Children’s learning from others: Conformity to unconventional counting

Abstract

The current study investigated whether children’s conformity to a majority testimony influenced their willingness to revise their own erroneous counting knowledge. The content of the testimonies focused on conventional rules of counting, by means of pseudoerrors (i.e., unconventional counts) occurring during a detection task. In this work measurements were taken at two different time points. At time 1 children aged 5 to 7 years (N = 88) first made independent judgments on the correctness of unconventional counting procedures presented by means of a computerized detection task. Subsequently, they watched a video in which four teachers (unanimous majority) or three (non-unanimous majority) made correct claims about the counts and children had to decide whether the informants were right or not, and justify their answers. Our participants conformed significantly more when the correct testimony was provided by a unanimous majority than by a non-unanimous majority. In addition, in two of the three pseudoerrors presented, there was no difference in the children’s tendency to conform to unconventional counts as age increased. At time 2, which was taken to test whether the effect of the testimony was maintained over time, the responses of the 32 children (16 from each age group) who had endorsed the claims of the unanimous majority at time 1 revealed that teachers’ testimonies only had a lasting influence on elementary school children’s understanding of conventional counting rules.

Keywords

Child development counting pseudoerrors detection task conformity unanimous majority non-unanimous majority

Introduction

The aim of this work is to study the extent to which teachers’ testimony can modify children’s erroneous ideas on the nature of conventional counting rules since testimony has proved to be an important source of learning.

Children learn not just by their own experience or first-hand observation of the world. They also obtain much of their knowledge from the direct or indirect testimony of others. Testimony, broadly understood to be communication transmitted through any means or support (oral, written, images, video, etc.) is an important source of information, either because such knowledge cannot be acquired by direct experience (e.g., future events, historical information, religious claims, or the existence of invisible elements such as viruses, gases or irrational numbers), or because the learner has to commit an enormous amount of resources, time and effort to the process (Harris, 2012; Harris, Koening, Corriveau, & Jaswal, 2018; Lane, Harris, Gelman, & Wellman, 2014; Mills, 2013; Schmidt, Rakoczy, Mietzsch, & Tomasello, 2016).

Nonetheless, not everything children learn from others has the same value, and this is the case of conventional rules. In fact, if we refer to the social practices that children build while interacting with the cultural environment in which they develop, conventional rules play a key adaptive role (e.g., how we eat, how we dress, the way we greet others, etc.). However, apart from these types of social conventions, there exist other conventions not related to social interaction but to ways of doing things or “procedures” linked to learning at school. For example, in certain fields of learning such as mathematics, children learn formal or logical rules allowing them to work with quantities, but they also acquire conventional rules about how to count a set of items, or how to add numbers. These procedures are conventional in that there may be alternative (and equally correct) ways to reach the solution. Although these rules play a key role when starting to learn (Briars & Siegler, 1984), facilitating the acquisition of a procedure by means of repeated practice, their value is different to that of rules associated with social customs.

Children usually learn to count a row of objects from left to right (although this varies according to the direction of writing, see, for example, Göbel, McCrink, Fisher, & Shaki, 2018), to count the elements of the row consecutively (spatial adjacency), to say all the number words aloud consecutively (temporal adjacency) and, among other things, to start counting at one end of the row. However, the final aim of this procedure is for children to understand the underlying concepts and the meaning of what they do (why we count, the notion of cardinality, etc.).

In this learning process, children should come to understand that the procedure itself is modifiable, that is, the solution can be reached by other means, providing they respect the logical rules related to the how-to-count principles: the one-to-one principle (every item in a display must be tagged once and only once); the stable order principle (the tags, regardless of their nature, must be ordered in a stable list of unique tags); and the cardinal principle (the last tag used in a count represents not only the last item in the array but also the cardinality of the set) (see Gelman & Gallistel, 1978). Only the logical rules are relevant to establish whether the right solution has been reached, as, for example, counting the same element twice does not respect the logical rules and inevitably leads to a wrong answer. In contrast, when we count all elements of a row nonconsecutively, the conventional adjacency rule is broken but the logical counting rules are not broken. In other words, a child’s conceptual advance in this area depends on their understanding of the relative value of conventional counting rules compared with the essential nature of logical rules. However, as demonstrated in several studies, even at relatively advanced ages (8–9 years) this distinction is difficult and many children hold erroneous ideas on the nature of both types of rules (Escudero, Rodríguez, Lago, & Enesco, 2015; Kamawar et al., 2010; Lago, Rodríguez, Escudero, & Dopico, 2016; LeFevre et al., 2006; Rodríguez, Lago, Enesco, & Guerrero, 2013), although their performance may vary according to the circumstances or variables of a certain task.

Taken together, findings of previous research show that acceptance of the violation of the conventional counting rules (known as pseudoerrors or correct unconventional counts in the scientific literature) varies according to the conventional rule infringed, with those violating the rules of left-to-right direction and spatial adjacency being more accessible than those violating temporal and temporal–spatial adjacency (Lago et al., 2016). The degree of acceptance of pseudoerrors is greater when children are asked to simply accept or reject pseudoerrors than when they are required to justify their response (Escudero et al., 2015). Also, children’s performance is better on pseudoerrors with explicit cardinal values (after counting the elements, the counter says aloud “there are…”) than on pseudoerrors without explicit cardinal values (Rodríguez et al., 2013). These authors suggest the presence of the cardinal value relegates the procedure by which the correct cardinal value is obtained to a lesser role, helping the child to realize that unconventional procedures also lead to correct responses. Finally, it is also worth noting that although understanding of conventional counting rules tends to improve with age and school grade, this progress is not linear, and greatly depends on the development of other mathematical concepts. In this line, LeFevre et al. (2006) found that children with more developed numerical skills increased their performance on pseudoerrors over grade level, while the least skilled children actually decreased their acceptance over time.

Learning this type of curricular content (i.e., counting rules) is usually conducted in a testimonial context: the teacher teaches a way to “do things” and the children repeat and practice the procedure because they trust the person giving them the information. Teachers represent the epistemic authority of an expert, as well as being trustworthy, two dimensions that are essential for trust in an informational source (Koenig & Harris, 2007). Overall, there is converging evidence that from early ages children selectively trust the testimony of others, taking into account some epistemic aspects of the informants (Bernard, Proust, & Clément, 2015; Birch, Vautier, & Bloom, 2008; Chan, 2011; Einav, 2014).

Longitudinal and cross-sectional studies on conventional counting rules agreed in stating a slow developing rate during primary school. The question is, will the trust in the testimony of others (specifically in consensual judgments, since conformist transmission is a key component of social learning as stated by Harris, 2012; or Muthukrishna, Morgan, & Henrich, 2016), be sufficient to encourage an increase in the slow pace of development of the comprehension of conventional counting rules? A previous study with preschoolers and second graders (Enesco, Rodríguez, Lago, Dopico, & Escudero, 2017), showed that when children have a strong prior knowledge, they are reluctant to endorse counterintuitive testimonies about conventional rules of counting, even when the information is provided by a majority of teachers. Children faced two conflicting claims regarding the acceptability of pseudoerrors: one offered by three teachers in agreement, the other one offered by a lone teacher (dissident). Most children endorsed claims that considered pseudoerrors as incorrect counts, irrespective of the source of information (majority or dissenter). Specifically, the children were more inclined to endorse the majority claims when they rejected the pseudoerrors (the mean percentage was 73.1 of the trials) than when they accepted the pseudoerrors (the mean percentage was 29.2 of the trials). In other words, the general findings revealed that children did not side with the teachers in majority but weighed the arguments of the informants. This tendency persisted two weeks later, showing that children do not accept the violations of the conventional rules of counting and that their adherence to these rules is little susceptible to influence and revision. However, a control pseudoerror unanimously justified as a correct count by the four teachers showed a different pattern, since around 62% of the participants sided with the unanimous majority, although only 30% of them still accepted it as a valid count when asked two weeks later, just as it happened with the rest of the pseudoerrors. Thus, Enesco, Rodríguez, Lago, Dopico, and Escudero (2017) concluded that confronting two different perspectives (one of them coinciding with the child’s own view) did not favor the child’s willingness to revise their prior ideas.

This hypothesis is in line with the findings from studies based on the Asch-style paradigm indicating that children are more likely to accept the testimony of a unanimous majority than that of a partial majority (Haun, van Leeuwen, & Edelson, 2013). This acceptance may be the result of simple conformity, understood as the tendency to discard one’s own knowledge or judgment to conform to the majority opinion (e.g., Haun et al., 2013), as is the case of perceptive judgments, and/or a revision of their own knowledge, as could be our case of judgments on unconventional counts. Taking these findings into account, here we focused on the influence of a unanimous or non-unanimous majority on children’s conformity, not on children’s endorsement of the majority depending on the nature of their statements: counterintuitive or not. To this end we used an Asch-style paradigm in which the majority’s testimony was always counterintuitive, but as in the study of Enesco et al. (2017), we retained the general content of the informants’ claims and the equivalent role played by the members of the majority and the dissident as teachers, to make them equally reliable.

Several studies have used an Asch-style paradigm with child-friendly procedures (e.g., testimonies are presented on video or in illustrations) and the results found are divergent. Whereas Corriveau and Harris (2010) found that the rate of conformity to the consensus in 3-and 4-year-olds, when exposed to the erroneous opinion of a majority of adults in simple perceptual judgments tasks ranged from 19% to 29% of trials, meaning that children tended to favor their own perceptive judgments on line lengths, Haun and Tomasello (2011) reported a higher rate of conformity. These authors asked 4-year-olds to label familiar animals according to their size (daddy, mommy, or baby) after exposure to the erroneous judgment of three peers. Broadly speaking, children conformed with their peers in between 27% and 37% of the trials, depending on condition. Overall, the level of conformity tended to be higher when the majority was unanimous and when the response was public rather than anonymous, which led the authors to conclude that the children had not actually revised their judgment, but only the public expression of it. By contrast, Rakoczy, Ehrling, Harris, and Schultze (2015), using an adapted version of the judge–advisor paradigm, found that children aged 3 to 6 years consistently used advice to revise their limited initial knowledge, and made greater adjustments when advised by an apparently knowledgeable informant. Specifically, the percentage weight of advice was 80% with a knowledgeable advisor and 58% with an ignorant advisor. These scores show that children revised, but not completely, their initial judgments in light of the information provided by the advisors.

Turning to the issue in hand, the general findings on children’s ability to discriminate between conventional and logical counting rules clearly show that it is limited and develops slowly, although children may be flexible in certain circumstances. Accordingly, taking as our reference studies on testimony, which in recent years have demonstrated that children are able to use it to systematically and selectively acquire factual knowledge from preschool years (Rakoczy, Ehrling, Harris, & Schultze, 2015), we have designed a situation in which the incorrect judgments of children aged 5 to 7 years on the violations of the conventional counting rules conflict with the correct claims of a unanimous or non-unanimous qualified majority. Our approach is close to the paradigm of studies on conformity although our study presents important differences with previous research regarding the type of “claims:” (a) judgments are made on the significance of counting, which is the foundation for many mathematical concepts and procedures, not on aspects such as labeling familiar and exotic animals or foods, or deciding which is the longest line; (b) children always faced correct (not erroneous) majority claims (unanimous or not) that run counter to their own knowledge, because three or four teachers accepted as correct the counting pseudoerrors; and (c) the teachers always justified their decision in a way that was accessible to the children. Additionally, children were always asked to justify their answers, before and after the teachers’ testimony.

The novel aspect, and the first aim of the current study, was to examine whether children’s tendency to conform allows them to question their own erroneous conception of conventional counting rules (tested with a counting detection task) and whether they use testimony differently depending on whether it is unanimous or not. We expect: (a) that children will be more flexible in their judgments on the violations of conventional rules after listening to the testimonies of the majority (unanimous or not), and will accept the counting pseudoerrors; (b) that this flexibility will be greater in the unanimous than in the non-unanimous condition, because observing another’s dissent might increase children’s own independence, reinforcing their confidence in their response; and (c) that before listening to the teachers’ testimony, the children will justify their response by alluding to the violation of conventional counting rules, and that, after exposure to the testimony, they will justify their responses in terms of the information provided by the teachers and not on the basis of social motivation (e.g., deference to the informants as knowledgeable adults).

Our second aim is to determine whether children’s conformity changes with ages. We expect to find no differences between the groups in the tendency to conformity if it is structurally similar to that of adults. However, we expect that the older the children are, the more they will question their erroneous conception of the conventional rules, due to the judgments made by the majority and their own learning experience, which enhances their understanding of the meaning of counting.

Our third and last aim is to evaluate whether the effect of conformity, that is supposed to promote the revision of the prior knowledge, is maintained over time. For this purpose, we presented the children who demonstrated conformity to the testimonies of the unanimous majority at time 1 (that is, children that accepted as correct one or more pseudoerrors only after hearing the teachers’ testimony), a similar detection task at time 2, two and a half months later, to establish whether they would still accept the pseudoerrors as correct counts. We anticipate that this effect will continue over time, especially in the older age group for the same reasons stated above.

Method

Participants

A total of 88 children were individually tested: 44 preschoolers (aged between 5 years 3 months and 6 years 2 months), and 44 first grades (ranged in age from 6 years 4 months to 7 years 2 months). Participants were preschoolers and first graders because counting is a crucial curricular content in those educational levels and children’s comprehension of the skill is still incomplete at those ages, as previous research has shown (e.g., Escudero et al., 2015; Kamawar et al., 2010). Furthermore, we were interested in potential age differences in children’s tendency to conform.

Children were distributed into four groups, depending on the age and the condition to which they were randomly assigned (unanimous majority vs. non-unanimous majority): (a) 22 preschoolers in the unanimous majority condition (12 girls and 10 boys, mean age (M_age ) = 5 years 9 months, standard deviation (SD) = 3.11 months, range = 5 years 4 months to 6 years 1 month); (b) 22 preschoolers in the non-unanimous majority condition (7 girls and 15 boys, M_age = 5 years 10 months, SD = 2.77 months, range = 5 years 3 months to 6 years 2 months); (c) 22 first graders in the unanimous majority condition (9 girls and 13 boys, M_age = 6 years 11 months, SD = 3.30 months, range = 6 years 4 months to 7 years 2 months); and (d) 22 first graders in the non-unanimous majority condition (6 girls and 16 boys, M_age = 6 years 9 months, SD = 2.79 months, range = 6 years 4 months to 7 years 2 months).

The analysis was carried out with 80 participants. Eight children (two preschoolers and six first graders) systematically accepted the pseudoerrors before listening to the teachers’ claims and were excluded from the analysis because their data were inadequate to assess conformity. There were 42 preschoolers, 20 in the non-unanimous condition and 22 in the unanimous condition. The 38 first graders were equally distributed in both conditions.

All participants came from the same educational center, a private school on the outskirts of Madrid (Spain). They spoke Spanish as their first language, were from the Spanish ethnic majority group (white) and, according to information provided by school staff, were of middle socioeconomic status. No one presented learning difficulties or other school adjustment problems. Written parental consent was obtained for children to take part in the study and all children took part voluntarily.

Materials and Procedure

Two measurements were taken at two different time points two and a half months apart. At time 1, all the children were tested individually by two female experimenters in a single semistructured interview lasting around 20 minutes, in a quiet room near their own classroom. At time 2, individual semistructured interviews were conducted with the children who had showed conformity to the majority opinion, in one or more pseudoerrors, at time 1.

At time 1, the children were shown a counting detection task using a video recording of a computer program,¹ in which four characters (Mara, Eli, Eva, or Tina) counted different sets of objects (ranging from 7 to 9 items) that were always in a row and the same instructions were given for the five trials (see Table 1). The computer game started with a character (Rosa) presenting the game: “We are going to play counting things with some girls. I am going to put some objects on the table for them to count. You must pay attention and tell me if they have done it right or wrong.” Then the character (for instance, Eva) counted the objects aloud, touching them with a finger so the objects move slightly as a sign they have been counted. When the counting was over, Rosa asked each participant “Has she done it right, or has she done it wrong?” Following the child’s response, the interviewer asked the child to justify her/his answer (for instance, Why?, Why can’t she do that?, Why do you think that?).

Table 1.

Description of the counts performed by the characters at time 1.

Trials	Rule transgressed	Set size	Trial description
Pseudoerror 1	Temporal adjacency	8	They counted the eight sweets by twos, saying aloud only the even numbers (2–4–6–8)
Pseudoerror 2	Beginning from one end	7	They started their counting from the fourth sock. Then, they continued with the rest of socks, from left to right
Pseudoerror 3	Spatial adjacency	9	They skipped the seventh dummy and counted it at the end (with the label “9”)
Erroneous count	Logical rule: one to one correspondence	7	They split the Spanish number word “4” in three tags “cu-a-tro” to refer to the fourth, fifth and sixth books
Correct conventional	No violations	9	They counted all racquets from left to right, consecutively

When each one of the counts of the detection task was over, the children watched a video with the teachers’ testimony about that same counting trial. This sequence (counting detection task video followed by teachers’ testimony video) was repeated 5 times, one for each one of the counting trials presented in the interview. The teachers were four adult females and real persons. The interviewer addressed the children, saying: “Look, now you’re going to see some teachers. They’re all math teachers. They also saw Eva counting and we’ve asked them whether Eva has done it right or has done it wrong. Listen carefully.” In the unanimous condition, the four teachers endorsed the pseudoerrors, while in the non-unanimous condition three teachers accepted the counts and a lone teacher rejected them. The dissident was a different person in each trial and always sat at one end of the group. In pseudoerror 3, the dissident’s testimony was presented first, while in the other two, the majority’s opinion was given first. In both conditions, the teachers justified their decisions. To make their justification easier for the participants to understand, they were based on those given by children of similar ages to those taking part in our study (see Table 2). The teachers changed position consistently, with their spatial location varying from trial to trial. The teachers were shown sharing their thoughts (speaking quietly to each other) before presenting their final judgment, although only one of them expressed aloud the judgment of the majority (“We all think that…”).

Table 2.

Description of the arguments offered by the teachers after each counting trial in unanimous and non-unanimous conditions at time 1.

	Majority arguments (*)	Dissident arguments (*)
Pseudoerror 1	She counted right, because she counted by twos and there is a total of 8. Counting by ones is the same as by twos	She counted wrong, because she left out four numbers: 1, 3, 5 and 7. You have to count all the numbers without skipping
Pseudoerror 2	She counted right. Although she started from this one (pointing to the fourth one), she counted them all	She counted wrong, because you can never start to count from the middle of the row
Pseudoerror 3	She counted right, because although she first skipped this object (pointing at it) she counted it later	She counted wrong, because she said 9 (the last number) here (pointing at it), and it should have been the last, there (pointing at the last element on the right). You cannot go backwards when you count
Erroneous count (**)	She counted wrong, because she said “cu-a-tro” [fo-ur] for several objects. You have to say “cuatro” [four] in only one object
Correct conventional(**)	She counted right, because she counted all the objects without skipping any

Note: (*) the argument stated by the majority was the same in both conditions (unanimous and non-unanimous). The dissident argument only appeared in the non-unanimous condition. At time 2, children were not shown the teachers’ testimonies; and (**) in erroneous and correct conventional counts, there was unanimous agreement among the four teachers, in both conditions.

In order to ensure that participants had understood the situation, the interviewer gave them a brief summary of the informants’ arguments (“Let me remind you what the math teachers said.”). In the unanimous condition, the interviewer said: All the teachers said that…Are the math teachers right or not? And in the non-unanimous condition: These three teachers said that…and this other teacher said that…Who do you think is right: these three teachers or this one? The experimenters simultaneously pointed at them on the screen to clarify which teacher/teachers was/were being mentioned.

In both experimental conditions, the children were asked to justify their choices, and no feedback was given after their responses. All the interviews were recorded on an audio recorder for subsequent transcription and analysis. Each interview consisted of five trials: one correct conventional count; one erroneous count; and three pseudoerrors without explicit cardinal values (see Table 1). The order of presentation of the trials, at time 1, was partially counterbalanced and children were randomly assigned to one of the resulting three orders.

The children’s responses before hearing the teachers’ testimony were coded by assigning 1 when the count was accepted and justified as correct, and 0 when rejected or the response was not duly justified. After exposure to the teachers’ testimony, duly justified judgments endorsing the claims of the unanimous or non-unanimous majority were coded as 1 (= presence of conformity), otherwise children’s responses were coded as 0 (= absence of conformity) (see Table 3).

Table 3.

Categories for children’s justifications when judging pseudoerrors.

Code	Description	Example
Before listening to the testimony
Correct—duly justified	Stating the correctness of the counting by alluding to the logical rules and/or the cardinal value	Right: “Even though she counted quietly, there still are 8”
Conventional rules	Spatial adjacency	Wrong: “She has skipped one and counted it at the end”
	Temporal adjacency	Wrong: “She has counted every other number instead of saying them all in order”
	Beginning at one end of the row	Wrong: “She has not started from the first one [item located at the left side of the array]”
Other	Arguments that do not fit into above categories	Wrong: “She counted the number nine twice”
After listening to the testimony
Correct—conformity duly justified	Accepting the pseudoerror with the same or a similar argument to the majority	“She counted all the numbers, but she said them very quietly”
Conformity based on authority	Accepting the pseudoerror by focusing on the authority of the majority	“There are three people who say that” “They are math teachers and know more”
Rejection with similar argument	Rejection of the pseudoerror with the same or a similar argument to the dissident	“She has said that you never, ever can start from the middle”
Other	Arguments that did not fit into above categories	“She didn’t count everything”

All the participants performed very well in the correct conventional count and in the erroneous count: 96.6% and 97.7% correct answers, respectively. These trials were excluded from subsequent analysis because they were used as a control for children’s understanding of instructions as well as to make the characters’ counting performance more realistic.

Independent intercoder agreement for 25% of the interviews was consistently greater than 96% and disagreements between coders were discussed until a consensus was reached.

At time 2, we tested the children who had accepted 1, 2, or 3 pseudoerrors (previously rejected in their independent judgments) after listening to the teachers’ unanimous testimony (32 of 41 participants, 16 from each age group). The number of children in the non-unanimous condition showing conformity to the majority testimony was 9 (out of 39 in this condition), and their data have not been included in the analysis. Although the counting detection task was the same as at time 1, the objects, the quantities (8, 9, and 7 elements for pseudoerrors 1, 2, and 3, respectively), and the characters performing the different counting trials were changed. In addition, the teachers’ testimony was omitted and the participants were only asked to judge the different counts (conventional, erroneous, and pseudoerrors) and to justify their responses. The presentation order and the coding criteria for the responses were the same as those for time 1.

Results

Preliminary univariate analysis revealed that there were no differences between conditions in the number of children who accepted (or not) each pseudoerror before listening to the teachers’ testimony. These results were the same regardless of whether the two age groups were analyzed together or separately.

According to the first two objectives and given that the dependent variable was dichotomous for an individual child (presence [= 1] or absence [= 0] of conformity), this study used binomial logistic regression to predict the probability that a child who first rejected an unconventional count would accept it after hearing the teachers’ correct testimony. Each pseudoerror was modeled separately, and Condition (non-unanimous, unanimous), Age-Group (preschoolers, first graders), Gender (girls, boys), and Order of presentation of the trials (1, 2, 3) plus the interaction Condition by Age-Group, were included as predictor variables. Because the analysis was based on the number of responses in which children first rejected a pseudoerror but then accepted it after hearing teachers’ opinion, models for pseudoerrors 1, 2, and 3 included 60, 65, and 60 children, respectively, and not the 88 children initially interviewed.

The predictive power of different models for each type of pseudoerror was tested (see Table 4). Following a stepwise method, model 1 included all the predictor variables. When no significant Age-Group by Condition interaction was found, a second model was fitted that included all the predictor variables except the interaction. The predictive power of model 2 was significantly better than that of the baseline model for the three pseudoerrors (χ2 (4, N = 60) = 25, p = 0.000; χ2 (4, N = 65) = 19.12, p = 0.001; χ2 (4, N = 60) = 29.21, p = 0.000, for pseudoerrors 1, 2, and 3, respectively). Likewise, the resulting chi-square comparing model 2 to model 1 was not significant and there was a slight improvement in model fit based on the Bayesian information criterion for the three pseudoerrors (model 2). Thus, this second model was preferred.

Table 4.

Fit indexes for binary logistic regression models.

Model	Negative log likelihood	Accuracy of classification	R² Nagelkerke	χ ²	Parameters	p	Akaike information criterion	Bayesian information criterion
Pseudoerror 1. Temporal adjacency
Baseline model	71.53	71.7%	–	–	1	–	72.53	73.31
Model 1	45.97	83.3%	0.50	25.56	6	0.000	51.94	56.64
Model 2	46.53	83.3%	0.49	0.56	5	0.454	51.53	55.42
Pseudoerror 2. Starting from one end
Baseline model	88.24	58.5%	–	–	1	–	89.24	90.05
Model 1	67.98	73.8%	0.36	20.26	6	0.001	73.98	78.86
Model 2	69.12	73.8%	0.34	1.14	5	0.286	74.12	78.19
Pseudoerror 3. Spatial adjacency
Baseline model	73.30	70%	–	–	1	–	74.30	75.08
Model 1	42.83	81.7%	0.57	30.47	6	0.000	48.83	53.50
Model 2	44.10	81.7%	0.55	1.27	5	0.260	49.10	52.99

The binomial logistic regression for pseudoerror 1 indicated that Condition and Group were significant predictors of children’s conformity to the teachers’ testimony (χ2 (4, N = 60) = 25, p < 0.001). The two predictors accounted for 48.9% of the variability of children’s conformity to the teachers’ testimony. Condition and Group were significant at 1% level (Condition Wald = 11.84, p = 0.001; Group Wald = 7.62, p = 0.006). The odds ratio (OR) for Condition showed that children in the unanimous testimony condition were 26.99 times more likely to conform than children in the non-unanimous testimony condition (95% CI 4.13–176.32). The OR for Group indicated that older children tended to conform 9.34 times more than preschoolers (95% CI 1.91–45.65). The model correctly predicted 93% of cases where there was no conformity and 58.8% where there was conformity, giving an overall percentage correct prediction rate of 83.3%. The univariate analysis showed that first graders were significantly more likely to conform (44.4%) than were preschoolers (15.2%), χ2 (1, N = 60) = 6.28, p = 0.012, and that children in the unanimous majority condition were more prone to conformity (50%) than were children in the non-unanimous majority condition (6.7%), χ2 (1, N = 60) = 13.87, p < 0.001.

The binomial logistic regression for pseudoerror 2 revealed that Condition was the unique significant predictor of children’s conformity to the teachers’ testimony (χ2 (4, N = 65) = 19.12, p = 0.001). Condition accounted for 34.3% of the variability of children’s conformity to the teachers’ testimony. Condition was significant at 1% level (Wald = 13.05, p < 0.001). The OR of conformity was 10.78 times greater for children in the unanimous testimony condition than for children in the non-unanimous testimony condition (95% CI 2.97–39.17). The model correctly classified 73.7% of cases where there was no conformity and 74.1% where there was conformity, giving an overall percentage correct prediction rate of 73.8%. The results of univariate analysis suggested that children in the unanimous majority condition were significantly more likely to conform (66.7%) than children in the non-unanimous majority condition (20%), χ2 (1, N = 65) = 14.49, p < 0.001.

Finally, the results for pseudoerror 3 showed that Condition was also the unique significant predictor of children’s conformity to the teachers’ opinion (χ2 (4, N = 60) = 29.21, p < 0.001). This predictor accounted for 54.6% of the variability of children’s conformity to the teachers’ testimony. Condition was significant at 1% level (Wald = 12.99, p < 0.001). The OR showed that the odds that children change their response after listening to the teachers’ testimony were 53.58 higher for children in the unanimous testimony condition than in the non-unanimous testimony condition (95% CI 6.15–466.91). The model correctly predicted 76.2% of cases where there was no conformity and 94.4% where there was conformity, giving an overall percentage correct prediction rate of 81.7%. The results of univariate analysis indicated that there was a significantly higher tendency to conform to the teachers’ testimony in the unanimous majority condition than in the non-unanimous majority condition (63% vs. 3% of the children), χ2 (1, N = 60) = 25.4, p = 0.012.

The analysis conducted revealed that Condition was a significant predictor of children’s conformity to the teachers’ testimony for pseudoerrors 1, 2, and 3. These results partially confirm the hypothesis of greater flexibility in children’s judgments on the violations of conventional rules after listening to the testimonies of the majority, but only for the unanimous majority condition. As may be seen in Table 5, children deferred to the teachers’ judgment more frequently when the correct testimony was provided by a unanimous than by a non-unanimous majority. In the unanimous testimony condition participants were readily able to accept the pseudoerrors as correct counts, but not when there were conflicting testimonies, as happened in the non-unanimous testimony condition. The data also showed that overall children were not more likely to conform to unconventional counts as age increased, probably because it was not new information for children. Group only was a significant predictor of children’s conformity on pseudoerror 1, because first graders were more capable than preschoolers to understand the logical justification provided by the unanimous majority for this unconventional count, which allowed them to overcome their erroneous knowledge and accept the unconventional count as correct. Gender, Order of presentation of the trials and the interaction of Condition by Group were not significant for any pseudoerror.

Table 5.

Percentages of children showing conformity to the majority testimony in each type of pseudoerror.

Condition	Type of pseudoerror	Preschoolers	First graders
Non-unanimous majority	1. Temporal adjacency	0	10.52
	2. Beginning from one end	5	31.6
	3. Spatial adjacency	0	5.3
Unanimous majority	1. Temporal adjacency	22.7	52.6
	2. Beginning from one end	50	47.4
	3. Spatial adjacency	45.5	36.8

All these results point to the fact that the understanding of the relative value of conventional counting rules develops very slowly, as evidenced not only in children’s performances related to the number of pseudoerrors accepted spontaneously, but also in the fact that they did not blindly follow the correct judgment of the majority in all the pseudoerrors. The overall percentage of children who initially rejected the pseudoerrors and conformed, in one or more cases, to the majority, was higher in the unanimous condition than in the non-unanimous one (78.1% vs. 23.1% of the children, respectively). More specifically, 41.5% of the children in the unanimous condition conformed in two or three pseudoerrors, and 36.6% in only one. The corresponding rates for the non-unanimous majority condition were 2.6% and 20.5%, respectively.

We analyzed the children’s justifications when judging the pseudoerrors, before and after listening the teachers’ testimony, because they would offer some insight into the thought processes underlying the incorrect responses. In the first case, as expected, the children, regardless of the condition they were assigned to, rejected the pseudoerrors essentially because they violated three conventional rules (see Table 6): spatial adjacency; temporal adjacency; and beginning at one end. In the second case, following exposure to the testimonies, the children’s justifications suggested the importance of informational motivation. As it is illustrated in Table 6, children took into account the arguments given by the teachers in the vast majority of the trials. The trials in which children justified their conformity by the fact that the informants were teachers, or the number of people who agreed in the statement, were notably scarce.

Table 6.

Percentages of children’s justifications when judging the pseudoerrors.

	Preschoolers	First graders
Before hearing the testimony
Correct responses	28.6	30.7
Spatial adjacency	19	18.4
Temporal adjacency	23	23.7
Beginning at one end	16.7	23.7
Other	12.7	3.5
After hearing the testimony
Same or a similar argument to the majority	41.3	59.6
Same or a similar argument to the dissident	39.7	36.8
Authority of the majority	4.8	2.6
Other	14.3	0.9

Finally, and in accordance with the third aim of our study, to determine whether the conformity effect was maintained over time, we conducted a mixed analysis of variance on the number of pseudoerrors accepted, using time point as the repeated measure: 2 (Age-Group: preschoolers, or first graders) × 2 (Time Point: time 1 vs. time 2). There were significant main effects of Age-Group, F(1, 30) = 8.85, p < 0.01, η_p ² = 0.23, and of Time Point, F(1, 30) = 7, p < 0.05, η_p ² = 0.19, as well as a significant interaction between these two factors, F(1, 30) = 8.65, p < 0.01, η_p ² = 0.22. Bonferroni post hoc tests revealed that while at time 1 there were not significant differences in the number of pseudoerrors accepted by preschoolers and first graders after hearing the teachers’ testimony, at time 2 primary school children significantly outperformed preschoolers in terms of the number of pseudoerrors spontaneously accepted, F(1, 30) = 12.50, p < 0.01, η_p ² = 0.30. The findings indicated, on the one hand, that both groups were equally permeable to correct testimonies (M = 1.63, SD = 0.72 and M = 1.63, SD = 0.62, respectively for preschoolers and first graders) and, on the other hand, that they differed in that only in older children the effect of testimonies had a lasting effect (M = 0.44, SD = 0.89 and M = 1.69, SD = 1.08, respectively for preschoolers and first graders). These data suggest that primary-school children’s tendency to conform not only makes them endorse correct unanimous testimonies about conventional rules of counting, abandoning their own erroneous prior knowledge, but also that this is not a temporary change because it is maintained over time.

Discussion and Conclusions

The results from the current study confirm the difficulty children have in accepting pseudoerrors as correct counts before hearing the teachers’ testimony. These data suggest that during the development of counting knowledge, children incorporate non-essential features (conventional rules), which leads them to reject numerous correct unconventional counting procedures. The mean percentage of rejected pseudoerrors before listening to the testimony reached 70.4% of trials, which is comparable to the 73.7% obtained by Rodríguez, Lago, Enesco, and Guerrero (2013) with children of similar ages. Furthermore, the results are consistent with those obtained in other studies regarding the justifications provided by the children when rejecting the pseudoerrors (e.g., Lago et al., 2016; Rodríguez et al., 2013). They also confirm that understanding of the optional nature of the conventional rules of counting continues to develop beyond the early years of primary education, as suggested by several authors (e.g., Escudero et al., 2015; LeFevre et al., 2006; Kamawar et al., 2010).

Does the teachers’ testimony foster the children’s learning of the optional nature of conventional counting rules? The effect of the teachers’ testimony resembles that of the presence of the cardinal number in that it seems to weaken the power of the conventional rules, which helps children to distinguish the essential from the nonessential counting aspects. We partially confirmed our predictions about the influence of the teachers’ testimony on children’s decisions: our participants revised their judgments on the unconventional counts when the testimony came from a unanimous majority; however, they rarely changed their judgments when there was a dissident (non-unanimous majority).

Did children revise their initial judgment simply because they were asked to make the same judgment twice, as suggested by Rakoczy et al. (2015)? In the current research the answer appears to be no, because our findings revealed a significant difference between conditions. This difference is attributable to the type of majority (unanimous or not) and not to the effect of repetition since in both conditions children were asked to make the same judgment twice and, furthermore, they had to justify their responses.

In the unanimous and non-unanimous majority conditions, the children’s tendency to conform reached 78% and 23% respectively, when we included the participants who changed their judgment on at least one pseudoerror after hearing the teachers’ testimony. Consequently, and in line with prior research, our results show that children vary greatly in their conformity to others’ testimony and do not blindly accept all of them. Instead, children are selective when deciding which information to endorse (Einav, 2014), depending on several factors such as their ability to mentally represent the information (Lane et al., 2014), the strength of their prior knowledge (Chan & Tardif, 2013), some epistemic aspects of the informants (e.g., past accuracy), whether the informants generate information that is consistent or not with their own knowledge (Corriveau & Harris, 2010), experience (Jaswal, Croft, Setia, & Cole, 2010), or the normative domain (Flynn, Turner, & Giraldeau, 2018). In general, the percentage of trials in which children conformed to the testimony of the unanimous majority (42.3%) is consistent with those obtained by other authors (Asch, 1956; Haun & Tomasello, 2011). Besides, these percentages, together with the lack of differences across age groups, seem to indicate that children’s tendency to conform is structurally similar to that of adults, as suggested by Corriveau and Harris (2010) and Haun, van Leeuwen, and Edelson (2013).

Our participants conformed five times more when the correct-logical view on pseudoerrors was provided by a unanimous majority than by a non-unanimous majority, and this trend was not linked to a specific type of pseudoerror. In the non-unanimous condition, the children’s performance on pseudoerrors barely improved, as had been found in a previous study (Enesco et al., 2017), in which the majority of teachers defended either a wrong perspective (similar to the child’s own one) or a correct (logical) perspective. The fact that the non-unanimous majority did not have any effect on our older participants, contrary to what previous research on conformity transmission suggests (see, for example, Morgan, Laland, & Harris, 2015; Muthukrishna et al., 2016), can hardly be explained by the number of informants in agreement. Earlier works have shown that a majority formed by three adults is a sufficiently large number to be effective (Corriveau, Fusaro, & Harris, 2009). Maybe, the reason for this different pattern of response can be related to the kind of knowledge assessed in our study. Children have strong prior ideas about the counting procedure, and they feel fairly sure about the “correct” (conventional) way to perform this activity. In contrast, previous research comparing total and partial majorities have mostly assessed children’s decisions in uncertain or unknown situations where any suggestion could be reasonably right. As it has been also observed in Enesco et al. (2017), children of similar ages were much more likely to endorse a wrong than a correct majority, and overall, their decisions were based on the justifications provided by the informants and not on the number of informants. Additionally, considering the fact that, in the present study, the justifications for the unanimous and non-unanimous majorities were identical, and that in this latter condition (i.e., non-unanimous) the majority was formed by three teachers, we believe that the reason the children tended to discredit the testimony when there was a lack of unanimity was due to the fact that the explanations provided by the dissident were similar, on most occasions, to those of the children before exposure to the testimony, and this made them reaffirm their position. In other words, if children faced a situation in which there are four experts, three of which emit a different judgment to their own, but one that coincides, why should they revise their own ideas?

In overall terms, our findings lend support to those of other studies suggesting that in certain conditions children may be less inflexible when judging unconventional counts (Escudero et al., 2015; Lago et al., 2016; Rodríguez et al., 2013). Our findings also provide further evidence in support of the idea that trust in the testimony of others is key as a source of learning and this trust is enhanced when it is placed in a unanimous majority (Bernard et al., 2015; Corriveau et al., 2009; Morgan et al., 2015).

However, our findings only partially coincide with the proposal that, from an early age, children tend to reject the claims of a unanimous majority when they are inconsistent with their own perceptions and beliefs, since, on at least one occasion, 78% of the children revised their judgments on the pseudoerrors in the unanimous majority condition (Einav, 2014; Haun & Tomasello, 2011; Lane et al., 2014; Seston & Kelemen, 2014; Sobel & Kushnir, 2013). As has just been stated, one possible explanation may lie in the nature of the knowledge evaluated in this study, which is considerably different from the type of knowledge assessed in previous studies (e.g., specific information about the labels of unknown objects). Counting is a complex procedure that children acquire early in an informal context but without a full understanding of its meaning. Becoming aware of the logical properties of counting is a long process that—for many children—requires the support of formal instruction. However, for such instruction to be effective, it is necessary to work on children’s prior ideas about number, making explicit their underlying misconceptions. In some way, this is what our participants faced in the experimental setting. We used a child-friendly procedure in which the teachers always offered an explicit and accessible justification for the different counting procedures, and this may have facilitated children’s revision of their prior ideas, provided that the teachers were not in conflict. In other words, the unanimous majority setting might have favored an informational and non-social motivation to accept the testimony. If social pressure was responsible for children’s deference to the teachers’ opinion, they would accept teachers’ majority testimony even when it was non-unanimous. However, the participants did not endorse the correct testimony of a majority of teachers over that of a lone dissident teacher, but they endorsed the same testimony when it was provided by a unanimous majority of teachers.

Is the conformity effect maintained over time? The results at time 2 showed that when the first graders revised their strong prior knowledge the changes were overall maintained, but in the case of the preschoolers, acceptance for pseudoerrors fell from 54.2% to 14.6% of the trials. This may be because older children can recognize, and verify, the correct evidence given by the informants in the procedures they usually employ to solve different kinds of mathematical problems. These different response patterns seem to indicate that the primary-aged children’s tendency to conform is attributable to informational rather than normative motivation (Deutsch & Gerard, 1955), as the children revised their beliefs on the meaning of conventional counting rules. Our findings are similar to those obtained by Rakoczy et al. (2015) in a sample of younger children, but not to those of Haun and Tomasello (2011), who suggested that children actually only changed the public expression of their judgment and not the “real” judgment.

The nature of the children’s justifications also seems to support this idea. The children’s enhanced performance on accepting the pseudoerrors cannot be considered to stem from their simply accepting the teachers’ opinions due to social motivation, since only 6.3% of the children who sided with the majority, unanimous or not, justified their decision by the number of informants (“because there are more of them” or “because they are all in agreement”). The child-friendly situation used may also have avoided the children trying to please the informants by accepting their testimonies, which were not presented in person but on video. Finally, the preschoolers’ failure to maintain their acceptance of pseudoerrors at time 2 could be at least partly due to the fact they did not elaborate further the reasoning provided by the majority although pseudoerrors were distinctive information for them (i.e., violate their expectations about the conventional rules of counting) (Howe, Courage, Vernescu, & Hunt, 2000). Another complementary explanation is related to more general developmental aspects, such as the ontogeny of the normative mind. Young children’s rigidity in applying different types of norms has been also documented in other domains such as gender or moral development. For example, a recent research has shown that young children, but not adults, conceive instrumental norms (how to do a practical task, like pinning a nail into a piece of wood) as categorical imperatives that must be applied regardless of individual goals (Dahl & Schmidt, 2018). In any case, these suggestions should be considered with prudence as long as we do not have further evidences.

In conclusion, this study has shown that school-aged children are capable of revising their initial judgments on unconventional counts when a unanimous majority provides arguments justifying their validity, and that this effect is maintained over time in first graders. Nevertheless, it is perhaps premature to suggest that a conceptual change occurred in the development of the conventional counting rules because, among other things, they successfully applied them on repeated occasions, both in formal and informal educational settings, without the need to question them. Future research should go further, for example, in exploring the long-term effects of these changes including a larger sample, and maybe a control group (who never sees the teachers’ testimonies), as well as new data resulting from new pseudoerrors. We also consider it necessary to establish why some children do not benefit from the testimonies of the unanimous majority, to determine whether they need a broader exposure to accept the unconventional counts or whether cognitive limitations with respect to the executive functions prevent them from inhibiting a strong habitual response.

From an educational perspective, it is critical to understand how conformity to conventional norms affects children’s learning and, more specifically, children’s comprehension of conventional counting rules. Conventional rules involve several advantages and disadvantages for learning to count. On the one hand, they guide children’s behavior when executing the counting procedure, leading to a correct performance. On the other hand, it has been shown that many elementary students assume that conventional rules have to be applied unconditionally. A full understanding of counting implies the recognition of which rules must be followed and which ones can be modified. So, it is necessary to find additional ways to promote among children the comprehension that any count is correct as long as the logical rules are preserved. The paradigm of learning from others based on the conformity to the testimony of a unanimous majority may be an important framework to integrate other findings and minimize the effect of conventional counting rules. The possibility to experiment with different counting procedures and to reflect on which is more appropriate will help children to develop a deeper conceptual understanding (see Siegler, 2003). It seems essential that teachers promote an abstract conception about counting among students, since it has been proved to be one of the foundations of arithmetic (Dowker & Sigley, 2010; Foster, Anthony, Clements, Sarama, & Williams, 2018).

Footnotes

Acknowledgments

We thank the Colegio de Educación Infantil y Primaria El Tejar in Majadahonda (Madrid) and all the children for their cooperation. We are also very grateful to the journal editor and reviewers for their valuable comments on previous versions of this manuscript.

Funding

This research was funded by a project grant (PSI2012-31477) from the Ministerio de Economía y Competitividad.

ORCID iD

Ana Escudero

Note

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