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The Child's Conception of Space
“A massive and most important study. . . . All teachers of young children should know the general outline of the work.” ―Evelyn Lawrence, British Journal of Educational Studies The nature of space, whether an innate idea, the outcome of experience in the external world, or an operational construction has long been a source of philosophical and speculative psychological discussion. This book deals with the development of the child’s notions about space.
- GenresPsychologyEducation
512 pages, Paperback
First published January 1, 1956
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About the author
Jean Piaget
266 books688 followersJean Piaget (1896 - 1980) was a Swiss philosopher, natural scientist and developmental theorist, well known for his work studying children, his theory of cognitive development, and his epistemological view called "genetic epistemology." In 1955, he created the International Centre for Genetic Epistemology in Geneva and directed it until his death in 1980. According to Ernst von Glasersfeld, Jean Piaget was "the great pioneer of the constructivist theory of knowing."
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September 26, 2025
In this arresting text, the Swiss psychologist Jean Piaget and his collaborator Bärbel Inhelder do for space what he has already done for number, viz., to trace the stages of the psychological genesis of the concept in the maturing child. For, just as the child arrives at an adequate concept of number only after going through several stages in which a large mass of perceptions and intellectual schemata are assimilated and integrated (such as relative size, order, seriation, arrangement, quantity, conservation, reversibility of operations such as addition, subtraction, multiplication and division etc.), so too with space. Modern mathematicians regard space as endowed with an ascending hierarchy of properties: topological, differentiable, metric and so on. The authors contend that the development of these ideas in the young child follows the logical order, rather than the historical order in which they were originally discovered. For Euclidean geometry came first, then in the nineteenth century, differential geometry including non-Euclidean spaces and only around the turn of the twentieth century did mathematicians adumbrate the axioms of topology (then known as analysis situs), at the hands of Poincare, Hausdorff and company.
Piaget and Inhelder follow the same method as in Piaget’s earlier work on arithmetic and number, an experimental approach based on interviews with children of various ages. The interviewer poses questions to the child about props selected to illustrate a concept and asks him to predict what will happen if it is viewed from another perspective or submitted to a possible operation, or to make a drawing to depict his understanding. Drawings are important because they reflect what the child has internalized, meaning he must already have gained the relevant concepts and be familiar with them, as opposed to perception, in which, presumably, the child experiences the very same visual field as the adult. The phenomena of which he takes note, of course, depend upon what concepts he has available to him at his current stage of development. From the recordings of these interviews, many of which are reproduced and discussed in the text, the psychologist can surmise a series of conceptual stages the child goes through through a process of induction and analyze the reasons why the development unfolds in the order it does.
Topology comes first, as it involves only qualitative properties such as proximity, relative position etc. A child of three to five years fails to discern the difference between a square and a disk, for instance. Mathematically, we would say that the two are homeomorphic, or deformable the one into the other via a continuous bijective transformation with continuous inverse. In a series of nicely conceived and analyzed experiments, Piaget and Inhelder outline perceptual or sensori-motor space versus representational space, the child’s progress in recognizing shapes (haptic perception), elementary spatial relationships as they are represented in drawings (pictorial space), linear and circular order, the folding and unfolding of knots and the relationship to the surroundings and lastly, the idea of points and the idea of continuity. In each case, they propose a series of three or four stages (with substages) and analyze the transitions between them and the reasons that promote or hinder a transition and development of the mature concept. For instance, they see three stages in the recognition of shapes: I, characterized by the ability to recognize familiar objects but not to pick out their shapes (2 ½ to 4 ½ years), substage IA characterized by passive tactile grasping and exploration while IB involves the beginning of abstraction of shape, at least topological properties thereof (both square and circle are viewed alike as closed shapes); IIA (4 ½ to 5 ½ years) in which one can observe progressive differentiation of shapes according to their angles and even dimensions (triangle versus square and circle, ellipse versus circle); IIB (5 to 5 ½ years), more active though not systemic exploration, distinguishing rhombus, trapezoid, crosses and stars; III (commencing at 6 ½ years), characterized by methodical exploration and concrete operations, ability to recognize, distinguish and represent in drawings all shapes.
Next after topology comes projective geometry. Here, the child has to consciously form the concept of a straight line and to understand that the aspect in which an object appears depends on the perspective from which it is viewed. The authors investigate such things as the projection of shadows, coordination of perspectives (ability to foresee how an object will look from another point of view), geometrical sections, rotations and developable surfaces.
The final stage in the genealogy of the concept of space is the Euclidean, or metrical; here, the child becomes conscious of quantitative properties of geometrical figures and of the space in which they are embedded. These can either be measured directly or estimated. Clearly, projective and Euclidean space are interdependent, as both involve the coordination of multiple points of view. The authors focus on transformations of Euclidean space (affine and similarity transformations). To recognize that two triangles are congruent, for instance, one needs not only the concept of a straight line, but also that of parallelism and slope, or, alternately, interior angle, and then understand that one has to compare the sides respectively angles in succession. The authors do not purport to go into all of the metrical relationships involved in differential geometry (which would lead into difficult questions of curvature, motions of rigid bodies etc.), but limit themselves to the next elementary step: the construction of the concept of Cartesian coordinates. This represents a challenge to the child because he must learn not only to pay attention to internal features of an object, but also to how it sits in relation to other things. In the conclusion, Piaget and Inhelder map out their understanding of the intuition of space and how it relates to psychological development; in particular, the respective roles played by perception, operation and intelligence. The child learns by envisioning and performing concrete operations and seeing what results. At the stage of mature intelligence, these concrete operations become abstract and capable of being entertained in a hypothetico-deductive manner; i.e., the child may ask himself, what would it look like if I did this? Moreover, the operations themselves are reversible and can be integrated into a group-theoretical schema. With this, we connect—however cursorily here—with Piaget’s overall psychology of intelligence.
This reviewer finds the questions Piaget and Inhelder raise throughout fascinating and their technique excellent. In every case, the proposed sequence of mental stages is well grounded in observation and sufficient documentation of actual reports, in children’s own words, is provided. Unless one has had close contact with children, many of the observed phenomena will appear surprising. Nevertheless, the authors explain fairly well and convincingly why they can be expected to occur, given the conceptuality available to the child at its respective stage of development. Moreover, the overall plan of the work appears to be designed to convey with some plausibility how the various concepts build upon themselves in order to issue in the completed concept of space. In consequence, this text turns out to be quite rewarding and suggestive of new possibilities and perspectives.
Just two may be noted here: 1) the authors’ psychological observations are highly pertinent to the philosophy of space, in general (whether or not one subscribe to the doctrine pejoratively referred to as ‘psychologism’). They certainly help one to reconstruct the history of mathematics from the logical point of view. For one can see how new concepts, when introduced, facilitate greater understanding and unleash a wealth of novel possibilities. Read Riemann’s inaugural lecture of 1854 in this light, for example, in which he sketches for the first time the general concept of a manifold in possibly more than three dimensions (which was long a stumbling block; Herbart and even Gauss apparently had difficulty conceiving of spaces of more than three dimensions, as for them, space was what is given to us in experience, namely, three-dimensional). Now, since the rise of topology and modern differential geometry early in the twentieth century, mathematicians have imagined a cornucopia of novel spaces, often radically different from the space of our ordinary experience. It would be very promising if a historian of mathematics were to engage in a systematic study of modern concepts of space since the beginning of the twentieth century, with an eye to a psychological analysis of the concepts and their genesis in the imagination, along the lines sketched for us by Piaget and Inhelder in the simplest of cases. What, for instance, was the crucial insight that allowed Hermann Weyl to formalize his conception of a Riemann surface, and thereby to found the rich field of differentiable manifolds in general? Or what motivated Alexander Grothendieck’s introduction of scheme theory in algebraic geometry, and later topos theory? To date, we have at most anecdotal evidence preserved in the corporate memory of the mathematical community, on the level of a folk tradition. Contemporary philosophy of space and time is perhaps too entranced with Einstein’s general theory of relativity, innovative and revolutionary as it was, adequately to take into account this vaster field of phenomenological experience.
2) Must not the authors’ psychological analysis of concept formation apply to other domains; granted, the case of geometry is comparatively easy to analyze, but would we not expect similar rules to govern the acquisition of non-geometrical knowledge? It seems the field is wide open: potentially physics, algebra and other non-geometrical parts of mathematics, even the child’s self-identity and social awareness etc. could all profitably yield to insights by following Piaget and Inhelder’s genealogical methodology. For this reviewer, it prompts the following reflection, as well: do we not, even as adults, appear as much as children in the eyes of God as the children interviewed here appear in the psychologist’s eyes? (Or, would it be too generous and Socratic to speak thus, as if we could excuse thereby the mysterium iniquitas, putting it down solely to ignorance?)
Piaget and Inhelder follow the same method as in Piaget’s earlier work on arithmetic and number, an experimental approach based on interviews with children of various ages. The interviewer poses questions to the child about props selected to illustrate a concept and asks him to predict what will happen if it is viewed from another perspective or submitted to a possible operation, or to make a drawing to depict his understanding. Drawings are important because they reflect what the child has internalized, meaning he must already have gained the relevant concepts and be familiar with them, as opposed to perception, in which, presumably, the child experiences the very same visual field as the adult. The phenomena of which he takes note, of course, depend upon what concepts he has available to him at his current stage of development. From the recordings of these interviews, many of which are reproduced and discussed in the text, the psychologist can surmise a series of conceptual stages the child goes through through a process of induction and analyze the reasons why the development unfolds in the order it does.
Topology comes first, as it involves only qualitative properties such as proximity, relative position etc. A child of three to five years fails to discern the difference between a square and a disk, for instance. Mathematically, we would say that the two are homeomorphic, or deformable the one into the other via a continuous bijective transformation with continuous inverse. In a series of nicely conceived and analyzed experiments, Piaget and Inhelder outline perceptual or sensori-motor space versus representational space, the child’s progress in recognizing shapes (haptic perception), elementary spatial relationships as they are represented in drawings (pictorial space), linear and circular order, the folding and unfolding of knots and the relationship to the surroundings and lastly, the idea of points and the idea of continuity. In each case, they propose a series of three or four stages (with substages) and analyze the transitions between them and the reasons that promote or hinder a transition and development of the mature concept. For instance, they see three stages in the recognition of shapes: I, characterized by the ability to recognize familiar objects but not to pick out their shapes (2 ½ to 4 ½ years), substage IA characterized by passive tactile grasping and exploration while IB involves the beginning of abstraction of shape, at least topological properties thereof (both square and circle are viewed alike as closed shapes); IIA (4 ½ to 5 ½ years) in which one can observe progressive differentiation of shapes according to their angles and even dimensions (triangle versus square and circle, ellipse versus circle); IIB (5 to 5 ½ years), more active though not systemic exploration, distinguishing rhombus, trapezoid, crosses and stars; III (commencing at 6 ½ years), characterized by methodical exploration and concrete operations, ability to recognize, distinguish and represent in drawings all shapes.
Next after topology comes projective geometry. Here, the child has to consciously form the concept of a straight line and to understand that the aspect in which an object appears depends on the perspective from which it is viewed. The authors investigate such things as the projection of shadows, coordination of perspectives (ability to foresee how an object will look from another point of view), geometrical sections, rotations and developable surfaces.
The final stage in the genealogy of the concept of space is the Euclidean, or metrical; here, the child becomes conscious of quantitative properties of geometrical figures and of the space in which they are embedded. These can either be measured directly or estimated. Clearly, projective and Euclidean space are interdependent, as both involve the coordination of multiple points of view. The authors focus on transformations of Euclidean space (affine and similarity transformations). To recognize that two triangles are congruent, for instance, one needs not only the concept of a straight line, but also that of parallelism and slope, or, alternately, interior angle, and then understand that one has to compare the sides respectively angles in succession. The authors do not purport to go into all of the metrical relationships involved in differential geometry (which would lead into difficult questions of curvature, motions of rigid bodies etc.), but limit themselves to the next elementary step: the construction of the concept of Cartesian coordinates. This represents a challenge to the child because he must learn not only to pay attention to internal features of an object, but also to how it sits in relation to other things. In the conclusion, Piaget and Inhelder map out their understanding of the intuition of space and how it relates to psychological development; in particular, the respective roles played by perception, operation and intelligence. The child learns by envisioning and performing concrete operations and seeing what results. At the stage of mature intelligence, these concrete operations become abstract and capable of being entertained in a hypothetico-deductive manner; i.e., the child may ask himself, what would it look like if I did this? Moreover, the operations themselves are reversible and can be integrated into a group-theoretical schema. With this, we connect—however cursorily here—with Piaget’s overall psychology of intelligence.
This reviewer finds the questions Piaget and Inhelder raise throughout fascinating and their technique excellent. In every case, the proposed sequence of mental stages is well grounded in observation and sufficient documentation of actual reports, in children’s own words, is provided. Unless one has had close contact with children, many of the observed phenomena will appear surprising. Nevertheless, the authors explain fairly well and convincingly why they can be expected to occur, given the conceptuality available to the child at its respective stage of development. Moreover, the overall plan of the work appears to be designed to convey with some plausibility how the various concepts build upon themselves in order to issue in the completed concept of space. In consequence, this text turns out to be quite rewarding and suggestive of new possibilities and perspectives.
Just two may be noted here: 1) the authors’ psychological observations are highly pertinent to the philosophy of space, in general (whether or not one subscribe to the doctrine pejoratively referred to as ‘psychologism’). They certainly help one to reconstruct the history of mathematics from the logical point of view. For one can see how new concepts, when introduced, facilitate greater understanding and unleash a wealth of novel possibilities. Read Riemann’s inaugural lecture of 1854 in this light, for example, in which he sketches for the first time the general concept of a manifold in possibly more than three dimensions (which was long a stumbling block; Herbart and even Gauss apparently had difficulty conceiving of spaces of more than three dimensions, as for them, space was what is given to us in experience, namely, three-dimensional). Now, since the rise of topology and modern differential geometry early in the twentieth century, mathematicians have imagined a cornucopia of novel spaces, often radically different from the space of our ordinary experience. It would be very promising if a historian of mathematics were to engage in a systematic study of modern concepts of space since the beginning of the twentieth century, with an eye to a psychological analysis of the concepts and their genesis in the imagination, along the lines sketched for us by Piaget and Inhelder in the simplest of cases. What, for instance, was the crucial insight that allowed Hermann Weyl to formalize his conception of a Riemann surface, and thereby to found the rich field of differentiable manifolds in general? Or what motivated Alexander Grothendieck’s introduction of scheme theory in algebraic geometry, and later topos theory? To date, we have at most anecdotal evidence preserved in the corporate memory of the mathematical community, on the level of a folk tradition. Contemporary philosophy of space and time is perhaps too entranced with Einstein’s general theory of relativity, innovative and revolutionary as it was, adequately to take into account this vaster field of phenomenological experience.
2) Must not the authors’ psychological analysis of concept formation apply to other domains; granted, the case of geometry is comparatively easy to analyze, but would we not expect similar rules to govern the acquisition of non-geometrical knowledge? It seems the field is wide open: potentially physics, algebra and other non-geometrical parts of mathematics, even the child’s self-identity and social awareness etc. could all profitably yield to insights by following Piaget and Inhelder’s genealogical methodology. For this reviewer, it prompts the following reflection, as well: do we not, even as adults, appear as much as children in the eyes of God as the children interviewed here appear in the psychologist’s eyes? (Or, would it be too generous and Socratic to speak thus, as if we could excuse thereby the mysterium iniquitas, putting it down solely to ignorance?)
October 27, 2025
THE FIRST VOLUME OF THEIR STUDIES OF SPACE AND MEASUREMENT
Jean Piaget (1896-1980) was a Swiss developmental psychologist known for his epistemological studies with children. His theory of cognitive development and epistemological view are known as "genetic epistemology." Bärbel Inhelder (1913-1997) was also a Swiss psychologist, best known for her work with Piaget.
They wrote in the Preface to this 1948 book, “The study of the concept of space, or rather, of the innumerable ideas involve in the concept of space, is for many reasons an indispensable part of child psychology. To begin with, it is clear that if the development of various aspects of child thought can tell us anything about the mechanism of intelligence and the nature of human thought in general, then the problem of space must surely rank as of the highest importance…
“[W]e must apologize for the lengthiness of the text that follows. Even so… it by no means exhausts our original program of research, but is followed by a second volume [‘The Child’s Conception of Geometry’] dealing with metric concepts and problems of Euclidian measurement…. The problem of space is one of such complexity that we were compelled to undertake a large number of experiments before feeling competent to offer any general solution.”
They explain, “during the development of representational space, representational activity is… reflected or projected back on to perceptual activity… Knowing nothing of the stages which led up to this transformation, the adult assumes that perception involves co-ordinate systems or vertical-horizonal relations right from the outset, when in fact such systems are extremely complicated and are only fully developed by the age of 8 or 9.” (Pg. i4)
They note, “Even in the case of everyday images, is it at all possible to imagine a landscape, a house, or other familiar object without the aid, as essential components, of the schema of the various roads traversed, the actions performed, or the changes of position commanding different perspectives? And this assistance is made even more necessary by the fact that such images are nearly always multi-sensorial and refer to complete actions.” (Pg. 41-42)
They state, “Our study of drawing and haptic perception showed that the simplest of topological relationships such as proximity and separation are also the first to emerge in the course of psychological development. This order of appearance is also maintained when space is treated automatically by geometricians. In the realm of psychology it would be true to say that the relation of proximity expresses the most fundamental characteristic of the actions by which the subject generates the notion of space.” (Pg. 80)
They report, “children can easily recognize a straight line, for they distinguish circles from squares at sight, and can easily follow a straight line by arranging matches on an existing line or along the edge of the table. On the other hand, they have no clear idea of what constitutes a straight line, in the sense of a symbolic image going outside the field of perception, or as regards organizing a new construction within the framework provided by perception. Verbally, they do not know what the word ‘straight’ means, only understanding the word ‘line,’ while being unable to draw a straight line in practice. But above all, when it is a matter of forming straight lines with matches, even parallel with the edge of the table… they fail utterly and completely.” (Pg. 159-160)
They summarize, “The main conclusion to be drawn from this discussion is therefore, that global or comprehensive coordination of viewpoints is the basic prerequisite in constructing simple projective relations. For although it may be granted that such relations are invariably dependent upon a given viewpoint, nevertheless… a single ‘point of view’ cannot exist in an isolated fashion, but necessarily entails the construction of a complete system linking together all points of view… This is the prime and fundamental difference between projective space and topological relationships dealt with in Part One.” (Pg. 244)
They explain, “in other fields children find great difficulty in effecting perceptual transpositions of any complexity… the child does not seem unable to transpose dimensional relations perceptually, except for the fact that his errors are larger than those of adults. Rather it is that his perception is subject to a perceptual activity which intelligence controls in a rather crude fashion, in the sense that the gradations between squares and rectangles are still too coarse.” (Pg. 359)
They note, “Here we touch on one of the worst misconceptions which has plagued the theory of geometrical concepts. From the fact that the child breathes, digests and possesses a heart that beats we do not conclude that he has any idea of alimentary metabolism or the circulatory system. At the very most he may have noticed that his movements in breathing, or felt his pulse. But such perceptual-motor awareness does not lead to any understanding of the internal phenomena of which these movements are only the outward and visible sign. Similarly, from the fact that he can stand up or lie flat ,the child at first derives only a strictly empirical awareness of the two postures and nothing more. To superimpose upon this a more general schema he must at some point go outside the purely postural field and compare his own position with those of surrounding objects, and that is something beyond purely empirical knowledge.” (Pg. 378)
They state, “If one takes the trouble to follow the detailed psychological development of the idea of spatial measurement, one encounters a vicious circle, almost as difficult to comprehend as that met with in the problem of time measurement. And here the psychologist is, as so often, grappling with the difficulties and preoccupations of the physicist rather than with those of the mathematician.” (Pg. 475)
This book will be of great interest to those studying Piaget’s (and Inhelder’s) ideas.
Jean Piaget (1896-1980) was a Swiss developmental psychologist known for his epistemological studies with children. His theory of cognitive development and epistemological view are known as "genetic epistemology." Bärbel Inhelder (1913-1997) was also a Swiss psychologist, best known for her work with Piaget.
They wrote in the Preface to this 1948 book, “The study of the concept of space, or rather, of the innumerable ideas involve in the concept of space, is for many reasons an indispensable part of child psychology. To begin with, it is clear that if the development of various aspects of child thought can tell us anything about the mechanism of intelligence and the nature of human thought in general, then the problem of space must surely rank as of the highest importance…
“[W]e must apologize for the lengthiness of the text that follows. Even so… it by no means exhausts our original program of research, but is followed by a second volume [‘The Child’s Conception of Geometry’] dealing with metric concepts and problems of Euclidian measurement…. The problem of space is one of such complexity that we were compelled to undertake a large number of experiments before feeling competent to offer any general solution.”
They explain, “during the development of representational space, representational activity is… reflected or projected back on to perceptual activity… Knowing nothing of the stages which led up to this transformation, the adult assumes that perception involves co-ordinate systems or vertical-horizonal relations right from the outset, when in fact such systems are extremely complicated and are only fully developed by the age of 8 or 9.” (Pg. i4)
They note, “Even in the case of everyday images, is it at all possible to imagine a landscape, a house, or other familiar object without the aid, as essential components, of the schema of the various roads traversed, the actions performed, or the changes of position commanding different perspectives? And this assistance is made even more necessary by the fact that such images are nearly always multi-sensorial and refer to complete actions.” (Pg. 41-42)
They state, “Our study of drawing and haptic perception showed that the simplest of topological relationships such as proximity and separation are also the first to emerge in the course of psychological development. This order of appearance is also maintained when space is treated automatically by geometricians. In the realm of psychology it would be true to say that the relation of proximity expresses the most fundamental characteristic of the actions by which the subject generates the notion of space.” (Pg. 80)
They report, “children can easily recognize a straight line, for they distinguish circles from squares at sight, and can easily follow a straight line by arranging matches on an existing line or along the edge of the table. On the other hand, they have no clear idea of what constitutes a straight line, in the sense of a symbolic image going outside the field of perception, or as regards organizing a new construction within the framework provided by perception. Verbally, they do not know what the word ‘straight’ means, only understanding the word ‘line,’ while being unable to draw a straight line in practice. But above all, when it is a matter of forming straight lines with matches, even parallel with the edge of the table… they fail utterly and completely.” (Pg. 159-160)
They summarize, “The main conclusion to be drawn from this discussion is therefore, that global or comprehensive coordination of viewpoints is the basic prerequisite in constructing simple projective relations. For although it may be granted that such relations are invariably dependent upon a given viewpoint, nevertheless… a single ‘point of view’ cannot exist in an isolated fashion, but necessarily entails the construction of a complete system linking together all points of view… This is the prime and fundamental difference between projective space and topological relationships dealt with in Part One.” (Pg. 244)
They explain, “in other fields children find great difficulty in effecting perceptual transpositions of any complexity… the child does not seem unable to transpose dimensional relations perceptually, except for the fact that his errors are larger than those of adults. Rather it is that his perception is subject to a perceptual activity which intelligence controls in a rather crude fashion, in the sense that the gradations between squares and rectangles are still too coarse.” (Pg. 359)
They note, “Here we touch on one of the worst misconceptions which has plagued the theory of geometrical concepts. From the fact that the child breathes, digests and possesses a heart that beats we do not conclude that he has any idea of alimentary metabolism or the circulatory system. At the very most he may have noticed that his movements in breathing, or felt his pulse. But such perceptual-motor awareness does not lead to any understanding of the internal phenomena of which these movements are only the outward and visible sign. Similarly, from the fact that he can stand up or lie flat ,the child at first derives only a strictly empirical awareness of the two postures and nothing more. To superimpose upon this a more general schema he must at some point go outside the purely postural field and compare his own position with those of surrounding objects, and that is something beyond purely empirical knowledge.” (Pg. 378)
They state, “If one takes the trouble to follow the detailed psychological development of the idea of spatial measurement, one encounters a vicious circle, almost as difficult to comprehend as that met with in the problem of time measurement. And here the psychologist is, as so often, grappling with the difficulties and preoccupations of the physicist rather than with those of the mathematician.” (Pg. 475)
This book will be of great interest to those studying Piaget’s (and Inhelder’s) ideas.
January 29, 2022
I want to read this book
April 16, 2008
I actually have the hardcover. It's interesting to know which is more important between a square and a circle. There are many more to read.
Displaying 1 - 4 of 4 reviews