CHAPTER NINE

Quantity, Logic, Time, Movement, and Velocity

C H A P T E R 8 dealt with what might be called the first wave of Piaget's experimental assault on the development of intelligence in the postinfancy years. There was to be a second wave, of larger scale and longer duration. This second wave of research had a small beginning in the period 1935-1940 with a few scattered journal articles (e.g., Inhelder, 1936; Meyer, 1935; Szeminska, 1935; Piaget and Szeminska, 1939) and moved into high gear in 1941 with the publication of the first (Piaget, 1952b) of a long series of full-length books on various cognitive-developmental problems: number, quantity, logic, space, time, and so on.

QUANTITY There are several studies which touch on one or another aspect of the child's grasp of quantity notions (Apostel, Morf, Mays, and Piaget, 1957; Fischer, 1955; Inhelder, 1936; Piaget, 1960a; Piaget and Szeminska, 1939; Szeminska, 1935). The earlier papers are primarily of historical interest, since their contents have for the most part been incorporated into the systematic book on the subject by Piaget and Inhelder, Le Developpement des quantites chez I'enfant (1941). This book is divided into four sections, each three chapters long. In addition, there is the customary chapter of summary and conclusions at the end of the book. The first section deals with what is probably the best-known segment of the quantity work: the so-called conservation of matter, weight, and volume of an object in the face of changes of shape. The basic technique is a simple one (ibid., p. 7). The experimenter gives the subject a ball of clay and asks him to make another exactly like it—"just as big and just as heavy." After the child has done this, the experimenter retains one of the balls as a standard of comparison and changes the appearance of the other by stretching it into a sausage, flattening it into a cake, or cutting it into several pieces. T h e experimenter then attempts to find out whether the child thinks the 298

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amount of clay, the weight, and the volume have changed or have remained invariant (i.e., conserved) as a result of the transformation. The method of inquiry varies with the type of quantity notion investigated. For matter (amount of clay, or global quantity), the child is simply asked if the standard and the altered piece of clay both possess the same amount of clay (la mime chose de pate is the usual expression here); occasionally, this is concretized in terms of "just as much to eat," or something of that sort. In the case of weight, a scale balance is used (the book is not clear as to whether it is used with all subjects) and the experimenter asks if variable and standard weigh the same (la mime chose lourd), or would keep the scale arms horizontal if placed on opposite pans. The assessment is more indirect in the case of volume. A glass container with water in it is used as the common measure. The experimenter shows that each ball of clay, when placed in the container, causes the water level to rise to the same height. He then alters one of the balls and asks if it wall still make the water rise to that same height. Note the analogy between these kinds of problems, given to preschool children and older, and the problem confronting the infant in acquiring the object concept (Chapter 4). In the latter case, the acquisition consists of discovering that the sheer existence of an object remains invariant, is conserved, despite changes of position in space (particularly, whether it is in or out of the infant's visual field). In the present experiments and in most of the conservation studies,1 a similar but much more subtle acquisition is required: to discover that certain attributes of an object remain invariant in the wake of substantive changes in other attributes. The principal findings of these studies are as follows. First, each type of quantity concept (matter, weight, and volume) shows about the same developmental trend: (1) no conservation; (2) an empirically founded, "on and off" sort of conservation, i.e., the child tentatively hypothesizes conservation for some transformations but denies it for others; and (3) a logically certain, almost axiomatic assertion of conservation in the case of all transformations for the type of quantity concept in question. The other major finding is that, despite this apparent similarity among tasks, of the conservation of matter, weight, and volume are not achieved of a piece. For Piaget's subjects, conservation of matter seems to become common at 8-10 years of age, of weight at 10-12, and of volume only at 12 years and after.2 1 We shall shortly describe one particular conservation task, involving the dissolving of sugar in water, whose formal properties are even more like those of the object concept problem. 3 The only statistic to be found in the quantity book bears on this horizontal decalage among the three types of quantity concept. Of 180 children aged 4-10 years, 55 showed no conservation of any kind, 67 showed conservation of matter alone, 38 of matter and weight but not volume, and only 20 of all three (ibid., p. 12). Although this is not explicitly stated, one gets the impression that developmental reversals (e.g., conservation of volume achieved before conservation of weight) were rare or absent.

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Piaget's interpretation of these findings is complex and detailed, but the key elements appear to be the following. There are probably two developing schemas which together contribute to the acquisition of matter conservation, i.e., conservation of the global "amount of stuff" in the piece of clay. There is first the general capacity to multiply relations (ibid., pp. 24-25), already described in Chapter 5 apropos of concreteoperational Grouping VII. Consider the case where the ball of clay is transformed into a sausage. Conservation of matter will be a likelier outcome if the child notices both length and thickness changes and can apprehend that what the clay has gained in length it has lost in thickness (leaving total quantity invariant). The second schema, closely related to the first, is called atomism (ibid., pp. 28-29). Again, the belief in conservation becomes more probable if the child can conceive of the clay as a whole composed of tiny parts or units which simply change their location vis-a-vis one another when the whole undergoes a transformation of shape. Conservation of matter is here the expression of the fact that the total sum of these parts remains the same, whatever their spatial distribution. What prevents the child, once in possession of these intellectual tools, from immediately extending the invariance of matter to that of weight and volume? In the case o£ weight, subject protocols suggest the following difficulty (ibid., pp. 36-40), While readily granting that the total number of tiny units of clay always remains the same (and thereby granting conservation of matter), the subject may yet believe that the weight of each unit varies with its location in the whole. Egocentric prenotions about the nature of weight (weight is the sensation of pressure on my hand when I hold an object, etc.) seem to pose a specific obstacle to the conservation of weight, even when the child is fully in possession of the schemas necessary for conservation of matter. A parallel obstacle exists in the case of volume. Piaget believes that nonconservation of volume (where the other two conservations are established) results specifically from an implicit belief that each tiny unit of clay varies in the amount of space it occupies, compresses and decompresses, alters its density, as a function of its position in space following transformation of the whole (ibid., pp. 65-66). The conservation of volume is a late achievement because, as we shall see, the requisite schemas relating to density and compression-decompression of matter are themselves late achievements. One of the major conjectures of the first section, then, is that there are certain schemas concerning the physical characteristics of objects whose aquisition at least facilitates the formation of the quantity conservations. However, it remains for experiments other than those just described to bring the development of these schemas more fully to light. The second section describes such an investigation in the case of the atomism schema; the third section does the same for those of density and compression-

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decompression. In the atomism study, the following task was administered to about 100 children of 4-12 years of age (ibid., pp. 83-84). The subject was shown two identical glasses containing equal quantities of water, and established their equivalence of weight on a scale balance. The experimenter then put two or three pieces of sugar in one of the glasses and marked the height to which the water rose. A number of questions were asked of the child, both before and after the pieces of sugar completely dissolved in the water. As in the earlier studies, the primary intent of the questions was to find out what the child thought would remain invariant as the sugar slowly changed state. Its qualities (sweet taste)? Its existence as a substance? Its weight? Its volume? The first developmental stage here is an interesting one. The younger children appear to think that the sugar becomes completely annihilated as an existent when it dissolves, much as the infant regards an object as no longer existing which has passed out of the visual field (see footnote 1). Curiously enough, however, many of these same children do believe that a (disembodied) sugar taste will somehow be left behind in the water— a kind of "conservation of taste" reminiscent of the "conservation of smile" in Lewis Carroll's Cheshire cat! But the majority feel that even this poor vestige will disappear in a day or two and the water will again become tasteless. Stage 2 is a complex one, comprising various transitional phases. Its essential criterion is the assertion that sugar-as-existent does indeed remain invariant after the sugar has dissolved. Furthermore, the more stable and definite this belief in conservation, the more likely it is to entail an atomistic rather than some other rationale: that is, the sugar is not really "gone"; it still exists as very tiny, invisible particles spread throughout the water. As in the previous conservation experiment, this belief in the continued existence of the sugar in the form of microscopic grains does not automatically bring with it the conservation of its weight and volume. The children of stage 2 are quite ready to assume that the tiny grains of sugar (the belief in the existence of which leads to belief in the conservation of substance) are by their very diminutiveness not endowed with either weight or volume. These invariances are achieved later in childhood; first weight, then volume, just as in the preceding conservation study. There are several experiments which deal with the development of conceptions of density and compression-decompression. In one, the experimenter heats a piece of popcorn until it pops and asks the child, first, whether or not the amount of matter and the weight have remained invariant, and second, why the volume has changed. In another, the child is shown several objects of different density and is asked various questions about amount of matter, weight, volume, and their interrelations (e.g., why this object is smaller but heavier than that object, etc.). Piaget

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draws two major conclusions from the results of these and other experiments described in the third section. First of all, the data here point towards a general conclusion about the evolution of quantity concepts, a conclusion which the previous studies had also suggested. These concepts begin by being confused and undifferentiated in the young child's cognition and only gradually emerge from this undifferentiated totality as separate, stabilized quantity concepts (e.g., ibid., pp. 134, 183-184). Thus, in the beginning there really is no concept of amount of matter, or weight, or volume, distinct and separate from each of the others. A little later, as we have seen, amount of matter differentiates from this conglomerative concept to become a rational affair for which conservation can be predicated, for which subquantities always sum to the same total quantity, and so forth. However, weight and volume are at this point still undifferentiated and, a posteriori, are not yet separably rational concepts which can submit to reversible operations. Still later, these two also articulate from one another, and each in its turn goes on to become a genuinely quantitative construct. The popcorn experiment illustrates quite clearly the earlier stages of this differentiation process. The younger subjects immediately assume that the piece of corn weighs more after it has popped because it is "bigger." Weight and volume (for these children, a kind of global "bigness") are apparently not seen as distinct and different properties which can vary independently, although usually correlated in nature. It is only after the child recognizes the logical independence of these properties that any idea of a genuine quantification of either becomes possible (e.g., ibid,, p. 315). The second major conclusion is more specific to the experiments at hand, although also obviously relevant to those described earlier: a genuine grasp of the concept of volume and of its relation to weight requires the development of a schema of substance density and related concepts concerning compression and decompression of matter. Piaget's data suggest that about the time the child becomes capable of managing volume problems he also shows the following sort of conception about the nature of matter (e.g., ibid., pp. 130-133, 183). Substances are composed of numerous tiny parts or elements with empty spaces in between. Substances (and of course the objects made from them) can vary as to how tightly these elements are compressed or packed together, i.e., how much of the total volume is really substance and how much is essentially empty space. Objects which are heavy for their size are thus composed of tightly packed elements; lighter ones are more loosely packed, with lots of empty spaces in between. The transformation of volume in the popcorn study is readily explained by children who think in these terms. The tiny elements of which the corn is composed have simply decompressed and are therefore farther apart from each other than they were before. Piaget believes that through the auspices of this underlying

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schema about the nature of matter the child finally works out a coherent and consistent notion of volume, one which permits him to contrast and relate it to weight (density problems of the third section) and to establish it as an invariant under certain transformations of form and state (conservation problems of the first and second sections). The first three sections of the quantity book deal with the child's understanding of the concepts of amount of matter, weight, and volume, either directly, or in terms of other concepts (atomism, density, etc.) thought to underlie this understanding. The fourth section concerns the child's developing capacity to perform certain basic logical and mathematical operations on these concepts, particularly operations involving the addition of asymmetrical and symmetrical relations. Such operations are, of course, very general (we shall encounter them again in other Piaget volumes). They are studied here as "forms" specifically applied to the quantity notions as "contents." There are a number of experimental questions here. Can the child successfully seriate objects of valuing weights, particularly when volume and weight are not correlated across the object series? Does the child recognize the transitivity principle as it applies to equal and unequal weights and volumes? To illustrate the general tenor of these experiments, let us examine more closely some of the paradigms for investigating transitivity of weight. Where the transitivity principle applies to inequalities of weight, one such paradigm entails giving the child three objects of different weight (but weight uncorrelated with volume) and asking him to seriate them by weight (heaviest, middle, lightest), with the added condition that he can compare the weight of only two objects at a time. It turns out that young children have considerable difficulty in solving problems of this kind. For instance, the child may think that it suffices to establish A < C and A < B alone in order to conclude that A < B < C or A < C < B; and conversely, A < B < C is not for him a logically necessary conclusion from the knowledge that A < B and B < C. The experimental technique and results are the same in the case of equalities of weight, e.g., that A = B and B = C does not in itself guarantee that A = C for the young child, and A + B is not necessarily equal to C + D after establishing A = B = C = D. In general, it can be concluded that formal compositions of this sort cannot be managed until the quantitative concepts in question become stable entities endowed with conservation. LOGIC The investigations to be reported in this section are actually of narrower compass than the title "Logic" suggests (are there any Piaget experiments which do not have something to do with logic?). These studies

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were designed specifically to uncover the origins of classification and seriation operations and their genesis from early to middle childhood. This work is systematically set forth in Piaget and Inhelder, 1959; however, there are previews of parts of it in earlier publications (Piaget, 1952b, 1957-1958; Inhelder, 1955), and a book review provides an excellent brief summary (Donaldson, 1960). The Piaget and Inhelder book is bounded by an Introduction and a Conclusion, both oriented towards theory, with ten experimental chapters in between.3 The first eight of these chapters concern classification behavior; the last two deal with seriation. Perhaps the most interesting and important research in La Genese des structures logiques elementaires is that reported in the first four chapters, and it is this research that we shall cover in most detail. Two types of experimentation are described here. The first concerns free classification: the child is given a potpourri of objects (geometric forms, cut-outs of people, animals, plants, and the like—the precise collection varying with the experiment) and told to put those together which "are similar" or "go together." The second type of study deals with the child's understanding of the relation between a class and its subclasses. From these experiments the authors conclude that there are three rough stages in the developing mastery of elementary classification operations. In stage 1 (21/2-5 years), the child tends to organize classifiable material, not into a hierarchy of classes and subclasses founded on similarities and differences among objects, but into what the authors term figural collections. The sorting behavior of this stage has several distinguishing characteristics. First, it is a relatively planless, step-by-step affair in which the sorting criterion is constantly shifting as new objects accrue to the collection. The expression Piaget uses to describe it—a sorting which proceeds de proche en proche {ibid., p. 285)—is wonderfully descriptive. Second, and partly in consequence of this inch-by-inch procedure bereft of a general plan, the collection finally achieved is not a logical class at all but a complex figure (hence figural collection). The figure may be a meaningful object, e.g., the child decides (often post hoc) that his aggregation of objects is "a house." Or instead, it may simply be a more or less meaningless configuration. It should be made clear here that figural factors are not the sole determinants of the child's groupings at each and every step in the sorting sequence. Frequently, at least part of the child's collection
8 There is one striking difference between this book and Piaget's earlier ones, namely, its marked (for Piaget) bent towards quantitative presentation of findings. Two examples: the book virtually begins by stating the total number of subjects tested (2159); as a whole it contains 35 data tables (most of its predecessors contained none). The authors make it clear that this effort at quantitative presentation is intended to disarm the criticism that Piaget's elaborate theoretical architecture is generally founded on the sand of small N's (Piaget and Inhelder, 1959, Preface)—a criticism they had apparently been catching from all quarters!

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is founded on a Minilarity-of-attribut.es basis. What often happens is that the child begins by putting similar objects together, as though a genuine classification were in progress, and then "spoils" it by incorporating his "class" into a nonclass, configurational whole. One 3-year-old, for instance, put some circles into a pile, then put some squares together next to the circles, then continued the process with other groups of objects. What appeared to be a sequence of constructed logical classes revealed its true configurational colors when the child looked at what he had made and said: "Un train, tsch, tsch, tsdi!" (ibid., p. 40). Piaget asserts that there are two related difficulties at the root of the young child's inability to compose genuine classes (e.g., ibid., pp. 50-52). The first difficulty is that the child, in his alternating reliance on similarity and configurational criteria, indicates that he cannot yet differentiate two essentially different kinds of colligation: the formation of a logical class and the construction of an infralogical whole. The differentiation and separate development of these two kinds of operations will be among the more important achievements of the concrete-operational period (see our Chapter 5). As in the case of the three-year-old mentioned above, partial groupings based on attribute similarity do occur at this age, but they are conceptually fragile and unstable affairs, forever in danger of turning into infralogical totalities. The second underlying difficulty is just as important, in Piaget's view. The stage-1 child also cannot differentiate, and hence cannot coordinate, class comprehension (the sum of qualities which define membership in a logical class) and class extension (the sum total of objects which possess these criterial qualities). In a genuine classification, these two class properties must always be in strict correspondence: the definition of the classification basis determines precisely which objects must constitute its extension, and the nature of the objects in a given collection places tight constraints on the definition of the class they together form. But for the young child, there seems to be no such strict correspondence. For example, the comprehension of the "class" he begins to construct does not determine a unique extension, as it must in true classification. Thus, he begins by putting squares together, but he does not include all the squares present, or he contaminates his collection with nonsquares. In the same way, we have seen that the extensions he does end up with frequently determine not a class but an infralogical whole, and thus they effectively take him out of the realm of logical classification altogether. One can partially summarize the young preoperational child's difficulties in classification this way. Several years of experience in applying sensory-motor schemas to reality have provided him with ample cognitive equipment to "see" similarities between objects and gather them into collectivities on the basis of these similarities. But the mere possession of this ability leaves him farther from a genuine grasp of classification than

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might be suspected. For one thing, he has yet to learn to form similarityof-attributes groupings which remain untainted by the ever-intrusive configurational factors. That is, he needs to distinguish logical from infralogical operations. But in addition, he has yet to manage the essential coordination between class comprehension and important extensional notions like "some" and "all." The latter notion figures in any classification; the former becomes of particular importance where hierarchies of classes are to be dealt with. As Piaget shows in his analysis of the subsequent two stages, a full and complete grasp of the comprehension-extension relation matures surprisingly late. Furthermore, special testing is required to diagnose hidden gaps in this understanding. The child's ability to bandy about classification-relevant phrases (e.g., "dogs are animals," "some of these are red," etc.), either under ordinary questioning or in spontaneous discourse, is likely to be a most unreliable guide.4 Figural collections give way to nonfigural collections around by^-l-S years (stage 2). The child now forms groups of objects on a similarity-ofattributes basis alone, tries to assign every object in the display to one or another group, and can even partition major groups into their constituent subordinate groups. In short, he now appears to be in command of genuine classificatory operations. Why then, does Piaget still call his productions "collections"—albeit nonfigural ones—rather than "classes"? In what way can he be said to fall short of possessing a full-fledged concreteoperational structure o£ classifications? What is still lacking, Piaget contends, is a subtle and hard to diagnose but nonetheless crucial ability to grasp and keep constantly in mind the inclusion relation obtaining between a class and its subclasses, to recognize that a subclass A is included in class B but does not exhaust it (hence, to recognize that A — B — A') and to keep this A — B relation firmly in mind across all manner of changes in the spatial distribution of class and subclass or in one's distribution of attention regarding them. The capacity to do this entails the precise coordination of class comprehension and extension discussed earlier; one must at every step compare the different but overlapping extensions of class and subclass, e.g., B includes the extension of A but adds to it that of A'. For Piaget, mastery of the inclusion relation, with all that its mastery implies, is the sine qua non of a concrete-operational (stage 3) as opposed to late preoperational (stage 2) cognition of logical classification: In the case of inclusion, the subsuming class B continues to subsume . . . whether the subsumed parts [i.e., subordinate subclasses] A and A' are ac* As Donaldson aptly puts it in her review of the Inhelder and Piaget book: "It might not be too inadequate a summary o£ the book to say that it consists in an attempt to show that, in the absence of special inquiry, the child's ability to handle language may grossly mislead us as to his ability to handle classificatory systems" (Donaldson, I960, p. 182).

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tually brought together (as a collection of continuous elements or by an act of abstract "colligation") or are dissociated under the form A =B —A' (in space or by abstraction). Contrariwise, the essence of a collection as opposed to a class is that it exists only when its elements are continuous in space (even if the basis of their being together is no longer figural), and consequently ceases to exist qua collection when its sub-collections are separated from each other: the result is that, when the sub-collections are reunited under the form A -f- A\ the subject indeed sees them as constituting the whole B (thus A -j- A' = B), but when the sub-collections are dissociated, in space or even simply in thought, the child no longer sees them as constituting the supraordinate collection and is thus shown to be incapable of the operation A = B — A'. An operation being reversible by definition, we conclude that if the inverse operation A = B — Af is still inaccessible to the subject, the union A + A' — B does not at stage II yet constitute a direct operation, but simply an intuitive union by momentary differentiation of the collection B into sub-collections A and A' (ibid., pp. 55-56).

The book describes several experiments designed to illustrate the older child's better management of the inclusion relation. Indirect evidence comes from a series of experiments (Chapter 7) which indicate his greater flexibility and mobility in ascending and descending a class hierarchy, in shifting criteria and reclassifying a previously classified array of objects, in anticipating what a hierarchy will contain in advance of constructing it, and the like. However, the most direct evidence is provided by two ingenius experiments reported in Chapters 3 and 4. One involves the child's ability to handle the class quantification concepts "some" and "all" as applied to classes and subclasses in a hierarchy. The child is presented with a series of objects which can be partitioned into several sets of classes and subclasses. For example, the display might consist of 2 red squares, 2 blue squares, and 5 blue circles (ibid., p. 65, Fig. 7). This display yields these classes and subclasses: the class of blue objects (B) with subclasses of blue circles (A) and blue squares (A'); the class of squares (B) with subclasses of blue squares (A) and red squares (A'). The questions posed the child are of two forms: Are all the B's A (or A'), i.e., are all the blue ones circles, are all the squares red, etc.? Are all the A's (or A''s) B, i.e., are all the circles blue, are all the red ones squares, etc? A simplified statement of the results of this study is the following. The younger child appears to construe both types of questions as asking whether A and B are of identical extension, i.e., A =B7 As a general rule, this simplification of the task leads him to a correct response to the first type of question; for example, he compares the extension of blue objects (B) and circles (A) and rightly concludes that not all B is A because there are also some blue squares {A'). However, this strategy leads to an incorrect response to questions of the second type. If asked whether all the circles are blue, he gives the astonishing answer that they are not, because there are also blue squares! In effect, the child is interpreting the

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second type of question, not as asking whether all A is some B, but as asking whether all A is all B, The second experiment is a replication, with variations, of one Piaget carried out years ago and reported in the book on number (Piaget, 1952b). The experimental paradigm is this. The child himself constructs a class hierarchy out of a set of objects or pictures of objects before him, e.g., the class of flowers (B) with subclasses of primroses (A) and other flowers (Af). The experimenter then makes sure the child understands the basic properties of his simple hierarchy: that a bouquet of all theflowers(B) would contain the primroses (A) and the others {A'), i.e., B — A + A''. This established (and children of stage 2 generally have little difficulty here), the experimenter poses "quantification of inclusion" questions concerning the relative extensions of B and A. Are there more primroses or more flowers here? Would a bouquet of all the flowers be bigger, smaller, or the same as a bouquet of primroses? The stage-2 child fails these questions, usually because he makes a quantitative comparison, not between A and B, but between A and A', e.g., "there are more primroses (.-4) because there are only a few of the others (Af)." The following is a more or less typical behavior protocol:
THE (5;6). "If I make a bouquet of all the primroses and you make one of all the flowers, which will be bigger?—Yours.—(the experimenter takes 4 primroses and 4 other flowers and repeats the question.)—The same (A — A').—If you gather all the primroses in a meadow will any flowers remain?— Yes.—And if you gather all the flowers will any primroses remain?—Yes . . . no.—Why?—Because you take all the flowers.—And if one gathers all the yellow primroses will any primroses remain?—Yes, there will still be the violet ones.—And if one gathers all the primroses, will there be any yellow primroses left?—No, because you take all the primroses and there aren't any left." The questions on quantification of inclusion still remain insoluble (Piaget and Inhelder, 1959, p. 108).

The following is a somewhat simplified account of Piaget's interpretation of these two experiments. The stage-2 subject cannot yet quite dominate the logical inclusion operation, epitomized by the logical equation A = J5 — A' (see again the Piaget quotation cited on p. 306). In the case of the first experiment, he is unable to recognize that the "all" of A does in fact correspond to the "some" (although not the "all") of B, as the equation A — B — A' precisely expresses. He does not clearly understand that "included in" is not synonymous with "equals," and this indicates that his mastery of the structure of a class hierarchy is still incomplete. In the-second experiment, he is unable to keep in mind the class B (with A a subclass in it) when his attention is directed to A itself. Again, this can be construed as an inability to perform the operation A = B — A'. In effect, what he needs to be able to think, and cannot, is this: "I recognize that A is still a subportion of B, and hence of lesser extension than

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B, even though I have momentarily abstracted it from B in order to compare their extensions." Piaget believes that the child can recognize that A and A' comprise B when he focuses attention on the whole B (thus, he can perform B — A + A'), but "loses" B (and the fact that A — B — Ar) when he isolates A as a comparison term. With B momentarily inaccessible as an object of thought, the child cannot do other than compare A with its complement A'. We shall deal with the remainder of the book very briefly. Its fifth chapter adduces further evidence on the development of the inclusion relation by means of special experiments on the child's management of complementary, singular, and null classes. The seventh chapter does the same with studies of the capacity to manipulate class hierarchies in a planful and flexible manner: to anticipate a hierarchy in advance of actually constructing it, to construct new hierarchies with old materials by changing the classification basis, and the like. The sixth chapter describes investigations of class multiplication abilities by means of matrix tasks (like those of Raven Progressive Matrices). The authors conclude that class multiplication and class addition are approximately synchronous developmental attainments; indeed, the former may even appear to be more precocious because of certain facilitating perceptual properties inherent in matrix tasks. The eighth chapter shows that a tactokinesthetic rather than visual presentation of classifiable materials yields the same three developmental stages: figural collections, nonfigural collections, and genuine classification. The last two chapters deal with the ability to serialize rather than classify. The ninth chapter describes experiments on simple additive seriation which are essentially variations on studies done earlier (Piaget, 1952b). An interesting finding here is that children can apparently construct a series of sticks of graded lengths in a drawing before they can produce it in reality, i.e., before they can actually arrange the sticks in order of length. Finally, the tenth chapter reports a study concerning the multiplication, not of classes, but of asymmetrical relations. Its apparent complexity notwithstanding, the ability to multiply several asymmetrical series together also appears to emerge in rough developmental concordance with the other abilities we have been describing. NUMBER There are a number of publications which report the work of Piaget and his associates in this area. The basic reference is The Child's Conception of Number (Piaget, 1952b), and the present account is taken almost completely from this source. There exists an excellent detailed summary of this book (National Froebel Foundation, 1955); briefer and less cor:

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plete synopses are also available, including several by Piaget himself (e.g., 1953a, 1956). In addition, there are miscellaneous other publications which deal with one or another aspect of the number research (e.g., Inhelder and Noelting, 1957; Piaget, 1937c; Piaget and Inhelder, 1941, 1959; Piaget and Szeminska, 1939; Szeminska, 1935). And finally, there is some interesting more recent research in the area by the Geneva group and others (e.g., Apostel, Mays, Morf, and Piaget, 1957; Greco, Grize, Papert, and Piaget, 1960). Some of this work will be described in Chapter 11. More than was the case with the quantity and logic research, the number experiments require some stage-setting preamble before their intended significance can emerge clearly. In particular, two preliminary questions need answers. (1) Precisely what sorts of mathematical skills or knowledges did Piaget have in mind to study? (2) What is his working conception of the nature of number and of arithmetic operations? And particularly (3) what are the basic skill components which these operations are thought to entail? For the first, the simplest answer is that Piaget was and is much more interested in a kind of "number readiness" than in arithmetic achievement as such. He wanted to probe and diagnose for developing numberrelevant capabilities considerably more subtle and basic than those involved in the familiar elementary operations of counting, of rote addition, subtraction, etc., i.e., the mundane arithmetic behaviors one tends to associate with the traditional primary-school classroom. The capabilities he wanted to study have to do more with the essential and fundamental properties of the number system, the underlying assumptions about the nature and behavior of numbers which the ordinary adult tacitly makes—tacitly because they are so ingrained and "obvious"—in his routine arithmetic operations. This way of approaching the problem is, of course, not unique to the area of number for Piaget. For example, in the quantity studies an analogous attempt was made, first, to isolate a similarly tacit, because obvious, assumption about quantity concepts, namely, their conservation in the face of shape changes, and then to show that young children do not necessarily make this assumption, i.e., that conservation of quantity is in fact something which needs developing. But to isolate the proper underlying assumptions and capabilities in a given area, those which will pay off in developmental study, it is necessary to do a kind of job analysis of that area. In the case of number, this involves an attempt to find out what it is that numerical operations really entail in the way of component skills and beliefs and what prior acquisisitions these operations imply. Piaget has made such an analysis of number, and it is this analysis which has largely set the course for his experimentation (Piaget, 1952b, pp. viii, 94-95, 156-157, 182-184, 243; see also our Chapter 5, p. 198). According to Piaget, number is essentially a fusion or

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synthesis of two logical entities: class and asymmetrical relation. If one enumerates a set of objects and thereby arrives at its cardinal-number value ("there are 10 objects here"), one is in effect treating the objects as though they were all alike, just as one would do if one assigned them to a common class. Just as we disregard object differences in classifying a set of objects, so also do we disregard object differences in assigning the set its cardinal value. Thus, number clearly has a class component to it. Is a number simply a class, then? No, because although the enumerated objects are, as just stated, treated as equivalent to one another in so far as their being assigned a cardinal number is concerned, there is also a sense in which they are regarded as different from one another—not the case in class operations. In the process of discovering their cardinal value by enumeration, one has to order the objects: count this object first, then the next, then the next, and so on. It obviously makes no difference what the order of enumeration is, but there must be some order; one has to count them in some sequence and keep track of which have already been enumerated so as not to count the same objects more than once. This ordination5 process partakes, not of class, but of relation operations. The objects arranged in the order in which one enumerated them form a true series, a set of as)mmetrical relations, exactly analogous to a series of sticks of graded lengths. Here, however, the object differences are not of length but of ordinal position ("first object counted," "second object counted," etc.). Numerical units have, therefore, a peculiar status; they appear to be both class elements and asymmetrical relation elements at one and the same time. In one respect they are all equivalent, just as class elements are: in another respect they are all different, like the terms of an asymmetrical series. In order to count them, they must be counted seriatim; once counted, they are again all indistinguishable, just "10 objects." If this analysis of the nature of number is accurate, then it suggests for Piaget that developmental study of the fundaments of numerical operations must have a very broad base. It will, of course, include investigation of the child's understanding of ordination, cardination, and their interrelations. It will also, of course, deal with the child's grasp of the essential additive and multiplicative properties of numbers. And it will also study the genesis of mathematical notions related to the above, e.g., the operation of one-one correspondence as the basis for both cardinal equivalence of sets and for multiplication. But it should also include study of class and seriation operations themselves, both directly and as they figure in the above numerical operations. Actually, Piaget sees
6 Piaget uses the terms ordination and cardination to refer to operations concerning ordinal and cardinal numbers, respectively. These are useful nouns (although not in English usage with these meanings, so far as the writer can discover) and will be retained here.

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classes, relations, and numbers as cognitive domains which develop synchronously in a tightly intertwined, mutually dependent way.6 Cognition of the extensional aspects of classes (see the previous section), for example, requires the prior acquisition of certain notions in the number area (e.g., "none," "some," "all"). But reciprocally, and more to the present point, Piaget strongly adheres to the view that a genuinely operational (in the concrete-operational sense) grasp of number necessitates a similarly operational management of classes and relations. When the child is capable of reversible seriation operations and of genuine classification (inclusion relation and all), then and only then will he be in a position to really understand what numbers are and how they behave. The "very broad base" of Piaget's number research becomes immediately apparent when one skims the number book. The first chapter deals with conservation of quantity (a direct carry-over from the quantity research) as a prelude to a conservation of cardinal number—the latter mediated by the operation of placing two sets of objects in one-one correspondence (second, third, and fourth chapters). In the fifth and sixth chapters Piaget considers logical seriation, both per se and in its numerical guise (ordination), and then plunges into a detailed study of the ordination-cardination relationship. The seventh chapter likewise begins by leaving number for logic, this time logical classes, and then returns to number by analyzing the relation between it and class. The last three chapters deal with additive and multiplicative arithmetic operations, but these also are discussed from the standpoint of the logical operations from which they derive. The first two experiments reported in the book make the transition from conservation of quantity to conservation of number. The first is clearly a conservation-of-quantity study: instead of balls of clay molded into different shapes, the task involves water poured into different-shaped vessels; the question then is simply whether there is the same amount of water in the two vessels. In the second experiment, the vessels contain beads instead of water, and the problem could be construed as either conservation of quantity or of number, depending on how the question is asked (same amount of beads? same number of beads?). The task in the first experiment is said to deal with the concept of conservation of continuous quantity, the second with that of discontinuous quantity. In both investigations, as in the earlier one involving the balls of clay, there was the expected three-stage sequence; (I) no conserva• It is hard to stress enough the unity which Piaget sees in the development of cognition. The construction of number, of quantity, of logic, of space, etc,—all are believed to proceed apace and lean upon each other in diverse ways for their development. It would be tiresome to keep specifying each and every liaison of mutual dependence, and we shall not attempt it, but the fact of such dependence should constantly bt kept in mind.

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tion; (2) conflict between conservation and nonconseivation, with perception and logic alternately getting the upper hand; and (3) a stable and logically certain conservation, based primarily on a coordination of vessel-height and vessel-width relations. This second experiment also incorporated a feature of more specific import for the study of number development. The beads were initially added to the two containers on a one-one correspondence basis, i.e., add a bead to container A at the same time that one adds a bead to container B, add a second to both, add a third to both, and so on. The fact that the two containers were filled in this way did not guarantee conservation of number for the younger children in the face of contradictory perceptual impression, even though the one-one correspondence method is a mathematically certain way of establishing cardinal equivalence of sets without counting. The next several experiments were focused more directly on the role of this kind of correspondence in insuring cardinal equivalence. In a typical one the child was presented with a row of objects and asked to take the same number from a pile near at hand. The developmental sequence here is of some interest. In stage 1, the child is content simply to make a rough figural approximation to the row, e.g.. he makes a row of about the same length as the model, but of different density, and hence, of different cardinal value. In stage 2, the child spontaneously makes use of the method of one-one correspondence: he places one object opposite each one in the model row and thus exactly reproduces its cardinal value without counting.7 However, the experimenter has only to destroy the optical correspondence by spreading out or closing up one of the rows for the child to give up his earlier belief in cardinal equivalence. Like the stage-child, he now falls prey to perceptual illusion, e.g., the longer row is thought to contain more objects by virtue of its length. In the final stage, one-one correspondence is also used to establish the initial numerical equality, but now the equality is maintained after the optical correspondence is destroyed. The implication is clear: once more a concept must fight its way into stable, operational existence through a cobweb of illusion-producing perceptions. As with the quantity and logical notions, a genuine concept of cardinal number is by no means guaranteed by the ability to mouth appropriate numerical terminology in the presence of sets of objects.
It is obvious that most of the problems described in this section could handily be solved by judicious use of simple counting operations; the present task is a case in point. But it is a moot question as to just how useful such operations are to the child of 4-7 years (most of Piaget's subjects in these studies were within this age range). In the majority of protocols that Piaget cites for this kind of experiment, counting did not appear to be involved at all; when it was, it was surprisingly unhelpful in producing a certain and stable cognition of cardinal value in the teeth of illusion-giving perceptual impression (e.g., ibid., p. 59). In Chapter 11 we shall return to this interesting business of the role of counting in the young child's arithmetic understanding.
7

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In experiments of this kind, the one-one correspondence by which cardinal equivalence is established can proceed by any pairing of elements from the two sets. That is, any object from the model row can be placed opposite any object in the child's row in the process of setting the two series in one-one correspondence. The next experiment inquires about the child's performance when particular elements of the one set must correspond to particular elements of the other. In the first case, there is cardinal correspondence between unseriated sets of elements; in the second case, there is ordinal correspondence between two asymmetrical series of elements. In this study, the child was shown 10 dolls of differing heights and 10 miniature walking sticks, also graded in height. He was first told to arrange dolls and sticks "so that each doll can easilyfindthe stick that belongs to it" (ibid., p. 97): in other words, to seriate both sets of elements and place the matching elements of the two series in ordinal correspondence. Once this was achieved, the experimenter closed up one of the series, so that each doll was no longer opposite its own stick, and the child was asked to find the stick which belonged to some particular doll singled out by the experimenter. Other questions and procedures were also used with the same materials, but we shall not pursue them here. There were several important findings. First of all, the youngest children found it impossible even to construct a given series in the first place. They seemed to have at their disposal no rational procedure for doing this, for example, by selecting the shortest doll, then the next-toshortest, etc., until the whole series was constituted. Piaget interprets this failure as an inability to grasp the reversibility inherent in seriable elements, i.e., to grasp that a given element n is at one and the same time longer than element n — 1 and shorter than element n + 1. (Thus it was that, in another study, children of this level were quite unable to insert new elements in their correct places within an already-constructed series.) However, once capable of seriating, the child was equally capable of establishing the correct ordinal correspondences, i.e., assigning to each doll its correct walking stick. But here, as in previous experiments, there was a stage where destruction of the perceptual correspondence (spreading out one of the series, mixing up its elements, and so on) sufficed to render the child incapable of reconstructing it, of finding the correct stick for a given doll. A number of children repeatedly made a particular error here, an error relevant to the line of investigation next undertaken: aware that they had to count in the second series to find therightstick for a particular doll in the first, they kept choosing the n — lth stick where the nth stick was called for. It appeared as though they were somehow mixing up the ordinal number of the sought-after stick (nth) with the cardinal number (n — 1) of those smaller than it. This ability to differentiate and coordinate the ordinal and cardinal

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aspects of number was the subject of the next several experiments. These experiments are quite space-consuming even to summarize, so we shall simply describe the sort of arithmetic understanding they investigated. In general, they tested for the child's ability to distinguish between and see the relationship between ordinal position in a series and the cardinal values determining this position and determined by it. Can the child deduce the various cardinal values associated with a selected ordinal position (nth) in a series, i.e., the cardinal number of elements prior to it (n — 1), prior to and including it (n), and following it (total minus «)? Conversely, can he deduce the ordinal position of an element in the series, given information about these various cardinal values? A simple way to characterize Piaget's results in these studies—a rather flaccid characterization in view of the qualitative richness of his observations and the subtle interpretation he himself gave them—is to say that ordination and cardination are not at all well coordinated in the young child's mind. The child's ability simply to make a vocal enumeration of series elements (Piaget took pains to insure that the child was not tested on elements too numerous for him to count) did not at all guarantee a grasp of this important relationship—a relationship so essential to a real understanding of number. The last four chapters of the book report experiments which measure the child's burgeoning awareness of the basic additive and multiplicative properties of numbers. As mentioned earlier, however, Piaget's analysis of these properties proceeds in tandem with analyses of the corresponding class-and-relation logical operations. Thus, the first of these four chapters relates the experiment on the additive composition of classes reported in the preceding section of the present chapter (the one in which the child must compare the class extension of a class B with that of one of its subclasses A). The second chapter deals with the additive composition, not of class, but of number itself. It was discovered in one study, for example, that young children have difficulty in conserving an arithmetic whole when the additive composition of its parts is varied, i.e., understanding that 8 objects partitioned into sets of 4 and 4 are numerically equivalent to 8 objects distributed 1 and 7, grasping the fact that the increase from 4 to 7 is compensated by a corresponding decrease from 4 to 1, leaving the whole invariant. The ninth chapter proceeds to experiments involving the use of one-one correspondence across several sets of (unseriated) objects as a vehicle for making elementary arithmetic multiplication. For instance, if n flowers are set in one-one correspondence with n other flowers and with n little flower vases, how many flowers should go in each vase if each vase is to have an equal number of flowers, and how many vases would be needed if each vase could hold but one flower? The last chapter returns full circle to the first with the study of the child's ability to make use of a measuring unit in determining quantities of water in

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various ditf:erent-shaped vessels. Since number, unlike class, involves iterable units, it is closely linked with the infralogical operation of measurement, of which the likewise iterable measuring unit is the cornerstone (see again our Chapter 5, p. 199). TIME It would not require any foreknowledge of Piaget's work to suspect that concepts of time, movement, and velocity might be rather closely related, both logically and in terms of their psychological development. It therefore comes as no surprise that Piaget looks upon his book on movement and velocity (Piaget, 1946b) as simply the sequel or continuation of his book on time (Piaget, 1946a). These two books are the prime sources of information on his extensive theoretical and experimental work in these areas. Indeed, they are almost the only sources; there exist only a few others of direct relevance (1942c, 1955c, 1957d, 1957g). Probably the best available summary of his thinking on temporal development is to be found in the concluding chapter of the time book itself; there is an analogous chapter in the volume on movement and speed, but it is likely to be less helpful to the average reader. Since these three areas are so tightly interlocked in Piaget's logical and developmental analysis, our procedure will depart somewhat from that of previous sections. We shall begin the present section by outlining Piaget's general conception of the development, not only of time concepts, but of those of movement and speed as well. This done, the remainder of the section will consist of a summary of some of the experiments he has carried out on the development of time concepts. The next section will then be given over primarily to the empirical aspects of his work on movement and velocity, with only the minimum of superimposed theory necessary to round out the picture for these two concepts. The first thing that needs to be said about time, movement, and velocity constructs (or perhaps by now it does not need to be said) is that they are literally constructs; not apriorities in the child's mind, they require a slow and gradual ontogenetic construction. Like other notions already discussed and yet to be discussed, they are put together step by step through the formation of their constituent logical operations (actually, infralogical operations here, as we saw in Chapter 5). Second, this ontogenetic construction is one in which each and every stage is marked by an extraordinarily close interdependence among the three types of concepts—a particularly striking example of the developmental unity and interdependence among areas mentioned in footnote 6. Piaget believes that the young child initially confuses successions of events in time and the temporal intervals these successions engender with their analogues in space, that is, with the successions of points traversed

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in a movement and the spatial distances between the points. Given a single movement which proceeds across points A, B, C, D, in that order, the child will correctl) state that C was traversed "after" A, that it took "more time" to make the itinerary AC than the itinerary AB; in short, he will act as though his general conception of temporal succession and temporal duration were the same as ours. But appearances are deceptive here, because when the child has to deal with temporal successions and intervals in two movements at once, two movements, moreover, which proceed at different velocities, he makes astonishing errors. To take but one example: he is unwilling even to admit simultaneity of starting and stopping, let alone equality of duration, of two simultaneous movements whose velocities and, therefore, distances traversed are different (Piaget, 1946a, ch. 4). The child acts as though each movement had its own "time"—Piaget speaks of it as "local time" (e.g., ibid., p. 273)—and that the "times" indigenous to different movements can therefore not be coordinated. What needs to be constructed intellectually is a "homogeneous time" (ibid., p. 273) which is the common medium for all movements—synchronous or asynchronous, same speed or different speed— and which is by that fact differentiated from the spatial order and intervals comprised in any single movement. The time which needs construction is one which "constitutes a coordination of movements of different velocities" (ibid., p. 273), and therefore must be tested for in situations other than those of the one-movement, ABCD type illustrated above. But to "coordinate movements of different velocities" surely requires some rational conception of movement and velocity to begin with, and this, k turns out, is precisely what the young child lacks. Initially, both are evaluated solely in terms of the end or termination point of the motion through space involved. In the case of movement, one object will be said to have made a longer journey, to have moved farther, if it ends up ahead of another, even though the former's itinerary was straight and the latter's was zigzag, and hence of greater total distance (Piaget, 1946b, ch. 3). The child compares only the positions of the termination points, neglecting the points of departure and the spatial intervals between. And velocity is likewise reduced to a schema of "passing" or of "being ahead," rather than being conceived as a specific relation between time and distance. When the child sees one object catch up to or end up ahead of another, he will conclude that it moved faster; but when the experimenter arranges things so he cannot actually see the "passing" (e.g., the two movements of unequal velocity take place inside tunnels), he is quite incapable of inferring a difference in velocity from the perceptually obvious fact that different distances were traversed in the same time (ibid., ch. 6). Concepts of both movement and speed, then, are initially in such a state of development as to be of little service to the construction of operational time. But the paradox is that they cannot reach such a state

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themselves without the assistance of that very conception of time which appears to depend upon their development (Piaget, 1946a, p. 274). In the experiment cited above, for example, an inability to coordinate the two movements within a common temporal framework will, of course, make impossible the multiplication: (more distance) X (same time) = (greater speed). This sort of situation obtains everywhere in the genesis of intellectual operations as Piaget analyzes them: to achieve concept A requires prior developments in concepts Bf C, D, etc., and conversely, a kind of developmental circle. Although Piaget is not as specific and clear here as one might wish, the presumption is that the circle just avoids being a vicious one by virtue of the fact that development proceeds by very small increments: tiny advances in one area (via the usual mechanism of decentration with progressive equilibration, etc.) pave the way for similarly small advances in another; these advances then redound to the developmental advantage of the first area, and so the spiral continues through ontogenesis. Putting aside the question of whether development, here or elsewhere, does in fact proceed by such cross-fertilizations, there can be no question but that basic time, movement, and velocity concepts do develop more or less contemporaneously. But here we need to answer the question put to the number research. What is it that develops? What, in Piaget's view, are the crucial abilities that the child gradually acquires in these areas? The answer, as with number, almost amounts to a recital of the tables of content of the two books. In the case of time, there is first of all a conceptual grasp of temporal order of succession and of the temporal intervals between succeeding temporal points—analogous to the ordinal and cardinal aspects of number, respectively. Other achievements include an understanding of temporal simultaneity, additivity and associativity of temporal intervals, the measurement of time through the construction of the temporal unit, and finally, what Piaget calls "lived" time (ibid., p. 205 ff.), including the concepts of age and of internal, subjective time. In the case of movement, there are the concepts of spatial order, composition of displacements in space (distances), and relative movements. And for velocity, there is the notion of the time-distance relation and its ultimate measurement in a variety of situations: in successive versus simultaneous movements, for uniform versus accelerated motion, and in the case of relative velocities. The first investigation described in the book on time was an omnibus affair which assessed the child's understanding of a variety of time concepts. The apparatus consisted of a pear-shaped bottle (I) whose narrow end opened into a thin cylindrical bottle (II) with measuring lines which was placed below it. The narrow end of I had a spigot attachment permitting the experimenter to start and stop the flow of a colored liquid from I into II. In the beginning, I contained all the liquid and II was

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empty. The child was given a picture of the apparatus and asked to draw in the level of the liquid in I. The experimenter then turned on the spigot until the liquid rose to the first measuring line in II. To fix terminology, we shall say that the liquid had now risen to level IIx in II and dropped to level Ij in I. The child was given a second picture of the apparatus and asked to draw in the liquid levels as they were now. Then a second quantity of water, equal to the first, was allowed to flow into II, thus constituting new levels I2 and II2; and the child drew in these levels on a third picture. This procedure was repeated until all the liquid had flowed from I into II. The child now had before him a collection of pictures which together formed an ordered, equal-interval series of levels in the two vessels. A number of problems were posed with these pictures as experimental materials and the principal findings were the following. (1) As earlier work on seriation would lead us to predict (Piaget, 1952b), the younger children had difficulties in establishing the temporal order of the pictures and in finding the appropriate II level for a given I level when the I and II halves of each picture were separated from each other. It was as though they had no clear conceptual grasp of succession and order in time, i.e., time as a straight-line affair with events occurring in ordered sequence along it. (2) The younger subjects would not concede that the corresponding drops in I and rises in II took the same amount of time to occur (many of them made this error even when referred, not to the drawings, but to the vessels themselves). Thus, for example, the child might maintain that the time I± — I2 was not equal to the time IIj — II2, because the water level rose at a faster rate in II (because of its thinness) than it dropped in I. Moreover, this belief in inequality of temporal durations was sometimes maintained even when the child would admit the simultaneity of starting and stopping. It appeared as though—and subsequent research amply confirmed this—temporal order, simultaneity, and duration are very poorly coordinated notions for the preoperational child. (3) The younger subjects seemed to lack any genuinely metric conception of time; they were unable to grasp the idea of a temporal unit by means of which synchronous and successive temporal intervals in different movements could be compared. Thus, they could not compare the temporal duration of lx — I2 with that of llx — II3, of It — I 3 with that of II 2 — II4, and so on, even when the equality of the successive intervals lt — I2, ^2 — ^3, etc., had been impressed upon them by the experimenter. The details of this experiment (and we have by no means covered them all) occupy the first major section of the book. The second section reports investigations which follow up and extend its various findings. The first of these illuminates with particular clarity both the young child's undifferentiation between time and space and, deriving from it, his inability to establish the necessary relation between ordered points in time and the

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temporal intervals between them when two movements are involved. There were a number of procedural variations in this study. In one, an object hops from point ^ t o Dx as a second hops a shorter distance from A2 to B2 (starts and stops obviously synchronous; distances Ax — B1 = A2 — B2, Bx — C1~ B2 — C2, etc.); immediately thereafter, the second object adds a hop from B2 to C2 while the first object remains stationary at Dx. The children of stage 1 completely failed to dissociate temporal order and interval from spatial order and interval here: the object which went the lesser distance in the greater total time (the second object) was thought both to have stopped sooner (order) and to have traveled for the shorter time (interval). A number of experimental checks and controls which were introduced suggested that this spatial-temporal undifferentiation, rather than other factors, was at the root of these curious responses. But the behavior of the stage-2 children was, if anything, even more startling: correct interpretation of temporal order coupled with incorrect judging of temporal interval, or the converse—duration correct and order incorrect. Parallel results were obtained in other variations of the task, e.g., different departure times with simultaneous arrivals. There was a similar experiment which called for the assessment of orders and intervals in the case of simultaneous movements which took place at different velocities (this was the study briefly cited in the introduction to this section). In this situation, the object which went faster and farther, i.e., ended up ahead of its counterpart on a parallel path, was judged by the preoperational subjects to have stopped later and/or to have been of longer temporal duration. An interesting example of the checks and controls mentioned above was included here. The two simultaneous movements (unequal velocity and distance) were made to take place, not in parallel this time, but in opposite directions and terminating at the same point; thus, neither object passed the other in space. This variation in procedure produced a decided increase in correct responding in the younger subjects, particularly with regard to simultaneity of arrival. The next experiment is perhaps the best known of the time studies; certainly, it is as ingenious as any. A vessel of water has two identical branching tubes at its bottom end (like an inverted Y) with a spigot which releases water through both tubes at once, obviously in equal quantities per unit of time. When the tubes drain into separate containers of identical shape and size, the preoperational child readily believes in the simultaneity of starts and stops and the equality of durations for the two outflows. When the containers are of different size and shape, however— shades of the conservation-of-quantity research—he forthwith tumbles into all the difficulties discussed previously: inequality of starts and stops, inequality of durations, etc. Variations on this experimental procedure

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also adduced evidence that, the younger subject has difficulties in seeing temporal intervals as forming a hierarachy of inclusions (duration A less than duration B less than duration C, and so forth) and in applying the transitivity rule to them (A = B, B = C, hence A = C; A < B,B < C, hence A < C). There are still other experiments reported in the long middle section of the book. One study showed that young children have trouble adding successively occurring temporal intervals to form a total interval; for instance, they will not necessarily infer equal total duration for two movements from the knowledge that their component durations were equal. Similarly, temporal intervals appear not to be associative in the early stages- Thus, the child cannot establish equalities such as (A + B) + C = A + (B + C), when A, B, and C are durations which occur in sequence. Piaget's data indicate that the additivity and associativity properties develop synchronously: if the child can deal with one, he can generally manage the other (ibid., p. 171). We shall conclude by citing a series of experiments on the concept of age taken from the final section of the book. The young child's notion of age appears from these studies to have two related idiosyncrasies. (1) Age is not differentiated from size (especially height). Bigger things are older than smaller things, and things which have stopped growing have stopped getting any older (a consummation devoutly to be wished!). In one study, for example, the child was shown a picture of two trees of obviously different species, one bigger than the other. He was then asked which he thought was the older tree, or whether it was not possible to tell. The younger children said the bigger one was older; the older children said that one could not tell without knowing when they were planted. (2) Because of its association with size, age bears no necessary relation to date of birth. If A is born after B but eventually outstrips it in size, it is "older." The child was shown two series of pictures representing the year-by-year growth of two trees. One tree (a pear tree) was planted one year after the other an (apple tree) but grew faster and eventually became the larger of the two, bore the most fruit, and so on. Which tree was older? The following is an example of how young children deal with this problem: Joe (1:6) succeeds in seriating the apple trees by saying "one year, two years, three years, etc.—Look, when the apple tree is two years old we plant this pear tree. Which is the oldest?—The apple tree.—And the year after this?—Still the apple tree.—And the year after, here are photos taken on the same day (P4 = i?3). Which is oldest?—The pear tree.—Why?—Because it has more pears. . . .—And here (P 5 and R4)?—The pear tree.—How old is it?—(Joe counts one by one) 4 years old.—And the apple tree?—(Counts with his finger) 5 years old.—Which of the two is the oldest?—The pear tree. —Why?—Because it's four years old.—Are you older when you're 4 or when

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you're 5?—When you're 5.—Then which is the oldest?—/ don't know the pear tree because it has more pears" (ibid., p. 229).

MOVEMENT AND VELOCITY Piaget conceives of movement as a displacement in space in reference to an ordered set of fixed spatial positions or placements (e.g., Piaget, 1946b, pp. 258-259). An object A is said to have "moved" if, formerly in one placement in an ordered series of placements ABCDE, it is now found in a different position in the same series, e.g., BCDAE. This conception led him to begin his research on movements or displacements with preliminary studies of the child's understanding of spatial order. In one of these studies, the apparatus consisted of three different-colored wooden balls (A, B, and C) which could be slid along a wire behind a screen. Problems of the following sort were put to the child. The objects disappear behind the screen in the order ABC; in what order will they emerge on the other side of the screen (ABC)} In what order will they reemerge on the first side (the inverse CBA)? If the wire is rotated 180° behind the screen, in what order will they emerge on the other side (CBA)} If rotated 360° (ABC again)? We shall as usual eschew a detailed account of stage-by-stage development here in favor of reporting the general sense of the findings. The youngest subjects have no difficulties with the first question but tend to fail the others, including the second one (inverse movement of balls in order CBA). Interestingly, children of this level are not averse to predicting that the middle element B may emerge in front in the case of the inverse movement. Piaget feels that it is only when the child has an operational conception of order, with direct and inverse orders rigorously coordinated, that the relation "between" (B "between" A and C) becomes a symmetrical one, something invariant for both ABC and CBA orders (ibid., p. 15). In sharp contrast to the hesitations and errors of the earlier stages, the stage-3 subjects (about age 7) go so far as to discover a rule for finding the correct order for any rotation of the wire: direct order ABC for even-numbered 180° rotations; inverse order CBA for odd-numbered ones. The next series of investigations dealt with movement proper, in particular with the distances generated by the displacement of objects through space. With distance, as with time and velocity, the spatial order of terminations (which object passed the other and ended up ahead of it) is the dominating criterion for the preoperational child. Centering exclusively on order of terminations, he neglects positions of departure and—what is really criterial for distance—the sum of spatial units comprised between departure and termination. In one study, the child was presented with two strings, one above the other, which looked like this:

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B
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The strings were said to be tramway tracks and little balls running along them were to be the tramway cais. The child was given a number of tasks to perform, including the following. The experimenter moved his car tor a distance of several segments on track JB and asked the child to make just as long a trip (le nierne long chemin) on A. The younger subjects tended to move their cars only far enough to be directly above the experimenter's and hence made a trip of lesser total distance. In this case the experimenter did several things to induce a differentiation between distance traversed and order of termination, e.g., he moved his car one segment from the point of departure (thus straight up and still directly below A's point of departure) and asked the child to make his car take just as long a journey. Also included in the set of problems: the child was given a little piece of cardboard equal in length to a B segment and asked if this would help him in any way to make sure a given set of A and B distances were equal. The results of the study suggested a gradual evolution from a rigid and unyielding dependence on superposition of cars as the criterion for equality of distances, through a beginning differentiation between superposition and actual distance traversed (but with inability to see total distances as composed of distance units), to a ready use of the unit measure as the only certain method of assessing distance, and, implied in this, a conception of distance as the sum of tiny distance units. Just as with time, number, and quantity (recall the schema of atomism described in the first section of this chapter), the distance to which a movement gives rise is not really a rational concept until it is construed as fractionable into arbitrarily small, additive unit distances. The additive composition of distances was more intensively studied in the next investigation. In one part, a funicular railway train (bead) ascended and descended a mountain (cardboard model) along a fixed itinerary traversing points O (base), A, B, C, and D (summit). The child's task was to add part distances so as to compare total ascent with total descent. For example, if the train made in succession the itineraries OC, CB, BD, DO, has its total ascending distance been greater than, less than, or equal to its total descending distance? As in the previous study, the child had measuring aids at his disposal. And as before, the data showed a progressive mastery of measurement operations with increasing age. There was one incidental finding which demonstrated with particu-

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lar force the primitive state of the distance concept in young children. Most of them asserted that the total ascent OD was greater than the total descent DO, presumably because egocentric notions of more effort = more distance intruded. For some of these subjects, actual measurement of the distances sufficed to dispel this illusion, but for others it did not. There was one other experiment on movement which we shall only mention. The study dealt with relative movements: a snail moves along a board while the board itself moves along a table in the same or opposite direction as the snail's movement. To be able to compose such sets of interdependent movements so as to arrive at the "net" movement of the snail in relation to the table is a late achievement, one requiring not concrete but formal operations. See our remarks on this task in Chapter 6, p. 217. The remainder of the book describes Piaget's research on the concept of velocity. The first several experiments all bear on a single point: that children initially reduce velocity to an intuition of order and changes of order, i.e., that object traveled faster which, initially behind another, caught up to it and ended up ahead. We shall summarize these experiments by describing the typical reaction of the younger subjects to the velocity problem each experiment presented. When two parallel and simultaneous movements of unequal speed and distance take place inside tunnels, so that the child cannot see the faster one gaining on the slower one, the child thinks they traveled at equal speeds. When two simultaneous movements of unequal velocity and distance begin at a common point and end at a common point (the longer and faster one taking an angled or sinuous intinerary and the shorter and slower one following a straight-line path), the child believes the velocities were equal. When simultaneous movements proceed along concentric circles (the movement along the larger circle being of course faster), the child asserts equality of speed. When one object starts its movement at the same instant as a second but from a position considerably behind it, the young child will say it traveled faster if it ends up in front of the second when they both stop, but not if it ends up parallel to or just behind the second (in all three cases its actual speed was considerably greater than the second's). If two objects make parallel movements of equal distance, one starting before the other in time but both terminating simultaneously (termination points superimposed), the child either thinks the speeds were equal or else that the one which started first went faster, since it initially "passed" the (stationary) second one in the beginning of its movement. The next study dealt with relative velocities, an analogue of the experiment on relative movements mentioned earlier. Eight cardboard bicyclists move along an endless ribbon at uniform speed. A little man counts the bicyclists as they pass him. He is immobile during the first 15 seconds (time enough for all 8 bicyclists to go by him once); during the next

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I5-second interval, however, he himself moves slowly, either in the same direction as the bicyclists or in the opposite direction. The subject was then asked to tell whether more, fewer, or the same number of bicyclists would pass the little man in the second interval, relative to the first. As with relative movements, the ability to compose relative speeds appears to be a late developmental achievement. Until age 8 or so, children tend to assume that the number of bicyclists which pass the man per unit of time will be the same whether the man himself is moving or not. And not until about age 11 could the child systematically deduce the correct result prior to experimentation and, in addition, give a rational explanation for it. The book continues with reports of experiments entailing the quantification of velocities and velocity differences, either by establishing simple proportions among times and distances or by actually making estimates of the arithmetic value of the distance-over-time ratio (Piaget, 1946b, p. 185). Since it is, of course, necessary first to possess a stable concept of velocity before its measurement has any meaning, the major devel opmental changes in these experiments take place between middle childhood and adolescence, i.e., in transition from concrete to formal operations. The first study deals with the estimation of the velocities of movements which occur, not synchronously, but successively. An object makes a rectilinear movement, its time is recorded by a stop watch, and its itinerary is traced on paper. A second movement is then made and similarly recorded. This movement may be of same duration and different distance, different duration and same distance, or different duration and different distance, relative to the first. The child's task is simply to judge whether the velocities were equal or different, and if different, which was faster. As might be suspected, children who could easily solve the velocity problems involving simultaneous movements described earlier had great difficulties here, where the movements were successive, where the only remnant of a movement consisted of a line on paper and a stop-watch number. However, older subjects did solve these problems, and the developmental progression was an orderly one: solution of problems where either times or distances were equal; then solution where times and distances were unequal but in simple proportion; and finally solution to any and all problems posed. A second investigation concerned what Piaget calls conservation of uniform velocity (1946b, p. 210). A toy car travels a certain distance on the first day of a trip, a man rides a bicycle for half this distance during the same day, and both intineraries are recorded by parallel lines on paper. The questions asked of the child included these. How far will the car go the second day, the third day, etc., in traveling at the same speed and for the same time? How far will the man go on these same days, traveling at his speed (half the car's) during the same time intervals? On

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the last day, the car goes at its usual speed but travels only half the day; how far will it go? Does the (absolute) distance between the termination points of car and man remain the same, day after day, or does it regularly augment? The youngest subjects are unable to reproduce constant distances for the car in its daily progress. Older ones manage this but make a systematic error in the case of the man: from the second day on they keep constant the first day's difference in distance between man and car, thus making the man travel at a speed equal to that of the car. Still older subjects succeeed in constructing the itineraries correctly but are unable to predict, in advance of actual construction, that the difference in distance between man and car will regularly increase from day to day (last question above). The ability to deduce this in advance of experience appears to come in at around 10 or 11 years of age. The last experiment described in the book was designed to measure the child's understanding of uniformly accelerated movement: whether the child can predict that an object rolling down an inclined plane will increase in speed all the way to the bottom, whether he can recognize that, by virtue of this increase in speed, the object will cover increasingly more distance per unit of time as it continues to roll, and so on. Again, developmental progress continues into early adolescence. One curio to close the chapter: some children thought the distance per unit of time would grow shorter and shorter as the object neared the bottom because its speed was increasing!