the infinite challenge levels of conceiving the endlessness of numbers by rums falk the hebrew...

COGNITION AND INSTRUCTION, 28(1), 1–38, 2010Copyright C© Taylor & Francis Group, LLCISSN: 0737-0008 print / 1532-690X onlineDOI: 10.1080/07370000903430541

The Infinite Challenge: Levels of Conceivingthe Endlessness of Numbers

Ruma FalkThe Hebrew University of Jerusalem

To conceive the infinity of integers, one has to realize: (a) the unending possibility of increas-ing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greaterthan that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set isitself infinite (immeasurable gap). Three experiments probed these understandings via competitivegames and choice tasks accompanied by in-depth interviews. Participants were children 6 to 15 yearsold and adults. The results suggest that roughly from about age 8 on, children grasp potential andactual infinity. However, for several additional years their conception of actual infinity is incompletebecause the immeasurable gap between a finite and an infinite set is not entirely internalized. Evenmany adolescents and adults fail to appreciate this gap. Distinguishing between number concepts andtheir names facilitates conceiving aspects of infinity. Educational implications of these findings arediscussed.

Problems of the infinite have challenged human intellect and imagination as no other singleproblem in the history of thought (Kasner & Newman, 1949; Love, 1989). The quest for compre-hending infinity goes a long way back in history. The ancient Greeks were awed and fascinated bythe mysterious and often paradoxical nature of the concept, and the subject still holds tremendousappeal for scholars. This is evidenced by many writings including some eloquent books on thecultural history of the infinite and its paradoxes, with allusions to literature, poetry, spirituality,and references to time-space and cosmic issues (e.g., Barrow, 2005; Maor, 1987; Nickerson,2010; Peter, 1957/1976).

Many parents have observed their young children (of pre- and early-school years) goingthrough obsessive counting sprees and then struggling with the question of the end of numbers(Gelman, 1980). Some adults still remember being advised as children to count sheep whenhaving difficulty falling asleep. This attempt has often failed and resulted instead in wonderingabout the length of the process and its possible termination. Erlich (1978) described MartinBuber’s autobiographical account of his adolescent preoccupation with perceiving the limits ofspace and time that had almost driven him to the brink of suicide. And Popper (1976) relatedthat the problem of infinity was one of the early genuine philosophical problems that had worried

Correspondence should be addressed to Ruma Falk, 3 Guatemala St., Apt. 718, Jerusalem 96704, Israel. E-mail: [email protected]

2 FALK

him around age 8, “I could neither imagine that space was finite (for what, then, was outside it?)nor that it was infinite” (p. 15). And he still had serious misgivings after the matter had been“explained” to him.

In an informal survey of adults’ childhood memories concerning their thoughts about infinity,we1 learned that most of them tried to cope either with the endlessness of numbers or withspatial problems like the limits of earth, sea, sky, horizon. Others wondered about life–death andeternity, and a few puzzled over devices such as two plane-parallel mirrors with the never-endingline of reflections of a person situated in between, or toys like the Russian doll with its seriesof progressively smaller nested dolls (Barrow, 2005, p. 61; Zippin, 1962, pp. 6–7). This studyfocuses on the development of infinity notions in the numerical area. We investigate the basicunderstanding of the unending upward and downward progression of the counting numbers andof the cardinality of the set of numbers as (infinitely) greater than that of any finite set. We do notdeal with conceptions concerning the mathematics of infinite sets or with the measuring aspectsof numbers (see Tall, 1980).

THE BREAKTHROUGH TO MATHEMATICAL ABSTRACTION

In studying the development of the initial steps of coping with numerical infinity, one gets aglimpse into the emergence of intuitions that are not based on worldly observations but areabstracted by the cognitive system. Infinity is paradigmatically an intangible, distilled notion: Nomaterial representation can fully display it, and no real-life experience can accomplish going onforever. Evans (1983) wrote: “Once they [children] have accepted the possibility that there is nolargest number, they have moved from a concrete, at least potentially experienceable mathematicsinto the realm of abstraction” (p. 172). I tend to agree with Evans, interpreting “abstraction” inaccordance with some of its dictionary definitions as “formation of an idea by mental separationfrom particular instances or material objects.” Indeed, whatever “infinity” means, it is devoid ofany real-life embodiment. Mathematics is considered by some authors the science (or art) of theinfinite (Zippin, 1962, p. 3).

The experimental study of the conception of infinity has specific difficulties. When confrontedwith the beginning of a process that can go on forever, one has to infer its endlessness; theinfinitude of the process can never be actually experienced. Despite fascinating artistic attemptsby Escher to depict infinity through the continuous traversal of an endless loop (Schattschneider,1994), the infinite was not pictured. Representations of recursive relations can only allude tothe interminable continuation but not show it. At the same time, many adults do conceive theinfinity of numbers without perceiving it. Hence, the experimental study of this important andintriguing development calls for specific methods of probing. Tasks of counting appear to becandidates for starting an experiment, but the endlessness of the process can obviously never beactuated.

1Since many of my graduate students over the years helped in collecting data and discussing the issues, I often usethe plural first-person pronoun in reporting the research.

INFINITE CHALLENGE 3

ASPECTS OF INFINITY

What does it mean to “understand infinity”? What do we, who believe we possess the concept,think we understand? This paper focuses on three major aspects of infinity: everlasting process,boundless amount, and immeasurable gap (Falk & Ben-Lavy, 1989; Falk, Gassner, Ben-Zoor,& Ben-Simon, 1986; Monaghan, 2001). Paralleling the history of the subject, one has first torealize that the process of counting, or increasing numbers, is interminable, namely, to conceivethe so-called “potential infinity,” in accordance with Aristotle’s view of infinity (Maor, 1987,p. 54; Stewart, 1996, pp. 63–64). Secondly, though equivalent from a mathematical point ofview, the set of numbers has to be considered infinite, in toto: This is the so-called “actualinfinity.” These two notions are not necessarily equivalent from a psychological point of view.Actual infinity, according to which an infinite set is an entity, or object, rather than just a process(Monaghan, 2001; Moreno & Waldegg, 1991), departed from the Aristotelian and Gaussian claim(that had dominated scientific thought for thousands of years) that infinity has only potentialexistence (Barrow, 2005, pp. 26–31; Dubinsky, Weller, McDonald, & Brown, 2005, Section 2.1).Guillen’s (1983, p. 44) explanation of the distinction between potential and actual infinity (thoughgrammatically false) maintained that the former is a verb, whereas the latter is a noun. Thirdly,one might maintain that a set is (actually) infinite and still believe that it only slightly exceeds ahuge finite set. A stable conception of actual infinity presupposes a grasp of the immeasurablegap, or infinite distance between an infinite set and any conceivable finite set.

A more advanced aspect of actual infinity—though one we will not treat here—concerns thevery definition of an infinite set as a set that can be put into one-to-one correspondence witha proper subset of itself (e.g., Peter, 1957/1976, pp. 97–98; Reid, 1992, p. 169). This bizarreequivalence results in many disturbing paradoxes (some discussed by Galileo) and has alwaysbeen a source of serious psychological difficulties. The apparent violation of Euclid’s ancientaxiom that the whole is greater than its part seriously obstructs the cognitive acceptance ofthe mathematical theory of actual infinity. A lot has been written on the obstinate paradoxesthat grow out of this defining feature of an infinite set (e.g., Aczel, 2000, pp. 53–56; Dubinskyet al., 2005; Falk, 1994; Rucker, 1982, pp. 5–6, 73–75; Salmon, 1970; Waldegg, 2005). Ourbeliefs and principles have been abstracted throughout our lifetime from experience with thefinite. Consequently, we have inbuilt objections to departures from the seemingly immutablerules, and tremendous difficulty in coping with the rules of the mathematics of the infinite. We donot deal with people’s responses to these compelling contradictions of the infinite in the presentresearch, but rather with the first intuitive commonsensical conceptions of numerical infinity. It isimportant, however, to keep in mind the centrality of the conflicts incurred by having to abandonour finitist habits of thought.

PREVIOUS STUDIES AND SOME OF THEIR LIMITATIONS

According to Monaghan (2001, pp. 244–245), most pre-university students (16–18 years old)viewed infinity primarily as a process (i.e., potential) and only some expressed an object (i.e.,actual) view. Although participants’ language reflected a process interpretation, Monaghan notedthat the border between process and object may not be well defined in their minds. The researchthat we know of addressed mostly participants’ notions of potential infinity. Some studies posed

4 FALK

geometric or physical problems, others mixed spatial and material problems with numerical ones,and a few were sheerly numerical.

Evans (1983) and Gelman (1980) asked children aged 5–9 questions such as “How far cana rocket ship go?”, “How long is forever?”, or “What would happen if you dropped a ball intoa bottomless pit?” Fischbein, Tirosh, and Hess (1979) asked children aged 11–15 about thepossibility of infinitely subdividing a line segment and other geometric figures (following thetradition of Piagetian inquiries with preoperational children). In the upward direction of contin-ually increasing some measure, finitist responses prevailed among the youngest, and childrenprogressed with age to perceiving the process as limitless. Going downward in continual division,most children supported the idea that the divisibility process comes to an end. Smith, Solomon,and Carey (2005) listed additional works confirming the finding that many students can imagineonly a limited number of divisions of matter or geometric figures before total disappearance ofthe amount in question. In my opinion, answers to questions like the above have no bearing onparticipants’ ideas of numerical infinity (Falk et al., 1986; Monaghan, 2001). These responsesconfound reasoning about the extension of large or small numbers with physical and spatialproblems such as the limits of space or the rocket’s mechanical constraints. Those who claimedthat dividing the line segment would eventually end had often related to the physical dimensionsof the dividing points and the perceived decreasing subsegments that leave no space for furtherdivisions. At the same time, rewording the problem in terms of “ideal” points with no length, andusing phrases such as “forever” or “bottomless” beg the question and risk betraying the expectedanswer.

Smith et al. (2005) and Tirosh and Stavy (1996) focused on children’s ideas about the finitenessor infiniteness of dividing objects and numbers. In their thought experiments, both physicalparticles or geometric objects and numbers were successively cut into halves and the questionwas whether this could go on indefinitely or would the divided matter disappear or the numbersget to zero. On the whole, both studies concluded that children’s understanding of numbers asinfinitely divisible was positively related to their acknowledging the same with respect to materialbodies. They differed about the order of the two developments. Smith et al. found (with childrenaged 8–12) that realization of the infinite divisibility of matter preceded that of numbers, whereasTirosh and Stavy found the reverse order (among 12–18-year olds) noting that participants whogave finitist responses to physical and geometric questions had been obstructed by technicalconsiderations referring to the practical difficulties in dividing small particles (p. 681). Langford(1974), who had used similar questions of the two types (not fully described in the text), foundnumerical questions to be easier than spatial ones (among 9–15-year olds). Again, withoutknowing whether particles can or cannot be endlessly divided, I think that inferences aboutunderstanding the infinite divisibility of numbers, based on responses to problems of materialand geometric nature, should be taken with caution. Children, who claim that real objects cannotbe indefinitely divided, might be practically right, but this does not necessarily pertain to theirideas of the divisibility of numbers. Since numbers suggest an abstract mathematical context,and shapes or objects suggest the real world, the study of children’s conception of mathematicalinfinity should better address directly the numerical sphere (Falk et al., 1986; Monaghan, 2001).

Evans (1983) and Gelman (1980) interviewed young children (ranging from preschool to thirdgrade) about the limits of the process of increasing numbers. They started by asking every child,“What is the biggest number you can think of?” Then they followed the child’s response by in-depth inquiry asking questions such as “What happens if I add one to the number you gave me?”


They proceeded in that style to find out whether the child believed that such successive iterationscan go on forever. Slightly more than half of the first and second graders and above 80 percentof the third graders acknowledged, in the course of the interview, that continued additions wouldsteadily yield greater numbers and that the counting numbers are not bounded. The key to theirunderstanding (according to Hartnett, Gelman, & Meck, 1987) was their grasp of the successorprinciple that applies to any number (see Sarnecka & Carey, 2008, for the role of the successorfunction in young children’s mastery of the cardinal principle). These thought experiments areinstructive, and their conclusions make sense. The only problem lies in the initial question. Theperfect answer to the question about the biggest number one can think of is that there is nosuch number, because for every number there exists a greater one. One can hardly expect youngchildren, even those who realize that numbers grow with no end, to be able to respond this waywhen asked to provide the largest number. There must be another way to find out directly whetherchildren understand this principle. Furthermore, the recurrent what-ifs concerning addition of yetanother number might be suggestive of the desired answer. Many children have apparently heardfrom parents or siblings that there is no end to numbers. Hence, the aforementioned questionsmight elicit recitation of what they have heard without having internalized its meaning. Evans(1983) and Gelman (1980) were aware of this problem. Formulating a question that would bediagnostic of the child’s own ideas without being suggestive is not simple.

Falk et al. (1986) and Falk and Ben-Lavy (1989) devised several tasks that partially avoidedthese problems. These two pilot studies probed children’s understanding of potential infinityand of two aspects of actual infinity: the numerosity of the set and the extent of its excess overa very large finite set. The present research largely extends these preliminary experiments andincorporates their findings.

THE AIMS OF THE RESEARCH

One major purpose was to establish diagnostic tasks that would reveal how children of differentages conceptualize the potential and actual infinity of numbers. To be optimally indicative of achild’s understanding, a task should be meaningful (i.e., pose a motivating problem about whichthe child can competently think); require a simple, clear decision (that can easily be coded asright or wrong); and be novel for the respondent (to eliminate cliches learned by rote). A relevant-involvement methodology that does not use academic terms, speaks to children in a language theyunderstand, and makes it worth their while to try hard by rewarding good outcomes has beenexpediently employed in studying children’s concepts of randomness and probability (Epstein-Kainan, 2000; Falk, Falk, & Levin, 1980; Falk & Wilkening, 1998). In the present case, insteadof dealing with urns and beads, the child has to manipulate familiar symbolic entities, namely,numbers. Notions of infinity have not yet been investigated by such a method.

An easy-to-understand competitive game in which the children’s success depends on theirapprehending the potential infinity of numbers was devised. That game had a triple purpose:to examine by the same method the understanding of one’s ability to indefinitely (a) increase(positive) integers, (b) decrease (negative) integers, and (c) decrease (positive) fractions. In (a)and (b) the set of possible numbers is unbounded from one side and is ordered and discrete,in (c) the set is bounded between 1 and 0 and there are infinitely many fractions between anytwo. A fraction has neither a predecessor nor a successor, and the question is whether or not one

6 FALK

can continually move downwards without reaching zero. The hypotheses to be tested were thatchildren’s performance in the downward direction would lag behind that of the upward direction,because negative numbers and fractions are encountered later than natural numbers; and thatdecreasing fractions would be the hardest to manage because of their density (Smith et al., 2005,p. 103) and their getting infinitesimally close to zero.

I did not find a way to contrive a convincing competitive game for studying participants’ graspof actual infinity. Two other tasks were designed for that end: First, participants were requiredto compare sizes of sets; and second, they had to make operational (nonverbal) assessments ofdistances between set sizes. The purpose was to test their understanding of the infinity of the setof numbers in the first case, and that of the infinite leap in size when moving from finite sets tothe set of numbers in the second case. Previous studies of intuitions of actual infinity (Moreno& Waldegg, 1991; Tirosh & Tsamir, 1996; Tsamir & Dreyfus, 2002, 2005; Tsamir & Tirosh,1999) dealt with questions of comparison between two infinite sets (e.g., natural numbers vs.perfect squares) in order to tackle the issue of the bizarre equivalence between a set and its propersubset. Tsamir (1999) contrasted comparisons between two finite sets with comparisons betweentwo infinite sets. Our crucial comparison, for assessing participants’ grasp of actual infinity, wasdifferent: One had to compare an enormous finite set with the set of all numbers. Conceiving theinfinite gap to the numerosity of all numbers was hypothesized to be more elusive than conceivingthe infinity of numbers, because, in addition to realizing that the cardinality of numbers exceedsthat of any finite set, one has also to realize that this excess is unlimited. So far, participants’ graspof that gap has not been investigated. Another objective was to ascertain the order of developmentof the concepts of potential and actual infinity in present-day children given the background of thehistorical order of their emergence. Presumably, actual infinity—which came to be recognizedmuch later—would prove to be harder to conceive by individuals.

In all cases, the initial responses were planned to require simple decisions or operationswith minimal verbal explication. Then the responses—whether binary choices or quantitativeassessments—were to be accompanied by verbal interrogation designed to delve into their un-derlying reasons.

OVERVIEW

Experiment 1 investigated children’s understanding of the potential infinity of the process ofincreasing and decreasing numbers by means of three competitive games. Experiment 2 testedthe understanding of actual infinity of the set of numbers by means of comparison of sets. AndExperiment 3 probed the conception of the infinite gap between an actually infinite set and finitesets by means of a task of location of sets.

Experimenters and Participants

The experimenters were the author and approximately 30 graduate students of psychology andeducation at the Hebrew University of Jerusalem, who have participated, over several academicyears, in an elective research-seminar on the Development of Mathematical Concepts. Runningexperiments was part of their course requirements. The design was thoroughly worked out with


the students, and the standard procedure was rehearsed, as well as the way to fill the results intoready-made, uniform forms. The work was supervised by the author. The participants were 748children of different socioeconomic backgrounds, not diagnosed as having learning difficulties:362 in Experiment 1, 80 in Experiment 2, and 306 in Experiment 3 (there were fewer participantsin Experiment 2 because the comparison task of that experiment was, in effect, repeated in thecourse of responding to Experiment 3). The children ranged in age from 6 to 15. Another 122adults aged 16–55 (considered to have achieved the Piagetian stage of formal operations), withno less than 10 years of school education, including some undergraduate students who had nottaken academic courses in mathematics, took part in Experiment 3. Participants were recruitedthrough acquaintances and through schools; parents’ permission was secured for the children.

Design

Each participant was examined individually, in just one of the three experiments. Only Experiment1 comprised three subexperiments in the form of three competitive games: Game 1 in the upwarddirection with natural numbers, Game 2 in the downward direction with negative integers, andGame 3 downward with positive fractions. The three games were presented to each child inthat order. However, not all the children could play all the games, in particular, those involvingnegative numbers and fractions posed difficulty to children who dropped out of these games (356children completed Game 1; 270 Game 2; and 210 Game 3).

Method

A standard form for verbatim recording responses (whether choices, assessments, or verbal ex-planations, all in Hebrew) had been prepared for each of the three experiments. The experimentalsession started with simple tasks, namely, binary or quantitative choices. Then participants werequestioned in detail about their reasons. We presented the questions in a flexible manner, con-tingent on previous responses (as in Gelman, 1980; see also Ginsburg, Kossan, Schwartz, &Swanson, 1983, on protocol methods). Obviously, the interviews that followed participants’ ar-ticulations varied. The children, but not the adults, had been promised prizes contingent on theirperformance. The prizes (stickers, erasers, and various trinkets) for winning competition trials orfor making correct comparisons or assessments were handed out at the end of the session.

Afterward, the experimenter judged the performance and scored each response—positivelyor negatively—on several predetermined variables (hard-to-judge cases were scored “?”). Thesescores and the participant’s age were hidden from a second experimenter (judge) who consideredthe written protocol and rescored the performance on the same variables. The agreement betweenthe two judges was always higher than 80% and often higher than 90% for all the variables in thethree experiments. In cases of disagreement, the experimenters and the author reconsidered theprotocol and usually agreed about the scoring (problematic cases were still scored “?”).

The scores on the various tasks in the three experiments are summarized quantitatively intables of percents and in figures, that is, in descriptive-statistics exhibits. Our evaluation ofthe results—which, having been based on large numbers, are apparently rather stable—rests oncommonsensical, studious consideration of observed frequencies in conjunction with participants’

8 FALK

verbal explanations. Most importantly, we note recurrent patterns of findings within the researchand compatibility with other studies, so as to show the accumulation of information by replication(Guttman, 1977; Ross, 1985). Results that confirm the study’s initial predictions increase theprior probability of the hypotheses, and every successful replication lends more credibility to theassertions (Falk, 1998).

Participants’ words provided the basis for (positive or negative) scoring of their performance,and at the same time showed the diversity of their ways of thinking beyond the homogeneouscategorical coding. As will be shown, the abundant verbal responses often revealed unanticipatedways of reasoning and a few exemplified exceptionally sophisticated or clear thinking. Repre-sentative excerpts of these verbalizations and the lessons derived from them are thus importantconstituents of the results. Some are quoted in the article’s text, and the reader is referred fromdifferent sections to (no less instructive) numbered quotations (Quotes) in the Appendix. A quotedchild is identified throughout by gender (B = boy, G = girl) and age in years; adults are identifiedby gender (M = male, F = females) and age.

EXPERIMENT 1: POTENTIAL INFINITY—COMPETITIVE GAMES

There is no smallest among the small and no largest among the large; But always something stillsmaller and something still larger.

—Anaxagoras (ca. 500–428 B.C.), quoted in Maor (1987, p. 2)

Competitive Game 1 affords whoever understands the endlessness of the process of increasingnumbers an opportunity to utilize that understanding to their advantage. One has to realize thatthere exists a larger number than any number suggested by the opponent, large as it may be.Most importantly, one has to be aware of this possibility to start with, before mentioning anynumbers (as opposed to learning by prolonged trial and error while playing). This is what we takeoperationally to indicate understanding of the principle of potential infinity.

The same applies to the competitions in Games 2 and 3, where one has to suggest a smallernumber than any negative number or than any positive fraction the opponent may suggest. Again,to achieve a full positive score, the child has to indicate understanding of the principle of winningthese games beforehand. In Game 3, one should also realize that however minuscule the fractionnamed by the opponent, a still smaller one can be suggested without getting to zero.

Procedure

The rules of Game 1 were presented as follows: “Let’s play a game. Each of us will say a number,the one whose number is greater wins. Would you like to be first or second in choosing a number?”After recording the child’s choice, and before saying the numbers, the child was asked to justifyits choice. Then the numbers were named and the winner determined. This was repeated, ifnecessary, a few times, no more than four (so as to be sure the game had been understood and thechild’s performance was consistent). All the choices, explanations, and pairs of numbers werefilled in the form.


After that, the experimenter said, “Now let’s say larger and larger numbers by turns. I start with[e.g.] 10.” The alternate chain of increasing numbers—for example, 10, 100, 300, 1000, million,trillion, googol, and so on—ended either when the child failed to suggest a larger number, orwhen interrupted by the experimenter after progressing smoothly for a while. This sequence ofnumbers provided a starting point for an open-ended interview concerning questions such as howlong can such a game go on, whether it would eventually end, or if and how one can win.

The same procedure, with minor but crucial differences, was employed in the next two gameswhere the winner was the player whose number was smaller. When children chose to go secondin Game 2, the experimenter’s choice was zero. When they preferred to go first, if they named apositive number, the experimenter responded by zero; and if they chose zero, the experimenterresponded by minus 1. The rules of Game 3 forbade using zero and negative numbers. To seewhether the child could manage the task with fractions, the experimenter’s choice, as a first playerwas 1; and when second, it was 1/2 if the child’s number was 1, and 1 if the child’s number wasgreater.

The experiment, comprising the three games and the interviews, lasted about 50 minutes.

Results and Discussion

Scoring. In each of the three games, children were scored on three variables labeled game,principle, and infinity. The three scores were determined independently of each other.

A positive score for game was awarded when the child chose to play second and knew how toexploit the second position for winning. Because young children often rashly jump ahead to playfirst, we were “forgiving” of an initial choice to go first. A child, who saw after one trial how thegame works and then chose several times to go second and managed to suggest a larger number,was scored positively.

A positive score for principle was based on the child’s ability to justify a priori the choice toplay second. The judges had to decide whether the idea had been conveyed that whatever numberis mentioned first, one can come up with a larger or smaller number (depending on the game).

To score positively on infinity, a child had to be able to progress in the game of alternatelyincreasing or decreasing numbers, and then to express, in some way, the idea that (but for practicallimitations) this game can go on forever. In Game 3, one had also to display an understanding ofthe possibility of indefinitely reducing fractions while staying above zero all along.

We hypothesized that winning Game 1, that is, scoring positively on game, would be theeasiest and winning Game 3 the hardest. Within each of the three games, children were expectedto fare better on game than on principle and infinity, where appropriate responses depended onverbal understanding and proficiency.

Quantitative Summary. Table 1 presents the percents of positive scores on each of the threevariables in the three games according to the child’s age group. The numbers of children whowere able to make an attempt of the task in the first place (out of which the above percents werecomputed) are also reported. Children’s success in playing Game 1 is manifest. Seventy percentor more of the children from age 6 on opted to go second and were able to suggest a highernumber than the one named by the experimenter. Apparently, at the start of school, most children

TAB

LE1

Per

cent

sof

Pos

itive

Sco

res

inT

hree

Com

petit

ive

Gam

esby

Age

Gro

upan

dby

Var

iabl

e,in

Exp

erim

ent1

(In

Par

enth

esis

:Num

ber

ofP

artic

ipan

ts)

Upw

ard

Dow

nwar

d

Gam

e1,

Nat

ural

num

bers

Gam

e2,

Neg

ativ

ein

tege

rsG

ame

3,Po

siti

vefr

acti

ons

Age

grou

pG

ame

Pri

ncip

leIn

finit

yG

ame

Pri

ncip

leIn

finit

yG

ame

Pri

ncip

leIn

finit

y

6–7

(92)

69.6

(92)

53.8

(80)

43.4

(83)

47.7

(44)

43.2

(37)

51.5

(33)

33.3

a (9)

42.9

a (7)

28.6

a (7)

8–9

(84)

89.3

(84)

72.6

(73)

72.8

(81)

73.4

(64)

58.2

(55)

80.7

(57)

61.0

(41)

46.9

(32)

51.6

(31)

10–1

1(1

05)

90.2

(102

)82

.2(9

0)76

.7(1

03)

82.6

(86)

84.6

(78)

72.3

(83)

83.7

(86)

71.8

(71)

66.7

(87)

12–1

5(8

1)88

.5(7

8)86

.7(7

5)83

.3(7

8)85

.5(7

6)84

.0(7

5)79

.2(7

7)77

.0(7

4)74

.6(6

7)68

.4(7

6)To

tal(

362)

84.3

(356

)73

.9(3

18)

69.3

(345

)75

.6(2

70)

72.2

(245

)73

.6(2

50)

74.8

(210

)67

.2(1

77)

63.7

(201

)

Not

e.So

me

child

ren

eith

erdi

dno

tres

pond

toce

rtai

nta

sks

ordi

dno

tgiv

ea

clea

ran

swer

,hen

ceth

edi

ffer

entn

umbe

rsof

resp

onde

nts

ina

give

nag

egr

oup.

a Bas

edon

less

than

10pa

rtic

ipan

ts.

10


have already mastered the basic procedure of forming successors, or increasing numbers, thathad been practiced so often while counting (this accords with Evans’, 1983, and Gelman’s, 1980,results obtained by a different method).

Across the three games, from age 8 on, more than 60% of the respondents consistently electedto be second and managed to win by naming either a larger or a smaller number than theexperimenter’s. This result is, however, qualified by noting that the denominators for computingthe percents of positive scores are somewhat smaller for Games 2 and 3 due to children whowere unable to deal with either negative numbers or fractions. For example, in Game 2, 73.4%,that is, 47 of the 64 8–9-year olds who could play the game scored positively on game. These 47children were only 56.0% of the 84 children in that age group, 20 of whom could not play thegame that involved negative numbers. The drops in the numbers of respondents between gameswere related to the reductions in the percents of correct responses (out of the actual numbersof respondents). The correlation coefficients, across ages, between these two differences werefor Game 1 and Game 2:.93; Game 1 and Game 3:.93; Game 2 and Game 3:.84. These positiverelationships mean that the increased difficulty in moving from game to game in different ages canbe quantified by either difference. At the same time, a portion of the younger children who scoredpositively in the two downward-directed games had not yet learned about negative numbers orfractions. They coped on the basis of what they had encountered in mundane contexts (e.g.,negative temperatures, or halving an apple).

As predicted, children’s overall ability to articulate the principles of winning and of the infinitepossibility of increasing or decreasing numbers lagged behind their proficiency in playing all threegames (see the Total-line in Table 1). As found in many other developmental studies (e.g., Falket al., 1980), the children seemed to perform better in choice tasks than to be able to explainwhat they had done. Notably, the few exceptions, where their verbal explanations surpassed thatof playing the games, were limited to the downward direction. These were cases in which—byanalogy to natural numbers—children understood the principle of the infinite process, but wereimpeded in winning the game by their inferior skill in dealing with negative numbers or fractions.

As hypothesized, children generally displayed a better grasp of potential infinity in the upwardthan in the downward direction: Their scores on the three variables of Game 1 were generallyhigher than the corresponding scores in Game 2, which in turn were higher than those of Game3. This was replicated in most of the age groups with only a few exceptions that, as we learnedfrom the explanations, were clearly the result of learning that took place due to the experiencewith parallel tasks in Game 1.

The overall conclusion that emerges from Table 1 is that from ages 8–9 on, most childrenunderstand potential infinity in both directions. The understanding in going upward precedes thatof going downward, and decreasing negative numbers is easier to handle than decreasing positivefractions. Acting correctly generally antecedes the ability to verbalize the principle.

Verbal Performance. A massive amount of verbal responses was collected. Many of thesearticulations were unexpectedly enlightening. The examples in this section and the additionalQuotes in the Appendix were selected to convey the flavor of children’s wide spectrum of ideas.

We tried to play with children below age 6, but realized that they were too young for thesegames (Quote 1). These children were not included in the numerical results. Typical excerptsfrom the responses to each of the three games, separately, are first presented. Then we have a

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look at some examples of responses to more than one game by the same child. Problematic orbizarre articulations are lastly presented.

Responses to single games. In Game 1, G8 chose to play alternately first and second:“Because I like to play once first and then second.” And when asked to justify her choice togo second: “Because I love the number 2, because it is small.” This would typically count asa Piagetian “egocentric” response. She scored negatively on game and principle. G9, who hadscored + on game, explained her choice to play second, “I don’t like being first very much; 2 ismy lucky number.” She scored negatively on principle. G7 scored negatively on infinity: “Thereis no end to numbers, mother said.” B9, who was asked whether the continuous game wouldever end, said, “Yes, when we get to the greatest number in the world. . . . Yes, there is a numbergreater than all numbers, but I don’t know how you say it.” See also Quotes 2, 3, and 4.

Some of the positively scored responses to Game 1 were aptly phrased in the children’slanguage. G6: “I don’t want to play first! I know why—because I’ll know what to say after youhave said; I can say a bigger one.” B10 succinctly explained, “One number chases another, andthere is no end to numbers.” And B11: “It is always possible to invent more numbers by increasingthe number of digits.” See also Quotes 5, 6, and 7. Sometimes, spontaneous understanding wastriggered by the game: G8, who was asked how long the sequence of alternating numbers will goon, said, “Until 2,000 or until the last number . . . I don’t know what it is.” When asked what ifthe experimenter would say the last number, she answered, “Then I’ll say last number + 1. Onecannot end this game because there is no last number; one can always find another number.” Thefollowing are short but telling justifications of the choice to go second. B12: “Because I’ll havethe advantage of being able to say a larger number than yours.” G12: “Because I’ll know withwhat I have to cope.”

Many children, who had played Game 1, could not play Game 2 for lack of familiarity withnegative numbers, as B7 who chose to go first: “Because now I have such a small numberthat no one can overtake—zero!” Others, who were familiar with negative numbers, still failedto conceive potential infinity in that direction. G11, who had scored positively on game andprinciple, said after some steps of the decreasing alternating sequence, “I think that somewhereone will finish because all the time we get further and further away from zero, until there wouldn’texist any numbers so far away.” See Quote 8 for partial learning triggered by the experiment.

Good responses to Game 2 often indicated a fair grasp of the symmetry in going up and down.B7: “I want to be second, just as in the first game. We’ll only have to put a minus. If I hearwhat you say, I can say a greater number and make it minus.” G11: “A bigger minus is a smallernumber.” G11: “I want again to be always second. It is the opposite of what we said before, butdownwards. There are infinitely many minuses, and I’ll be able to say a smaller number thanyours.” See Quotes 9 and 18.

The overall inferior performance in Game 3 (Table 1) was also noticeable in some of theverbalizations. Apparently, the density of the fractions, the fact that a fraction has neither apredecessor nor a successor, and the lower bound set by zero, all contributed to children’sdifficulty. B8 started well in the decreasing sequence, but when asked how long it will go on:“Until zero. That is the end; it is the smallest.” G9: “We can count all the numbers between 0and 1, because there only is room there for a finite number of numbers.” G10: “I think we’lleventually reach zero, because the numbers get smaller and smaller until they’ll disappear andbecome zero.”


At the same time, even some children who had not learned fractions, and were familiar fromdaily use only with a half and a quarter, managed to play by continuously halving or quarteringthe current numbers. G8 chose to go second. This resulted in the following ordered pairs insuccessive trials: (1, 1/2); (1/4, 1/4 of 1/2); (1/4 of 1/4, 1/4 of 1/4 of 1/4). See also Quotes 10 and23.

Several children from age 8 on articulated the idea that the greater the number by which youdivide a positive quantity, the smaller the result. B8: “This game will never end because you canalways divide by a greater number and get a smaller number” (see Quotes 14 and 33). There alsowere explanations of the endlessness of decreasing positive fractions. B10: “There are infinitelymany numbers between 0 and 1, because one takes all the usual numbers and turns them intofractions. We’ll never reach zero, we’ll only want to reach it more and more.” B11: “We cannotreach zero, we only approach it. There is an infinity of numbers between 0 and 1 even though itlooks like a small interval.” B11: “There is no end to how much one can divide the 1.” And B12:“One cannot win. We’ll not reach zero. When one writes a decimal fraction one can always addmore zeros: 0.000. . ..” See Quotes 11, 12, 13, and 19.

Responses to several games. Some children consistently scored, either positively or nega-tively, in all the games, and others usually performed better in Game 1 than in the more advancedgames. The following examples represent these different patterns: B11 conceived the symmetrybetween positive and negative numbers, but failed on infinity in both cases. In Game 1: “I thinkwe’ll eventually reach the end when we run out of numbers, though it may take a long time,maybe even a year.” And in Game 2: “Just as we’ll eventually reach the end of big numbers, sowe’ll get to the end of negative numbers. It is the same, only one had put a minus in front of it.”B12 hesitated about whether or not the numbers have an end in Game 1, and responded to Game2 by “We can finish that game because it is decreasing more and more and it becomes smallerthe further away we get from zero.” In Game 3: “If we play a long time the numbers will endbecause they get smaller and smaller.” See also Quote 15.

For positive scores across games, see Quote 16 by a boy of 6 who intuitively understoodpotential infinity in the two directions. G7 in Game 1: “I want to be second, this way I’ll hearyour number and say a greater number. You can write now that I always want to be second.” InGame 2: “I want to be first and I’ll say the smallest number, zero; just a moment! Do we playunder zero? Then I want to be second.” She responded to the experimenter’s zero by minus 1.“You can write, as before, that I always want to go second.” And G10 in Game 1: “I want to playsecond, because I’ll know what you said and I’ll be able to plan which bigger number to say”; inGame 2: “One cannot finish this game either, because you simply take all the numbers, as before,and transfer them backwards behind zero”; in Game 3: “Here you take all the numbers, as before,but you make kind of fractions out of them.” See also Quotes 17, 18, and 19.

Unlike the children who had carried over the principle from Game 1 to the other games, otherssucceeded in Game 1 and less so in Games 2 and 3. B9 claimed in Game 1 that “there is noend to numbers; we can play it forever.” But in Game 2: “No, downstairs there is some kind ofan end. We can still go a lot, but there certainly is a smallest number.” See Quotes 20, 21, and22. Conversely, in only a few cases the child succeeded better in the later games. G14 scorednegatively on game in Game 1 and said in Games 2 and 3: “I want to be second because I learnedmy lesson in the first game that one has to go second in order to win.”

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Conflicting responses. Scores of “?” often reflect not only our doubts but also those of thechildren who were vacillating between contradictory ideas, possibly signifying a transition stage.G6: “There are a lot of numbers. . . . For me the biggest number is only 100, but I think othersmay have larger numbers.” And G8: “We can go on as long as we wish, we constantly increasethe numbers.” When asked if there is a largest number: “I don’t know—there should be.” SeeQuotes 23, 24, and 25 for internal contradictions and expressions of conflict.

Unanticipated responses. Some surprising articulations were instructive in disclosing a re-freshing variety of ideas. B9 chose to go second in Game 2 and responded to zero by “minuszero.” And B7 responded to zero by “half a zero. 1/2 zero is smaller than zero because it is not awhole zero but only part of it.” These children adopted sensible rules, but they had a problem withthe concept of zero. A “scientific” method of refuting alternative hypotheses was employed byB13 who chose in all three games to go second and then first: “I can always find a bigger/smallernumber [depending on the game] than yours. I wanted to show that whoever chooses to play firstwill always lose.” Several children, who clearly saw through the game situation, responded intheir singular style. After having Game 1 explained to her, G10 naively asked, “Do we say itinside our head?” For a more explicit criticism of the obvious task see Quote 26.

The Role of Names. The question of the existence of numbers has often been confoundedwith that of their names. Kasner and Newman (1949, p. 23) related that Kasner’s 9-year-oldnephew had suggested the name “googol” for a very big number, namely, 1 followed by 100zeros. He was certain that this number was not infinite and equally certain that it had to have aname. Likewise, Fynn’s (1974) Anna “knew full well that numbers have the capacity for goingon and on” but she soon ran out of words to express very large numbers, so she “inventedone, ‘squillion’ . . .. Anna was beginning to have a need for such a word” (p. 75). These twochildren appear to be on the verge of apprehending the endlessness of numbers; at the same time,they feel more secure when gigantic finite numbers are identified by names. Our participants’preoccupation with the issue of names was an eye-opening discovery, which admittedly had notbeen anticipated. It came out in many of their responses.

The absence of names failed some children in the competitive games, and the unfamiliarnames that we used intimidated them. G7 said in response to squillion, “I do not recognize alarger number, so perhaps this is the largest number.” G8 expressed her dilemma as follows: “Iknow more numbers but I don’t know how you say them.” B8: “I don’t know a number greaterthan a milliard, I don’t know how you name them. We don’t know the largest number. There is acertain number but its name is unknown, nobody knows.” See also Quotes 27, 28, and 29 (by agirl of 15).

Other children managed to recover from the initial surprise of hearing an unknown number.G7 responded to 2 billions by “What kind of a number is 2 billions?” And when asked if shewasn’t familiar with it, “OK, then I say 4 billions; it’s bigger because you said 2 so I say 4.”B8, who was taken aback upon hearing the experimenter’s googol, said, “I don’t recognize sucha number. Is this the largest number? Just a moment, I’ll try to say a larger number—googolplus 1.” See Quotes 5, and 23 (the latter demonstrates an attempt to cope with unknown fractionnames). That one does not need to know the name in order to recognize a number’s existence,was spelled out by B9 who responded to a squillion by “Two squillions. Not that I know what asquillion is, but it doesn’t matter.” See also Quotes 30, 31, 32, and 33.


Several rather mature explanations of one’s ability to always generate numbers, independentlyof knowing their names, were given by children of a wide range of ages. G9 maintained that “thenumbers cannot end. Many numbers have not yet been invented. We don’t know how to say them.Even a math professor doesn’t know what is the biggest number because he can invent more andgo on all the time.” G10: “You can never finish such a game because you can always add moreand more and invent more names as long as you wish.” B11: “Numbers have no end; there isonly an end to the names that I know.” And another B11: “It is always possible to invent morenumbers by increasing the number of digits.” See Quotes 7, 19, 34, and 35.

We learned from these recurrent references to the question of number names that one majorkey to recognizing the endlessness of numbers is the ability to separate numbers from their labels.Those who couldn’t do it were stymied in the process of increasing numbers when running out ofnames. On the other hand, those who could detach the two grasped the arbitrariness in assigningnames to numbers. This helped recognizing the existence of an unlimited succession of growingnumbers whether one can or cannot name them.

Comparison With Other Sources. The words of many children supported tracing theemergence of the notion of potential infinity to the prolonged early experience of counting andincreasing numbers by continuous addition as described by Evans (1983), Gelman (1980), andHartnett et al. (1987). G10: “One can go on counting and counting and never reach the end.” Seealso Quotes 17 and 20. These children have experienced the repeatability of adding one to eachnumber while counting, but not the endlessness of the process. The latter is eventually inferredby induction (Fischbein, 1987, pp. 51–52; Peter, 1957/1976, p. 51). Being able, first, to namenumbers and, second, to form a successor to any numeral, they extend the chain indefinitely. Asmaintained by Johnson-Laird (1983), “we can think of this procedure working iteratively andapplying to its own output” (p. 445). They reason by recurrence and extrapolate the familiaractivity indefinitely. A recent debate—that has some bearing on the issue of grasping potentialinfinity—concerns whether very young children learn the natural numbers by inductive inference(or bootstrap) based on applying the successor principle to small, known cardinal numbers(Margolis & Laurence, 2008; Rips, Asmuth, & Bloomfield, 2006, 2008). The details of thisimportant discussion are out of the scope of this work. It suffices, however, to note that theresponses of some of our participants—who were older than the ones around which the debaterevolves—articulated the idea that adding one to any given number leads to an unending sequence(e.g., Quotes 6, 19, and 21).

The longer one counts, the bigger the numbers one gets to know. On the whole, our data agreewith the suggestion that (lack of) familiarity with big numbers is related to (not) grasping theirendlessness. Yet, it is important to note that the overlap between mastering large numbers andapprehending their endlessness is only partial; there are exceptions. See Quotes 5, 30, 31, 32, and33 for the words of children who were not versed in large numbers and nevertheless understoodthe principle of the unending progression. Conversely, Quotes 2 and 29 belong to children whowere not intimidated by large numbers but were not sure of their endlessness. Familiarity withbig numbers is thus neither a necessary nor a sufficient condition for conceiving potential infinity(in general agreement with Hartnett et al., 1987).

In a similar vein, we found that familiarity with negative numbers or fractions is not a necessarycondition for functioning in Games 2 or 3, respectively. This tallies with Smith et al.’s (2005)

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report in studying children’s understanding of infinite divisibility of numbers. They found thatmost of the children who had first denied the existence of numbers between 0 and 1 changedtheir mind when asked about 1/2; and some of those who were acquainted with only a fewfraction names still maintained that one can keep on dividing numbers, as in Quotes 10, 16, and23.

Our quantitative results in Table 1 (Game 1 vs. 3) are compatible with Langford’s (1974) reportthat a majority of children, starting from age 9, regarded the operation of addition as amenable toindefinite iteration; and that in the case of division, this understanding was put off to later ages.Comparison of our results of Game 3 with those of Tirosh and Stavy (1996, Table 3), shows thattheir 17–18-year-olds achieved, in their numerical tasks, a higher rate (82%) of infinite responsesthan did our eldest children in the age group 12–15 (around 70%). The levels of understandingthe infinite divisibility of numbers of Smith et al.’s (2005) children aged around 10–12 (61% intheir Table 1) and our children of about the same ages were not far removed from each other.In general, our participants performed somewhat better. This can reasonably be accounted forby the competitive-game situation that was more motivating than a thought experiment, and bythe priming effect of the first two games. Similar verbal responses were obtained in comparing1/n for different ns. Many of Smith et al.’s students articulated the general rule, “The smallerthe denominator the larger the fraction” (p. 119), as did our respondents in Quotes 14 and 33.Likewise, some comparable justifications of never getting to zero were obtained in both cases:“There always has to be something left when you divide it” (Smith et al., p. 117) and Quotes 11,13, and 19. Hence, these two sets of results reinforce each other.

Tirosh and Stavy (1996), who had studied children’s understanding of infinite divisibilityof numbers as well as that of material or geometric objects, reported that the numerical tasksyielded higher percentages of correct responses than the physical or spatial ones. They found,like Langford (1974), that, in the material case, concrete connotations limited participants’ abilityto extrapolate. Moreno and Waldegg (1991) found that the geometric context seems to preventaccess to conceiving actual infinity. Though we focused merely on numbers, our respondentsoften turned, of their own accord, to geometric images. In most cases, the spatial analoguesinterfered with their ability to grasp the unending decrease of positive fractions. G9: “If we’ll goon we’ll finish the game because there is little space between 0 and 1.” And G10: “We’ll reachzero finally because each time there is a smaller distance and in the end no gap will remain.”Relating to points on a line segment had a similar adverse effect. G10: “One can finish counting allthe numbers between 0 and 1 because an interval is many points combined, and if each point is anumber, we could count all the points.” See also Quotes 15 and 21. Quotes 17 and 18 demonstratehow children overcame the spatial stumbling block.

Infinite Process and Infinite Set. The games in this experiment had been designed to probechildren’s cognitive status of potential infinity, but the words of quite a few children related toactual infinity, thereby indicating that the distinction between the two is rather tenuous. See, forexample, Quotes 9, 24, and 35. Indeed, Monaghan (2001) noted that the process-object duality isnot clearly differentiated in children’s minds. Likewise, Moreno and Waldegg (1991) and Waldegg(2005) reported that students identify the two assertions that each number has a successor andthat there are infinitely many numbers. Hence it seems that conceiving the endlessness of theprocess automatically turns into regarding the set as infinite.


Falk et al. (1986) noted that some children understood the conservation of the numerosityof the set of whole numbers when they objected to the possibility of increasing it by addingsome numbers. A unique property of an infinite set is its unchanged cardinality in the faceof several transformations (Falk, 1994; Moreno & Waldegg, 1991). In particular, infinite sets(whether countable or not) can be “thickened” or “thinned” by adding or subtracting a finite andsometimes an infinite number of elements without in any way affecting their cardinality (Kasner& Newman, 1949, p. 55). G8 maintained that the sequence of increasing numbers can go onwithout end because there are infinitely many numbers, and when asked about adding anothernumber, she said: “Infinity is not a number, this will not change. It is the same.” See also Quotes 16and 36.

The verbalizations concerning actual infinity were incidental to the study of potential infinity.In the next experiments, we look directly at participants’ notions of actual infinity.

EXPERIMENT 2: ACTUAL INFINITY—COMPARISON OF SETS

. . . and their camels were without number as the sand by the seaside for multitude.—Judges, 7: 12

It is not clear whether the biblical author meant that one just loses count or that the sand atthe seaside is infinite. In conversation, people often sway between the very big and the infinite(Barrow, 2005, chap. 4). Dictionary definitions of colloquial coinages like jillion, squillion, andzillion speak about a very large but indefinite number and are vague about the question offiniteness, thus obfuscating the picture. Maor (1987, p. 16) cautioned that large numbers, big asthey are, have nothing to do with infinity. Even the sand grains in the universe—a symbol ofplenty—can be assigned a finite number that can be exceeded, as has long ago been shown byArchimedes in The Sand Reckoner.

In Experiment 2, we did not mention the question of finiteness. The children had to comparesizes of sets. The crucial comparison from our point of view was between what the child considersa very big (finite) set and the set of all natural numbers (the “smallest” infinite set).

Procedure

We compiled a list of nine sets of familiar objects whose sizes we intended to compare with eachother. Two sets at a time were paired off for comparison. The question was always “what are theremore of, X or Y?” where X and Y were replaced in each trial by the elements of the respectivetwo sets. In a given comparison, we randomly named the greater set either first or second. The setchosen as bigger was paired with other sets, but not in a transparent order. The “winning” finiteset—assuming transitivity—was finally compared with that of all numbers.

The following is an example of a sequence of binary comparisons involving six of the sets, inwhich the set considered bigger by the child is italicized, and the last step involves the set of allnumbers: hairs on your head versus fingers on two hands; fingers on two hands versus days ina month; grains of sand on earth versus hairs on your head; hairs on your head versus days ina month; leaves on a tree versus hairs on your head; all numbers versus grains of sand on earth.

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TABLE 2Results of Comparing Numbers with Grains of Sand by Age Group, in Experiment 2

Age Group Number of Children Percent of Correct Responses

6–7 15 33.38–9 15 80.010–11 29 65.512–15 21 85.7Total 80 67.5

All the children’s choices and explanations were recorded. We took specific care to obtain a clearexplanation of the last choice. The experiment lasted 15–20 minutes.


Quantitative Summary. The finite set considered biggest by all the children was that ofgrains of sand. In general, children compared the sizes of finite sets as we did. The focus of ourinterest was the comparison between numbers and grains of sand. To score positively, one had tomaintain that there are more numbers than grains of sand and to display understanding that theformer set is infinite whereas the latter is finite. Table 2 presents the percents of children whoscored positively according to age group.

The percents in Table 2 and those of positive scores on infinity in Game 1 (Table 1) broadlysuggest that conceiving actual and potential infinity go hand in hand. In both cases, up fromaround age 8, a majority of the children coped well with questions of both kinds. The numbersdo not tell us whether these were two independent understandings or in effect the same one.Children’s verbal comments shed some light on this issue.

Verbal Explanations. The children’s justifications of their choices usually enabled unequiv-ocal scoring. G6 scored negatively: “There are more grains of sand than anything else, becausethey are so little. Numbers have a beginning, but you cannot start counting sand.” B9 scorednegatively despite a seemingly correct start: “There is no end to numbers, but there are moregrains of sand because the grains are so small that there are more of them on earth.” See Quote37 for a failure of an older girl. These are examples of unforeseen inferences: “The elements aretiny—ergo the set is large.”

Children generally felt confident when answering correctly. G9: “There are more num-bers than grains of sand, because numbers you can invent and invent and go on inventingand nobody could stop you.” B11: “Numbers are more because I can count all the grainsof sand with numbers and there always will remain more numbers.” See Quotes 38 and 39.


Additional Insights. Although children had not been asked to name numbers in this exper-iment, some spontaneously reverted to the subject of number names while comparing the twosets. G11: “There are numbers until infinity but no names anymore. Since all grains of sand canbe given names, there are in fact more numbers.” B15 philosophized about this issue: “Thereare more grains of sand than names of numbers that I recognize. I can count grains of sand onlyby means of numbers that I know. I do not mean to say that there are more grains of sand butjust that when I count the grains I use my knowledge about numbers” (see again Quote 38).These responses indicate that the issue of naming, which was found to preoccupy some childrenin thinking on potential infinity (Quotes 19, 28, and 34), affects children’s thinking about thecardinality of the set of numbers as well. The subject also appeared (unsolicited) in Experiment3 (Quote 43).

Just as some children in Experiment 1 spoke in terms of actual infinity (e.g., Quote 35), sothere were in this experiment children who responded in terms of potential infinity, namely, anever-ending process. B11: “There is an end to grains, but the numbers have no end because youcan count milliard and 1, 2, 3, . . . so up to infinity.” See also Quote 39. Such expressions suggestthat apprehending the two types of infinity evolve together.

EXPERIMENT 3: ACTUAL INFINITY—LOCATION OF SETS

Above everything, we must realize that “very big” and “infinite” are entirely different. . . . There isno point where the very big starts to merge into the infinite. You may write a number as big as youplease; it will be no nearer the infinite than the number 1 or the number 7.

—Kasner and Newman (1949, p. 34)

The figure of speech “almost infinite”—liberally used in daily discourse to describe a hugeamount—constitutes a contradiction in terms. In a way, it is worse than using “infinite” in thesame sense because it seems to suggest that you only have to add one or two items to reach infinity,whereas there is absolutely no predecessor to the infinite. One cannot “approach infinity.” Anenormous finite set is nowhere near “the infinite.”

The conclusion from Experiment 2 was that from about age 8 on, most children believe thatthere are more numbers than grains of sand. We do not know, however, how many more numbersthan grains they think there are: Do they realize that the gap between the two sets is infinite, ordo they think that there are almost as many grains of sand as numbers? In the procedure devisedfor this inquiry, participants were not asked about the numerical size of that gap; their task was toevaluate the distance between the sizes of these two sets in comparison with the distance betweentwo extremely discrepant finite sets.

Method

Participants had to represent the relations among gaps between different set sizes on a long linearstrip. The straight line served as a means for designating one’s estimates and not as an object forstudying infinity of the spatial-geometric dimension or of the measuring property of numbers.Because this assignment is rather demanding (one has to fathom the unfathomable gap), we ran

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this experiment also with a group of adults, for whom it was assumed that the tasks of the previoustwo experiments would have been rather trivial. Obviously, the gap between, say, grains of sandand numbers is infinite, hence it is also infinite in relation to gaps between finite sets—discrepantas they may be—and it cannot be shown on a line.

In several studies, relevant to the present task, researchers asked participants to estimate givenArabic numerals and numerosities of given sets by representing their magnitudes as points ona number line (e.g., Laski & Siegler, 2007; Siegler & Booth, 2004). All these studies founda tendency of young children to generate estimates that (roughly) fit a logarithmic function.This means that differences in smaller magnitudes are exaggerated relative to larger magnitudes.The larger the magnitudes, the more compressed they are represented. The same was found fortasks of comparing numbers. With age and education, the logarithmic scale gradually tended tobe replaced by a linear scale, so that larger magnitudes were differentiated more clearly. In thepresent experiment, in which an infinite cardinality is involved, even using a logarithmic mappingfunction would not justify representation on a finite point.

Instruments. We wrote the names of the nine sets used for binary comparisons in Experiment2 on the wide handles of nine forked plastic clips. The sets were: fingers on one hand; fingers ontwo hands; fingers on hands and feet; days in a month; agorot in a shekel (Israeli currency: thereare 100); leaves on a tree; hairs on your head; grains of sand on earth; and all numbers. The clipswere to be fastened on a linear metal strip of a 5-meters-long tape measure that was marked withunits of length on only one side. The tape was kept coiled in its case, and could be released andpulled out, or switched back in, by pressing a button.

Procedure. Two training stages, intended to teach the task, preceded the final test trial.Firstly, the participant was asked to order the three finger-clips according to set size. Then

the experimenter drew a strip of tape out of the case, and the participant, who faced the blankside, had to attach the clips to the tape with spaces whose relations would match those of thedifferences between the set sizes (see Figure 1). Most children had no difficulty with this stage.Despite seeing no units of measurement on their side of the tape, they managed to make thedistance between fingers on hands and feet and fingers on two hands about twice as big as thatbetween fingers on two hands and on one hand. A few of the youngest children had to be helpedin adjusting the distances until they understood the idea of representing the relations between thedifferences.

Secondly, some other clips—bearing names of bigger finite sets, and presented indisorder—had similarly to be fastened on the tape in a way that would pro rata represent the gapsbetween the sets’ sizes. Because we all don’t know the number of hairs on one’s head, nor forthat matter the number of leaves on a tree, we helped the participants to estimate roughly therelations among the gaps between sizes and to proportionate the generated distances accordingly.For example, when the leaves clip was placed rather close to fingers on hands and feet, we askedthe participants how the distance between these two sets compares with the distance betweenfingers on hands and feet and fingers on two hands. They always knew that the former distanceshould be greater and they moved the leaves clip much further. The purpose was for them to


FIGURE 1 Training for Location of Sets, in Experiment 3.

understand the principle of proportional representation of gaps, so that we could see what theywould decide when it came to locating the set of all numbers.

In the test trial—the target of the inquiry—only a very small set, a very large set, and an infiniteset had to be placed. All the clips were removed from the tape and only fingers on one hand,grains of sand on earth, and all numbers were presented and had to be located. We tried to makeup for the inherent weakness of this task by appropriate instructions. The main drawback wasthat, in order to respond correctly, one had to refuse to locate the numbers clip. A refusal might goagainst the grain for participants who tend to cooperate and comply with the instructions. Hence,it could incur an upper-biased rate of failures. To counteract this liability, we let participants(starting from the second training stage) pull out as long a strip of tape as they wished. Thenwe asked, each time before locating a big set (leaves, hairs, sand, numbers), whether this can orcannot be done, whether the length of the tape suffices, or does one need an extension (and howlong). The purpose was to legitimize requests for extensions or claims that no length in the worldwould do.

All verbal responses were recorded. We particularly asked for explanation of the responseto the test trial and the reasons for the decision concerning the set of numbers. The experimentlasted around 50 minutes.

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Scoring. The experimenter, whose side of the tape was numbered, recorded three measure-ments (in cm) for each participant. These were the readings of the mid base of the three clips:fingers, m(f); grains, m(g); and numbers, m(n). When extensions were required, the extra lengthwas included in the measure, and whenever a participant maintained that no length whatsoeverwould suffice, the registered measure was ∞.

A participant’s performance score, was the critical ratio (CR), defined as:

CR = m(n) − m(g)

m(g) − m(f ). (1)

CR measures how big one thinks is the gap between all numbers and grains of sand in relationto the gap between grains of sand and fingers. Note that CR could be negative if one believesthat there are more grains than numbers, or zero if the set sizes are considered equal. A positiveCR means that there are more numbers than grains but not infinitely more. The correct responseresults in CR = ∞. Scoring CR > 0, whether finite or infinite, is equivalent to claiming correctlythat there are more numbers than grains of sand, in Experiment 2.

In addition, participants were scored (positively or negatively) on verbal understanding. Thiswas mostly based on the explanation of the placement of the numbers clip. Verbal expression ofthe impossibility of representing the immeasurable gap by any finite distance was required forscoring positively. This judgment was made by the experimenter (as far as possible) independentlyof the participant’s CR, which was concealed from the second judge who based the scoring onlyon the written explanation.

Quantitative Summary. Altogether, out of 428 participants, 103 produced CR ≤ 0, 165produced CR > 0, and 160 produced CR = ∞. The distributions (in percents) of these three maincategories of CRs in each of the five age groups are presented in Figure 2. Whereas Experiments 1and 2 indicated that from about age 8 on most children conceive both potential and actual infinity,Figure 2 shows that understanding of the infinite gulf between the finite and the infinite develops

FIGURE 2 Percentages of three CR values in each age group, in Experiment 3 (N = 428).


later, if at all. Infinite CRs exceeded 50% of the cases only starting from adolescence (ages 12–15),and even adults’ performance was far from perfect. The percents of positive verbal-understandingscores as a function of age are presented later in comparison with the other experiments.

The overall modal CR was finite positive. It is meaningful to distinguish within this categorybetween 0 < CR ≤ 1 and CR > 1 (see Equation 1). The former inequality means that there aremore numbers than grains, but not very many more, certainly no more than the number of grainsminus 5; whereas the latter means that the difference is greater than between grains and fingers,but finite. The participants divided about equally between these two kinds with no increase withage of preference for the second. All in all, the immensity of the leap to infinity appears to bebeyond the grasp of many people of all ages.

Verbal Explanations. We focused mainly on participants’ words concerning the very possi-bility of locating the numbers set. On the whole, the explanations matched the location decisions.Examples of a few exceptions are included in the subsection Conflicting cases.

Nonpositive CR. Younger children who would have scored negatively in Experiment 2produced many nonpositive CRs. G6 placed grains further than numbers (CR = –.42) andexplained, “There are more grains of sand—they are teeny-weeny” (cf. Quote 37). B8 (CR =0) first placed numbers last on the tape, adjacent to grains, but then: “Oh, I was mistaken, thereare as many. May I put both on the same place?” G8 located grains at the end of the tape, butimmediately decided that this wasn’t far enough and asked permission to place the clip at theend of the corridor. Then she took the numbers clip and walked along the imaginary straight lineextending from the tape; mumbling to herself “Don’t exaggerate!” she placed numbers betweenfingers and grains (CR = −.22). And B9 asked for an extension of 300 m for grains and only 100m for numbers (fingers were located at 5 cm).

Contrary to our expectations, comparison of sets was not a cinch even for 20% of the adultswhose CR was negative or zero. M23: “Grains of sand on earth are googol times numerous thannumbers and one can locate them according to a ratio of googol.” Another M23 placed grainsand numbers together at the tape’s end, and explained, “Grains and numbers are the greatest ofall the sets about which you have asked, and they are big to the same extent.”

O < CR ≤ 1. These participants would have scored positively on comparison of sets (Exper-iment 2). The task of location, however, revealed the extent of their deficient grasp of the endlessdistance to the set of numbers. Their words are more telling than any statistical test.

B10: “All the numbers are almost the same as grains of sand” (CR = .063). F22: “Had I gota tape of a km or more, I would have placed grains of sand in a distance of km, and a bit afterthat, say 10 cm, I would have placed all numbers” (CR = .0001). M26: “All numbers comein fact immediately after grains of sand—it is roughly like grains plus one” (CR = .048). Onthe other hand, he betrayed his conflict in adding that “it is best to lay numbers aside and saythat they are out of the game.” See also Quotes 40 and 41. These participants’ words in factdescribe an attempt to adjust upward their estimate of the size of the set of grains because thereare more numbers; but the extent of their adjustment is greatly deficient. This accords with theanalyses of many psychologists (e.g., Epley & Gilovich, 2002; Tversky & Kahneman, 1974) ofthe process by which people make estimates: They start with an initial anchor (either provided bythe experimenter or spontaneously self-generated) and then adjust their response in a direction

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that seems appropriate. The adjustment, however, is typically insufficient, rendering the finaloutcome too close to the anchor. In the present case, the adjustment from the anchor of the set ofgrains was infinitely insufficient.

CR > 1 and finite. Although these CRs indicate understanding that the gap between the set ofnumbers and a huge finite set is greater than between extremely disparate finite sets, participants’grasp of the interminable leap to the infinite is still flawed. The tremendous distance between thesizes of the two farthermost finite sets is nil relative to the immeasurable distance to an infiniteset. Many of these participants did not display any conflict in locating numbers on a finite point,and often they didn’t even use up all the available length of the tape.

B8, who had not placed the numbers clip at the end of the tape, explained, “There is an infinitenumber, so I put them almost at the end” (CR = 1.29). M17 generated CR = 1.15 and explained,“The difference between the amount of grains of sand and fingers is less than between numbersand sand.”

Infinite CR. The impossibility of locating the numbers set anywhere on that line was justifiedin diverse styles, mostly rather cogently. B8: “All numbers? There is no end to it. We could make10 rounds around earth and even more and the numbers wouldn’t end.” B11: “For grains of sandI need around 100,000 additional strips, for all the numbers one can put very many strips andit wouldn’t suffice. No matter how much additional length you’ll give me, I wouldn’t be able tolocate that set” (see also Quote 42). As in the previous experiments (e.g., Quotes 19 and 38),several children referred to the subject of number names even in this task; see Quote 43.

A few participants gave vent to their frustration of being unable to locate numbers by sharplycriticizing the impossible task. G11 angrily pulled out the tape measure to the end and toreit: “Numbers are the largest set, it extends to infinity. Only a fool will try to count them. It isimpossible to locate.” And B14 grimaced at having to place all numbers and threw the clip away:“There is no place for that clip. Even if you take a tape that will circle earth 20 times there wouldbe no place on it.”

Many adolescents and adults expressed the impossibility of locating numbers quite com-pellingly. We couldn’t have said it better. B13: “If I put fingers on a distance of 1mm, thengrains of sand could not be located on that specific meter but only on a meter with some billionsof centimeters. All numbers cannot be located; physically there does not exist anything that isinfinite, only in theory.” M24: “One cannot locate the numbers on this axis. Relative to infinity,the amount of sand is zero.” M27: “OK [grinning], all the numbers cannot be located on themeter. It is not a question of scale, and it doesn’t matter what is the length of the tape. There isno such thing as the number of all numbers. Just as infinity is not a number, it is neither a place.There doesn’t exist a point that can represent infinity.” M30: “Obviously all numbers have noplace here. One cannot grasp it in one’s mind and surely not in the senses.”

Conflicting cases. We had doubts concerning scoring when participants claimed that bothgrains and numbers cannot be located because they are both infinite (a CR of ∞/∞ is undefined).In these cases, the scoring depended on the exact verbalism. F21 said “impossible” both aboutlocating grains and numbers, but when pressed to explain: “I would have put grains of sand on themoon. The sand is enormous but, after all, finite, but numbers—as I have learned—are infinite.”


Hence, her CR was considered ∞, and she scored positively on verbal understanding. In contrast,another F21, who had refused to locate the two sets, said, “What is important is that they shouldbe close to each other, because grains of sand are immense and numbers are infinite” (CR > 0;negative verbal-understanding score).

In a few cases a participant produced a finite CR but nevertheless scored positively on verbalunderstanding. G8 (CR = 2.72) said about grains “in Hondolulu” and she explained, “it’s very faraway, that’s what we children use to say, in Hondolulu.” About numbers she said, “in the middleof nowhere; there are infinitely many numbers, so there is also an infinite distance to numbers.”When asked why she placed the clips the way she did, she (impatiently) answered, “Because Icannot show an infinite distance on this meter. Even the end of the longest meter in the worldwouldn’t do. You cannot show the end of numbers even in imagination.” Apparently, the inherentflaw of the impossible location task might be blamed for the inconsistency between the finite CRand the positive verbal score. The same sentiment was shared by B12 (CR = 1.09): “I used ascale but it was not enough because you cannot get something that does not end into a scale.”When asked, “Why, then, did you place it?” he answered, “Because you asked, but it is againstmy understanding.”

The aforementioned examples suggest that some of the reported finite CRs (in Figure 2) shouldperhaps be discounted from the cases of failures to conceive the infinite leap to the set of numbers.However, because the instructions had unequivocally permitted refusal to locate a given set, wefeel that there may be only very few such cases that we do not know of. The interview that hadprobed into one’s reasons for the location most probably identified cases of a grasp of the infinitegap despite producing a finite CR. Consequently, we regard the verbal-understanding scores assomewhat more valid than the CR indices, and we use the verbal scores in comparing the resultsof the three experiments.

INTEGRATING THE EXPERIMENTS

Judging by the results of Experiments 1 and 2, one could have concluded that there is nodevelopmental difference in the emergence of the concepts of potential and actual infinity. Amajority of the children, starting at 8–9 years of age, scored positively both on conceiving theendlessness of increasing numbers and on comparing sets. Experiment 3, however, changes thepicture by showing that a pivotal feature of an actually infinite set—its immensurable distancefrom big finite sets—is not grasped by most children until much later. Many responses wouldhave been counted as understanding actual infinity by the criterion of Experiment 2 but fail thatof Experiment 3 (e.g., Quote 41).

Recall that a correct comparison of the sizes of the sets of grains and numbers in Experiment2 is equivalent to producing a positive CR (either finite or infinite) in Experiment 3. Indeed, inall age groups the rates of positive scores in Experiment 2 and of positive CRs in Experiment 3are neither far removed from each other nor systematically different (cf. Table 2 and Figure 2).In this respect, Experiment 3 replicates and extends the findings of Experiment 2. However, thepercents of infinite CRs in Experiment 3 are consistently lower than those of correct comparisonsin Experiment 2 in all age groups.

Table 3 compares the rates of positive verbal scores in the three experiments, according toage group. The table shows similar rates of comprehending the endlessness of the process of

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TABLE 3Percents of Correct Verbal Responses by Age Group and Type of Infinity, in Experiments 1–3

Type of Infinity

Potential Actual

Age Group Processa Setb Gapc

6–7 43 33 118–9 73 80 1910–11 77 66 3912–15 83 86 60Total 69 68 35

aBased on the infinity scores in Game 1 (Experiment 1, N = 345).bBased on positive scores in Comparison of Sets (Experiment 2, N = 80).cBased on verbal-understanding scores in Location of Sets (Experiment 3, N = 428). In the adult age group the percentwas 52.

increasing numbers and the infinity of the set of numbers. There is no systematic advantage toeither understanding. However, in all age groups, conceiving the infinite leap from a finite toan infinite set clearly lags behind nominally acknowledging the potential and actual infinity ofnumbers.

Figure 3 compares positive verbal-understanding rates based on infinity scores in Game 1 ofExperiment 1 with rates of positive verbal-understanding scores in Experiment 3, for each singleyear of age. The advantage of understanding potential infinity over understanding the infinite gapof actual infinity is conspicuous. In all 10 ages, the difference between the percents is in favor ofunderstanding the endlessness of the process. The probability that, for independent respondents,a difference in the predicted direction will recur 10 times by chance is 1/1024. The differenceis significant in every sense of the word. The same pattern of differences is replicated, for each

FIGURE 3 Percents of verbal understanding as a function of each year of age and type of infinity. “Process” is basedon infinity scores in Game 1 (Experiment 1, N = 345), and “Gap” is based on verbal-understanding scores in Locationof Sets (Experiment 3, N = 428).


year of age, in comparing the results of playing Game 1 in Experiment 1 (game scores) with theresults of the performance variable CR of Experiment 3 (Equation 1). This recurrence of resultsin each of 10 independent comparisons between two independent tasks reinforces and validatesthe conclusion that conceiving potential infinity does not mean grasping the infinite gap betweena finite and an infinite set.

We learn from the three experiments, in combination, that children of 8–9 years and olderbelieve that numbers go on indefinitely and that there are infinitely many numbers. Their under-standing is, however, wanting because they do not internalize the endless distance to the amountof numbers. The latter understanding develops later and is often imperfect even in adulthood.

GENERAL DISCUSSION

A Never-Ending Process

Using diagnostic tasks, which circumvented the need to ask direct questions about infinitude (but,as noted, had their own shortcomings), we gained information concerning participants’ views ofpotential and actual infinity. In particular, three competitive games and their accompanying inter-views provided a rich resource for learning about the emergence of children’s ideas concerningthe endless growth (decline) of numbers.

The principle that for any number, large as it may be, there is a larger number is the essenceof the potential-infinity construct. A majority of the children, from age 6 on, proved to possessthis principle to start with and apply it for finding a winning response (the same was true for thedownward direction among a somewhat smaller but still considerable portion of the children).Many observations in the course of playing the games verified the account of previous studies(e.g., Evans, 1983; Gelman, 1980) that children’s repeated experience of forming successors whilecounting eventually leads, by induction, to conceiving the unending succession. Interestingly, afailure may also have its benefits: Children’s words often indicated that not only success incontinuously increasing numbers, but also the failure to produce a largest number, and thus reachan end, contributes to the idea that the progression is interminable. In a similar vein, Falk andKonold (1997) showed that people’s failure to find a pattern in a sequence of stimuli brings aboutthe notion that the sequence is random.

Weak and Strong Concepts of Infinity

It is of interest to compare the various steps in the individual’s understanding of infinity with thehistorical progression of infinity cognitions. Mathematicians were generally suspicious of infinitysince the days of Zeno’s paradoxes in the fifth century B.C. One century later, Aristotle tried tosolve the conflict between the disturbing paradoxes and the realization that time and numbersseem to go on endlessly by introducing the distinction between potential and actual infinity. Herecognized the former as a process and rejected the latter, arguing that there could not be anyobject that is infinitely large (Moore, 1995). This approach held as orthodoxy for many yearsthrough the Middle Ages and it was defended by Gauss in 1831 (Maor, 1987, p. 54). Accordingto Moore, despite the commonly accepted mathematical view of actual infinity, developed by

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Cantor in the late nineteenth century, infinity is still a slippery concept. Although the presentresearch did not deal with the concept on that level, but addressed participants’ commonsensicalintuitions, one could perhaps expect on the basis of the historical background, that, psycho-logically, potential infinity would be easier to intuit and would precede the intuition of actualinfinity.

This expectation was not borne out by the first two experiments. The competitive games andthe comparison of sets showed that the ideas of potential and actual infinity emerge in parallelquite early at about the same age. Children spoke interchangeably in either language, irrespectiveof the subject of the inquiry. However, the third experiment showed that, although many youngchildren succeed in passing the tests of Experiments 1 and 2, until adolescence most of them failthe test of appreciating the infinite distance between a big finite set and an actually infinite set. So,in this respect, the results match the historical development. Conceiving the unending distanceto the cardinality of an infinite set is indeed harder and it develops later than that of potentialinfinity. Despite the old (discredited) zoological adage that ontogeny recapitulates the phylogeny,Beyth-Marom, Fidler, and Cumming (2008) reported that research on the perception of differentconcepts in mathematics, physics, and biology has repeatedly shown that people’s initial, intuitivebeliefs are similar to earlier, now discredited, scientific theories (see also Harper, 1987; Moreno& Waldegg, 1991). Although this is not generally accepted, and without endorsing any positionon the issue, the partial parallelism that was found between the succession of participants’ notionsand that of historical ideas is worth noting.

The Greeks’ “horror infinity” has been explained by the absence of an algebraic language; andthe extended taboo on actual infinity was attributed to the disturbing paradoxes, which defy ourintuition (Maor, 1987, p. 3 and p. 58). The story goes that even Cantor exclaimed, “I see it but Idon’t believe it!” after having proved the one-to-one correspondence between the rational and thenatural numbers. Our results suggest that the misgivings concerning actual infinity over the yearscould also have been related to the difficulty to cope with the unbridgeable gap to an infinite set.

Three levels of conceiving the extent of numbers were identified. (1) finite: The process ofincreasing (decreasing) numbers is limited; there has to be a largest (smallest) number, and the setof numbers is bounded. (2) weakly infinite: Numbers are countless; there are more of them thanmembers of a big finite set, but not too many more, their excess is measurable. (3) strongly infinite:Not only are the numbers innumerable, but the cardinality of the set of numbers immeasurablyexceeds that of any finite set. The second and the third levels can, respectively, be thought ofas the weak and the strong concepts of actual infinity. Evidently, the strong concept outlaws theexpression “almost infinite.”2

The three levels are roughly age-correlated. The lowest is typical of young children, whosecounting skills are suboptimal, and the highest abounds mainly among adolescents and adults.Yet, by no means can these be translated into developmental stages that match exact ages. Eachlevel was identified across a wide span of ages, and many participants of the same age functioned

2In his article on “The Relativity of Wrong” Asimov (1989) gives an example suggesting that in some sense consideringan immense finite cardinal number almost infinite is not utterly (psychologically) wrong: “Newton’s theories of motionand gravitation were very close to right, and they would have been absolutely right if only the speed of light were infinite.However, the speed of light is finite” (p. 42). He further explains that although the difference between infinite and finiteis itself infinite, at the speed at which light actually travels, it takes it 0.0000000033 seconds to traverse a meter, and itwould take 0 seconds if light traveled at infinite speed. Because the inverse of the huge finite speed is almost zero, thatspeed might reasonably seem at first to be almost infinite.


on different levels. Nor could many individuals be assigned a definite level of understanding.As with other concepts, and even more so, children and adults sway between conflicting ideasand they experience dissonances and internal contradictions (Acredolo & O’Connor, 1991; Falk& Wilkening, 1998). Mastering the strong concept of actual infinity does not come in one fellswoop (Siegler, 1986, pp. 9–11). The gray area of uncertainty and simultaneous adherence toincompatible beliefs usually drags. Development is “better depicted as a series of overlappingwaves than as a stair-step progression” (Siegler, 1995, p. 426). On this noisy background, thefinite, the weakly infinite, and the strongly infinite are the main landmarks on the road to the ideaof infinity.

Numbers and Number Words

Gelman and Gallistel (1978, chap. 7) outlined the evolvement of the number concept as startingwith establishing one-to-one correspondence between objects of a set and the tags assigned tothem while counting, and then realizing the special significance of the final tag in the seriesas representing the numerosity of the set. Hence, this last name is a major constituent in theprocess of quantifying the set size. A short, but comprehensive survey of the diverse positionsconcerning the relationships between number words and number concept is included in Baroody,Li, and Lai’s (2008) article. They reviewed the vast literature concerning the role of numbernames in the acquisition of the first number concepts by toddlers. Their conclusions supportedthe view that language, in the form of the first few number words, acts as a catalyst for grad-ually constructing the first cardinal concepts. In a study of numbers without words, Gordon(2004) reported that members of an Amazonian Piraha tribe, whose language had no words fornumbers greater than two, could not perceive exact numerosities, and tasks requiring cognitivemanipulations of numbers were beyond them. Likewise, Pica, Lemer, Izard, and Dehaene (2004)studied numerical cognition in an Amazonian indigene group whose language has a very smalllexicon of number words and found that they failed in exact arithmetic with numbers largerthan four or five. Without getting into the debate about whether language determines the na-ture and content of thought, plenty of evidence in the literature points to a strong link betweenthe concept of a number and its label. However, none of these studies has tackled the problemof people’s ability to disentangle the tie between numbers and their respective numerals. Nei-ther was the connection with acknowledging the numbers’ endlessness discussed as was donehere.

Conceivably, because of the important role of the label in forming the number concept (agreedamong many scholars), after having made the connection, undoing the tie between a numberand its name is not easily accomplished (see Falk, 1994, on nominal realism in connection withnumbers and their names). In our three experiments, participants referred time and again tothis connection, both when obstructed in perceiving the infinity of numbers because of lack ofnames and when succeeding in the infinity task by disengaging numbers from their names andrealizing that labels can be changed or invented. In an informal conversation, after playing thegames of Experiment 1, a boy of 11 said, “That there are no more words does not mean thatthere is an end to numbers; in fact, words may also be endless if we invent longer and longernames.”

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The ability to acknowledge the existence of unnamed numbers proved an important key toconceiving their infinitude. We encountered many cases in which the endlessness of numbershad been inferred despite limited familiarity with number words. It appears that having a conceptof numbers as existing independently of number words is an important step in conceiving theirinfinitude. Apt educational efforts could possibly consolidate the conception of numbers whilerelaxing their dependence on specific verbal labels.

Didactic Implications

The lesson for teaching is twofold: Initially, the labels are essential for acquiring the concept ofa (finite) cardinal number because the last one in counting equals the number of items in the set.Hence names of numbers have to be properly learned. As advocated by Baroody et al. (2008),“It may help young preschoolers construct a robust and general cardinal concept of the intuitivenumbers by labeling a wide variety of examples . . . with a number word” (p. 266). At the sametime, students would better not fixate on these words, and teachers should not din the names toomuch into their pupils. The focus should rather be on learning the notations in the decimal or anyother-based system. This will facilitate increasing (decreasing) numbers indefinitely. Knowingnumbers’ names becomes less important the higher they get.3

When students stumble in attaching the right words to (big) numbers, the teacher canexplain—while correcting their mistakes—the extent of arbitrariness in the naming of num-bers. A number can be named differently in different languages, or even in the same language,whereas its mathematical definition is unique. Every number can be increased (decreased) in-definitely via, for instance, multiplication by positive (negative) powers of 10, without namingit anew. All this can be understood by rather young children, as exemplified in many of thechildren’s response (e.g., Quotes 7 and 34).

The tasks and interviews of the research had been designed to unveil participants’ existentconception of numerical infinity, not to teach them. Yet, unavoidably, some learning took placebecause the experimental situations triggered expressions of latent knowledge. In particular, therepeated competitive situations in the games of Experiment 1 encouraged the emergence ofinsights. Such games can be used as stimulating, entertaining tools in the classroom, in particularwith young children who would learn while playing that you can always increase (decrease) anygiven number. Comparison and location of sets can equally well be exercised and discussed withchildren of a wide range of ages.

All in all, educational programs should take into account the findings that many children areable to comprehend certain aspects of infinity. Methods for materializing this potential should bedevised. Beyth-Marom et al. (2008) advocated integration of theory, research, and application inthe area of statistical cognition. Analogously, our descriptive results concerning conceptions andmisconceptions of infinity should be combined with the normative mathematical theory to formprescriptive guidance for education.

3For years I have been embarrassed about confusing between trillions, billions, and their meaning in different countries.Now I realize that, though I should perhaps not be proud of it, being able to deal with symbolic representations of numbersis more important.


IN CONCLUSION

By way of summary, several innovative methodological contributions, empirical findings, andnewly gained insights are:

• Previous studies had investigated children’s thoughts either about the limits of one’s abilityto go on increasing numbers (Evans, 1983; Gelman, 1980), or about the limits of decreasingfractions by repeated divisions (Langford, 1974; Smith et al., 2005; Tirosh & Stavy, 1996). Thisresearch examined—by means of the same procedure, in a within-subjects design—children’sability to increase indefinitely positive numbers and decrease both negative numbers andfractions. The results outlined similarities in the three types of responses, as well as the orderof difficulty of the three tasks.

• Rather than probing by thought experiments and hypothetical questions, a relevant-involvementmethod, based on competitive games was developed for investigating conceptions of potentialinfinity. Most importantly, choosing to play second and explaining that choice do not dependon naming numbers; this eliminated the problems raised by shortage of numbers’ names. Themethod does not test children’s familiarity with big numbers but the very understanding of theprinciple of potential infinity, that is to say, that for every number there exists a larger (smaller)one. Indeed, some of our younger children’s verbalizations were surprisingly analogous tothe mathematical definition of a sequence that tends to infinity (or, for that matter, to minusinfinity or to zero). They maintained, in fact, that for any preassigned number, however large(small), there is, farther away in the sequence, a larger (smaller) number (e.g., Quotes 6 and11).

• Children had to compare a huge finite set with an infinite set, in contrast to previous studieswhere either two finite or two infinite sets had been compared. We gained some insights abouttheir underlying reasoning both from failures (e.g., there are more grains of sand than numbersbecause the grains are so tiny) and from successful comparisons (e.g., one can count all thegrains with numbers, and still more numbers remain).

• Studying the perception of the immeasurable gap between a finite and an infinite set yieldedinstructive conclusions: The finding that participants’ performance in comparison of sets wassuperior to that of location of sets highlights the theoretical difference between the weak andthe strong conceptions of actual infinity and shows that they do not develop hand in hand.

• Participants’ preoccupation with the issue of numbers’ names was particularly enlightening.We learned that one has to be able to distinguish between a number and its representationin order to conceive the endlessness of numbers. Past studies had looked at the formationof the tie between a number and its tag in the process of acquiring number concepts, butnot at severing that tie. And studies that had investigated word–object relationships, and thedistinction between symbol and referent, did not deal with the numerical context. The findingthat children were capable of increasing (decreasing) numbers with unknown names wassignificant. Although they ran out of names of numbers, they understood that the numbersthemselves do not terminate.

Finally, I find it imperative to highlight the pivotal role of reasoning by induction as a ve-hicle for moving from finite counting to realizing the endlessness of numbers. Many authorshave expounded the principle of induction as both mathematical and psychological. Kasner

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and Newman (1949, p. 35) maintained that the principle of mathematical induction “affirmsthe power of reasoning by recurrence,” and they quoted Poincare: “If a property be true of thenumber one, and if we establish that it is true of n+1, provided it be of n, it will be true of all thewhole numbers.” They go on to explain that in fact this is a psychological truism: “Mathematicalinduction is not derived from experience, rather is it an inherent, intuitive, almost instinctive prop-erty of the mind.” Peter (1957/1976) commented that “this is a most important lesson, namelythat the infinite in mathematics is conceivable by means of finite tools” (p. 51). Fischbein (1987,pp. 51–52) noted that Poincare had emphasized the characteristic intuitive leap involved in induc-tion, and attributed the conviction attained by that reasoning to “the power of the spirit” (therebybegging the question via renaming the phenomenon). The children in our study have often voicedthe principle of induction in their own nonacademic style; Poincare’s “property” in their casewas the existence of a bigger number (e.g., Quotes 6, 19, 20, 21). They displayed no hesitation,and considered their conclusion self-evident. We have, in effect, witnessed that intuitive leap inaction.

ACKNOWLEDGMENTS

The study was partly supported by the Sturman Center for Human Development, The HebrewUniversity of Jerusalem. Thanks are due to many students of the Hebrew University for theirsignificant contributions to the experiments and their interpretations. I am grateful to RaymondNickerson, the editor, David Tall, and another anonymous reviewer for helpful comments andsuggestions. However, I alone am responsible for the views expressed in the article. Yael Orentook care of the figures. I received invaluable help from Raphael Falk throughout all the stagesof working on this project.

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APPENDIX

Participants are identified by gender (children: B = boy, G = girl; adults: M = male, F = female)and by age. The experimenter’s words appear in parenthesis and procedural details or commentsare bracketed.

The Quotes

Experiment 1:

1. B5: I want to play first. (Why?) Because once I played with Dana and I was first and Iwas the king; she was just a queen. I say 2 (5), you win. [On trial 2] I want to be second.(Can you explain why?) Because I like to be second.


2. B7 [after getting successfully to a billion in the continuous part]: (How much can wego on?) Until it ends, until we reach the biggest number. I haven’t yet learned it; I thinknobody in my family knows it.

3. G7: We can go on like that until the night, until we reach the biggest number. The biggestnumber is a milliard. (What if we add 1 to a milliard?) It will get bigger. (So, is that thebiggest?) No, but there is a biggest number.

4. G9: I think we can finish the game when we get to the end of numbers. I think there isan end, but I don’t know where it is.

5. G9 [in the increasing sequence]: (500) 600 (20,000) 30,000 (million) 2 millions (billion).What is that? (A very large number) Aha! [ironically] 2 billions (trillion) 2 trillions(googol) 2 googols. (Will the numbers end up for us?) No, because each time you makeup a number, I can add to it. (Did you recognize these numbers?) No. (How did youmanage to find bigger numbers?) Because you said. [This example demonstrates thatfamiliarity with big numbers and understanding potential infinity are not the same.]

6. G10: I want to be second because so I’ll hear what you say and I’ll say something bigger.. . . I think one cannot ever finish that game because each time you can add one andanother one, and there is no end to it.

7. G11: No, the game will not end. Perhaps in words there is an end, but when you write anumber, you can always write another zero and another zero, so that it will not end forthe life of me.

8. G6 [in Game 2, exemplifies some insight gained while playing]: I want to play first: zero(–1). What is this? I am in first grade, not in the army! [Trial 2] I want to play secondbecause I learned from you. (–10) –1. [She realizes that she lost, and in Trial 3 wantsagain to go second] Now I understand what is better, (–10) –100. [Trial 4] (–1,000)–2,000. [After a few steps in the continuous part] This game is confusing, I am not atall familiar with such numbers and I don’t know how to go on.

9. G12 [Game 2]: We cannot end this game. It is the same as before even though it isminus. Whoever has not learned about minus thinks that zero is the smallest number,but if one did learn, one knows that there are infinitely many numbers also down there.

10. B9 [Game 3]: I want to be first; I’ll say the smallest number, 1 (1/2). Oh! Fractions areallowed, then I want to be second, (1/3) 1/4 [This boy hasn’t yet learned about fractionsin school].

11. G10: For each number there is a smaller number that is still above zero.12. G12: One cannot reach zero because it is not a fraction.13. G12: The fraction decreases continually, but it does not vanish.14. B14: One cannot reach zero, because you can add more digits to the number in the

denominator and the fraction will decrease.15. G10 [Game 1]: At some time we’ll end the game when we’ll use all the numbers.

Everything has eventually an end, so surely the numbers have. [Game 3] I think we’llreach zero at the end, because the numbers are so tiny that finally nothing will remain.There are not so many numbers between 0 and 1 because there is little space there.

16. B6 [Game 1]: (How long will the game go on?) Until no end; the numbers will neverstop, because if there is, say, a milliard, one can add to it another and another milliarduntil infinity. (What happens if we add 1 to infinity?) Nothing, because infinity is not

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a number. [Game 3] I want to play second: (1) 1/2; (1/4) a half of a quarter. I’ll take asmaller and smaller grain, each time still smaller.

17. B11 [Game 1]: One can count and count and never get to a number that is the last one.[Game 2] Here too one cannot finish the game, because one can also count backwardswithout an end. [Game 3] We’ll never reach zero, we’ll only get closer to it each time.Even though the interval looks small it includes all the numbers in the world, becauseeach number you can turn into 1 divided by that number.

18. B12 [Game 2]: Big deal! I’ll always be second and say a smaller number, because toeach big number you can add a minus and it becomes small. [Game 3] Between 0 and1 there is no end to the numbers, because you can take each number and divide 1 byit. You see the interval as small because we are relatively big, but in it there are lots ofthings.

19. G13 [Game 1]: There is always a larger number. Not always all of them have names.There are numbers that were not given a name because they are so big, but you canalways add one to them. [Game 3] We cannot end the game, because between zero andany number there is always a number in the middle. One does not reach zero; betweenzero and the smallest number that may be mentioned there is always another number.

20. G7 [Game 1]: There is no end to numbers, each time comes a bigger and a bigger one,and when you count it does not end. [Game 3] We’ll have to stop at a certain numberwhen we’ll not be able to say a smaller one.

21. G10 [Game 1]: Each number that you say I can add one to it. One cannot finish thegame. [Game 2] Like in the previous game, but now in minuses. [Game 3] We are gettingcloser and closer to zero and then we’ll reach it, because there will not remain any spaceanymore.

22. G11 [Game 1]: There is always a number which is bigger. [Game 2] I don’t know whetherthe negative numbers have an end. Perhaps because they get smaller and smaller theyare going to end. [Game 3] If we’ll play a long time we’ll reach zero, because we areconstantly approaching it and finally there will not remain any gap between zero andthe number before it.

23. G10 [Game 3]: (1/2) 1/4; (1/3) a quarter of 1/3, I don’t know how you call it because wehaven’t yet learned it, but there are more like these. They get smaller and smaller. Youcan cut half from half and from this you can cut a quarter, and so it continues until youreach zero.

24. B11: The largest number that I know is trilliard billion. Just a moment! The largestnumber is infinity and I know how to write it, ∞. (Can we add to it?) No. it is impossible;there is a contradiction here, there are infinitely many numbers and we can always addone, only I don’t know how to call it.

25. B12 [Game 3]: There also are infinitely many numbers between 0 and 1, because we takeall the numbers and make 1 divided by them. Then just as there is no end to numbers,there will be no end to fractions. I think that in the end we’ll reach zero, because weapproach it constantly.

26. B11: I want always to be second so that I can win. I don’t like to play, it’s boring. Oneshould play a different game: Each one will write a number on a note and then we’ll seewhose number is greater. This would be fair.


27. G10 [in the continuous sequence of Game 1]: (120) 900 (1,000) 1,200 (2,000) 3,000(million) million and a hundred (2 millions) 10 millions (billion). What is that? (A verylarge number.) I don’t know, I cannot go on because I don’t know any more names ofnumbers.

28. B10 [when confronted with a squillion]: That’s it! I don’t know any more, there is anend, no more numbers have been invented; perhaps there are numbers for which noname has yet been invented.

29. G15 [after coping well with big numbers in the alternately increasing sequence; whenasked if one can win that game]: Yes. (How is that?) Because if I will not know the nameof a certain number, and then you’ll say a greater number than my last number, you’llwin.

30. G12 [Game 1]: (Can we end the game?) No, because there is no end to numbers. (Aminute ago you said that I had named a number that you didn’t recognize.) This is right,but I can deal even with numbers that I don’t know. Whatever you say, I’ll say 2 or 3more; this is the reason I chose to play second.

31. B13 [when confronted with a trillion]: What on earth is a trillion? (A number) 100trillions, what is a trillion I don’t know, but what difference does it make?

32. B15 [when confronted with googol]: I don’t know what a googol is, but it makes nodifference. (Why is that?) Because I’ll say 1 1

2 times a googol, which is surely greater.33. B11 [Game 3, after coping well with the decreasing sequence]: (Shall we go on?) OK (1

divided by 7 googol) I don’t know . . . well. . . 1 divided by 8 googol. (Are you familiarwith the number googol?) No, but you said 7 googols, so 8 googols is bigger and 1/(8googols) is smaller.

34. G12 [Game 1]: There is always a continuation. I don’t know their names, but at least inwriting there is no end.

35. G13 [Game 1]: We cannot finish the game because there are infinitely many numbers,only I don’t know how you call all of them.

36. B10 [Game 1]: It will go on forever and ever. There is no end to numbers, there is aninfinity of numbers. (What will happen if we add 1 to infinity?) It will always remaininfinity [giggles].

Experiment 2:

37. G14: The grains of sand are very tiny and there are more of them than numbers if wecount them.

38. B11: There are a lot of grains of sand. I don’t know their number, but there are surelygreater numbers than the number of grains, but I cannot read or write them because wehaven’t yet learned.

39. G11: Numbers are infinite. You can count and count without end; and grains of sandyou count and that’s it!

Experiment 3:

40. B9: Numbers and sand—-I don’t know. Numbers have no end; I think I should put themclose to each other [CR = 0.19].

41. M49: There are a few fingers, so I put it near the beginning. Grains are many. So I putit towards the end; then I decided to increase the distance between grains of sand and

38 FALK

numbers because after all numbers are infinite, so the gap should be greater. Grainsshould be somewhere on the way, but closer to numbers [CR = .61].

42. F17 [asked permission to pull the tape to the end, but before completing pulling]: theclip should be out of the tape. (Suppose we extend the tape in additional 10 m?) Nomatter how much you extend it, even if the tape will reach the moon, because there isno end to numbers [CR = ∞].

43. B11: Numbers have no end. I can always add more and more and invent names: trilliards,culliards, shilliards, without end. If there is no end to numbers, then also their distancehas no end and one cannot determine where it is [CR = ∞].