Visually Scaling Distance from Memory: Do Visible Midline Boundaries Make a Difference?

Abstract

We examined how 4- to 5-year-old children and adults use perceptual structure (visible midline boundaries) to visually scale distance. Participants completed scaling and no scaling tasks using learning and test mats that were 16 and 64 inches. No boundaries were present in Experiment 1. Children and adults had more difficulty in the scaling than no scaling task when the test mat was 64 inches but not 16 inches. Experiment 2 was identical except visible midline boundaries were present. Again, participants had more difficulty in the scaling than no scaling task when the test mat was 64 inches, suggesting they used the test mat edges (not the midline boundary) as perceptual anchors when scaling from the learning to the test mat.

1. Introduction

The ability to scale distance is fundamental for using maps, models, and diagrams to understand and represent the external world. For example, we might use a model to represent our solar system or use a map to plan a route. In order to successfully use these scaled representations, individuals must understand how distances in two differently-sized spaces map onto one another. Scaling skills appear early in development, with studies showing that children as young as 3.5 years can visually scale distance along a single dimension when the distances are relatively small, though precision increases substantially with age (Frick & Newcombe, 2012; Huttenlocher, Newcombe, & Vasilyeva, 1999; Huttenlocher, Vasilyeva, Newcombe, & Duffy, 2007; Plumert, Hund, & Recker, 2019; Vasilyeva & Huttenlocher, 2004). Spatial scaling has also been implicated in successful science, technology, engineering, and mathematics (STEM) performance (e.g., Boyer & Levine, 2012; Frick, 2018; Hodgkiss, Gilligan, Tolmie, Thomas, & Farran, 2018; Möhring, Newcombe, Levine, & Frick, 2016; National Research Council, 2012), underscoring the importance of understanding the cognitive processes underlying scaling skills. At present, relatively little is known about how children and adults visually scale distances when the edges of the spaces are not easily viewable at the same time. The goal of this project was to further understand the cognitive processes involved in visual scaling by examining whether midline boundaries that subdivide spaces make it easier to scale distance.

Recent work has focused on the strategies that might be used to visually scale distance between two spaces that differ in size. One possibility is a proportional reasoning strategy, which involves visually coding distance relative to two reference points (e.g., landmarks or edges) in one space and then mapping this relative distance onto another space using the corresponding reference points (Gilligan, Hodgkiss, Thomas, & Farran, 2018; Huttenlocher et al., 1999; Huttenlocher et al., 2007; Spetch & Parent, 2006; Uttal, Sandstrom, & Newcombe, 2006). For example, 4-year-old children use relative distance to search in the middle of two landmarks when the distance between the landmarks is doubled, indicating that they code the location as halfway between the landmarks (Uttal et al., 2006; see also Frick & Newcombe, 2012; Möhring, Newcombe, & Frick, 2016). Another possible strategy is a mental transformation strategy, which involves mentally shrinking or expanding (i.e., transforming) the original space to match the size of the other space (Möhring et al., 2016; Vasilyeva & Huttenlocher, 2004). As is the case for mental transformations such as mental rotation and image scanning, larger transformations take longer and produce greater error (e.g., Bundesen & Larsen, 1975; Kosslyn, 1975; Shepard & Metzler, 1971). Consistent with this notion, Möhring, Newcombe, and Frick (2014, 2015) found that scaling errors increased linearly as the scale ratio increased for preschool children and adults. Similarly, Möhring et al. (2016) found that the speed and accuracy of adults’ discrimination responses varied linearly with the scaling magnitude, providing additional support for the mental transformation strategy. These findings suggest that both proportional reasoning and mental transformation strategies can be used to visually scale distances.

One question that has not received much attention is how the size of the space might affect the scaling of distance, especially when the spaces involved are larger than those that can fit on a computer screen or piece of paper. Recently, Plumert et al. (2019) investigated spatial scaling using mats placed on the floor of a divided testing room that varied in length from 16 to 128 inches. Children and adults first watched an experimenter place an object on a learning mat on one side of the room and then attempted to place a replica object on a test mat on the other side of the room that was either identical (no scaling task) or different in size (scaling task). Any decrement in performance on the scaling relative to the no scaling task reflects the impact of performing a scale translation. Across three experiments, both children and adults were less accurate on the scaling than no scaling task when the test mat was 64 inches in length or larger. In contrast, accuracy was not different for scaling and no scaling tasks when the test mat was 32 inches or smaller, indicating that test mat size, not the direction of scale translation, was the critical factor. What accounts for these results? Plumert et al. (2019) described the robust pattern of findings as a perceptual anchoring effect, noting that when making a scale translation from the learning to the test mat, the visually available test mat serves as a perceptual anchor. This perceptual anchor is less well-grounded when both edges of the test mat are not 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. These findings indicating that the size of the test mat matters are consistent with earlier literature showing that children have difficulty accurately scaling configurations of locations in large spaces such as classrooms (Liben, Moore, & Golbeck, 1982; Siegel, Herman, Allen, & Kirasic, 1979; Uttal, 1994, 1996).

The findings from Plumert et al. (2019) underscore the importance of size (especially the size of the test space) in mapping distances between spaces and provide preliminary support for the perceptual anchoring effect. However, these experiments leave open the question of how children and adults may combine perceptual anchoring with available perceptual structure to visually scale distance. One important type of perceptual structure is visible midline boundaries that divide larger spaces in half, such that the edges of each half of the space can be simultaneously viewed without making head or eye movements. At present, however, it is unknown whether visible midline boundaries facilitate scale translations. In particular, to what extent do children and adults take advantage of a midline boundary to reduce the size of the space, thereby facilitating accurate scaling with larger test spaces?

There is a large literature showing that visible boundaries impact performance in spatial memory tasks (Cohen, Baldwin, & Shermin, 1978; Kosslyn, Pick, & Fariello, 1974; Plumert & Hund, 2001; Simmering & Spencer, 2007). In one early study, Kosslyn et al. (1974) found that adults and children overestimated the distances between locations separated by opaque boundaries that divided a large room into regions and underestimated distances between locations that were in the same region. Similarly, Plumert and Hund (2001) found that older children and adults remembered groups of objects as closer together than they really were when salient boundaries subdivided those objects into groups during learning. Huttenlocher, Hedges, and Duncan (1991) proposed the category adjustment model as one explanation for these subdivision effects. According to this account, subdivision effects result from combining fine-grained coding of individual locations (i.e., precise metric details about the location, such as its distance and direction from a reference point) with category information represented by prototypes at the centers of spatial regions or spatial groups (see also Sampaio & Wang, 2010, 2017). Relying on spatial prototypes results in bias toward category centers (and away from category boundaries), especially when fine-grained coding is less certain. Despite biases in spatial estimates, subdivision is helpful because it improves precision overall by reducing variability in estimates.

Other work has revealed important changes in subdivision effects during development that depend on the size of the space. In a classic set of studies, Huttenlocher, Newcombe, and Sandburg (1994) asked 4-, 6-, and 10-year-old children to search for an object they previously saw hidden in a 5-foot long sandbox. Four- and 6-year-old children exhibited biases toward the center of the sandbox, suggesting they treated the entire box as one region, whereas 10-year-old children exhibited biases toward the centers of the two halves of the sandbox, suggesting they subdivided the box into two regions. However, when 4-, 6-, and 10-year-olds were asked to reproduce a location on a small rectangle (7.87 inches long) drawn on a piece of paper, all three age groups exhibited biases toward the centers of the two halves of the rectangle. This indicates that young children subdivide smaller spaces before they subdivide larger spaces. More recently, Frick and Newcombe (2012) examined subdivision effects in a spatial scaling task involving small one-dimensional fields presented on a piece of paper (i.e., “strips” that were 10.24 inches long). Participants viewed maps with marked locations (i.e., eggs in farm fields) next to the fields and were asked to mark the egg locations in the new fields. On some trials, the maps were the same size as the fields (no scaling trials), and on other trials (scaling trials), the maps were smaller. Three- and 4-year-olds’ responses were biased toward the middle of the field, whereas 5- and 6-year-olds and adults exhibited bias away from the middle of the field, demonstrating the emergence of subdivision effects for small spaces during childhood.

The goal of the present investigation was to examine whether a visible boundary at midline impacts how adults and 4- to 5-year-olds scale distance in smaller and larger spaces. Given previous research showing that adults are highly sensitive to boundaries in spatial memory tasks, we used adults as our primary group for testing the hypotheses of the study. We also tested 4- to 5-year-olds because we know that children this age can scale distance along a single dimension and previous work has shown that the size of the space impacts how young children subdivide spaces (e.g., Huttenlocher et al., 1994). The ages, materials, and tasks were identical to those used in Plumert et al. (2019). 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 to the other section of the room (divided by a curtain) and attempted to place another picture of the object on a test mat of the same size. The scaling task was identical except that the learning and test mats were different sizes. The test mat for both the scaling and the no scaling tasks was always the same size for a given condition, making it possible to directly compare performance across the scaling and no scaling tasks. Any decrease in performance on the scaling relative to the no scaling task reflects the impact of performing a scale translation.

The present experiments compared performance on the scaling and no scaling tasks using learning and test mats that were 16 inches and 64 inches. This combination of mat sizes has not been tested previously and is important for understanding scaling processes. The 64-inch test mat is a critical size because it is wide enough to disrupt perceptual anchoring, which should lead to decrements in scaling performance. However, introducing a visible boundary at its midline would create two halves that were 32 inches wide, a distance that should not disrupt perceptual anchoring. The 16-inch mat is a useful size for comparison purposes because we would expect no disruption of perceptual anchoring for scaling, but 4- and 5-year-olds may exhibit different patterns of spatial bias across 64- and 16-inch mats.

The goal of Experiment 1 was to test whether adults and children have more difficulty in the scaling than in the no scaling task with a test mat size of 64 inches but not 16 inches. This served as a replication test of the pattern of findings from Plumert et al. (2019), providing further support for the importance of perceptual anchoring effects in scaling. The goal of Experiment 2 was to test whether visible boundaries that divided the learning and test mats in half would reduce scaling error, especially for the 64-inch test mat. Given that Plumert et al. (2019) found no difference between the scaling and no scaling tasks when test mats were 32 inches or smaller, we reasoned that adding midline boundaries to the 64-inch test mat might improve scaling performance (demonstrated in smaller absolute error) if participants used the visible midline boundary as a perceptual anchor when making the scale translation. From a proportional reasoning strategy perspective, a visible midline boundary could be especially beneficial for scaling because one could estimate the relative distance using only the edges of one half of the space (i.e., one edge of the mat and the visible midline boundary), not the entire space. However, it is possible that midline boundaries would not improve scaling performance if participants continued to rely on the outer edges of the test mat as perceptual anchors when making the scale translation. From a mental transformation strategy perspective, a visible midline boundary would not likely be beneficial because it would be difficult to shrink or expand only one half of a space during the scaling process. Doing so would disrupt the spatial features of the space overall and violate a variety of object properties.

In addition to analyzing absolute error, we analyzed directional error to examine the pattern of bias evident in placements, providing additional evidence regarding whether adults and young children subdivided the spaces with and without visible midline boundaries in the scaling and no scaling tasks. We expected that adults would demonstrate clear subdivision effects regardless of mat size or boundary presence in both the scaling and no scaling tasks, replicating previous findings from the memory and scaling literature (Frick & Newcombe, 2012; Huttenlocher et al., 1991). In contrast, we expected that children would treat the large mats as one spatial category in both the scaling and no scaling tasks regardless of boundary presence, but might exhibit subdivision effects when the mats were small and included visible midline boundaries, replicating previous spatial memory and scaling findings (e.g., Frick & Newcombe, 2012; Huttenlocher et al., 1994).

2. Experiment 1

2.1. Method

2.1.1. Participants.

Forty-eight 4- to 5-year-old children and 48 adults participated, including 24 children and 24 adults in each of the two experimental conditions. An a priori power analysis using G*Power indicated that a sample of 96 participants would be adequate to detect main effects and the predicted Condition x Task interaction at .80 power, using effect size estimates (η_p² = .08) from Plumert et al. (2019). The mean ages were 5 years 0 months (range = 4 years 6 months to 5 years 11 months; 20 girls, 28 boys), and 19 years 10 months (range = 18 years 3 months to 23 years 7 months; 23 women, 25 men). Data from two additional 4-year-olds and one additional 5-year-old were excluded because they did not complete the task. Ninety-eight percent of the children were European American and 2% were Hispanic. Four percent of mothers had completed their high school education or less, 17% had completed some college education, and 79% had a 4-year college education or beyond. 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. Eighty-six percent of adult participants were European American, 6% were Hispanic, 4% were Asian American, 2 % were African American, and 2% were Native Hawaiian or Pacific Islander. Adults were recruited through psychology courses at a large public university and received course credit for participating.

2.1.2. Apparatus and materials.

The experimental space was an 11.5 ft. x 10.5 ft. laboratory room with floor-to-ceiling white canvas curtains around the perimeter. The curtain also divided the room into two identical sides that were 11.5 ft. x 5.25 ft. each. One side was used during learning and the other side was used during test (see Figure 1). Each side contained a single light-brown, vinyl mat centered on the floor at all times. The size of the mats varied throughout the session and depended on the experimental condition: large mats were 64 in. long x 8 in. wide and small mats were 16 in. long x 2 in. wide. A measuring tape on the underside of each mat (not visible to participants) was included to measure locations to the nearest 1/4 inch along the horizontal dimension. An X on the floor of each section indicated where participants should stand (21 inches from the center of each mat). The setup of the experimental space was identical to that used by Plumert et al. (2019).

Figure 1.

An aerial view of the experimental room illustrating learning and test details for the no scaling (Panel A) and scaling (Panel B) tasks when scaling up in Experiment 1.

Ten pairs of laminated circles containing pictures of objects were used for learning and test, including an apple, ball, butterfly, chicken, fish, ladybug, penguin, present, star, and tiger. We used the present and tiger during practice trials and the eight remaining objects during test trials. Objects placed on the large mats were 4 in. in diameter, and objects placed on the small mats were 1 in. in diameter, so that the diameter of the objects was half the width of each mat. The pairings of objects and locations were randomized across participants. These materials were identical to those used by Plumert et al. (2019).

2.1.3. 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 used a small learning mat and a large test mat, and the no scaling task used large mats for learning and test. In the scaling down condition, the scaling task used a large mat for learning and a small mat for test, and the no scaling task used small mats for learning and test. It is important to note that the test mat size was identical across scaling and no scaling tasks in each condition to facilitate direct comparison of performance across scaling and no scaling trials (a within-participants variable); however, the size of the test mats differed across the scaling up and scaling down conditions (a between-participants variable). We used proportional errors (relative to the test mat size) in all analyses to support comparisons involving test mats of differing sizes (see below for more details). This mixed model design allowed us to test the variables of interest in the most efficient manner possible.¹ All aspects of the design and procedure were identical to those used by Plumert et al. (2019).

All participants were tested individually in the laboratory in a single session lasting approximately 30 minutes. One experimenter was with the participant at all times and gave instructions throughout the session. A second experimenter changed mats when needed, placed objects during learning, and measured participants’ placements at test. The session began by familiarizing participants with the no scaling task then the scaling task. For children, this included one demonstration and one practice trial for each task (described below). Adults did not need the demonstration trials because verbal instructions were sufficient. They completed a single no scaling practice trial followed by a single scaling practice trial. Following the practice trials, participants completed 8 scaling trials and 8 no scaling trials.

2.1.3.1. Demonstration and practice trials.

One experimenter pulled back the curtain that divided the room, exposing the mats from both sections. The participant stood at the opening between the two sides of the room and faced the mats. Both mats were the same size (either large or small), depicting a no scaling trial. The mats’ size depended on the experimental condition. The experimenter directed the participant’s attention to the mats by saying, “Can you look at both of these mats and see that they look exactly the same?” The participant was 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.”

The experimenter then closed the curtain dividing the room and asked participants to stand on the X marked on the floor of the learning space. The experimenter instructed participants to look at the object (a picture of either a present or a tiger) that was on the learning mat. The presentation order of the objects was randomized. Demonstration and practice locations were randomly selected from seven possible positions halfway between adjacent test locations. Children first watched 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 walked children over to the test space and asked them to stand on the X while the experimenter placed the object in the correct location.

Next, children completed a no scaling practice trial using the same objects and mats. Children walked over to the learning space and saw an object 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 could put the identical object in the same place on the other mat. Children walked over to the test space and stood on the X. They were allowed to step off of the X to place the object. Incorrect placements were moved to the correct location. Adults also completed a no scaling practice trial.

Children then watched a scaling demonstration trial to help them understand 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 both mats, which now differed in size. Children stood at the center of the learning mat 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 experimenter closed the curtain, and asked children 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 walked children over to the other space and asked them to stand on the X while they watched the experimenter place the object in the correct location. Following the demonstration trial, children completed a scaling practice trial. Children walked over to the learning space where a new object was located in a new location on the learning mat. They were told to remember the location and walk over to the other space and put the object in the same place on the other mat. Children then placed the object on the test mat. Incorrect placements were moved to the correct location. Adults also completed a scaling practice trial.

2.1.3.2. Test trials.

Participants completed 16 test trials, which included 2 blocks of 8 trials. Each trial block consisted of eight locations equally spaced on the mats (7 inches apart for the large mat and 1.75 inches apart for the small mat). The locations varied along the horizontal axis and were centered along the vertical axis. 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 task blocks was counterbalanced across participants, and the order of locations was randomized within each task block for each participant.

2.1.4. Coding and measures.

2.1.4.1. Reversals.

Participants’ placements were measured to the nearest ¼-inch along the horizontal dimension using the ruler on the underside of each mat. As in other studies of spatial scaling (Frick & Newcombe, 2012; Möhring et al., 2014, 2015; Plumert et al., 2019), the 4- and 5-year-olds occasionally made reversal errors. We used the coding scheme developed by Plumert et al. (2019) to identify three types of reversal errors: opposite-side mirror reversals, same-side mirror reversals, and side reversals (see Figure 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 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 (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. It is important to note that a placement was classified as a reversal only if it was closer to the reversed location than to an adjacent (reversed) location. This meant that placements had to be less than .875 inches from the reversed location for the small mat and less than 3.5 inches from the reversed location for the large mat. As in Plumert et al. (2019), 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 reversal errors for 6.25% of locations for 4- and 5-year-olds (48 out of 768) and 0.52% for adults (4 out of 768). These corrected placements were used in all analyses of absolute and directional errors.

Figure 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 each mat. Visible midline boundaries were present only during Experiment 2.

2.1.4.2. Outliers.

After all reversals were corrected, we classified placement values that were larger than the mean ± 2SDs for each age group, location, and condition as outliers and omitted these values from all analyses. We omitted 4.95% of locations for 4- to 5-year-olds (38 out of 768) and 4.95% for adults (38 out of 768). In addition, we omitted one location for an adult because an experimenter error occurred on that trial.

2.1.4.3. Absolute error scores.

We used absolute (unsigned) error to measure the accuracy of placements. These scores were calculated by first determining the absolute distance between each remembered location and the corresponding actual location for each trial, then dividing by the length of the mat to create a proportion score to facilitate comparison across conditions that included different sized test mats. We averaged the error scores for the eight locations for the scaling task and again for the no scaling task. These absolute error scores reflected the degree to which participants placed objects near their actual locations on the test mat (see Plumert et al., 2019).

2.1.4.4. Directional error scores.

We used signed error to measure directional bias in placements. Participants received a directional (signed) error score, reflecting the degree to which they placed objects in the inner (Locations 4 and 5) and outer (Locations 1 and 8) locations closer to the middle of the mat than they really were. We focused on the inner and outer locations because these are most sensitive to bias in placements (for theoretical rationale, see Huttenlocher et al., 1991). For locations on the left side of the mat, signed error was calculated by subtracting the actual location from the corresponding remembered location. For locations on the right side of the mat, signed error was calculated by subtracting the remembered location from the corresponding actual location. As such, positive values reflected bias toward the center of the mat, whereas negative values reflected bias away from the center of the mat. The signed errors were divided by the length of the mat to create proportion scores.

2.2. Results

2.2.1. Absolute error.

The goal of this experiment was to establish that children and adults had more difficulty with the scaling than non-scaling task on the 64-inch test mat but not on the 16-inch test mat when no boundaries were present, showing that it is more difficult to make a scale translation when the two edges of the test mat are not easily viewable at the same time (Plumert et al., 2019). All main effects and interactions that reached traditional levels of statistical significance (p < .05) are reported below.

Figure 3 shows the pattern of absolute proportional error across age groups, conditions, and tasks. Absolute error scores were entered into an Age (children, adults) x Condition (scaling up, scaling down) x Task (scaling, no scaling) mixed model Analysis of Variance (ANOVA) with age group and condition as between-participants factors and task as a within-participants factor. There was a significant main effect of task, F (1, 92) = 5.52, p < .05, η_p² = .06, indicating that children and adults exhibited larger absolute error in the scaling task (M_error = .051, SD = .029) than in the no scaling task (M_error = .046, SD = .027). Although the Task x Condition interaction did not reach statistical significance, F (1, 92) = 2.57, p = .11, η_p² = .03, we conducted planned comparisons to test our predictions about the effect of task on each condition (see also Wei, Carrol, Harden, & Wu, 2012). As expected, the task main effect was significant for the scaling up condition, F (1, 46) = 8.17, p < .01, η_p² = .15, indicating that children and adults exhibited larger absolute error in the scaling task (M_error = .054, SD = .031) than in the no scaling task (M_error = .046, SD = .027). In contrast, the task main effect did not reach statistical significance in the scaling down condition, F (1, 46) = .27, p = .61, η_p² = .006, indicating that absolute error did not differ significantly across the scaling (M_error = .048, SD = .026) and no scaling (M_error = .046, SD = .026) tasks. These proportional error scores translate into an average error of 3.46 inches in the scaling task and 2.94 inches in the no scaling task for the 64-inch test mat, and an average error of .77 inches in the scaling task and .74 inches in the no scaling task for the 16-inch test mat.

Figure 3.

Mean absolute proportional error in scaling and no scaling tasks for children and adults when placing objects on 64-inch test mats (Scaling Up Condition) and 16-inch test mats (Scaling Down Condition).

As expected, there was a significant main effect of age, F (1, 92) = 244.54, p < .001, η_p² = .73. Four- and 5-year-olds (M_error = .070, SD = .018) exhibited significantly greater overall absolute error than adults (M_error = .027, SD = .006). The Age x Condition interaction also was significant, F (1, 92) = 4.62, p < .05, η_p² = .05. Simple effects tests indicated that for children errors did not differ across conditions, F (1, 46) = 2.46, p = .12, η_p² = .05 (scaling up M = .074, SD = .015; scaling down M = .066, SD = .021), whereas for adults errors were larger in the scaling down condition (M = .029, SD = .006) than in the scaling up condition (M = .025, SD = .005), F (1, 46) = 4.81, p < .05, η_p² = .10.

2.2.2. Directional error.

We analyzed patterns of bias in directional errors to examine whether children and adults subdivided the test mats in the absence of visible boundaries. We expected that adults would demonstrate clear subdivision effects regardless of mat size, but that children would treat the mats as one spatial category regardless of mat size. Directional error scores were entered into an Age (children, adults) x Condition (scaling up, scaling down) x Task (scaling, no scaling) x Location (inner locations, outer locations) mixed model ANOVA with age group and condition as between-participants factors and task and location as within-participants factors. Figure 4 shows the pattern of directional error for the inner and outer locations across age groups and conditions. All statistically significant findings involving the location factor are reported, given that they were the focus of this analysis as indicators of subdivision effects.² The analysis revealed a significant main effect of location, F (1, 90) = 12.11, p < .01, η_p² = .12, and a significant Location x Age interaction, F (1, 90) = 5.35, p < .05, η_p² = .06. As expected, adults’ placements of the inner and outer objects differed significantly depending on location, F (1, 47) = 82.64, p < .001, η_p² = .64. Adults’ placements of the inner locations were biased away from the middle of the mat (M = −.022, SD = .018), whereas their placements of the outer locations were biased toward the middle of the mat (M = .012, SD = .018), indicating a clear subdivision effect that confirmed our predictions (Huttenlocher et al., 1991). In contrast, children’s placements of the inner and outer objects did not differ across locations, F (1, 47) =.41, p = .53, η_p² = .009. Overall, children’s placements of the inner locations (M = −.001, SD = .066) and outer locations (M = .006, SD = .032) were highly variable and were not indicative of a clear subdivision effect (Huttenlocher et al., 1994).

Figure 4.

Mean directional proportional error for the inner and outer locations for children and adults when scaling up from 16 to 64 inches and scaling down from 64 to 16 inches. Positive values reflected bias toward the center of the mat, whereas negative values reflected bias away from the center of the mat.

2.3. Discussion

The goal of this experiment was to test whether adults and children have more difficulty in the scaling than in the no scaling task with a test mat size of 64 inches but not 16 inches. We focused on test mat size rather than the direction of scale translation (i.e., scaling up or scaling down) here because previous findings demonstrate that the size of the test mat rather than the direction of the scale translation is the critical factor (see Experiment 3 from Plumert et al., 2019). As expected, children and adults exhibited larger errors in the scaling task than in the no scaling task when the test mat was 64 inches, whereas errors were similar in magnitude across the scaling and no scaling tasks when the test mat was 16 inches. As such, the present findings set the stage for examining the role of boundaries in making scale translations and replicate the key pattern of findings from Plumert et al. (2019), suggesting that it is more difficult to make a scale translation when the edges of the test space are not easily viewable at the same time. As expected, absolute placement errors were larger for children than for adults, indicating that the precision of spatial coding undergoes significant improvement between early childhood and adulthood.

We also examined whether children and adults subdivided the space, as evidenced by their patterns of bias in directional error. For both test mat sizes, adults’ placements of the inner locations were biased away from the middle of the mat and their placements of the outer locations were biased toward the middle of the mat, demonstrating that they subdivided the mats into two halves, even in the absence of visible boundaries. These findings are consistent with well-established subdivision effects in adulthood (Huttenlocher et al., 1991). In contrast, children’s placements of the inner and outer locations were highly variable and did not evidence a clear pattern of subdivision, suggesting that they may have treated the entire mat as one spatial category rather than subdividing it into two regions via the midline. These results are consistent with previous memory findings showing that young children do not subdivide relatively large spaces in the absence of visible boundaries (Huttenlocher et al., 1994).

The goal of Experiment 2 was to test whether visible midline boundaries make it easier to visually scale larger spaces. Given that Plumert et al. (2019) found no difference between scaling and no scaling tasks for test mats 32 inches and smaller, we expected that including visible boundaries (essentially creating two side-by-side 32-inch regions) might reduce error when using a 64-inch test mat if children and adults used the midline boundary and one edge of the mat to perceptually anchor the scale translation. However, no such reduction in error would be expected if people rely only on both (outer) edges of the test mat to perceptually anchor the scale translation. Consistent with other work, we expected that boundaries would improve accuracy overall (Plumert & Hund, 2001). We also expected that adding a visible midline boundary might lead children to subdivide the 16-inch test mat into two regions, consistent with other work showing that young children subdivide small spaces (Huttenlocher et al., 1994). We again expected adults to subdivide the test mats into two regions regardless of size.

3. Experiment 2

3.1. Method

3.1.1. Participants.

The participants were forty-eight 4- to 5-year-old children and 48 adults. None had participated in Experiment 1. There were 24 children and 24 adults in each of the two experimental conditions. The mean ages were 4 years 11 months (range = 4 years 6 months to 5 years 7 months; 23 girls, 25 boys) and 19 years 7 months (range = 18 years 3 months to 22 years 11 months; 26 women, 22 men). Data from three additional 4-year-olds were excluded because they did not complete the task. Eighty-four percent of the children were European American, 12% were African American, 2% were Asian American, and 2% were Hispanic/Latino. Two percent of mothers had completed their high school education or less, 13% had completed some college education, and 85% had a 4-year-college education or beyond. Eighty-eight percent of adult participants were European American, 6% were Asian American, and 6% were Hispanic. Children and adults were recruited in the same manner as in the previous experiment.

3.1.2. Apparatus and materials.

The same experimental room was used as in the previous experiment. The large and small mats were identical, with the exception that a visible black boundary divided each mat into two equal sized halves. The boundary was .125 in. wide on the small mat and .5 in. wide on the large mat. The objects were identical to those used in the previous experiment.

3.1.3. Design and procedure.

Participants were randomly assigned to one of two conditions: scaling up or scaling down. The conditions were identical to those used in the previous experiment, except that a midline boundary was added to both the learning and test mats. All aspects of the procedure were the same as in the previous experiment.

3.1.4. Coding and measures.

The coding and measures were identical to those used in the previous experiment. Again, we corrected for mirror reversal errors. We corrected reversal errors for 8.98% of locations for 4- and 5-year-olds (69 out of 768) and 0.13% for adults (1 out of 768). These corrected placements were used in all analyses of absolute and directional errors.

After all reversals were corrected, we classified placement values that were larger than the mean ± 2SDs for each age group, location, and condition as outliers and omitted these values from all analyses. We omitted 5.86% of locations for 4- to 5-year-olds (45 out of 768), and 4.16% for adults (32 out of 768). In addition, we omitted one additional location for one 4-year-old, two 5-year-olds, and one adult because an experimenter error occurred during those trials.

3.2. Results

3.2.1. Absolute error.

The goal of this experiment was to determine whether visible boundaries that divided the learning and test mats in half (essentially creating two side-by-side 32-inch regions) would reduce error when scaling using the 64-inch test mat. As such, we were especially interested in whether there was a difference in absolute proportional error across the scaling and no scaling tasks when the test mats were 64 inches, in comparison to when the test mat was 16 inches. All main effects and interactions that reached traditional levels of statistical significance (p < .05) are reported below.

Figure 5 shows the pattern of absolute proportional error across age groups, tasks, and conditions. Absolute error scores were entered into an Age (children, adults) x Condition (scaling up and scaling down) x Task (scaling, no scaling) mixed model ANOVA with age group and condition as between-participants factors and task as a within-participants factor. The main effect of task was marginally significant, F (1, 92) = 3.83, p = .053, η_p² = .04, indicating that children and adults exhibited larger absolute error in the scaling task (M_error = .037, SD = .024) than in the no scaling task (M_error = .034, SD = .024). As in Experiment 1, although the interaction did not reach statistical significance, F (1, 92) = 1.27, p = .26, η_p² = .01, we again conducted planned comparisons to test our hypotheses about the effect of task on each condition. For the scaling up condition, the main effect of task again was significant, F (1, 47) = 4.21, p < .05, η_p² = .08, indicating that children and adults exhibited larger absolute errors in the scaling task (M_error = .036, SD = .023) than in the no scaling task (M_error = .030, SD = .022). In contrast, the task main effect did not reach statistical significance in the scaling down condition, F (1, 47) = .42, p = .52, η_p² = .01, indicating that absolute error was similar in the scaling (M_error = .038, SD = .024) and no scaling (M_error = .037, SD = .026) tasks. The proportional error scores translate into an average error of 2.30 inches in the scaling task and 1.92 inches in the no scaling task for the 64-inch test mat, and an average error of .61 inches in the scaling task and .59 inches in the no scaling task for the 16-inch test mat.

Figure 5.

Mean absolute proportional error in scaling and no scaling tasks for children and adults when placing objects on 64-inch test mats (Scaling Up Condition) and 16-inch test mats (Scaling Down Condition) with a visible midline boundary.

As would be expected, there was again a significant main effect of age, F (1, 92) = 171.31, p < .001, η_p² = .65. Four- and 5-year-olds (M_error = .053, SD = .018) exhibited significantly greater overall error than adults (M_error = .018, SD = .004).

We also conducted cross-experiment comparisons of absolute error to compare the effects of scaling and no scaling without and with a boundary. We expected that visible boundaries would reduce error overall. Absolute error scores were entered into an Age (2) x Condition (2) x Task (2) x Experiment (1: no boundary, 2: boundary) mixed model ANOVA. Only statistically significant effects involving the experiment factor are reported here to avoid redundancy with the previous analyses. As expected, the analysis revealed a significant main effect of experiment, F (1, 184) = 46.06, p < .001, η_p² = .20. Overall, participants were more accurate when the mat included a boundary dividing it in half (Experiment 2: M_error = .035, SD = .024) than when it did not include a boundary (Experiment 1: M_error = .048, SD = .028). The analysis also revealed a significant Age x Experiment interaction, F (1, 184) = 4.29, p < .05, η_p² = .02, and a significant Age x Condition x Experiment interaction, F (1, 184) = 5.48, p < .05, η_p² = .03. For adults, simple effects tests revealed that only the main effect of experiment was significant, F (1, 92) = 81.41, p < .001, η_p² = .47, indicating that they were more accurate when the task space included a boundary dividing the mat in half (Experiment 2: M_error = .018, SD = .004) than when it did not include a boundary (Experiment 1: M_error = .027, SD = .006). For children, there was a significant main effect of experiment, F (1, 92) = 20.92, p < .001, η_p² = .19, and a significant Condition x Experiment interaction, F (1, 92) = 4.11, p < .05, η_p² = .04. Children in the scaling up condition were much more accurate when the task space included a boundary dividing the mat in half (Experiment 2: M_error = .049, SD = .018) than when it did not include a boundary (Experiment 1: M_error = .074, SD = .015), F (1, 46) = 26.50, p < .001, η_p² = .37. Although the effect of the boundary was not as strong, children in the scaling down condition also were more accurate when the task space included a boundary dividing the mat in half (Experiment 2: M_error = .056, SD = .019) than when it did not include a boundary (Experiment 1: M_error = .066, SD = .021), F (1, 46) = 20.45, p < .001, η_p² = .18. These findings confirm that visible midline boundaries reduced absolute error for adults and children.

3.2.2. Directional error.

We analyzed patterns of bias in directional errors to examine whether children and adults subdivided the mats with visible boundaries. We expected that adults would demonstrate clear subdivision effects regardless of mat size. In contrast, we expected that children might exhibit subdivision effects with the small mats given that visible boundaries were present. Directional error scores were entered into an Age (children, adults) x Condition (scaling up, scaling down) x Task (scaling, no scaling) x Location (inner locations, outer locations) mixed model ANOVA with age and condition as between-participants factors and task and location as within-participants factors. All statistically significant effects involving the location factor are reported here given their importance for evaluating subdivision effects.

Figure 6 shows the pattern of directional proportional error for the inner and outer locations across age groups and conditions. The analysis revealed a significant main effect of location, F (1, 92) = 11.95, p < .01, η_p² = .12, a significant Location x Condition interaction, F (1, 92) = 7.28, p < .01, η_p² = .07, and a significant Location x Age x Condition interaction, F (1, 92) = 10.17, p < .01, η_p² = .10. Simple effects tests examined the effects of location, condition, and their interaction for each age group separately. For adults, only the main effect of location was significant, F (1, 46) = 62.66, p < .001, η_p² = .58. As in Experiment 1, adults’ placements of the inner locations were biased away from the middle of the mat (M = −.013, SD = .012), whereas their placements of the outer locations were biased toward the middle of the mat (M = .006, SD = .013), indicating a clear subdivision effect (Huttenlocher et al., 1991). In contrast, for 4- and 5-year-old children, simple effects tests indicated that the Location x Condition interaction was significant, F (1, 46) = 9.47, p < .01, η_p² = .17. The location main effect was significant for the scaling down condition, F (1, 23) = 7.65, p < .05, η_p² = .25. Children’s placements of the inner locations were biased away from the middle of the mat, whereas their placements of the outer locations were biased toward the middle of the mat, suggesting they subdivided the small test mat when it included a visible midline boundary. However, the location main effect was not significant for the scaling up condition, F (1, 23) = 2.19, p = .15, η_p² = .09, indicating no clear pattern of directional error for the large test mat even with a visible midline boundary.

Figure 6.

Mean directional proportional error for the inner and outer locations for children and adults when scaling up from 16 to 64 inches and scaling down from 64 to 16 inches with a visible midline boundary. Positive values reflected bias toward the center of the mat, whereas negative values reflected bias away from the center of the mat.

3.3. Discussion

The goal of Experiment 2 was to determine how visible midline boundaries affected performance on the scaling and no scaling tasks. We expected that including midline boundaries would reduce absolute error when scaling up to the 64-inch test mat if children and adults used the midline boundary and one edge of the test mat to perceptually anchor the scale translation. However, it was possible that absolute errors would not be reduced if children and adults relied on both (outer) edges of the test mat to perceptually anchor the scale translation. As in Experiment 1, children and adults again exhibited larger errors in the scaling task than in the no scaling task when using the 64-inch test mat. In contrast, absolute error did not differ in the scaling and no scaling tasks when using the 16-inch test mat. These findings replicate the pattern evident in Experiment 1 and in previous work by Plumert et al. (2019), which did not include visible midline boundaries. This pattern of results suggests that children and adults rely on the entire test space rather than the midline boundary to perceptually anchor the scale translation. Absolute errors again were smaller for adults than for children, demonstrating an increase in the precision of spatial coding with development. Absolute errors also were smaller when visible boundaries were present than when they were absent, indicating that visible boundaries facilitate memory precision overall (Plumert & Hund, 2001).

We also examined directional errors to provide details about spatial subdivision effects. As expected, adults’ placements of the inner locations were biased away from the center of the mat, whereas their placements of the outer locations were biased toward the center of the mat, reflecting clear subdivision effects. This time, the 4- and 5-year-olds also showed evidence of subdivision when the small mats included visible midline boundaries, underscoring the importance of midline boundaries and smaller mat sizes in facilitating subdivision for young children. These findings confirm the importance of visible boundaries for coding location, and highlight the powerful effect of visible midline boundaries in small spaces for supporting young children’s spatial subdivision skills.

4. General Discussion

The goal of this project was to test whether visible midline boundaries make it easier to scale larger distances. Across two experiments, we found that children and adults exhibited larger absolute error in the scaling task than in the no scaling task when using the 64-inch test mats, regardless of the presence or absence of visible midline boundaries. In contrast, absolute error did not differ in the scaling and no scaling tasks when using the 16-inch test mats, regardless of the presence or absence of midline boundaries. As such, the present findings underscore the importance of test mat size in visual scaling. We replicated the pattern of absolute error across two additional experiments here that was evident across three experiments in Plumert et al. (2019). The consistent pattern of findings across five studies that large test mats (64 inches or larger) lead to larger errors for scaling than for no scaling trials suggests that making head or eye movements to view the entire test mat disrupts the process of making the scale translation, providing strong support for the perceptual anchoring effect (Plumert et al., 2019).

Our findings also help clarify the nature of perceptual cues used for anchoring. In particular, the present findings suggest that children and adults rely on outside edges of the entire space rather than visible midline boundaries to anchor the scale translation. One question these results raise is why didn’t children or adults use the midline boundary as a perceptual anchor during the scale translation? Spatial subdivision exists within a hierarchy where one space is subdivided into two (or more) smaller regions. These regions operate as spatial categories during memory processes, leading to characteristic patterns of bias evident in directional errors and an overall improvement in accuracy evident in absolute errors. However, the regions do not seem to be treated as independent of one another when visually scaling distance. That is, even when marked by visible midline boundaries, the two halves of the mats are not treated independently as two smaller mats side by side. Instead, the size of the entire mat affected scaling such that children and adults made larger absolute errors in the scaling than in the no scaling task for the 64-inch mats regardless of the presence or absence of visible midline boundaries. This suggests that subdivided regions remain nested within the next higher level of the spatial hierarchy, making it difficult to ignore the higher level when visually scaling distance. Additional work is needed to further understand the role of visible boundaries in scaling tasks, focusing especially on comparisons with memory tasks that do not involve scaling.

As expected, absolute errors were smaller for adults than for children, demonstrating an increase in spatial precision with development. This finding is consistent with a large body of work showing profound improvements in spatial memory and scaling over development (Frick & Newcombe, 2012; Hund & Plumert, 2002; Huttenlocher et al., 1994, 1999; Plumert et al., 2019; Spencer & Hund, 2003; Vasilyeva & Huttenlocher, 2004). The findings also are consistent with the spatial precision hypothesis, which contends that improvements in the fidelity of coding spatial location across early childhood lead to differences in response patterns over development (Schutte & Spencer, 2009; Spencer, Simmering, Schutte, & Schöner, 2007).

Absolute errors also were smaller when visible midline boundaries were present than when they were absent, suggesting that boundaries improve spatial precision overall. These findings are similar to those of Plumert and Hund (2001), where memory precision for children and adults was significantly greater when visible boundaries subdivided the task space into regions than when the visible boundaries were absent. In fact, in that study, memory precision varied as a function of boundary salience, with opaque walls leading to the most precise responses, followed by visible lines, then no boundaries. Together, these findings suggest that visible boundaries support spatial memory precision for children and adults.

We also examined bias in directional errors to provide details about spatial subdivision effects. As expected, adults’ placements of inner locations were biased away from the mat center, whereas their placements of outer locations were biased toward the center of the mat, indicating that they subdivided the large and small mats into two regions regardless of the presence or absence of visible midline boundaries. These findings demonstrate that adults use both visible and mentally imposed midline boundaries to subdivide spaces into smaller regions to facilitate memory and scaling, a finding that is common in the literature overall (Engebretson & Huttenlocher, 1996; Hund & Plumert, 2002; Huttenlocher et al., 1991; Plumert & Hund, 2001; Sampaio & Wang, 2009). When visible midline boundaries were absent, 4- to 5-year-old children showed no evidence of spatial subdivision. However, when visible boundaries were included in Experiment 2, children showed evidence of subdivision with the small mats but not the large mats. These findings underscore the importance of both small mat size and visible boundaries for facilitating spatial subdivision for young children. Overall, these sensitivities are consistent with previous research demonstrating the effects of visible boundary salience on spatial memory (Hund, Plumert, & Benney, 2002; Kosslyn et al., 1974; Plumert & Hund, 2001; Simmering & Spencer, 2007) and the importance of task space size for spatial subdivision (Frick & Newcombe, 2012; Huttenlocher et al., 1994). In fact, our findings suggest that the inclusion of visible midline boundaries in small spaces may be critical for supporting spatial subdivision skills among young children in the scaling paradigm used here.

What implications do the present results have for understanding spatial scaling strategies? From the perspective of a proportional reasoning strategy, it may be somewhat surprising that including visible midline boundaries did not erase the difficulty with scaling up to a larger test mat. As noted earlier, a proportional reasoning strategy involves visually coding distance relative to two reference points (e.g., landmarks or edges) in one space and then mapping this relative distance onto the other space using the corresponding reference points. If participants were using a proportional reasoning strategy, they could have used one edge of the mat and the visible midline boundary to estimate the distance, thereby improving accuracy. This finding was not supported by our data. Instead, the present findings may provide support for mental transformation strategies during spatial scaling. Recall that using a mental transformation strategy involves mentally shrinking or expanding the original space to match the size of the other space (Möhring et al., 2016; Vasilyeva & Huttenlocher, 2004). This mental transformation process is similar to mental rotation and image scanning processes, in which larger transformations take longer and produce greater error (e.g., Bundesen & Larsen, 1975; Kosslyn, 1975; Möhring et al., 2014; 2015, 2016; Shepard & Metzler, 1971). Although our findings did not focus on changes in response times or errors across various scaling ratios, they are consistent with the notion that participants mentally transform (shrink or expand) the entire learning mat to match the test mat’s size to perform our visual scaling task because the presence of visible midline boundaries did not improve accuracy for scaling up. We think that it may be difficult to shrink or expand only one half of a space during the scaling process because doing so would disrupt the spatial features of the space overall and violate a variety of object properties. Additional research is needed to further clarify how children and adults use mental transformation strategies in combination with perceptual anchoring to facilitate scaling, including clarity with regard to effects of scaling ratio.

There were several limitations to the current investigation. One limitation of our findings was that we used planned comparisons to explore our predictions regarding differences between scaling and no scaling tasks for each test mat size. Another limitation was the lack of diversity in our sample, which limits the generalizability of findings. Future research should include more diverse samples of child and adult participants when testing differences between scaling and no scaling performance across a variety of test mat sizes. Nonetheless, the present results underscore the importance of test mat size for visual scaling, providing further support for the perceptual anchoring effect (Plumert et al., 2019). Our findings also clarify the utility of perceptual cues for anchoring, demonstrating that children and adults rely on the edges of the test mat, not midline boundaries, during the scaling process. This pattern is consistent with a mental transformation strategy for scaling, suggesting that children and adults scale the entire space during the translation process. Our findings also underscore that midline boundaries have important effects overall, improving accuracy for adults and children and facilitating spatial subdivision of small spaces by young children. In sum, our findings add to the growing body of research regarding the cognitive processes involved in visually scaling distances (Frick, 2018; Frick & Newcombe, 2012; Huttenlocher et al., 1999; 2007; Möhring et al., 2014, 2015, 2016; Möhring, Newcombe, Levine, & Frick, 2016; Plumert et al., 2019; Vasilyeva & Huttenlocher, 2004; Szubielska & Möhring, 2019; Szubielska, Möhring, & Szewczyk, 2019).