When Days Are Numbered Calendar Structur

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When Days Are Numbered: Calendar Structure and

the Development of Calendar Processing

in English and Chinese

Melissa K. Kelly and Kevin F. Miller

University of Illinois at Urbana–Champaign Ge Fang

Institute of Psychology, Chinese Academy of Sciences, Beijing, People’s Republic of China and

Gary Feng

University of Illinois at Urbana–Champaign

Unlike English, Chinese uses a numerical system for naming months and days. This study explored whether this difference in naming affects the development of simple calendar calculation. Eight- and 10-year-old children as well as undergraduates in China and the United States were asked to name the day or month that comes a specified time before or after a given day or month. In each age group Chinese speakers primarily used calculation based on calendar names to solve these tasks, while English speakers primarily resorted to reciting the names. The magnitude of these differences was substantial; on difficult tasks Chinese fourth graders performed at speeds comparable to those of English-speaking adults. Implications for models of how linguistic structure affects cognition are discussed. © 1999 Academic Press

Key Words:symbolic development; cross-cultural research; language and cognition; calendar processing.

This research was supported by NIMH Grants K02MH01190 and R01MH50222 to the second author. The authors thank the students and staff of Leal Elementary School in Urbana, Illinois, and of the Beijing Institute of Foreign Languages Elementary School in Beijing, China. Renee Baillar-geon, Judy Deloache, Gregory Murphy, and Brian Ross provided helpful comments on an earlier version of this article. Special thanks are due to Shiou-yuan Chen and Xiuhong Cao for help with coding strategy data.

Address correspondence and reprint requests to Kevin F. Miller, Department of Psychology, University of Illinois at Urbana–Champaign, 603 E. Daniel, Champaign, IL 61820-6267. E-mail: kevinmil@uiuc.edu.

0022-0965/99 $30.00

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Thinking about domains such as time, number, and language is intimately connected with the symbol systems— calendars, numbers, orthographies, etc.— used to represent those domains. In each of these domains, children master a conventional set of symbols that are then used in the course of mundane problem solving. The difficulties that children and adults have in thinking about a domain such as time may result from the complexity of time as a concept and/or from general cognitive limitations, but they may also reflect the idiosyncrasies of particular calendar systems. The research described here examined whether cross-linguistic variation in calendar systems affects the developing ability to solve simple problems involving calendar time.

Symbol Structure Effects on Cognition

The structure of symbol systems can affect cognition in different ways at different levels of competence (Miller & Paredes, 1996). Looking at early symbolic development, studies have shown that the organization of a symbol system may facilitate or retard children’s learning of that system. For example, Miller, Smith, Zhu, and Zhang (1995) found that differences between Chinese and English in the organization of number names make the acquisition of number names differentially difficult for Chinese and English speakers. Chinese number names present children with a clearer base-10 structure than do English number names, and the simplicity and consistency of this structure are directly reflected in the relative ease with which children acquire number names in Chinese and English.

Effects of symbol structure need not be limited to initial acquisition. Although many processes become automatic with practice, there is some evidence that the structure of a symbol system can still affect the problem solving of competent symbol users. Studies of mental abacus calculators, who perform mental calcu-lation using an image of an abacus (Hatano, Miyake, & Binks, 1977; Hatano & Osawa, 1983; Stigler, 1984), have reported that abacus calculators show a different pattern of errors than non-abacus users, indicating a greater likelihood of making mistakes that reflect the misplacement of a bead on an abacus. Miller and Zhu (1991) found reflections of the structure of number naming systems in a task in which adults were required to name the reverse of a two-digit number presented on a computer screen. Although ordinary naming of two-digit numbers had long since become automatic, adult English speakers had difficulty reversing two-digit numbers ending in 1 (e.g., saying “14” when shown “41”), reflecting the idiosyncratic rules that English employs for naming teen numbers. Chinese speakers showed no particular difficulty reversing such numbers, a result that had been predicted based on the linguistic structure of number names in the two languages.

Abacus calculation and reverse number naming both involve unusual skills that are unfamiliar to ordinary speakers of a language. Studies of such skills provide some evidence that the psychological impact of the organization of

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symbol systems is not limited to initial acquisition, but can continue to affect the processing of proficient symbol users. These studies do not, however, indicate whether such effects occur on familiar and routine problem solving. The research reported here used a familiar task in order to see how the structure of calendar systems affects the developing ability of native speakers of different languages to perform simple calendar calculations.

Time Symbols and Concepts

By age 5 years at the latest, children spontaneously use a strategy of counting in rhythm to measure durations of events (Levin & Wilkening, 1989). Conven-tional time and calendar systems provide children with a consistent set of units to use for this purpose. Yet research on children’s mastery of calendars has shown that acquisition and use of these conventional systems are not easy tasks. Friedman (1983, 1984, 1990) has conducted the most systematic studies of developmental changes in children’s ability to reason about conventional time. Friedman (1990) described a model of mental representations of time in which an initial list-based representation is gradually supplemented by an analog spatial representation that incorporates information about distance between different months or days. According to Friedman, children initially learn lists of the names of the days of the week and the months of the year; thus, determining which day comes 3 days after Monday requires them to recite the list in order to obtain a precise answer. As children get older they begin to develop and use a visual image of the calendar that permits them to determine the relative distance between dates without calculating the exact distance.

Results from Friedman’s (1986) developmental studies indicate that it is not until after the age of 10 that children begin to report using imagery to help solve calendar problems. Results of tasks asking children to identify which of 2 days or months comes first after a given day or month going forward or backward in time clearly indicate that through 4th-grade children utilized a verbal list strategy while from 10th grade on children utilized an imagery strategy as well. Additionally, Friedman reported that children younger than 4th graders were not capable of solving calendar problems requiring them to think backward in time. He attributed this in large part to the difficulty of using a verbal list strategy in reverse.

Where adults are required to produce an exact answer (i.e., determining which day comes 4 days after Tuesday), even adults persist in using a verbal list strategy (Friedman, 1983). It is important to note that Friedman’s (1990) model was based on results from English speakers. It is possible that a calendar system in a different language would yield different mental representations and support different methods for problem solving. Different calendar systems may present children with different problems along the way to acquisition, and they may also affect the difficulty with which children acquire efficient strategies for calendar calculation. The calendar systems used in modern English and Chinese differ in the ease with which they lend themselves to the use of numerical strategies for calendar problem solving.

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English and Chinese Calendar Systems

The seven day names.Modern Chinese and English both use the same 7-day week and the same 12-month solar calendar. Although the calendars are isomor-phic, in the sense that each weekday and month in one calendar has a counterpart in the other language, Chinese and English differ in the ways in which names for the individual elements are formed, as shown in Fig. 1. English names for the days of the week derive from the ancient gods who were believed to rule each particular day (Boorstin, 1985; Zerubavel, 1985). The seven planets of ancient astronomy (including the sun and the moon, but not the Earth, Neptune, Uranus, or Pluto) were also named for the same gods, but the relation is obscured in English because planet names are derived from the Latin name for these gods whereas day names for the most part come from the equivalent Germanic god. Chinese names for the days of the week are generally formed by combining the word for week,xing qi(literally “star period”), with the cardinal number for the day of the week, beginning with Monday. Thus the Chinese term for Thursday is xing qi si ( “week four”). The one exception is Sunday, which uses the characterri(“sun” or “day”) instead of the number 7.1

The 12 month names.Names for the months of the year in English (Boorstin, 1985; Grove, 1986) are based on a melange of names for gods, Caesars, and Latin number names (see Fig. 2). Although later month names are derived from a numerical system, these numbers are those used in a different language (Latin), and they no longer correspond to the conventional numbering of the months (deriving from an early Roman 10-month calendar). Thus, although there is a derivational structure to some English month names, it is not one that either children or adults are likely to be able to use in problem solving. Chinese month names follow a simple numerical structure consisting of the cardinal number corresponding to a particular month, starting with January, plusyue(“month”). Thus the Chinese name for March issan yue(“three month”).2

This structure is used consistently for all 12 month names.

The numerical structure of the Chinese calendar lends itself to the use of arithmetical strategies for performing simple calendar calculations. If one wanted to determine what month comes 7 months after January (“one month”), one could add the number of months (7) to the original date (1) and determine the answer (8, or August). This calculation is more complicated for problems that cross the boundary of the year or week (e.g., the month 7 months after October is not “month 17,” but rather 17 modulo 12, or May (5)). Similar procedures hold for the days of the week as well, although the Chinese name for Sunday violates the overall numerical structure, as described previously.

1_{Chinese week names can also be formed by replacing}

xing qiwith either li bai(“worship service”) orzhou(“cycle”), but the basic structure of a numerical week, with an exception for Sunday (eitherri,“sun/day,” ortian,“heaven/sky”), remains the same.

2_{Because Chinese is a classifier language, there is no confusion between}

san yue(“three month,” or March) andsan ge yue(“three [classifier] month,” or three months).

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FIG. 1. Names for days of the week in Chinese and English. English names derive from the seven planets of ancient astronomy. Chinese names generally are a numerical list, consisting ofxing qi(literally “star period”) plus a number (1– 6). The name for Sunday substitutesri(“sun” or “day”) for the number 7.

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FIG. 2. Names for months in Chinese and English. English names derive from a mixture of Roman gods, the Roman emperors Julius and Augustus Caesar, and Latin number names held over from an earlier 10-month year. Chinese names are a numerical list, consisting of a number (1–12) plusyue(“moon” or “month”).

KELLY

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Some prior research has suggested that Chinese speakers can take advantage of the numerical structure of the Chinese calendar system. Jiang and Fang (1997) studied the ability of Chinese elementary school children and adults to determine the distance between two days of the week and reported that Chinese children showed a different set of strategies than those reported in Friedman’s research, with a greater tendency to use numerical calculation. Huang (1997) studied performance on a month calculation task by Chinese college students who had grown up in rural settings, where the traditional and nonnumerical lunar calendar is still widely used. These adults also showed evidence of using the numerical structure of the solar calendar when solving problems presented in that format. The present study provides the first direct comparison of the development of English speakers’ and Chinese speakers’ ability to perform simple calculations involving the months of the year and the days of the week. In general, we expected that Chinese speakers would take advantage of the numerical structure of their calendar, while English speakers would rely on the list structure of the English calendar system. This in turn would lead Chinese speakers to be faster and show different error patterns than American children of the same age. Users of numerical and list-based strategies should also differ in the relative difficulty of problems that cross the week or year boundary relative to those that do not, because this requires extra calculation with numerical strategies. Finally, they should differ in the relative difficulty of tasks that require them to calculate forward (into the future) versus backward (into the past). The difference between addition and subtraction of small numbers is likely to be much smaller than the difference between reciting a list in a familiar forward order and reciting it in a backward order (e.g., counting months backward).

Comparing calculation with both days of the week and months of the year is important for two reasons. First, the Chinese system for naming months of the year is consistently numerical, while the system for days of the week includes an exception (the name for Sunday). Children may be more likely to use a calcu-lation strategy that works consistently, or they may be more successful when they do so. Second, English has a conventional numbering for the months of the year, starting with January, that is familiar to adults who engage in writing checks and filling out forms. There is no corresponding conventional ordering for the days of the week. Thus adults and older English-speaking children might adopt a numerical strategy when dealing with problems involving months of the year, but they would be unlikely to do so with days of the week.

METHOD

Participants

A total of 196 participants (28 second graders, 32 fourth graders, and 31 adults in China; 32 second graders, 33 fourth graders, and 35 adults in the United States) took part in one individual session lasting about 30 min.

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university in Beijing, China. The second-grade participants (14 females, 14 males) ranged in age from 7;10 to 9;2 (M 58.39,SD5 .36) years of age. The fourth-grade participants (16 females, 16 males) ranged from 9;11 to 10;11 (M5

10.4,SD5.29) years of age. Adult participants in China (15 females, 16 males) were recruited from Beijing Normal University and were paid the equivalent of about 1 U.S. dollar for their time. All Chinese participants were native speakers of Mandarin and were recruited from schools serving populations almost entirely members of the dominant Han ethnic group.

Children in the United States came from the second and fourth grades of a public school serving an academic community in a university town in the Midwest. The second-grade participants (20 females, 12 males) ranged in age from 7;6 to 8;9 (M57.99,SD5.36) years of age. The fourth-grade participants (19 females, 14 males) ranged in age from 9;2 to 10;7 (M5 9.93, SD5 .32) years of age. Adult participants (19 females, 16 males) in the United States were introductory psychology students at the University of Illinois who participated for course credit. Children were recruited from a school with a population approximately 75% Caucasian, 20% African-American, 2% Hispanic, 3% Asian, and 1% Native-American. Adult participants were recruited from a subject pool with a population approximately 70% Caucasian, 11% Asian, 7% Hispanic, and 6% African-American.

There were 12 additional U.S. participants who were dropped from the study. Eleven U.S. second graders were dropped; 3 could not read the calendar names well enough, 3 did not know the months of the year, 3 lost interest and did not complete the task, and external noise interfered with data collection for 2 participants. One U.S. fourth grader was dropped for not knowing the months of the year.

Stimuli

Stimuli consisted of the names of the months of the year and the days of the week presented in six trial blocks. Each trial block consisted of stimuli from one of the six conditions: weekday naming, month naming, weekday forward, week-day backward, month forward, and month backward. Stimuli in the two naming conditions were presented alone on the screen. As a mnemonic cue, in the last four calculating conditions a line was presented to the left or to the right of stimuli to help the participants remember which task they were performing. The line was to the right of the day or month name for forward tasks and to the left for backward tasks.

Each participant was presented with the weekday naming task first, the month naming task second, and then the four calculating tasks. The two naming conditions consisted of one block of weekday names and another of month names. Within each block every weekday and month name was presented twice. Thus, the weekday naming task block had 14 trials and the month naming task block had 24 trials. Order within the blocks was random, but all participants saw the same two blocks.

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The four calculating conditions consisted of two blocks of weekday names and two of month names. However, within each block each weekday and month name was presented only once; weekday conditions had 7 trials and month conditions had 12 trials, with each of the four tasks preceded by a practice block of 4 trials. Including naming and practice trials, each participant experienced 92 trials altogether. Order within the blocks was random and the order of the blocks was counterbalanced across participants, but the participants all saw the same four blocks. Participants were assigned to one of four task orders crossing list (months first or days first) with task order (forward calculation first or backward calculation first), counterbalanced across participants within each grade 3 sex combination.

Apparatus

Stimuli were presented on Macintosh Powerbooks (Models 170 and 180). Reaction times were collected using microphones attached to a PsyScope button box which allowed timing to the millisecond by a program generated using PsyScope (Cohen, MacWhinney, Flatt, & Provost, 1993).

Procedure

Adult participants were tested individually in a small experimental room. School-aged participants were tested individually in extra rooms at their schools. All participants were asked by the experimenter to recite the days of the week and the months of the year before testing began. During the computerized tasks, the experimenter recorded any errors or instances in which the microphone did not register or incorrectly registered a response.

All participants began with the weekday naming task followed by the month naming task, which served as warm-up tasks and ensured that children were familiar with the names of the days of the week and months of the year. Participants were asked to read the word on the screen as quickly as possible without making mistakes. Each trial in the naming conditions began with 500 ms of blank screen followed by a fixation point, which remained on the screen until the participant pressed the center button on the button box. A 250-ms blank screen was immediately followed by the stimulus, which remained on the screen until a voice response was recorded.

Each of the four trial blocks (Months/Days3Forward/Backward) began with instructions presented on the computer screen and also read out loud by the experimenter. The experimental task was to help a farmer figure out when he should expect his flowers to sprout or blossom (forward conditions) or to decide when he needed to plant flowers (backward conditions). Participants were told that the farmer’s flowers take 4 days to sprout and 7 months to blossom. In the forward calculation tasks participants were asked to figure out the day/month the flowers would sprout/blossom if they were planted on a given day/month. In the backward calculation tasks participants were asked to figure out the day/month

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the farmer should plant the flowers in order to have them sprout/blossom on a given day/month. A reminder card which said either “4 Days” or “7 Months” depending on the condition was left in sight of the participant to reduce confusion between trial blocks. As a further cue, stimuli in the forward conditions were presented on the left side of the screen, with a horizontal line on the right representing the answer. In the backward conditions stimuli were presented on the right side of the screen, with a horizontal line on the left representing the answer. Feedback was provided on each trial by replacing the horizontal line with the correct answer as soon as the participant responded.

Following instructions for each task, the participant was asked to explain the task to the experimenter and then was given two practice problems. Strategy data were collected for these problems by asking participants to describe how they solved them. Participants then received four more practice trials on the computer before the experimental trials began. At the end of the experiment, participants were asked to describe any additional strategies that could be used to solve the calculation tasks.

Each trial in the forward and backward conditions began with 500 ms of blank screen followed by a 1-s fixation point followed by 500 ms of blank screen. The stimulus then appeared and remained on the screen until a voice response was recorded. At this point the correct answer replaced the blank line next to the stimulus on the screen and remained there until the participant pressed the center button on the button box.

RESULTS

Reaction Times

Naming times.Times for naming the months of the year and the days of the week were predicted not to vary between countries. Figure 3 shows naming times for days of the week (left) and months of the year (right) by country and age group. Reaction times for correct responses were analyzed by a Country (2: United States, China) by Grade (3: 2, 4, Adult) by Gender (2: M, F) by Task Order (4) by List (2: Days, Months) MANOVA, with List a within-subjects factor. The only significant effects were a general improvement in naming speed with age (F(2, 140)512.07,p,.001) and a finding that months were named faster than days (F(1, 140) 5 18.18,p, .001). Speed of naming days of the week and months of the year did not differ between the U.S. and Chinese participants, nor were there significant interactions involving Country for naming time. Although the Country by Age by List interaction was not significant, it is noteworthy that Chinese second graders were quite a bit faster at naming months

FIG. 3. Mean reaction times for the naming tasks by country and age. Time to name weekdays is shown on the left; time to name months is shown at the right.

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than were their American peers. Acquiring a regular system of names such as those embodied in the Chinese calendar system that builds on a familiar list (the numbers) is much easier than learning the English calendar system (Miller, Kelly, & Fang, 1999), so this difference is not surprising. The focus of the current study is whether differences in using the calendar systems persist after children in both countries have become familiar with their language’s calendar names. For the other age groups and tasks, there were no differences in the speed with which participants could access the name of calendar elements. Thus, differences in calendar calculation should not be the result of overall differences in the acces-sibility of the day and month names in the two calendar systems.

Days of the week.Figure 4 presents results for calendar calculation using the days of the week, with the forward (left) and backward (right) results presented separately. Reaction times for correct responses for the weekday calculation tasks were analyzed in a Country (2: United States, China) by Grade (3: 2, 4, Adult) by Gender (2: M, F) by Task Order (4) by Direction (2: Forward, Backward) MANOVA, with Direction a within-subjects factor. Main effects of Country (F(1, 139) 5 56.21, p , .001), Grade (F(1, 139) 5 59.25, p , .001), and Direction (F(1, 139)556.81,p, .001) were found. Chinese participants were consistently faster than U.S. participants, the forward task (calculating days ahead rather than days before) was always faster than the backward task, and older participants were faster than younger participants. As Fig. 3 suggests, there were also significant interactions between Country and Grade (F(1, 139) 5

12.85,p,.001), between Country and Direction (F(1, 139)527.46,p,.001), and between Grade and Direction (F(1, 139) 5 14.53, p , .001). These interactions, as well as those discussed later, were assessed by a series of tests of simple effects, with Bonferroni adjustment ofa levels to reflect the number of

comparisons made (Keppel, 1982; Stevens, 1986). The difference between the forward and backward conditions was particularly pronounced for the U.S. participants, and these differences were much larger for the two younger groups of U.S. participants than for the adults. The improvement between U.S. second and fourth graders was much larger than for the corresponding Chinese groups, with the difference between second- and fourth-grade Chinese participants not reaching significance after Bonferroni adjustment. This pattern is consistent with our expectation that Chinese speakers would tend to use arithmetic to perform the calendar tasks, because reciting a list in an unfamiliar backward order is more difficult than switching from addition to subtraction.

Months of the year.Figure 5 presents results for calendar calculation using the months of the year, with the forward (left) and backward (right) results presented

FIG. 4. Mean reaction times for the weekday calculation tasks by country and age. Time to calculate the day 4 days after the date presented is shown on the left; time to calculate the day 4 days before the date presented is shown on the right.

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separately. As Fig. 5 shows, the overall pattern of results was similar to that found for the days of the week (shown in Fig. 4), with two interesting differences that may reflect the longer length of the list of months and the greater consistency of Chinese month names compared with weekday names. First, the magnitude of between-Country differences was much larger than for the days of the week. Second, the Chinese fourth graders were at least marginally faster than were the U.S. college students, despite the approximately 10-year difference in ages. Reaction times for correct responses for the month calculation tasks were analyzed in a Country (2: United States, China) by Grade (3: 2, 4, Adult) by Gender (2: M, F) by Task Order (4) by Direction (2: Forward, Backward) MANOVA, with Direction a within-subjects factor. Main effects of Country (F(1, 137) 5 115.71, p, .001), Grade (F(1, 137) 5 45.83,p , .001), and Direction (F(1, 137)536.89,p, .001) were found. Chinese participants were consistently faster than U.S. participants, the forward task was always faster than the backward task, and older participants were faster than younger participants. As with days of the week, there were significant interactions between Country and Grade (F(2, 137) 5 12.46, p , .001), between Country and Direction (F(1, 137)5 26.09,p,.001), and between Grade and Direction (F(2, 137)5

4.81,p,.01). In addition, there was a significant three-way interaction between Country and Grade and Direction (F(2, 137)55.19,p,.01). Calculation time decreased with age much faster for the U.S. participants, although as noted above the Chinese fourth graders were marginally faster than the U.S. adults. The difference between forward and backward directions was much larger for the U.S. participants than it was for their Chinese peers. The difference between forward and backward directions was particularly large for the younger U.S. participants, averaging 13.55 and 7.34 s for the second and fourth graders, respectively. These differences were consistent with the prediction that Chinese participants at all ages would rely on a fast calculation-based strategy, while U.S. participants were more likely to use a list-based strategy. Reciting a list of months or days of the week backward is quite difficult, particularly compared with the difference between addition and subtraction of small numbers.

For the month task, we also found unexpected interactions involving gender, with a significant interaction between Gender and Direction (F(1, 137)56.06, p , .05) and Country by Gender by Direction (F(1, 137) 5 7.90, p , .01). Separate MANOVAs considering each gender separately replicated all of the effects described above; the same pattern of Country effects and the same interactions were found when just the males and just the females were considered separately. Separate analyses of the two month calculation tasks by country and

FIG. 5. Mean reaction times for the month calculation tasks by country and age. Time to calculate the month 7 months after the date presented is shown on the left; time to calculate the month 7 months before the date presented is shown on the right.

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age showed that there was a significant gender effect favoring males on the month backward task in both countries (F(1, 175) 5 5.94, p , .05) and a significant Country by Gender effect for the month forward task (F(1, 174) 5

5.46,p,.05), with males faster than females in China and with females faster than males in the United States. Although statistically significant, the magnitude of the effects involving gender was relatively small when compared with the effects of language. These gender differences generally correspond to gender differences in strategy use and accuracy that will be presented in a later section of this paper, at which point we will further discuss the issue of gender differences on these tasks.

Within-boundary vs between-boundary calculations.A special feature of cal-endars—in contrast with numbers—is that they comprise repeating or modular lists. This has important consequences for using arithmetic strategies in the calendar calculation task. For example, adding 7 months to the 10th month on a month forward task would produce an answer of “17th” month. One must either convert the answer into its modulo-12 result (5th month) or else solve the problem by instead subtracting the 129s complement of 5 (that is, the number— 7—that when added to 5 results in 12). In either case, problems that cross the boundary of the week or year should be more difficult for users of calculation strategies than are other problems. There is no intrinsic reason to predict that reciting a list that crosses the same boundary would be significantly more difficult than would reciting the list that does not cross the boundary. We therefore predicted that cross-boundary problems would be more difficult for Chinese than for American participants, when compared with their performance on within-boundary problems.

Effects of crossing boundaries were assessed with a series of paired-samplet tests (with Bonferroni adjustment) for each age group within a country compar-ing calculation time for trials that required crosscompar-ing the year or week boundary with those that did not. For the Chinese participants, trials that crossed bound-aries were significantly slower than those that did not for the month list for all age groups, but no significant boundary-crossing effect was found for the U.S. participants at any age. The same boundary effect was found for the day list on the backward calculation task for the Chinese fourth graders and adults but not for the Chinese second graders, and none was found for the U.S. participants. For forward calculation using the day list, the difference between boundary-crossing and within-boundary problems was not significant for any group. Because of the longer list of months and the potential confusion between base-10 numbers and

FIG. 6. Accuracy rates on the calendar calculation tasks by country, age, and task. The top graphs show performance on the month task; the bottom graphs show performance on the weekday task. The left graphs show accuracy rates for calculation of the target item a given interval after the date presented; the right graphs show accuracy rates for calculation of the target item a given interval before the date presented.

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a modulo-12 system of names, it is not surprising that the effect of boundary crossing was greater for months than for weekdays. The finding that Chinese participants but not Americans were disrupted by problems that involved cross-boundary calculations is consistent with the prediction that Chinese speakers would use the numerical structure of their calendar in performing calendar calculations.

Accuracy.Looking at the accuracy data revealed some very interesting diver-gences between the two calendar lists in the accuracy with which participants performed the calendar calculation tasks. These differences reflected a slight but significant difference between the Chinese representations for the days of the week and months of the year, specifically the problems introduced by using a nonnumerical value for Sunday (xing qi ri). Figure 6 shows the error rate for each task by country and age.

For the month calculation tasks (top), American participants made consistently more errors than did Chinese participants. Individual proportions of correct responses (arcsine transformed) for the month calculation tasks were analyzed in a Country (2: United States, China) by Grade (3: 2, 4, Adult) by Gender (2: M, F) by Task Order (4) by Direction (2: Forward, Backward) MANOVA, with Direction a within-subjects factor. As Fig. 6 suggests, Chinese participants were significantly more accurate than were U.S. participants (F(1, 176)510.23,p,

.001), and accuracy improved with age (F(1, 176)₅11.85,p,.001). Accuracy was also higher on the forward calculation task than on the backward calculation task (F(1, 176)513.28,p,.001). As with the reaction time data, an unexpected interaction between gender and task was found (F(1, 176)5 11.81,p, .01). Overall, females were more accurate than males on the forward calculation task (17% errors vs 25% errors for the males) and slightly less accurate than males on the backward calculation task (29% errors vs 27% errors for the males). This gender difference in accuracy suggests that at least part of the gender difference in reaction time on the forward task may reflect a speed–accuracy tradeoff. The speed differences between countries on the month tasks cannot be attributed to such a strategy, however, because the faster participants were also more accurate. For the weekday calculation tasks (bottom of Fig. 6) a more complicated picture was found. For these tasks, the Chinese participants were not generally more accurate than their American peers, and the Chinese adults showed a particularly high error rate, performing worse on the weekday calculation tasks than did U.S. adults. In contrast to the U.S. adults, who were more accurate on the weekday task than on the month task, Chinese adults showed the opposite pattern. Individual proportions of correct responses (arcsine transformed) for the weekday calculation tasks were analyzed in a Country (2: United States, China) by Grade (3: 2, 4, Adult) by Gender (2: M, F) by Task Order (4) by Direction (2: Forward, Backward) MANOVA, with Direction a within-subjects factor. U.S. participants were more accurate than were Chinese participants (F(1, 176) 5

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(F(1, 176)54.14,p,.05), which reflected the fact that only for adults were the U.S. participants more accurate than were the Chinese speakers. Accuracy improved somewhat with age (F(1, 176)₅11.85,p,.001) and was higher for the forward than for the backward direction (F(1, 176)521.53,p,.001). There was also a Gender by Task interaction (F(1, 176)55.02,p,.05). This reflected significantly more accurate performance by females on the day forward task (11% errors for females vs 16% for males) but not on the day backward task (21% errors for both genders).

Looking at the kinds of errors produced, invalid or impossible answers were quite rare, produced by only one American second grader and five Chinese second graders. In the United States, this occurred when one child read “sum-mer” instead of “September” in the month naming task. In China, these errors all occurred in one of the calculation tasks. One said “day eight,” one said “nineteen month,” and three said “thirteen month,” none of which are valid day or month names. The Chinese errors would all have been the correct answer if the Chinese week and year were not modular numerical systems (i.e., the correct answers should have been “day one,” “seven month,” and “one month,” respectively). They thus imply an overly general transfer of arithmetic strategies to the calendar calculation task.

Looking at the accuracy data provides evidence that transferring calculation techniques from numbers to calendars can lead to problems as well as to the large increases in speed already discussed. For both months and weekdays, the fact that these are modulo-12 and modulo-7 lists, respectively, led to some instances in which straightforward calculation does not produce a correct answer. In the case of the weekday list, the inconsistency provided by the use of a nonnumerical name for Sunday may account for the relatively high error rate among Chinese-speaking adults for the weekday calculation tasks.

Strategy use.Strategy data were collected for sample problems administered before the reaction time data were collected. Figure 7 shows the types of strategies used by country and age. The percentage of subjects who mentioned or were observed using each strategy at least once is shown. The overt reports of strategies are consistent with the data on calculation time and accuracy in suggesting that the structure of calendar names has an enormous effect on the strategies used in simple calendar calculations. By far the most common strategy for the U.S. participants involved reciting the days of the week or months of the year from the stimulus date to the target. This strategy was used on occasion by about half of the Chinese second and fourth graders but was virtually nonexistent among Chinese adults, while more than 90% of U.S. college students continued to use this strategy. An indicator of the relative difficulty of accessing the calendar names is shown by the much greater likelihood that U.S. participants would resort to holding up their fingers while counting off the days or months. This strategy was used by most U.S. participants at all ages but by only 14% of Chinese second graders and by even fewer of the older participants. Another

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indication of the difficulty of working with the English calendar system is the number of U.S. participants who resorted to strategies involving counting for-ward (by the 129s or 79s complement) in order to solve the backward-direction problems.

Whereas more than 90% of U.S. participants at all ages used a list-reciting strategy at least once, more than 90% of Chinese participants at all ages used arithmetic to solve the calendar tasks at least once. Only half of U.S. college students reported using a calculation strategy, and they were five times more likely to use calculation on the month task (47%) than on the weekday tasks (9%). This is not surprising, because English-speaking adults are familiar with the numerical equivalents of month names, but there is no corresponding con-ventional numbering for the days of the week in English. That only half of the U.S. college students ever used a calculation strategy was somewhat surprising, but it provides additional evidence that the presence of a numerical system of calendar names can have a direct impact on the cognitive processes people use in thinking about time.

For Chinese speakers, the likelihood of using calculation also differed between the month and weekday lists, but only for the second- (81% used calculation for months vs 63% for weekdays) and fourth-grade (91% for weekdays vs 72% for months) participants. All Chinese-speaking adults used calculation on the month task, and 94% reported doing so on the weekday task as well. Although we do not have data on strategies used for individual problems, the relatively high error rate for Chinese adults on the weekday task suggests that eschewing the calcu-lation strategy on weekday problems might be an adaptive strategy choice on the part of children.

Differences in strategy use between males and females also appear to account for at least part of the gender differences in speed and accuracy reported above. In both countries, female participants were more likely than males to use a list strategy and less likely to use a calculation strategy on all tasks. In general, list strategies are substantially slower than calculation strategies, but they are less likely to produce errors due to problems in transferring between a base-10 and modulo-7 or modulo-12 list. Thus the differential likelihood of using calculation versus list-recital strategies may account for the unexpected gender differences in speed and accuracy in this study.

Several other strategies were reported by small numbers of participants. A

FIG. 7. Percentage of participants who mentioned using a particular technique at least once by country and age. Strategies are: (1) reciting list, reciting a list of days/months; (2) fingers, using fingers to keep track of days/months; (3) reverse, reciting a list forward to solve backward-direction problems (e.g., finding the month 7 months before the stimulus by counting 5 months forward from the stimulus); (4) visualize, any strategy involving visualization of the months or days; and (5) memorize pair, any strategy involving direct retrieval of an answer. Because participants could report using more than one strategy, totals can exceed 100%.

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relatively small number of participants reported working with a visual image of the calendar system or remembering specific pairs of items, but these were never dominant strategies at any age. It is not surprising that the use of analog strategies was so low. Unlike the tasks used by Friedman (1983) involving estimation of relative durations, an analog strategy is unlikely to produce the exact answer required for the tasks employed here.

DISCUSSION

We predicted that differences in the structure of the calendar systems in Chinese and English would affect the online processing of even competent calendar users. Reaction time, strategy use, and error distribution results indi-cated there were clear differences between English and Chinese speakers. Chi-nese speakers of all ages performed the calendar tasks faster than English speakers of the same age. Furthermore, while the Chinese speakers showed a pattern of strategy use which remained relatively consistent across ages, the pattern for English speakers was of increasing the number of strategies used with increasing age. The different kinds and numbers of strategies used by speakers of the two different languages were reflected in differences in the errors made by the two groups and the speed with which the problems were solved. In general, Chinese speakers were much more likely than English speakers to apply calcu-lation strategies to solving simple calendar calcucalcu-lation tasks.

Online Processing Effects

Effects on reaction time.Chinese speakers at all ages were faster at solving the calendar problems than same-age English speakers. In fact, Chinese second and fourth graders solved the backward calculation tasks nearly as quickly as did American adults. Chinese speakers have two advantages over English speakers in making these calendar calculations: they have more direct access to information about sequencing and distance between weekdays and months because of the numerical nature of the names and, for the same reason, they have access to a much simpler strategy for solving the problems, arithmetic. The differences in calculation time across languages, particularly when English-speaking adults and Chinese-speaking children are compared, are quite remarkable given the perva-sive and substantial age differences in speeded tasks typically reported (see Kail, 1991, for a review of studies showing such differences in a variety of domains). Effects on strategy use.Siegler described the development of strategy use from the novice level to the expert level as a progression from a single-strategy stage to a multiple-strategy stage, with a final narrowing of strategies to the expert level, which primarily involves reliance on memory (Siegler, 1996; Siegler & McGilly, 1989). Our participants reported using strategies which run the gamut of Siegler’s strategy use descriptions. American second graders almost always used only one strategy. The majority of the American fourth graders also used only one strategy, but a few were beginning to try different ways to solve the

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problems. On the whole, American children relied exclusively on a verbal list strategy to solve the calendar problems. When they attempted to use another type of strategy they almost always used it incorrectly. While there was a decrease with age in the use of overt behaviors, like counting on fingers, to keep track of the weekdays and months, over half of the adults still used them. American adults also often used an intermediate approach to solving the problems. Although they all reported using a verbal list strategy, half of them also reported using other strategies like numerical equivalents. Among American participants, adults were far more flexible in their approach to the problems than the children. On the other hand, Chinese participants, including most of the youngest participants, had found numerical strategies that work well with the Chinese numerical calendar system, and these strategies persisted into adulthood.

Applying calculation to weekday tasks led to a comparatively large number of errors. In this case, the fact that Chinese children were significantly less likely to use calculation on the weekday than on the month tasks may reflect an adaptive strategy choice in light of the greater difficulty of using this strategy for weekday problems. Adults were much less likely to change strategies for the weekday task, although their error rate suggests this might not have been a wise choice. Further evidence of the difficulty that English speakers have in calendar problem solving was provided by the use of special strategies to shortcut reciting the list, particularly counting forward by the 12₉s or 7₉s complement to solve backward problems. The existence of this strategy underscores both the difficulty of solving calendar problems using the English calendar system and the flexi-bility of strategy discovery and strategy choice.

Conclusions.Looking only at the English speakers, the developmental pattern that emerges from this study is consistent with the model proposed by Friedman (1990). Effective use of a mental model comprising both a serial list and a visual representation to come up with a particular solution would almost certainly be facilitated by the use of multiple strategies. The developmental trend in the use of strategies mirrored the trend Friedman reports for mental representations of calendars. In both our study and Friedman’s, young children almost exclusively used a verbal list strategy. Adults were able to use a broader range of strategies. The picture is quite different for Chinese speakers. For them, there is no movement from one strategy to multiple strategies. Instead, Chinese speakers from an early age converge on a dominant strategy that persists into adulthood. Thus the developmental course of calendar calculation appears to be quite different for English speakers and Chinese speakers, with these differences closely related to the surface structure of the calendar system used in each language. Despite the fact that Chinese and English share a 12-month year and a 7-day week, differences in the two languages’ names for these days and months make different strategies more or less accessible to the speakers of the two languages. English speakers show quite a bit of flexibility and creativity in developing a variety of strategies for solving the tasks that we presented, while

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Chinese speakers generally assimilate calendar problems to familiar calculation procedures.

The exact nature of the relationship between language and thought has been a matter of debate and study for some time. Such research is often framed in terms of the claim of the Sapir–Whorf hypothesis that linguistic structures provide a template within which nonlinguistic experience is framed. Research looking for linguistic influences on nonlinguistic concepts and perceptions has yielded a very mixed and inconsistent picture of such influences (e.g., Lakoff, 1987; Lucy, 1992). In addition to there being an unsettled pattern of empirical effects, though, the questions posed by the Sapir–Whorf hypothesis may miss some of the most important ways in which linguistic structure contributes to cognitive develop-ment (Miller & Paredes, 1996). Symbol systems such as calendars are learned in order to serve as tools for solving basic problems such as those presented in this experiment. How such a system is organized has consequences for the ability of its users to perform the tasks for which it was acquired in the first place. This study demonstrated that the structure of calendar systems has a significant impact on the speed with which adults can perform basic calendar calculations, even after many years of familiarity with the calendar names of their native language. Results of this study are consistent with other work (e.g., Nunes, 1992) suggesting that the psychological impact of acquiring a cognitive tool lies less in general effects of learning some symbolic system than in the influence of specific features of particular symbol systems on the way that children think about problems relevant to that system. For example, Carraher, Carraher, and Schli-emann (1985) found that working-class children in Brazil distinguished problems to which they would apply informal oral arithmetic methods from those to which they would apply the formal algorithms learned in school, with different kinds of errors and misunderstandings associated with the use of each kind of algorithm. Thus, Nunes has argued that many of the changes in children’s mathematical thinking that follow upon learning school mathematics are limited to their understanding of the kinds of problems to which they apply such procedures, rather than leading to more global changes in mathematical reasoning.

Much of human cognitive ability rests on a foundation of culturally and linguistically determined symbolic tools. The research presented here shows for one specific domain that the structure of those tools can have a substantial effect on both the extent of developmental change within a domain and the specific course that it takes. When developmental psychologists study the acquisition of skills that use such symbolic tools, they must be sensitive to the difficulty in distinguishing phenomena that reflect the structure of the language the child is learning from universal aspects of cognitive development.

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