Is scaling up harder than scaling down? How children and adults visually scale distance from memory

Cognition 185 (2019) 39–48

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Original Articles

Is scaling up harder than scaling down? How children and adults visually scale distance from memory

T

Jodie M. Plumerta, , Alycia M. Hundb, Kara M. Reckerc ⁎

a

The University of Iowa, USA Illinois State University, USA c Coe College, USA b

ARTICLE INFO

ABSTRACT

Keywords: Visual scaling Spatial cognition Cognitive development Memory

In three experiments (N = 288), we examined how the direction of the scale translation impacts how 4- to 5year-old children and adults visually scale distance from memory. Participants first watched an experimenter place an object on a learning mat and then attempted to place a replica object on a test mat that was either identical (no scaling task) or different in scale (scaling task). In Experiment 1, both children and adults had difficulty scaling up from 16 to 128 in. (1:8 scaling ratio) but not scaling down from 128 to 16 in. (8:1 scaling ratio), suggesting that scaling up was harder than scaling down. In Experiment 2, we reduced the scaling ratio from 1:8 to 1:2 and found that children and adults had no difficulty scaling up from 16 to 32 in. or scaling down from 32 to 16 in.. In Experiment 3, we kept the scale ratio the same (1:2) but increased the size of the test mat and found that participants had difficulty with both scaling up from 32 to 64 in. and scaling down from 128 to 64 in.. We conclude that scaling up is not harder than scaling down. Rather, visually scaling distance is more difficult when participants cannot view both edges of the test mat simultaneously while making the scale translation. Across all experiments, 4- to 5-year-olds were less accurate than adults in their placements overall, but they exhibited the same patterns of performance on the scaling and no scaling tasks, suggesting that visual scaling processes are age-independent. The General Discussion focuses on how visual scaling emerges from a complex interplay of cognitive processes and visual constraints.

1. Introduction The ability to scale distance is a fundamental spatial skill involved in a wide variety of complex human activities, from reading maps of large-scale spaces to understanding structures in cell diagrams. In all of these activities, individuals must understand how distances in two differently-sized spaces or objects map onto one another. In some cases, these mappings involve scaling down such as representing our solar system with a model, but in other cases these mappings involve scaling up such as representing a chemical compound using a diagram. Recently, studies have shown that informal skill in scaling proportions is related to formal knowledge of fractions and that spatial scaling predicts science and mathematics achievement (Boyer & Levine, 2012; Frick, 2018; Hodgkiss, Gilligan, Tolmie, Thomas, & Farran, 2018; Möhring, Newcombe, & Frick, 2015). These findings support the claim that spatial scaling skills play a critical role in many STEM disciplines (National Research Council, 2012). However, the cognitive processes underlying spatial scaling are not yet well understood. The goal of this

⁎

investigation was to further examine the processes involved in visually scaling distance by systematically examining how the direction of the scale translation impacts spatial scaling in young children and adults. Previous work has shown that young children can visually scale distances between two differently-sized spaces, albeit precision increases substantially with age (Frick & Newcombe, 2012; Huttenlocher, Newcombe, & Vasilyeva, 1999; Huttenlocher, Vasilyeva, Newcombe, & Duffy, 2007; Vasilyeva & Huttenlocher, 2004). For example, Huttenlocher et al. (1999) showed 4-year-old children the location of a dot on a small rectangular map (8 in. long) and then asked them to use the map to find the corresponding location in a larger rectangular sandbox (60 in. long) by making a pointing response. In general, 4-yearold children pointed accurately to the locations in the sandbox, and they preserved the spatial ordering of the locations. In a related study, Huttenlocher et al. (2007) found that children as young as 3.5 years can scale distance when using a map to place objects in the sandbox, indicating that the ability to scale distance along a single dimension develops quite early in childhood. However, there is some evidence that

Corresponding author at: Department of Psychological and Brain Sciences, W311 SSH, University of Iowa, Iowa City, IA 52242, USA. E-mail address: [email protected] (J.M. Plumert).

https://doi.org/10.1016/j.cognition.2018.12.013 Received 20 July 2018; Received in revised form 18 December 2018; Accepted 20 December 2018 0010-0277/ © 2018 Elsevier B.V. All rights reserved.

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this ability undergoes improvement until at least age six (Frick & Newcombe, 2012). Frick and Newcombe (2012) compared how accurately 3- to 6-year-olds and adults marked a location in a corresponding space that was either the same size (unscaled trials) or a different size (scaled trials) as the original space. They found that the difference between scaled and unscaled trials decreased with age. Overall, placement errors were not different for 6-year-olds and adults. What strategies might children and adults use to visually scale distance? One possibility is to visually code distance relative to two landmarks or edges in one space and then map this relative distance onto another larger or smaller space using the corresponding landmarks or edges (Huttenlocher et al., 1999, 2007; Spetch & Parent, 2006; Uttal, Sandstrom, & Newcombe, 2006). Previous work has shown that 4-yearolds rely on relative distance to search in the middle of two landmarks when the distance between the landmarks is doubled, indicating that they coded the location as halfway between the two landmarks (Uttal et al., 2006). This suggests that young children may use an informal proportional reasoning strategy to map relative distances between spaces. Note that a formal understanding of proportions (e.g., Piaget & Inhelder, 1967) is not necessary for visually scaling relative distance. Rather, one could visually estimate a proportional distance in one space (e.g., a quarter of the way across), and then visually map this proportional distance onto another space. According to Möhring, Newcombe, and Frick (2016), the accuracy or speed of scaling should not be impacted by the magnitude of the scale translation when using a relative distance strategy since proportional or relative distances can be encoded regardless of the absolute size of the space. Another possible strategy for visually scaling distance is to mentally transform the size of one space to match the size of the other space (Möhring et al., 2016; Vasilyeva & Huttenlocher, 2004). This would involve visually coding a distance in one space and then mapping that distance onto another space by mentally shrinking or expanding the original space to fit the new space. As with other mental transformations (e.g., mental rotation, image scanning), larger transformations should take a longer time to perform and should result in greater error (e.g., Bundesen & Larsen, 1975; Kosslyn, 1975; Shepard & Metzler, 1971). To test this hypothesis, Möhring, Newcombe, and Frick (2014) asked preschoolers and adults to encode the location of a target on a map and then point to the same location on a larger referent space. They found that both children and adults were less accurate in their responses when the scale transformation was larger. In a follow-up study, Möhring et al. (2016) asked adults to discriminate whether two marked locations in spaces that differed in size were in the same relative location. They found that the speed and accuracy of responses varied linearly with the scaling magnitude, supporting the idea that mental transformation processes may underlie visual scaling. To date, much of the focus has been on understanding how the magnitude of the scale translation impacts scaling performance in children and adults. Far less well understood is how the direction of the scale translation (i.e., scaling up vs. scaling down) impacts scaling performance, and what scaling up and scaling down can reveal about the processes underlying visual scaling. One exception is a classic study by Siegel, Herman, Allen, and Kirasic (1979) in which 5-, 7-, and 10year-old children learned the layout of a town in a small model and then reproduced the layout in a larger room by placing the buildings in their locations (scaling up), or learned the layout in a large room and reproduced the layout in a smaller model by placing pictures of the buildings (scaling down). The scale ratio between the model and room was 1:6. Although overall accuracy increased significantly with age, all age groups were relatively more accurate when scaling down from the room to the model than when scaling up from the model to the room, suggesting that the direction of the scale translation matters. This finding is intriguing, but to our knowledge has received very little attention in recent studies of children’s visual scaling. Another exception is the study by Möhring et al. (2016) mentioned earlier that used a discrimination task to test the effects of scale

magnitude. On each trial, adults viewed two shapes (e.g., two squares) displayed together on a computer monitor, each with a dot inside, and were asked to judge whether the dots were in the same or different (relative) location. Over trials, the space that stayed a constant size was either always larger or smaller than the spaces that varied in size. Regardless of whether the constant space was larger or smaller, discrimination accuracy decreased and response time increased linearly as a function of scale magnitude. One could argue that finding no difference across the constant-small and constant-large conditions means that scaling up and scaling down led to no differences in performance. However, because both shapes were present side-by-side during the entirety of each trial, it is not clear which space participants used as the reference space since they could scan back and forth between the two shapes to make their judgment. Moreover, it is not clear whether the findings would be similar if the spaces were much larger. Why might scaling up be harder than scaling down? As mentioned earlier, scaling distance along a single dimension involves mapping distances between two differently-sized spaces using the corresponding edges of the two spaces (Uttal et al., 2006). When scaling up to a larger space, people may have difficulty mapping the two spaces because the edges on the larger space are now farther apart and may not be simultaneously viewable. We hypothesize that this disrupts the scaling process because people must make head or eye movements in order to make the scale translation. This may be particularly problematic if people are attempting to expand an image of a mentally represented space to fit a visually available space. Support for this idea comes from a study by Kosslyn, Ball, and Reiser (1978) in which they asked people to “zoom in” on a part of a mental image of a face such that the rest of the face overflowed their image (was no longer “visible”). This resulted in slower reaction times when making judgments about parts of the face that overflowed the mental image. People may experience similar disruption when scaling up to larger spaces that overflow the visual field (i.e., the edges are not simultaneously viewable). Much of the recent work on scaling has focused on very small spaces with edges that are easily viewable from a single viewpoint (Frick & Newcombe, 2012; Möhring et al., 2014, 2015, 2016). For example, Möhring, Newcombe, Levine, and Frick (2015) asked 4- and 5-year-old children to indicate the location of an egg in a field by pointing to a 18 cm × 22 cm space on a touch screen while viewing a map of the same or different scale (from 1:1 to 1:4). Analysis of localization errors revealed an effect of scale ratio best explained by a linear function, indicating that localization errors increased as scaling ratio increased (see also Möhring et al., 2014). These findings support the idea that localization is harder when scale ratios are larger, but as the authors note, leaves open questions related to spaces larger than those used in their study. Older work on children’s spatial scaling has clearly shown that younger children have difficulty scaling up to much larger spaces (Liben, Moore, & Golbeck, 1982; Uttal, 1994, 1996). For example, Uttal (1994, 1996) asked 5-year-olds to learn a configuration of objects on a small map and then reproduce that configuration of locations in a large room. Although children generally preserved the configuration of the set of objects, they had difficulty correctly scaling the distances among the objects when placing them in the large room. Although scaling a configuration of objects is undoubtedly more complex than scaling the location of a single object, this work suggests that the size of the test space impacts young children’s scaling accuracy. The goal of the present investigation was to further examine how young children and adults visually scale distance along a single dimension by systematically testing how the direction of the scale translation impacts spatial scaling. We focused on 4- and 5-year-olds because we know that children as young as 3.5 years can scale distance along a single dimension, thereby allowing us to systematically test how both young children and adults respond to variations in the direction of the scale translation. The distances we used (16–128 in.) were much larger than the distances used most recently by Frick and Newcombe (2012) and Möhring et al. (2014, 2015, 2016), allowing us 40

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to test how children and adults scale up and down when spaces span a wide range of the visual field. Across three experiments, we used a within-subjects design to compare how children and adults localize objects in a memory task with no scaling and a memory task with scaling. In the no scaling task, participants watched an experimenter place a picture of an object on a (learning) mat on the floor. They then walked over to the other side of the room (divided by a curtain) and attempted to place another picture of the object on a same-size (test) mat. In the scaling task, participants watched an experimenter place a picture of an object on a (learning) mat on the floor. They then walked over to the other side of the room and attempted to place another picture of the object on a differently-sized (test) mat. Although the learning mats differed in size across the two tasks, critically, the test mat for both the scaling and the no scaling tasks was always the same size for a given condition. Thus, any difference in placement accuracy across the two tasks at test reflects the impact of scaling up or scaling down on performance. We sought to answer two major questions related to how children and adults scale distance. The first was whether scaling accuracy is influenced by the direction of the scale translation. To address this question, we tested how young children and adults visually scale from a smaller space to a larger space (scaling up) and from a larger space to a smaller space (scaling down), both when the scale ratio is small (1:2) and large (1:8). Given that Siegel et al. (1979) found that scaling up from a small space to a large space is more difficult than scaling down from a large space to a small space, we expected that scaling up might be more difficult than scaling down in our task, especially when the scaling ratio was large. The second question was whether the patterns of scaling performance differed between young children and adults. Consistent with Frick and Newcombe (2012), we expected that the difference between the scaling and the no scaling tasks would be greater for 4- and 5-year-old children than for adults. This would suggest that although the ability to visually scale distance along a single dimension appears early in development, visual scaling abilities continue to undergo improvement between early childhood and adulthood. However, given that Siegel et al. (1979) found that the direction of the scale translation did not interact with age, we expected that scaling up would be relatively more difficult than scaling down for both children and adults. This would suggest that some of the basic cognitive processes underlying visual scaling are age-independent.

A

Test

Learning (no scaling task)

B

Test

Learning (scaling task) Fig. 1. An aerial view of the experimental room illustrating learning and test trials for the no scaling (panel A) and scaling (panel B) tasks when scaling up.

participants prior to their participation in this research.

2. Experiment 1

2.1.2. Apparatus and materials The experiment took place in an 11.5 ft. × 10.5 ft. room. A white canvas curtain surrounded the periphery of the room from floor to ceiling. In addition, a canvas curtain divided the room into two equally sized enclosures (each 11.5 ft. × 5.25 ft.). One enclosure was used during learning, and the other enclosure was used to during test (see Fig. 1). Extra large (128 in. long × 16 in. wide) and small (16 in. long × 2 in. wide) light-brown, vinyl mats were used as the referent spaces. We used a measuring tape located on the underside of each mat (not visible to participants) to place the objects during learning and to measure object placements during test. Each enclosure had a single mat centered on the floor at all times. An X marked on the floor of each enclosure was used to show participants where to stand during learning and test. The X was 21 in. away from the center of each mat. Ten laminated circles with pictures of objects were used to help participants learn the locations: an apple, ball, butterfly, chicken, fish, ladybug, penguin, present, star, and tiger. The present and tiger were used during practice trials, whereas the other eight objects were used during test trials. The diameter of the circles was half the width dimension of each mat. Thus, objects placed on the extra large referent mats were 8 in. in diameter, and objects placed on the small referent mats were 1 in. in diameter. With the exception of size, the objects depicted on the circles were identical for the extra large and small mats.

2.1. Method 2.1.1. Participants The participants were forty-eight 4- to 5-year-old children and 48 adults. There were 24 children and 24 adults in both the scaling up and scaling down conditions. The mean ages were 4 years 11 months (range = 4 years 4 months to 5 years 6 months; 24 girls, 24 boys) and 19 years 6 months (range = 18 years 1 months to 22 years 11 months; 24 women, 24 men). Children were recruited from a child research participant database maintained by the Department of Psychological and Brain Sciences at the University of Iowa. Parents received a letter describing the study followed by a telephone call inviting their child to participate. Ninety-two percent of the children were European American, 4% were Asian American, 2% were African American, and 2% were Hispanic/Latino. Two percent of mothers had completed their high school education or less, 26% had completed some college education, and 72% had a 4-year-college education or beyond. Children received a small gift for participating. Adults participated to fulfill research credit for an introductory psychology course. Ninety percent of adult participants were European American, 8% were Asian American, and 2% identified themselves as Other. This research was carried out with human subjects approval from the Institutional Review Board at the University of Iowa. Informed consent was obtained from all 41

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2.1.3. Design and procedure All participants were tested individually in the laboratory in a single session. The session began with familiarizing the participants with the no scaling and scaling tasks. For children, this consisted of one demonstration and one practice trial for each task (described below). Adults were not given the demonstration trials because verbal instructions were enough for them to understand the tasks. They completed a single no scaling practice trial followed by a single scaling practice trial using the same practice instructions and procedures experienced by the children. Following the practice trials, children and adults completed 8 no scaling trials and 8 scaling trials in two blocks, presented in counterbalanced order. Participants were randomly assigned to one of two conditions: scaling up or scaling down. In the scaling up condition, the scaling task involved learning locations on the small mat and placing them on the extra large mat (1:8 scaling ratio), and the no scaling task involved learning locations on the extra large mat and placing them on another extra large mat. In the scaling down condition, the scaling task involved learning locations on the extra large mat and placing them on the small mat (8:1 scaling ratio), and the no scaling task involved learning locations on the small mat and placing them on another small mat. Importantly, for each participant, the test mat was the same for both the scaling and no scaling trials (see Fig. 1), allowing for direct comparison across the scaling and no scaling tasks. Two experimenters were present throughout the session. One experimenter was with the participant at all times and gave instructions throughout the session. The other experimenter changed mats when needed, placed objects in their correct locations during learning, and measured participants’ placements at test.

the object in the correct location. If placements were not close to their actual locations, they were corrected. Children then completed a scaling demonstration trial to ensure that they understood this more complex task. While children were not watching, the learning mat was changed. Then, the experimenter pulled back the curtain that divided the room to expose the mats from both enclosures, which now differed in size. Children stood at the curtain opening between the two sides of the room and faced both mats. The experimenter highlighted that the mats were now different sizes and told children that they would see an object on one mat and should try to remember exactly where it goes because they would be asked to put a different-sized picture of the object in exactly the same place on the other mat. The curtain was closed, and children were told to stand on the X and look at the object that was on the learning mat so that they could “remember exactly where it goes.” The experimenter then showed children another object that was identical to that object except in size, and explained to them that they would need to put the new object in the same place on the other mat as the current object was on the current mat. The experimenter then walked children over to the other enclosure and had them stand on the X to watch while the experimenter placed the object in the correct location, demonstrating how to complete the scaling task. Following the demonstration trial, children completed a scaling practice trial. Children walked over to the learning enclosure where the object was located in a new location on the learning mat. They were told to remember the location and walk over to the other enclosure and put the object in the same place on the other mat. If placements were not close to their actual locations, they were corrected. 2.1.6. Test trials Participants completed a total of 16 test trials (two blocks of 8 trials). Each block of trials consisted of eight locations equally spaced on the mats (14 in. apart for the extra large mat and 1.75 in. apart for the small mat). For one block of trials, participants completed the scaling task, and for the other block of trials, they completed the no scaling task. To allow for comparison between the tasks, true test locations were the same for each block of trials. The order of tasks was counterbalanced across participants, and the order of locations was randomized within task blocks for each participant.

2.1.4. Warm-up At the beginning of the experiment, one experimenter pulled back the curtain that divided the room, exposing the mats from both enclosures. Both mats were either extra large or small (depicting a no scaling trial), depending on the experimental condition. The participant stood at the curtain opening between the two sides of the room. The experimenter directed the participant’s attention to the mats and said, “Can you look at both of these mats and see that they look exactly the same?” The participant was then told that he or she would “see an object on one mat and should try to remember exactly where it goes because you will have to put another object in exactly the same place on the other mat.”

2.1.7. Coding 2.1.7.1. Reversals. Participants’ placements were measured to the nearest ¼-inch using the ruler attached to the underside of each mat. As in other studies of spatial scaling (Frick & Newcombe, 2012; Möhring et al., 2014, 2015), the 4- and 5-year-olds occasionally made reversal errors. There were three types of reversal errors: opposite-side mirror reversals, same-side mirror reversals, and side reversals (see Fig. 2). Opposite-side mirror reversals were on the wrong side of the mat but in the correct location relative to the opposite edge of the mat (Panel A). For example, a child might place an object in the leftmost location on one side of the mat that was supposed to be placed in the rightmost location on the other side of the mat. Same-side mirror reversals were placements that were on the correct side of the mat, but had the outermost location substituted for the innermost location

2.1.5. Demonstration and practice trials One experimenter closed the curtain dividing the room and asked participants to stand on the X marked on the floor and look at the object that was on the learning mat. The object was either a picture of a present or a tiger. The order of presentation of the objects was randomized across participants. Demonstration and practice locations were randomly selected from seven possible positions halfway between adjacent test locations. Children first completed a no scaling demonstration trial to ensure that they understood the task. They were told to “look at the object and remember exactly where it goes.” The experimenter then showed children an identical object and explained to them that they would need to put the new object in the same place on the other mat. The experimenter then walked children over to the test enclosure and had them stand on the X while the experimenter placed the object in the correct location, demonstrating to children how to complete the no scaling task. Next, children completed a no scaling practice trial (using the same objects and mats). Children walked over to the learning enclosure to see that the object was located in a new location on the learning mat. They were told that it was their turn, and that they should try to remember the location so that they can put the identical object in the same place on the other mat. Children then walked over to the test enclosure and stood on the X. They were then allowed to step off of the X and replace

A.

B.

C.

Test (reversal error)

Test (reversal error)

Test (reversal error)

Learning (true location)

Learning (true location)

Learning (true location)

Fig. 2. An example of the three types of reversal errors: opposite-side mirror reversals (Panel A), same-side mirror reversals (Panel B) and side reversals (Panel C). The dashed line indicates the midline of the mat and was not visible during the experiment. 42

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These scores were calculated by determining the absolute distance between each remembered location and the corresponding actual location for each trial. We then averaged the difference scores for the eight locations for the scaling task and again for the no scaling task. Consistent with Weber’s Law, the magnitude of error was scaled to the length of the test mat, meaning that both children and adults made proportionally larger errors on the extra-large mat than on the small mat. Therefore, scores were transformed into proportion scores by dividing each participant’s absolute error for each task by the total length of the mat. Scaling the errors to the size of the test mat allowed us to make direct statistical comparisons of error across the scaling up and scaling down conditions (which involved different sized test mats).

Table 1 Number of child reversal errors in scaling and no scaling tasks for scaling up and scaling down conditions in Experiment 1–3. Experiment and condition

Scaling

No scaling

Total

Experiment 1 Scaling up Scaling down

13 10

5 10

18 20

Experiment 2 Scaling up Scaling down

5 10

8 9

13 19

Experiment 3 Scaling up Scaling down

8 2

9 6

17 8

Grand total

48

47

95

2.2. Results To test for differences in error across the scaling and no scaling tasks for each condition (scaling up and scaling down) and age, proportional error scores were entered into an Age (children, adults) × Condition (scaling up, scaling down) × Task (scaling, no scaling) mixed model ANOVA with age and condition as between-participants factors and task as a within-participants factor.2 As would be expected, there was a significant main effect of age, F (1, 92) = 288.03, p < .0001, ηp2 = 0.76. Four- and 5-year-olds (Merror = 0.086, SD = 0.024) exhibited significantly greater overall proportional error than adults (Merror = 0.034, SD = 0.013). There was also a significant effect of task, F (1, 92) = 19.02, p < .0001, ηp2 = 0.17, which was subsumed under a significant Condition × Task interaction, F (1, 92) = 7.50, p < .01, ηp2 = 0.075. Simple effects tests revealed a significant effect of task for the scaling up condition, F (1, 46) = 26.78, p < .0001, ηp2 = 0.37, but not for the scaling down condition, F (1, 46) = 1.24, ns. As shown in Fig. 3, both children and adults had more difficulty reproducing locations of objects in the scaling task (Merror = 0.061, SD = 0.03) than in the no scaling task (Merror = 0.046, SD = 0.03) when scaling up from 16 in. to 128 in.. There was no significant difference between the scaling task (Merror = 0.058, SD = 0.036) and the no scaling task (Merror = 0.055, SD = 0.032) when scaling down from 128 in. to 16 in.. These proportional error scores translated into an average absolute error of 7.82 in. in the scaling task and 5.88 in. in the no scaling task for the scaling up condition, and an average absolute error of 0.93 in. in the scaling task and 0.88 in. in the no scaling task for the scaling down condition.

(Panel B). Side reversals were placements that preserved the correct location relative to the other locations on one side of the mat, but were shifted to the other side of the mat (Panel C). For example, these reversals included placements that had the innermost location on the right side of the mat in the outermost location on the left side of the mat. A placement was classified as a reversal if it was closer to the reversed location than to an adjacent (reversed) location. This meant that placements had to be less than 0.875 in. from the reversed location for the small mat and less than 7 in. from the reversed location for the extra large mat. As in previous work, we corrected for these reversal errors by calculating where the object would have been placed relative to the true location without the reversal. We corrected 4.95% of locations for 4- and 5-year-olds (38 out of 768) and 0% for adults (0 out of 768). These corrected placements were used in all analyses. Table 1 shows the number of reversal errors children made in the scaling and no scaling tasks in the scaling up and scaling down conditions (note that adults made no reversal errors in the three experiments). We analyzed the percentage of child reversal errors in Experiment 1 in a Condition (scaling up, scaling down) × Task (scaling, no scaling) mixed model Analysis of Variance (ANOVA) with the first factor as a between-participants factor and the second as a withinparticipants factor. This analysis yielded no significant effects, indicating that reversals were distributed relatively evenly across the tasks and conditions. 2.1.7.2. Outliers. After all reversals were corrected, we classified placement values that were larger than the mean ± 3SDs for each age group, location, and condition as outliers and omitted these values from all analyses. We omitted 1.56% of locations for 4- and 5-year-olds (12 out of 768) and 1.04% for adults (8 out of 768). In addition, we omitted one location for a 4-year-old because an experimenter error occurred on that trial.

2.3. Discussion The goals of this experiment were twofold: (1) to investigate whether both young children and adults had more difficulty scaling up than down when the scale difference between two spaces was very large (1:8), and (2) to determine whether young children had more difficulty with the scaling than the no scaling task compared to adults. With respect to the first question, we found that both children and adults exhibited more error on the scaling than on the no scaling task when scaling up but not when scaling down (i.e., an interaction between task and condition). With respect to the second question, we did not find a larger difference between the scaling and no scaling tasks for children than for adults (i.e., no interaction between age and task). Rather, adults were more accurate overall than the 4- and 5-year-old children

2.1.7.3. Absolute error scores. Participants received an absolute error score for each test trial, reflecting the degree to which they placed objects near their actual locations on the test mat. We focused on absolute (unsigned) error rather then directional (signed) error in order to test how the direction and extent of the scale translation impacted the overall accuracy of placements in the scaling and no scaling tasks.1 1 Analyses of directional error for Experiment 1 (and subsequent experiments) revealed that adults always exhibited bias toward the centers of the two halves of the mat, regardless of whether the task involved scaling or no scaling. The 4and 5-year-olds were far less systematic, sometimes exhibiting bias toward the centers of the two halves of the mat and sometimes exhibiting bias toward the center of the entire mat. There were no discernable patterns based on task or size of the learning and test mats. In general, these findings are consistent with well-established findings showing that adults subdivide spaces such as those used here into two halves, whereas children under the age of 10 years tend to

(footnote continued) treat such spaces as a whole (e.g., Huttenlocher et al., 1994). 2 Analysis of raw absolute error scores in separate Age (children, adults) × Task (scaling, no scaling) ANOVAs for the scaling up and scaling down conditions revealed an identical pattern of results to the analysis of proportional error scores in Age (children, adults) x Task (scaling, no scaling) x Condition (scaling up, scaling down) ANOVAs in all experiments reported in this paper. 43

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experiment. A medium mat (32 in. long × 4 in. wide) and a small mat (16 in. long × 2 in. wide) were used as referent mats. The mats and objects were identical except for scale. Objects placed on the medium mats were 2 in. in diameter, and objects placed on the small mats were 1 in. in diameter. The test locations were 3.50 in. apart on the medium mats and 1.75 in. apart on the small mats.

0.1 0.09 0.08

Proportional Error

0.07 0.06 0.05

No Scaling (1:1)

3.1.3. Design and procedure Participants were randomly assigned to one of two conditions: scaling up or scaling down. In the scaling up condition, the scaling task involved learning locations on the small mat and placing them on the medium mat (1:2 scaling ratio), and the no scaling task involved learning locations on the medium mat and placing them on another medium mat. In the scaling down condition, the scaling task involved learning locations on the medium mat and placing them on the small mat (2:1 scaling ratio), and the no scaling task involved learning locations on the small mat and placing them on another small mat. All aspects of the procedure for the scaling and no scaling trials were the same as the previous experiment.

Scaling (1:8)

0.04 0.03 0.02 0.01 0

Adults

Children

Scaling Up Condition

Adults

Children

Scaling Down Condition

Fig. 3. Mean proportional error in scaling and no scaling tasks for children and adults when scaling up from 16 to 128 in. and scaling down from 128 to 16 in..

when reproducing locations in the scaling and no scaling tasks. Given that the scale difference between the two mat sizes was the same regardless of whether participants reproduced locations on the extra large or small test mat (1:8), the absolute scale difference between the learning and test mats cannot account for why children and adults exhibited significantly more error when scaling up than when scaling down. However, previous research suggests that decreasing the scale difference between learning and test spaces affects how children reproduce locations (DeLoache, Kolstad, & Anderson, 1991; Marzolf & DeLoache, 1994; Möhring et al., 2014, 2015; Vasilyeva & Huttenlocher, 2004). For example, Vasilyeva and Huttenlocher (2004) found that 4and 5-year-old children made smaller scaling errors when the learning map and test rug were more similar in scale. Likewise, Möhring et al. (2015) found that 4- and 5-year-olds’ localization errors decreased as the scale ratio decreased. Together, these studies suggest that the scale difference between two spaces may play an important role in scaling. We conducted a second experiment to examine whether decreasing the scale difference between the learning and test mats would facilitate participants’ ability to scale distance, especially when scaling up. Children and adults again performed scaling and no scaling tasks, either scaling up with a ratio of 1:2 or scaling down with a ratio of 2:1.

3.1.4. Coding and measures The coding and measures were identical to those used in the previous experiment. Table 1 shows reversal errors for the scaling and no scaling tasks in the scaling up and scaling down conditions. We again analyzed the percentage of child reversal errors in a Condition (scaling up, scaling down) × Task (scaling, no scaling) mixed model ANOVA. This analysis yielded no significant effects. We corrected for reversal errors in the same manner as in Experiment 1. We corrected reversal errors for 4.12% of locations for 4- and 5-year-olds (32 out of 768) and 0% for adults (0 out of 768). These corrected scores were used in all analyses. After all reversals were corrected, we classified placement values that were larger than the mean ± 3SDs for each age group, location, and condition as outliers and omitted these values from all analyses. We omitted 1.43% of locations for 4- and 5-year-olds (11 out of 768) and 0.13% for adults (1 out of 768). The data analysis strategy was identical to that used in Experiment 1. 3.2. Results To test for differences in error across the scaling and no scaling tasks for each condition (scaling up and scaling down) and age, proportional error scores were entered into an Age (children, adults) × Condition (scaling up, scaling down) × Task (scaling, no scaling) mixed model ANOVA with age and condition as between-participants factors and task as a within-participants factor. As would be expected, there was a significant main effect of age, F (1, 92) = 256.008, p < .0001, ηp2 = 0.74. Four- and 5-year-olds (Merror = 0.072, SD = 0.018) exhibited significantly greater overall proportional error than adults (Merror = 0.027, SD = 0.007). In contrast to Experiment 1, there was no effect of task, F (1, 92) = 0.052, ns, and no Task × Condition interaction, F (1, 92) = 0.871, ns. As shown in Fig. 4, both children and adults reproduced locations very similarly in the scaling and no scaling tasks when scaling down from 32 in. to 16 in. and when scaling up from 16 in. to 32 in.. The average proportional error in the no scaling and the scaling tasks was 0.048 (SD = 0.029) and 0.051 (SD = 0.024) for the scaling up condition and 0.051 (SD = 0.031) and 0.05 (SD = 0.029) for the scaling down condition. These proportional error scores translated into an average absolute error of 1.54 in. in the no scaling task and 1.62 in. in the scaling task for the scaling up condition, and an average absolute error of 0.82 in. in the no scaling task and 0.79 in. in the scaling task for the scaling down condition.

3. Experiment 2 3.1. Method 3.1.1. Participants The participants were forty-eight 4- to 5-year-old children and 48 adults. There were 24 children and 24 adults in both the scaling up and scaling down conditions. The mean ages were 5 years 1 month (range = 4 years 2 months to 5 years 11 months; 25 girls, 23 boys) and 19 years 2 months (range = 18 years 2 months to 22 years 10 months; 25 women, 23 men). Data from two additional 4-year-olds were excluded due to failure to complete the task and experimenter errors during task administration. Children and adults were recruited and consented in the same manner as in the previous experiment. Ninety percent of the children were European American, 2% were African American, 4% were Asian American, and 4% were Hispanic/Latino. Four percent of mothers had completed their high school education or less, 21% had completed some college education, and 75% had a 4year-college education or beyond. Eighty percent of adult participants were European American, 6% were Asian American, 2% were American Indian/Alaska Native, 8% were Hispanic/Latino, and 4% were reported as Other.

3.3. Discussion

3.1.2. Apparatus and materials The experimental room was the same as that used in the previous

The goal of the second experiment was to test whether decreasing the scale difference between the learning and test mats would facilitate 44

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as in the previous experiments. Eighty-nine percent of the children were European American, 4% were Asian American, 2% were African American, 2% were American Indian, and 3% were reported as Other. Seven percent of mothers had completed their high school education or less, 17% had completed some college education, and 76% had a 4-year college education or beyond. Seventy-one percent of adult participants were European American, 8% were Asian American, 4% were African American, 2% were American Indian, 10% were Hispanic/Latino, and 5% were reported as Other.

0.1 0.09 0.08

Proportional Error

0.07 0.06 0.05

No Scaling (1:1) Scaling (1:2)

0.04 0.03

4.1.2. Apparatus and materials The same experimental room was used as in the previous experiments. Three different sized mats were used in this experiment (medium: 32 in. long × 4 in. wide, large: 64 in. long × 8 in. wide, and extra-large: 128 in. long × 16 in. wide). The mats and objects were identical except for scale. Objects placed on the medium, large, and extra-large mats were 2, 4, and 8 in. in diameter, respectively. The test locations were 3.50 in. apart on the medium mats, 7 in. apart on the large mats, and 14 in. apart on the extra-large mats.

0.02 0.01 0

Adults

Children

Scaling Up Condition

Adults

Children

Scaling Down Condition

Fig. 4. Mean proportional error in scaling and no scaling tasks for children and adults when scaling up from 16 to 32 in. and scaling down from 32 to 16 in..

participants’ ability to scale up. Children and adults again completed the scaling and no scaling tasks, but the scale ratio for the scaling task was reduced to 1:2 or 2:1. Unlike Experiment 1, children and adults no longer exhibited more error on the scaling task than on the no scaling task when scaling up than when scaling down. Thus, when the scale difference between the learning and test mat was reduced, scaling up was no longer more difficult than scaling down. Again, scaling condition did not interact with age, indicating that the pattern of responding in the scaling up and scaling down conditions did not differ significantly by age. Like the previous experiment, 4- and 5-year-old children exhibited significantly larger error overall than adults when reproducing locations in both the scaling and no scaling tasks. These findings suggest that reducing the scale difference between the learning and test mats made scaling up easier. However, it is important to note that while reducing the scale difference, we also reduced the absolute size of the larger test mat. Why might reducing the size of the larger test mat eliminate the difference between scaling up and scaling down? To understand why this might be the case, it is important to note that the scaling task required participants to use a minimum of two reference points (i.e., the edges of the mat) to map distances between the two spaces. When people are scaling up, they may have difficulty mapping distance from a smaller to a larger test mat if the edges on the larger space are not simultaneously viewable (as in Experiment 1), but not if the edges on the larger space are simultaneously viewable (as in Experiment 2). We further tested this hypothesis by asking participants to either scale up from 32 in. to 64 in. or to scale down from 128 in. to 64 in.. As in Experiment 2, the scale ratio between the learning and test mats was 1:2 and 2:1, but the size of the test mat in both conditions was larger (64 in. rather than 32 in.) and the same across conditions. If the size of the test mat is the critical factor, then participants should have difficulty both with scaling up and scaling down.

4.1.3. Design and procedure Participants were randomly assigned to one of two conditions: scaling up or scaling down. In the scaling up condition, the scaling task involved learning locations on the medium mat and placing them on the large mat, and the no scaling task involved learning locations on the large mat and placing them on another large mat. In the scaling down condition, the scaling task involved learning locations on the extralarge mat and placing them on the large mat, and the no scaling task involved learning locations on the large mat and placing them on another large mat. All other aspects of the experiment were identical to the previous experiments. 4.1.4. Coding and measures The coding and measures were identical to those used in the previous experiments. Table 1 shows reversal errors for the scaling and no scaling tasks in the scaling up and scaling down conditions. Again, an analysis of the percentage of child reversal errors in a Condition (scaling up, scaling down) × Task (scaling, no scaling) mixed model ANOVA yielded no significant effects. As in the previous experiments, we corrected for reversal errors. We corrected 3.26% of locations for the 4- and 5-year-olds (25 out of 768) and 0% for adults (0 out of 768). These corrected scores were used in all analyses. After all reversal errors were corrected, we classified placement values that were larger than the mean ± 3SDs (rounded to the nearest 0.25 in.) for each age group, condition, and location as outliers and omitted these values from all analyses. We omitted 1.3% of locations for 4- and 5-year-olds (10 out of 768) and 0.91% for adults (7 out of 768). In addition, for three 4- and 5-year-olds and one adult, we omitted one location because an experimenter error occurred on that trial. The data analysis strategy was identical to that used in the previous experiments. 4.2. Results

4. Experiment 3

To test for differences in error across the scaling and no scaling tasks for each condition (scaling up and scaling down) and age, proportional error scores were entered into an Age (children, adults) × Condition (scaling up, scaling down) × Task (scaling, no scaling) mixed model ANOVA with age and condition as between-participants factors and task as a within-participants factor. As would be expected, there was again a significant main effect of age, F (1, 92) = 203.27, p < .0001, ηp2 = 0.69. Four- and 5-year-olds (Merror = 0.064, SD = 0.017) exhibited significantly greater overall proportional error than adults (Merror = 0.026, SD = 0.007). There was also a significant main effect of task, F (1, 92) = 40.33, p < .0001, ηp2 = 0.31, but no significant Condition × Task interaction, F (1, 92) = 0.254, ns. There was also a significant Age × Task interaction, F (1, 92) = 4.29, p = .041,

4.1. Method 4.1.1. Participants The participants were forty-eight 4- to 5-year-old children and 48 adults. There were 24 children and 24 adults in both the scaling up and scaling down conditions. The mean ages were 4 years 11 months (range = 4 years 3 months to 5 years 8 months; 26 girls, 22 boys) and 19 years 8 months (range = 18 years 4 months to 22 years 5 months; 23 women, 25 men). Five additional 4-year-olds and two additional 5year-olds were excluded because they did not complete the task. Children and adults were recruited and consented in the same manner 45

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and adults no longer had difficulty scaling up or down when we reduced the scaling ratio to 1:2 or 2:1 (i.e., when they learned the locations on a 16-inch mat and replaced them on a 32-inch mat or learned the locations on a 32-inch mat and replaced them on a 16-inch mat). In our third experiment, we found that children and adults had difficulty with both scaling up and down when we kept the scale ratio the same (1:2 or 2:1) but increased the size of the test mat (i.e., when they learned the locations on either a 32-inch or a 128-inch mat and replaced them on a 64-inch mat). Across all three experiments, scaling condition did not interact with age, indicating that the patterns of responding to scaling up and scaling down did not differ between early childhood and adulthood. Contrary to our expectations, we did not find a larger difference between the scaling and no scaling tasks for children than for adults (i.e., no interaction between age and task). Rather, the 4- and 5-year-old children were less accurate overall than adults when reproducing locations in the scaling and no scaling tasks. We began this investigation with the hypothesis that scaling up would be more difficult than scaling down, especially when the scale ratio between the two mats was larger. Although the results of Experiment 1 were consistent with this prediction and with the Siegel et al. (1979) study showing that it was easier to scale down from a room-sized layout to a small-scale model than vice versa, Experiments 2 and 3 revealed that the absolute size of the test space determined whether children and adults had difficulty with scaling up or down. How might we explain the discrepancy between these two sets of findings? The small-scale model used by Siegel et al. (1979) was 32 × 40 in. and the room layout was 180 × 240 in.. In the present investigation, we observed no difference between scaling up and scaling down for test spaces 32 in. and smaller. The Siegel et al. (1979) room layout exceeded our 32-inch test space size, but the model space was very close to our 32-inch test space size. When children in the Siegel et al. study were scaling down from the room to the model, they would be able to keep both edges of the model in view at the same time while placing the objects, whereas this was not the case when they were scaling up from the model to the room. Had Siegel et al. used a larger model space, they might have seen similar placement errors when scaling up and scaling down. Likewise, the small test space sizes used in Möhring et al. (2016) might help explain why they found no difference between their small-constant and large-constant conditions in their computer-based discrimination paradigm. A major question our findings raise is why do larger test spaces disrupt the scaling process? In our investigation, participants first had to form a representation of the object location on the learning mat. They then had to use that representation to place the corresponding object on the test mat. When the two mats were identical in size (the no scaling task), participants needed to place the object in the same absolute location on the test mat. To do so, participants could simply use an absolute distance strategy which would involve coding the distance from one edge of the learning mat and then transferring that distance to the test mat. When the two mats differed in size (the scaling task), participants needed to place the object in the same relative location on the test mat. To do so, participants could use a mental transformation strategy to shrink or expand the mentally represented learning mat to fit the visually available test mat (Möhring et al., 2016).4 In this process, the visually available test mat serves as the perceptual anchor during the scaling translation. We hypothesize that this perceptual anchor is less well-grounded when the edges of the test mat are not

0.1 0.09 0.08

Proportional Error

0.07 0.06 0.05

No Scaling (1:1) Scaling (1:2)

0.04 0.03 0.02 0.01 0

Adults

Children

Scaling Up Condition

Adults

Children

Scaling Down Condition

Fig. 5. Mean proportional error in scaling and no scaling tasks for children and adults when scaling up from 32 to 64 in. and scaling down from 128 to 64 in..

ηp2 = 0.045.3 As shown in Fig. 5, both children and adults had more difficulty reproducing locations of objects in the scaling task than in the no scaling task when scaling up from 32 in. to 64 in. and when scaling down from 128 in. to 64 in.. The average proportional error in the no scaling and the scaling tasks was 0.038 (SD = 0.02) and 0.051 (SD = 0.029) for the scaling up condition and 0.036 (SD = 0.02) and 0.047 (SD = 0.024) for the scaling down condition. These proportional error scores translated into an average absolute error of 2.41 in. in the no scaling task and 3.25 in. in the scaling task for the scaling up condition, and an average absolute error of 2.29 in. in the no scaling task and 3.01 in. in the scaling task for the scaling down condition. 4.3. Discussion The results of this experiment clearly show that increasing the size of the test mat overall while keeping the scale ratio constant between the learning and test mat resulted in difficulty with both scaling up and scaling down relative to the no scaling task. Again, although children exhibited larger error overall than adults, they exhibited the same pattern of performance as adults in the scaling up and scaling down conditions (i.e., scaling condition did not interact with age). Combined with the results of Experiment 2, these findings confirm that scaling up is not more difficult than scaling down. Rather, it appears that the process of scaling distance from one space to another space is disrupted when children and adults cannot simultaneously view both edges of the second space while making the scale translation. We discuss this pattern of findings in more detail in the General Discussion. The results of all experiments are summarized in Table 2 below. 5. General discussion The goal of this investigation was to further examine the cognitive processes underlying the ability to scale distance along a single dimension. We asked two questions: (1) Is scaling up more difficult than scaling down? and (2) Do patterns of scaling performance differ between young children and adults? In our first experiment, we found that both children and adults had difficulty scaling up with a 1:8 ratio (i.e., when they learned the locations on a 16-inch mat and replaced them on a 128-inch mat), but no difficulty scaling down with an 8:1 ratio (i.e., when they learned the locations on a 128-inch mat and replaced them on a 16-inch mat). In our second experiment, we found that children

4 Participants could also use a relative distance strategy to perform the scaling task. Larger test spaces may also disrupt use of a relative distance strategy because participants may be slower and less accurate in estimating proportions when both edges of the space are not simultaneously viewable (e.g., estimating one third of the distance across a room). Hence, the perceptual anchoring hypothesis is agnostic with respect to the mental transformation and relative distance strategies.

3

However, simple effects tests failed to reveal the source of the interaction. When broken down by task, both children and adults exhibited significantly greater proportional error in the scaling task than in the no scaling task. When broken down by age, children exhibited significantly more error than adults in both the no scaling task and in the scaling task. 46

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Table 2 Summary of results for Experiment 1–3. Experiment

Task

Findings

No scaling

Scaling

Learning mat

Test mat

Learning mat

Test mat

Experiment 1 Scaling up (1:8) Scaling down (8:1)

128 16

128 16

16 128

128 16

Significantly more error in scaling than no scaling task No significant difference between scaling and no scaling tasks

Experiment 2 Scaling up (1:2) Scaling down (2:1)

32 16

32 16

16 32

32 16

No significant difference between scaling and no scaling tasks No significant difference between scaling and no scaling tasks

Experiment 3 Scaling up (1:2) Scaling down (2:1)

64 64

64 64

32 128

64 64

Significantly more error in scaling than no scaling task Significantly more error in scaling than no scaling task

Note. All mat sizes are in inches.

Our second hypothesis that children would be less accurate across the board in the scaling tasks than in the no scaling tasks was not supported. Like adults, when the test mats were 32 in. or smaller, the 4to 5-year-olds performed very similarly on the memory and the scaling tasks, but when the test mats were 64 in. or larger, the 4- to 5-year-olds exhibited more error on the scaling task than on the no scaling task (as did adults). As would be expected, the 4- to 5-year-olds were less accurate overall than adults in their placements across all conditions and tasks, which is consistent with a large body of work showing improvement in metric coding of location over development (Hund & Plumert, 2002; Huttenlocher, Newcombe, & Sandberg, 1994; Plumert, 2008; Plumert, Hund, & Recker, 2007; Schutte, Simmering, & Ortmann, 2011; Spencer & Hund, 2003). Children also made reversal errors in both the scaling and no scaling tasks, whereas adults never made such errors. This is consistent with other studies that have reported reversal errors on scaling tasks with children in this age range (Frick & Newcombe, 2012; Möhring et al., 2014, 2015). The fact that the size of the test mat impacted young children and adults very similarly lends support to the idea that visual scaling processes interact with the properties of the visual system and the perceptual structure of the space. Overall, our findings are consistent with other work showing that visual scaling skills are in place early in development (Frick & Newcombe, 2012; Gilligan, Hodgkiss, Thomas, & Farran, 2018; Huttenlocher et al., 1999; Vasilyeva & Huttenlocher, 2004), but add to the literature by showing that the basic cognitive processes involved in visual scaling appear to be age-independent, at least for the distances used here. Nonetheless, the accuracy and precision of metric coding undergoes developmental change, regardless of whether the task involves coding relative or absolute distance. A final issue concerns the fact that we did not find effects of scale magnitude on error. For example, a comparison of Experiments 1 and 2 for the scaling down condition clearly showed that children and adults exhibited almost no difference between the scaling and no scaling tasks when they went from 128 in. to 16 in. (8:1) and from 32 in. to 16 in. (2:1). In fact, the difference between the no scaling and scaling tasks was 0.05 in. for the 8:1 scale ratio and 0.03 in. for the 2:1 scale ratio. Thus, they performed nearly identically across two very different scale ratios.5 This is consistent with the Frick and Newcombe (2012) study showing no difference between scaling performance with 1:2 and 1:4 scaling ratios, but inconsistent with other studies showing poorer performance with larger scaling ratios (Möhring et al., 2015, 2016; Vasilyeva & Huttenlocher, 2004). We suspect that these inconsistencies

easily viewable at the same time, leading to more error in making the scale translation. Making even small eye or head movements in order to view both edges of the test mat may be enough to disrupt the mapping between the mentally represented learning mat and the visually available test mat. As other work on visual attention has shown, head or eye movements require people to hold information in visual working memory, adding to the demands of the task and reducing the precision of responses (Van der Stigchel & Hollingworth, 2018). This perceptual anchoring hypothesis is supported by constraints on peripheral vision imposed by the structure of the human visual system (Rosenholtz, 2016). To show how these constraints operated here, we calculated the viewing angles to the mat edges for all of the mat sizes. Based on national averages for height, we assumed an eye height of 60 in. for adults and 37 in. for 4- to 5-year-olds. We also took into account standing position (21 in. from mat center) and the width of the mat (mat width was scaled to mat length by a factor of 1:8). These calculations indicated that for adults and children, the viewing angles to the edges of the 16-inch mats were within or close to the macular region of the visual field (8–18° of visual angle), and the viewing angles for the 32-inch mats were well within the near-peripheral region (18–60° of visual angle). Though not within the foveal region, these eccentricities easily support simultaneous viewing of both edges of the mat. In contrast, the viewing angles to the edges of the 64- and 128-inch mats were near or within the mid-peripheral region (60–120° of visual angle), making it very difficult to hold both edges of the mat within view at the same time. An obvious question our findings raise is why didn’t the size of the learning mat matter? In the most extreme case, we found that participants were equally good at placing the objects when they went from a 128-inch learning mat to a 16-inch testing mat (8:1 scaling ratio) as when they went from a 16-inch learning mat to a 16-inch testing mat. Thus, it appears that mat size did not impact the original encoding of the location during learning. Rather, it appears that mat size impacted the subsequent scaling of the location at test. As noted above, we hypothesize that the process of bringing the two differently-sized spaces into register with one another is be disrupted when head movements are required to view both edges of the perceptual anchor. These findings are consistent with previous research highlighting the importance of a continuous perceptual standard on young children’s ability to encode relative information (e.g., Duffy, Huttenlocher, & Levine, 2005; Sophian, 2000). An important question for future research is whether this pattern of results also holds in scaling tasks that do not involve a significant memory component such as those used by Möhring et al. (2014, 2015, 2016). Further research using eye tracking would be helpful for determining how looking behavior contributes to the scale translation process in scaling tasks with and without a significant memory component.

5 Note, however, that we did not design our studies to test for a linear effect of the scale magnitude, since we did not include an intermediate scale magnitude (e.g., 64 in. to 16 in.). Inclusion of an intermediate scale magnitude might have revealed a linear effect of scale magnitude on error.

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arise at least in part from differences in task demands. Performance differences based on scale ratio are most evident when cognitive demands are high, such as in discrimination tasks where the visuospatial details require extreme precision from adults (e.g., Möhring et al., 2016) or in scaling tasks involving large two-dimensional spaces that are challenging for young children (Vasilyeva & Huttenlocher, 2004). Further research is needed to understand why the extent of the scale translation matters in some studies but not in others. In conclusion, the results of this investigation add to a growing body of research on spatial scaling in children and adults and to our understanding of spatial skills more generally. Our results show that scaling up is not inherently more difficult than scaling down, but that it is harder to scale up and down when the edges of the test space are not easily viewable at the same time. On a theoretical level, these findings show that visually scaling distance involves a complex interplay of cognitive processes and visual constraints that appear to operate similarly in children and adults. Critically, this investigation revealed a perceptual anchoring process that likely operates when visually scaling distance in both small and large spaces, but whose effects are only seen with the challenges of visually scaling distance in large spaces. As such, this work adds to our understanding of the basic processes underlying fundamental spatial skills that are critical for both everyday functioning and STEM learning. On a practical level, this suggests that children and adults may more accurately understand scaled distances depicted in smaller rather than larger representations. For example, it may be easier to appreciate the relative distances between planets in our solar system when they are represented in a small model on a table than when they are represented in a large model in a museum, especially at closer viewing distances. As such, these results have potentially important implications for how scaled representations are designed in learning contexts.

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Acknowledgements This research was supported by grants awarded to Jodie M. Plumert from the National Institutes of Health (R03-HD36761) and the National Science Foundation (BCS-0343034). We especially thank Jessica Flathau, Aldrin Roman, Hanxi Tang, and Breanna Williams for their help with data collection. We also thank the children and adults for their participation in these studies. Declarations of interest None. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.cognition.2018.12.013. References Boyer, T. W., & Levine, S. C. (2012). Child proportional scaling: Is 1/3 2/6 3/9 4/12? Journal of Experimental Child Psychology, 111, 516–533. Bundesen, C., & Larsen, A. (1975). Visual transformation of size. Journal of Experimental Psychology: Human Perception and Performance, 1, 214–220. DeLoache, J. S., Kolstad, V., & Anderson, K. N. (1991). Physical similarity and young children’s understanding of scale models. Child Development, 62, 111–126. Duffy, S., Huttenlocher, J., & Levine, S. (2005). It is all relative: How young children encode extent. Journal of Cognition and Development, 6, 51–63. Frick, A. (2018). Spatial transformation abilities and their relation to later mathematics performance. Psychological Research. https://doi.org/10.1007/s00426-018-1008-5. Frick, A., & Newcombe, N. (2012). Getting the big picture: Development of spatial scaling

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