Children's Analog and Digital Clock Knowledge

Williamt J. Friedman and Frank LaycockOberlin College

FRIEDMAN, WILLIAM J., and LAYCOCK, FRANK. Children's Analog and Digital Clock Knowledge.CHILD DEVELOPMENT, 1989, 60, 357-371. Understanding the clock system requires knowledge ofseveral distinct components, including reading displays, transfomiing times, and understandingtheir temporal referents. 2 experiments were conducted te determine the ages at which children canread and transform times given in analog and digital displays, can link times to activities, and canjudge the order of hours in tbe day. Altogether, 240 children from first to fiflii grades were tested.Digital time reading was well developed by the first grade. Analog time reading was equivalentonly for whole-hour problems, with some other times proving difficult even for the oldest children.However, diere was no overall digital advantage for tasks requiring the addition of 30 min, and therelative difficulty of analog and digital displays varied by problem type. Reported methods indicatedthat children used a number of different processes in solving the problems. In spite of the gradualdevelopment of reading and transformation skills, even die youngest children knew the times ofmany activities and understood the order in which daily activities occur. However, clock times werenot incorporated in the earliest representations ofthe order of daily events.

The clock system is important in the livesof chilciren from the preschool years onward,but it is probably tiie least studied of the ma-jor symbol systems that confront children.This relative lack of research may be due inpart to a failure to segregate clock knowledgefrom otiier aspects of time knowledge thathave been studied (Friedman, 1982). In thisstudy we attempt to raise and answer a num-ber of basic questions about the developmentof clock knowledge in children.

In spite of the apparent integrity of theclcx:k system, tiiere are a number of distinctcomponents of clock knowledge. First, thereis the obvious ability to Icwk at a clcxrk andread tiie time. Second, tiiere is the ability tooperate on clcx:k times to extract relation-ships. Just as arithmetic operations are key tothe understanding of number, so the ability todetermine what the time will be in 15 min orhow long it is until 2:00 are essential parts ofunderstanding the clock system. Third, thereis tiie understanding ofthe meanings of times,such as the knowledge of "where" a givenclcx:k time falls within a day or what ac^vitiestypically occur at that time.

This multiplicity of components allowsus to study what happens when children con-front an ecologically important but complex

symbol system. If the representations andprocesses tiiat eventually c;apture differentcomponents make different capacity demandsor require different amounts of practice, wemay find considerable disparities in the agesat which the components are acquired. Thus,what we intuitively cx)nsider to be a singlecontent would be mastered piecemeal.

The first component, reading clock times,itself appears to involve multiple processes,and these processes seem to differ consider-ably for analog and digital clock displays. An-alog clcx:ks represent hours on a continuum inwhich numerals and the sucx^eding intervalsare interpreted as discrete steps frvm 1 to 12.Minute values are indicated by a separate butsuperimposed scale ranging from 1 to 60. Inprinciple, people who caa read clcwks mightstore an associated time name for each config-uration and use retrieval processes (Siegler &Shrager, 1984) to read the display. But thereare 720 different times to the exact minute, sosuch a representation, while allowing rapidperformance, would require considerablememory resources. Instead most individualtimes are probably read by means of a self-terminating series of operations, operationsthat correspond to the prcxedures that chil-dren are taught These include attending totile numeral to which the hour hand points or.

We wish to thank the children and staff of the Jefferson Elementary School, Rotteickm, NY, theTisdale Schcml, Ramsey, NJ, and the Eastwood and Prospect Schools, Oberlin, OH, and SusanBrudos, Denise Dahlin, Cadty Jordan, Glare Macdonald, Nazeem Mulugarine, Garolyn Schult,Jeanene Smoker, and Tracy Swan. Robert Siegler's comments on an earlier version of this articleand JefiErey Bisanz's review are also gratefully acknowledged. Requests for reprints should be sentto: William J. Friedman, E>epartment of Psychology, Oberlin College, Oberlin, OH 44074.

[Child Development, 1989, 60,357-371. © 1989 by the Society for Research in Child Development, Inc.ll ^ reserved. 0009-3920«9«0024)008)H>1.00]

358 Child Development

if necessary, has passed most recenUy to getthe hour v^ue, determining the minute valueby counting clockwise by 5s from the top oftiie clock iace until the 5-min mark pointed toor immediately preceding the minute handpointing is reached, and, if necessary, count-ing the remaining hash marks by Is.

The processes, of course, should vary ac-cording to the specific times that are dis-played and m i | ^ also be expected to changein c;ertain ways witii prac^ce. For example,children prol»bly quickly leam to associatetime names with certain "landmark" point-ings such as "o'clcx;k" with the minute handpointii^ to the 12 and "30" with the minutehand pointing to tiie 6. These associationseliminate the need to count 5-min marics.Witii repetition, chilciren probably come toreco^ize additional 5-min multiples directiyand te use tiieir values as starting points forother 5-min and 1-min increments. Such asso-ciations presumd[>ly become a component ofwhat remains for most times a sequence ofoperations.

Reading digital clocks would ^pear toinvolve qpiite different processes. Becausehour and minute values are shown as numer-als, tiiey can be read like any other 1- and 2-digit nmnbers, as long as one possesses thisskill and knows that ^ e c»lon is used to parti-tion the hour value and the minute value, that:00 displays are read as "o'clock," and that azero in the 10s place to the right of tiie colonis read as "oh." Performanc^e seems mainly amatter of retrieving number names, so thereshould be little variation from problem toproblem and little change in tiie processeswitii practice. In general, number retrievalseems to make fewer prc«essiiig demandsthan tiie mental prociedural sequences that wehave assumed underlie most analog clcwkreading problems. Therefore, even given sim-ilar exposure and training, we would expectdigital clock reading to reach skilled levelsbefc»e ansk)g cicxik reading, with the possibleexception of whole-hour analog times.

The seccmd component of ckick knowl-edge may show less cMvergence between ana-log and ciigital d o c ^ . Operations on clocktimes, such as adciing some interval to thepresent time, prc^xibly depend to a consider-able extent on mental procechiral sequencesfor both disi^y types, with direct retrievalpUyii^ a minor role. Given the large numberof pc»sible stioting times and intervids to beadded or subtracted, prestorage of answersseems infeasible. However, the particular

involved may be infiuenced by

the type of display. For example, imaginedminute htuid movements may be used to addor subtract some interval from an analog dis-play, whereas digital displays may be morelikely to invoke numerical addition or sub-trac:tion. Also, people who have had consider-able exi^rience witii an^og cdcxiks mig^t usemental im^es of minute hand movements toperform many operations on digit^ly dis-played times. Finally, for both display types,certain problems may come to be solved byretrieval-like processing. Problems involvinga^Ung or subtracting half an hour fromwh<rfe-hour or half-hcmr times may be en-countered frequency enou^ that operationsbec»]me aub»n^ized, direct associations de-velop, or chilcken leam to api^y their mathe-matical knowledge about adding halves. Ac-cordingly, we m i ^ t expect ^\a.t as childrengrow older they come to solve such problemsrapidly, whereas most other problems, bothanalog and digital, continue to require rela-tively long solution times.

The third component of clock knowledgeis understanding tiie referents of clock times.There are at least two levels at which thiscx>rresponcienc^ mi|^t be understood. One isknowing tlw usual times of activities. This in-form^on could be coded by representationsas simple as paii«d associations between timenames and activities. The otiier is knowingthe temporal "location" ofa clock time witiiina day or the daily order of clock times. Thisinfonnation would seem to require richer rep-resentations, such as associative networics, se-rial structures, or images (see Friedman, inpress).

Given this distinction, children couldleam isolated links between particnilar timesand ac^vities before they possess an overallscheme for the orfer of cloc^ times, just astiiey might laiow the montfi of their birthdbybefore undei^tandBi^ the ord^ of die months.However, it is diMcuIt to predict when tem-poral structures ccHne to be represented be-cause previous research has shown some r ^ -resentations to be present by early cdiildhoed(e.g., Friectoaan, 1977; Muto, 1^2; N^son,1 S@), whereas otiiers are acquired betweenmidcHe c iltfiicxKl aad adolescence (Fried-man, 19^). The age of acqtusition may de-pend on the number of rej^itions ofthe tem-pcmil pattem th^ have been e3^)erienced andthe number c}f elements in Ae stnictaire. Chil-dren, oi covu^e, have encountered hundredsof r^etitions of the dhutly cycle by the timethey b ^ i n school. However, it is unclearwhether cloc^ times are incorporated in theirfirst representations of daily time pattems or

whetiier the times are added to what are ini-tially activity-based representations.

The present study was designed to pro-vide information about when children cometo understand the three components of theclock system. Unlike other studies which fo-cus principally on analog clock reading skills(Case, Sandieson, & Dennis, 1986; Siegler &McGilly, in press; Springer, 1952), we testedthe same samples of children on comparableanalog and digital clcuk tasks. These compari-sons allow us to separate reading skills, whichaccording to our analysis are likely to differsubstantially for the two clock types, from op-eration skills, which are likely to involve com-plex procedural sequences for both. In ad-dition, we determined the ages at whichchildren can link clcwk times to daily activi-ties and the ages at which they can order aseries of times defined by hour readings and aseries defined by activities. These tasks allpertain to the meaning of times, but somecould be answered by acc;essing isolated asso-ciations, whereas others require the ability tointerrelate a set of times.

Experiment 1In the first experiment, we examined age

diflferences in children's ability to use analogand digital displays. We tested 160 childrenfrom first through fifUi grades in two differentschool district. To measure detection ability,we presented each child with pictures of ana-log and digital clocks and asked them to tellthe time. In the transformation tasks, weasked them to tell what the time would be in30 min. When the hce was displayed it wasnamed, so that transformation ability could beseparated from detection ability. To leamabout the processes underlying perfonnanceon tiie detec^on and transformation tasks, weanalyzed children's errors and reported meth-ods in addition to tiieir accuracy. To aboutone-tiiird of the subjects we also gave an op-tional transformation task in which they wereasked to dfrectiy compare their methods forsolving analog and digital problems.

METHOD

SubjectsThirty-two children from each of grades

1-5 participated in the study. Half at eachgrade came from a suburban school in NewJersey and half from a school near a large cityin upstate New York. The sample was evenlydivided by sex at each grade. Mean ages fortiie five grades are 6.6, 7.7, 8.6, 9.7, and 10.6.

Friedman and Laycock 359

ProcedureEach child was given the following tasks,

which were administered in the order listedbelow, nearly all in a single session of about30 min. The optional transformation task wasonly given to tiie children from one schoolbecjause of time limitations, and in this schoolonly 61 out of 80 children had time for thistask during the autiiorized session length (8,12, 11, 15, and 15 children from grades 1-5,respectively). One child in each sex, gradeand school combination was assigned to eachof eight stimulus orders. On any given task,half of the children received digital problemsbefore analog problems. For half of the chil-dren, the order of analog and digital presenta-tion was the same in the detection and trans-formation tasks. The optional transformationtask always had the opposite order for analogand digital problems as the main transforma-tion task. In each task, half of the subjectsreceived digital problems based on the hour4:00 and half on the hour 8:00. Analog prob-lems in each task had the reverse assignment.

Detection task.—Three analog and threedigital stimuli were presented: 4:00,4:30, and4:43 for one display type, and 8:00, 8:30, and8:43 for the other. The tiiree problem typeswere selected to represent increasing levelsof complexity in the application of analogreading procedures. The X:00 problems use adirect hour hand pointing and a minute handlandmark. The X:30 problems require appli-cation of the passed hour rule for the hourhand and recognition of a minute hand land-mark or the use of 5-min multiple incre-ments. The X:43 problems involve passedhour, 5-min multiple increment, and 1-minincrement procedures. Order of the threeproblem times varied randomly across theeight stimulus orders. Stimuli were 8 X 9-cmcards. Analog stimuli consisted of 6-cm-diam-eter clock faces, with 5-min marks labeledwith numerals and hands of 2.5 and 1.8 cm.The usual hash marks indicated 1-min values.Digital stimuli were 1.5-cm-hi^ blcwk num-erals separated by a colon and enclosed in 4.5X 2.4-cm rectangles.

General instructions for the detectiontasks were, "I'm going to show you sometimes using pictures of clcxjks. For each onetell me what time it says. Also tell me as yougo along how you are figuring out the answer.Please tell me everything you are thinking foreach one." On each trial children were askedto tell the time that would be shown, and thenthe stimulus card was presented. Childrenwere next asked to explain how they had fig-ured out the answer, with prompts to

360 Chad

age them to expand their explam^ons. Theexperimenter transcribed their responses.

Transformation task.—TTiree analog andthree di^E^ stimuli were used: 4:1^, 4:23,and 4:50 for cuie ciisi^y type and 8 ^ , 8:23>and 8:50 for the other. The X:30 problemswere chosen because diey cmi be solvedlai^ly by ^ ; ^ y i i ^ the knowlec^e t b ^ hc^an hour added to a half hour time cM»Bpletesthe hour. The other two problem types werepresumed to have aitah^ or c i ^ i ^ ad^^ui-tages. Tile X:23 problems seemed easier forad<^H>n € >ex»blons than minute hand opera-tions, amil ac^ltion may be more likely &»- dig-ital ];m^ems. However, the X:!SO proHemsmi|^t be ciU£Bcnilt to sc^ve by adiition be-cause the minute value changes u^en thehour is ccHnpleted, but easier to solve by con-sidering tiie movement of the minute hand.General instructions for tiie task were, "Inthis part I'm gfmtg to ask you questions aboutsc»ne times. Hetuse tell me the answers andtell me how ycwi are figuring tiiem out. IHeasetell me everytibtng you are thinking for eachone. Remein^r, 30 minutes n^aas the siunethin^ as hdf an hour." On each tri^, childrenwere shown die stimulus, ^ d , "Iliis clcx^says [time given]," and asked what it

say in 30 min. The experimenter re-coided the answer and any overt behavior orverbalization. Next, children were adced fol-low-up questtcms as in the detection task.

Optional transformation task.—Twoprobleois involvii^ adding a half hour andtwo involviiig subtrac^ng a half hour werep r e ^ t e d , in tirat order. One problem of eacdipair was Euial<^ fmd one was d^^ital; one w£^based on a 4:00 time, the otiier on an 8:00time: fbr die a^iition pair, 4:40 and 8:40; forthe subtractlcm psur, 4:20 and 8:20. Theseprd^^ms were similar to the X:50 problemsof die niuun transfonmdicHi task in their aE >ar-ent ameoabil^ to solution by isia^j^mg min-ute hand movexQKit We were interested in

>A^edier tiie digital p r^ i^ns wotildimage medMlon or whe&er t ^ two

display types would lead to cUSeretrt pro-cesses, l l ie optional transfomudton tmk al-lows a mcHe d^ict ccnnparison of medHxIsused cm analog and ^^tal pttMems &im themain txaiofenmtflfHi tuks becaiue chfidrenwere a^ced to coomare tiie f^^^oadbes theytoc4c to aa^og aad a^^^ preseateMot^ oS^esame i»o1>lem.

Before eac^ stimulus was ^eseirted, sub-jects were instructed to "tell me as you goalong whether you are thitUdi^ ^mut a clockface with hands moving or thinking about

numbers in your mind or cJoing it some otherway." The order of die italicized i&rases var-ied ranckvnly. For tiie addition pair, questionswere ofthe sfoaei fcmn as in the imun treuisfOT-m i ^ n tEG . POT the sul:Araction pair, sul^jectswere told, "Tbis dock says [timegjven]," imd asAied what it had said 30 minago. l l ie ei^jwiimenter recorded the imswerand any veiita^xattons. Follmv-up questionswere used as m tlie rwm trans&ffms^on task.After each pair erf i^t^ems, c hilEben wereadkeci, "C^e of d i ^e pnstnlems had a clockfiice Euid the c^ier ^ d nivrobers. Did you do

the same way?" This was followed by; i , such as, "Can you tell me more?"

"Did you tiiioUc about ac^ii^ numbers forbotii, or a clock face and moving hands forbotii?"

RESULTS

DetectionProportion correct.—The meem propor-

tion coi^K^ for deteeticm {ffobfetns is shownin tiie top half erf Table 1. Tlie response"Don't know" was scored as an error in thisand subsequent analyses. For digital prob-lems even first graders are able to read thedisplay as a time, and performfuic^ is essen-tiaUy parfec^ by second ^ade. Of die analogset, oi^y tiie X:00 p r o e m s are dxiut as accu-r ^ as die ctigital ones. The X:30 pare mastered by seomd grade, but X:43lems remain cUIBcult tiirou^ diird grade andlater.

Because of die high levels of ^(aad tiius limited vadance) after first grade fortwo of tile joudog profc^bms and ail d * ^ ci^-tal problems, an overall ANOVA was notapsax^a^. A s^>antte repeated-measuresAM)VA was performed for the fiist gradealong widi school, sex, and order (digitalproblems first or second) as between-^sul^iect&cbHS, and analog versus c i i g ^ and time(X:00, X:30, and X:43) as wi&in-sul^ect fee-tors. Sex was si iSBcEmt alone, F(l,24) =iZ.QS, p < .02^ ^xi in &e diree-u^y iisterac-tion widi ani^og versus cH^tal uid time,F(2,48) == 5.69, p < .01. These ^Eects reflectthe fiu^ that boys were more accnirate overallat this ag0 aad thiU tiieir accur^^ was nu>reun^rm across d^g^al problems tiban girls'. Ofgreater ii^rest aze tlw clear &icUiigs du^ cUg-ital deletion was superior to ma^ deteo-tion, FilM) = 43.24, p < .001, tiiat time dis-played was a signtflcffiri: somce of vsaismce,Fit^m) = Um, p < .001, ffl^ diat diis timeeSeci. was attrAmt^te to the analog, not digi-tal ccmd^on, analog versus c%ital x time,F(2,48) = 20.27, p < .001. Two nonoi^ogonal

Friedman and Laycock 361

TABLE 1

MEAN PROPOBTION CORRECT FOR EACH GRADE ON THE DETECTIONAND TRANSFORMATION TASKS IN EXPERIMENT 1

GRADE

Detection:12345

Mean

Transformation:12345

Mean

:00

.72

.97

.881.001.00

.91

;30

.16

.59

.72

.97

.91

.67

ANALOG

:30

.37

.91

.97

.97

.88

.82

ANALOG

:23

.06

.16

.34

.53

.44

.31

:43

.03

.34

.69

.78

.81

.53

:50

.03

.41

.53

.75

.72

.49

;00

.751.001.001.001.00

.95

:30

.12

.50

.78

.97

.91

.66

DIGITAL

:30

.75l.OO1.001.001.00

.95

DIGITAL

:23

.09

.28

.41

.50

.72

.40

:43

.72

.971.001.001.00

.94

:50

.03

.25

.47,59,75

.42

planned comparisons revealed significant dif-ferences between tiie analog X:00 and X:30means, F(l,90) = 4.10, p < .05, and betweendie analog X:30 and X:43 means, F(l,90) =3.99, p < .05.

A separate ANOVA tested grade differ-ences in the analog X:43 problems alone. Itrevealed a highly significant grade efiFect,F(4,155) = 22.02, p < .001, and a highly sig-nific:ant linear grade trend, F(l,155) = 78.09,p < .001. The linear trend shows that accu-racy improves continuously during this agerange.

Reported methods.—A rater assigned thechildren's reports on their methods of solu-tion to one oftwo categories for the digital setand four categories for the analog set. Interob-server s^reements with random samples of100 responses were 98% for the digital setand 72% for the analog se t The first rater'sassignments were used in our analyses. Themethod categories are listed in Table 2,which also shows the proportion of correctand incorrect reports assigned to each cate-gory for each stimulus time. The digital cate-gory "direct reading" included desc^bing orexplaining the display or simply pointing outthat one's answer was die time shown or thatone could tell by looking at i t The analogcategory "direct reading" included statingthat one just knew the answer, explaining

some con-ect clock fact such as that 60 or 12equals a whole hour or that 6 equals 30 min,or simply describing where the handspointed. The category "appropriate multistepoperation" included explaining a procedurefor deriving the minute hand value, such ascounting each number by 5s, often startingat 6, or counting hash marks from the 8. "In-correct dynamic" included erroneous rulesabout the clock, such as using a minute handrule to read the hour hand position.

For the digital detection problems, be-tween 90% and 95% of children who gavecorrect answers reported that they read thedisplay directiy. An example is, "It just says8:30." The few children who made mistakesmosdy reported some version of "by lookingat it" or merely described the display.

Of children who were correct on the ana-log detection problems, about 80% reportedsome sort of direct reading method for theX:00 and X:30 problems, whereas the over-whelming majority reported a muitistep oper-ation for the X:43 problems. These methodswere reported by fewer tiian a third of chil-dren who gave incorrect answers, though it isnotable that many who were incorrect on theX:43 problems refened to appropriate opera-tions. Most children who gave erroneous an-swers did not explain how they arrived atthem or cited some incorrect rule.

362 Chfld Devek^nnent

TABLE 2

PROPOBTION OF REPORTED METHODS ASSIGNED TO EACH CATECXJBY!N THE ANALOG AND DIGITAL DETECTION TASKS IN EXPERIMENT 1

ANSWER CoBREcrr

REPORTED METHOD :00 :30 :43

ANSWER INCORRECTT

:00 :30 :43

Analog detection:Direct reading 92Appropriate multistep

operation 03Incorrect dynamic 02Don't know or otiier 03

Digital detec^on:Direct reading 90Don't know or other 10

.79 .07 .28 .24 .20

170202

9505

.87

.00

.06

.93

.07

.07

.29

.36

.62

.38

.03

.38

.34

.75

.25

.31

.15

.35

.60

.40

Given the considerable consistency inthe mediods reported for arriving at conectanswers, there was littie room for age ciiffer-ences. Nevertheless, we performed PearsonX tests of association between grade and re-ported m^iu4$ foi^ cc»Tect items on each ofthe three {»dNiem types for tiie analog anddigital sets (left half of Table 2). These testsmust be interpreted with ciaution because ofthe U^e propcHtion of cells with small ex-pected frequencies. Only two of these tests,diose for analog detection X:00 said X:30,reached conventional levels of si^ficance,x'*(12, JV = 1^) = 25.11, p < .02, and x^{12, N= 132) = 27.96, p < .01, respectively, andneither showed continuous age trends. Ineach case the dominant method in Table 2was tiie domiimnt method at each age. Tlius,it appears thirt tJie main age trends are frommediods associated with inconect answers tomediods associated widi conect answers,with Hide chfo^e in tiie distribution of meth-cxis among correct answers.

TransformationProportion correct.—The bo^>m half of

Table 1 shows the accuracy results for thetrans&mnation task. For both the analog andd i g ^ sets, chddren's trans£»mation per-fonnaitce lagged well behind dieir detectionpei'lisBuuwe (shown on tiie top half of TEA>le1). Both sete c^ problems showed neaily iden-tical overall pfqpcntiCHis correci, and for bothsets t ^ X:30 pnx^ms were the easiest. Theywere die only probl^ns where half or more ofdie cldldivn were accurate be&are tiiird orfinirth grade. The X:23 probl^ns remeined(^fflcok even for the oldest children, espe-cially in the awikog set The only appa^ntaiuk^^dlgital ^SE^eace is du^ in die digitalset X:23 and X:50 showed approximately

equal prcqrortions correct, but in the analogset the X:50 problems were easier.

A repeated-measures ANOVA was per-formed with school, grade, sex, and order(digital problems first or second) as between-sulijec: &ctors, and analog versus digital andtime (X:30, X:23, and X:50) as witiiin-subgectfectors. Of the main efifec^, school, F(l,120)= 7.72, p < .01; grade, F(4,120) = 14.32, p <.001; sex, F(l,120) = 12.77, p < .01; order,F(l,120) = 6.16, p < .02; and time, F(2,240) =51.05, p< .001, were si^ficant, whereas an-alog versus digital was not. Schools A and Bhad means of .55 and .43, males were moreaccurate than fonuJes (.56 vs. .41), and chil-dren who received the digitel set first weremore accurate overall (.54 vs. .44). Gradechanges and time d^^^nc^es are shown inthe bottom half of Table 1.

Four two-way interactions were ^cant. Most interesting was tiie interaction be-tween analog versus distal and time, Ft2,24O)= 6.70, p < .01. Because the over^l me^utsfor analog and d l g ^ X:30 are essentiallyidentkatl, dus shows &at as ej^tected, die an-alog and cj^gitd sets d t l ^ in die relative c ffi-culty of the X:23 a i^ X^50 ^wblems. Timeinteracted with grade, F{B^4Q) = 2.32, p <.(B, wppSBmaiiy because X:30 problems reachI ^ ^ levels of accuKicy at earlter ^ e s tiianX:23 and KM prdbtems. The intera«*ion be-tween grade and analc^g versus c^f jal,F(4,120) = 2.62, p < .04, probably reactedfirom tile sli ^ stly more accurate aiuiktg per-formaxK^ at g i ! ^ s 2 and 4, as contrasfeect widiscHne x4sat more accnmite responses on theci^ital set at gcoAe 5. t l ie interaction betweenandk>g versus ciUgj^ and school, F( l , l^ ) =5.41, p < .03, refiec^d a small crossover In

Friedman and Laycock 363

TABLE 3

PROPORTION OF REPORTED METHODS ASSIGNED TO EACH CATEGORYIN THE ANALOG AND DIGITAL TRANSFORMATION TASKS IN EXPERIMENT 1

ANSWER CORRECT ANSWER INCORRECT

REPORTED METHOD

Analog transformation:Just knewCount hy 5s or 10s. . . .AdditionCompletes the hour ..Movement of minute

handNo answer or other . . .

Digital transformation:Just knewCount by 5s or 10s. . . .AdditionCompletes the hour ..Analog referenceNo answer or other . . .

:30

.17

.16

.22

.32

.10

.03

.14

.09,35.22.12.08

:23

.00

.31

.51

.00

.04

.14

.00

.17

.70

.00

.03

.09

:50

.01

.55

.21

.00

.10

.13

.01,19.60.00.12.07

:30

.04

.11

.04

.00

,04.77

.02

.09

.09

.00

.09

.71

:23

.00

.37

.04

.00

.03

.56

.01

.10

.17

.00

.06,67

:50

.00

.26

.06

.00

.06

.62

.00

.06

.02

.00

.09

.84

the relative difficulty ofthe analog and digitalsets for the two schools, A three-way interac-tion between order, analog versus digital, andtime was also significant, F(2,240) = 3.11, p <.05.

Reported methods.—Method reports forthe transformation problems were assigned toone of six categories for each of the analogand digital sets (see Table 3). Interobserveragreements with random samples of 100 re-sponses were 79% for the analog set and 82%for the digital set Most of the categories areself-explanatory. For the category "count by5s or 10s," we presumed that children werecovertiy counting 5-min marks on the clockface in the analog condition and counting bydecades in the digital condition. Overt behav-iors like pointing and lip movement werevery rare, tiius preventing us from confirmingthese interpretations. The category "analogreference" involved referring to a clock facein explaining one's method.

Reports accompanying correct answersare considered first. Similar methods were re-ported on the analog and digital sets for theX:30 and X:23 problems. Methods suggestingdirect retrieval were common on both sets onthe X:30 problems. These include simply re-porting that a half and a half make a whole, orcomplete hour, or that one just knew the an-swer. Addition was also a common reportedmethod on the X:30 problems. Counting by 5sor 10s and addition were the most commonmethods for botii the analog and digital X:23problems, though addition was favored some-

what more strongly in the digital condition.This difference was especially pronounced onX:50 problems, with the majority of reportsfor the analog set indicating counting by 5s or10s and the majority ofthe reports for the digi-tal set indicating addition.

Only a minority of reports accompanyingincorrect answers could be classified in thesecategories. However, it is notable that manychildren whose answers were incorrect on theanalog transformation X:23 and X:50 prob-lems reported relevant operations, countingby 5s or 10s.

As for the detection tasks, we computedPearson x tests of age differences in themethods associated with correct answers. Thedistribution of methods was significantiy asso-ciated witii grade for the analog X;30 andX:50 problems, x C20, N = 107) = 41.83, p <.01, and x^(16, N = 78) = 27.02, p < .05,respectively. For the X:30 problems, therewas a developmental decrease in counting by5s and 10s and increases in "addition," "com-pletes the hour," and "movement of minutehand." This may reflect a trend away frommultistep procedures and toward more directsolutions to half-hour problems. There wereno clear age trends for the X:50 problems,though there was some tendency for countingby 5s and 10s te decrease with age and "addi-tion" and "movement of minute hand" to in-crease with age.

Optional TransformationTwo procedures were used to analyze

the optional transformation data. First, a re-

364 Devefo^nent

TABLE 4

FRBQUENCY OF METHODS FOR ANALOG AND DIGITAL PROBLEMSIN THE OPTIONAL 'DUNSFORMATIQN TASK IN

METHOD CATEGORY

Primarily numericalPrimarily operation on

clock &ce or imageComhination

PLUS 30

Analog

15

519

Digital

34

56

MINUS 30

Analog

13

422

Digital

28

54

peated-m^uures ANOVA of proportion cor-rect included the &ctors grade, sex, analogversus digital, and plus versus minus transfor-mation. Only tiie ^Etde eSect was significant,F(4,43) = 5.57, p < .01. The absence of asigni&iant Euialog versus distal effect wasconsistent witii the results ofthe main trans-fonnation task, tiiou^ the absence of a sexeffect was nc^ Witiun the limits of inferencefrom a null finding, and bearing in mind tiieconfouiK^g between plus versus minus andtask fflder, it appears te subtracting one halfhoiM- (mean |»opOTtion conect = .65) is nomote difficult tiian adding one half hour(mean = .64).

The second analysis was intended to de-termine whetiier we could d^rentiate theaEy tfoaches taken on tiie amdog £md dl0taltasla. Jue^s were instructed to read ail of tiieavaili^le Infbnriation for each probtem (in-d u i n g answers, tiie metiH)dB reported andanswa's to tibe follow-up questicms), Mid thento ass i^ the problem to one of 12 metiiodcateg(»ies. Tliree of tiiese categwies were forprinmrily numerical operations (such as addi-titm, sid>tcaction, or counting), and three in-volved openUions on a clock &ce or ^xam oia ciodc fece (e.g., overt or covert stepwsemovnnent, or d^ect 180° movement of theminute htuid). The remaining six ct^gorieswere: a combination of numerical and clock&ce <^>erEitions; a (^ttegory in n^ich Ae r^jercoidd not distb^uish wUch of tiie two wasused; aitd categories f<w generalizing tiie an-swer frran a previmjs problem, reportu}[g aruk, uactoss^dsle resp(»ises, and insuSl^ntnuHerit^. One of tbe first two raters as^^eiredto overuse the "ccxid>in^on" cat^goiy, so atiUnI rater was ad(M- Final classifiet^ons in-chi4ed only responses wh«« two or tiiree oftiie three rftters agreed on tiie as»gnment to(me (^ the 12 cate^Eories. With 12 c^goriesdie ej^>ected [Hopcniion of ^reement £or twoor tiiree out of the three niters is .24. TTiea c ^ rates of f«Fe«nent for the four prob-lems rtuiged from .80 to .92..

Table 4 presents the results for collapsednumerical and cottf^»s^ clock hce categoriesand the c(»nbination categcny. Excluded arerespcmses tiiat &iled to meet tiie iaterob-server apv^nnent criterion or Ml in tiie othercateg(»ies. For tiie d ^ ^ {Htiblems, primarilynumerical metiiods accounted for a^^noxi-mately tiiree quarters of the responses. Forthe aiial(^ iwdblems, numerical methodswere also ctnnmon, but about half fell in the

ifcHy. The clock fece cate-accounted fbr about one tentii of the re-

sponses on e«£h m£^em. There was no evi-dence of age di^rences for any of the fourproUems: Chi square analyses of method xgrade were n<^ significfuit.

It may be that tiw presence of a digitalabsence of a clock fece) on tiie

lems produces a bias to use addi-tion, sutoacticm, or unit counting, ti^about (Hie-qiBtifer (^tiie responds ganalog inuigecy. When a clodc fece is pro-vided alcmg witii the time name on the asialogprol^ms, imi^Uiied oper^ons on tiie clockfewe pr«(kHniiiiite. These opra^ons occurredmrai^y in the cf^^ory combining clock feceand numerical me^Kls, probably becausediSdren usually coui^ed the 5-min marksaround the doidc fece. "SWien diiidren le-pofted coi^iiyag, it was often diSicult to deter-mine lA^etiua' t ^ y m^n t a purely numericalmethod, such as ojuntiD^ by 10s, or whetiiertiiey were sequentiftlly attemiing to marks onthe cbc^ fece. Thus, tito num«ricai cf^gory

^ imlades scane chil Sren who sA leastp used c^ock fece oprat^ons, whereastite OHxdsiOd^on cal^f^oty ^robali^y indtnilesmate c(ffln^dete r^iorts c^cotmting around Ilieclock fece.

DISCUSSION

Tiae pftttwn of results for the two dis£^ytyx>es c l ^ y siMjrted tiie distinctitHi be-tween rratcbs^ sldils aaid operation sidlls. Dig-ital time reading was very accurate by first

Friedman and Laycock 365

grade and virtually perfect by second grade,whereas analog performance varied greatiyaccording to the display time. Whole-hourtimes were accurately read by first grade andhalf-hour times by second grade, but the X:43problems did not reach comparable levels un-til the third grade, and they remained difiRcultfor at least some children through the fifthgrade. Mastery of transformation tasks laggedbehind detection by several years: not untiltiie fourth or fifth grades were children accu-rate on most of the problems. However, incontrast to detection, there was no overall dif-ference in the .difficulty of analog and digitalproblems. In general, the problems that weexpected would involve multistep mental se-quences—analog X:43 detection problemsand analog and digital X:23 and X:50 transfor-mation problems—were relatively late acqui-sitions. By contrast, problems presumed toinvolve substantial retrieval components—analog X:(X) and X:30 detection problems andall digital detection problems—were solvedat relatively early ages. Analog and digitalX:30 transformation problems appeared to fallbetween the two sets in difficulty, perhapsrefiecting a developmental shift from proce-dural sequences alone to a greater reliance ontiie principle that 2 half hours equal a wholehour.

Reported methods also supported ouranalysis of the processes involved in detec-tion and transformation. Most children re-ported reading the times directiy for digitaldetection in general and analog detection ofwhole-hour problems. These retrieval-likeprocesses (see also Siegler & McGilly, inpress) cannot be precisely characterized onthe basis of our findings or previous ones, butthey are clearly distinguishable fixim the pro-cedural approaches taken on X:43 analog de-tection and most transformation problems.The methods reported for the analog detec-tion X;43 problems showed the use of the 5-min-multiple and 1-min-increment proce-dures that children are commonly taught.Procedural approaches were also evident inthe analog and digital transformation tasks.Counting by 5s or 10s, either numerically oraround the clock perimeter, and additionwere frequentiy reported.

It is interesting that children often useddifferent procedures for the analog and digitaltransformation tasks, even tiiough the startingtimes were always given aloud. The X:50problems were somewhat easier than theX:23 problems in the analog but not digitalconditions. This suggests that tiie presence ofthe clock fece allows children to exploit the

180° spatial relation between pairs of times, atleast when they are 5-min multiples. The op-tional transformation task also showed thatchildren often performed clock fece opera-tions when an analog stimulus was present,whereas digital stimuli evoked mainly numer-ical approaches.

Experiment 2In the second experiment, we repeated

the detection and transformation tasks but didnot ask the children to describe their meth-ods. This change allowed us to see whetherthe results of Experiment 1 were likely tohave been distorted by the thinking-aloud in-structions or the follow-up method probes.Second, we added a number of tasks concem-ing the meaning of clock times. Two of theseasked children to specify activities that typi-cally take place at a particular clock time or toreport the time at which particular activitiesusually occur. These tasks could be solved ifchildren simply possess associative pairs be-tween time names and activities, so we pre-dicted that they would be early achieve-ments. We also asked children to order sets offour cards, each of which represented a partic-ular time. In one condition the cards werepictures of daily activities, in another theywere clock times, and in a third they con-tained two activity pictures and two clocktimes. These tasks assess children's knowl-edge of when particular times occur relativeto other times. We were interested in deter-mining whether clock times were incorpo-rated in children's first representations ofthestructure of daily activities or were a later ad-dition.

METHOD

SubjectsSixteen children from each of grades 1--5

participated in the study. They came from asmall, socioeconomically mixed college townin rural Ohio. The sample was evenly dividedby sex at each grade. Mean ages for the fivegrades are 7.1, 8.6, 9.1, 10.3, and 11.2.

ProcedureEach child was given four tasks, in the

order listed below, in a single session of lessthan 30 min. The detection and transforma-tion tasks were administered first according totiie same ordering scheme as in Experiment 1and followed identical procedures, exceptthat children were not asked to think aloud orto explain their methods.

Order task.—Children were asked to or-der four sets of four cards each. One set con-

366 CliUd Devdopment

TABLE 5

MEAN PROPORTION CORHECT FOR EACH GRADE ON THE DETECTIONAND TRANSFOKrfATION T A S K S IN EXPBIUMENT 2

GRADE

Detection:12345

Mean

Transformation:12345

:00

.94

.88

.94

.94

.87

.91

:30

.19

.37

.941.00.88

.67

ANALOG

:30

.75

.381.00.87.94

.79

ANALOG

;23

.06

.06

.56

.44

.56

.34

;43

.33

.06

.63

.56

.87

.49

:50

.06

.19

.63

.56

.81

.45

;00

1.001.001.001.001.00

1.00

:30

.31

.50

.56

.88

.94

.64

DIGITAL

;30

.94

.941.001.001.00

.98

DIGITAL

:23

.06

.06

.62

.31

.75

.36

;43

.941.00.94

1.001.00

.97

:50

.13

.06

.37

.31

.75

.33

tained depictions of a child having break&st,children arriving at school, a child eatinglunch, and a child arriving home from school.The second set consisted of digital cards simi-lar to those used in the detection and transfor-mation tasks. They showed the times 9:00,11:00, 1:00, and 3:00. The last two sets eachcontained two activity cards and two digitalcards. One used the cards break&st, 9:00,lunch, and 3:00. The other used arriving atschool, 11:00, 1:00, and arriving home. Allcards were approximately 6.5 x 8.0 cm. Halfof the children received the activity set firstand the digiial set second; the other half re-ceived tiie reverse order. The mixed setswere presented third and fourtii in counter-balanced order.

At the start of the activity and digitaltasks, children were shown the four cards in aconstant random order and told, "These areall times/things tiiat happen a ^ r you get upin the moming and before dinner. They hap-pen between waking juid dinner time." Next,children leamed tiie card mean ing until theymet A e oiterion of cme perfect identifica^pnof all four. Then the experimenter said, "Iwant you to put tiiem in owler. That n^ans Iu^nt you to put tiiem in a line, with tiie firstthing tiiat happens here, and the next thingthat hai^>ens here [eto.]." Prior to the mixedsets, children were instructed, "The next

bunch have some times and some things thathappen. All of them are still between wakingand dinner time. I want you to put them inorder." The experimenter recorded the ordersproduced for each se t

Mappmg task.—For each of five times(10:00, 2:00, 6:00, 8:00, and 4:00, in tiiat con-stant random order), children were asked,''What do you ususdly do at [time]?"Unless tiiey specified A.M./P.M. or a part of tiieda.y (e.g., moming), tiiey were asked, "Which

[time] is tiiat?" In the second part ofthe task, children were asked, "At what timedo you ?" for each of five activities inthe following constant random order: eat din-ner, go to school, go to bed, get up in themoming, and eat lunch.

RESULTS

DetectionProportion correct.—The top half of Ta-

ble 5 shows the mean proportion correct fortile detection problems. The levels and pat-tems of accmacy are similar to those shown inTal^e 1 (Exp^iment 1), except for the some-what greater grade-to-grade fluctuations onemight e]q>ect in the present smaller sample.An ANOVA was performed on the flrst grad-ers' data, as fbr Experiment 1, except thatschool was not a factor. Again, analog versus

Friedman and Laycock 367

TASK

digital, F ( l , l l ) = 10.14, p < .01, time, F(2,22) TABLE 6= 18.19, p < .001, and their interaction, », c « ^ ,-.v{ooo\ - QRQ ^ ^ m .,,«™ .,11 .i,r«;fi^™«* MEAN SPEARMAN RANK-ORDER CORRELATIONF(2,22) = 9-63, p < .01, were a I significant. COEFFICIENT FOR EACH GHADE ON THEAn interaction between time and order of tiie QRDER TASKS OF EXPERIMENT 2analog and digital tasks was also marginally ==^^^^=^^^^^^^=ss=^^^=^^^significant, F(2,22) = 4.05, p < .04. Plannedcomparisons between the X:00 and X:30 andthe X:30 and X:43 analog means failed to rep- GRADE ActivityIicate the significant differences found in Ex-

1 QA4'

periment 1. A second ANOVA examining the „ , X *grade differences in the analog X:43 problems g , •««,revealed highly significant grade and linear 4 gg*grade effects, F(4,74) = 8.15, p < .001, F(l,74) 5'.'.'.'.'.'.'.',.', LOO*= 21.66, p < .001, respectively. This findingreplicates tiie results of Experiment 1.

Digital Integrated

!55*91"91

TransformationProportion correct.—The bottom half of

Table 5 shows the mean proportions correctin the transformation task. Again, the levelsand pattems are consistent with those seen inExperiment 1 (Table 1). The ANOVA repli-cated tiie grade effect, F(4,60) = 16.22, p <.001, but sex, order, and time were not signifi-cant, as they had been in Experiment 1. It isimportant to note that analog versus digitalwas not significant in either study, showingthat digital clocks are easier for detection butnot for transformation.

Two of the original three two-way in-teractions were significant: time x grade,F(8,120) = 2.10, p < .05; and grade x analogversus digital, F(4,60) = 4.07, p < .01. Al-though the analog versus digital x time in-teraction was not replicated (p < .07), wetested its most interesting component, the an-alog versus digital difference in X:23 versusX:50 problems, and this comparison was sig-nificant, F(l,75) = 5.43, p < .03. This con-firms the finding that X:50 problems are eas-ier than X:23 problems in the analog but notdigital conditions. No other effects weresignificant.

Order TaskTable 6 shows mean Spearman rank-

order correlation coefficient between the or-ders that children produced on the order tasksand the correct orders. Scores on the two inte-grated tasks were averaged because they re-vealed similar age trends. These results indi-cate that children know the order of familiardaily events at least by first grade, whereasdigital clock times cannot be interrelated bymost children until second grade. The inte-grated sets of activities and times were or-dered at above chance levels by first gradebut continued to increase in accuracy untilthird grade.

* p < .001 by one-tailed t test that r = 0,

Eighty percent ofthe first graders' errorson the digital task involved placing the cardsin numerical order (1, 3, 9, 11), but this re-sponse accounted for fewer than half of theerrors by older children. Because the numeri-cal order is correct for some intervals (halfdays beginning at noon or midnight) and be-cause the acceptable interval was specified interms of activities (waking and dinner) itcould be argued that the first graders' deficitwas in translating the boundary activities intoclock times and not in constructing a correctorder for moming and aflemoon. However, asthe mapping task reveals, half or more of evenfirst graders can give reasonable clock timesfor waking and dinner, so this is unlikely toaccount for the very low levels of accuracy inthe first-grade group.

A repeated-measures ANOVA comparedthe three order scores as a function of grade.Task, F(2,150) = 52.16, p < .001, grade,F(4,75) = 15.14, p < .001, and their interac-tion, F(8,150) = 10.41, p < .001, were all sig-nificant. A planned comparison showed theactivity set to be significantiy more accuratethan the other two sets, F(l,150) = 29.20, p <.01, and the significant interaction supportsthe description that the activity set did notshow the developmental increases of theother two.

Mapping TaskResponses to the mapping task were

scored according to a schedule of times weconsidered plausible (e.g., get up: 6:00-9:00A.M.; dinner: 4:30-7:30 P.M.). For a responseto be scored correct, children did not have tospecify A.M./P.M. or a part ofthe day. However,if they did so in the activities-to-times subtaskthe combination needed to be correct. Inter-

368 Develoinnent

TABLE 7

MEAN PROPORTION GORRECT FOB EACH GHADE ON THE MAPPING TASKS IN EXPERIMENT 2

GRADE

STIMULUS TIME

2:00 4:00 6:00 8:00 10:00

Time-to-activities task:1 192 633 81**4 1.00**5 1.00'*

Mean 73

.19

.75**

.75**

.75**

.75**

.64

.37

.69*

.81**

.88**

.75**

.70

.62

.81**

.87*'

.81**1.00**

.82

.75**

.69''

.94**

.88**1.00**

.85

STIMULUS ACTIVITY

Arrive atDinner School

Activities-to-times task:1 50** .63**2 69** .81**3 87** 1.00**4 88** 1.00**5 87** 1.00**

Mean 76 .89

NOTE,—Entries are subject to rounding error.* p < .06 by binomial test i^unst estimated chance level for the task.** p < .01 by binomial test against estimated chance level for the task.

Go toBed Get Up

.93

Lunch

.87**

.94**

.88**

.94"l.OO'*

.56**

.87'*

.94**1.00**1.00**

.37*

.88**

.75**l.OO**

.94**

.79

observer agreement was .95 for the subtask inwhich the stimuli were times and .97 fbr thesul^ask in which activities were presented.The first rater's assignments were used.

Table 7 shows tiie results fw the two sub-tasks. The significance levels refer to one-tailed binomial tests using estimates ofchance levels of perform^ice for the null hy-potiiesis. These estimates were obtained byscorii^ as coirect or incorrect rand<»n combi-nati(»is of mismatched responses to problemsfbr half of the children (e.g., an answer to thedinner question was scored as if it has beengiven to the lunch question). Ghance levelestimates were .41 correct for tiie times-to-activities subtask and .14 for the activities-to-times subtE^k. Even by first gr^e , most chil-dren knew tiie times of several activities andcould name likely activities jbr some, but notall, times. Second graders had mc^e com-plete knowledge of time-activity correspon-dences. A sepmr^e tabulation of tiie numberof d i i l ^ n correctly ^)edfyi!ig A.MJP.M. eitiierspqai&uieously or in respcmse to tiie fi^low-upquesticm in the times-to-activities subtask (in-terobserver agreement: .94) showed tiiat amean of half or more children were correctby second

DISCUSSION

£j^)eriment 2 replicated the main find-ings for accuracy in the first experiment,^ a i n , except for andog times to the minute,clock r e ^ n g occurred well before childrencould perform operations, and the digkal ad-vantf^ for reading did not extend to thetntBsfbnnation task. We also repeated tiiefinding that relative problem difBculty diJ fersfor analog and d % ^ transformation prob-lems. Children seemed to bike advantage ofspatial sdlutions to the analog X:50 prob^ms,rendering them easier than adding 30 min totile non-5-min multiple times.

Among the new findings, the youngestgroup could link many times Emd activities.This ability precedes £uxnir»te ordering ofclock times or inte^^ated sets of activities anddock times, prol»^ly because individud as-sociative purs are acquired before representa-tions of order that indude CI<K^ times. But notall represent^joi^ of ord^ are late acquisi-tions: consistent witii previous studies (Fried-man, 1977; Muto, lS^ ) , first graders werevery £ux:urate in cvdiering canls representingdaily activities. Tlie delay between order-ing daily activities and ordering clock times

Friedman and Laycock 369

shows that hours are not the first code for rep-resenting daily order.

General DiscussionThe gradual acquisition demonstrated in

this study shows the importance of distin-guishing different components of clockknowledge and of describing the multiplerepresentations and processes that underlie it.Children apparentiy bring to the task of clockleaming knowledge of a number of associa-tions between times and activities as well asrepresentations of when daily activities occurrelative te one anotiier. By first grade theyapply number reading skills to digital dis-plays and to certain landmark analog configu-rations. In the next year or two, children de-velop representations of the relative times ofoccurrence of hours ofthe day. By third grademost children have acquired considerablesldll in reading the large number of analogtimes to the minute, though these problemsremain difficult at least through HfUi grade.The ability to operate on clock times appearsrelatively late for both analog and digital dis-plays, with some success by tiiird grade butslow improvement in the following years.

We suggested earlier that procedural se-quences, retrieval, associations, and orderrepresentations may all play a role in clockknowledge, and we are now in a position toconsider these claims. The clearest evidenceis for the role of procedural sequences in ana-log detection and both analog and digitaltransformation. The reported methods paint apicture that is consistent with that of Case etal.'s (1986) analysis of adults' and Siegler andMcGilly's (in press) analysis ofchildren's ana-log detection. Not only is tiie relative diffi-culty of different problems quite similar inthe three studies (e.g., whole hours are easierthan 5-min multiples, which are easier thantimes to the minute), but there are strikingsimilarities in strategy reports. For example,all three studies found strategies involving in-crementing the minute value by 5-min multi-ples.

These findings support the role of pro-cedural processing in analog detection. Caseet al.'s (1986) and Siegler and McGilly's (inpress) results show a number of effects thatappear to be the same for children and adults,such as a right-sector advantage for certaintimes and slower responses for minute read-ings tiiat are farther fit>m 5-min multiples. It isalso worth noting that slow responding ontimes to the minute, both in relative and abso-

lute terms, is preserved into adulthood: Sieg-ler and McGilly's second and tiiird graderstook about twice as long to solve times to theminute as whole hour times, our fiftii graders(in data not reported here) took more thanthree times as long, and Case et al.'s adultsubjects took about twice as long (still about 3sec). These results suggest that the processesunderlying analog detection remain funda-mentally unchanged after childhood.

The present study also showed tiie likelyinvolvement of procedural processing in thetransformation tasks because reported meth-ods seemed to refiect multistep operations.We also found evidence that the operationsoften differed for analog and digital displays,even though the starting times were givenaloud for both. Digital displays seemed toevoke mainly numerical approaches, such asaddition or incrementing by decades, thou^analog mediation was occasionally reported.Analog problems evoked a mixture of ap-proaches, including many tiiat capitalized onthe spatial properties of the display.

There is less evidence conceming therole of retrieval and associations in clockknowledge. The most direct evidence for re-trieval-like processes came from the frequentreports of directiy knowing the answer ontiiese problems. Siegler and McGilly (inpress) reported similar findings for tiieirwhole-hour (analog) problems. They believethat performance on certain problems isbased on accessing associations between par-ticular configurations and time names. How-ever, neither their methods nor ours are ade-quate to precisely characterize the underlyingprocesses.

Indirect evidence for the involvement ofassociative processes in time knowledgecomes frota the finding that clock times canbe linked to activities before they can be or-dered themselves. This seems to indicate thatsome type of representation containing rela-tively littie relational information is involvedin the early knowledge.

The case for qualitatively different repre-sentations of order is purely circumstantial.Preschool children possess reliable knowl-edge of the order of daily activities (Fried-man, 1977; Friedman & Bmdos, 1988; Muto,1982; Nelson, 1986), 10-year-olds to adoles-cents represent week and month order (Fried-man, 1986), and in between we find that sec-ond and third gradei^ leam to order the hoursof the day, to relate them to landmark activi-ties, and to distinguish A.M. times from P.M.

370 Devek^nnent

times. It seems likely that clock times, likedaily activities, months, and days ofthe week,can be "localized" t h r o i ^ processes more di-rect than pn)cedural sequences and on tiie ba-sis of represent^ons richer thfui associativepfurs, but such a conclusion must await fur-tiier research.

In tiiis study we used age differences todistinguish between different components ofclock knowledge. But one can also ask whatdeveli^mental £u^rs are responsible for thegnuiual acquisition of tiiis knowledge. We as-sumed tiiat associ^ve links and retrieval-based processes would be acquired at rela-ti %Iy early ages, procedural sequences at rel-atively late ages, and order rei»esentations at£^s determined by tiie number of elementsrepresented and the number of exposures tothe sequence. Two other accounts of the de-velt^Hnent of clock knowlec^ are also com-patible witii many of our findings.

Case et al. (1986) believe tiiat under con-ditions of spontaneous acquisition, children'sunderstanding of the clock system will belimited by tiieir status in a series of age-related stages, each characterized by tiie num-ber of dimensions or variables that can beevaluated in a problem. Aside from tiie diffi-culty of determining objectively what is a "di-mension" in some new domain, tiie model isgenerfdly successful in predicting the order inwhich diildren can extract infixrmation abouttiie hour hand, hour hand and 5-min multi-ples, and these two plus 1-min increments. Aswe saw earlier, the developmental sequenceof whole hour times, 5-min multiples, andtimes to tiie minute has been amply demon-strated. Their model is even stronger in that itpredicts not only the sequence but tiie ages atwhich different levels of proficiency areachieved. The model may be of somewhatlimited value for the present results becauseit is not explicit about when associations be-tween times and activities should be present.It could be argued, tiiou^, that these arecomp^ble witii "polar dimensional (global)thoi^it," in which case the m£^[ring per-fomiance of oax first graders would supporttiieir theory. Perhaps most troublesome is thedifficulty of knowing at what age childrenshould be able to orcbsr daily activities, clocktimes, days of the week, or months. It is notobvious how Case et al.'s mot^l anctlyzeswhat we have called oider r^H%sentations,nor how it would e x f ^ n the fact tiiat chil-dren appear to leam to order elements ofthese stmctures at ages rangii^ from about 4years for daily activities to 10 years for days ofthe week and months.

Siegler and McGilly (in press)the strategy-choice model (Sieger &1984) to analog clock reading skill. Theybelieve that children reEui clocks by uaing amixture of retrieval and backup str^egies.Altiiou^ tiieir study does not test age <^Eer-ences, tiie general model predicts that olderchildren will rely more on retrieval becausethrotij^ pfitctice they will develop morepeaked associations to particular problems(see Sieger & Shrager, 1^4). However, tiiesubstantial differences between adults' re-sponse times for whole hour times and timesto the minute (Case et al., 1^6) make it un-likely that retrieval replaces procedural ^ -proadies on all prt^ems. More likely, giventhe large number of clodc times to the minute,retrieval merely streamlines one or more con-stituents of wlmt for many problems r^nains apartially procedund process. Tliis would beanEd< cms to the rouUnization of sum retrievalin muhiplace addition but adults' continuedreliance on proceduml sequence. In addi-tion, as for Case et al.'s model, it is imclearhow well tiie strategy choice/distribution ofassociations model can explain the develop-ment of the ability to relate cMferent times toone anotiier.

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