Why is teaching estimation strategies important?Abstract
Estimating is part of numerical cognition and is correlated with mathematical performance, however the relation of mathematical performance with the strategies used has been less researched. This study investigated the relationships between strategies in the number line estimation task, from zero to one hundred, quantitative reasoning, and arithmetic performance. A total of 57 fourth-grade students from a Brazilian school participated. The strategies in the NLE were classified as: 1) no visible procedure, 2) counting, and 3) proportional judgment. Results indicated that strategies 2 and 3 led to greater accuracy than strategy 1. Although estimation accuracy was correlated with quantitative reasoning and mathematical performance, the strategies used were not associated with performance. However, the strategy used by the students varies with the number position in the number line, suggesting different cognitive demands according to the different number magnitude. The analysis of three individual cases revealed distinct patterns of strategy use, suggesting pathways to foster numerical estimation skills.
Keywords:
Numerical cognition; Number estimation; Number line; Mathematical competence; Quantitative reasoning
Resumo
Realizar estimativas faz parte da cognição numérica e se correlaciona com o desempenho matemático, mas a relação de tal desempenho com as estratégias de estimativa utilizadas é menos pesquisada e pouco conhecida. O estudo atual investigou as relações entre estratégias utilizadas na resolução da tarefa de estimativa na reta numérica de 0 a cem, com o desempenho em raciocínio quantitativo, e desempenho aritmético. Participaram 57 estudantes do 4º ano de uma escola brasileira. As estratégias utilizadas pelos estudantes na tarefa de estimativa na reta numérica foram classificadas: 1) nenhum procedimento visível, 2) contagem e 3) julgamento proporcional. Os resultados indicaram que as estratégias 2 e 3 levaram a maior precisão do que a 1. Embora a precisão tenha se correlacionado com raciocínio quantitativo e aritmética, as estratégias utilizadas não foram associadas ao desempenho. No entanto, a estratégia adotada pelos participantes variou de acordo com a posição do número a ser estimado na reta numérica, o que pode refletir diferentes demandas cognitivas conforme a magnitude dos números apresentados. A análise de três casos individuais evidenciou padrões distintos no uso das estratégias, sugerindo caminhos para estimular a estimativa numérica.
Palavras-chave:
Cognição numérica; Estimativa numérica; Reta numérica; Desempenho matemático; Raciocínio quantitativo
1 Introduction
The ability to make numerical estimates is an important ability for everyday and school activities and has been related to basic and complex mathematical abilities. Simple questions, such as how long is left in the class or how many minutes it will take to travel a certain distance, require the ability to make estimates.
The most common task used to evaluate numerical estimates is the Number-Line Estimation Task (NLE), organized by Siegler and Booth (2004). There is ample evidence that the better the accuracy in this task, the better the students’ performance in numerical comparisons and arithmetic tasks, as well as in other mathematical areas (Gilmore et al., 2018; Schneider et al., 2018). The meta-analysis conducted by Schneider et al. (2018) indicated that NLE is a highly efficient diagnostic tool to evaluate possible difficulties in numerical understanding, and the students’ performance in this task is correlated to counting, arithmetic, and with mathematical performance in general, more specifically, with fraction in older children, that is, in different areas of mathematical knowledge (Schneider et al., 2018). The authors also highlight that the task evaluates the understanding of numerical magnitude on a continuous level, an essential ability for every mathematical knowledge.
Understanding of numerical magnitude is evident in interventions such as board games that involve moving forward and backward along a number line (Siegler; Ramani, 2009), which resulted in improvements in numerical estimation and mathematical performance, as well as a better use of prior mathematical learning. Link et al. (2015) developed a training program on the number line for 1st-grade German children, which, after training, showed significant progress in addition with one number, as well as in numerical estimation. Such works indicate the possibility of stimulating numerical estimation and foreseeing its effects in mathematical learning beyond estimation precision itself. However, in these studies, the role of the strategies used to estimate numbers on the number line was not considered. As seen by Fidelis et al. (2021), the analysis of strategies used by students has been frequently studied in other areas of mathematical knowledge, including in Brazil.
The estimation strategies reflect the cognitive resources that children, teenagers, and adults use to estimate a certain amount and represent it. For instance, the counting strategy, one of the most commonly used strategies for representing amounts in NLE, is often expressed through continuous and spaced traces on the number line, sometimes followed by oral counting. The strategy of proportional judgement implies observing the size of the number line, separating it initially in a proportional manner, into two parts, and, after, finding the space in which a given number should be indicated based on the evaluation of the two other parts.
The estimation strategies reveal a certain understanding of number properties, and more effective strategies can be stimulated by adding traces on the number line that delineate the center and the quarters of the number line (Peeters; Verschaffel; Luwel, 2014). We also highlight that the number line itself has been used as a pedagogical support for children with low performance in mathematics (Sutherland et al., 2023).
Duro and Dorneles (2018) analyzed the strategies used by thirty students from 3rd to 6th Grades and identified several strategies: counting, the most frequent in the group (51.8%), followed by strategies based on points delineated by students (12.8%), with some proportional evaluation, up to combined strategies (10.5%). The results indicated that students did not make estimates easily and many used laborious and imprecise strategies to do so. Siegler and Booth (2004) had already shown that precision in estimating numbers on the number line increases over time and with the development of mathematical knowledge. There is evidence that children with difficulties in mathematics have more problems developing economic and precise strategies (Ashcraft; Moore, 2012), indicating that less efficient strategies lead to lower performance in estimation and mathematics. There is also recent evidence that the choice of more precise estimation strategies is related to better performance in mathematics (Xing et al., 2021). The NLE has been widely used to evaluate the ability to implement numerical estimations, and the study of strategies used in these tasks has grown recently (Jung et al., 2020).
This relationship between the precision in estimation and mathematical performance has been explored in the context of representation models of numerical magnitude, mainly the logarithmic and linear models (Siegler; Opfer, 2003; Friso-Van Den Bos et al., 2015). In earlier phases of mathematical knowledge, children tend to represent numbers in a logarithmic form, overestimating smaller numbers and underestimating larger ones; for example, the distance between 15 and 20 will be greater than the distance between 75 and 80. With the advancement of schooling and the numerical experience, this representation becomes progressively more linear, with the numbers being distributed more proportionately along the line and keeping a similar distance between them. This transition from a logarithmic to a linear model reflects not only a change in estimation accuracy but also in the strategies used to construct an NLE. Strategies based in counting, such as marking at every 10 mark until reaching the target number, or reference points explicit marked, such as tracing the point 50 to, from it, try to locate the number 75, can indicate a representation that is still grounded in more concrete processes, while the use of proportional evaluation or procedures with no visible marks can reflect a more automatized processing. Hence, investigating the strategies used by children when making estimates in the number line can provide valuable clues about the developmental stage of its magnitude representation and their relationship with broader mathematical performance.
Another area of mathematical development that has been much studied and correlated with mathematical performance is quantitative reasoning. Nunes and Bryant (2021, p. 151), define quantitative reasoning as “the ability to represent and to reason about quantities and relations between quantities mathematically (...)". and highlight that one does not need to use numbers to think about quantities. These relationships are divided into two large groups: part-whole relations, also called additive relations, and fixed ratio relations, also called multiplicative relations. The former approach the relationships of the part and the whole between quantities and is subjacent to the activities of adding and subtracting, the latter encompasses situations involving a ratio between two amounts, as well as situations in which a third amount is formed by the two previous ones, as well as situations of multiple proportions (Nunes; Bryant, 2021).
It is important to distinguish quantitative reasoning from arithmetic, considering that this last one is related to numbers, their classification, and behavior in the four operations, while quantitative reasoning refers to the capacity to reflect on the relations between amounts. A brief example of additive relations: if Maria had six balloons and released two, how many does she have now? Before making an arithmetic operation, children need to reflect on the relation between numbers four and six to understand and solve the situation in this reflection. Is it this understanding of additive and multiplicative relations that differentiates children who, when faced by the problem above, ask: Is it more or less? Or: it is eight! Many studies (Magina; Santos; Merlini, 2014; Ching; Nunes, 2017; Nunes; Bryant, 2021; Nogues; França; Dorneles, 2023) have pointed out the need to develop quantitative reasoning in the classroom to guarantee a robust understanding of mathematical knowledge, avoiding a repetition of these contents, exclusively based on memory. Both quantitative reasoning and arithmetic are needed for mathematical learning. Nonetheless, as Nunes and Bryant (2021) stress, arithmetic has its place guaranteed in education, whereas quantitative reasoning is not.
In this article, we have established relationships among the strategies used in the number line, quantitative reasoning, and arithmetic performance. The 4th Grade students (N=57) from a public Brazilian school were evaluated in three tasks: Number-Line Estimation Task (NLE), quantitative reasoning (QR), and school performance task (SPT). Later, we analyzed the strategies used by three research subjects to examine their relationship with students’ performance and to identify intervention possibilities for teaching these strategies.
Hence, this article aims to explore the relationship between the strategies students use at NLE and their performance on QR and SPT.
2 Method
2.1 Sample
The study is part of a research project registered at Plataforma Brasil and approved by the Research Ethics Committee of the researchers’ university, under the number 82570618.9.0000.5347. 55. The students were evaluated in three tasks described later, with the informed consent signed by their legal guardians. Students’ intellectual level was evaluated using Raven's Progressive Matrices (Angelini; Custodio; Duarte, 1999), and those with an average or higher score were included in the sample. Therefore, the research had 57 participants ranging from 9 to 11 years old (m=9.8 years old, SD=0.5), most of them were girls (63.2%).
2.2 Instruments used
All tasks were collectively applied in two sections: in the first, the NLE and QR tasks were applied and, in the second session, the SPT was applied, which evaluated arithmetic performance.
The Number Line Estimation Task (NLE) asks participants to indicate the position of a number along a line extending from 0 at the start to 100 at the end. Estimates for 22 numbers were asked (2, 3, 5, 8, 12, 17, 21, 26, 34, 39, 42, 46, 54, 58, 61, 67, 73, 78, 82, 89, 92, 97). As an evaluation criterion, we used the percentage of absolute error (Siegler; Booth, 2004), and the strategies used to estimate it were also recorded and considered. The strategies were classified into three types: Strategy 1, with no visible procedure; Strategy 2, counting in the numerical series; and Strategy 3, proportional judgment, with the indication of numbers in the middle and the quarter of the number line. The strategies were observed in the moment as the task was applied and confirmed a posteriori in the analysis of the written material. The task of Quantitative Reasoning (QR) consists of 15 problems of additive and multiplicative reasoning adapted from Nunes (2009) and Nunes et al. (2015). The School Performance Test (SPT) is a standardized test developed for the Brazilian context (Stein, 1994) that is frequently used in research to evaluate the mathematical performance of elementary and middle school children. The arithmetic subtest, the only SPT applied, encompasses 35 written arithmetic problems. Both QR and SPT, the total score of correct answers was considered.
3 Results
Next, we present our main results, starting with the descriptive analysis of strategy use. Strategy 1, with no visible procedure, was the most used, 84.36% of times, followed by Strategy 2, counting, with 14.6%, and, last, Strategy 3, proportional judgement, used 1.04% of times. Table 1 below presents a descriptive analysis of performance on each task by the strategy used in the number line estimation
First, when comparing the accuracy at NLE according to the type of strategy, the Kruskal-Wallis test indicated a significant difference in the accuracy of numerical estimate between the strategies used (H(2)=10.616, p<0.01). The post-hoc comparisons by Bonferroni reveal that Strategy 2 resulted in a significantly higher precision than Strategy 1 (p<0.05), and Strategy 3 had a higher precision than Strategy 1 (p<0.05). We could not see a significant difference between Strategies 2 and 3 (p=0.118). The same analysis was conducted for the performance on the other evaluated tasks. The Kruskal-Wallis test did not indicate significant differences in performance in quantitative reasoning among groups that used different number line estimation strategies (H(2)=4.689, p=0.096). Similarly, there were no significant differences in strategy type and arithmetic performance (H(2)=2.175, p=0.337). These results suggest that, though certain strategies of number line estimation are associated with a greater precision in the estimation task, they do not seem to be directly related to the performance in quantitative or arithmetic reasoning. Next, we conducted correlational analysis, which indicates a positive and meaningful association between the accuracy in number line estimation and the performance in quantitative reasoning, as well as between quantitative reasoning and arithmetic performance. Nonetheless, we perceived no meaningful correlation between accuracy in number line estimation and arithmetic performance, as indicated by the data in Table 2.
These data suggest that individuals with lower performance in quantitative reasoning tend to show greater accuracy in number line estimation and better performance in arithmetic tasks. This result corroborates previous studies that also indicate a consistent relationship between abilities of quantitative reasoning, accuracy in tasks of numerical estimation, and formal mathematical performance, showing that a more consolidated understanding of numerical magnitudes favors not only more precise estimations but also a more efficient use of strategies in arithmetic contexts (Jung et al., 2020; Nogues; Dorneles, 2022).
Later, to investigate whether the power of associations varies depending on the strategy used, we conducted separate correlation analyses for each strategy in the NLE task. Thus, providing a more detailed view of how different ways of estimating influence mathematical performance. The results indicate that, considering the participants who used Strategy 1, there were significant, though weak, correlations of accuracy in number line estimation with quantitative reasoning (r=0.123, p<0.001) and with arithmetic performance (r=0.061, p<0.05). Among those who used Strategy 2, we could perceive a weak significant correlation between accuracy in number line estimation and quantitative reasoning (r=0.166, p<0.05). Among participants who used Strategy 3, no significant correlations were observed among the variables analyzed. These findings enable us to interpret that, despite the small magnitude, there is a tendency that the power of associations among number line estimation, quantitative reasoning, and arithmetic performance varies according to the strategy used. Furthermore, the fact that number line estimation is associated with quantitative reasoning but not with arithmetic performance reinforces the idea that these abilities involve different cognitive demands. Quantitative reasoning is more related to the ability to understand and relate magnitudes, which can facilitate a better numerical estimation. Arithmetic performance can depend more heavily on declarative knowledge and the mastery of specific procedures, which do not always affect accuracy in estimation tasks.
To further explore these results and examine possible differences in cognitive profiles, students were classified according to their performance on the arithmetic subtest: low (up to 14 correct answers), average (15-17), or high (over 18). Generally, the results indicated that there is a significant difference in number line estimation accuracy among students, as measured by their arithmetic performance (H(2)=0.176, p=0.916). However, when comparing precision according to the type of strategy used separately in each group, we could see that, in the group with lower arithmetic performance, there is a significant difference only between the use of strategies 1 and 2 (H(2)=16.833, p<0.001), that is, students who used Strategy 2 (m=0.941, DP=0.062) had a better accuracy than those who used Strategy 1 (m=0.901, DP=0.100). In the other groups, average performance (H(1)=2.909, p=0.088) and high performance (H(2)=3.376, p=0.185), there were no meaningful differences in the type of strategy used for accuracy in number line estimation. These results suggest that, specifically for students with low arithmetic performance, the use of a more structured strategy, such as counting on a line, can act as a support for the estimation task, helping them to make more accurate estimates. This can happen because counting provides an external and concrete reference that helps organize the number line. In contrast, the lack of a visible procedure can lead to less structured processing of the number line, resulting in greater variability and less precise estimates. On the other hand, in the groups of average and higher arithmetic performance, the lack of differences among strategies suggests that these students can have a more stable mental representation of the number line, enabling them to make relatively precise estimations, regardless of the type of strategy used. This fact can be related to a greater conceptual mastery of numerical relations and a greater familiarity with the structure of the number line, making them less dependent on explicit strategies to reach good estimations.
Next, to understand in more detail the use of different types of strategy, the number line was separated into four regions according to the target number to be estimated: a) numbers lower than 25; b) numbers between 25 and 50; c) numbers between 50 and 75; d) numbers between 75 and 100. The percentage data of each strategy used and its respective average precision by region in the line can be seen in Table 3 below:
Based on these results, we can affirm that Strategy 1 is the most widely used across all regions, with use percentages ranging from 76.52% to 88.73% and consistent precision ranging from 0.900 to 0.918. Strategy 2, though less used, had a competitive precision, with values between 0.894 and 0.943; the region between 75 and 100 stands out, reaching a higher precision (0.943). Strategy 3, despite being the least used (with percentages close to 0% in most regions), showed greater precision in the regions where it was applied, reaching 0.956 in the region between 50 and 75. These results suggest that, though Strategy 1 has been more used, Strategies 2 and 3 can have advantages in terms of precision in specific regions of the number line.
To evaluate the association between the region in the number line and the type of strategy used, we conducted a chi-squared test. The results indicated a significant association between those variables (χ2(6)=60,12,p<0,001), though with a small effect size (Cramér's V = 0,155). This result suggests that participants' strategies vary according to their estimated position on the number line, reflecting different cognitive demands depending on the magnitude of the numbers presented. The analysis of frequency revealed that Strategy 1, with no visible procedure, was broadly used across all regions, mainly for estimates of greater magnitudes. Strategy 2, counting, was more frequent for smaller values, while Strategy 3, proportional judgment, was used almost exclusively in the intermediate range of the line (50-75). These findings help explain the previously observed patterns: though Strategy 3 was the most precise, it was not correlated with mathematics performance, possibly because it was applied only within a restricted interval of the number line. On the other hand, Strategies 1 and 2, which had lower accuracy, were more broadly distributed along the line and showed correlation with quantitative reasoning and arithmetic performance. Hence, these results suggest that the type of strategy used can depend on the magnitude of the numbers presented, as well as participants’ individual abilities.
4 Qualitative analyses of specific cases
To complement the quantitative analysis, we conducted a qualitative analysis to better understand the strategies participants used and to identify possible cognitive profiles when performing number line estimation. This approach allowed us to observe individual patterns that would not have been evident from inferential statistics alone. The data were analysed based on patterns in the observed answers. We should mention that most students completed the task using Strategy 1. However, some students stood out by using proportional judgment, counting, or by altering strategies during task execution. From that, we selected three students based on their answer consistency and strategy use. Next, we will describe each of them in detail.
4.1 Student A
For the first item asked at the task (target number: 58), student A used Strategy 3, proportional judgment, as he did the corresponding mark in the middle of the line (50) and later counted one by one until reaching the target number (58). After the second item of the task, he used Strategy 2, counting, in the five following items, the target numbers were, in order: 92, 5, 8, 17, 26. This behaviour suggests that, at first, the counting strategy was considered more reliable or accessible, possibly because it allows greater control over the marking of the target number. However, after the seventh item, he started to mostly use Strategy 1, with no visible procedure, suggesting an effect of familiarity with the task, as the student felt more confident to make direct estimations without the explicit support of auxiliary procedures. Even so, we can see an exception in target number 54 (one of the last task items), in which the student used Strategy 3 again. The proximity of this number to the middle of the line might have led to a return to the proportional judgement, showing the influence of internal reference points in decision-making.
Regarding accuracy, the student showed more consistent performance when using Strategies 2 and 3 (accuracy ranging from 0.87 to 0.96), which reinforces the hypothesis that these strategies offer greater control and adjustment in number line estimation. Contrariwise, the change for Strategy 1 resulted in greater oscillation in its accuracy (precision varying between 0.74 and 1), using this strategy in 15 out of the 22 target numbers to be estimated. However, it achieved accuracy with values over 0.95 (which is considered a great accuracy) in only 6 of them. This pattern suggests that, though Strategy 1 might reflect progress in the student’s confidence in his estimates, it cannot guarantee the same level of accuracy as the more structured Strategies 2 and 3.
When analysing precision by region on the number line, we can see that the highest levels were found for numbers between 50 and 75 (average accuracy=0.95), followed by numbers smaller than 25 (average accuracy=0.89). In contrast, the regions between 25 and 50 and between 75 and 100 had slightly lower values (average accuracy=0.86). Finally, when considering the student’s general performance, we can see a high level of mathematics performance (QR=9; SPT=20) and a reasonable ability in number line estimation (NLE-average=0.89). These results suggest that, despite strategic variations and oscillations in accuracy during the task, the student has a cognitive repertoire for numerical estimation. The gradual transition between strategies and the tendency to use familiar numerical references reinforce the importance of internalizing the structure of the number line in the development of the estimation capacity.
4.2 Student B
The second case analysed, regarding student B, shows a different pattern of strategy use compared to student A. This student used mostly Strategy 1, with no visible procedure, and Strategy 2 (counting) only twice for target numbers 5 and 8. The predominant use of Strategy 1 suggests that the student trusts his mental representation of the number line and can make direct estimates without explicit anchoring or counting support. Moreover, his average accuracy was 0.94, indicating that, despite the lack of a visible procedure, his answers were consistently close to the correct values. This performance reinforces the hypothesis that the student had a good internalization of the numerical structure and a strong sense of numerical magnitude. On top of that, his performance in quantitative reasoning (QR=10) was slightly above average, while his performance in arithmetic was higher (SPT=21), indicating a mathematical profile of good performance.
When analysing the precision by region on the number line, we can see that the higher values were found for numbers lower than 25 (average accuracy=0.95), followed by the region between 25 and 50 (average accuracy=0.94). In contrast, the regions between 50 and 75 and between 75 and 100 had slightly lower values, with average precisions of 0.93. These results suggest that the student’s accuracy was slightly better for smaller numbers, which can indicate a more refined mental representation for this numerical range or greater familiarity with these values.
The low variation in the strategies used might indicate a more automatized answer pattern, possibly resulting from a consolidated knowledge of the representation on the number line. However, the punctual choice of Strategy 2 for target numbers 5 and 8 can indicate that, for very small numbers, the students might have felt the need to have more control, using counting to ensure precision. This behaviour may indicate that, despite a high general precision, there is a strategic refinement depending on the position of the number in the line. The predominance of Strategy 1 can suggest an advanced internationalization of the numerical system; however, it might also reflect a reduced use of alternative strategies that could be useful in different contexts. This result matches the study by Xing et al. (2021), which shows a relation between more advanced estimation strategies and greater mathematical ability.
4.3 Student C
Student C results show a different pattern from previous ones in the use of strategies and precision in estimates, suggesting particularities in the mental representation. This student alternated the use of Strategies 1 (with no visible procedure) and 2 (counting) most of the time, using Strategy 3 (proportional judgment) for target numbers 4, 67, and 89. This alternance between Strategies 1 and 2 might indicate a gradual transition among strategies based on counting and more automatized strategies, with no consistent mastery of any of them. The limited use of Strategy 3 reinforces the hypothesis that he/she has not yet completely internalized the number line as a line of proportional reference, depending more on less efficient strategies to locate numbers.
In general, this student was precise in reasonable-to-low estimates (average = 0.86). No strategy used guaranteed a consistent accuracy: in the case of Strategy 2 (counting) the student did not have a spacing pattern between markings that was adequate to the size of the number line, which hindered its accuracy. Similarly, when he/she used Strategy 3, he/she marked reference points on the number line so that he/she could later count based on them. Nevertheless, the placement of these marks was irregular and inaccurate, also affecting the precision of estimates. This data suggests that the student has not yet consolidated the efficient use of Strategy 3 and may be in a transition phase in the development of the proportionality notion in the number line.
The detailed analysis of the regions on the number line shows that the best accuracy was in the region between 50 and 75 (average = 0.9), followed by the region of numbers smaller than 25 (average = 0.87), the region between 75 and 100 (average = 0.85), and, last, the region between 25 and 50 (average= 0.8). This standard can be related to how he perceives and mentally organizes the numeric sequence. In the regions of numbers below 50, the strategy that prevailed was counting, leading to overestimating numbers; that is, the marks done corresponded to numbers higher than the target ones. This overestimation of smaller numbers is common in children who still represent the numbers in a more logarithmic than linear way, mainly when the knowledge of the number system is not yet fully consolidated (Friso-Van Den Bos et al., 2015). In this logarithmic representation, it is as if the child was using a magnifying glass over the small numbers, enlarging the space among them, while the higher numbers were squeezed in the end of the line. For instance, when trying to mark the number 10 in a line of 0 to 100, the child could position it close to the middle of the line, closer to 25 than to 0, while numbers such as 75 or 100 were placed very close to each other, with almost no perceptible difference. This pattern can indicate that the student had not yet consolidated the proportional relation between the numbers and their location on the number line. As previously stated, children who still have not developed a linear notion of numbers frequently distribute the initial values in line in a more spaced way and compress the higher values. This difficulty in maintaining a proportional scale throughout the line can lead to a tendency to overestimate smaller values, corroborating the hypothesis that this student might be in the initial phase of spatial representation of numerical magnitudes. On the other hand, in the regions of numbers over 50, we found the opposite phenomenon: a tendency for overestimation, with markings corresponding to values smaller than the target number. This change can be explained by the predominance of the use of Strategy 1 in these regions, suggesting that, as the numbers got higher, the student started to trust more in the intuitive evaluation, though inaccurate. This transition can indicate an attempt to use the number line more automatically, with insufficient refinement to guarantee the precision of estimates.
Furthermore, this student presented low mathematical performance, in the QR (8 correct answers) and SPT (14 correct answers), which illustrates the relationship between abilities of numerical estimation and formal mathematical knowledge, described in the literature review. Studies indicate that children with mathematical difficulties tend to present less accuracy in the number line (Ashcraft; Moore, 2012; Xing et al., 2021), with the use of less precise strategies, which can be related to less exposure to activities involving magnitude and proportion and less development of quantitative reasoning, as it was clear in this student case. Moreover, the use of less sophisticated strategies, such as counting, can reflect lower fluency in numerical processing, resulting in less precise estimations. The case of student C enables us to think about the relationship between mathematical performance and the ability to make estimates, as, despite predominantly using the strategy with no visible procedures, this student had a low overall average in his/her accuracy. This case is opposed to student B, in which the same strategy prevailed but with greater accuracy in their estimates, as well as higher performance on the other mathematical tasks.
The comparative analysis of the cases of students B and C shows that using the strategy with no visible procedures does not guarantee good precision in numerical estimation. In fact, this strategy may indicate two opposing scenarios: on the one hand, it can reflect an efficient internalization of the numerical system and a more precise representation of the number line; on the other hand, it can indicate a superficial or incomplete understanding of the system, in which the student makes estimates in a intuitive and random manner, not mobilizing more elaborate cognitive processes, which helps explain its high frequency in the same sample. In the case of student C, though predominantly using this strategy, we observed limited precision and difficulties in the estimation and arithmetic tasks, suggesting that the strategy with no visible procedure did not correspond to a consolidated understanding of the numerical system. Conversely, student B demonstrated an opposite standard: he/she used mostly the strategy with no visible procedure, but a higher precision in estimates, aligned with a higher performance in quantitative reasoning and arithmetic. These results reinforce the hypothesis that the performance in estimation tasks does not depend only on a strategic choice but on the quality of the subjacent number knowledge that supports a more precise and efficient representation of numbers.
Table 4, below, summarizes the main characteristics of the three cases analysed, highlighting the predominant strategies used at NLE, the precision pattern in the region of line, and the performance in quantitative reasoning (QR) and arithmetic (SPT).
We can observe that, despite the predominance of Strategy 1, the results in cases B and C were contrasting. Student B had high general accuracy and higher mathematical performance suggesting that, in this case, the lack of visible procedures corresponds to an efficient internalization of numerical estimates. In contrast, student C, though mostly using the same strategy, showed low precision and inferior performance, indicating that the strategy of no visible procedure, on its own, does not guarantee efficiency, depending on the quality of the number's interior processing. Student A, in turn, demonstrated a learning path during the task, starting with more visible strategies and later migrating to Strategy 1, even amid oscillations in precision, highlighting the importance of strategic flexibility in the development of estimation competence.
These qualitative findings reinforce the evidence from the study's qualitative analysis, demonstrating that accuracy in number line estimations depends not only on the choice of strategy but also on the quality and efficiency with which it is applied. The contrasts indicated in each case analysed point out the need to investigate in more detail the relations between the type of strategy used and the numerical knowledge involved in its application, indicating that the mastery of more advanced strategies does not occur homogeneously between children and can reflect different trajectories of learning and numerical development.
5 Discussion
This study aimed to investigate the relationship among the strategies used in the Number Line Estimation Task (NLE), the precision of estimate, quantitative reasoning, and the arithmetic performance of Elementary School students. The quantitative analysis indicated significant differences in the accuracy of numerical estimates according to the strategy used. Strategy 3 (proportional judgment) and Strategy 2 (counting) led to greater accuracy than Strategy 1 (with no visible procedure). However, only strategies 1 and 2 showed a significant correlation with arithmetic performance, while strategy 3, despite being more accurate, was not directly associated with measures of quantitative reasoning and arithmetic.
When we deepened the analysis by categorizing students by arithmetic performance, we found that only in the lower-performing group were there significant differences in estimation accuracy across strategies. In this group, Strategy 2 led to more precise estimations than Strategy 1, suggesting that the use of more structured strategies could benefit students less familiar with the number line. In the average and high-performance groups, no significant differences were observed in the precision of estimates among strategies, which can indicate that these students have more consolidated numeric knowledge, regardless of the procedure used.
The qualitative analysis of the three individual cases complemented these findings, showing different profiles of strategy use and their impacts on estimation accuracy, as well as their relationships with arithmetic performance. Student A demonstrated a learning effect during the task, starting with more explicit strategies, as well as counting and proportional evaluation, and, gradually, using the strategy with no visible procedure. Despite this transition, precision varied, being greater for estimates made with more structured strategies. This student demonstrated a high mathematical performance, suggesting that his strategic adaptation can be related to a more consolidated and flexible numerical knowledge. In turn, student B predominantly used Strategy 1, no visible procedure, using only two estimates through counting. His/her high average precision suggests that he/she has a more refined mental model of the number line, enabling him/her to locate numbers more accurately. His/her mathematical performance, higher than average, reinforces this interpretation, indicating that greater familiarity with numeric concepts can be associated with a more efficient, automated use of the number line. On the other hand, student C demonstrated greater variability and difficulty with numerical estimates. Alternating between the strategies of counting and a lack of visible procedure, he/she had lower global precision. Furthermore, he/she had a standard systematic mistake: in the numbers under 50, there was an overestimation, possibly due to a logarithmic organization on the number line, while the numbers over 50 showed an underestimation, with a more automated use, though inaccurate, of strategy 1. His lower mathematical performance suggests that difficulties in internalizing the number line can impact estimation and other mathematical abilities. The description of the strategies used by the three students reinforces the importance of investigating not only the accuracy of number line estimation but also the strategies used to estimate, mainly considering the possibilities of intervention in the classroom that stimulate the use of more efficient strategies and the internalization of the number line as a linear model of magnitude. Such examples illustrate the discussion about the relation between estimation capacity and arithmetic performance, combining qualitative and quantitative results. Furthermore, the results led us to consider the teaching of estimation strategies in schools. We can observe a possible learning in students A and B throughout the task development, towards the use of strategy 1, a strategy that, supposedly, expresses higher internalization of estimation understanding, with no need for counting support or approximation strategies. In general, our results follow the tendency of studies in the area, which point out that children with lower arithmetic performance use less economic and precise estimation strategies (Ashcraft; Moore, 2012; Jung et al., 2020), while children with average and higher performance had greater stability and precision in the use of strategies. The main limitation of the work is that fact that the same strategy (Strategy 1) had a double facet: it reflected an efficient internalization of the numeric system and a precise representation in the number line, in some cases and, on the other hand, in others, indicated an understanding, even if fragile, of the number system that was reflected in intuitive and imprecise estimates. We suggest that future studies plan differentiation mechanisms that can be observed during the application of instruments and can differentiate the two groups using the same strategy
6 Final remarks
To conclude, we seek to answer the question in the title. We highlight three reasons to teach estimation strategies. The first refers to the set of evidence about the relation between children’s performance in numerical estimation and mathematical performance, a relation that has still not resulted in a group of studies that stimulate the teaching of numerical estimation in schools. The description of subjects A and B indicates the search for accurate and economic strategies in a short task, which suggests that the teaching of estimation can go through interventions that stimulate economic and precise strategies. The second reason is related to the lack of studies focused on the strategies used by students to solve estimation tasks. Though we do have studies that indicate that stimulating the making of estimates contributes to improving mathematical performance in different areas, little is known about the impact of the specific stimulation of more efficient and economic strategies. We believe that the teaching of estimates in schools should be incentivized, including the work with different strategies, as those are related to the estimation accuracy, even though this relation is not sufficiently explained. Our study sought to highlight the importance of stimulating the development of precise estimation strategies, especially among children who do not develop them spontaneously. As we mentioned at the beginning of this article, there is evidence that children benefit from guided stimulation towards numerical estimation and that more precise strategies are correlated with better academic development. The last reason is that the exercise of certain abilities in specific areas of school learning can lead to positive effects in other areas, even those that are not directly related. An example is the stimulation of phonological awareness that was also correlated with mathematical performance. In this sense, the possibility of making interventions in numerical estimations can serve as a proxy for the development of other areas, including estimates in different domains, such as time and space, among others.
Agradecimentos
The authors thank FAPERGS (Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul) for financial support (Grant/Process No. 10/2024 AUXÍLIO RECÉM-DOUTOR ou RECÉM-CONTRATADO – ARD/ARC).
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Data Availability:
Os dados gerados ou analisados durante este estudo estão incluídos neste artigo publicado.
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Editor-in-Chief:
Prof. Dr. Roger Miarka
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Associate Editor:
Profa. Dra. Celi Espasandin Lopes
Os dados gerados ou analisados durante este estudo estão incluídos neste artigo publicado.
