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Attribution License (Fraction Comparisons
Learning rational numbers, particularly the concept of fractions, is challenging for students (Lortie-Forgues, Tian, & Siegler, 2015). The acquisition of fraction knowledge in education is important because it constitutes part of the core knowledge for algebraic, probabilistic, and proportional reasoning (Siegler & Lortie-Forgues, 2014). Indeed, fraction knowledge is predictive of mathematic achievement (Jordan et al., 2013; Siegler, Thompson, & Schneider, 2011; Torbeyns, Schneider, Xin, & Siegler, 2015). Notably, fractions not only are difficult for children to master but also remain challenging for adults (for a review, see Vamvakoussi, 2015). Adults and children typically use different strategies to compare the magnitudes of fractions as a function of the type of fraction comparison problems with which they are confronted (see, e.g., Fazio, DeWolf, & Siegler, 2016; Schneider & Siegler, 2010; Siegler et al., 2011; Siegler & Pyke, 2013). In addition, adults tend to represent fractions as discrete magnitudes when simpler strategies are not pertinent (Fazio et al., 2016). Taken together, these findings are in line with the overlapping wave theory proposed by Siegler (1996). Thus, difficulties in learning fractions are due to different factors, such as applying whole-number properties to rational numbers or applying procedures used in fraction addition (or subtraction) to fraction multiplication (or division; Siegler, Fazio, Bailey, & Zhou, 2013; Torbeyns et al., 2015). An illustration is given in fraction comparison problems in which participants consider a fraction as two separate natural numbers rather than one rational number (Stafylidou & Vosniadou, 2004) in a certain context. For instance, when deciding whether 1/7 is larger than 1/3, one could erroneously answer that 1/7 is larger than 1/3 because 7 is larger than 3 when considering the fractions as two separate natural numbers.
The natural number bias (Vamvakoussi, Van Dooren, & Verschaffel, 2012), defined as a robust human tendency to rely on natural number knowledge when working with rational numbers (Ni & Zhou, 2005) could be in part at the root of systematic errors when comparing fractions with common numerators (e.g., 1/7 vs. 1/3). Because the concept of a natural number coexist with the concept of a rational number (Chi, 1992, 2005; diSessa, 2008; diSessa & Sherin, 1998; Posner, Strike, Hewson, & Gertzog, 1982; Vosniadou, 2013; Vosniadou & Skopeliti, 2014; Vosniadou, Vamvakoussi, & Skopeliti, 2008), the concept of a natural number might interfere with the concept of a rational number in certain contexts, resulting in a natural number bias. This bias persists from childhood to adulthood in fraction comparison problems likely because (as with other cognitive biases) it is highly efficient in certain contexts, such as when the two fractions to compare have the same denominator (e.g., 4/6 vs. 7/6). In this context, applying natural number knowledge (i.e., that 4 is smaller than 7) when comparing 4/6 vs. 7/6, leads to the correct answer that 7/6 (1.16) is larger than 4/6 (0.66). However, it is worth noting that previous studies have demonstrated that natural number knowledge is crucial for mathematical achievement (Siegler & Lortie-Forgues, 2014) and for the development of conceptual and procedural knowledge of fractions in particular (e.g., Rinne, Ye, & Jordan, 2017).
Van Dooren, Lehtinen, and Verschaffel (2015) have argued that the systematic difficulty in comparing fractions with common numerators (as revealed by higher error rates or longer response times) at any age might be rooted in part in our spontaneous tendency to rely on a well-known strategy (i.e., the “greater the whole number, the greater the fraction” strategy) in a context in which this strategy is not appropriate. Consistent with this assumption, studies have reported that children (Gabriel, Szucs, & Content, 2013a; Meert, Grégoire, & Noël, 2010a), adolescents (Van Hoof et al., 2013), educated adults (Meert, Grégoire, & Noël, 2009, 2010b; Vamvakoussi, Van Dooren, & Verschaffel, 2012) and experts in mathematics (Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013) require more time to compare fractions with common numerators (i.e., a context in which the “greater the number, the greater the fraction” strategy is misleading) than with common denominators (i.e., a context in which the “greater the number, the greater the fraction” strategy is appropriate). While educated adults tend to rely on “the greater the number, the greater the fraction” strategy to compare fractions with common components and thus continue to be affected by the natural number bias (Bonato, Fabbri, Umiltà, & Zorzi, 2007; Meert et al., 2009), they are less affected by it when comparing fractions with different components, presumably because this context might trigger to a lesser extent this misleading strategy (Faulkenberry & Pierce, 2011; Gabriel, Szucs, & Content, 2013b; Schneider & Siegler, 2010; Sprute & Temple, 2011).
According to Alibali and Sidney (2015), several factors could affect the natural number bias observed in fraction comparison tasks such as the strength and preciseness of rational number representations. They speculate that the strength of these rational number representations could be dependent on the participant’s experience with fractions, the delay from the time they last activated this representation and the characteristics of the problem to solve that might trigger one sort of representation over the other. Cognitive development and schooling also affects fraction magnitude abilities as rational and whole number representations become increasingly more precise over time (see Siegler et al., 2011). However, the fact that even expert mathematicians display a natural number bias in certain context contradict these findings (Obersteiner et al., 2013).
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.
Inhibition of Misleading Strategies During Logico-Mathematical Problem Resolution
According to dual-process theories of human thinking (Evans, 2008; Evans & Stanovich, 2013; Kahneman, 2011), systematic reasoning errors (or reasoning biases) in different domains are rooted, in part, in our spontaneous tendency to rely on heuristics (i.e., rapid, parallel, automatic, and effortless strategies). Studies have provided convergent evidence that inhibitory control is necessary at any age to avoid using a misleading heuristic in a context in which it interferes with the strategy (or the strategies) leading to the correct answer (see for reviews Bjorklund & Harnishfeger, 1995; Borst, Aïte, & Houdé, 2015; Diamond, 2013; Houdé & Borst, 2015). For example, studies have demonstrated that the ability to overcome systematic errors in classical developmental tasks such as in Piaget’s A-B (Diamond, 1998 but see Munakata, 1998 for a working memory account of the A-not-B error), class-inclusion (Borst, Poirel, Pineau, Cassotti, & Houdé, 2013; Perret, Paour, & Blaye, 2003) and number-conservation (Houdé & Guichart, 2001; Houdé et al., 2011) tasks is not exclusively rooted in the acquisition of knowledge of increasing complexity (as Piaget assumed, Piaget, 1983) but is also dependent on the progressive ability to inhibit misleading heuristics.
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 (Ahr, Houdé, & Borst, 2016; Borst, Ahr, Roell, & Houdé, 2015; Lanoë, Vidal, Lubin, Houdé, & Borst, 2016), in mathematics (Lubin et al., 2016; Lubin, Vidal, Lanoë, Houdé, & Borst, 2013) and in science (Potvin, Masson, Lafortune, & Cyr, 2015).
Some studies have already demonstrated that the inhibition of contextual or perceptual cues is needed to compare fractions. For example, Siegler and Pyke (2013) provided evidence that 6th and 8th 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 (Henik & Tzelgov, 1982; Szűcs & Soltesz, 2007). In addition, switching costs have been reported during arithmetic problem solving (i.e., multiplication of whole numbers), with participants displaying poorer performance when switching between two strategies than when using the same strategy in two consecutive trials (Lemaire & Lecacheur, 2010). Switching cost has been interpreted as possibly reflecting the role of working memory and/or inhibitory control in strategy choice. However, no study to date has provided evidence that the inhibition of a misleading strategy contributes to the ability to compare fractions with common numerators across development using a negative priming approach.
The Negative Priming Paradigm Adapted to Misleading Strategies
The negative priming paradigm was originally designed by Tipper (1985). The negative priming paradigm rests on the logic that if a distractor (or a strategy) is inhibited on a given item (i.e., the prime), then its activation on the following item (i.e., the probe) should be disrupted (i.e., longer response times and higher error rates) in comparison to a control condition in which the prime does not require inhibition of that strategy (e.g., Borst, Moutier, & Houdé, 2013; Tipper, 2001 but see Neill, Valdes, & Terry, 1995, for a non-inhibitory episodic retrieval account of negative priming). Although neuroimaging studies that investigated the neural underpinning of negative priming used different procedures and different negative priming tasks, there is tentative evidence that the negative priming effect relies on a similar brain area regardless of the nature of the task (see for a review Frings, Schneider, & Fox, 2015). Importantly, negative priming elicits activation in the right dorsolateral prefrontal cortex (Egner & Hirsch, 2005; Krueger, Fischer, Heinecke, & Hagendorf, 2007; Ungar, Nestor, Niznikiewicz, Wible, & Kubicki, 2010) and spatially adjacent regions such as the medial and inferior prefrontal cortex (Wright et al., 2005). These prefrontal areas are closely related to inhibitory control processes (see, e.g., Aron, Robbins, & Poldrack, 2014). Lesions studies provided convergent evidence for the role of the frontal lobe in the negative priming effect, with no negative priming effect observed in patients with frontal lobe lesions (Metzler & Parkin, 2000; Stuss et al., 1999).
Negative priming effects were originally reported in attentional tasks to reveal the inhibition of distractors (e.g., Tipper, Weaver, Cameron, Brehaut, & Bastedo, 1991) but now have been reported in numerous tasks in various domains to reveal the inhibition of overlearned misleading heuristics (see for a review Houdé & Borst, 2015). Note that the negative priming paradigm adapted to misleading strategies reveals that a misleading heuristic must be inhibited in order to solve a given task but provides no information on the strategy that is activated once the misleading heuristic has been inhibited (Borst, Moutier, & Houdé, 2013). In the present study, we used a similar negative priming approach to determine whether the selection of the appropriate strategy to compare fractions with common numerators relies, in part, on the inhibition of a misleading heuristic as previously evidenced in other school learning studies (e.g., Ahr et al., 2016; Borst, Ahr, Roell, & Houdé, 2015; Lanoë et al., 2016; Lubin et al., 2013; Lubin et al., 2016; Potvin et al., 2015). Note that Meert et al. (2009) reported that adults required more time to compare the magnitude of two natural numbers when preceded by fractions with common numerators than when preceded by fractions with common denominators. The authors interpreted this priming effect as an evidence that participants inhibited the selection of the larger denominator when comparing common numerator fractions. Importantly, the priming effect was only observed when the denominators of the common numerator factions in the prime matched the natural numbers in the probe which does not provide evidence that the “greater the whole number, the greater the fraction” strategy was inhibited. Moreover, it is difficult to determine whether the priming effect reported in this study reflect a positive or a negative priming effect. Indeed, the priming effect could evidence that participants are faster to compare natural numbers when preceded by common denominator fractions in which they can directly compare the components of the fractions to compare their magnitudes.
The Present Study
In the present study, we designed a variant of the negative priming paradigm originally designed by Tipper (1985). Our paradigm was designed in such a way that on test trials, the misleading strategy (i.e., the “greater the whole number, the greater the fraction” strategy) that participants were required to resist (i.e., inhibit) to solve the prime item became an appropriate strategy for the probe item. Conversely, in control trials, the strategy necessary to complete the prime item was unrelated to the strategy required to solve the probe item. For instance, in the test trial, participants had to determine on the prime which of the two fractions with the same numerator (e.g., 7/4 vs. 7/3, hereafter referred as common numerator items) was the largest. This item typically required participants to resist (inhibit) using the “greater the whole number, the greater the fraction” strategy (which would lead participants to erroneously state that 7/4 > 7/3 because 4 > 3). After the prime, participants had to determine on the probe which of the two fractions with the same denominator (e.g., 8/3 vs. 5/3, hereafter referred as common denominator items) was the largest, a context in which the “greater the whole number, the greater the fraction” strategy was appropriate (8/3 > 5/3 because 8 > 3). In the control trial, participants first decided (on the prime) which of the two fractions (4/3 vs. 4/5, hereafter referred as common-denominator items) had a denominator larger than the numerator (here 4/5). Then, on the probe, as in the test trial, they determined which of the two fractions with the same denominator (2/6 vs. 5/6) was the largest, a context in which the “greater the whole number, the greater the fraction” strategy was appropriate (2/6 < 5/6 because 2 < 5). Regarding the appropriateness of the method, we note that by construction, the negative priming paradigm requires that the participants succeed above the chance level of performance on the control and test primes for the priming effects on the probes to be observed. Thus, we tested 9th 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 9th 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., Aïte et al., 2016; Borst, Moutier, & Houdé, 2013; Lanoë et al., 2016).
Method
Participants
Thirty-six typically developing adolescents (mean age ± SD: 14.6 ± 0.4 years, 17 girls) were recruited from middle schools in Caen (Calvados, France). All of the adolescents were in Grade 9 and had not repeated a year. Thirty-three adults (21 ± 1.7 years, 17 women) without cognitive impairment participated in the study. They were all students from a state university with no selection of the students. Among the 30 adults, 9 were enrolled in a BA in science and 21 in a BA in humanities and social sciences (4 in law, 6 in history, 4 in sociology and 7 in psychology). The two groups differed significantly in age, t(67) = 21.4, p < .0001, Cohen’s d = 4.91, but not in sex repartition, χ2(1) = 0.38, p = .53.
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, M = 72.3 ± 19.2% of errors in common numerator items) and 2 adults (6%, M = 63.4 ± 4.8 of errors in common numerator items) were excluded from the analysis because they performed at chance levels of performance. Moreover, we excluded one adolescent and one adult because their average RT exceeded the average RT of the group by more than 3 SD. Thus, analyses were conducted on 20 adolescents (mean age: 14.6 ± 0.3 years, 9 girls, M = 16.3 ± 13.5% of errors in common numerator items) and 30 adults (mean age: 21 ± 1.8 years, 14 women, M = 16 ± 16.2% of errors in common numerator items). The gender distribution did not differ between the two groups, χ2(1) = 0.01, p = .90.
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.
Materials
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 vs. 5/4) and common numerator items (e.g., 4/2 vs. 4/5), participants were asked to determine which of the two fractions was the largest. Importantly, participants could rely on the “greater the whole number, the greater the fraction” strategy to determine which of the two fractions was the largest for the fractions with common denominator items, whereas they could not use it for the common numerator items. In the denominator-numerator items, we asked participants to determine which of the two fractions possessed a denominator larger than the numerator. Critically, the inhibition of the “greater the whole number, the greater the fraction” strategy was not needed to solve the denominator-numerator items. Note that the two fractions presented in the denominator-numerator items had the same numerator (e.g., 4/2 vs. 4/5) as the common numerator items because the two types of item served as primes in the control and test trials of our negative priming paradigm, respectively. We designed fractions in each item using numbers between 2 and 9 for numerators and denominators (see Appendix - Fraction design). We presented 24 items (eight of each type) for the practice trials and 96 (32 common denominator items, 16 common numerator items, and 16 denominator-numerator items) for the experimental trials.
Procedure
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 Figure 1). Trials were presented in a random order, with the exception that no more than two test or control trials could occur successively. Each trial started with the presentation of a fixation point followed by a presentation of the type of comparison participants needed to perform on the next item. The question was displayed at the top of the screen in blue and red 24-pt Arial font for half the participants and red and blue font for the other participants, respectively. Then, two fractions were displayed with a question mark in between the two until participants provided an answer. Participants pressed the left (or right) button of the mouse to indicate that the fraction on the left (or the right) of the question mark was the larger one (or had a denominator larger than the numerator). Response times (RTs) were recorded from the onset of the fractions to the button press. Immediately after participants pressed one of the two response buttons, a fixation cross appeared, followed by a presentation of the type of comparison to perform on the next trial. Finally, two fractions and a question mark were displayed until a response was provided. In between each trial, a 450 x 450 pixel blurred jpeg image was displayed in the center of the screen to limit the transfer of processes from one trial to the next. No judgment was required on these images.
Figure 1
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 Lemaire and Lecacheur (2010)’s study.
Results
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 Table 1).
Table 1
Means and Standard Deviations (in Parentheses) of ERs (%) and RTs (ms) in Adolescents and Adults
| Item | Trial | M (SD)
|
|||
|---|---|---|---|---|---|
| Adolescents
|
Adults
|
||||
| Included (n = 20) | Excluded (n = 14) | Included (n = 30) | Excluded (n = 2) | ||
| 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) | |||
Note. See Appendix for the comparison of included and excluded adolescent ERs and RTs.
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 d). When testing the absence of an interaction effect (which were the effects of interest in our study), Bayesian statistics were used to provide an estimate of the evidence in favor of the null (BF01) or the alternative (BF01) hypothesis. The Bayes factor indexes the extent to which observed data are supported by one model over another. For example, a Bayes Factor (BF01) 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 (Jeffreys, 1961). We also provided for the critical interaction hypotheses on the probe items an interpretation of the Bayes Factor according to Jeffreys (1961)’s cutoffs. Bayesian Analyses were performed using the JASP software (Version 0.8.3.1) with a default Cauchy prior width of r = 0.707. Note that BFs reported for the interaction reflect the BF of the full interaction model over the main effects model only. Note that because the distributions of the ERs were not normal (all ps <.01) but the assumptions of homoscedasticity were confirmed (all ps > .30), we ran non-parametric ANOVAs when analyzing ERs on the primes and on the probes.
Primes Analyses
Non parametric one-way (Kruskal-Wallis) and repeated measure (Friedman) ANOVAs on the prime ERs revealed a main effect of the condition, χ2(1) = 6.74, p < .01, BF10 = 19.86, but no effect of age, χ2(1) = 1.07, p = .30, BF01 = 2.44, and no two-way interaction, χ2(1) < 1, BF01 = 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, F(1, 48) = 112.35, p < .0001, = .70, BF10 = 5.36 x 1012 - with participants requiring more time to perform common numerator items (M = 2846 ± 150 ms) than denominator-numerator items (M = 1519 ± 83 ms), and a main effect of age, F(1, 48) = 5.05, p = .03, = .09, BF10 = 1.16 - with adults (M = 1950 ± 131 ms) faster than adolescents (M = 2415 ± 161 ms) - but no two-way interactions, F < 1, BF01 = 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 Appendix.
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 (M = 2942 ± 1151 ms) than to solve common numerator items in which both fractions were improper or both fractions were not improper (M = 2673 ± 916 ms), F(1, 48) = 8.62, p = .005, = 0.15, BF10 = 11.24, but no effect of age, F < 1, BF01 = 2.23, and no interaction between age and the type of fraction, F < 1, BF01 = 5.84, were observed. A similar ANOVA of the ERs revealed no effect of the type of fraction, χ2(1) < 1, BF01 = 1.53, of age, χ2(1) < 1, BF01 = 2.25 and of interaction between age and the type of fraction, χ2(1) < 1, BF01 = 5.04 for the interaction.
Probes Analyses
Non parametric one-way (Kruskal-Wallis) and repeated measure (Friedman) ANOVAs on the probe ERs revealed no effect of the condition, χ2(1) < 1, BF01 = 2.81, no effect of age, χ2(1) = 1.41, p = .24, BF01 = 1.97, and no two-way interaction, χ2(1) < 1, BF01 = 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, F(1, 48) = 7.46, p < .01, = .13, BF10 = 4.25, but no effect of age, F(1, 48) = 1.12, p = .29, BF01 = 1.48, and no two-way interaction, F < 1, BF01 = 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: M = 2730 ± 863 ms for adolescents and 2421 ± 1010 ms for adults) than those preceded by a denominator-numerator item (2538 ± 859 ms for adolescents and M = 2287 ± 960 ms for adults): t(19) = 2.08, p < .025, Cohen’s d = 1.07 for adolescents and t(29) = 1.77, p < .05, Cohen’s d = 0.73 for adults (see Appendix for the effect of improper fractions on the negative priming effects reported).
Primes Versus Probes Analyses
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 vs. common denominators) x 2 (age: adolescents vs. adults) mixed-design ANOVA on ERs and RTs. To conduct this analysis, we included ERs and RTs for primes in the test trials (i.e., common numerator items) and for probes in the control trials (i.e., common denominator items). Note that we did not include ERs or RTs for probes in the test trials because performance was potentially affected by the prime. Participants required more time in determining which of the two fractions was largest when the two fractions had the same numerator (i.e., primes in the test trials) than when the two fractions had the same denominator (i.e., probes in the control trials), F(1, 48) = 31.11, p < .0001, ηp2 = .39, BF10 = 5621.25 but we found no main effect of age, F(1, 48) = 1.74, p = .19, BF01 = 1.32, and no significant interaction between the type of fraction and age, F(1, 48) = 1.96, p = .17, BF01 = 1.88. Non parametric ANOVAs on the ERs revealed a main of type of trials, χ2(1) = 7.53, p = .006, BF10 = 3.69, but no main effect of age, χ2(1) < 1, BF01 = 3.28, and no two-way interaction, χ2(1) < 1, BF01 = 3.02.
Discussion
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 (Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013) and educated adults (Meert et al., 2009; Vamvakoussi et al., 2012), we found that adolescents and adults required more time and were less accurate in comparing common numerator items than common denominator items, with adults being generally more efficient than adolescents. We suspect that formal instruction and familiarity with rational numbers might be the cause of the difference observed between adolescents and adults in the present study. One could argue that adults and adolescents might be slower at performing common denominator than common numerator items because they are more familiar with the former than the latter. Although this might be a possibility when rational numbers are first introduced in the mathematical curricula in 4th 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., Szűcs & Soltesz, 2007), might also be important to compare such fractions.
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., Siegler & Pyke, 2013). A limitation of such a correlational approach is that none of the classical inhibitory control tasks assess individual differences in the ability to inhibit a misleading strategy. Thus, the lack of correlation between performance in a classical inhibitory control task and a fraction comparison task does not necessarily reflect that inhibition plays no role in the choice of strategy used to solve the fraction comparison task. On the other hand, the negative priming paradigm has been widely used to demonstrate that inhibitory control is required within a given task to overcome misleading heuristics, and a growing number of studies have successfully applied this paradigm to demonstrate the role of inhibitory control over misleading heuristics in various domains including various school learnings (see, e.g., Aïte et al., 2016; Ahr et al., 2016; Lubin et al., 2013; Lubin et al., 2016; Borst, Moutier, & Houdé, 2013). In addition, the negative priming approach alleviates some of the shortcomings of pure correlational approaches (e.g., lack of causal evidence, directionality of the association reported).
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 (Houdé & Borst, 2014, 2015). Indeed, studies have reported that the amplitude of negative priming effects decreases with age between childhood and adulthood, such as in a verb-inflection task (Lanoë et al., 2016), in a revised version of Piaget’s class-inclusion task (Borst, Poirel, Pineau, Cassotti, & Houdé, 2013) or in a spatial perspective-taking task (Aïte et al., 2016), which has been interpreted as reflecting the increasing ability to inhibit a specific misleading heuristic in each of these domains.
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 (Lubin et al., 2013; Pritchard & Neumann, 2009). A possible explanation as to why some studies but not others reported a difference in the amplitude of the negative priming effects with age might be due to the type of inhibitory control needed in different contexts. Indeed, some authors argue that we should draw a distinction between intentional inhibitory control and automatic inhibitory control (Lechuga, Moreno, Pelegrina, Gómez-Ariza, & Bajo, 2006). According to this view, whereas intentional inhibitory control is an executive function mediated by the prefrontal cortex (Aron, 2007), automatic inhibitory control relies on automatic processes sustained by brain structures that mature earlier (Lechuga et al., 2006), which would explain the lack of a difference in the negative priming amplitude with age in some contexts. This automatic inhibitory control could potentially appear following intense training (Jasinska, 2013). Indeed, much as one can automatize the activation of a strategy after an intense training, one can potentially automatize the inhibition of a strategy. We argue that the inhibition process (to resist the “greater the whole number, the greater the fraction” strategy) involved in the comparison of fractions with common numerators might be automatized in some students as early as ninth grade, presumably due to intensive schooling on fractions.
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 n equal parts (e.g., a pizza can be divided in 6 equal parts, each of them representing 1/6 of the pizza). Thus, children start by manipulating and comparing fractions with common denominators before fractions with common numerators or with no common components. Starting the curricula with the manipulation of fractions with common denominators, a context in which the “greater the whole number, the greater the fraction” strategy is an efficient strategy, might promote the implicit transfer of the properties of whole numbers to fractions. This might explain why inhibition of the “greater the whole number, the greater the fraction” strategy is available at all ages to compare fractions with common numerators. Our finding complements other findings reported in literacy (Ahr et al., 2016; Borst, Ahr, Roell, & Houdé, 2015; Lanoë et al., 2016), mathematics (Lubin et al., 2016; Lubin et al., 2013) and science (Potvin et al., 2015) and provides additional evidence that inhibition might be a core mechanism of learning at school (for reviews see Houdé & Borst, 2015).
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 Appendix for the details of the analyses).
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., Fazio et al., 2016; Schneider & Siegler, 2010; Siegler et al., 2011; Siegler & Pyke, 2013). Children’s difficulties with fractions have motivated interventions to improve fraction knowledge (Fujimura, 2001; Gabriel et al., 2012; Moss & Case, 1999). The challenge for fraction instruction is to help children understand that magnitudes are a property of not only whole numbers but all real numbers (Siegler et al., 2011). In line with previous studies showing that pedagogical interventions based on inhibitory control learning are more efficient at overcoming systematic errors in logical reasoning (Houdé et al., 2001; Rossi et al., 2015) or in mathematical reasoning (Rossi, Lubin, Lanoë, & Pineau, 2012; Lubin, Lanoë, Pineau, & Rossi, 2012) than more traditional pedagogical interventions, we suspect that students’ understanding of fraction can be improved by using pedagogical interventions based in part on learning to inhibit the “greater the whole number, the greater the fraction” heuristic when comparing fractions with common numerators.
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., Allport, Styles, & Hsieh, 1994) or greater switching costs when switching between two tasks than within the same tasks (e.g., Philipp, Kalinich, Koch, & Schubotz, 2008). Thus, it is unlikely that the negative priming effect reported in the present study is a byproduct of the difference in switching costs between the control and test trials.
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., Schneider & Siegler, 2010).
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 (Borst, Moutier, & Houdé, 2013; Lanoë et al., 2016). Note that the Bayes statistics on the interaction effects on the probe responses times and error rates do not entirely alleviate the sample size issue given that they only showed that evidence for the null hypothesis was moderate (i.e., three time more likely to occur than the alternative hypothesis).
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 Appendix for the analyses of the effect of the improper fractions).
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 (Jordan et al., 2013; Namkung & Fuchs, 2016; Siegler et al., 2012). In addition, our findings provide converging evidence that inhibitory control might be one of the core mechanisms of cognitive development and school learning (for a review, see Houdé & Borst, 2015).