Where Does the Ordered Line Come From? Evidence From a Culture of Papua New Guinea

Abstract

Number lines, calendars, and measuring sticks all represent order along some dimension (e.g., magnitude) as position on a line. In high-literacy, industrialized societies, this principle of spatial organization—linear order—is a fixture of visual culture and everyday cognition. But what are the principle’s origins, and how did it become such a fixture? Three studies investigated intuitions about linear order in the Yupno, members of a culture of Papua New Guinea that lacks conventional representations involving ordered lines, and in U.S. undergraduates. Presented with cards representing differing sizes and numerosities, both groups arranged them using linear order or sometimes spatial grouping, a competing principle. But whereas the U.S. participants produced ordered lines in all tasks, strongly favoring a left-to-right format, the Yupno produced them less consistently, and with variable orientations. Conventional linear representations are thus not necessary to spark the intuition of linear order—which may have other experiential sources—but they nonetheless regiment when and how the principle is used.

Keywords

spatial concepts visual culture order number cognitive diversity open materials

Two millennia ago, Aristotle proposed a scheme for understanding nature: a ladder on which different beings were ordered, with the simplest on the bottom rung and the most sophisticated on top (Archibald, 2014). This ancient scheme embodies the principle of linear order, according to which the order of elements with respect to some dimension—such as simplicity, magnitude, or sequence—is represented by their positions in a linear arrangement. Today, the principle of linear order is instantiated in calendars, volume sliders, number lines, and scientific graphs, and even implicitly in written text (Ingold, 2007; Tversky, 2011). It is a taken-for-granted cornerstone of visual culture, at least in industrialized societies.

The principle of linear order also appears to occupy a privileged place in human cognition. Within high-literacy, industrialized societies, the ordered line is deployed spontaneously as an organizational strategy (Bergen & Lau, 2012; Fuhrman & Boroditsky, 2010; Tversky, Kugelmass, & Winter, 1991) and serves as a powerful reasoning tool (De Soto, London, & Handel, 1965). In fact, adults in such cultures operate with implicitly linear mental representations of number (e.g., Dehaene, Bossini, & Giraux, 1993; Restle, 1970), time (e.g., Ishihara, Keller, Rossetti, & Prinz, 2008; Núñez & Cooperrider, 2013; Price, 2009), and other ordered concepts (Gevers, Reynvoet, & Fias, 2003; van Dijck & Fias, 2011). Implicit notions of linear order are even evident in children before they can read (McCrink & Opfer, 2014; Nuerk et al., 2015).

Linear order thus appears to be a fixture of conventional representational practices and of human cognition. But what are the origins of this principle, and how has it become such a fixture? One possibility is that representational practices exhibiting linear order are ubiquitous because of the centrality of the principle in cognition. Conversely, the principle may be central in cognition because of the ubiquity of representational practices that exhibit it. Current evidence hints at a more complex relationship that goes beyond these chicken-or-egg extremes. For instance, in cultures whose members share an intuition about the orientation of ordered lines, this intuition mirrors representational practices, such as writing (e.g., Bergen & Lau, 2012; Shaki, Fischer, & Petrusic, 2009; Tversky et al., 1991; Zebian, 2005).

The precise relationship between conventional representations and intuitions about linear order is hard to disentangle in high-literacy, industrialized cultures. In such societies, it is difficult to rule out influence from representational conventions, even in preliterate children (Nuerk et al., 2015). The scant work that has been done with more isolated indigenous groups has focused on one instantiation of linear order in particular—the number line—and has presented a mixed picture. Dehaene, Izard, Spelke, and Pica (2008a) had Mundurucu (Brazil) participants perform a number-line estimation task in which they were asked to indicate the appropriate location of numerical stimuli (e.g., sets of dots) on a graphical line. Many Mundurucu participants mapped the stimuli on the line according to their magnitude, a mapping consistent with the principle of linear order. But a significant subset of participants produced bimodal responses, mapping small numerosities to one endpoint and large numerosities to the other (Dehaene, Izard, Spelke, & Pica, 2008b). Núñez, Cooperrider, and Wassmann (2012) performed the same task with unschooled Yupno (Papua New Guinea) participants. Strikingly, all responded in a categorical fashion, using only the endpoints of the line. Responses by participants in both cultural groups thus call to mind another strategy for organizing abstract relations: spatial grouping. Both linear order and spatial grouping have been argued to be fundamental spatial organization principles in both graphical practices and reasoning (De Soto et al., 1965; Tversky, 2011).¹ However, the possibility that grouping might rival or even eclipse linear order in some cultures has not been investigated.

Key questions about the origins of linear order thus remain open. First, it is unclear whether members of cultures that lack representational practices involving linear order do indeed have intuitions about the principle. The number-line estimation task cannot provide satisfying answers to this question because of its constraints: It imposes a particular spatial organizational strategy (i.e., a line), in a particular orientation (i.e., left to right), and assesses whether people can make sense of it. Imposing this graphical representation could spark intuitions about the ordered line; on the other hand, first establishing the extremes of the set may foreground the relationship between each additional stimulus and those extremes, inviting categorical responding. Second, to the extent that intuitions about linear order are evident in cultures that lack conventional representations involving the principle, further questions remain about when and how the principle might be used. Would people in such cultures use linear order to the same extent as people in the United States, or would they instead favor spatial grouping? And if they were to use linear order, would their lines take a consistent spatial form?

To address these questions, we examined spontaneous spatial representations of abstract relations in the Yupno and in U.S. undergraduates. In U.S. society, as in all contemporary industrialized societies, conventional representations exhibiting linear order abound. Yupno culture, by contrast, lacks calendrical systems, linear measurement tools, writing, or other conventional practices involving ordered lines, and many Yupno adults have not had formal schooling (for additional information, see the Supplemental Material available online and Wassmann, 1993). We used an open-ended task in which participants were asked to arrange sets of cards differing in the size or numerosity represented. This task allowed us to see when people recruited linear order, if at all, and how they implemented it. In Study 1, we used a basic version of the spatial-arrangement task; in Studies 2 and 3, we altered the procedure to highlight the possibility of construing the sets categorically. These different versions of the task allowed us to probe the strength of intuitions about linear order in the two groups.²

Study 1

Method

Participants

Yupno adults (n = 13) participated in the task in exchange for a small gift. They were tested in the village of Gua, in the Yupno valley of Papua New Guinea. At the time of data collection, the valley was not connected by road to any urban centers, and it did not have electricity, telecommunications services, or tourist infrastructure. Area residents make their living by subsistence farming (see the Supplemental Material for further ethnographic details). One Yupno participant was eliminated for failing to understand the instructions. Thus, the final sample consisted of 12 participants (6 women). The majority of these participants (n = 9) had no formal schooling whatsoever, and all were “unschooled” according to the cutoff used in previous research in this population (i.e., no education beyond sixth grade; Núñez et al., 2012).³ Undergraduates (n = 16) at the University of California, San Diego (UCSD), participated in exchange for course credit (12 women).

Materials

The arrangement task was adapted from Tversky et al. (1991). In the original task, participants placed stickers on square pieces of paper. In our version, to avoid providing a rectilinear frame, we instead asked participants to arrange circular cards on a large circular sheet. Further, our version of the task used cards rather than stickers, to allow for adjustments and rearranging.

Each set of cards represented a domain that was either categorical or orderable. The categorical stimuli were of two types: animals (a picture of a pig or dog) and shapes (a blob or squiggly line). The orderable stimuli were also of two types: size (a circle with a size that varied across stimuli) and numerosity (from 1 to 10 dots).⁴ Each size card showed a centered blue circle, which ranged in diameter from 1 cm to 5.5 cm, in increments of 0.5 cm. Each numerosity card showed a pseudorandom arrangement of black dots, each 1 cm in diameter. Half of the sets consisted of four cards, and half consisted of five cards. For the categorical stimuli, four-card sets always involved two tokens of each category (e.g., two pigs, two dogs), and five-card sets involved two tokens of one category and three of the other. The orderable stimuli consisted of both consecutive sets (e.g., a four-card numerosity set with cards depicting 1, 2, 3, and 4 dots) and nonconsecutive sets (e.g., cards depicting 1, 2, 5, and 9 dots). All stimuli were printed on circular cards (8 cm in diameter) and laminated.

Procedure

Participants were tested individually. They sat in front of a large blue circular sheet (105 cm in diameter) and were instructed: “I am going to give you a mixed-up group of cards. Look them over and lay them out in an organized way.” The instructions were developed in collaboration with an academic linguist with good Yupno proficiency and a native speaker of Yupno with good English proficiency; they were expressly designed not to bias participants toward any particular spatial arrangement. For the Yupno participants, prerecorded instructions spoken in Yupno by a native speaker were played from a laptop, and a native speaker of Yupno with good English proficiency was present to facilitate communication. For the U.S. participants, the instructions were presented orally by the experimenter.

The experimenter, positioned to the left and slightly in front of participants, handed the shuffled sets of cards to them, one set at a time. Participants arranged 12 sets of cards, presented in two blocks of 6 sets each. Each block consisted of two categorical sets (one of animals and one of shapes), a size set, a brightness set (see note 4), and two numerosity sets. The sets were presented either in that order or with the positions of the categorical and numerosity sets switched. Whether participants began with categorical sets and ended with numerosity sets or vice versa was counterbalanced across participants (see the Supplemental Material for the card sets and lists used). Participants’ arrangements were documented with video or still photographs, taken from a position facing the participants.

Between the two blocks, participants completed a brief task based on one used previously to induce interval representations (in which exact spatial distance represents conceptual distance; Tversky et al., 1991). The task did not induce interval arrangements in either group, however, and was not used in our later studies.

Analysis

Before analysis, the images of the final arrangements were cropped so that the participants were not visible. These cropped images were then coded in a random order to determine the organizational strategy used (Fig. 1); for linear arrangements, the orientation of the line was also coded. An arrangement was considered to exhibit spatial grouping if the cards were put into distinct piles or if adjacency relations suggested category boundaries (e.g., a row with the three pigs on the left and the two dogs on the right). An arrangement was considered to exhibit linear order if the cards were arranged in a path that preserved the order of the elements perfectly (or if two adjacent cards were in transposed order, which occurred rarely). An arrangement that met none of these criteria was coded as “other.” Finally, arrangements exhibiting linear order were coded for their orientation relative to the participant’s body. The eight possible orientations were rightward, rightward-away, away, leftward-away, leftward, leftward-toward, toward, and rightward-toward. For example, a line increasing (larger size, greater numerosity) to the participant’s right was coded as “rightward,” a line increasing in a direction both away from and to the right of the participant was coded as “rightward-away,” and a line increasing toward the participant’s body was coded as “toward.”

Fig. 1.

Representative arrangements illustrating the two spatial organization strategies: spatial grouping (top) and linear order (bottom). Categorical stimuli (left) can be grouped but not ordered, whereas orderable stimuli (right) can be either grouped or ordered. Each example shown here was produced by a different participant in Study 2. The examples of spatial grouping were produced by Yupno participants, and the examples of linear order were produced by U.S. participants.

The first author (K. C.) coded all the arrangements. Reliability was assessed by having a second, naive coder analyze the entire data set. Reliability was 95% for organizational strategy and 92% for directionality. Disagreements were adjudicated by the second author (T. M.).

The adoption of a particular strategy (e.g., linear order) was analyzed on a trial-by-trial basis using mixed logit models, with fixed effects of cultural group (Yupno vs. U.S.), stimulus type (e.g., size vs. numerosity), and their interaction, and random effects of both participants and items, when appropriate. The different types of categorical stimuli were collapsed in all analyses. All binary predictors (e.g., cultural group) were centered. For models that analyzed all three levels of stimulus type (i.e., categorical, size, and numerosity), size trials acted as a baseline with which the two other stimulus types were compared.

All models used the maximal converging random-effects structure justified by the experimental design (Barr, Levy, Scheepers, & Tily, 2013; see the Supplemental Material for model specifications). Models were implemented with the lme4 package (Bates, Maechler, Bolker, & Walker, 2015) in the R statistical software environment (R Core Team, 2015). Reported proportions are model estimates, which account for individual- and item-specific variability.

Results

Spatial grouping

We first analyzed the use of spatial grouping, an organizational strategy that may be used for both categorical (animals, shapes) and orderable (size, numerosity) domains. The Yupno and U.S. participants did not differ overall in their use of grouping (p > .32). This strategy was employed equally often for size stimuli (M = .04, 95% confidence interval, CI = [.02, .06]) and numerosity stimuli (M = .03, 95% CI = [.02, .04]), p > .34, but significantly more often for categorical stimuli (M = .68, 95% CI = [.63, .73]), b = 5.12, SEM = 0.95, p < .001. However, a significant interaction, b = −2.76, SEM = 1.37, p = .044, revealed that this increased use of the grouping strategy for categorical stimuli was more pronounced among the U.S. participants (categorical: M = .77, 95% CI = [.72, .83]; size: M = .01, 95% CI = [.01, .02]; numerosity: M = .00, 95% CI = [.00, .00]) than among the Yupno (categorical: M = .56, 95% CI = [.47, .64]; size: M = .08, 95% CI = [.04, .11]; numerosity: M = .07, 95% CI = [.05, .09]). No other effects approached significance (all ps > .3). Thus, although both groups largely reserved the grouping strategy for categorical stimuli, this selective use was stronger among the U.S. participants.

Linear order

We next analyzed the use of the ordered line, a strategy that could be applied only to the two types of orderable stimuli, that is, size and numerosity stimuli. There were no effects of cultural group (p > .56) or stimulus type (p > .63). Ordered lines were the dominant response for both stimulus types, for both the Yupno (size: M = .88, 95% CI = [.76, 1.00]; numerosity: M = .86, 95% CI = [.77, .95]) and the U.S. participants (size: M = 1.00, 95% CI = [.99, 1.00]; numerosity: M = .99, 95% CI = [.97, 1.00]; Fig. 2). Thus, although the U.S. participants used linear order numerically more often than did the Yupno, this effect did not reach significance. Analyses of the orientations of these ordered lines are presented later, after Study 3.

Fig. 2.

Frequency of use of the linear-order strategy for arranging the size and numerosity stimuli. Results are shown separately for the two cultural groups (U.S., Yupno) in each of the three studies. Error bars represent ±1 SE (one error bar is too short to be visible). The proportions shown are estimates from a model of all three studies.

Discussion

Despite living in a culture that lacks representational practices exhibiting linear order, Yupno participants used this organizational principle spontaneously, to an extent comparable to that of U.S. participants. Such practices, therefore, are not necessary to spark the intuition that order can be represented in a linear arrangement. We also sought to compare the strength of intuitions about linear order in the two groups by allowing a competing strategy—grouping—but it was used rarely for the orderable stimuli. To create a tension between the strategies, and thus conduct a better test of whether intuitions about linear order are stronger in U.S. adults than in the Yupno, in Studies 2 and 3 we manipulated the procedure to highlight the possibility of spatial grouping.

Study 2

In Study 2, participants again arranged sets of cards, but they began by arranging pieces of fruit. Fruit is commonly grouped by category in everyday life in both cultures (e.g., in marketplaces), and being asked to arrange it should call to mind this organizational strategy. We reasoned that if the Yupno have a weaker intuition about linear order than U.S. adults do, they might produce fewer ordered lines after being primed to group by category.

Method

Participants

Yupno adults (n = 19) participated in exchange for a small gift. Five were eliminated for failing to understand the instructions. Thus, the final Yupno sample consisted of 14 participants (4 women). Most (n = 8) had no exposure to formal schooling whatsoever, and all were classified as unschooled by the previously used cutoff (Núñez et al., 2012). UCSD undergraduates (n = 19) participated in exchange for course credit. Two were eliminated because of irregularities in the experimental procedure. Thus, the final U.S. sample consisted of 17 participants (15 women). None of the participants in Study 2 had participated in Study 1.

Materials

As in Study 1, the stimuli were either orderable or categorical. The orderable stimuli were identical to those in Study 1 (i.e., size and numerosity stimuli). The categorical stimuli included the same set of animals from Study 1; the abstract shapes were replaced with more ecologically valid concrete objects (rocks and carrying bags). We used eight card sets, each consisting of four cards. Of the orderable stimuli, half were nonconsecutive and half were consecutive; the nonconsecutive sets always preceded the consecutive sets.

Procedure

The procedure was the same as in Study 1, except for the addition of a fruit-arranging task at the beginning. For the fruit task, participants were instructed: “I am going to give you a mixed-up group of objects. Look them over and lay them out in an organized way.” The experimenter then handed the participant the fruits in a disorganized handful. Culturally common fruits were used: oranges, passion fruits, and tree tomatoes for the Yupno; apples, oranges, and avocados for the U.S. participants. There were two trials, each involving two fruits each of two types. For example, a Yupno participant might receive two passion fruits and two tree tomatoes on the first trial and then two oranges and two passion fruits on the second.

After the fruit-arranging task, participants arranged the eight sets of cards, one after another, in a fixed order: one objects set, one animals set, two size sets, two brightness sets, and two numerosity sets (see the Supplemental Material for the card sets).

Analysis

Coding and analysis were conducted as in Study 1. Reliability was 94% for organizational strategy and 95% for directionality.

Results

Spatial grouping

There was no effect of cultural group on frequency of use of the grouping strategy; the Yupno and U.S. participants used grouping to the same degree (p > .21). Once again, grouping was used equally often for the size stimuli (M = .17, 95% CI = [.13, .21]) and the numerosity stimuli (M = .18, 95% CI = [.13, .22]), p > .94, although for both types of stimuli it was used more than 4 times as often as in Study 1. As before, grouping was used significantly more often for the categorical stimuli (M = .78, 95% CI = [.75, .82]), b = 3.50, SEM = 0.65, p < .001. Although this increased use for categorical stimuli was, once again, more pronounced for the U.S. participants (categorical: M = .83, 95% CI = [.79, .87]; size: M = .10, 95% CI = [.07, .14]; numerosity: M = .08, 95% CI = [.05, .10]) than for the Yupno participants (categorical: M = .72, 95% CI = [.66, .78]; size: M = .24, 95% CI = [.17, .31]; numerosity: M = .30, 95% CI = [.22, .38]), this effect was only marginally significant, b = −1.57, SEM = 0.93, p = .09. No other effects approached significance (all ps > .21).

Linear order

There were no effects of cultural group (p > .20) or stimulus type (p > .27) on frequency of use of the linear-order strategy. Ordered lines were the dominant response for both stimulus types, for both the Yupno (size: M = .61, 95% CI = [.46, .76]; numerosity: M = .63, 95% CI = [.48, .78]) and the U.S. participants (size: M = .82, 95% CI = [.73, .90]; numerosity: M = .90, 95% CI = [.83, .97]; Fig. 2). Thus, although the U.S. participants used linear order numerically more often than the Yupno, this effect did not reach significance. Analyses of the orientations of these ordered lines are presented later, after Study 3.

Study 3

In Study 3, we further probed the strength of intuitions about linear order. In addition to the categorical-priming task from Study 2, we included a procedural change to further highlight the possibility of construing the sets categorically. In Studies 1 and 2, participants received all the cards in a set at once, which invited them to consider the set’s overall ordinal structure. In Study 3, after first receiving the extremes of a set, participants received one card at a time. This procedure foregrounded the relationship of each additional card to the extremes, rather than the overall structure of the set. In this way it was similar to the number-line estimation task (e.g., Núñez et al., 2012), which begins with an introduction of the line’s endpoints as salient anchors.

Method

Participants

Yupno adults (n = 13) participated in exchange for a small gift. Two were eliminated for failing to understand the instructions. Thus, the final Yupno sample consisted of 11 participants (5 women). Most (n = 8) had no formal education whatsoever, and all were classified as unschooled by the previously used cutoff (Núñez et al., 2012). UCSD undergraduates (n = 15) participated voluntarily in exchange for course credit. One was eliminated because of irregularities in the experimental procedure. This left a final sample of 14 U.S. participants (8 women). No participants in Study 3 had participated in either previous study.

Materials

The materials were similar to those used in Study 2 (i.e., categorical objects and animals sets and orderable size and numerosity sets). However, there was just one six-card set for each category, and the orderable sets were nonconsecutive (e.g., the numerosity set depicted 1, 2, 3, 5, 6, and 7 dots).

Procedure

Participants began with two trials of the fruit-arranging task used in Study 2. All participants then arranged four sets in the following fixed order: objects, animals, size, and numerosity. For each set, participants were first given two cards and were instructed, as before, to “lay them out in an organized way.” They were then given the four remaining cards one at a time. With each additional card, they were instructed: “I am going to give you another card. Carefully look over the whole group and lay them out in an organized way.” For the categorical sets, the first two cards were tokens of the two categories; the remaining cards loosely alternated between the categories. For the orderable sets, the first two cards were the set’s extremes; the remaining cards loosely alternated between small and large magnitudes (see the Supplemental Material for the card sets and the order in which the cards within a set were presented).

Analysis

Participants’ final six-card arrangements were analyzed in the same way as in Studies 1 and 2. Reliability was 95% for organizational strategy and 86% for directionality.

Results

Spatial grouping

There was no effect of cultural group on frequency of use of the grouping strategy (p > .21). Once again, spatial grouping was used equally often for size stimuli (M = .42, 95% CI = [.36, .48]) and numerosity stimuli (M = .32, 95% CI = [.26, .38]), p > .37, although the overall use of grouping doubled from Study 2. Again, grouping was used significantly more for categorical stimuli (M = .76, 95% CI = [.73, .79]), b = 1.59, SEM = 0.57, p = .005. Although this increased use of grouping for categorical stimuli was, yet again, more pronounced for the U.S. participants (categorical: M = .82, 95% CI = [.80, .85]; size: M = .32, 95% CI = [.26, .39]; numerosity: M = .21, 95% CI = [.16, .26]) than for the Yupno participants (categorical: M = .68, 95% CI = [.64, .71]; size: M = .55, 95% CI = [.49, .61]; numerosity: M = .45, 95% CI = [.39, .51]), this effect was only marginally significant, b = −1.84, SEM = 0.99, p = .06. No other effects approached significance (all ps > .21).

Linear order

The use of linear order differed between the cultural groups; the U.S. participants produced significantly more linear arrangements (M = .79, 95% CI = [.63, .95]) than the Yupno participants did (M = .22, 95% CI = [.04, .41]), b = −21.34, SEM = 6.48, p < .001 (Fig. 2). Thus, although the Yupno participants produced ordered lines on the majority of trials with orderable sets in Studies 1 and 2, this was no longer the case in Study 3. Instead, grouping was the dominant strategy for the Yupno, whereas the ordered line remained the dominant strategy for the U.S. participants. There was also a marginal effect of stimulus type, b = 4.22, SEM = 2.30, p = .07; linear order was used more for the numerosity stimuli (M = .60, 95% CI = [.39, .80]) than for the size stimuli (M = .48, 95% CI = [.27, .69]). This, in turn, was driven by a marginal interaction between cultural group and stimulus type, b = 8.61, SEM = 4.54, p = .06: The Yupno produced more ordered lines for the numerosity stimuli (M = .36, 95% CI = [.02, .69]) than for the size stimuli (M = .09, 95% CI = [.00, .28]), whereas the U.S. participants produced ordered lines equally often for the size stimuli (M = .79, 95% CI = [.55, 1.00]) and the numerosity stimuli (M = .79, 95% CI = [.55, 1.00]). We next present analyses of the orientations of ordered lines in all three studies.

Analyses Combining All Studies

To confirm that the two cultural groups were differentially affected by task procedures, we analyzed the data from all three studies together. The mixed logit model was analogous to the models for the individual studies, with additional fixed effects for study (1–3) and its interactions, and with Study 1 acting as a baseline against which the other studies were compared. Because the frequency of use of linear order did not differ between size and numerosity stimuli in any of the studies, stimulus type was not included as a fixed effect in the analysis of linear order.

Spatial grouping

Use of grouping did not differ between the size stimuli (M = .15, 95% CI = [.11, .19]) and the numerosity stimuli (M = .11, 95% CI = [.08, .14]), p > .32, but was significantly greater for the categorical stimuli (M = .73, 95% CI = [.69, .77]), b = 5.04, SEM = 1.04, p < .001. Yet again, this increase was marginally more pronounced for the U.S. participants (categorical: M = .82, 95% CI = [.78, .86]; size: M = .07, 95% CI = [.04, .10]; numerosity: M = .04, 95% CI = [.02, .07]) than for the Yupno participants (categorical: M = .62, 95% CI = [.55, .69]; size: M = 0.25, 95% CI = [.18, .31]; numerosity: M = .19, 95% CI = [.14, .25]), b = −2.19, SEM = 1.22, p = .07. There was also a highly significant influence of Study, b = 1.29, SEM = 0.50, p < .01. grouping was more common in Study 2 (M = .36, 95% CI = [.31, .42]) than in Study 1 (M = .29, 95% CI = [.24, .34]), and more common yet again in Study 3 (M = .57, 95% CI = [.50, .63]). No other effects approached significance (ps > .24).

Linear order

Use of linear order decreased reliably in later studies (Study 1: M = .94, 95% CI = [.90, .97]; Study 2: M = .75, 95% CI = [.68, .81]; Study 3: M = .54, 95% CI = [.40, .68]), b = −5.16, SEM = 1.59, p = .001. More important, this influence of task procedures differed significantly between the cultural groups, b = −7.41, SEM = 2.75, p = .007. To investigate this interaction, we analyzed the two cultural groups separately. Among the Yupno, the use of linear order was affected significantly by the procedural changes across the three studies (Study 1: M = .87, 95% CI = [.79, .94]; Study 2: M = .61, 95% CI = [.49, .72]; Study 3: M = .21, 95% CI = [.05, .37]), b = −7.17, SEM = 2.91, p = .014. The U.S. participants, by contrast, were largely unaffected by the procedural manipulations, continuing to rely on linear order in every study (Study 1: M = .99, 95% CI = [.98, 1.00]; Study 2: M = .86, 95% CI = [.79, .94]; Study 3: M = .80, 95% CI = [.64, .95]), p > .36. The context in which participants were asked to arrange the cards thus had a selective impact on the Yupno, dampening their use of the ordered line.

Directionality of ordered lines

We next analyzed the direction of increase (i.e., larger size, greater numerosity) in participants’ ordered lines. This analysis collapsed the data across all the studies because of the limited number of ordered lines produced by the Yupno in Study 3. Overall, the Yupno produced ordered lines that were highly variable in their orientation, and significantly more variable than those produced by the U.S. participants, χ²(1, N = 276) = 92.5, p < .001, Wallraff rank-sum test of angular distance (Fig. 3). Indeed, among the U.S. participants, almost all the ordered lines (94.9%) were along the left-right axis, and they were tightly concentrated around a rightward orientation, with a bootstrapped 95% CI spanning only 5° (M = 3.3°, 95% CI = [6.3°, 1.1°]; circular dispersion: κ = 1.8). By contrast, nearly half of the Yupno participants’ lines (46.3%) were not oriented along the left-right axis. Their lines exhibited a broad range of orientations, with a 95% CI spanning 38° (M = 21.4°, 95% CI = [3.4°, 41.3°]; circular dispersion: κ = 0.8). Thus, the U.S. participants produced ordered lines that were highly consistent in a rightward orientation, whereas the Yupno produced ordered lines that were oriented more freely around the entire 360° range of possible directions.

Fig. 3.

Directionality of the ordered lines produced across all three studies. Each circular plot is divided into eight sections, according to the scheme used for coding orientation (rightward, rightward-away, away, etc.). Within these sections, each spoke represents an actual ordered line. The spokes are plotted around the centers of the sections, with Gaussian noise to avoid overplotting.

General Discussion

In high-literacy, industrialized societies, the principle of linear order is a cornerstone of both visual culture and everyday cognition. To illuminate the origins of this principle, we examined how abstract relations are organized spatially in two cultural groups, one with diverse and ubiquitous representational practices exhibiting the principle (U.S.) and one without such practices (Yupno). Members of both groups organized sets of orderable cards into lines, which suggests that familiarity with conventional linear representations is not necessary to spark intuitions about linear order. But the two groups did differ in when they used the principle and how they implemented it: The U.S. participants were steadfast in their use of linear order, regardless of task procedures, and their ordered lines were consistently oriented rightward; the Yupno participants, in contrast, relied on linear order primarily when grouping was not a salient alternative, and their lines were oriented idiosyncratically.

If the notion of linear order is not rooted ultimately in visual culture, then where does it come from? Although it is tempting to consider the principle an a priori intuition, possible experiential sources abound. Rudiments of linear order are present in natural phenomena, such as animal tracks or human footprints, which embody an unfolding sequence. Similarly, when groups of people walk on narrow paths, they form lines that can be construed as ordered. It may also have been possible for participants to discover the utility of linear order during the arrangement task itself: Putting a series of cards in a line may facilitate a series of pairwise comparisons, which are necessary to arrive at a full ordering of a set of elements. Thus, the fact that the Yupno and the U.S. participants produced similar arrangements does not necessarily mean that they drew on similar—let alone a priori—mental representations. Rather, the Yupno may have deployed an ad hoc organizational scheme, whereas the U.S. participants may have drawn on an internalized convention.

Lifelong experience with graphical practices appears to regiment intuitions about linear order. By regimentation, we mean a process in which culture reinforces certain habits of thinking and behaving, with consequences for when and how those habits are called upon. For example, experience with conventions for driving on one side of the road may encourage broad use of that principle (e.g., when one is walking on the sidewalk) and entrench a particular way of implementing it (e.g., walking on the right). The same consequences can be seen in the case of linear order. U.S. participants make broad use of this principle across contexts; in fact, it may be so entrenched that it springs to mind whenever orderable elements are in play, even in implicit tasks (e.g., Dehaene et al., 1993). Yupno participants, who have not had a lifetime of exposure to linear order, abandon this principle when categorical construals are highlighted. This pattern sheds light on earlier results regarding Yupno participants’ performance on the number-line estimation task (Núñez et al., 2012). That task, much like the procedure used in Study 3, introduces salient anchors—the line’s endpoints—and thus invites a categorical construal. In both the number-line task and our card-arrangement task in Study 3, the Yupno overwhelmingly favored spatial grouping. Graphical practices also regiment intuitions about linear order by determining how it is implemented. Ordered lines in the visual culture of the United States—and wherever Latin script predominates—proceed rightward. As has been demonstrated in prior studies (e.g., Bergen & Lau, 2012; Tversky et al., 1991), we found that U.S. participants strongly favored this direction. In contrast, the Yupno, without this regimenting influence, produced lines oriented in all directions.

We also found a pattern—weak but present numerically across all three studies—in which U.S. participants used spatial grouping for categorical stimuli more than the Yupno did. A possible interpretation of these results is that spatial organization strategies in U.S. participants are regimented generally such that linear order is selectively applied to orderable stimuli and spatial grouping is selectively applied to categorical stimuli. In the case of grouping, this regimentation may be due to experience with Venn diagrams and other graphics that colocate elements of the same category. Without the regimenting influence of this visual culture, the Yupno appear to be more flexible in their spatial organization strategies.

Together, these findings hint at how the ordered line has come to occupy a privileged place in both visual culture and cognition. Unlike other representational conventions, such as writing (Gelb, 1963), tree diagrams (Lima, 2014), and number lines proper (Núñez, 2011), the ordered line likely did not have to be invented. Rudiments of linear order in the natural world, as well as expediencies of organizing information, could have motivated its initial manifestations. But such early uses would likely have been limited and evanescent. Since the Middle Ages, however, the uses of linear order have multiplied as a result of the growing importance of quantitative information in accounting, navigation, timekeeping, and other domains (Crosby, 1997); at the same time, ordered lines have become more durable and portable, enshrined in books, other artifacts, and now digital forms (e.g., Buringh & Van Zanden, 2009). As high-literacy, industrialized cultures have become saturated with an ever-increasing number of manifestations of linear order, the principle has, in turn, become increasingly regimented as a powerful tool of everyday cognition.

Footnotes

Acknowledgements

We thank the Yupno residents of Gua for their participation and assistance, James Slotta for his help with the Yupno language, and Danielle Jacques and Natalie Allen for their research assistance.

Action Editor

Ayse K. Uskul served as action editor for this article.

Declaration of Conflicting Interests

The authors declared that they had no conflicts of interest with respect to their authorship or the publication of this article.

Funding

The National Geographic Society (Grant 9131-12) funded the fieldwork on which the reported studies are based. The National Science Foundation’s Spatial Intelligence and Learning Center (Grant SBE-0541957) funded the postdoctoral research of K. Cooperrider.

Supplemental Material

Additional supporting information can be found at

Open Practices

All materials have been made publicly available via the Open Science Framework and can be accessed at https://osf.io/8dvsz. The complete Open Practices Disclosure for this article can be found at https://journals-sagepub-com.web.bisu.edu.cn/doi/suppl/10.1177/0956797617691548. This article has received the badge for Open Materials. More information about the Open Practices badges can be found at .

Notes

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