Adaptive processing of fractions — Evidence from eye-tracking
Introduction
Number magnitude representation is commonly considered as one of the most basic number representations (e.g., Dehaene, Piazza, Pinel, & Cohen, 2003). Over the last decades research on numerical representations has been extended to multi-digit integers (e.g. Dehaene et al., 1990, Korvorst and Damian, 2008, Nuerk and Willmes, 2005, Poltrock and Schwartz, 1984, Verguts and De Moor, 2005; for a recent review see Nuerk, Moeller, Klein, Willmes, & Fischer, 2011), but also the case of negative numbers (e.g. Fischer, 2003, Ganor-Stern et al., 2010, Shaki and Petrusic, 2005) and decimal numbers have been addressed (Desmet, Grégoire, & Mussolin, 2010). However, the magnitude representation of fractions has been investigated only recently.
Generally, a common fraction is composed of two natural numbers and a line in between, the vinculum. The numerical magnitude of fractions does not follow a linear relationship of the components. Therefore, neither the numerator nor the denominator provides reliable information about the size of a fraction. Instead the relation between numerator and denominator codes the magnitude of the fraction. For instance, consider the comparison of 4/7 and 5/6, the numerator of the first fraction is smaller than the numerator of the second fraction (i.e. 4 < 5), but the denominator of the first is larger (i.e. 7 < 6). In this case, the fraction with the larger numerator is numerically larger. But the fraction with the larger numerator can also be the smaller one (e.g., in 4/9 and 3/5 with 4 > 3, but 4/9 < 3/5). Especially children have problems in understanding this relationship when first learning the concept of fractions and initially rely on their knowledge about natural numbers reflecting the so-called whole number bias (Ni & Zhou, 2005).
Processing of fraction magnitude has been investigated primarily using magnitude comparison tasks in which participants have to decide, which one of two fractions is the numerically larger/smaller one. At least three different types of strategies comparing fraction magnitudes can be differentiated (see Faulkenberry & Pierce, 2011, for an overview):
First, one way to compare fractions is to compare only the magnitudes of the fraction components (i.e., numerator and denominator). Participants apply such a component-based comparison strategy, when it is easily applicable such as (i) when comparing the magnitude of a fraction to a fixed standard (e.g., 1/5, 0.2 and 1; Bonato, Fabbri, Umiltà, & Zorzi, 2007) and (ii) when comparing fractions with common denominators (e.g., 3/7 vs. 5/7; Meert, Grégoire, & Noël, 2009).
Second, the magnitude of fractions can be compared by considering the integrated overall representation of the fractions' magnitudes on a mental number line (e.g., Schneider & Siegler, 2010). Such holistic strategies are mostly used when participants have to compare fractions with common numerators or without common components (Meert et al., 2009, see also Schneider and Siegler, 2010, Siegler et al., 2011, Sprute and Temple, 2011) and are corroborated by recent fMRI data indicating populations of neurons specifically tuned to the overall magnitude of fractions (Ischebeck et al., 2009, Jacob and Nieder, 2009).
Third, Meert, Grégoire, and Noël (2010) found evidence suggesting that some fractions may also be processed in a hybrid way, combining both the above processing strategies. They suggested that even when participants compare fraction pairs holistically, component-based comparisons nevertheless influence the processing of fractions.
To summarize, the mode of fraction magnitude processing (i.e., holistic, componential, or hybrid) seems to be influenced by the fraction pair types involved in the task at hand. However, in this study we want to go beyond an examination of fraction processing for different fraction types. Instead, we hypothesize that even the same fraction type is not always processed in the same way. In the following we argue that variations in fraction processing may depend on facilitations or obstacles imposed by a particular experimental context.
Processing a fraction is a complex cognitive process. Therefore, it is not surprising that participants may try to adapt their strategy such that fraction processing becomes easier and less demanding, when it is possible. In fact, there are first indications from RT and error data that participants may adapt their comparison strategies, depending on the experimental context. For instance, in the studies of Meert et al., 2009, Meert et al., 2010 (see also Ganor-Stern, Karasik-Rivkin, & Tzelgov, 2011, for context effects in comparison of unit fractions) RT and error rates increased for fractions sharing either numerators or denominators when they were presented together with filler items in the same block. A possible explanation suggested by Meert et al. (2010) is that mixing different fraction pair types results in a hybrid processing style. In the no filler condition (only items with identical numerators or denominators) participants were able to identify the larger fraction by focusing on the magnitude of the fraction components. However, this was not a beneficial strategy after filler items have been added to the stimulus set. For the latter, processing and comparing the overall magnitudes of the two fractions seemed to be more beneficial. Thus, in the condition with filler items, participants seemed to process both the magnitude of the components and the overall magnitude of the fractions. Yet, while we agree that this is a viable interpretation of the data pattern, this account has not been tested systematically.
Evaluating this interpretation would be desirable, because the data pattern observed by Meert et al., 2009, Meert et al., 2010 may also be interpreted within the context of cognitive costs due to switching between different processing strategies (i.e., holistic vs. componential) depending on fraction pair type and experimental context (see Luwel, Schillemans, Onghena, & Verschaffel, 2009 for a similar interpretation). In the blocked presentation format, participants may choose to primarily rely on the processing of either the components or overall fraction magnitude, because the type of the next fraction is 100% predictable and informative as to whether there is a specific decision relevant component required or not. For instance, when the item set consists exclusively of fraction pairs with common denominators, participants can solve the task by exclusively focusing on the numerators, which in turn drives componential processing of this fraction type. On the other hand, when only fractions without common components are presented within one block, holistic processing of fraction magnitude is most beneficial. However, when fraction types are not presented in a blocked but mixed manner participants can no longer anticipate the most beneficial strategy a priori, but need to figure out whether there is a relevant component and if so to consider this component for the comparison process, whereas they have to consider the fractions' overall magnitudes, when there are no common components. So, in the case of mixed presentation of different fraction types, participants basically need to switch between the alternatives of componential and holistic processing on a trial by trial basis, which in turn prolongs RT. While this interpretation of general switching costs due to participants' adaptation to both fraction type and experimental context can well explain the results of Meert et al., 2009, Meert et al., 2010, it could not be tested directly in the studies of Meert et al., 2009, Meert et al., 2010 because filler items were not presented in a separate block. Therefore, this will be done in the current study to systematically evaluate influences of adaptation to fraction type and experimental context.
Recent studies usually used regression analyses on RT data to identify different processing strategies. However, how participants adapt to different experimental contexts might not be detected easily relying on RT data only. Therefore, we also recorded participants' eye fixation behaviour while engaged in a fraction magnitude comparison task, because eye-fixation location and fixation duration indicate, which part of a stimulus is processed at the moment, with processing duration being reflected by the time the eye fixated upon the respective part of the stimulus (e.g., Kennedy et al., 2004, Rayner and Pollatesk, 1989; see Brysbaert, 1995, Moeller et al., 2009, Moeller et al., 2011 for applications in numerical cognition research).
In number comparison tasks, analysing the number of fixations was informative about the processing strategies participants used to compare multi-digit numbers (e.g., Meyerhoff et al., 2012, Moeller et al., 2009). Similarly, an evaluation of participants' eye fixation behaviour should be informative as to the way their processing of fraction magnitude (i.e., holistic vs. componential) depends on (i) different fraction types as well as on (ii) the experimental context. As it possible to differentiate between the processing of numerator and denominator, evaluating participants eye-fixation behaviour provides more direct evidence on the differential processing of fraction components as can be achieved by overall performance measures such as RT and/or error rate. In case the decision is primarily based on processing of the magnitudes of the fraction components, the respective relevant component should be fixated preferentially (such as the numerator in fraction pairs with common denominators). Additionally, this type of participants' eye fixation behaviour should be most pronounced, when identification of the relevant components is corroborated by the experimental context, for instance by blocking items of the same type (e.g., a block of numerator relevant items only). Based on the above considerations the present study examined the processing of fraction magnitude by investigating the processing of different fraction pairs (i.e., same numerator, same denominator, and mixed pairs) under different blocking constraints (i.e., fully blocked, semi-blocked, and fully-random) with particular interest being paid to participants' eye fixation behaviour. Our corresponding hypotheses regarding participants' processing of different fraction types as well as adaptation to experimental context will be described in the following.
In previous studies overall distance was not matched between the different fraction pair type groups. By matching overall distance, our stimulus set allowed for an unbiased direct comparison of RT between the different fraction pair types. Based on the results of Meert et al. (2009), we expected componential processing to result in faster responses and fewer errors for same numerator and same denominator pairs than for mixed pairs (differing numerators and denominators), because comparing the magnitudes of one component should be sufficient to solve the task.
In addition to this categorical analysis, we also ran regression analyses as in previous studies (e.g., Meert et al., 2009, Meert et al., 2010) for each pair fraction type separately to compare our results to those of previous studies. When RT is predicted best by the distances between the components this would indicate componential processing of fraction magnitude. On the other hand, when RT is predicted best by overall numerical distance this would be consistent with the notion of holistic processing of fraction magnitude. In line with our hypothesis for the direct comparison of fraction pair type groups, we expected RT to be predicted best by componential distances for the fraction pair type groups with common components, whereas the comparison of fractions without common components should be predicted best by their overall distance.
We recorded participants' eye movements in addition to their reaction times. This allows us to examine and specify processing of both components of a fraction.
The general eye movement hypotheses are as follows: Mimicking the results of magnitude classification via manual button press, we hypothesized that the number of fixations should be highest for fraction pairs without common components, because these are most difficult and difficulty is usually associated with an increase in number of fixations (cf. Rayner & Pollatesk, 1989). However, more particularly, we expected that the overall fixation pattern is the result of a combination of a common fixation pattern for fraction processing in general and specific adaptations to this pattern depending on the respective fraction pair type.
First, the assumed general fixation pattern is depicted in Fig. 1A with more fixations on denominators than on numerators reflecting the more difficult processing of the inverse relationship between the magnitude of the denominator and that of the whole fraction. A typical observation in reading research is that the difficulty of processing a specific word influences eye fixation measures such as the distribution of fixations or reading times (e.g., Liversedge and Findlay, 2000, Rayner, 1998, Starr and Rayner, 2001). For instance, the number of fixations as well as their durations depends on word frequency, the predictability of a word from context or the number of different meanings a word conveys (see Rayner, 2009, for a review). Most importantly, however, participants' fixation behaviour is specifically influenced by the violation of semantic expectancies. In reading, prolonged fixation durations for words were observed when semantic expectancies, as derived from the prior read text, for instance, are violated and the resulting interference needs to be resolved (e.g., Morris, 1994; see also Rayner, 2009).
Interestingly, there is now also first evidence for similar effects of participants' fixation behaviour in number processing research. Moeller et al. (2009) (see also Huber, Mann, Nuerk, & Moeller, 2013) studied participants' eye fixations in a two-digit number magnitude comparison task. When comparing two-digit number pairs, number pairs can be either unit-decade compatible, in which case the tens and the unit digit of the overall larger number are larger than the tens and unit digit of the overall smaller number (e.g., 42_57, 4 < 5 and 2 < 7) or incompatible, in which case the overall larger number contains the smaller unit digit (e.g., 37_52, 3 < 5, but 7 > 2; see Nuerk, Weger, & Willmes, 2001). In unit-decade incompatible number pairs, this causes interference, which needs to be resolved by additional processing indicated by additional fixations or reading time. And indeed Moeller et al. (2009) observed that participants fixated the unit digits reliably more often when comparing unit-decade incompatible number pairs as compared to compatible pairs. Hence, because a separate comparison of the unit digits lead to a decision bias incompatible with the overall decision, processing incompatible numbers was not only generally more difficult but the interference was resolved by additional processing of the unit digits, as indicated by more fixations. Following this rationale of additionally required processing indicated by additional fixations, we expected a similar effect for the processing of denominators. The magnitude of a denominator is inversely related to the magnitude of the whole fraction, which obviously violates the standard numerical allocation of larger digit to larger magnitude. Thus, applying established findings from reading research and first evidence for number processing to the case of fraction processing, we expected more fixations on denominators than on numerators reflecting additional processing, because of the inverse relationship of denominator magnitude and overall magnitude of the fraction.
Second, we expected (i) that for fraction pairs with matching numerators the denominators should be fixated relatively more often, because the latter are of primary relevance for the magnitude comparison process. On the other hand, (ii) we hypothesized that numerators should be fixated more often for fraction pairs with identical denominators, again reflecting the particular relevance of the former for the respective magnitude comparison. Finally, (iii) for fractions without common components, we expected that both components should be fixated more often (see Fig. 1B for an illustration of these specific processing aspects).
Third, Fig. 1C shows the hypothetical additive combination of the general overall fixation pattern1 (Fig. 1A) and the respective specific fixation patterns (Fig. 1B). This would then result in (i) a stronger imbalance, favouring fixations on denominators for fraction pairs with common numerators, (ii) levelling of the proportion of fixations on denominators and numerator for fraction pairs with common denominators, and (iii) a balanced increase of fixations on denominators and numerators for fraction pairs without common components.
Moreover, in this study, we wanted to go beyond differentiation of processing styles for different fraction types and aimed at examining whether fraction processing is adaptive to the experimental context more directly. Therefore, we generated three conditions in that we varied stimulus presentation constraints. We expected participants to adapt their fraction processing styles in the three conditions.
- (i)
In a first fully blocked condition the three fraction types (same numerator, same denominator, and mixed pairs) were presented in a blocked design.
- (ii)
In a second semi-blocked condition we collapsed same numerator and same denominator pairs into one block and the mixed pairs in another block.
- (iii)
In a third random condition, all three fraction pair types were presented in randomized order within the same block.
In line with the results of Meert et al., 2009, Meert et al., 2010 we expected both RT and error rates to increase from fully blocked over semi-blocked to fully randomized presentation of the to-be-compared fractions. More specifically, we hypothesize that this increase should be more pronounced for fraction types with common components as compared to mixed pairs. Only the former should benefit from blocked presentation corroborating fast identification and preferential processing of the relevant components. Finally, we expected that adaptation for the case of fraction pairs with common components should result in a decrease of RT over the time course of the blocked condition.
With respect to participants' adaptation of their eye fixation behaviour to the experimental context the following hypotheses were pursued:
First, for the distribution of fixations across experimental contexts (i.e., blocking conditions) we expected that preferential fixating of the decisive digits should be most pronounced in the fully blocked condition, weaker in the semi-blocked condition and should diminish in the random condition.
Second, we hypothesized that participants not only adapted their general fixation behaviour (as indicated by the overall number of fixations), but also the location of their initial fixation on the to-be-compared fraction. As only in the fully blocked condition the relevant digits of the next trial can be anticipated a priori, we expected participants to preferentially fixate the relevant fraction components first in the fully blocked but not in the semi-blocked and random condition.
Third, in line with the prediction for RT we suggest that adaptation of participants' eye fixation behaviour not only takes place between blocks but also within blocks. Accordingly, this should be reflected in systematic modulation of participants' fixation behaviour over the time course of a blocking condition. When fraction processing is indeed adapted to experimental context preferential fixating of the decisive digits should be more pronounced towards the end of a condition. Two ways of adaption are possible: Participants could adapt to the experimental context either by fixating increasingly on relevant digits or by fixating decreasingly on irrelevant digits. We expected participants to adapt their eye fixation behaviour when comparing fraction pairs with common components in the blocked condition only.
Section snippets
Participants
A total of 36 students of the University of Tuebingen participated in the study (20 female, 16 male). Average age was 24.6 years with a standard deviation (SD) of 1.81 years (range: 22 to 30 years). All participants reported normal or corrected-to-normal vision.
Apparatus
Eye-fixation behaviour was recorded by an EyeLink 1000 tracking device (SR-Research, Kanata, Ontario, Canada). Following a 9 point calibration at the start of the experiment and drift corrections before each trial the spatial resolution of
Reaction times and errors
Using a 3 × 3 ANOVA, we tested whether participants' RT differed across fraction pair types and conditions. Indeed, we found a reliable main effect of pair type [F(2, 64) = 49.39, p < .001, η2p = .61, GG = .59]. NREL was processed fastest (M = 1679 ms), followed by DREL (M = 1941 ms) and MP (M = 2452 ms), with all three paired comparisons significant (ps < .001). Additionally, mean RT differed between conditions [F(2, 64) = 12.36, p < .001, η2p = .28]. Mean RT (M = 1833 ms) was shortest in the fully blocked condition as
Discussion
In this study we aimed at examining the processes underlying fraction magnitude processing. We paid particular attention to two research questions: How specifically do participants (i) process different fraction pair types and (ii) adapt to different experimental contexts. To address these two questions we not only considered participants RT and error data but also investigated their eye-fixation pattern. In the following the implications of the current results for each of the two research
Conclusions
The current eye movement study provides converging evidence that participants' processing of fractions is moderated by fraction pair type and experimental context. First, participants processed different fraction pair types differentially. By focusing on the relevant components participants were able to process fraction pairs with common components faster than fraction pairs without common components. Second, participants also adapted their processing of fractions to the experimental context.
Acknowledgements
This research was supported in part by a project within the ScienceCampus (WissenschaftsCampus) Tuebingen (Cluster 1/TP 1) supporting Stefan Huber. Moreover, part of this research was funded by the German Research Foundation (DFG) for a project within the Research Group FOR 738/2/TP02 supporting Korbinian Moeller. Moreover, we are grateful to Kathrin Festl, Lena Geiger, Julia Lichte and Andrea Schwager for their help in data acquisition.
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