for the first numerical operations). In that case we will speak of “pseudo-empirical abstractions.” While the results are read off from [p. 31] material objects, as is the case with empirical abstraction, the observed properties are actually introduced into these objects by the activities of the subject. Consequently, we are in the presence of a variety of reflecting abstraction that operates with the aid of observables that are external, on the one hand, but that are constructed by reflecting abstraction, on the other. By contrast, the properties covered by empirical abstraction were already in the objects before the subject engaged in any act of observation.

We will therefore have to contend with two problems in this book: (1) What are the mechanisms of reflecting abstraction? (2) What are its relationships (complex because they are not at all symmetrical) with empirical abstraction? Indeed, while reflecting abstraction becomes more and more autonomous (it works unaccompanied in pure logic and mathematics), empirical abstraction makes progress only through its dependence on reflecting abstraction.

In Part One of this book, Chapters 1 through 4 will cover elementary arithmetic constructions and Chapters 5 through 7 will cover logical structures. Part Two will be dedicated exclusively to relations of order and Part Three, concerning spatial constructions, will raise questions about the relationship between the two forms of abstraction to a greater extent than the first two parts did.

N.B. Although the research we report here was not done with any pedagogical purpose in mind, we find it difficult not to notice that knowledge of the schoolchildren's reactions described in this book might be of some utility to educators. (We are thinking in particular of the surprising difficulties that children have in understanding the meaning of the ultra-simple multiplications in Chapter 2, etc.)

ACKNOWLEDGMENTS

The studies published in this volume were financed by the Swiss National Fund for Scientific Research, the Ford Foundation in New York, and the Foundations Fund for Research in Psychiatry in New Haven. We would like to express our thanks for their support.

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¹ Jean Piaget, Introduction à l'épistémologie génétique, Vol. 1, La pensée mathématique (Paris: Presses Universitaires de France, 1950), pp. 72–73.

^2* In the second edition of the Introduction, published by Presses Universitaires de France in 1973, this same material can be found in Vol. 1, pp. 75–77. See also the editor's introduction to current volume, “Reflecting abstraction in context.”

^3* All but two of the empirical studies reported in this volume were carried out in 1971–1972.

^4* Réfléchissement has no cognate in English, so I will follow the lead of an earlier writer on the subject—Henry Markovits, Intelligence and abstraction réfléchissante in Piaget's theory of cognitive development, Canadian Psychological Review, 18, 74–77, 1978–and translate it as “projection.” Rita Vuyk, in her valuable reference work Overview and critique of Piaget's genetic epistemology 1965–1980 (2 vols., London: Academic Press, 1981) uses the phrase “projective reflection.”

^5* Piaget uses the term réflexion, which does have a cognate in English. Following Markovits' lead, I will translate this as “reflection,” keeping in mind that the term has a technical meaning for Piaget. Vuyk prefers “reconstructive reflection.”

^6* Piaget is referring to the six substages of the sensorimotor period of development, which extend from birth to roughly 2 years of age. See Jean Piaget, Origins of intelligence in children (translated by Margaret Cook, New York: Humanities Press, 1952); The construction of reality in the child (translated by Margaret Cook, New York: Basic Books, 1954); and Play, dreams, and imitation (translated by CM. Gattegno and Frances M. Hodgson, New York: Norton, 1962) for detailed accounts.

^7* For Piaget, to “thematize” something is to know it consciously and in an easily verbalized form.

^8* Abstraction réfléchie.

^9* Pensée réflexive.

^10* “True representation” in Piaget's sense requires the semiotic function, as manifested in pretend play, deferred imitation, mental imagery, and the language ability sufficient for the production of sentences of two or more words. “True representation” does not develop until substage 6, from 18 to 24 months of age, which completes the sensorimotor period (see also note 6 above). For a later and somewhat different structural perspective on preoperational thinking, see Jean Piaget, Jean-Blaise Grize, Alina Szeminska, and Vinh-Bang, Epistemology and psychology of functions (translated by F. Xavier Castellanos and Vivian D. Anderson, Dordrecht: Reidel, 1977).

^11* The voluminous literature on concrete operations includes several theoretical treatments: Jean Piaget, La réversibilité des opérations et l'importance de la notion de “groupe” pour la psychologie de la pensée, in H. Piéron & I. Meyerson (Eds.), Onzième congrès international de psychologie, Paris, 25–31 juillet, 1937: Rapports et comptes rendus (pp. 433–435), Paris: Alcan, 1938; Jean Piaget, Le méchanisme du développement mental et les lois du groupement des opérations, Archives de Psychologie, 28, 215–285; Essai de logique opératoire (2nd ed., Ed. J.-B. Grize), Paris: Dunod, 1972. The truly vast empirical literature on concrete operational thinking begins with Jean Piaget and Alina Szeminska, La genèse du nombre chez l'enfant, Neuchâtel, Delachaux et Niestlé, 1941, and Jean Piaget and Bärbel Inhelder, Le développement des quantités physiques chez l'enfant, 1941, Neuchâtel, Delachaux et Niestlé, and continues through the books on space, time, movement and speed, geometry, and classification and sedation, which are cited later in this book.

CHAPTER ONE

Abstraction, differentiation, and integration in the use of elementary arithmetic operations

(with Alina Szeminska)

[p. 31]

Our materials for this study were two s...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Reflecting Abstraction in Context
• Part One: Logico-Arithmetical or Algebraic Abstraction
• Part Two: The Abstraction of Order
• Part Three: The Abstraction of Spatial Relationships
• General Conclusions
• Author index
• Subject index