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For children, adolescents and educated adults, comparing fractions with common numerators (e.g., 4/5 vs. 4/9) is more challenging than comparing fractions with common denominators (e.g., 3/4 vs. 6/4) or fractions with no common components (e.g., 5/7 vs. 6/2). Errors are related to the tendency to rely on the “greater the whole number, the greater the fraction” strategy, according to which 4/9 seems larger than 4/5 because 9 is larger than 5. We aimed to determine whether the ability of adolescents and educated adults to compare fractions with common numerators was rooted in part in their ability to inhibit the use of this misleading strategy by adapting the negative priming paradigm. We found that participants were slower to compare the magnitude of two fractions with common denominators after they compared the magnitude of two fractions with common numerators than after they decided which of two fractions possessed a denominator larger than the numerator. The negative priming effects reported suggest that inhibitory control is needed at all ages to avoid errors when comparing fractions with common numerators.

Learning rational numbers, particularly the concept of fractions, is challenging for students (

The natural number bias (

According to

Taken together, findings from these previous studies tentatively suggest that participants might be more error prone when comparing two fractions with common numerators because they apply a well-known strategy consisting of choosing the fraction with the greater whole number when the fractions have common components (i.e., the “greater the whole number, the greater the fraction” strategy). While this strategy is efficient for comparing fractions with common denominators, it can leads to systematic errors when comparing fractions with common numerators. In this context, the present study aimed to determine whether the inhibition of the well-known “greater the whole number, the greater the fraction” strategy is needed to compare fractions with common numerators in adolescents and in educated adults as in other types of logico-mathematical problems.

According to dual-process theories of human thinking (

Importantly, the failure to inhibit a misleading strategy partly explains not only systematic errors in logical Piagetian problems but also systematic difficulties faced by children in literacy (

Some studies have already demonstrated that the inhibition of contextual or perceptual cues is needed to compare fractions. For example, ^{th} and 8^{th} graders with better conceptual and procedural knowledge of fractions displayed better performance in an anti-saccade task, a typical inhibitory control task. More generally, comparing the magnitudes of numbers sometimes requires inhibiting contextual or perceptual cues, such as the physical size of the number (

The negative priming paradigm was originally designed by

Negative priming effects were originally reported in attentional tasks to reveal the inhibition of distractors (e.g.,

In the present study, we designed a variant of the negative priming paradigm originally designed by ^{th} graders who are able to successfully compare the magnitude of common numerator items (i.e., the prime in the test condition in our negative priming paradigm). We also tested young adults because we used age as a proxy of the skill to compare fractions, young adults being more skilled than 9^{th} graders.

We reasoned that if adolescents and educated adults must inhibit the “greater the whole number, the greater the fraction” strategy to compare fractions with common numerators, then they should require more time (and/or commit more errors) when comparing fractions with the same denominator that are preceded by primes when the participants compared fractions with the same numerator than they did when preceded by primes in which participants decided which of the two fractions had a denominator larger than the numerator (revealing a typical negative priming effect). Notably, testing adolescents and adults allowed us to determine (a) whether educated adults must still inhibit the “greater the whole number, the greater the fraction” strategy when comparing common numerator items and (b) whether the efficiency of inhibiting the “greater the whole number, the greater the fraction” strategy in that context increases with age. If inhibitory control is required to compare common numerator items even in educated adults, then we should find negative priming effects in adults. Finally, if the efficiency in inhibiting the misleading heuristic (i.e., the “greater the whole number, the greater the fraction” strategy) increases with age, then the amplitude of the negative priming effect should be smaller in adults than in adolescents, as reported in previous negative priming studies on misleading strategies (e.g.,

Thirty-six typically developing adolescents (mean age ± ^{2}(1) = 0.38,

By definition, negative priming effects can be observed only if participants accurately compared common numerator items (i.e., the prime in the test condition). Therefore, 15 adolescents (41% of the sample, ^{2}(1) = 0.01,

Written consent was obtained after a detailed discussion and explanations were given. Parental written consent was obtained for the adolescents, who were tested in accordance with national and international norms that govern the use of human research participants. All participants reported normal or corrected-to-normal vision and were native French speakers. All adolescents attended the same middle-school, and all adults attended the same university in Caen, France, serving a diverse population with a wide range of socioeconomic statuses and cultural backgrounds.

Each item consisted of a visual presentation of two fractions on a computer screen (in 50-point black Arial font on a white background). Between the two fractions, a question mark was displayed. We designed three types of items: common denominator items, common numerator items and denominator-numerator items. In the common denominator items (e.g., 2/4

We tested participants individually in a quiet room on a laptop with a screen resolution of 1366 x 768 pixels. Stimuli were presented using E-prime 2.0 (Psychological Software Tools, Inc., Pittsburgh, USA). Participants started by completing 24 practice trials (8 for each type of item) to familiarize themselves with the three types of items and the response pad. Feedback was provided regarding whether their answers were correct. To prevent familiarizing participants with the prime-probe sequences, the three types of items were presented in blocks, starting with the common denominator items and finishing with the common numerator items. Next, participants performed 32 experimental trials (16 test and 16 control trials) consisting of pairs of primes and probes during which no feedback was given on the correctness of the responses. In the test trials, participants performed a common numerator item (as a prime) and then a common denominator item (as a probe). In the control trials, participants performed a denominator-numerator item (as a prime) followed by a fraction with a common denominator item (as a probe; see

Experimental design. Participant responds by pressing the left or the right button to indicate her or his fraction choice. Items were presented in French.

Given that we were interested in potential priming effects, we standardized the sequence of responses between the primes and the probes in both the test and control trials. The four possible pairs of response sequences occurred two times in both the test and control trials (i.e., ‘left’ on the prime problem then ‘left’ on the probe problem, ‘left’ then ‘right’, ‘right’ then ‘left’, and ‘right’ then ‘right’). Note that participants needed to switch from one strategy to another between the prime and probe items in both types of trials; thus, any difference in probes’ RTs between the control and test trials does not simply reflect switching costs as the ones reported in

All analyses of RTs included only data from trials in which participants responded correctly on the prime and the probe. Outliers were defined as RTs greater than 2 SDs from the mean for such participants on the probe items (or the prime items) in the test (or the control) trials. Outliers occurred on 6.7% of trials in adolescents and 6.8% of trials in adults. After removing outliers, participants’ errors and RTs were averaged separately for the prime and probe items on the test and control trials (see

Item | Trial | ||||
---|---|---|---|---|---|

Adolescents |
Adults |
||||

Included ( |
Excluded ( |
Included ( |
Excluded ( |
||

ERs (%) | |||||

Prime | Test | 16.3 (13.5) | 72.3 (19.2) | 16 (16.2) | 63.4 (4.8) |

Prime | Control | 12.5 (10.1) | 15.6 (22.9) | 6.4 (9.9) | 3 (4.2) |

Probe | Test | 9.7 (8.2) | 27.7 (22.1) | 7.9 (11.5) | 12.5 (17.7) |

Probe | Control | 12.2 (12.9) | 28.1 (23.7) | 8.7 (11.8) | 31.3 (8.8) |

RTs (ms) | |||||

Prime | Test | 3080 (960) | 2612 (1082) | ||

Prime | Control | 1751 (693) | 1288 (487) | ||

Probe | Test | 2729 (863) | 2420 (1010) | ||

Probe | Control | 2538 (859) | 2287 (960) | ||

Negative Priming | 192 (392) | 134 (426) |

When computing the error rates (ERs) on the probe, we included only data from trials in which participants responded correctly on the prime. We conducted separate 2 (type of trials: test or control) x 2 (age: adolescents or adults) analyses of variance (ANOVAs) of the ERs and the RTs for the prime and probe items. For each of the analyses, we report the effect size either in the ANOVA (partial eta squared) or in terms of the difference of the means (Cohen’s _{01}) or the alternative (BF_{01}) hypothesis. The Bayes factor indexes the extent to which observed data are supported by one model over another. For example, a Bayes Factor (BF_{01}) of 3 in favor of the null hypothesis means that the observed data are 3 times more likely to have occurred under the null than the alternative hypothesis (

Non parametric one-way (Kruskal-Wallis) and repeated measure (Friedman) ANOVAs on the prime ERs revealed a main effect of the condition, χ^{2}(1) = 6.74, _{10} = 19.86, but no effect of age, χ^{2}(1) = 1.07, _{01} = 2.44, and no two-way interaction, χ^{2}(1) < 1, BF_{01} = 0.17. Adults and adolescents committed more errors when comparing fractions with common numerators (test trials: 16.2 ± 2.1%) than when deciding which fraction had a denominator larger than the numerator (control trials: 9.5 ± 1.2%).

A two-way mixed design ANOVA of the RTs revealed a main effect of the type of trial, _{10} = 5.36 x 10^{12} - with participants requiring more time to perform common numerator items (_{10} = 1.16 - with adults (_{01} = 21.71. Analyses on the type of strategy used by the participants to solve common numerator items and the effect of improper fractions on this comparison can be found in the

To determine whether participants required more time to compare two fractions in which one of the two was an improper fraction in common numerator items and whether age modulated this effect, we performed two additional ANOVAs on the prime RTs and ERs of the test trials. A two-way ANOVA of the RTs revealed that participants required less time to solve common numerator items in which one of the two fractions was an improper fraction (_{10} = 11.24, but no effect of age, _{01} = 2.23, and no interaction between age and the type of fraction, _{01} = 5.84, were observed. A similar ANOVA of the ERs revealed no effect of the type of fraction, χ^{2}(1) < 1, BF_{01} = 1.53, of age, χ^{2}(1) < 1, BF_{01} = 2.25 and of interaction between age and the type of fraction, χ^{2}(1) < 1, BF_{01} = 5.04 for the interaction.

Non parametric one-way (Kruskal-Wallis) and repeated measure (Friedman) ANOVAs on the probe ERs revealed no effect of the condition, χ^{2}(1) < 1, BF_{01} = 2.81, no effect of age, χ^{2}(1) = 1.41, p = .24, BF_{01} = 1.97, and no two-way interaction, χ^{2}(1) < 1, BF_{01} = 11.46, suggesting a strong evidence in favor of the null hypothesis (i.e., the observed data are more than 11 times more likely to have occurred under the null than the alternative hypothesis). Given that we found no main effect of the type of trials on the ERs, any negative priming effect reported on RTs is unlikely to be due to speed/accuracy trade-offs.

A similar ANOVA on the RTs revealed a main effect of the type of trial, _{10} = 4.25, but no effect of age, _{01} = 1.48, and no two-way interaction, _{01} = 3.20. Note that the Bayes Factor of the interaction suggests moderate evidence in favor of the null hypothesis (i.e., it is about 3 times more likely that there is no difference than a difference in the amplitude of the negative priming effects between adolescents and adults). Importantly, we found that adolescents and adults required more time to perform common denominator items preceded by common numerator items (test trials:

Finally, to verify whether adults and adolescents had more difficulty with common numerator items than with fractions with common denominator items, we conducted a 2 (type of fraction: common numerators _{p}^{2} = .39, BF_{10} = 5621.25 but we found no main effect of age, _{01} = 1.32, and no significant interaction between the type of fraction and age, _{01} = 1.88. Non parametric ANOVAs on the ERs revealed a main of type of trials, χ^{2}(1) = 7.53, _{10} = 3.69, but no main effect of age, χ^{2}(1) < 1, BF_{01} = 3.28, and no two-way interaction, χ^{2}(1) < 1, BF_{01} = 3.02.

In the present study, we investigated (a) whether inhibitory control is required to compare common numerator items in adolescents and educated adults and (b) whether the efficiency to inhibit the misleading heuristic (i.e., the “greater the whole number, the greater the fraction” strategy) that leads participants to be biased when comparing common numerator items (i.e., the natural number bias) increases between adolescents and adults. In line with previous studies on adolescents (^{th} grade, by Grade 9 and later, students are most likely equally familiar with both types of problems. Thus, we suspect that the difficulty to compare common numerator items in adolescents and adults might be related in part to their tendency to rely on the “greater the whole number, the greater the fraction” strategy whenever fractions possess common numerators, even in a context in which this strategy is misleading such as with fractions with common numerators.

The present study offers an opportunity to go a step further in explaining which process might allow an individual to overcome errors when comparing fractions with common numerators. Indeed, we found that adolescents and adults required more time to compare common denominator items after they had just succeeded in determining which of the two fractions with common numerators was larger than after they succeeded in determining which of the two fractions had a denominator larger than the numerator. Taken together, our results suggest that adolescents and educated adults must inhibit the “greater the whole number, the greater the fraction” strategy when comparing common numerator items. Although the ability to inhibit the “greater the whole number, the greater the fraction” heuristic might contribute to the ability to compare fractions with common numerators, inhibiting contextual or perceptual cues, such as the physical size or the location of the number (see, e.g.,

One could wonder why we did not include a classical inhibitory control task in the present study to show that inhibitory control was involved to solve common numerator items, as in previous studies (see, e.g.,

By comparing the performance of adolescents and adults on a negative priming paradigm adapted to fraction comparison, our findings shed light on the development of fraction comparison skills by showing that inhibitory control is needed at all ages to avoid errors when comparing fractions with common numerators. Previous studies have argued that the amplitude of negative priming effects might reflect the ability to inhibit a specific heuristic in a given context (

In contrast with these findings, the amplitude of the negative priming effects in our study did not differ between adolescents and adults, which might suggest that the inhibition of a componential processing strategy might be already largely efficient in ninth graders. This finding is consistent with the fact that we did not find that knowledgeable adolescents were less efficient than adults at comparing common numerator items. Note that other studies have also reported no difference in the amplitude of negative priming effects with age (

We note that using the “greater the whole number, the greater the fraction” strategy to compare fractions with common elements might be a byproduct of the mathematics curricula used in primary school. Indeed, in primary school, simple fractions are first introduced as tools for solving problems that cannot be solved by relying on whole numbers, such as when children need to understand that the “whole” can be divided into

The lack of difference in the amplitude of the negative priming effects should be taken with caution given that it can be generalized only to adolescents who succeeded in inhibiting the “greater the whole number, the greater the fraction” strategy when comparing common numerator items. The comparison of adolescents included and excluded from the final sample revealed that excluded adolescents had greater difficulty than included ones solving common numerator and common denominator items. Thus, excluded adolescents might have a lower understanding of fractions in general. That said, we note that they displayed above-chance levels of performance only when performing common denominator items, which suggested that they might be more prone to use the “greater the whole number, the greater the fraction” misleading strategy in this specific context (see

Our findings suggest that difficulty in inhibiting the “greater the whole number, the greater the fraction” heuristic might contribute to the difficulty comparing fractions with common numerators, but other factors also contribute to such difficulty. Indeed, a number of studies have provided evidence that a lack of conceptual and procedural knowledge on fractions contributes to the specific difficulties in comparing common numerator fractions (e.g.,

A potential limitation of the present study is that participants were asked to perform a different task on the prime items in the control trial (i.e., determine which of the two fractions has a denominator larger than its numerator) than for all other items (i.e., determine which of the two fractions is the largest). We asked participants to perform a task of a different nature on the prime in the control trial to prevent the strategy being used in the prime items priming the strategy to use on the probe items. However, one could argue that the switching costs between the prime and the probe might differ between the test and the control trials. If so, the negative priming effect could be a consequence of the difference between the switching costs induced by switching between two strategies while performing the same tasks (i.e., comparing the magnitude of fractions) in the test trials and the ones induced by switching between two tasks (i.e., deciding which of the two fractions has a denominator larger than its numerator or which of the two fractions is the largest) in the control trials. However, previous studies reported either no difference in switching costs when switching within the same task or between two tasks (e.g.,

In addition, one could argued that providing feedbacks during the practice trials could constitute a minimal form of education to the participants and thus change their spontaneous choice of strategy. That said, we note that even if feedback was provided in the practice trials, 41% of adolescents and 6% of adults were excluded because they performed at chance in the negative priming paradigm. Thus, the comparison of fractions with common numerators remains challenging even after receiving feedbacks, which is consistent with the difficulties faced by students at school in this academic learning.

Moreover, the use of single digit numerators and denominators in the common denominator and common numerator items might have encouraged participants to rely more systematically on the “greater the whole number, the greater the fraction” heuristic. Indeed, previous studies demonstrated that the use of a holistic processing strategy increases when fraction comparison tasks include fractions with two digits components and a wider range of numbers (see, e.g.,

The sample size constitutes another clear limitation of the present study. Indeed, the sample size might be too small to detect a significant age effect on the amplitude of the negative priming effect. Thus, the lack of difference in the amplitude should be interpreted with caution despite the fact that the sample size (after exclusion of participants) was determined pre-hoc on the basis of previous negative priming studies that used similar sample sizes and reported significant effects of age on the amplitude of the negative priming effect (

In light of these potential limitations, future studies are needed to investigate whether similar negative priming effects can be observed (a) when different prime items are designed in the control trials (b) when no feedbacks are provided in the practice trials, and (c) when a wider range of numbers including two digits number is used to generate the fractions and whether these negative priming effects would vary with age on a larger sample size. Moreover, we need to better understand the role of improper fractions on the emergence of the negative priming effects in the context of fraction comparison (See

Finally, the degree to which our findings can be generalized to the general population must be considered, given that the participants performed no standardized math test. We note that our sample of adolescents and adults might be representative of their respective parent populations because adolescents were all in Grade 9 in a middle school serving a diverse population (with none of them repeating a year), and adults were undergraduates in a public university serving a diverse population who all obtained a high-school diploma before attending the university.

In conclusion, our study provides the first evidence that inhibitory control is needed in adolescents and educated adults in order to compare fractions with common numerators, as revealed by a negative priming paradigm consistent with studies showing that fraction knowledge is associated with executive functions (

Fractions were designed so that the average distance based on the magnitude of the fractions did not differ between the common numerator items (

Prime |
Probe |
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Control | Test | Control | Test | ||||

6/7 | 6/3 | 7/2 | 7/8 | 7/3 | 9/3 | 3/7 | 5/7 |

5/4 | 5/8 | 2/7 | 2/9 | 7/3 | 5/3 | 8/6 | 2/6 |

5/3 | 5/7 | 3/7 | 3/2 | 4/3 | 8/3 | 5/3 | 8/3 |

6/2 | 6/9 | 5/7 | 5/6 | 2/9 | 4/9 | 4/6 | 3/6 |

4/9 | 4/2 | 6/5 | 6/3 | 7/5 | 6/5 | 8/7 | 9/7 |

7/8 | 7/5 | 9/7 | 9/4 | 4/5 | 3/5 | 9/8 | 6/8 |

7/2 | 7/9 | 4/5 | 4/7 | 2/6 | 5/6 | 3/4 | 6/4 |

5/6 | 5/2 | 9/7 | 9/8 | 7/4 | 2/4 | 4/8 | 6/8 |

3/2 | 3/4 | 2/3 | 2/4 | 4/7 | 6/7 | 7/8 | 4/8 |

6/4 | 6/7 | 4/9 | 4/3 | 3/2 | 7/2 | 8/5 | 9/5 |

7/9 | 7/6 | 8/6 | 8/5 | 7/2 | 5/2 | 9/5 | 3/5 |

8/3 | 8/9 | 7/4 | 7/3 | 8/2 | 9/2 | 3/2 | 7/2 |

6/8 | 6/5 | 5/3 | 5/8 | 3/6 | 5/6 | 4/9 | 2/9 |

4/3 | 4/8 | 6/8 | 6/7 | 5/9 | 4/9 | 2/9 | 6/9 |

5/9 | 5/4 | 3/2 | 3/6 | 9/4 | 6/4 | 8/6 | 7/6 |

4/5 | 4/3 | 8/7 | 8/9 | 7/4 | 5/4 | 5/7 | 3/7 |

To determine the type of strategy used by adults and adolescents in common numerator items, we computed the correlation between the RTs and either the distances between the magnitudes of the fractions or by the distance between the two denominators of the two fractions in the pairs. We reasoned that if participants computed the magnitudes of the fractions to compare them, RTs should be correlated with the distance of the magnitudes of the fractions but not with the distance between the denominators and vice versa if participants compared the denominators of the fractions. In adults, we found a significant correlation between the RTs and the distance between the magnitudes of the fractions,

Mean RTs for each of the 6 different numerical distance between the denominators of the fractions (A for adults and B for adolescents) and for each of the 6 different numerical distance between the magnitude of the fractions to compare (C for adults and D for adolescents). Dot in red was considered an outlier (steam-and-leaf plot procedure) and was excluded from the correlational analysis.

Taken together these results suggest that adults required less time to perform common numerator items as the distance between the magnitudes of the two fractions increased but not as the distance between the denominators of the two fractions increased. This pattern of correlations suggests that adults most likely computed the magnitudes of the two fractions to compare them, which is in line with

Six pairs of fractions on the prime of the test trials contained one proper and one improper fraction. Given that comparing such pairs could be performed without accessing the magnitude of the fractions but by choosing the fraction in which the numerator was higher than the denominator, participants might rely on different types of strategies when performing these pairs than pairs of fractions in which both of the fractions were improper or both were not improper. To determine whether participants required more time to compare two fractions in which one of the two was an improper fraction in common numerator items and whether age modulated this effect, we performed two additional ANOVAs on the prime RTs and ERs of the test trials. A two-way ANOVA of the RTs revealed that participants required less time to solve common numerator items in which one of the two fractions was an improper fraction (_{10} = 11.24, but no effect of age, _{01} = 2.23, and no interaction between age and the type of fraction, _{01} = 5.84, were observed. A similar ANOVA of the ERs revealed no effect of the type of fraction, χ^{2}(1) < 1, BF_{01} = 1.53, of age, χ^{2}(1) < 1, BF_{01} = 2.25 and of interaction between age and the type of fraction, χ^{2}(1) < 1, BF_{01} = 5.04 for the interaction.

The authors have declared that no competing interests exist.

The research procedure was consistent with the principles of the research ethics published by the American Psychological Association.

We wish to thank Lucie Besnehard and Thomas Augier for their valuable help with the data collection and Margot Roëll for the English editing of the manuscript. We also would like to thank the middle schools and the teachers, the children and their parents for participating.

The authors have no funding to report.