Complex fraction comparisons and the natural number bias: The role of benchmarks

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Highlights

  • Adults show a “smaller components—larger fraction” bias in complex fraction comparison.

  • The strength of the bias depends on problem type and mathematics experience.

  • Proximity to salient benchmarks—specifically 0 and 1—can help people overcome the bias.

  • Large inter-individual variability challenges the notion of a general bias.

Abstract

People are often better at comparing fractions when the larger fraction has the larger rather than the smaller natural number components. However, there is conflicting evidence about whether this “natural number bias” occurs for complex fraction comparisons (e.g., 23/52 vs. 11/19). It is also unclear whether using benchmarks such as 1/2 or 1/4 enhances performance and reduces the bias (e.g., 11/19 > 1/2 and 23/52 < 1/2, hence 11/19 > 23/52). We asked 107 adults to solve complex fraction comparisons that did or did not afford using benchmarks, and we assessed response time and accuracy. We found a reverse bias (i.e., smaller components—larger fraction) that was greater among participants with lower mathematics experience. Fractions' proximity to 0 or 1 facilitated performance and decreased bias; effects of other benchmarks were nonsignificant. These results challenge the generality of the natural number bias in fraction comparison and highlight its variability.

Introduction

Numerous studies have documented people's difficulties with rational numbers (Behr, Wachsmuth, Post, & Lesh, 1984; Carraher, 1996; Lortie-Forgues, Tian, & Siegler, 2015; Siegler, Fazio, Bailey, & Zhou, 2013; Stafylidou & Vosniadou, 2004). One source of these difficulties is that people rely on natural number principles to solve rational number problems, even though these principles do not always apply. This overextension of natural number principles is called the “whole number bias” or “natural number bias” (Ni & Zhou, 2005; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Vamvakoussi, Van Dooren, & Verschaffel, 2012). For example, when comparing the magnitudes of two fractions, people tend to consider the fraction that has the larger natural number components (numerator and denominator) to be larger. Such natural-number-based reasoning is successful for some problems (hereafter: “congruent” problems, e.g., 3/5 > 1/2, with 3 > 1 and 5 > 2) but not others (hereafter: “incongruent” problems, e.g., 1/4 > 2/9, although 1 < 2 and 4 < 9).

The natural number bias has been documented in many different rational number tasks (Obersteiner, Van Hoof, Verschaffel, & Van Dooren, 2016; Vamvakoussi, Van Dooren, & Verschaffel, 2013; Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015; Van Hoof, Verschaffel, & Van Dooren, 2017). In this study, we focus on the most extensively studied of these tasks, fraction magnitude comparison. Understanding how people compare fraction magnitudes is important because numerical magnitude is a core aspect of all numbers, including fractions (Siegler, Thompson, & Schneider, 2011), and because performance on fraction magnitude comparison is a predictor of later mathematical achievement (Siegler et al., 2012). Despite the many studies of fraction comparison, however, we still know little about the cognitive mechanisms underlying the bias, and about factors that contribute to its occurrence and strength.

Some researchers have argued that the natural number bias in fraction comparison arises because automatic activation of natural number magnitudes interferes with more effortful activation of fraction magnitudes (Vamvakoussi et al., 2012; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). However, recent evidence has challenged the generality of this view, and suggested that the bias may vary depending on the affordances of the problems, people's mathematics experience, and, related to these factors, the strategies people use to solve the problems (Alibali & Sidney, 2015). Moreover, empirical evidence for the natural number bias in fraction comparison is largely limited to simple fraction comparison problems. There are conflicting results regarding the existence and direction of the natural number bias in studies that used complex fraction comparison problems (reviewed by Gómez & Dartnell, 2015). These studies included complex problems of various types, including comparisons in which the fractions do not share common components (i.e., numerators or denominators), comparisons in which the numerical differences between the fractions are small, comparisons involving unfamiliar fractions, and comparisons in which the fractions have two-digit components.

This study investigates the natural number bias in complex fraction comparison in adults, and considers factors that may influence its occurrence, direction, and strength. We focus specifically on factors related to problem type and mathematics experience that may affect how strongly people activate holistic fraction magnitudes (e.g., the numerical value of a fraction like 7/13, rather than, for example, the magnitudes of the components 7 and 13). As we argue below, increased activation of holistic fraction magnitudes may reduce the natural number bias. Therefore, studying factors that affect how strongly people activate magnitudes may advance our understanding of the mechanisms underlying the bias.

The natural number bias in fraction comparison—better performance on congruent than on incongruent problems—has been observed in primary and secondary school students (Gómez, Jiménez, Bobadilla, Reyes, & Dartnell, 2015; Meert, Grégoire, & Noël, 2010b; Van Hoof et al., 2013; Van Hoof, Verschaffel, et al., 2017) and in adults (DeWolf & Vosniadou, 2011; Obersteiner et al., 2013; Vamvakoussi et al., 2012). Thus, the bias emerges early and persists into adulthood.

The persistence of the bias suggests that it cannot be fully explained by insufficient conceptual knowledge of fractions (Vamvakoussi & Vosniadou, 2004). Instead, several researchers have advanced a dual-process account (Gillard, Van Dooren, Schaeken, & Verschaffel, 2009; Vamvakoussi et al., 2012). Dual-process theories (e.g., Kahneman, 2000) distinguish between a cognitive system that processes information quickly and intuitively (“System 1”), and another system that performs slower, analytical reasoning (“System 2”). Some researchers argue that in fraction comparison, people's faster and more correct responses to congruent than incongruent comparison problems may be an instance of dual processes. Whereas activation of magnitudes of natural numbers appears to be highly automatic and may thus be considered a System 1 process, activation of magnitudes of fractions is – in most cases – much more demanding, deliberate, and slower, and may therefore be considered a System 2 process. In congruent fraction comparison problems, quick comparison of natural number magnitudes (provided by System 1) will always yield a correct response. In contrast, in incongruent fraction comparison problems, such reasoning will always yield an incorrect response. In such problems, people will give responses that are inaccurate, unless they activate analytical reasoning (provided by System 2) before responding.

At face value, the dual-process account seems to be a suitable framework to explain why people are generally more accurate on congruent fraction comparison problems (failure of System 2 to prevail in incongruent problems) and also why people who are equally accurate on both congruent and incongruent problems take longer to solve incongruent problems (success of System 2 in the face of interference from System 1 processes). However, the evidence to date does not allow general conclusions as to whether the dual process account can fully explain the natural number bias, and, more fundamentally, whether the natural number bias is a general phenomenon in fraction comparison. Indeed, the strength of the reported bias varies substantially across studies, for reasons that are not well understood. The reported accuracy advantage for congruent over incongruent problems ranges from 4 percentage points (91% vs. 87%, 5th- and 7th-graders in Meert et al., 2010b) to 49 percentage points (85% vs. 36%, 5th-graders in Gómez et al., 2015), and the reported response time advantage ranges from 252 ms (2048 vs. 1796 ms, adult mathematicians in Obersteiner et al., 2013) to 333 ms (2316 vs. 2649 ms, 11th-graders in Van Hoof et al., 2013).

One limitation of previous research is that in most studies, the fractions in each pair had common numerators (e.g., 3/5 vs. 3/7) or common denominators (e.g., 2/5 vs. 4/5) (Meert et al., 2010b; Vamvakoussi et al., 2012; Van Hoof et al., 2013). In these special types of problems, there is no need to activate overall fraction magnitudes, because it is sufficient to compare the non-common components. Studies of accuracy and response times (Meert, Grégoire, & Noël, 2010a; Meert et al., 2010b; Obersteiner et al., 2013; Schneider & Siegler, 2010) as well as eye movements (Huber, Moeller, & Nuerk, 2014; Ischebeck, Weilharter, & Korner, 2016; Obersteiner & Tumpek, 2016) suggest that people focus primarily on the non-common components in such pairs. Thus, performance differences between congruent and incongruent problems may reflect simple component-based reasoning, rather than interference between component and holistic fraction magnitude processing.

To study the generality of the natural number bias, it is necessary to use more challenging fraction comparison problems—and in particular, those without common components—that require reasoning about fraction magnitudes. Only a few studies have used such problems (Barraza, Avaria, & Leiva, 2017; DeWolf & Vosniadou, 2011, 2015; Obersteiner et al., 2013), and their conclusions are ambiguous, for two reasons. First, most of these studies included some simple fractions with single-digit components (e.g., 1/2, 1/4, 3/4, or 2/3), which are more familiar than more complex fractions (e.g., 17/52) (Barraza et al., 2017; DeWolf & Vosniadou, 2011, 2015; Vamvakoussi et al., 2012). It is easier to activate magnitudes of familiar fractions than unfamiliar ones (Liu, 2017). Thus, the performance differences between congruent and incongruent problems may be due, in part, to an imperfect distribution of familiar fractions between the congruent and incongruent problem sets. Such a confound is hard to test empirically, however, because it is difficult to clearly define when a fraction is familiar.

A second, and more important, reason why past research on complex fraction comparisons does not allow firm conclusions is that studies have yielded conflicting results (for an overview, see Gómez & Dartnell, 2015): the typical bias (better performance on congruent problems) was observed in two different samples of university students (DeWolf & Vosniadou, 2011; 2015, subsample of US students in Experiment 12); no bias was observed in a sample of expert mathematicians (Obersteiner et al., 2013); and a reverse bias (better performance on incongruent problems) was observed in two different samples of university students (Barraza et al., 2017; DeWolf & Vosniadou, 2015, subsample of Greek students in Experiment 13). Although it is not entirely clear what caused the varying bias patterns, participants in these studies may have differed in their ability to activate holistic fraction magnitudes. Two factors that may have affected this ability are the specific types of comparison problems used and participants' mathematics experience. We elaborate on these two factors in the following sections, and we also discuss some other factors that could enhance people's ability to activate holistic fraction magnitudes: the availability of benchmarks, and brief instruction (a “tip”) about the use of benchmarks.

Solving complex fraction comparison problems quickly (i.e., within seconds) requires reasoning about holistic fraction magnitudes. However, problems differ in how exact the mental representation of these magnitudes needs to be to yield a reliable solution. Fraction pairs with small numerical distances between fractions require more precise representations than fraction pairs with larger distances. Because mental representations of numerical magnitudes are approximate rather than exact (e.g., Dehaene, Dupoux, & Mehler, 1990; Gallistel & Gelman, 1992), fraction comparison problems become increasingly difficult as the distance between the fractions decreases (e.g., Meert et al., 2010a; Obersteiner et al., 2013; Schneider & Siegler, 2010).

Variations in numerical distance also seem to affect the strength of the natural number bias in fraction comparison. Barraza et al. (2017) and DeWolf and Vosniadou (2015, Experiment 2) reported a stronger bias in accuracy with small distances (average distances in the two studies: 0.13 and 0.14, respectively) compared to larger distances (0.24 and 0.38, respectively).4 When people struggle with making decisions based on holistic fraction magnitude representations (which may greatly overlap when fraction magnitudes are close), they may rely more strongly on the fractions' natural number components, and therefore be more prone to the natural number bias.

Some additional problem features that may affect the strength of the natural number bias are the distances between the fractions and benchmarks, and the positions of the two fractions relative to a benchmark. Benchmarks are familiar numbers (e.g., 0, 1, 1/2) that people may use as reference points when reasoning about less familiar numbers. When a fraction is very close to a benchmark (e.g., 1/99 ≈ 0; 59/60 ≈ 1; 29/60 ≈ 1/2), it is easier for people to activate the fraction's approximate holistic magnitude (Liu, 2017). Previous studies have not systematically considered whether the occurrence and strength of the natural number bias depends on the fractions' magnitudes relative to benchmarks. However, some studies have sought to control the potential influence of benchmarks using carefully constructed problem sets. For example, DeWolf & Vosniadou, 2011, DeWolf and Vosniadou, 2015 distributed problems in which the two fractions were on the same or different sides of 1/2 equally across the congruent and incongruent problem sets. Yet, other potentially salient benchmarks such as 1/4 or 3/4 have not been considered.

Although previous research has not systematically addressed people's use of benchmarks in a fraction magnitude comparison task, there is plenty of evidence that people do use benchmarks in another numerical magnitude task, number line estimation (e.g., Hurst, Leigh Monahan, Heller, & Cordes, 2014; Peeters, Degrande, Ebersbach, Verschaffel, & Luwel, 2016; Peeters, Verschaffel, & Luwel, 2017; Sullivan, Lushasz, Slattery, & Barth, 2011). In this task, participants are asked to indicate the correct position of a number symbol on a number line (or, vice versa, to indicate which fraction corresponds to a given position). Peeters et al. (2017) found that when the mid-point and/or the quartiles of the line were marked, adult participants made extensive use of these benchmarks to calibrate their estimates. In a condition without such marks, the majority of participants also referred to mental benchmarks, predominantly the endpoints, mid-point, and quartiles of the line.

These studies focused on benchmarks in number line estimation with whole numbers; there is considerably less evidence regarding fractions. One relevant study compared fifth-graders' performance and strategy use on fraction number line estimation for number lines with and without marks at varying positions (Siegler & Thompson, 2014). The data suggested that the benefit of the marks depended on the specific fractions (particularly their denominators), and that participants spontaneously made use of midpoint and quartile strategies even in the condition without marks.

In fraction comparison, benchmarks could also be helpful when the two fractions that are to be compared are on opposite sides of a benchmark, rather than particularly close to a benchmark. For example, in the comparison 11/19 vs. 23/52, both fractions are fairly close to 1/2, so reasoning about their proximity to 1/2 is not helpful. However, considering that 11/19 is larger than 1/2 and 23/52 is smaller than 1/2 may be helpful because it allows the conclusion that 11/19 must be larger than 23/52. Thus, when two fractions “straddle” a benchmark (i.e., are on opposite sides of a benchmark), activating their approximate magnitudes relative to the benchmark is sufficient for solving the comparison problem. The assumed underlying mechanism is that a benchmark between two fractions supports people in distinguishing their approximate mental representations of the two fractions' magnitudes, which would otherwise overlap. This mechanism should allow people to solve fraction comparisons successfully with less precise mental representations of fraction magnitudes than would be necessary without benchmarks. Because these approximate representations are nevertheless holistic rather than componential, reasoning based on benchmarks should be less strongly affected by the natural number bias.

If benchmarks promote accurate fraction comparisons, it would suggest that people represent fraction magnitudes on a mental number line that is structurally similar to an external number line (Toomarian & Hubbard, 2018). There is limited evidence that people do use 1/2 as a benchmark in simple fraction comparisons (Clarke & Roche, 2009; Fazio, DeWolf, & Siegler, 2016); however, no study has systematically compared performance between problems that do and do not afford the use of different benchmarks in complex fraction comparisons.

Mathematics experience may also influence the strength of people's natural number bias. For example, Obersteiner et al. (2013) found that mathematicians relied on holistic fraction magnitudes to compare complex fractions, and they showed no natural number bias in these problems. These findings suggest that high mathematics experience may enable quicker access to fraction magnitudes. This in turn should reduce component-based reasoning and therefore reduce the natural number bias.

Less is known about how mathematics experience is related to the natural number bias in non-experts. In a longitudinal study with fourth- through sixth-graders, Rinne, Ye, and Jordan (2017) clustered students according to their performance on fraction comparison problems. Over time, students shifted from biased, component-based reasoning to holistic fraction magnitude reasoning. Interestingly, the direction of the natural number bias also changed over time: Many students shifted from an initial typical bias (larger components—larger fractions) to a reverse bias (smaller components—larger fraction) before shifting to unbiased magnitude-based reasoning about fractions. Both the typical and the reverse bias may be explained by children's overgeneralization of natural number knowledge to fraction problems: initially, they may rely on knowledge that larger number symbols represent larger numerical magnitudes, whereas later on, they may overgeneralize their knowledge that larger fraction denominators can make smaller fractions. This latter type of overgeneralization could also explain reports of a reverse natural number bias in adults, discussed above.

Mathematics experience may also influence the natural number bias, because more experience may enable people to use more sophisticated and adaptive strategies in fraction comparison. Research suggests that people use a variety of strategies in comparing fractions (Clarke & Roche, 2009; Faulkenberry & Pierce, 2011; Fazio et al., 2016; Pearn & Stephens, 2004), although systematic analyses of strategy use on complex fraction comparison problems are not yet available. For simpler problems, Fazio et al. (2016) found that university students had many strategies at their disposal, and they were flexible in using them appropriately. This held true particularly for students from a highly selective university, who reported high SAT-math scores, and less so for students from a community college, who reported lower SAT-math scores.

Fazio et al. also found that people with higher and lower mathematics ability differed, not only in their ability to use the most efficient strategy for any given problem, but also in their ability to recognize efficient strategies. Participants first solved comparison problems and were then confronted with alternative strategies that were either correct or incorrect. They were then asked whether they preferred the alternative over the strategy they had initially used. Participants who were more accurate were also more likely to switch strategies when the alternative strategy was more efficient, whereas participants who were less accurate rarely switched strategies, presumably because they did not realize that their initial strategy was suboptimal. Thus, people with higher and lower mathematics experience may also differ in their responsiveness to instruction about strategy use in fraction comparison.

From a mathematics education perspective, it is also important to understand the effects of encouraging people to use benchmarks. Benchmark strategies are generally considered effective, and they are often recommended in standards for mathematics education (e.g., Common Core State Standards Initiative, 2010). If the present study shows that relying on benchmarks helps people overcome the natural number bias (presumably through increased holistic reasoning about fraction magnitudes), it would support calls to emphasize benchmark reasoning in fraction instruction.

Many factors may influence the occurrence and strength of the natural number bias in fraction comparison, including mathematics experience and features of the problems. In this study, we investigate these factors in complex fraction comparison problems, using a controlled experimental design. The study had three specific aims.

Our first and primary aim was to investigate how well adult participants solve complex fraction comparison problems and to determine whether they demonstrate a natural number bias in accuracy and response times. We expected that adults would display a natural number bias, that is, significant differences in accuracy and response times between congruent and incongruent problems (Hypothesis 1a). In view of conflicting evidence in previous studies, however, we did not make a directional prediction. We were also interested in whether participants' performance varies with mathematics experience. We expected that adults would solve most problems correctly, so for this research question, response time rather than accuracy was the dependent variable of primary interest. Yet, we expected that people with higher mathematics experience would be both more accurate and faster than people with lower experience (Hypothesis 1b). We also expected to find a stronger natural number bias for people with lower mathematics experience compared to people with higher mathematics experience (Hypothesis 1c).

Our second aim was to explore whether the availability of benchmarks would influence the occurrence of the natural number bias. We considered quarters and halves as potential benchmarks in our study, in light of previous evidence that people use these benchmarks in simple symbolic fraction comparison tasks and in number line estimation tasks (see 1.2). Moreover, these fractions are frequent in everyday life. In addition to these fractions, we considered 0 and 1 as benchmarks because in our experiment, all fractions were bounded by these numbers, and because previous research suggests that people are particularly quick in recognizing that fractions are close to 0 or 1 (Obersteiner et al., 2013). We hypothesized that, overall, participants would be more accurate and faster on problems that afforded benchmark strategies (hereafter, benchmark problems) than on problems that did not afford benchmark strategies (Hypothesis 2a; for definitions of problem types, see 2.2). Because benchmarks should facilitate the use of holistic magnitude representations, we expected that the natural number bias would depend on problem type, with a reduced bias in benchmark problems compared to no-benchmark problems (Hypothesis 2b). Finally, we expected that participants with higher mathematics experience would benefit more from benchmarks than participants with lower mathematics experience, because the former participants should be better able to adapt their strategy use to the affordances of the tasks, making the “bias-reducing effect” of benchmarks stronger for participants with higher compared to lower mathematics experience (Hypothesis 2c).

An additional, third aim of our study was to assess whether providing minimal instructional support (i.e., an explicit “tip”) about the potential usefulness of benchmarks would lead to better performance and reduce the potential bias. We hypothesized that participants who received a tip would perform better (Hypothesis 3a) and would show a smaller bias (i.e., a smaller effect of congruency; Hypothesis 3b) than participants who did not receive the tip. Furthermore, we expected that these effects would be particularly strong for problems for which the tip about benchmarks is most helpful, that is, in benchmark problems. Accordingly, for participants who received the tip, compared to those who did not, we expected a stronger effect of benchmarks (i.e., a problem type × tip interaction; Hypothesis 3c) and a stronger effect of benchmarks on mitigating the natural number bias (i.e., a problem type × tip × congruency interaction; Hypothesis 3d). Finally, we expected that the effect of the tip would be stronger for participants with higher mathematics experience (Hypothesis 3e), because they would be better able to adopt the suggested benchmark strategies.

Section snippets

Participants

Participants were 107 students (48 male, 59 female; M = 19.98 years, SD = 1.66) at a Midwestern university in the United States. Participants were recruited via signs posted in the psychology building and through a participant pool associated with an Introduction to Psychology course. They received extra credit points or compensation of $10 for their participation. Participants were divided into two groups based on their self-reported mathematics course work: a lower-experience group (less than

Results

Table 1 displays the estimated marginal means (MEM) and standard errors (SE) of the GEE procedure. Overall, participants were fairly accurate (MEM = 85%, SE = 1.0) and fast (MEM = 3707 ms, SE = 157) in solving the problems. These values are comparable to previous fraction comparison studies, which is remarkable because the problems were much more difficult than those in most other studies. These results may reflect the relatively high mathematical skills of our participants (as seen in their

Discussion

One aim of this study was to investigate whether adults show a natural number bias in complex fraction magnitude comparison and if so, whether this bias would be moderated by mathematics experience. A second aim was to explore whether the availability of benchmarks would influence the occurrence of the natural number bias. Overall, we found a reverse bias (i.e., smaller components—larger fraction) for both accuracy and response times. This overall bias was moderated both by mathematics

Conclusion

In conclusion, this study highlights variability in the natural number bias in fraction comparison problems. Our findings suggest that the occurrence of the bias depends on problem type and on prior mathematics experience. Together with previous research, our study suggests that the fractions' natural number components can bias people in different directions (towards “larger components—larger fraction”, or the opposite), depending on their experience and on their available strategies. Moreover,

Acknowledgement

This work was supported by a Feodor Lynen Research Fellowship granted to Andreas Obersteiner by the Alexander von Humboldt Foundation, Germany.

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    Present address: Freiburg University of Education, Kunzenweg 21, 79117 Freiburg, Germany.

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