Number Line Estimation: another view in the light of the ACE, Arithmécole, and ELFE data.

doi:10.21203/rs.3.rs-5317676/v1

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Number Line Estimation: another view in the light of the ACE, Arithmécole, and ELFE data.

https://doi.org/10.21203/rs.3.rs-5317676/v1

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Background. Numerous experimental or empirical papers using a number line estimation task have already been published. In this task, the participant must indicate the position of a number (e.g., 12) on a line bounded by two numbers (e.g., 0 and 100). However, the authors of these articles mainly sought to model the representation of numbers based on data obtained from participants.

Methods. This research draws on data from three French studies, each involving several thousand primary students, with an educational perspective. These data made it possible to investigate questions that have rarely been addressed, such as the singularity of this task and the asymmetry of the influence of the bounds. More importantly, they have enabled a systematic analysis of students' errors with the task presented in the form of multiple choice.

Results. Qualitative principal component and automated classification analyzes show some singularity of the number line task (in its multiple-choice exercise, at least). The asymmetry of the influence of the two bounds argues against the two main models in the literature—the logarithmic model and the linear model. The analysis of the approximately 100,000 students' erroneous choices reveals that they do not completely respect their distance to the correct choice: The expected order of choice through genuine estimation, that is first the correct number, then the nearest wrong number, then the middle wrong number and lastly the farthest wrong number, was not respected in more than half of the 46 number line estimation items. Furthermore, young children exhibited under-comprehension by confounding the magnitude of the target number with its position value, and educators or school authorities do not clearly understand the task or its modeling.

Conclusions. No model predicts all the observations, often original or curious, that we have made. Moreover, for theoretical reasons, we believe that the search for such a model is somewhat vain. In addition, we are sceptic about using the task for anything other than student assessment, given the distortions and difficulties that emerge from the children's erroneous choices, and from the practices of teachers and their supervisors also discussed in the article.

Psychology

Educational Psychology

number line

estimation

modeling

number magnitude

position number

multiple choice

The task of placing numbers on a number line, often referred to as number line estimation (NLE), is the mathematical task that has been most widely used in numerical cognition researches over the last two decades. It involves placing a number, often presented symbolically (e.g., 16), either on a bounded straight line with no other indication (e.g., [0, 30]), or choosing the mark corresponding to it from several marks with no indication. Although there are several meta-analyses of research with this task (e.g., Schneider et al., 2018; Ünal et al., 2024), they are far from including all studies. Even if the field of these studies is not yet closed, it is already impossible to cite them all, a fortiori to analyze them. Even with precise and restrictive criteria Qin et al. (2024) identified 1512 articles!

The present article, however, takes a new look at this placement task by means of a precise and in-depth analysis of errors as a function of distance from the (correct) target response. Most studies have simply analyzed errors in terms of averages of the absolute distance to the target response, or of pass/fail when a choice task was used. The present study, using data from three studies on large samples of students, attempts to remedy this shortcoming by analyzing students' incorrect answers or answer choices as a function of the target number.

Another original aspect of this study is that it does not use one or more modeling techniques, particularly graphical ones, for children's representation of numbers. Indeed, models do not necessarily reflect the reality of students' behavior. Not only because of the problem of inference from curves based on group data (Estes, 1956), but also because the models, even when individualized (e.g., Berteletti et al., 2010), offer no guarantee of reflecting the behaviors or processes actually followed by the children. More generally, the statistician Box (1976) repeats in his article that all models are wrong. Of course, some modelers, such as Zelner et al. (2021), are repelled by this last assertion, but others accept it (e.g., McElreath, 2020). It can also be argued that any statistic–even a simple average—is a model (Noël, 2013). But the lack of nuance in the levels of abstraction of the models seems as regrettable as the lack of nuance between significant and non-significant, which the modelers cited are unanimous in criticizing.

With regard to modeling, the number representations attributed to students have been postulated to be logarithmic (Dehaene, 2003 ; Dotan & Dehaene, 2016), linear (Ebersbach et al., 2008), nonlinear (Gunderson et al., 2012), bilinear (Stapel et al., 2015), multiple (Siegler & Opfer, 2003), transient with transition from one to another during development (He et al., 2016; Siegler & Booth, 2004), or as pertaining to more complex mixtures (DeVries et al., 2020; Moeller et al., 2009). The number line is said to be already observable in human newborns (Di Giorgio et al., 2019), and also in chicks (Rugani et al., 2015), monkeys (Rugani et al., 2024), and even insects (Giurfa et al., 2022). Boys generally perform better than girls on this task (Geary et al., 2021; Xu et al., 2023, for a review). This better performance by boys is thought to be due to their spatial skills rather than to the spatial anxiety of girls (Tian et al., 2022). But we cannot rule out the possibility that it stems more generally from boys' better numeracy performance from the first grade onwards (Fischer & Thierry, 2022) and, in any case, is consistent with it.

Our present research will not directly discuss such models, but will attempt to show that, insofar as they almost all reflect uniformly continuous representations, they are difficult to reconcile with certain specific student errors. Three questions will be investigated. Firstly, a general and basic question will consist of examining whether the estimation (placement) test is different from other tasks typically investigated in mathematics, such as mental arithmetic and numerical reasoning (Study 1). Such a singularity would in fact justify the countless research studies that have been carried out on it. Next, (2) we will test the hypothesis that placement errors differ according to the proximity of the bounds (left or right) of the target number. More precisely, we postulate that the placement errors tend to be overestimates for target numbers close to the left-hand bound, and underestimates for target numbers close to the right-hand bound. In the case of a positive outcome, however, the test of this hypothesis would suffer from a methodological weakness. For a very small target number, it would be difficult to estimate it on its left, and for a very large target number, it would be difficult to estimate it on its right. We have therefore also carried out the same analysis, eliminating erroneous placement distances that are not possible in both directions (Study 2). Finally, we will check whether the percentages of erroneous placement choices are perfectly correlated with their distance from the correct choice, as suggested by models of linear representation (Study 3).

To approach these questions, we have massive data from three French studies which included the placement task in the assessment tests. These studies all concerned primary schools students assessed in their respective classes. They are (1) "Arithmétique et Compréhension à l'École” (ACE1), described in Vilette et al. (2017), and ACE2, very similar to ACE1, except for the placement task, (2) “Arithmétique à l'école” (Arithmécole1 and Arithmécole2), for which other data were analyzed by Fischer et al. (2019), and (3) “Étude Longitudinale Française depuis l'Enfance » (ELFE), described in Fischer and Thierry (2022). These studies—a necessary condition for their inclusion—all report precisely the target numbers tested and the errors made by the students.

As we are using data from the ELFE longitudinal study in both our studies 1 and 3, we will describe it in general terms. In 2011, it recruited more than 18,000 newborns throughout Metropolitan France. When these children reached school age, their performance in the basic subjects—literacy (French) and numeracy—was assessed at the end of kindergarten middle section (2016), at the end of the first grade (2018) and at the start of the fourth grade (October 2020). In general, these children were spread across schools in France, rarely with more than one ELFE student per class. So as not to stigmatize this ELFE child, isolated in his class, the designers of the ELFE research asked the teachers to also assess, at the same time, the three students (if possible) in the class whose age was closest to that of the ELFE child. We will add the data from these students, recruited in addition^{^[1]} to the ELFE students assessed at the start of fourth grade, the only grade where the placement task was included in the ELFE study.

[1] Although teachers did not always assess three additional students, these additional students are in fact much more numerous than the genuine ELFE students in our sample.

Method

Participants. The participants are the 7713 students from the ELFE assessment at the beginning of 4^th grade with complete data in numeracy. Given the more than 18,000 children

originally recruited in the ELFE study, and the addition of students matched for age to the genuine ELFE students, this number may seem very small. But experimental attrition in the ELFE study is considerable. Firstly, because of the longitudinal nature and long duration of the study. Secondly, because the parents, teachers and directors of the schools receiving ELFE children had not only to give their consent, but also to establish liaison with the school in the case of the former (because of the strict anonymity of ELFE children), and to carry out the assessments and return the results in the case of the latter.

Materials. The target numbers to be placed (in bold), followed (in brackets) by the notations and numbers corresponding approximately to the positions of the three distractor marks, are:

(a) On a line bounded by 0 and 100: 12 (E1; 20, 45, 70), 82 (E2; 29.5, 53, 69), 68 (E3; 45, 51, 86), 26 (E4; 15, 50, 80).

(b) On a line bounded by 0 and 1000: 784 (E5; 300, 500, 900), 114 (E6; 232, 400, 600), 915 (E7; 400, 600, 800), 485 (E8; 300, 600, 800), 512 (E9; 260, 400, 580), 337 (E0; 250, 450, 600).

Before these 10 placement items, the students had to answer 10 mental arithmetic items and 10 numerical reasoning items. The mental arithmetic items, with their notation in brackets, are: 3 x 4 = ? (C1), 9 x 4 = ? (C2), 7 x 6 = (C3), 3 x ? = 18 (C4), 5 x ? = 30 (C5), ? x 4 = 40 (C6), ? x 3 = 24 (C7), 50 : 2 = ? (C8), 5 x 80 (C9), 64 : 2 (C0). The numerical reasoning items are series to be completed, for example the series 113, 109, 105, 101, ? (noted S8).

The number placement task being a choice task (of one mark among 4), we were able to analyze, in this research, the Success (correct choice = 1) versus Failure (any other choice = 0) scores. Numerical calculation and reasoning items were scored either 1 (correct answer) or 0 (other answer).

Procedure. The teachers of the classes concerned tested the students. For the most part, the students were able to work independently. However, in the mental arithmetic task, the teacher dictated the calculations, allowing a maximum of 5 s for the students to respond. In the placement task, the students had to circle the mark (vertical line) corresponding to the target number indicated on the left of the line (see two examples in Fig. 1).

To see whether the Placement (Estimation) items are special and stand out from the others, we performed a Principal Component Analysis (PCA: FactoMineR package for R software, v.2.11; see also Lê et al., 2008) on the 7713 students with complete numeracy data. In addition, we took advantage of the automated classification into three groups using the method of moving centers, made possible by the R kmeans function, to confirm the relevance of the groupings that appeared in the PCA.

Results

Figure 2, which results from the PCA, shows that of the two dimensions mainly identified, dimension 1, which explains more than 20% of the variance, perfectly separates the estimation items (E0 to E9) from the others.

The implementation of the automated classification into 3 groups confirms the grouping that appeared with the PCA: see Fig. 3.

Discussion

Axis 1 could reflect the difficulty of the items. The Estimation items are well separated and to the left of the others. In particular, the two leftmost items, E0 and E8, which contribute most to Axis 1 among the Estimation items, are also the most difficult Estimation items. But the item furthest to the right, S0, which contributes most strongly to axis 1 among the Series (and in absolute terms), is also the most difficult of the Series items. The average performance of the students, 69.45 in Calculation, 65.93 in Series and 60.61 in Estimation, does not correspond perfectly with this interpretation either. All this leads us to reject this interpretation. Whatever the interpretation of Axis 1, we can nevertheless conclude that the PCA suggests that the Placement test, as administrated (i.e., by multiple choice), does not measure exactly the same competence in arithmetic as the other two tests.

Method

Participants. Participants are the 2003 students (49.4% girls) in 1^st grade tested in June 2013 as part of an evaluation of the ACE1 program (Vilette et al., 2017). The age of the students was not reported systematically, but it should correspond to the classic age of French students at this grade level at the end of the school year, that is 7.0 years.

Materials and procedure. The students had to place the numbers 24 and 11 on a line bounded by [0, 30], and the numbers 35 and 82 on a line bounded by [0, 100], on four different sheets of paper. The ACE researchers then measured, to the nearest half centimeter, the placement mark left by the students.

Statistical analyses. The percentage of algebraic error is defined as [(number to be estimated) - (estimated number)]/scale. For example, a student who places 35 in a position corresponding to 29 will have a percentage error of -.06 (i.e., an underestimate of 6%). The methodological weakness of an analysis of this percentage, pointed out in the introduction, nevertheless leads us to calculate, in addition, an adjusted percentage by eliminating the estimated numbers whose absolute distance from the target number exceeds the absolute distance of the target number from the nearest bound. So, for our example, the absolute distance from 35 to the nearest bound (i.e., 0) being 35, the estimated numbers whose absolute distance exceeds 35 (which is impossible to the left of the target number, but possible to the right) will be eliminated from the analysis (any mark above 70 in our example). These algebraic percentages will then be subjected to planned Student's t-tests for one sample, compared to 0, for the target numbers on the left (11 and 35) and those on the right (24 and 82), giving a total of 4 planned comparisons. Our hypothesis of an overestimation of the target numbers in the left half, and an underestimation in the right half of the bounded line, should result in a positive difference for the 11 and 35 target numbers, and a negative difference for the 24 and 82 target numbers.

Results

The algebraic percentages of error are reported in Table 1. The predicted signs were clearly confirmed for both the simple and the adjusted percentages. Because of the method of adjusting the percentage, it was anticipated that the percentages would reduce in absolute value when moving from simple percentages to adjusted percentages. But above all, the difference between the adjusted mean percentages and 0 remains significant.

Table 1. Algebraic error percentages on target numbers

Target position	Left half		Right half
Target number	11	35	24	82
Simple percentage	+ .050^**	+ .225^**	- .147^**	- .091^**
Adjusted percentage	+ .033^**	+ .160^**	- .047^**	- .018^*

Note. The asterisks indicate the level of significance with a Student's t-test for one sample (mean compared to 0): ** indicates a level of significance of .001 and * indicates a level of significance of .05.

Discussion

Confirmation that children underestimate the numbers on the right-hand side of the boundary and overestimate those on the left-hand side is based on only 4 items—2 (numbers) x 2 (bounded lines)—chosen arbitrarily. Nevertheless, these numbers were chosen in the ACE research independently of our hypothesis. Moreover, these over- and underestimates are consistent with the simple hypothesis that children who have no precise idea of the position of the target number will respond towards the middle rather than clearly choosing one side or the other.

Are these results compatible with a logarithmic representation of the numbers? Such a representation should lead to an over-estimation of the numbers in the left half because the logarithmic intervals of two successive numbers are widest on the left, but also an over-estimation of the numbers in the right half (albeit lesser) because of their increasing compression as we approach the right boundary. For the bounded line [0, 100], figure 4 illustrates this well: 35 with the logarithmic scale is much further to the right than 35 with the linear scale, and 82 with the logarithmic scale is also further to the right than 82 with the linear scale. So while our results are compatible with a logarithmic representation for 35 (and 11, or more generally for the numbers in the left half), they contradict it for 82 (and 24, or more generally for the numbers in the right half).

There are two important points to note about the comparative illustration in figure 4. Firstly, Siegler and Booth (2004) commenting on their graph comparing the linear and logarithmic scales write that “relative to the linear function, the logarithmic ruler model exaggerates distances at the low end of the numerical scale and understates them at the high end” (p. 429). The numbers 35 and 82 illustrate this reduction in the distance between the two ends of the segment [0, 100]. But what is essential for our hypothesis is that even at the right-hand end, and certainly a little earlier, the logarithmic scale continues to overestimate the numbers compared with the linear scale. Secondly, to make the logarithmic scale coincide with the linear scale at the x = 0 bound, we need to use a model with log(A + x), where A > 0. This is, at best, a shortcut to the model we want, a shortcut that Gelman (2024) does not reject because, he notes, “we take a lot of shortcuts in statistics, without apologizing for it”. The only suitable A is A = 1, because for A < 1, the logarithm is negative, and for A > 1, we would have x = logA > 0 for the lower bound we want at 0.

Modeling using a logarithmic representation cannot therefore fully describe the actual behavior of many children. The following study of students' erroneous choices should make it possible to clarify this behavior.

Method

Participants. Participants were taken from ACE2 (June 2014) at the end of 1^st grade, Arithmécole1 (2015/16) and Arithmécole2 (2016/17) at the beginning (pretest) and end (post-test) of 2^nd grade, and ELFE at the beginning of 4^th grade. For the present study, all ELFE participants with complete data in the placement test were retained (N = 7991).

We therefore have a total of 2803 (ACE2) + 2295 (Arithmécole1) + 2433 (Arithmécole2) + 7991 (ELFE) = 15,522 students, who produced 170,282 number placements. However, the target numbers were often repeated from one research to another (see Fig. 6 to 11).

Material. In the ACE2 research, the students had to choose, each among four, 6 target numbers on a bounded line [0, 30] and 6 others on a bounded line [0, 100]. The target numbers, and the three numbers corresponding approximately to the locations of the distractors, are shown in Table 2 (see also Fig. 5 for two examples).

Table 2. Target numbers and numbers corresponding approximately to the distractor locations for the various items.

Research	Bounds	Number target	Number near	Number intermediate	Number far
ACE2	[0, 30]	24	18	12	6
		16	11	24	3
		8	2⁼	14⁼	23
		20	25	12⁼	28⁼
		10	4⁼	16⁼	24
		4	7	14	21
	[0, 100]	35	55	12	81
		82	69	53	29
		54	64	38	95
		14	3	31	63
		68	54	88	46
		40	59	16	80
Arithmécole2 (pretest)	[0, 100]	97	83	65	54
Arithmécole2 (pretest)	[0, 100]	3	14	31	63

Note. The two percentages on the same line with a superscript = indicate that their distances from the correct answer are approximately equal: we have then assigned a greater proximity to the choice minimizing the inter-versions compared with a ranking predicted by an authentic estimate.

The 6 items for the same bounds were presented on a single sheet of A4 paper in the order of the target numbers shown in Figure 6.

In the Arithmécole1 research, the students had to choose the 6 target numbers, 35, 82, 54, 14, 68, 40, each among four, on a bounded line [0, 100]. The 6 items, identical to those with the same bounds in the ACE2 study, were presented on a single sheet of A4 paper, both in the pretest and the post-test. In the Arithmécole2 study, the placement test was almost identical, with the target numbers 35, 82, 97, 3, 68, 40, in the pretest, and 35, 82, 54, 14, 68, 40 in the post-test.

For the ELFE research, the 10 placement items were described in study 1.

Procedure. All students were tested in their classrooms, either collectively (ACE2 and Arithmécole) or in small groups (ELFE). In the ACE2 and Arithmécole researches, the target numbers were also indicated orally by the experimenter. Participants were asked to circle the mark (vertical bold line) of the number circled on the left (Fig. 1 and Fig. 5). In all the researches their choices were recorded.

Processing. We systematically colored the highest, second-highest, third-highest and lowest choice percentages green, yellow, orange and red, respectively. This visualization technique makes it easy to detect inter-versions in the green, yellow, orange and red left-right order, which would be expected with an authentic estimate based on a linear representation (adapted to bounds).

Results. Figures 6 to 11 show the visualizations obtained for the 6 datasets analyzed. The choices of erroneous distractor marks are labelled as near, intermediate and far, depending on the relative (absolute) distance of the distractor mark from the correct mark.

Discussion

For the 46 percentage lines added together in figures 6 to 11, less than half were in perfect agreement with the expected left to right order: green à yellow à orange à red. The correct choice was not the most frequent choice in 9 lines. For example, for the target number 337, the mark for the nearest number was chosen by almost 44% of the students, whereas the correct mark was chosen by only 40% (see Fig. 11). But this last observation is not necessarily impressive. In fact, the mark of the target number 337 is difficult to distinguish from that of its close neighbor (see Fig. 1). The 1^st mark is not appropriate because, since 1000 is almost equal to 3x337, we can see (visually) that two transfers to the right of its distance from 0 do not arrive at 1000; we can also see that the third mark is almost in the middle (= 500), so much too far to the right to indicate 337. The 2^nd mark was therefore the obvious choice. But at the start of 4^th grade, the majority of students chose the 1^st mark!

On the other hand, the two lines in which the frequency of choosing the correct mark was relegated to 3^rd position, for the target numbers 14 (Fig. 6) and 485 (Fig. 11), merit in-depth analysis. For the target number 14, many first graders, having seen ‘4’ and heard ‘ka’ in ‘quatorze’, seem simply to have chosen the mark with the number 4 (by counting marks 1, 2, 3, 4) or in fourth position (see Fig. 5)! For the number 485, what is striking in its symbolic form (in any case, cf. He et al., 2021) is that it is close to 500, but smaller. It is therefore possible that many students have chosen a mark to the left of the middle. And, to be totally sure that their choice is indeed to the left of the middle (i.e., on the side of 0), they have chosen the mark furthest to the left, which does not lead them to the erroneous choice closest to the correct choice (see Fig. 1)! Another choice pattern is curiously different from a choice based on a genuine estimate. It concerns the target number 68 in the ELFE study (Fig. 11). In fact, the far choice was preferred to the near choice. While we might have expected the erroneous choices to be evenly distributed between the two marks corresponding to the intermediate and close choices, which were almost equidistant from the correct choice, the participants made the first of these choices 23 times more often than the second! The near choice was chosen so infrequently that it was even outnumbered by the far choice! Perhaps 68 seemed large to many of the students and made them hesitate between the marks associated with 68 (target) and 86 (intermediate), as shown by the percentages of the respective choices of these two marks (46.8 and 43.8), ignoring the choice of mark associated with 51 (near).

Even if these interpretations of some inter-versions are speculative, the interpretation of the target number as indicating the position rather than the magnitude has been largely supported by several other inter-versions:

• In the 5 figures (from Fig. 6 to Fig. 10) from the ACE2 and Arithmécole researches, for target number 40, the far distractor (≈80) was chosen more often than the intermediate distractor (≈59), which supports the hypothesis that position 4 was chosen.

• By first-graders (see Fig. 6), who had to place 4 on a line bounded by 0 and 30 and chose the erroneous mark in position 4, the furthest away (≈21), rather than the two nearer marks (≈14 for the intermediate mark, and 7 for the nearest mark)!

• By students in the early 2nd year of school (see Fig. 9), who had to place 3 on a line bounded by 0 and 100 and chose the wrong mark in position 3, further away (≈31) from the correct choice, than a nearer wrong mark (≈14)!

When the students have no way of ‘mathematically’ solving the question posed, they can substitute another approach that is within their reach. This leads to obvious failure, as in the case of the students at the end of the 1^st grade who substituted counting ‘1, 2, 3, 4’ for the estimation task. This may be due to a lack of understanding or confusion between the magnitude and order indicated by the target number. Even some successes may be underpinned by a phenomenon of under-comprehension when the latter is defined as success through partial processing of information associated with a mathematical concept (Pluvinage, 1983). For example, the most successful item for second graders (the target number 97: see Fig. 9) may have been chosen by some students on the basis of an impression of size or proximity to 100, but without considering the existence of numbers between 97 and 100, or that the third mark was not very far from 100 either. The target number 68 could have been chosen at random between the two large numbers by a majority of fourth-graders (see Fig. 11), although these students certainly did not check that an incorrect mark was (very slightly) nearer the correct answer than the incorrect answer overwhelmingly chosen.

Admittedly, in a class, and even more so in a large sample of students, there will always be one or two students who misunderstand a question. But our studies show that the same mistakes are made on a massive scale:

At the end of the 1^st grade, 326 students (out of 2803) incorrectly chose 4 in 4^th position, a choice very distant from the correct choice; similarly, 465 students incorrectly chose 40 in 4^th position.
At the start of the second grade, more than 400 students (out of 2433) each time, incorrectly chose 3 in 3^rd position and 40 in 4^th position, rather than in a position, also incorrect, but closer to their respective correct positions.
Finally, and most importantly, at the start of 4^th grade, over 3300 students (out of 7713) incorrectly placed 485 on a mark that was not the (incorrect) mark nearest the correct mark.

As a result, the educational value of this activity of placing numbers on a number line is questionable for many students, particularly for young students, before the second year of school (see also Fuson, 2009). It is therefore surprising that children, some of whom are not yet 5 years old, are asked to place numbers on a line bounded by 1 and 15 (James-Brabham et al., 2024). Admittedly, the authors avoid 0, a ‘dangerous’ number (Seife, 2002) that can confuse young children, but do they know the number ‘fifteen’? Having tested the numerical skills of over a thousand individual kindergartners, let me relate, from memory, the following anecdote: a boy of 3 or 4, when I asked him to show me ‘three’ with his fingers, replied: “What's three”?

But this question of the relevance of the number placement activity for young children can be raised above all in relation to certain researches on Chinese children. Perhaps encouraged by numerical performance of the latte, which is often superior to that of US children, including in the area of estimation on the number line (Siegler & Mu, 2008), Xu et al. (2013) tested 3- and 4-year-old Chinese children (age range: 3.3-4.7). With a line bounded by 0 and 10, and the 8 target numbers (all one-digit number except 5), they obtained a percentage absolute error (PAE)[2] of .25 with symbolic numbers in this age group. A simple simulation then shows that if the children had provided random responses, based on a prior differentiation between small (1, 2, 3, 4) and large (6, 7, 8, 9) numbers, their performance would have been significantly better (PAE » 0.15). As completely random responses would lead to a PAE of 0.325, the result of Xu et al. is compatible with the hypothesis that half (approximately) of the children respond completely at random and that the other half respond in a restrictively random way. Their random behavior would be restricted by the small/large opposition, that is, they respond at random on the left for small numbers, and on the right for large numbers. This hypothesis implies that the responses are not derived from a logarithmic representation. In fact, Xu et al. observed that a linear representation fits their data as well as, if not better than a logarithmic representation (R² = .88 vs. .79).

More recently Wu et al. (2024), with 70 participants aged 5 ½ years, mostly from the upper middle class of Wuhan (China), attempted to show that dragging the mark of a number from left to right to its final position on a touch screen generally facilitates performance compared to simply taping to the presumed position. The only condition where the difference in accuracy proved significant was when placing the numbers 11, 12, 13, 14, 16, 17, 18, 19 on a bounded line [0, 20]. However, the demonstration is flawed by an exchange between accuracy and speed, since the participants in the dragging condition made significantly less incorrect placements than those in the taping condition (the distance to the correct answer was on average 1.86 vs. 2.49), but also took much longer to reach their placements (9.43 s vs. 5.89 s). This difference is noteworthy because it allows participants in the dragging group to check, as they drag the mark, whether or not the experimenter's facial expression expresses satisfaction (it is not known whether the experimenter was aware of the authors' hypotheses). In addition, from the point of view of statistical methodology, we can also note that (1) the children who did not understand a preliminary question on the game with iPad/Smartphone were eliminated from the study, (2) of the children retained, the participants in the dragging group already showed, prior to the formal test, marginally superior spatial abilities to those of the participants in the taping group, p = . 066 with a two-tailed t-test (according to our calculation), (3) the significance of the difference between the dragging and taping groups is also due to the poor performance of the latter. A simulation shows, for example, that if children taped randomly in the 3^rd quarter for numbers 11 to 14, and in the 4^th quarter for numbers 16 to 19, their performance would be much better than that of children in the taping group, with a mean error of 1.50 versus 2.49.

In the numerous researches using a placement task, some have tempered the usefulness of the number line (Ellis et al., 2021; Huber et al., 2014; LeFevre et al., 2013; Ni, 2000). For example, James-Brabham et al. (2024), in a board game with online numbers, played with children from disadvantaged socio-economic backgrounds aged 61 months on average, report significantly lower gains on the numerical line test than on other numerical tests. More specifically, other researchers have highlighted the influence of positional numeration on number placement (Dotan & Dehaene, 2020; Nuerk et al., 2001; Patalano et al., 2023; Williams et al., 2022). The automatic activation (Zanolie & Pecher, 2014) and innateness (Núñez, 2011; Pitt et al., 2023) of the number line have also been challenged. Finally, Nuraydin et al, (2022) found no transfer of a fraction set including the number line on fraction comprehension or arithmetic.

Some teachers, who have learned from their training that mathematics is the science of rigor, may also be troubled by this task, which asks their students to place numbers approximately and, in part, on visual impressions. Furthermore, teachers generally avoid asking questions that students cannot—or cannot yet—answer. For example, in the case of 337 (see Fig. 1 for the item and Fig. 11 for the results), for which we proposed an adult solution in the previous discussion, students at the beginning of their 4^th grade were unlikely to arrive at a perfectly assured answer. The reason for this avoidance by teachers is obvious. Such questions lead, if not to random answers, at least to the kind of under-comprehension described at the start of this discussion. Finally, some pedagogues, skeptical of this new placement task, may also argue that, until 1990, students were never trained in such representations and that, without their support, they calculated rather better (cf. Chabanon & Pastor, 2019; Ninnin & Pastor, 2020).

But what is perhaps most disturbing, in France at least, is to observe that the Éducation Nationale (French national education authority) uses in its assessments bounded lines systematically graded (cf. Andreu et al., 2023). We illustrate this, in the Appendix, with the placement items in the 2023 second grade national assessment. In the presentation of this assessment, it is stated that the target skill is ‘Associating a whole number with a position’ and, in the resource sheet for student support, reference is made to estimation games on the Arithm'école ACE site (cf. ev22-ce1-maths-nombres-calculs-representer-ligne-numerique_0). Unfortunately, the graduations do not encourage placement by estimation. Worse still, they make it, if not impossible, at least more risky than counting. So we can see that placement by estimation on the number line can be circumvented and would even be counter-productive in second grade. One might think that, in later grades, we would end up with placements requiring real estimation, that is, where the graduation of the number line no longer enables a choice to be made by simple or complex counting. But this is not the case. At the beginning of fourth grade, the September 2023 assessment continues to evaluate the skill of ‘Placing a number on a graduated line’ (Bourgeois et al., 2024).

This inconsistency, resulting from a misunderstanding between cognitive psychologists and mathematics educators, may have led Vignali (2023) to try to find logarithmic versus linear representations of numbers in students with learning difficulties. However, as figure 12 shows, the logarithmic nature of the representations obtained by Vignali is doubtful. In fact, in her only illustration of a logarithmic representation, the numbers are compressed between 3 and 7 rather than between 7 and 9. Vignali's approach seems to ignore the fact that a model is not reality, and that a model may not be followed by anyone, mainly because of intra-individual variability (cf. Roth et al., 2024). In addition, a pragmatic explanation, probably closer to reality, is that the students (working in groups) who had spaced the first numbers too far apart (in particular 1 and 0), narrowed the following numbers from 3 to 7 when they realized this, and then used the whole space from 7 to 10 to space the last numbers evenly.

When the aim is to assess number knowledge rather than number representation, particularly cerebral representation (e.g., Berteletti et al., 2015; Cattaneo et al., 2009; Lavidor et al., 2004; Liu et al., 2019), the placement task appears to be very rich. To be successful, it requires estimation, number-measure comparisons, reasoning (Ruiz et al., 2024) and the notion of proportionality (Barbieri et al., 2020; Georges & Schiltz, 2021; Gunderson & Hildebrand, 2021; Smaczny et al., 2024). As a result, the number line placement task appears to be a useful measure of children's ability to assemble a collection of relevant knowledge to perform a new and complex numerically relevant task (LeFevre et al., 2013). The results of Schneider et al. (2018) also show that the numerical estimation task is a robust tool for diagnosing and predicting mathematical competence. This leads the latter authors to recommend its use in experimental and developmental learning studies.

Nevertheless, the massive student errors reported in the present study suggest that students need to be provided with aids to avoid under-comprehension. For a boundary [0, 100], for example, the reference numbers (e.g., 25, 50, 75; in connection with lengths: 1/4, 1/2, 3/4]) are certainly a major help. Indeed, (1) the developmental study by Rouder and Geary (2014) shows that improved performance on the placement task is largely the result of segmenting the line in the middle, (2) the fMRI study by Vogel et al. (2013) confirms the significant influence of landmarks (0, 50 and 100) on brain activation in the right intraparietal sulcus[3], (3) the research by Xing et al. (2021) shows that placements relative to a median landmark, inferred in addition to the two ends of the line segment, contribute strongly to the link between estimation on the number line and mathematical skills at the age of 6 to 8 years, and (4) in kindergarten, Xu and LeFevre (2016) have shown that sequential learning (e.g., what comes before and after the number 5) greatly improves performance on a version of the number line task where a reference point is presented in the middle of [0, 10] (i.e., at 5). We therefore suggest that teaching could teach students such benchmark numbers to enable them to compare the relative position of the target number when making placements on a number line (Namkung & Fuchs, 2016). With this in mind, a communication game, as implemented by Saxe et al. (2010) in 5^th grade, could be effective. It should be noted, however, that for fractions these cues are of little use (Siegler & Thompson, 2014). But the usefulness of representing fractions on a number line is not obvious.

[2] The PAE formula of Xu et al. (2013, p. 355) is considered to be a simple reproduction error.

[3] Von Aster and Shalev (2007) mention that the intraparietal sulcus (IPS) is “a cortical area that has been proved to be the domicile of the mental number line” (p. 869).

We were able to verify that choosing the place of a number on a bounded line in a multiple choice task has features that distinguish it from other numerical tasks such as mental arithmetic or reasoning. We then showed that a placement estimation task is influenced by the position of the number to be placed relative to the bounds of the line segment. The over-estimation induced by a logarithmic representation is incompatible with our observation of an under-estimation of numbers in the right half of the number line. Finally, and most importantly, we analyzed precisely the errors made by the students, using a choice task.

These errors are hard to explain by a continuous logarithmic or linear representation, or by more sophisticated mathematical models (Bayesian inference: Cicchini et al., 2022; cyclic power: Gross et al., 2018; structural analogy: Sullivan & Barner, 2014; estimation of proportions: Slusser et al., 2013; Zax et al., 2019). Systematic examination of students' choices has, on the other hand, proved more illuminating than theoretical mathematical modeling in explaining their errors. In this way, we were able to observe several types of difficulty or deviation.

The difficulties affect young children in particular. They may choose a position (e.g., the fourth) which corresponds to the order indicated directly (e.g. ‘four’) or indirectly (by phonetic analogy: ‘forty’), which reflects a non-distinction between the order and the magnitude indicated by the target number. Even when they succeed, they may do so without using all the information available to them (under-comprehension in the sense of Pluvinage, 1983). Children's responses often do not differ from a random choice, if not a total random choice, at least a random choice based on a prior dichotomization (small/large) of the range of bounds.

The deviations may concern re-educators who try to find logarithmic representations in children's productions. Furthermore, in French national assessments, the evaluation of estimation ability is totally biased, since counting is a considerably more efficient procedure than estimation (see Appendix).

Excessive elaboration of mathematical models cannot lead to a correct explanation, since all models are wrong (Box, 1976). On the contrary, the hypothesis of a simple and subjective dichotomization—small versus large—of numbers by young students, already put forward by Case and Okamoto (cited in Siegler & Booth, 2004, p. 430) and taking into account the position of the distractors and the proximity of the bounds, may suffice to explain most of our observations.

ACE Arithmétique et Compréhension à l’École

ELFE Étude Longitudinale Française depuis l’Enfance

PCA Principal Component Analysis

PAE Percentage Absolute Error

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The authors declare no competing interests.

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Number Line Estimation: another view in the light of the ACE, Arithmécole, and ELFE data.

Number Line Estimation: another view in the light of the ACE, Arithmécole, and ELFE data.

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Abstract

Figures

Introduction

Study 1: Is the placement (estimation) task singular?

Method

Study 2: Are algebraic placement errors influenced by the proximity of the bounds?

Method

Results

Discussion

Study 3: Are incorrect placement choices perfectly correlated with their distance from the correct choice?

Method

Discussion

General discussion

Conclusions

Abbreviations

References

Additional Declarations

Supplementary Files

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