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An Operational Momentum (OM) effect is shown by 9-month-old infants during non-symbolic arithmetic, whereby they overestimate the outcomes to addition problems, and underestimate the outcomes to subtraction problems. Recent evidence has shown that this effect extends to ordering operations for size-based sequences in 12-month-olds. Here we provide evidence that OM occurs for ordering operations involving numerical sequences containing multiple quantity cues, but not size-based sequences, already at 4 months of age. Infants were tested in an ordinal task in which they detected and represented increasing or decreasing variations in physical and/or numerical size, and then responded to ordinal sequences that exhibited greater or lesser sizes/numerosities, thus following or violating the OM generated during habituation. Results showed that OM was absent during size ordering (Experiment 1), but was present when infants ordered arrays of discrete elements varying on numerical and non-numerical dimensions, if both number and continuous magnitudes were available cues to discriminate between with-OM and against-OM sequences during test trials (Experiments 2 vs. 3). The presence of momentum for ordering number only when provided with multiple cues of magnitude changes suggests that OM is a complex phenomenon that blends multiple representations of magnitude early in infancy.

Number is a basic property of the environment to which human and non-human animals are spontaneously attuned (

In adult numerical cognition, it is widely accepted that numerical representations take the form of a ‘mental number line’. This model posits that numerosities are spatially represented along a continuum, such that (in Western cultures) small numbers are associated to the left side and large numbers to the right side of space, first evidenced by the so-called SNARC (Spatial-Numerical Association of Response Codes) effect (

A recent subfield that helps to unravel the developmental origins of the spatial-numerical associations is the Operational Momentum (OM) effect in arithmetic. The OM effect arises when participants systematically overestimate the outcomes of addition problems and underestimate those of subtraction problems (

Recently, the OM effect has been extended to ordering operations, as it arises even when 12-month-old infants represent increasing/decreasing relations among continuous magnitudes (i.e., size of a single object). In particular, when infants are habituated to an object that progressively increases or decreases in size, and are subsequently presented at test with ordinal sequences that exhibit greater or lesser size, they look longer to size changes whose direction violate the operational momentum experienced during habituation (i.e., the smaller sequence in the increasing condition and the larger sequence in the decreasing condition) (

These findings are at odds with earlier evidence suggesting that the dimension of size does not lead to SNARC-like effects. In fact,

Indeed, there is mounting evidence that, in adults, the coding of serially ordered information co-opts a spatially organized mental representation, as adults exploit a spatially oriented horizontal continuum to represent not only information learnt in a conventional fixed order, like numbers (e.g.,

In light of this evidence, in the present study we exploited the OM effect to investigate the mapping of order into space in infants younger than 7 months - i.e., 4-month-olds -, when exposure to culturally-shaped routines is even more limited. We chose to test infants at this particular age because recent research has shown that 4-month-olds are able to extract ordinal information from both numerical and size-based sequences, at least when confronted with increasing order. After habituation to increasing sequences composed of a single shape varying in size by a 1:2 ratio, 4-month-old infants looked longer to a newly-sized sequence of a reversed ordinal direction (

Building on previous research (

In Experiment 1 4-month-old infants were habituated with a series of increasing or decreasing size-based sequences and were then tested with new sequences in which the same ordinal direction was composed of larger and smaller sizes. If infants show an OM effect while ordering size, as they do at 12 months (

Methods were modelled after

Participants were 40 4-month-old infants (22 females, mean age = 4 months 21 days, range = 4 months 3 days to 5 months 0 days) assigned randomly and in equal number (

Stimuli were single colored shapes varying in area by a 1:2 ratio (range 5.5-84 cm^{2}) and were presented on a white background in the center of the computer monitor. There were five sets of stimuli: Three for the habituation phase and two for the test phase, each set being composed of a different shape displaying a unique color. The three habituation sets contained blue circles, green triangles, and red squares that were, respectively, 9, 18, 36; 11, 22, 44; and 13, 26, 52 cm^{2}. The two test sets contained three rainbow-colored bars that expanded/contracted along the horizontal axis and whose sizes were different in each set, with an overall area of 5.5, 11, 22 cm^{2} in the smaller set, and 21, 42, 84 cm^{2} in the larger set (see

The five sets of colored shapes as they were presented to infants in the increasing order condition of Experiment 1. Arrows represent passage of time; objects were presented serially, centered on the screen. Blue circles, green triangles, and red diamonds were presented in a fixed order during habituation, and the smaller and larger sets of rainbow-colored bars were presented, respectively, during the against-OM and the with-OM test trials, with presentation order counterbalanced across participants. For infants in the decreasing order condition the same habituation shapes were presented in reversed order (from the larger to the smaller), starting from the red diamonds and proceeding to the green triangles and the blue circles, and the smaller and larger sets of bars were presented, respectively, during the with-OM and the against-OM test trials.

Infants were habituated to increasing or decreasing sequences of blue circles, red squares, and green triangles, and were then tested with increasing or decreasing sequences containing newly-sized bars (see ^{2}), from the largest to the smallest shape for the decreasing condition (i.e., 52-26-13, 44-22-11, 36-18-9 cm^{2}). The use of a consistent fixed order of presentation of the sets across trials for each of the two habituation conditions provided infants with redundant cues to ordinality between, as well as within, trials (see ^{2}), with the order of presentation counterbalanced across participants. For infants in the increasing habituation condition the with-momentum test sequence comprised bars measuring 21, 42, 84 cm^{2}, and the against-momentum test sequence comprised bars measuring 5.5, 11, 22 cm^{2}; the opposite was true for infants in the decreasing habituation condition. For both conditions, both test trials exhibited an order (ascension, or descension) that was identical to that displayed in habituation.

Each infant was tested while sitting in an infant seat approximately 60 cm from the monitor where the stimuli were presented (24” screen size, 1920 X 1200 pixel resolution, refresh rate of 60 Hz) in a dimly lit room. A video camera was positioned just above the stimulus presentation monitor and was directed to the infant’s face. The live image of the infant’s face was displayed on a television monitor to allow the online coding of the infant’s looking times through the E-Prime 1.0 program by the experimenter, who was blind to the habituation condition to which the infant was assigned and to the order of test trials. The image of the infant’s face was also recorded via a Mini-DV digital recorder, and for half of the infants in the sample (

A cartoon animated image associated with varying sounds served as an attention catcher before the trial began. When the infant looked at the animated fixation point, the experimenter started the trial. Each trial consisted of a repeating cycle (6500 ms in total) that began with a black screen (500 ms) followed by the three shapes. Each shape appeared for 1750 ms on a white background, and was preceded by a 250 ms white inter-stimulus interval (ISI) to reduce the impact of looming/zooming cues. Each trial continued until the infant looked continuously for a minimum of 500 ms and ended when the infant looked away continuously for 2 s or looked for a maximum of 60 s. The three habituation stimulus sets were presented in a fixed order and repeated until the infant saw a maximum of 12 trials or met the habituation criterion, which was defined as a 50% decline in looking time on three consecutive trials, relative to the looking time on the ﬁrst three trials. Following the habituation phase, infants were given two test trials, a with-momentum sequence and an against-momentum sequence, with half of the infants seeing the with-momentum sequence first.

All statistics are two-tailed. The average number of trials received during habituation did not differ for infants tested in the the increasing and decreasing order conditions (^{2}_{p} = .48, in the absence of a main effect or interaction involving the factor order condition (

Overall mean total looking time (±SEM) to the first three and last three habituation trials, and to the with-OM and against-OM test trials displayed by infants in the increasing and decreasing order conditions in Experiment 1. Infants failed to show discrimination between the with-OM sequence and the against-OM sequence.

To determine whether during test trials infants looked longer to the ordered sequence that violated the momentum effect, as generated by the abstraction of the ordinal rule during habituation, mean looking times to the two test trials were entered into a three-way ANOVA with order condition (increasing vs. decreasing) and test order (with-OM first vs. against-OM first) as between-participants factors and test trial type (with-OM vs. against-OM) as within-participants factor. The analysis revealed no significant main effects or interactions (all

Overall, results show that infants did not manifest a differential looking behavior toward test sequences whose elements’ size violated the momentum effect, relative to those whose elements’ size followed the momentum. This pattern suggests that, unlike older, 12-month-old, infants (

In Experiment 2, a new group of 4-month-old infants was habituated to a series of either increasing or decreasing numerical sequences and then tested with new sequences in which the same ordinal direction was composed of larger or smaller numerosities. If infants experience an OM effect while ordering number, then they should look longer at the test series that violate the momentum created during habituation: Those habituated to increasing number should look longer to smaller test numerosities, and those habituated to decreasing number should look longer to larger test numerosities. Numerical sequences included numerosities that varied by a 1:3 ratio, as infants at this age succeed in numerical ordinal tasks when provided with this ratio (

The methods were the same as in Experiment 1, except as follows.

The final sample included 40 4-month-olds (20 females, mean age = 4 months 17 days, range = 3 months 28 days to 5 months 4 days) assigned randomly and in equal number (

The five stimulus sets used in the current experiment were created after ^{2}, and density was 0.06 elements per cm^{2} for all numerosities. Inversely, in the test sets, both the shape (i.e., hexagon) and the size of the envelope area were constant across numerosities so that density covaried with number, and non-numerical continuous variables were controlled both within and between sets by keeping cumulative surface area constant. In fact, item size was inversely correlated to number: The size of each single item in the smaller, medium and larger numerosity displays was, respectively, 0.45 cm^{2}, 0.15 cm^{2} and 0.05 cm^{2} in the 16-48-144 test set, and 1.44 cm^{2}, 0.48 cm^{2} and 0.16 cm^{2} in the 5-15-45 test set, and cumulative surface area was constant at about 7 cm^{2}. Overall, non-numerical variables that varied during habituation were held constant during test, and vice-versa.

A) The five sets of numerical displays as they were presented to infants in the increasing order condition of Experiment 2. Arrows represent passage of time; objects were presented serially, centered on the screen. The numerical sequences 8-24-72, 9-27-81 and 10-30-90 were presented in a fixed order during habituation, and the numerical sequences 5-15-45 and 16-48-144 were presented, respectively, during the against-OM and the with-OM test trials, with presentation order counterbalanced across participants. For infants in the decreasing order condition, the same habituation displays were presented in reversed order (from the larger to the smaller), starting from the 90-30-10 sequence and proceeding to the 81-27-9 and the 72-24-8 sequence, while the 45-15-5 and the 144-48-16 sequences were presented, respectively, during the with-OM and the against-OM test trials. B) The five sets of numerical displays as they were presented to infants in the increasing order condition of Experiment 3. Habituation sets were identical to those used in Experiment 2, whereas the two test sets included the same number of items as in Experiment 2 (5-15-45 and 16-48-144) but differed for cumulative surface area (8 cm^{2} and 26 cm^{2}, respectively).

All statistics are two-tailed. The average number of trials received during habituation did not differ for infants in the increasing and decreasing order conditions (^{2}_{p} = .46, with no main effect or interaction involving the factor order condition (

To determine whether infants’ looking times during test trials differed for the numerical sequence that violated the momentum effect and for the sequence that followed momentum, mean looking times to the two test trials were entered into a three-way ANOVA with order condition (increasing vs. decreasing) and test order (with-OM first vs. against-OM first) as between-participants factors, and test trial type (with-OM vs. against-OM) as within-participants factor. The analysis revealed no significant main effects or interactions (all

Overall mean total looking time (±SEM) to the first three and last three habituation trials, and to the with-OM and against-OM test trials displayed by infants in the increasing and decreasing order conditions of Experiment 2. Infants failed to show discrimination between the with-OM sequence and the against-OM sequence.

Results show that infants fail to exhibit an OM effect while ordering numerical sequences, as indexed by their comparable looking times at both ordered test sequences, the one violating the OM and the one following the OM generated during habituation. When considered together with the results of Experiment 1, this finding suggests that at 4 months of age ordering of both discrete and continuous quantities do not engender momentum-like effects. When the two test sequences differed only by numerical information, as non-numerical variables were strictly controlled, no OM effect was observed.

Research on infant numerical cognition has consistently shown that providing infants with multiple cues boosts their sensitivity to numerical differences (

In Experiment 3, a new group of 4-month-old infants was habituated with a series of either increasing or decreasing numerical sequences and then tested with new sequences in which the same ordinal direction was composed of larger and smaller numerosities. During test, however, they received multiple cues to magnitude differences between test trials, as not only number differed across trials, but also cumulative surface area. Unlike in Experiment 2, this dimension covaried with number across the with-OM and the against-OM test sequences. Again, if infants experience an OM effect while ordering number, then they should look longer at the test series that violate the momentum created during habituation: Those habituated to increasing number should look longer to smaller numerosities, and those habituated to decreasing number should look longer to larger numerosities. In Experiment 3, as in Experiment 2, numerical sequences included numerosities that varied by a 1:3 ratio.

The methods were the same as in Experiment 2 except as follows.

The final sample included 40 4-month-olds (16 females, mean age = 4 months 21 days, range = 4 months 7 days to 5 months) assigned randomly and in equal number (

The habituation sets were identical to Experiment 2. However, the test sets differed from Experiment 2, as the cumulative surface area was held constant at 8 cm^{2} in one set and 26 cm^{2} in the other set, so that the two sets differed not only in number but also in cumulative surface area (see ^{2} for the numerosities 5 and 16), the two medium numerosities (0.54 cm^{2} for the numerosities 15 and 48) and the two large numerosities (0.18 cm^{2} for the numerosities 45 and 144).

All statistics are two-tailed. Infants in both the increasing and decreasing order conditions required almost the same number of trials (^{2}_{p} = .51, with no main effect or interaction involving the factor order condition (

A three-way ANOVA was performed on mean looking times during test trials to determine whether infants’ performance at test revealed the presence of an OM effect. The ANOVA included the between-participants factors order condition (increasing vs. decreasing) and test order (with-OM first vs. against-OM first), and the within-subjects factor test trial type (with-OM vs. against-OM). Results revealed a significant main effect of order condition, ^{2}_{p} = .18, as overall looking times in test trials were longer for infants tested in the decreasing condition (decreasing: ^{2}_{p} = .24, as infants looked significantly longer to the against-OM than to the with-OM test trial (

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Paired-sample

Overall, results indicate that, unlike in Experiment 2, infants in the current experiment manifested a differential looking behavior at test toward the ordered numerical sequence that violated the OM relative to the one that followed the OM, by showing a significantly higher looking time at the against-OM sequence, which we suggest was violating the numerical magnitude expectations formed during habituation. These findings reveal that 4-month-old infants do experience momentum-like effects while ordering magnitudes, but - unlike 12-month-olds - they only experience an OM effect when cumulative surface area and number co-occur as cues to magnitude during the test phase. Nevertheless, a four-way ANOVA with experiment (Exp. 3 vs. Exp. 2) as an additional between-participants factor failed to confirm a significant difference between the current experiment and Experiment 2, with a significant main effect of test trial type, ^{2}_{p} = .15, but an Experiment x Momentum interaction, ^{2}_{p} = .04, which failed to reach significance. In contrast, the same ANOVA comparing Experiment 3 vs. Experiment 1 provided evidence for a reliable differential pattern of results, with a main effect of test trial type, ^{2}_{p} = .11, as well as a significant Experiment x Momentum interaction, ^{2}_{p} = .11. Similar results were obtained when the factors order condition and test order were omitted from the ANOVAs (Experiment x Momentum: Exp. 3 vs Exp. 2,

This study investigated whether OM effects recently described for ordinal operations in size-based sequences by 12-month-old infants (

To this aim, in three experiments we tested 4-month-old infants in an ordinal task in which they had to extract and represent increasing and decreasing order within size-based sequences (Experiment 1) and numerical sequences (Experiment 2 and 3), and then respond to sequences that exhibited greater or lesser sizes or numerosities. The presence of OM was inferred from infants’ longer looking times to sequences that violated the momentum generated during habituation relative to sequences that followed the momentum.

Infants’ performance during test trials suggests that momentum effects were absent during ordinal tasks when single magnitude cues were available to discriminate between with-OM and against-OM test sequences (Experiments 1 and 2), but apparent when redundant magnitude cues were available (Experiment 3). In fact, infants failed to show an increased looking behavior when presented with the against-OM single-shape sequence after having been habituated to either increasing or decreasing single-shape sequences (Experiment 1). Similarly, infants did not show increased attention towards the against-OM test sequence after habituation to either increasing or decreasing sequences in which number covariated with envelope and cumulative area, when the test trials differed only by pure numerical information (Experiment 2). It was only when test trials differed in both number and cumulative area (Experiment 3), and therefore when redundant magnitude cues were available, that increased looking time was observed for the against-OM sequence, suggesting that infants’ OM expectations were generated during habituation and later violated during the observation of the against-OM sequence.

Indeed, the habitation phases were identical in both Experiments 2 and 3, as infants in both studies were presented with the same exact habituation stimulus sets in which number covaried with envelope and cumulative area. Despite that, the momentum created during habituation was only expressed when infants received converging evidence on magnitude differences between the test sequences from multiple cues, suggesting that the covariation of cumulative area, envelope area,

A related phenomenon within the ordering literature has been observed previously, with infants younger than 11 months of age being able to extract numerical order only if additional cues such as spatial extent (

A notable finding to emerge from this study is that variations along the dimension of physical size by itself do not appear to engender momentum effects at 4 months of age. In fact, when size was the only available information that monotonically changed within ordered sequences for both the habituation and test phases, infants did not show any evidence of momentum (Experiment 1). However, note that changes in numerosities in Experiments 2 and 3 were indeed accompanied by corresponding changes in size of their enclosing area during habituation (smaller numerosities were embedded in smaller envelope areas, and larger numerosities in larger envelope areas), in addition to changes in overall surface area. Therefore, it is possible that, in these experiments, envelope area acted as an additional non-numerical cue that helped create OM during habituation, by enhancing perceived magnitude differences within the sequences. Performance during test trials could not be based on this dimension, as envelope area was constant across the with-OM and the against-OM test sequences. However, it is possible that the redundant ordering cues for numerosity (changing number), surface area (changing cumulative area) and size (changing envelope) in the habituation phase of both Experiments 2 and 3 were necessary to engender an OM effect (i.e., “anticipate larger/smaller”), and the corresponding expectations transfer only to stimuli whose magnitude is signaled by confounded cumulative surface area and numerical cues. Therefore, although linear changes in physical size by itself did not engender momentum effects at 4 months of age, the combination of different magnitude cues allowed the formation of OM expectations during habituation (i.e., number + cumulative surface area + size of envelope area), and triggered infants’ response to the violation of OM expectations during test (i.e., number + cumulative surface area).

Previous demonstrations of momentum effects during non-symbolic arithmetic in infancy have been interpreted as suggestive of attentional shifts along the internal oriented representational space for number (

Finally, there are two aspects of infants’ performance in the current study that merit further elaboration. First, in Experiment 1, where infants were required to order size, infant-controlled procedure resulted in different amounts of time required by infants to reach the habituation criterion depending on the order condition. In fact, infants exposed to the decreasing order needed significantly more time than infants exposed to the increasing order. Similarly, in Experiment 3, where infants were required to order arrays of discrete elements varying on numerical and non-numerical dimensions, overall looking times in test trials were longer for infants presented with decreasing order than for those presented with increasing order. Finally, visual inspection of

A second aspect of the current results that deserves consideration is that the strength of the OM effect generated during the ordering operation was comparable for both the increasing and decreasing orders, and that OM was present when infants were habituated to arrays of discrete elements but not to single shapes varying in size. This pattern of results, together with the previously reported asymmetry in infants’ ordinal abilities for both size (

The authors are indebted to parents and infants who donated their time to participate in the study. The authors also thank Carlo Toneatto for invaluable contribution in programming the experiment, Valeria Gariboldi and Gaia Ambrosoni for their help in testing infants. VMC was supported by a grant from the University of Milano-Bicocca (Fondo di Ateneo-Quota Competitiva 2015). KMC was supported by award R15 HD077518-01A1 from the National Institute of Child Health and Human Development. MDdH was supported by a grant from the Agence Nationale de la Recherche Scientifique, France (ANR-15-CE28-0003-01 NUMSPA), and by a Research Grant from the Fondation Fyssen, France.

The authors have no funding to report.

The authors have declared that no competing interests exist.