The Origins of Number Concepts

SCIENTIFIC | Established 1845 AMERICAN Merch 1973 Volume 228 Number 3 N ARTICLES 15 THE PROSPECTS FOR A STATIONARY WORLD POPULATION, by Tomas Frejka A projection lying between two extremes shows it arriving in 2100. THE VISUALIZATION OF GENES IN ACTION, by O. L. Miller. Jr. Electron micrographs of cell processes resemble diagrams arrived at indirectly. 34 THE FINE STRUCTURE OF THE EARTIIS INTERIOR, by Bruce A. Bolt New instruments confirm, among other things, that the core has a core. 24 | 50 INTERSTELLAR MOLECULES, by Barry E. Turner Twenty-six aré now known. They tell much about regions where stars are born. 70 THE ASYMMETRY OF THE HUMAN BRAIN, by Doreen Kimura The left hemisphere is dominant in speech. The right also has specialized tasks. 8! BICYCLE TECHNOLOGY, by S. S. Wilson This humane mechanism is a triumph of delicacy and efficient energy conversion. 92 THE MIGRATIONS OF THE SHAD, by William C. Leggett : A highly edible fish travels with remarkable precision in both space and time. / ryk / Bel, +" still THE ORIGINS OF NUMBER CONCEPTS, by Charles J. Brainerd Experiments with children show the order in which crucial concepts are learned. DEPARTMENTS Soest - Baan nd laa» | LT SCIENCE AND THE CITIZEN MATHEMATICAL GAMES THE. AMATEUR SCIENTIST BOOKS ‘ BIBLIOGRAPHY 44 110 114 121 128 LETTERS 50 AND 100 YEARS AGO THE AUTHORS 8 10 12 Philip Morrison (Book Editor), Trudy E. Bell, Jonathan B. Piel, David Popoff, John Purcell, James T. Rogers, Armand Schwab, Jr, C. L. Stong, Joseph Wisnovsky Samuel L. Howard (Art Director), Hil Arbel, Edward Bell aut oevraerwernt Gerard Piel (Publisher), Dennis Flanagan (Editor), Francis Bello (Associate Editor), soaan or epitcas saop.crıon DeraarmEnt Ke Donald H. Miller, Jr. érxtrat wanacen Richard Sasso (Production Manager), Leo J. Petruzzi and Carol Eisler (Assistant Production Managers), Pauline Ray Lipsher Sally Porter Jenks {Copy Chief), Dorothy Patterson, Candace E. Trunzo, Julio E. Xavier corr verattment Aovttrisine oineeron vo ASSISTANTS TO THE euerisner CIRCULATION manwacere wi Le à Harry T. Morris George S. Conn, Stephen M. Fischer William H. Yokel PUBLISHED MONTHLY BY SCIENTIFIC AMERICAN, INC, 415 MADISON AVENUE, NEW YORK, N.Y. 10017. COPYRIGHT © 1973 Br SCIENTIFIC AMERICAN, INC. ALL RIGHTS RE. SERVED. PRINTED IN THE U.S.A. NO PART OF THIS ISSUE MAY BE REPRODUCED BY ANY MECHANICAL, PHOTOGRAPHIC OR ELECTRONIC PROCESS, OF IN THE FORM OF A SECOND-CLASS POSTAGE PAID AT NEW YORK, N.Y., AND AT ADDITIONAL MAILING: OFFICES. AUTHORIZED AS SECOMD.CLASS MAIL BY THE POST MISSION OF THE PUBLISHER. NOR PHONOGRAPMIC OFFICE ES PER RECORDING, DEPARTMENT, YEAR, ALL OTTAWA, OTHER MAY IT BE CANADA, COUNTRIES. STORED AND EUROPEAN FOR IN A RETRIEVAL SYSTEM, PAYMENT SUBSCRIPTION OF POSTAGE COPIES TRANSMITTED IN MAILED CASH. C/O OR OTHERWISE SUBSCRIPTION INTERNATIONAL COPIED FOR PUBLIC OR RATE: SID PER DISTRIBUTORS, YEAR. US, PRIVATE USE POSSESSIONS KOEPOORTBRUG I, B WITHOUT AND 2000 WRITTEN CANADA; ANTWERF, BELGIUM.

MUM Ref The Origins of Number Concepts Experiments with children indicate that they first become aware of numbers in terms of ordered sequences and only later in terms of quantities. Such findings may give rise to a newer “new math” by Charles J. Brainerd ophers and mathematicians at least since the time of Pythagoras in the sixth century 8.c. Today it continues to be important not only to scholars but also to society as a whole, witness the triumph of the “new math” in the elementary schools of America during the past dec- © ade. The fact remains that comprehensive studies of the origin of number concepts in young children are relatively rare. I have recently participated in investigations at the University of Alberta that have developed some new information on the subject. Before I describe these studies it will be useful to review the main theories of the origin of number concepts. —> Pythagoras believed that what we call the positive integers or natural num- Richard Dedekind to “arithmetize” mathematical analysis. Both scholars derived rea) numbers—the combined set of all rational and irrational numbers that is from being god-given, natura] numbers were constructions of the human mind. The three most famous propagators of this suggestion were Gottlob Frege, Giuemploved in most classical mathematseppe Peano and Bertrand Russell. Obics—from the natural numbers. A third viously a theory was needed that would example is the proposal of Leopold trace the rise of the natura) numbers Kronecker to found all mathematics on the natural numbers. This Kronecker atbut how was such a theory to be contempted to accomplish solely with “finitary” methods, that is, methods invoking neither nonfinite entities nor proofs involving more than a finite number of steps. Still other mathematicians, in particufrom some more basic notion or notions, structed? If most or all of classical mathematics had evolved from the natural numbers, it was improbable that the required theory could be devised entirely within the bounds of classical mathematics, lar those who were conversant with con- First Frege, then Peano and finally temporary advances in symbolic logic, put forward the suggestion that, far Russell turned to symbolic logic as a potential source of the fundamental no- OO’ bers (1, 2, 3 and so on) were god-given AT BN UJ! “oo H° do we first learn concepts of number? The question has been a topic of debate among philos- \ / > entities that formed the ultimate foundation both of mathematics and of the universe, The Pythagoreans’ own discovery of such “incommensurable” quantities as the ratio between the diameter of a circle „and its circumference ultimately dis- "built on natural numbers. That the natural numbers provided the foundation ‘af mathematics, however, persisted as an “ “article of faith among mathematicians until well into the 19th century. "By then the attitude toward the nat- - ural numbers had begun to change. The centrality of natural numbers was no longer considered an accepted fact but + was viewed as a conjecture that required CARDINATION rigorous proof. The proofs usually took _ the form of a stepwise derivation of such CARDINATION determines the number of : well-known number systems as rational,- objects in a set by matching rather than by : real and complex numbers from the nat- „counting. In this example the objects corural numbers themselves. Two examples ' respond in number to the five fingers of a are the attempts of Kar! Weierstrass and hand; thus the set's cardinal number is 5. ORDINATION ORDINATION determines the number of objects in a set by counting “One, two, three” and so on rather than by matching. In this set, as in all finite sets, the last ordinal number, 5, is the cardinal number of the set.

CONCEPT OF ORDINATION in terms of “heavier than /lighter than” was tested by asking the subjects to lift clay balls that were identical in size but different in weight. In some tests the heaviest SECOND ORDINATION TEST, involvinx the relation “longer than/shorter than,” Presented the subjects with three lengths of wooden dowel; th e longest stick was half a centimeter longer and e shortest stick half a centimeter shorter than the third stick, 102 ball was at the right end of the row and the lightest at the left; in other tests the order was reversed. The comparisons in all the tests were with a third ball (middle). which was intermediate in weight. which was always placed between the other two. As with the weight teat, the order of Presentation was randomly left to right or right to left, Subjects compared 1 ong and short sticks with intermediate stick (color) but never directly compared long stick with short.

- tions necessary for :. theory of natural number. Frege was the first of the three to publish a specific theory. In Die cal pair of natural numbers be an ordinal be accepted without further analysis as ‘the foundation of mathematics. They peared in 1884, he proposed that the relation, then the complete series of natural numbers can be constructed stepwise with the aid of the rule of mathematical induction. Like Frege’s cardinal natural numbers could be reduced to the theory, Peano's states that the series of of natural numbers is an innate intuinotion of “class” and the operation of “correspondence,” by virtue of which classes are quantified. According to natural numbers presents a general prob- Fon, Present at birth in all normal mem. Grundlagen der Arithmetik, which ap- Frege, each natural number n was to be regarded as a “superordinate class” whose members, “subordinate classes,” each contain precisely n elements. Given two subordinate classes, A and B, the two are said to be members of the same superordinate class, that is, instances of the same number, if and only if a one-toone correspondence can be established between their respective elements. If instead the correspondence is many to one, then A and B are said to be instances of different numbers. In essence Frege’s theory states that the series of natural numbers presents a general problem of quantification, but that the general problem can be reduced to the more restricted notion of “cardilem of quantification, Unlike Frege's theory, however, Peano’s ordinal theory reduces the general problem to the more restricted notion of quantifying transitive, asymmetrical relations, or ordination. The commonest example of ordination is the counting of things. Just which of the two theories, the cardinal or the ordinal. is mathematically preferable is a question that has never been answered to everyone's satisfaction. Reasonable objections can be. lodged against both. For example, the cardinal theory is subject to the celebrated paradox, discovered by Russell in 1901, concerning the class composed of all those classes that are not members of themselves. Briefly. if Y is the class of all those classes that are nat members of themselves, then it can be shown that Y both is and is not a member of itself. nation,” or quantifying classes. The commonest example of cardination is the matching of things. Frege’s cardinal theory remained gen- With respect to the ordinal theory, as Russell pointed out, whereas Peano’s five axioms obviously are satisfied bv the erally unknown until Russell rediscovseries of natural numbers, they are ered it in 1901. Russell subsequently equally satisfied by other number syspublished the cardinal theory, with full acknowledgement to Frege, in his Printems, For example, the rational fractions (1, 1/2, 1/3, 1/4 and so on) satisfy the ciples of Mathematics (1903), in his joint axioms, as will any series of mathematiwork with Alfred North Whitehead, cal or empirical entities that has a begin- Principia Mathematica (1910-1913), and in his Introduction to Mathematical ning, no repetitions and no end and is such that every entity can be reached Philosophy (1919). Beweer the time Frege first published the cardinal theory and the time Russell rediscovered it Peano developed a second theory about the natura! numbers. This theory first appeared in his work Formulaire de Mathématiques cepted mathematical basis for choosing bers of the human species. _ N onmathematical scholars tend to view with profound indifference the tortures that mathematicians suffer over such basic issues as the nature of number, They have learned from centuries of hard experience that the mere fact that the foundations of some mathematical system or concept are not secure need not deter them from emploving the system in their work. On the contrary, mathematical notions whose foundations have been matters of continuous debate have often yielded the most mileage; the notion of an infinitesimal is perhaps the best-known example. An infinitesimal can be defined as a quantity that, although it is infinitely small, is nevertheless greater than zero. The calculus, as it was; originally formulated by Newton and Leibniz, involved infinitesimals; Newton called them fluxions. Intense disagreement followed over the question of whether such a curious item as an infinitesimal could be viewed as a legitimate mathematical concept. By the latter half of the 19th century the debate seemed to have been finally silenced by Weierstrass’ success in reformulating the calculus without theory, a choice that is natural enough 1972). Throughout this long period the calculus of infinitesimals has been regularly applied in the physical and biological sciences withott regard to its foggy mathematical status and without damage to the disciplines concerned. Just because neither the sciences nor society has suffered much as a result of scholarly indifference toward (more precisely ignorance of) the debate over inwhen one considers that the codiscoverfinitesimals, it should not be assumed between the cardinal and the ordinal theories the choice becomes a subjective matter. Typically the choice is detershall slightly reword here. First, 1 is a natural number. Second, any number that is the successor of a natural number mined by one’s degree of sympathy with : the successor of any natural number. - Fifth, if a series of natural numbers includes both the number 1 and the sucoffer a “psvchological” thesis: The series infinitesimals. Within the past decade, in a finite number of steps. In short, the however, Abraham Robinson’s work on nonstandard analysis has resurrected the domain of application of the ordina) theory is much wider than the series of nat- ‘ infinitesimal [see “Nonstandard Analural numbers. ysis,” by Martin Davis and Reuben Because there is no universally ac- Hersh; SCIENTIFIC AMERICAN, June, (1894) in the form of five axioms that I is itself a natural number. Third, no two natural numbers have the same successor. Fourth, the natural number 1 is not deny that the natural numbers are the invention of mathematical minds and one or another of three modern schools of mathematical thought: logicism, formalism and intuitionism. Those who Jean toward logicism favor the cardinal ers of the theory, Frege and Russell, _ thatthe debate over the origins of the were the principal founders of logieism.. number concept can be ignored with cessor of every natural number, then the Those whose sympathies are with for- equal safety. Unlike the infinitesimal, . series contains all natural numbers. malism lean toward the ordinal theory; number is not the exclusive property, or In essence Peano’s theory places the the fact that Peano's axioms seem to de- even largely the personal property, of natural numbers in an ordinal relation nude the number concept of innate the mathematician. Number has been a or, in the language of symbolic logic, a “meaning” probably explains this pref- concept of social importance since the “transitive, asymmetrical relation.” If we erence. As for intuitionists, they have dawn of recorded history, The signifiare willing to stipulate that the relation. in effectreturned to the Pvthagorean cance to society of number and numberR that obtains between every nonidenti- * position that the natural numbers must related skills has increased tremendous-

©@eeevcce ly with the rise of industrial civilization. In most Western nations today children receive considerable exposure to number concepts soon after they begin to speak. More or less haphazard at first, the exposure becomes simultaneously more intense and more systematic with the onset of formal education. During the first few years of elementary school roughly 50 percent of the curriculum is normally given over to inculcating the natural numbers and methods of manip- © e e eo 0e e 20666 ulating them, The child learns to add, subtract, multiply and divide the natural pumbers and finally to derive other num- 2000000000 oeoeosee ber systems from them. We expect our children to possess by pubescence a numerical competence much higher than that of an educated Greek or Roman adult of two millenniums ago. We have even gone so far as to develop labels that imply mental turpitude on the part of those otherwise normal children who C; fail to attain the standards of numerical competence that we deem desirable, for example “learning disability” and “underachievement.” Considering the emphasis our society puts on numerical competence, it might be expected that extensive mformation would long ago have been collected on how the concept of number naturally unfolds in the course of a child's mental growth. The truth is that we possess very few hard facts about the emer_ gence of numerical ideas in children’s “ € ~ « LS =: > } _ or thinking. It is the paucity of such data that gives social relevance to the mathematical debate on the origins of the number concept I have just outlined. In what follows I shall describe some recent studies in the area of developmental psvchology that were designed to investigate the relation, if any, between the actual emergence of numerical ideas in young children and the hypothetical patterns of emergence suggested by the two principal theories of the origin of number concepts. 5 . ~ _ ä ng ' 8 à When the mathematician says that some concept Y (for example natural number) is “founded on” or “based on” some other concept X (for example cardination or ordination), he means that X is more fundamental in a logical sense than Y. When the developmental psychologist makes the same statement, CONCEPT OF CARDINATION was tested by asking the subjects to decide, without counting, which of six matched sets of dots (s-f) were in one-to-one correspondence and which in many-to-one correspondence. The top and bottom row of dots in esch set differed in color. If both rows contained the same number of dots, the dots were speced so that the rows were not of equal length. Similarly, when one row contained more or fewer dots than the other, dots were spaced so that rows were of equal length or the row with fewer dots was longer. 104 he means that X plays a necessary part in the emergence of Y in the thinking of the growing child. From the psychologist’s viewpoint such a statement leads to two major predictions that can be tested experimentally. The first prediction is that X will invariably appear

PERCENT “earlier than Y in the course of the child’s mental development. The second is that improvements in Y will be a conse“quence of improvements in X. Different research strategies are used to examine these two predictions. The “X before Y” prediction calls for normative, or average-performance, studies. The normative study consists of four steps. First, tests are devised that measure the presence of the concepts in question in the child’s thinking. Then the children to be tested are drawn from the range of ages when the concepts in question are normally acquired. The third step is testing for the presence or absence of the concepts. The fourth is comparison of the test results to determine the presence or absence of interconcept correlations of the predicted “X before Y” kind. The “better X means better Y” prediction is tested by what is known as a transfer experiment, a method that is slightly more complicated than the normative method. Seven steps are involved; the first three are the same as the first three steps in a normative study. Taken together, thev constitute the “pretest” phase of the transfer experiment. The information collected in the pretest phase is then used to divide the subjects into experimental groups and control groups. In the fifth step the experimental groups receive training designed to improve their performance in one or another of the concepts in question. After that the tests administered in the pretest phase are given to the experimental and control groups. The seventh and final step compares the test results both to determine the degree of improvement that resulted from training and to find whether or not training in one specific concept tends to “transfer” and improve performance with respect to another concept for which no training was provided. Both normative and transfer experiments appear to be simple, but a major -pitfall awaits the unwary investigator. ‚In constructing the tests whereby the ! subjects’ grasp of various concepts will ‘be assessed, one must guard against doosely defined methods of assessment that bear only a vague or intuitive re- U fl STAGE OF DEVELOPMENT GRASP OF CONCEPTS, as revealed by the test scores, placed each of 180 subjects in one of three stages of development. The children at Stage I were totally incapable of either cardination (color) or ordination (gray}. qe at Stage Il were capable of many-to-one eardination (color) and of left-to-right or ination (gray) but failed te grasp one-to-one and right-to-left relations. Children at Stage III grasped cardination (color) and ordination (gray! fully. As the percent of children at each of the stages indicates, the concept of ordination appeared in the children’s thinking long before the concept of cardination. conclusive. Although the pitfall may seem obvious, it was not avoidedin most earlier investigations of the growth of concepts of cardination and ordination and develops some competence in manipulating natural numbers. The tests children’s number concepts. not excludfor ordination assessed the children’s ing Jean Piaget's historic studies in the capacity to quantify two familiar transilate 1930's and early 1940's. tive, asymmetrical relations: “heavier At the University of Alberta we recently undertook two large-scale studies than/lighter than” and “longer than/ shorter than.” of the growth of three arithmetic elements in the thinking of young children: IP the first instance each child was the concept of cardination (the “setshown three clay balls in a row on a matching” of Frege and Russell), | the concept of ordination (the sequential tabletop. All three balls were the same size; the balls at each end of the row quantification of Peano) and the children’s competence in manipulating natwere the same color [see top illustration on page 102}. Although the balls were identical in size, they were different in ural numbers. The studies were| designed to answer three related questions of the“X before Y” kind: Which of the two comes first, cardination or ordination? Which comes first, ordination or competence? the lightest at the right end; in other tests the order was reversed. In each case the child was asked to compare the ' measured. For example, in the present comes first, cardination or natural-num- weights of the left and middle balls and “instance it was essential that the tests ber competence? then the weights of the middle and right concerning the concepts of cardination The first of our studies was concemed balls. When that had been done, the and ordination be explicitly deduced exclusively with the question of whether -child was questioned about the quantifrom the mathematical conceptions of N cardination or ordination came first. The tative relation between the balls at each these notions. If that had not been done, _ subjects were 180 Canadian children be- end of the row. The child had been givthe findings of the experiments could . tween five and seven, the range af ages en no information about this relation, only have been suggestive rather than when the average child first acquires the so that a grasp of the ordination con- „semblance to the concepts that are to be natural-number weight. In some tests the heaviest ball would be at the left end of the row and Which

STAGE Mil (13) STAGE | (22) STAGE Ill (43) ‚STAGE H (44) STAGE ti (35) STAGE II! ORDINATION STAGE ! CARDINATION SHARP CONTRAST between the elementary grasp of cardination and of ordination is apparent in this pair of charts. Of 93 children with a top-level grasp. of ordination (left), only 13 percent (color simultaneously possessed a top-level grasp of cardination. Ninetytwo children (right) had no comprehension at all of cardination; 43 percent (coger) nonetheless had a top-level grasp of ordination. ! cept was necessary if the questions were to be answered correctly. In the second instance three lengths of quarter-inch dowel were arranged in a row from left to right in front of the child. The three sticks appeared to be could not do so when the order was reversed. The children at Stage III were capable of true quantitative ordination and answered correctly regardless of the order of presentation of the objects. identical in length, but actually the sticks at each end of the row, which tion of classes on the basis of the corre- ble of cardination. The children at Stage spondence between their respective Ii were capable of quantifying the classes, the cardination test was de- many-to-one correspondences but not signed to assess the children’s capacity “ the ‘one-to-one correspondences. The to quantify pairs of classes that: dis- children at Stage III were capable of played either one-to-one or many-to-one cardination on the basis of both kinds of correspondence. The test consisted of correspondence. dle stick. As with the balls, the short stick was sometimes placed at the left and sometimes at the right [see bottom illustration on page 102). The child was asked to compare the lengths of the left and middle sticks and thelengths of the middle and right sticks and was then questioned about the quantitative relation between the sticks at the ends of “the row. Again, if the questions were to be answered correctly, ordination was The two ordination tests were administered to the 180 children. It was at once apparent that the group comprised children at three distinctive stages of development. The children at what 1 shall call Stage I were totally incapable of ordination. Those at Stage II were capable af spatial ordering but not of true quantitative ordination. This is to say that they answered correctly when the order of lightest to heaviest or shortest to longest was from left to right but | - 106 All 180 children were given the cardination test. As with ordination, three stages of development were found. The children at Stage I were totally incapasix problems [see illustration on page104}. In each problem the child. was shown two parallel rows of dots; the dots in the top row were red and the dotsin the bottom row were blue. The minimum number of dots in a row was six and the maximum number 10. When each row had the same number of, dots as the other (one-to-one correspondence), the dots were spaced so that the rows were nonetheless of unequal length. Similarly, when one row. had more or fewer dots than the ‘other (many-to-one correspondence), the dots were spaced so that the rows were the same length or the row with fewer dots was the longer. This arrangement guarded ‘against the child’s making his: Jude: ment on the basis of gross perceptual cues. To further ensure that the six peirs Oz most important finding came from comparing the results of the two.tests. The difference in the numbers of children at each of the three stages of development had already suggested that ordination appears in a child’s thinking longbefore cardination. For example, in ordination 93 of the 180 children were functioning at Stage Il and 24 at Stage I, whereas in cardination only 15 children were at Stage IH and 92 at Stage I [see: illustration on preceding page]. When the same child’s performances in both tests were compared, the “X before Y” position of ordination became even more apparent. Only 12 of the 93 children functioning at Stage IT] of ordination were also at Stage III of cardination. Of the 92 children functioning at were quantified only by correspondence Stage I of cardination, however, 40 were the child was directed not to count the functioning at Stage III and 32 more at ., spectively half a centimeter longer and half a centimeter shorter than the midoperation.) An were painted the same color, were re- Because cardination is the quantificadots. (Counting, of course, is an ordinal

Stage II of ordination [see illustration on opposite page]. All things considered, one could not ask for a clearer answer to “the question of whether ordination or cardination is the first concept to appear in a child’s thinking. “50 For our next normative test we selected a second group of 180 schoolchildren, divided into equal numbers of kinder- 40 39 | garten pupils (five to six years old) and PERCENT first-graders (six to seven years old). The children were given the ordination and cardination tests described above and in addition were given a two-part test of natural-number competence. The first 30 part of the test assessed the children’s capacity to add the first four natural numbers; it consisted of the 16 prob- 20 lems 1 + 1 = ?, 1+2= ? through 4+ 4= ?. The second part of the test assessed the capacity to subtract natural 10 numbers of 8 or less whose difference corresponds to one of the first four natural numbers. It too consisted of 16 problems: 2 — 1 = ?,8 — 1 = ? through 8 — 4 = ?. The 90 six-to-seven-year-olds ABOVE were given both parts of the test but AVERAGE only the frst part was given to the 90 AVERAGE LEVEL OF NUMBER COMPETENCE children aged five to six, who had not yet been introduced to subtraction in kindergarten. The number-test items were present- BELOW AVERAGE . GRASP OF ORDINATION proved to be associated with superior number competence among 180 additional elementary-school ‘subjects. Of 119 children with a top-level grasp of ordination, nearly three-quarters showed above-average or average number competence. ed simultaneously in verbal and written form. For example, for the first problem 60 the child was given a sheet of paper with 1+ 1=? printed on it and was simultaneously asked, “How many apples are 53 one apple and one apple?” If the child 50 wrote or spoke the correct answer, or did both, the problem was scored as having been solved. 40 considered to be below average, average and superior numerical skill in children of this age level. On the basis of these consultations we established three levels . of performance. Twelve to 16 correct 31 8 PERCENT After the test scores had been collected we consulted with the superintendents, principals and staffs of several elementary schools to determine what they 20 ”, answers were scored as superior, six to 46 - 11 correct answers average, and zero to + “five correct answers below average. We „sorted our subjects accordingly into A Kar 3 10 “> Analysis of the study showed first of “+ all that the test results replicated our | earlier findings about ordination and “eardination. Among the second group of 180 children ordination emerged in the _ > Ba 180. Cardination emergedie the | ABOVE AVERAGE | AVERAGE BELOW AVERAGE LEVEL OF NUMBERGONPETENGE game three stages as before, and ordina- GRASP OF CARDINATION proved not to be associated with competence in sing num The next question was whether ordina- . dination, nearly half nonetheless showed an ahove-average or average number competence. tion emerged long before cardination. hers on the part of the same 180 subjects. Of the 95-children with no comprehension of car-

tion preceded or followed natural-number competence in order of emergence. To find the answer we compared each CARDINATION child’s stage of development in ordination (stages I, II or HI) with the child’s degree of natural-number competence (below average, average or superior). Of the 180 children 119 were functioning ORDINATION at Stage III, the top level of ordination, 39 were at Stage II and 22 were at Stage 1. Of the 119 Stage III children 46 displayed the highest level of number competence, 38 displaved average compe- NATURAL NUMBER tence and 35 were below average. The comparison demonstrated to our satisfaction that ordination precedes natural- CORRECT ANSWERS (PERCENT) | CARDINATION | | a number competence in order of emergence. | | A third question remained: What is = the first to emerge, natural-number competence or cardination? We compared each child’s level of cardination with the child’s degree of natural-number competence. Of the 180 children only 33 were functioning at Stage III, the top level of cardination. Thirty- ORDINATION 55 \ i | | | NATURAL | NUMBER eight were functioning at Stage IL and 95 at Stage I. Of these 95, 15 displaved superior number competence and 30 average competence. The remaining 50 100° children displaved the lowest level of 1 0 26 40 60 1 80 CORRECT ANSWERS (PERCENT) | | 8 CARDINATION | | | parison satisfied us that natural-number competence emerges before cardination. Taken together, our two normative studies showed an invariant sequence in J N natural-number competence. The comthe growth of the children’s concept of | number. Ordination was the first to ‘emerge, followed by natural-number competence and then by cardination. The sequence strongly suggests that natural-number competence is founded on | | ORDINATION sz 2 : | | | | NATURAL NUMBER | TRANSFER EXPERIMENT (left) tested 1 40 60 ji 80 whether training in cardination and ordina- CORRECT ANSWERS (PERCENT) i CARDINATION tion would affect a child's competence in using natural numbers. A test group of 240 * children was divided into four sets of 60. each set equally competent in cardination, |: ordination and use of numbers (gray bars). The:sets were then paired; the children in fi . sets a and c were specially trained in ordi nation and cardination respectively and the children in sets b and d, as controls, received only routine testing. After eight train- ‚100 ! I | ing or testing sessions all four sete were giv- | ORDINATION ! NATURAL en a final test (colored bars). The training in ordination nearly doubled the scores of the children in set a; a transfer effect ie also apparent in the improvement in naturalnumber scores for set a. The training in caril dination nearly tripled the scores of the chilil | NUMBER dren in set c. The children’s scores were nonetheless low and their training had no 100| significant effect on natural-number scores. I}: | 0 20 40... CORRECT ANSWERS (PERCENT)

a prior understanding of ordination and not on a prior understanding of cardination. * It is possible to interpret the sequence established by our normative tests in other ways. For example, it could be asserted that ordination, natural-number competence and cardination are acquired in relative isolation from one another and that the observed sequence does not necessarily imply any cognitive in the second experimental group was also superior to that of the control group, showing that cardination too improves through the correspondence of elements. The Pythagorean assumption was well Place. Most parents, however unaware improvement in cardination, indicating perimental group trained in ordination dependence among these concepts. It was for this reason that we went on to our third experiment: the transfer exour “better X means better Y” predicmath.” In the “new math” the logicist ‘theory that natural numbers are derived from the quantification of classes was at ‘last put into educational practice. Indeed, it would be more accurate to refer trol group. The number performance of. to the “new math” as the cardinal apthe children in the experimental group “proach to early mathematics instruction, trained in cardination, however, was not whereby the child is first introduced to significantly superior to that of their conl ithe central idea of the logic of classes trol group. These findings confirmed | ‘and the natura) numbers are introduced what had been suggested by our normative studies: The conception of natural _ ‘later as by-products of the process of quantifying classes. remain, the cardinal approach domiing of ordination and not from a prior Today, although pockets of resistance number derives from a prior understandnates educational circles in North Amerunderstanding of cardination. ica almost as completely as the Pythag|: is instructive to consider the ques- ‘orean approach did when Russell spoke tion of how children might best be against it. Most major publishers of eletaught numerical skills in the light of mentary textbooks either have adopted these findings. Whether we like it or not, or are in the process of adopting some number competence is socially pré- version of the cardinal approach, and scribed in modern civilization. One 77 continued resistance to the change has might even say that number compe. become very difficult. 1 tence has come to have significant surTaking into account the implications vival value. For example, some of the of our normative and transfer experimost important decisions made by the ments, it seems dangerous to rest conaverage person during a lifetime inv olve | tent with either the old Pythagorean apnumber-related ideas‘such as moncv. Yet | proach or the new cardinal approach. it is obvious that if we do not design | -The order of emergence of the various mathematical instruction to agree with | concepts in children’s thinking is ordination first, number second and cardinatence of young children, whereas cardination training does not. The conclusion aging a large number of the minds: we: tends to improve the number compeeducation unduly dificult for children. tion third. Moreover, ordination training the hard facts of mental growth, we run the risk of making early mathematics! Even worse, we run the risk of discourare attempting to instruct. | that a new “new math” emphasizing or- During the 19th century and most of dinal notions is called for seems inesthe 20th it was standard practice in the capable. nations of the West to begin the formal sion all four groups were given the mathematical education of young chil- Dex the 10th and final testing ses: entrenched in educational circles, however, and Russell's argument was igas a result of training. Our third finding was that the average; nored for nearly a quarter of a century. improvement in ordination performance! | | In the past decade the change that was much greater than the average |Russell called for has begun to take they may be of the details, remember the rallying cry of the period: “New that cardination is more difficult to ac! quire than ordination. Our last two findings concerned number performance; We found that the children in the ex! were superior in number performance to the children in the matching con! periment that would test the validity of tion. Our subjects were 240 children between the ages of five and six; each child was seen once a week over a 10week period. During the first meeting the child was given the three tests emploved in the earlier normative studies. The children were then divided into four groups of 60 each. The groups were carefully matched with respect to the children’s test performances; this matching ensured that any differences in performance that might later emerge would not be due to initial differences among the groups. Two of the groups were now selected as an experimental group and a control group for training in ordination and the remaining two as an experimental group and a control group for training in cardination. Over the next eight weeks both experimental groups were given a weekly test of the kind emploved in the normative studies. The training was reinforced by simple feedback. This is to say that each child was told whether or not the answers to the test questions were correct; there were roughly 15 minutes of feedback training in each of the eight sessions. The control groups were given the same kind of weekly test but without feedback. No number tests were given during the eight weeks. dren with the natural numbers. It was this practice that Russell decried in: his introduction to the revision of Principles of Mathematics published in 1937. Ras." game tests for ordination, number per- -‚formance and cardination that had been Her might such an approach be instituted? The first concepts to be introduced should be ordination and ordination-like ideas such as the logic from the discredited Pythagorean as: sumption that the natural numbers are “fected to statistical analvsis. The analysis _ wevealed five major findings {sec illustrasell argued that the practice was derived. could. be introduced as by-products of “the results of the final tests were subof relations. Later the natural numbers jadministered during the first session, and the quantification of transitive, asymmetrical relations. Still later cardination fin on opposite page]. First, the ordina- unanalyzable entities that we simply could be introduced as a generalization 08 performance of the 60 children in must accept as being given. It is safe, to: | of thle natural numbers. Such an ap‘thefirst experimental group was clearly assume that he would have preferred :a. proach to early mathematics education “saperior to that of the 60 children in the curriculum wherein the introduction.of | would provide a much better fit with matching control group, demonstrating natural numbers is postponed until the: what is now known about the emerthat ordination definitely improves as a child has been taught the rudiments of | genoe of number concepts in the young result of feedback training. Second, the the logic of classes and, in particular, child s mind than either the Pythagorean cardination performance of the children ‘taught the quantification of classes:fi u cardinal approach.

Readers interested in further reading on the subjects covered by articles in this issue may find the lists below helpful. THE PROSPECTS FOR A STATIONARY WORLD POPULATION REFLECTIONS ON THE DEMOGRAPHIC Coxprrions NEEDED TO ESTABLISH A US. STATIONARY POPULATION VISUALIZATION OF BACTERIAL GENES IN, Action. O. L. Miller, Jr., Barbara A, OF THE STARLEYS OF CovENTRY. Geof- Hamkalo and C. A. Thomas, Jr., in frey Williamson. Geoffrey Bles, 1966. Science, Vol. 189, No. 3943, pages | THE MIGRATIONS OF THE SHAD 392-395; Julv 24, 1970. MORPHOLOGICAL STUDIES OF Transconir! rıon. O. L. Miller, Jr., and Aimée H. ATLANTIC Coast MIGRATIONS OF AMER- Bakkenin Acta Endocrinologica, Sup! plementum 168, pages 155-178; CAN Snap. G. B. Talbot and J. E. Sykes in Fishery Bulletin, Vol. 58, VISUALIZATION OF RNA SYNTHESIS ON pages 473-490; 1958. Fish Micration. F. Harden jones. St. CHROMOSOMES. O. L. Miller, Jr., and Barbara A. Hamkalo in The Interna- Martin’ s Press, Inc., 1968. WATER TEMPERATURE AND THE MIGRA- 1972. | tional Review of Cytology, Vol. 33, TIONS OF AMERICAN SHAD. W. C. Legpages 1-25; 1972. gett and R. R. Whitney in Fishery | | GrowTH. Tomas se in Population Studies, Vol. 22, No. 3, pages 379- INTERSTELLAR MOLECULES 397; 1968. UN TAUX D'ACCROISSEMENT NUL POUR LES Pays EN VOIE DE DEVELOPPEMENT INTERSTELLAR MOLECULES AND DE: SE Croups. D. M. Rank, C. H. Townes EN L'AN 2000: REVE ov RÉALITÉ? Jean Pichat-Bourgeois and Si-Ahmed Taleb in Population, No. 5, pages 957-974, September-October, 1970 THE GROWTR AND STRUCTURE OF HvMAN POPULATIONS: A MATHEMATI- CAL INVESTIGATION. Anslev J. Coale. THe BEHAVIOR OF ADULT AMERICAN and W. J. Welch in Science, Vol. 174, ber 10, 1971. INTERSTELLAR MoLECULES. B, E. Turner xc TECHNIQUES. Julian J. Dodson, William C. Leggett and Robert A. ‘Jones in Journal of the Fisheries Re- No. 4014, pages 1083-1101; Decemin Radio Astronomy. edited by GL scarch Board of Canada, Vol. 29, No. Verschuur and K. 10, pages 1445-1449, October, 1972. 1. Kellerman. Springer-Verlag, Inc., in press. INTERSTELLAR MOLECULES AND THE |IN- POPULATION AND THE AMERICAN FuTuRE. United States Commission on TERSTELLAR Mepicm. Edited by M. A. Gordon and L. E. Snvder. Springer- INTRODUCTION TO THE THEORY OF SEISmo ocy. Keith E. Bullen. Cambridge University Press, 1963. THe Densrry DISTRIBUTION NEAR THE BASE OF THE MANTLE AND NEAR THE EARTH's CENTER. Bruce A. Bolt in Physics of the Earth and Planetary Interiors, Vol. 5, pages 1-11; 1972. OBSERVATIONS OF PSEUDO- AFTERSHOCKS FROM UNDERGROUND EXPLOSIONS. Bruce A. Bolt and A. Qamar in Physics of the Earth and Planetary Interiors, Vol. 6, pages 100-200; 1972. THE ASYMMETRY | OF THE HUMAN BRAIN FUNCTIONAL ASYMMETRY OF THE BRAIN. Ix DicHoric Listentnc. Doreen) Ki- Psychology, edited bv Paul H. Mussen. John Wiley & Sons, Inc., 1970. KATHEMATICAL AND BEHAVIORAL Fouxmura in Cortex, Vol. 3, No. 2, pages 163-178; June, 1967. SPATIAL LOCALIZATION IN LEFT AND DATIONS OF NuMBER. Charles J. Brainerd in Journal of General Psychology, in press. Ricut Visvar FieLps. Doreen Kimura in Canadian Journal of Psychology, Vol. 23, No. 6, pages 445-458; December, 1969. HEMISPHERIC MATHEMATICAL GAMES THE ART OF NUMBRING BY SPEAKING- SPECIALIZATION FOR Rops: VuLGarzy TERMED NEPEIRS Speech PERCEPTION. Michael! Studdert-Kennedy and Donald Shankweiler in The Journal of the Acaustical Society of America, Vol. 48, No. 2, | Bones. W. Levbourn. London, 1667. Part 2, pages 578-594; August, 1970. THE VISUALIZATION OF GENES IN ACTION THE ORIGINS OF NUMBER CONCEPTS Tee CHLD's CONCEPTION OF NUMBER. Jean Piaget. Humanities Press, 1964. CONCEPT DEVELOPMENT. John H. Flavell in Cermichael's Manual of Child Verlag, Inc., in press. Future. 1972. THE FINE STRUCTURE OF THE EARTH'S INTERIOR SHAD (ALOSA SAPIDISSIMA) DURING MiGRATION FROM SALT TO FRESH WATER AS OBSERVED BY ULTRASONIC TRACK- Princeton University Press, 1972, Population Growth and the American Bulletin, Vol. 70, No. 3, pages 659670, July, 1972. Pad IBLIOGRAPHY I WHEELS WITHIN WHEELS: THE STORY Warten TERCENTEN ARY MEMORIAL VOL- " UME. Edited by Cargill Gilston Knott. | Longmans, Green and Company, | 1915. Ricart HEMISPHERE SPECIALIZATION FOR DEPTH PERCEPTION REFLECTED IN THE AMATEUR SCIENTIST VisuaL FIELD DIFFERENCES. Margaret PORTRAIT OF a GENE. Oscar L. Miller, Jr., and Barbara R. Beatty in Journal of Cellular Physiology, Vol. 74, No. 2, Part 2, Supplement 1, pages 225232; October, 1969. Durnford and Doreen Kimura inNa- PROCEDURES IN EXPERIMENTAL PHysICcs. ture, Vol. 231, No. 5302, pages |394 | John Strong in collaboration with H. Victor Neher, Albert E. Whitford, C. 395; June 11, 1971. Hawley Cartwright and Roger Hay- BICYCLE recHoLoCY ELECTRON MICROSCOPIC VISUALIZATION | oF Transcription. O. L, Miller, Jr., Rowe Hicu: Tae Story or THE Br Barbara R. Beatty, Barbara A. Hamkalo and C. A. Thomas, Jr, in Cold CYCLE. Arthur Judson Palmer. "EL P Dutton & Co., Inc., 1956. wi Spring Harbor Symposia on Quantita-: HANDBOOK OF THE COLLECTION ALLus- _ ward. Prentice-Hall Inc., 1938. PALLIUM ARSENIDE Lasers. Edited bv C. H. Gooch. John Wiley & Sons, Inc. _ 1969. | NEw CLass or Diope Lasers. Morton B. Panish and Izuo Hayashi in Scientive Biology, Vol. 35, pages 505-512; / TRATING Cycres. C. F. Caunter, Her 7 | tific American, Vol. 225, No. 1, pages 1970. Majesty's Stationery Office, 1958. | 32-40; July, 1971.