A language compatibility effect in fraction processing

Introduction

Mathematics and language are fundamental symbol systems for human thinking, and the question of their relationship has been pursued using many different approaches (Dowker & Nuerk, 2016; Gelman & Butterworth, 2005). Evidence for their close association comes from psychometric studies finding significant shared variation in measures of mathematical and verbal achievement (LeFevre et al., 2010; Purpura & Logan, 2015). Neuroimaging studies have identified common neural correlates for mathematical and verbal processing (Dehaene et al., 2003). Co-morbidity has been documented for mathematical and language disabilities (Kovas et al., 2007). Language might be necessary for some mathematical abilities: speakers of languages with limited inventories of number words have difficulty reasoning exactly about large numbers (Gordon, 2004; Pica et al., 2004). Finally, congruence between how a language names numbers and how place value denotes them is associated with better numerical performance (Lewis et al., 2020; Miller et al., 1995; Nuerk et al., 2005).

Our approach to the relationship between these two symbol systems focuses on the match between what a language provides and what a mathematical task demands. Consider fraction expressions such as 3/7. Different languages serialise the numerator and denominator components in different orders. For example, English names the numerator first whereas Korean names the denominator first. Does the serialisation bias of a language affect how well its speakers perform different fraction tasks? Are English speakers advantaged when the numerator drives performance, and are Korean speakers advantaged when the denominator drives performance? Thus, we are interested in language compatibility effects, which are fundamentally about the interaction between language and mathematics (Contreras-Saavedra et al., 2020; Nuerk et al., 2001, 2005; Poncin et al., 2020).

To date, language compatibility effects have been shown for natural numbers. Prior research has focused primarily on simple effects of language on mathematical performance, where the speakers of one language show better conceptual understanding than those of another. For example, Chinese number words map more transparently to place-value expressions than English number words. This appears to confer an advantage when preschool children learn to count, especially in the teens which are highly irregular in English by comparison (Miller et al., 1995).

Simple effects of language do not just affect the mathematical development of children; they are also present in the mathematical processing of adults. A particularly striking finding is the inversion effect (Bahnmueller et al., 2018). To understand this effect, one must first understand the unit-decade compatibility effect (Nuerk et al., 2001), which is the finding that when judging which of a pair of two-digit numbers is greater, response times (RTs) are faster when comparison of the tens places and comparison of the ones places lead to the same judgement (e.g., 57 vs. 41, where both 5 > 4 and 7 > 1) versus when they lead to conflicting judgements (e.g., 52 vs. 39, where 5 > 3 but 2 < 9). This suggests that people process the ones places when making these judgements even though this is mathematically unnecessary—the judgements can be made solely on the basis of the tens places. Nuerk et al. (2005) investigated whether the unit-decade compatibility effect is larger among German speakers than English speakers because German names the ones place before the tens place when serialising place value expressions, putting additional focus on this task-irrelevant component, whereas English does not. ¹ The predicted inversion effect was found, and has since been extended to other languages (Moeller et al., 2015; Pixner et al., 2011) and linked to transcoding errors (Imbo et al., 2014; Van Rinsveld & Schiltz, 2016; Zuber et al., 2009) and arithmetic errors (Göbel et al., 2014; Lewis et al., 2020).

Here, we extend the investigation of language compatibility effects to a new number context, fractions. As noted above, English names the numerator before the denominator, for example, the fraction 3/7 is “three sevenths.” By contrast, Korean names these components in the reverse order “chil (7) bun-ul sam (3).” Prior studies have focused on whether this serialisation difference impacts conceptual understanding of fractions. Specifically, English leaves the part–whole relationship between numerators and denominators implicit, whereas Korean signals it explicitly with “bun-ul,” which roughly translates to “of parts.” This morpheme bun-ul originates from the Chinese character “分,” which roughly means “to divide.” Miura et al. (1999) found an advantage of explicitly naming this relationship: Korean children had better conceptual knowledge of fractions than their English-speaking counterparts. To control for cultural differences in the educational systems of the two countries, Paik and Mix (2003) trained English-speaking children to also explicitly name the relationship “of parts.” For example, they taught children to name ¼ as either “one of four parts” or “of four parts, one”. They found that this improved their conceptual knowledge scores. Note that the approach of Paik and Mix (2003) was purely verbal. To isolate the effect of language, Mix and Paik (2008) included a condition where fraction concepts were introduced with pictorial representations (e.g., pieces of pizza). In this condition, there was no difference in the conceptual understanding of the two language groups. This suggests that introducing explicit naming of part–whole relations may improve the verbal mapping process but may be irrelevant for conceptual understanding.

The current study extends beyond the search for a simple effect of language (e.g., whether Korean speakers have better conceptual understanding of fractions than English speakers) to look for a language compatibility effect, specifically an interaction between the order in which a language serialises the numerator and denominator components of fractions and the information processing demanded by a mathematical task. Two experiments tested the prediction that English speakers will perform better on tasks where the numerator drives performance, and Korean speakers will perform better on tasks where the denominator is critical. The focus on information processing, specifically RTs and error rates, is important for two reasons. First, it better aligns the study of language compatibility and fractions with studies of language compatibility and whole (and decimal) numbers as exemplified by inversion effects, which also utilise performance measures such as RTs and error rates (Bahnmueller et al., 2018; Göbel et al., 2014; Nuerk et al., 2005). Second, it has spurred the development of two new tasks that may be useful in future studies of language compatibility. These are the Fraction Span Task for measuring the memory consequences of language compatibility and the Fraction Identification Task for measuring the processing speed consequences.

In greater detail, the new tasks each have an English-compatible (EC) condition where the numerator drives task performance and a Korean-compatible (KC) condition where the denominator is critical for performance. They were administered to two groups, native English speakers and native Korean speakers. We looked for a crossover interaction of condition and group for evidence of the predicted language compatibility effect. The fraction span task was inspired by prior studies that have used the digit span task to look for simple effects of language on mathematical processing (e.g., Ellis & Hennelly, 1980). It required participants to encode fractions in working memory (WM) and rehearse them there. It is relevant for testing the verbal encoding hypothesis: that verbal encoding is necessary for the serialisation bias of a language to affect fraction processing. The fraction identification task was inspired by paradigms in the attention literature (e.g., Alvarez & Oliva, 2009; Most et al., 2001). It required participants to make speeded judgements about fractions. It is relevant for evaluating the attentional focus hypothesis, which is that the serialisation bias of a language affects which fraction component is attended first.

Experiment 1

Results

We analysed the data for each task using a mixed repeated measures ANOVA with between-subjects factor language group (i.e., English speakers, Korean speakers) and within-subjects factor compatibility (i.e., EC, KC). For both tasks, we predicted an interaction signalling a compatibility effect. In particular, we predicted that English speakers would perform better in the EC versus KC condition and that Korean speakers would show the reverse pattern.

For the fraction span task, there was a main effect of language group, F(1, 63) = 14.13, p < .001, η _p ² = .18, with English speakers making more errors (M = 2.27, SD = 1.79) than Korean speakers (M = 1.05, SD = 1.34). There was also a main effect of compatibility, F(1, 63) = 4.75, p = .03, η _p ² = .07, with participants making fewer errors in the EC condition (M = 1.46, SD = 1.66) than the KC condition (M = 1.89, SD = 1.72). As predicted, the group × compatibility interaction was significant, F(1, 63) = 14.39, p < .001, η _p ² = .19, indicating a language compatibility effect; see Figure 1. This was driven by the English speakers, who made fewer errors in the EC versus KC condition, t(32) = 3.91, p < .01, d = 0.68. The Korean speakers’ showed the reverse pattern, making fewer errors in the KC versus EC condition, but only descriptively: this difference did not approach statistical significance (p = .21).

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Figure 1.

The language group × compatibility interaction for the fraction span task in Experiment 1.

For the fraction identification task, the average accuracy was 98.50%. Accuracy was high in both the EC (98.60%) and KC (98.51%) conditions, and whether the correct response was same (98.44%) or different (98.56%). We therefore focused on the RT data. We trimmed RTs on trials where participants were incorrect or that were outside the interval [200, 1,500] ms, excluding 3.40% of the data. There was a main effect of language group, F(1, 63) = 4.05, p = .048, η _p ² = .06, indicating that English speakers were faster than Korean speakers. There was no main effect of compatibility (p = .79). Critically, there was no language compatibility effect: the predicted group × compatibility interaction did not approach statistical significance, F(1, 63) = 0.07, p = .79; see Figure 2.

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Figure 2.

The language group × compatibility interaction for the fraction identification task in Experiment 1.

Experiment 2

Experiment 2 had two goals. The first was to replicate the language compatibility effect found in Experiment 1 for the fraction span task, and also to strengthen it: although the predicted interaction was found, only the simple effect for English speakers reached statistical significance. To address this goal, we recruited Korean speakers from a university in South Korea. This minimised their possible exposure to English naming of fractions. The second goal was to redesign the fraction identification task to evaluate the necessity of verbal encoding of fractions for observing a language compatibility effect. To address this goal, we added a second delay interval of 5,000 ms to this task. The reasoning was that at the longer delay, it would be difficult to maintain the first fraction in visuospatial WM, and participants would be more likely to encode the first fraction in verbal WM and rehearse it while awaiting the second fraction, and that this would produce the predicted language × compatibility interaction.

Results

For the fraction span task, a mixed repeated measures ANOVA on the mean number of component recall errors revealed a main effect of language group, F(1, 68) = 10.75, p < .01, η _p ² = .14, with English speakers making more errors (M = 2.11, SD = 2.26) than Korean speakers (M = 1.09, SD = 1.90). There was no overall difference between the EC and KC conditions (p = .20). Critically, the predicted language group × compatibility interaction was significant, F(1, 68) = 12.46, p < .01, η _p ² = 0.16, again indicating a language compatibility effect; see Figure 3. The simple effects showed the same pattern as in Experiment 1. As predicted, English speakers made fewer errors in the EC versus KC condition, t(33) = 3.13, p < .01, d = 0.42. Korean speakers made fewer errors in KC versus EC condition, although this descriptive difference again failed to reach statistical significance (p = .10).

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Figure 3.

The language group × compatibility interaction for the fraction span task in Experiment 2.

For the fraction identification task, the average accuracy was again high (97.56%). This was true in both the EC (98.25%) and KC (97.42%) conditions, and whether the correct response was same (97.37%) or different (97.90%). We again focused on the RT data. As in Experiment 1, we trimmed RTs on incorrect trials and those outside the interval [200, 1,500] ms, resulting in the exclusion of 5.0% of the data. We analysed the data for the 500 and 5,000 ms delays in separate mixed repeated measures ANOVAs. At the 500-ms delay, there was no main effect of language group (p = .75) or compatibility (p = .16). Critically, the language group × compatibility interaction again failed to reach statistical significance, F(1, 68) = 2.89, p = .09, η _p ² = .04; see Figure 4a. For completeness, we confirmed that neither the simple effect for the English speakers (p = .07) nor for the Korean speakers (p = .83) reached statistical significance. The lack of a language compatibility effect at the 500-ms delay duration parallels the findings of Experiment 1, and is again inconsistent with the attentional focus hypothesis.

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Figure 4.

The language group × compatibility interaction for the fraction identification task in Experiment 2.

As predicted, the results were different at the 5,000-ms delay. There was no main effect of language group (p = .46) or compatibility (p = 1.00). However, the language group × compatibility interaction was statistically significant: F(1, 68) = 9.89, p < .01, η _p ² = .13; see Figure 4b. English speakers were faster when the fractions differed in the numerator versus denominator component, t(33) = 1.86, p = .07, d = 0.10, although this difference did not reach statistical significance. By contrast, Korean speakers were significantly faster when the fractions differed in the denominator versus numerator component, t(35) = 2.46, p = .02, d = 0.13. The finding of a language compatibility effect at the 5,000 ms delay duration supports the verbal encoding hypothesis.

General discussion

Languages differ in the order in which they name the components of structured mathematical expressions. For the case of rational numbers expressed as fractions, English names the numerator component first whereas Korean names the denominator component first. We predicted a language compatibility effect—that performance will be best when a speaker’s language privileges naming of the component that is most relevant for task performance (Moeller et al., 2015; Nuerk et al., 2005; Pixner et al., 2011; Van Rinsveld & Schiltz, 2016). We tested this prediction in a study where English and Korean speakers performed tasks that included EC and KC conditions. The predicted interaction was found for the fraction span task. The simple effect was statistically significant for the English speakers—they made fewer errors when the denominator of the previously rehearsed fraction was the same as the numerator of the next-rehearsed fraction. The reverse pattern was only a descriptive trend for the Korean speakers, a failure we return to below when discussing the limitations of the current study. We take the finding of a language compatibility effect for this task in both experiments as evidence for the verbal encoding hypothesis.

The predicted interaction was also found for the fraction identification task, but only at the longer 5,000 ms delay duration utilised in Experiment 2. Here, it was the simple effect for Korean speakers that was statistically significant—they were faster to notice when the denominators of the two fractions differed than when the numerators differed. By contrast, the simple effect for English speakers was only a descriptive trend, a failure we also discuss further below. Critically, no interaction was found in either experiment at the shorter 500 ms delay. We take this more complex pattern of findings as evidence against the attentional focus hypothesis and in support of the verbal encoding hypothesis.

We have been careful to avoid framing our investigation of language compatibility effects in terms of linguistic relativity or the Whorfian hypothesis (e.g., Brysbaert et al., 1998). Our interest is in how the overall cognitive system performs mathematical tasks, not in the question of whether speakers of different languages are led to fundamentally different conceptualisations of mathematics. For this reason, we have described our tasks as requiring fraction processing, not fraction understanding, and have focused on how the overall cognitive systems accomplishes this processing. Our findings support the verbal encoding hypothesis, which relies on a domain-general component of this system, verbal WM, to explain the language compatibility effects observed here. This hypothesis is consistent with prior studies that have found that verbal mapping processes (Mix & Paik, 2008) and recruitment of verbal WM (Göbel et al., 2014) are important for finding language compatibility effects. However, we grant that these processes and memory systems are peripheral from the perspective of linguistic relativity (Brysbaert et al., 1998).

South Korea is one of the highest achieving countries in mathematics in international comparisons, and consistently performs much better than the United States. Prior studies have documented superior performance among Korean students versus American students (Fuson & Kwon, 1992; Song & Ginsburg, 1987). However, the relevance of this fact for our study is unclear. In both experiments, there was a main effect of language group on the fraction span task, with English participants making more errors than Korean participants. However, in Experiment 1, the effect of language group on the fraction identification task was significant and went in the opposite direction, with English participants faster than Korean participants. More generally, we expected that there might be overall differences in the performance of the two language groups, and therefore designed our tasks to look for cross-over interactions of language group and language compatibility condition. Said differently, we looked for one pattern among the English participants (i.e., better performance in the EC condition than the KC condition) and the opposite pattern among the Koran participants (i.e., worse performance in the EC condition than the KC condition). Thus, our theoretical predictions and experimental task designs are independent of any mean differences in the performance of the two language groups.

That said, two findings from the fraction span task deserve further consideration. ² The first is that the English speakers made more errors on this task in both experiments than the Korean speakers. This was true on average and also true at the individual level: only 6% of the English speakers showed perfect performance in Experiment 1 compared with 25% of the Korean speakers. (The percentages were 3% and 11%, respectively, in Experiment 2.) One reason for this may be that fraction naming in English is more irregular than in Korean. For example, English uses the cardinal form to name the numerator of a fraction but the pluralized ordinal form to name the denominator (e.g., “two-sevenths”). Moreover, it offers highly irregular names for benchmark unit fractions such as “half” and “quarter.” These irregularities might enforce an additional processing load compared with Korean, which names both components using the same cardinal form (e.g., “two parts of seven”). The second finding is that participants made fewer errors in the EC condition than the KC condition in Experiment 1; however, no main effect of compatibility was found in Experiment 2. This pattern might be explained by the relative English exposure of the Korean speaking samples of the two experiments. All of the Korean speakers in Experiment 1 were undergraduates at a U.S. university. Although they had been educated in Korea through secondary school, and had therefore received their fractions instruction in Korean, they might have had enough exposure to English naming of fractions in college to be able to also benefit from the structure of the EC condition. By contrast, the Korean speakers in Experiment 2 were undergraduates at a Korean university, and presumably had little or no exposure to English naming of fractions. This would be consistent with the disappearance of the main effect of compatibility in Experiment 2.

A goal for future research is to more directly test the verbal encoding hypothesis. Following Lee and Kang (2002), participants could complete the fraction span task and the fraction identification task while maintaining a concurrent load. In one condition, the load could be verbal, for example, repeating a sequence of pronounceable nonwords. In the other condition, it could be visuospatial, for example, remembering the identity of an abstract shape and its location in one of the four quadrants of space. The verbal encoding hypothesis predicts that the language compatibility effects observed in the current study would be eliminated in the verbal load condition. This is because participants would be forced to use non-verbal strategies to perform the fraction span and fraction identification tasks, and therefore the fraction naming differences of English and Korean would have not affect their performance. By contrast, the language compatibility effects should replicate in the visuospatial load condition. ³

A contribution of the current study is the development of two theoretically motivated tasks for studying fraction processing. The fraction span task is a natural extension of the digit span task which has been used by mathematical cognition researchers to study language effects on natural number processing (Ellis & Hennelly, 1980; Murray & Jones, 2002; Naveh-Benjamin & Ayres, 1986; Stigler et al., 1986). This task was used here to investigate the predicted language compatibility effect for fraction processing. That it successfully elicited this effect is in fact surprising. Participants could have chosen to treat the stimuli not as four fractions, but as eight natural number components. This componential strategy would have even been supported by the natural number bias that has been documented for fraction understanding (Ni & Zhou, 2005). This strategy would have eliminated any language compatibility effect. Yet our participants showed this effect, and thus must have chosen to treat the stimuli as fractions. It is an interesting question for future research whether this language compatibility effect, found here for adults, is also present for children who are just learning about fractions.

The fraction identification task is also new in the literature, and it may open up the study of new research questions regarding fraction processing. For example, consider the question of how mathematical, linguistic, and visual features trade off during fraction processing. Can a language’s bias against a fraction component (e.g., for English, the naming of the denominator second) be offset by increasing the visual salience of that component (e.g., by rendering the denominator in a font size 50% larger than that of the numerator)? If so, this might have implications for differentiating the instructional materials for teaching children about fractions based on the serialisation bias of their native language.

That said, the fraction span and the fraction identification tasks are relatively novel, and further work is required to refine them. Although the current experiments found the predicted language group × compatibility interactions, the simple effects were not always statistically significant. Experiment 2 increased the number of sequences in the fraction span task by 50% compared with Experiment 1, and the number of trials in the fraction identification task as well, yet some of the simple effects remained unreliable. Future studies should further increase the number of stimuli in the EC and KC conditions. Perhaps not surprisingly, cross-language differences in fraction processing may be relatively small in size, and therefore tricky to reliably measure.

Supplemental Material