Some Remarks on the Psychology of Number

In the October, 1897, number of The Pedagogical Seminary Mr. D. E. Phillips has presented a very considerable bulk of data bearing upon the question of the development of the number sense, from the standpoints both of experimentation and collection of illustrative material from child life. He has also presented a theory in interpretation of these facts. This theory, as I understand it, separates the counting process, in which number undoubtedly has its origin and development, from the consciousness of quantity, holding that the number idea in the form of a series, is a distinct process psychologically and pedagogically from the ratio idea; or, as it is stated in the summary on page lxxxiv, "Number as measurement is not the whole of the development of number but only the complimentary side of the series idea; number as measurement can by no means explain all the mental phenomena of numbers."

The matter has, I think, sufficient psychological interest of its own to justify further consideration; and this consideration is specially urgent when we note that according as we separate the counting or series idea from the quantity idea, or connect it organically with the latter, we get two quite different educational procedures indicated. I propose, accordingly, a further discussion of the matter, based for the most part upon Mr. Phillips's own data and treatment.

The first point which appears to require examination is the question of the origin and development of the series idea, both in the concrete and the abstract. At the outset I

( 178) experience considerable embarrassment in interpreting Mr. Phillips's statement (p. xli) that "nearly all children, no matter how taught, first learn to count independently of objects, in which the series idea gets ahead," or again (p. xliii ) that "the naming of the series generally goes in advance of the application to things." My difficulty is in understanding what Mr. Phillips means by the "naming of the series." His argument demands that this naming of the number series be genuine counting or enumeration; the facts which he educes in support of his proposition relate to the mere repetition or rattling off of a series of number names. I find it difficult to convince myself that Mr. Phillips has not distinguished between two such totally different matters as naming a number series and repeating a series of names of numbers: but careful reading of his paper has not assured me that he is free from this confusion. For example on page xxxviii it is said: "M knew number names long before numbers themselves, and applied them to anything indiscriminately, the numbers seldom agreeing with the number of objects." Page xxxix: "Have children of the kindergarten count, and you will find that they leave out the difficult names and carry on the series the same, often repeating without being conscious of it.

M when learning to count things or impressions, would count eight, nine, for one object. The number series always get ahead." "This child learned the figures in order; i.e., 1, 2, 3, 4. He could tell them in this way up to forty, but if they were changed around he would not know but what it meant the same. He associated number with place or order only." "Most children I have taught," says an experienced teacher, "will learn the names independent of things. For sometimes they do not associate them as how many of anything, but merely the name of the series. . . . Children learn number names as they learn other words, by hearing them repeated. They use them without knowing their respective values. They apply them promiscuously calling three objects ten as the case may be. Very frequently they have the idea that the number name belongs to the object in whatever order among others it is placed" (italics mine). A number of other instances could be quoted, but it is probably enough to refer to the summary of data on page lvii,

( 179) where it is stated that three hundred and fourteen out of three hundred and forty-one returns say that children learn number names by rote as an abstract word (italics mine).

While, I repeat, it hardly seems credible that Mr. Phillips could confuse counting with the repetition of number names, yet the foregoing quotations, which might be largely multiplied, point in this direction. It must be borne in mind that these statements are not isolated, but are the substance of the evidence educed by Mr. Phillips to show that the formation of the number series precedes the application of number to objects. It is hardly necessary, I suppose, to argue that since this supposed evidence refers in reality only to the learning of certain words, it is just as irrelevant to the proof of Mr. Phillips's proposition, as would be the evidence that children learn to say the alphabet before they count objects. The repetition of number names is no more counting than is the saying of counting out rhymes like: "Eeny, meeny, miny, mo," etc., in fact, hardly as much so, for in the latter there is a genuine pairing off, or crude equating which approaches the number concept. It is barely conceivable that Mr. Phillips does not mean to attribute any numerical significance to this saying of number terms (though if such is the case, I utterly fail to see why it should be exhibited as indicating that counting goes in advance of application of number to things), but merely means to indicate the satisfaction which the child takes in any series or succession. I do not doubt this latter fact at all, nor do I doubt that interesting psychological results could be got from a careful study of the child's delight in successive recurrences. One group of Chicago kindergartners, for example, has been making a careful study of this matter and finds the child's interest in statements of: "This is the house that Jack built" type, so strong that a large number of rhymes and stories along similar lines have been constructed to the children's eminent satisfaction. Nor do I doubt that such phenomena are indirectly relevant to the psychology of number; but it must be clearly recognized that such facts have absolutely no power in support of the statement that genuine counting or formation of a real number series precedes the application of number to things, which is the point in ques-

( 180) -tion. Moreover, so far as these facts have any connection with the number sense at all, the essential thing is to see under what circumstances the mere interest in seriality of repetition becomes transformed into an interest in that particular form of seriality which we term counting. To confuse series interest in general with the series interest in its counted off form, is about as flagrant a case of the fallacy of accident as one could imagine.

This brings us to the question which is of fundamental importance. What is the origin and significance of the interest in series and succession? Secondly, under what circumstances does this assume the numerical form? I begin with the first point. Mr. Phillips's Psychology of Succession seems to me somewhat unnecessarily crude. On page xxxviii he says: "Succession of some kind is the earliest and most continuous thing in consciousness. Series of innervations, touches, sounds, sights, etc., are the constant things in consciousness; one series only gives way to another" Page xxxv: "Changes in consciousness are continually taking place produced by the varying impressions from all the senses. Consciousness is not one continuous impression, but an innumerable multitude of successive changes. . . . As already stated, without doubt different senses contribute to these ideas." Page xxxvi: "The series idea is established by a multitude of successive and rhythmical sensations conveyed through the different senses"

I call this psychology crude because it seems to confuse the de facto occurrence of changes in consciousness with a consciousness of such changes, this latter being necessary to any consciousness of a series; and because it implies that sensations, simply as sensations, come to us, individualizing themselves, and also arranged in series. Now one cannot, intelligently, watch a baby for fifteen minutes without seeing that there is a very great difference between distinctness and orderly sequence in sense stimuli and definiteness and sequence in conscious sensation. It is a matter of great difficulty, requiring several months of infant life to secure ability to pick out a distinct sound, color, touch, or taste at all. The child's consciousness certainly begins with a sense blur or confusion into which specification is only gradually introduced.

( 181)

Instead, then, of taking for granted the existence of a series, supposing that it is adequately motivated by the mere occurrence of different sensations, we have to discover the principle through which the original blur is specified into definite states of consciousness, and these serially ordered with reference to each other. Of course an adequate discussion of this point is out of the question here, but, in order that we may have an idea of the origin of succession to put beside Mr. Phillips's, I would make the following statement: It is the motor reaction or process of adjustment which breaks up the sensory blur into definite states, and it is the continuity of motor adjustment (corresponding to habit) which arranges these into a series. Fortunately for our discussion all the evidence that Mr. Phillips brings up points in this direction. For example, page xxxvi: The cases given of "reproducing or following a series" are all motor. It is not mere having of sensations; it is throwing, repeating strokes with the clock, rolling mud balls, nodding the head, arranging pebbles, etc. On page xli, it is expressly stated: "A movement of some muscle, as the toe or finger, or nodding of the head, can be observed in nearly all children when counting at first." On page xli, instances of such motor reaction are given; also on page lvi, under the caption of "Tallying by Beats"; on page lvii, in Miss Shinn's valuable record, things were enumerated by touching them, taking nuts out of a box, transferring objects: and once more on page lix; so also page lxi. So striking is the evidence, that Mr. Phillips himself, in his final summary of conclusions on page lxxxiv, says: "Some movement is perhaps unavoidable in the early stages of counting. If the motor element be a necessity of all thought it is even more so in following an abstract series." The term "following" perhaps throws some light on Mr. Phillips's confusion. It seems to indicate that the series is already there and that the movement simply parallels the members of the series as they come and go. But this confuses the adult standpoint with the child's. The child makes the series by the movements: that is to say, by movement he interrupts the vague continuum and introduces definiteness or individuality.

We thus get a basis for interpreting the importance (which Mr. Phillips does well in emphasizing) of rhythm

( 182) in the formation of an orderly series. This subject is also of course too large to go into, but Dr. Bolton's important investigations may be referred to. One passage given on page xxxi may be quoted: "The subjects were found to be unconsciously keeping time by tapping, nodding, etc., at every second or fourth click. When movement was restrained in one muscle it was likely to appear elsewhere. These movements are the condition of rhythmical grouping." It is the grouping of minor movements within a larger whole of movement which marks off a mere on-going series into those regular recurrences of stress and slur which make up what we call rhythm.

When interpreted, Mr. Phillips's own data confirm the statement quoted by him from McLellan and Dewey's Psychology of Number "That number is to be traced . . . back to adjustment of activity" (p. xxxiv). This is the point worked out in considerable detail in Chapter 3, pages 35 to 44 of that book. Perhaps this is as fitting a place as any to say something about the rational factor in number. Mr. Phillips objects to the doctrine of the Psychology of Number (see his pp. l—li) on the ground that it makes altogether too much of the rational processes of abstraction and generalization, and altogether too little of the instinctive, unreflective side. I shall not charge Mr. Phillips with lack of candor; but either the statement in the book is so confused and obscure that the point is not plainly made, or else Mr. Phillips has read it somewhat carelessly. He certainly is bound, in making his objection, to use the terms abstraction and generalization in the same sense in which they are used by the authors in question. However defective the statement may be, the point is certainly insisted upon that practical activity precedes the rational in the conscious sense of the latter term; it is pointed out that abstraction first exists practically in simple selection or preference as to ends and means, and that generalization is of the same practical sort, consisting in the adjustment of means to ends; and that only at a later period does this practical selection and adjustment come to consciousness in explicit rational form. The whole moral of the book pedagogically, so far as primary number is concerned, might be said to be this: "Do not teach number

( 183) either merely mechanically or merely rationally. Give the child something to do which involves the use of numerical considerations in a reasonable related way. Thus he will gain practical familiarity with them, by using them for some purpose and end, instead of in a meaningless, haphazard way, will be forming orderly practical habits of relating, which afterwards will become conscious in real generalization." I quote from page 30 of the book referred to: "The child may and should perform many operations and reach definite results by implicitly using the ideas they (that is numbers) involve, long before these ideas can be explicitly developed in consciousness. If facts are presented in their proper connection, as stimulating and directing the primary mental activities, the child is slowly but surely feeling his way towards a conscious recognition of the nature of the process. This unconscious growth towards a reflective grasp of number relations is seriously retarded by untimely analysis."

In view of these facts, and in view of the fact that Mr. Phillips himself, finds the origin of the series in a purely objective basis—the occurrence of sensations, while McLellan and Dewey find it in a practical basis—motor reactions, it seems to me that I am not unreasonable in feeling that Mr. Phillips ought to have welcomed these authors as allies, and as providing a scientific support for his own contention (otherwise unproved) regarding the relation between the spontaneous and the conscious phases of number processes, instead of turning his guns upon them to the extent of two or three of his valuable pages.[1]

We now turn from the consideration of the series in general to that particular form of the series which we call

( 184) counting. We have already disposed of the confusion between counting process and the mere repetition of number names and hence are ready to deal with the former question proper. In order to define the problem I may refer to an apparent misconstruction by Mr. Phillips of the doctrine of the Psychology of Number. On pages xxxiv, l, lxxiv, lxxx, lxxxi,he implies or states that the ratio idea is insisted upon in that book to the exclusion of the series idea, or at least to the very great subordination of the counting process. Nothing could be further from the mark. The doctrine of that work is that the counting process (the series idea) and the quantitative or magnitude idea, are logically correlative; hence the ratio idea. To subordinate the series (how many) to magnitude idea (how much) would destroy ratio, not exalt it. On page 3-45 of that book it is stated that counting is the fundamental process of arithmetic. On pages 3-63 to 3-65 the necessity of the counting process proceeding from the mechanical to the intelligent counting, is expressly insisted upon. So again on pages 44 to 53-. In fact it is hardly worth while to give specific quotations. The characteristic theory of the book is the emphasis laid upon the principle that counting, the formation of the series, and measuring go together. More than this it is expressly stated both in the theoretical and the practical part that the numbering process or formation of a series, comes to consciousness first, and that ratio, as a conscious idea, is a later outcome (here, once more, is a difference between Speer's method and that of the book in question).

I do not mention this for the sake of justifying the book, although if Mr. Phillips has gone so far astray in understanding it, it is possible that others have also. I mention it for its value in helping define the real question. This

( 185) is not whether counting precedes or follows the ratio idea; but what is the use or value of the counting process? What is its motive? Why do we count? I doubt if anybody would go to the point of absurdity of developing an arithmetical method which omitted the counting side. Number as number is enumeration. Certainly Mr. Speer, with all the emphasis thrown upon conscious comparison of magnitudes and the statement of result in ratio form, has used counting in the definition of the ratio—my objection being simply that he does not emphasize sufficiently this counting or discrete factor. But the important question, both psychologically and pedagogically is: How shall we utilize the counting process? Shall we isolate it, or shall we treat it as having a function in the definition of value? This is all there is to the whole question.

I will now state my conception of the origin and development of the counting process, comparing this statement from time to time with that given by Mr. Phillips. My general conception is as follows: At the outset the child learns to count (in the genuine sense as distinct from the mere repetition of a mere lot of words) by reference to separate things making up a vague totality. If we take the facts preceding conscious counting, this is evident. The child keeps tally by some movement of his own with some objective fact or event; he equates, so to speak, an action of his own with some action outside of himself. All the evidence that can be educed with reference to the formation of system to number names points the same way—the connection between counting and the toes, fingers, hands, etc. So also does the connection of number with space—noted by Mr. Phillips himself. It is equally evidenced by observation of children in cases of learning to count as distinct once more, from saying number names. I am glad to refer to Miss Shinn's record which completely bears out (though in considerably greater detail than my own) observations which I have made upon three children. For a considerable period there is interest in marking off objects by pairs, one against the other. During this period the idea of two is understood, that of one and the other, but nothing beyond. While this pairing off was marked from sixteen months onward as late

( 186) as the twenty-eighth month, three could not be counted. Mr. Phillips gives a good deal of concrete data bearing out the statement made in the Psychology of Number that counting involves the ordinal idea as well as the cardinal, evidence taking the form that the child will identify a given number like three or four with a particular stick or object. In one of my own observations in teaching a girl of three to count upon her hands, in beginning with the thumb it was numbered one. The next time beginning with the little finger the thumb was called five, whereupon she insisted that it was one. This confusion continued for a few days, when the child spontaneously remarked: "Why, you can begin at either end." Up to this time she had never been certain with reference to three; after this she counted intelligently up to ten.

But it hardly seems worth while to go over the specific evidence. The moment one ceases to confuse counting with repetition of number names the proposition that we learn to count, irrespective to application to things, becomes absolutely meaningless. When we count we surely count something. The difficulty in learning to count is the difficulty of distinguishing relations of order from those of mere place or form ( see for example the cases on p. xlix of Mr. Phillips's article, in which cases of the confusion of form and number are given; so also pp. lv, lvi). So great is this demand for something to count, that I have known of a number of children who have reached the period of spontaneous counting who insisted upon saying one one, two ones, three ones, four ones, and so on, and were quite confused when the number was given to them simply as an aggregate. In one case I remember a child of four, who, when something was said to her about seven, was at first confused, and then spontaneously said: "O, I suppose you mean seven ones." Both the positive evidence, then, and the negative (that is, the difficulties which children have in learning to count) show that counting arises in connecting, not merely with a series succession, with the successive ordering of units in a series, such units being represented by fingers or objects with which the child is familiar. The abstraction previously referred to, necessary to counting, consists in the fact that the child must be able to grasp relative place or order in a

(187) series as distinct from absolute place, from spatial form, or other purely qualitative considerations: a fact adequately proved by Mr. Phillips's own data.

I come now to the period of spontaneous delight in counting—in which counting is free from all explicit reference to objects. The data collected by Mr. Phillips on this point are wholly in agreement with my own observation and with reminiscences which I have collected of adults regarding the counting period in their own lives (the point referred to on p. 26 of the Psychology of Number). This is in no wise in contradiction to what has already been said concerning the original connection of learning to count with some spatial reference and distinction. After the child has learned to count, the power acquired becomes temporarily an end in itself. The power is set free and manifests itself spontaneously. This is simply in accordance with the general principle of play. This delight in counting comes in the play period of the child's life, as a rule, and evidences the same general principle. Powers originally acquired under some practical (not merely sensible nor merely intellectual) stress, when acquired, run free or on their own account for awhile. The child takes similar delight in walking, running, talking, on their own account, doing them for the mere sake of doing them. It would be futile to deny this. It is folly not to take sufficient account of them pedagogically, and I quite agree with what Mr. Phillips says about the usual failure to utilize this spontaneous delight pedagogically. The child is hampered by being made to work in a formal way upon small numbers and counting by ones, when, if left to himself, he would spontaneously occupy himself with larger numbers and count by twos, fours and fives, almost as easily as by ones. This is as mischievous and cramping as it would be to keep the child going through exercises in learning to walk after he has learned. But theoretically it would be equally futile to deny that this spontaneous interest in counting for its own sake was preceded by a somewhat slow and tedious process in which the child was learning to count (just as to walk) through ordering or adjusting things to each other in a succession of acts.

We now come to the next step, both intellectually and

( 188) pedagogically. The average child who enters school at six or seven is emerging from this counting period. What he now needs is some further use to which his ability to count may be put, in order to gain new power. Otherwise there is arrested development. Nothing has interested me more in Mr. Phillips's article than his statement that numerical precocities and prodigies are connected with a too protracted dwelling in the counting stage. Now, unless some arrest, similar in principle though less in extent, is to occur in every child when he goes to school, some function must be found with reference to which the child may utilize his ability to count. My contention is that the number sense becomes vitalized and truly educative at this point by being largely directed towards the definition of values in the form of measurement. It is referred to things again, but not for the sake of merely counting the things—that is a relapse to an out-grown stage of development and is an error which is responsible for much of the benumbing, cramping grind of our school work. The reference to things is now in the sense of defining magnitude, measuring area, using scales to determine weight, enumerating the value of commodity with reference to money units, etc., etc. Here the power of counting is used as a tool of control and construction and so becomes truly educative.

Now a word, more definitely, as to the relationship of the series idea to the ratio idea. In the first stage, learning to count, there is always somewhat which is counted off. This somewhat limits the process of counting; and in turn the counting definitely marks out the somewhat counted. This "somewhat" may be illustrated by the fingers of the hand, by a pile of sticks or stones or whatever. The moment the child begins to count, this totality presents itself as vaguely quantitative. Through the counting, this vagueness is transformed into relative definiteness. Mr. Phillips says: "It is absurd to think of a savage attempting to measure a given quantity without any concept of number." I should say it was. Number is absolutely essential to the measurement of quantity. It is by counting or enumerating the number of units that we find out how much of a quantity there is. The point is simply that the savage and the child begin with

( 189) vague concepts of plurality, corresponding to the series, and with an equally vague sense of the totality or unity (quantity) which is split up into this plurality. By the application of one factor to the other each is defined. It is absurd to suppose that any one measures without a concept (vague that is) of number; but it is equally absurd to suppose that the savage or the child has a perfectly clear, definite, and ready-made concept of number before he ever counts up any quantity, and then proceeds to measure the quantity or find out how much there is. We begin with an equally vague "many" and an equally vague "much." Each clears the other up through mutual application. It hardly seems consistent in Mr. Phillips to insist so much upon the child's primitive, crude consciousness of plurality in the form of a series, and not to bring out with equal clearness the part played by intensive concepts of degree—little, much, more, less, light, heavy, hotter, colder, etc.,—in the child's experience. It certainly is misleading in the outcome, because it throws the factor of plurality out of perspective and out of its bearings, and hence narrows and distorts the pedagogical province of number.

Of course, one cannot avoid reflecting his own attitudes and prepossessions in any series of questions which may be sent out. But it must equally be borne in mind that the data received in reply to such questions have value just in proportion to the extent in which the questions have covered the whole ground or only a part of it. It is accordingly exceedingly unfortunate, to say the least, that not a single one of the questions in the syllabus given by Mr. Phillips has to do with the child's spontaneous experiences and expressions with reference to questions of quantity and magnitude. If one cared for one-sided results, I would agree to get up a syllabus bearing on what children say and do regarding questions of more and less, and instinctive comparison of values and amounts, the replies to which would throw into as great relief the quantitative side as Mr. Phillips's data do the strictly numerical side. But this would not be well balanced. The investigation should cover both points. I should be very glad if Mr. Phillips would supplement his present inquiry by another along this line; for, till this be under-

(190) taken, his present data are necessarily so one-sided as to lend themselves to false interpretations. It would take up, not only such questions as children's spontaneous interest in measurements, but also in formal comparison of amounts. I suggest merely one line which I think would be instructive. Many children in the same family are given to comparing the amount of food given them, particularly when it is a dainty or a luxury. Comparisons of amounts of candy, cake, dessert, and the use of counting to find out who has the most, would, I think, be quite fruitful in showing the way in which the child spontaneously uses number. (This point was suggested to me by Superintendent Ames, of Riverside.)

In the second period, that of counting as spontaneous play, the reference to quantity is, of course, not so obvious. It reappears, however, in a somewhat subtle form. I am glad to be able to refer again to Mr. Phillips's own data. It is seen in the interest of children in the "last" number—an interest I think which is almost universal. Mr. Phillips refers to the child who wanted to know all the numbers there are, page xxxviii. In general I think it will be found, moreover, that when children are spontaneously counting, they generally set up in advance the amount to which they are going to count up—as a hundred, a thousand, a million, or in extreme cases a child sets out to count clear up to the "last number." Here the magnitude conception recurs as at once setting a limit to and motivating the counting process.

In the third stage the relation between counting and magnitude is so obvious that nothing further need be said. I may say, however, one word more about the ratio idea in itself. The theory and practice of the Psychology of Number are that when counting is used by the child to value some amount or other, the ratio idea is implied or involved. It need not, therefore, be consciously, or explicitly stated. In fact, I should say that for a considerable period it should not be. It is enough that the child gets a sense for the use and application of number for purposes of evaluation, without his consciously formulating the factors involved. Familiarity through use must precede conscious formulation. But our point further is that when number is so used the transition to

( 191) the conscious ratio idea, whether in the form of ratio proper, or percentage of fractions, is natural, and inevitable. The child is saved that break which is almost uniform in present school practice—and a questionnaire upon this point would be instructive. I mean the break between whole numbers and fractions; and between the four fundamental processes and ratio and percentage. When number is used to value quantity, this idea is practically implied from the outset; and when conditions require, and the child is sufficiently mature, he passes without mental friction to conscious recognition. In closing I wish to say that this is not a mere doctrinaire statement, but that it rests upon continuous experimenting and observation in a school where the child's number sense is developed strictly in connection with construction operation in manual training, cooking, sewing, and science work, where number relations are introduced as instrumental to practical valuations.

In conclusion, the discussion may be thus summarized: A conscious series arises through a process of motor adjustments. This is rhythmical, and the rhythm facilitates at counting by making regular breaks and groupings. The series becomes numerical when its parts are ordered with reference to place and value in constituting the whole group. Along with this goes a vague recognition of muchness or magnitude (corresponding to the whole group) which is made definite through counting. Then the power of counting becomes an end in itself; what is counted is now simply some total amount or numerical magnitude set as limit. Finally (unless there is to be arrested development), this counting power is reapplied to magnitudes of value in order to measure or define them. This implies ratio, which comes to consciousness as need demands.


  1. There seems to be reason for supposing that Mr. Phillips has been led astray by identifying the principles of these authors with that of Mr. Speer—a procedure which is not just on either side. Mr. Speer is entitled to full claim to originality of method as far as Dewey and McLellan are concerned. It is not just to him to consider his book simply an application of the theory of the Psychology of Number, nor is it just to the authors of that book to interpret their theory simply in the light of Mr. Speer's practice. There is one fundamental point, I am happy to say, upon which we are in complete accord, that is the correspondence between the number idea and the quantity idea. But both the theory and practice of handling this point are very different. Mr. Speer's method, comparatively speaking, emphasizes the how much or continuity side at the expense of the discrete or how many side; and the two books interpret the conception of relation in very different ways. To Mr. Speer, relating means (upon the whole) intellectual comparison; to the authors of the Psychology of Number relating means, primarily, practical adjustment. On pp. li—Iii, Mr. Phillips quotes a sentence from the Psychology of Number and in the same connection, as if it bore upon its interpretation, makes the following quotation: "Reasoning in arithmetic establishes equality of relation. Reasoning in any subject, equality or likeness of relations." This latter sentence is quoted from Mr. Speer. It would be pedantic to make a moral issue of this, but I should like to know by what rule of criticism one is justified in quoting from one author to interpret another. As matter of fact, this statement is quite unlike the doctrine of Psychology of Number.

Valid HTML 4.01 Strict Valid CSS2