Abstract
This study probed the cognitive mechanisms that underlie order processing for number symbols, specifically the extent to which the direction and format in which number symbols are presented influence the processing of numerical order, as well as the extent to which the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals or generalisable to other notational formats. Seventy adults who were bilingual in English and Chinese completed a Numerical Ordinality Task, using number sequences of various directional conditions (i.e., ascending, descending, mixed) and notational formats (i.e., Arabic numerals, English number words, and Chinese number words). Order processing was found to occur for ascending and descending number sequences (i.e., ordered but not non-ordered trials), with the overall pattern of data supporting the theoretical perspective that the strength and closeness of associations between items in the number sequence could underlie numerical order processing. However, order processing was found to be independent of the notational format in which the numerical stimuli were presented, suggesting that the psychological representations and processes associated with numerical order are abstract across different formats of number symbols. In addition, a relationship between the processing speed for numerical order judgements and mathematical achievement was observed for Arabic numerals and Chinese number words, and to a weaker extent, English number words. Together, our findings have started to uncover the cognitive mechanisms that could underlie order processing for different formats of number symbols, and raise new questions about the generalisability of these findings to other notational formats.
Keywords
In the world around us, number symbols occur in a variety of notational formats, including Arabic numerals (“4”) and written number words in different languages (e.g., “four, “cuatro”). Regardless of the format in which number symbols are presented, they convey a sense of magnitude, which is the quantity of elements in a set (e.g., Gilmore et al., 2018; Piazza, 2010), and order, which is the sequential position or rank of a number in relation to other numbers (e.g., Goffin & Ansari, 2016; Jacob & Nieder, 2008; Lyons & Beilock, 2011; Lyons et al., 2014). Furthermore, how we process magnitude and order information for number symbols has been found to be related to more complex mathematical skills (e.g., De Smedt et al., 2013; Holloway & Ansari, 2009; Lyons et al., 2016; Merkley & Ansari, 2016; Morsanyi et al., 2017; Sasanguie et al., 2012; Schneider et al., 2016). For example, there is behavioural evidence that order processing of Arabic numerals is significantly related to mathematics achievement in adults (e.g., Goffin & Ansari, 2016; Lyons & Beilock, 2011; Morsanyi et al., 2017; Sasanguie, Lyons, et al., 2017; Vogel et al., 2017; Vos et al., 2017) and children (e.g., Attout & Majerus, 2018; Lyons et al., 2014).
Since the literature has consistently reported a relationship between the processing of numerical order and mathematical achievement, it is important to investigate the cognitive mechanisms that underlie order processing for number symbols. This includes the influence of the direction of the number sequence (i.e., the extent to which the processes involved in making a judgement about numerical order are similar or different for ascending, descending, and mixed direction number sequences) and the notational format of the numerical stimuli (e.g., the extent to which the processes involved in making a judgement about numerical order are similar or different for Arabic numerals and number words) on order processing. It is also important to investigate whether the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals, or is generalisable to other notational formats, such as written number words. Addressing these issues will contribute to theoretical perspectives on the mechanisms that underlie numerical order processing, as well as contemporary models of numerical cognition, which generally address the representations and processes associated with numerical magnitude, rather than numerical order.
Assessment of numerical ordinality
The Numerical Ordinality Task is commonly used to assess the processing of numerical order. Participants are presented with two or three numerical stimuli and make a button press response to indicate whether the numbers are in order (e.g., 2 3 4) or not (e.g., 3 2 4). Where three numbers are presented at a time, numerical distance is defined as the difference between the maximum and median number, which is equal to the difference between the median and minimum numbers (e.g., 2 3 4, Distance = 1). Typically, participants are faster and more accurate when discriminating between ordered number sequences with a smaller numerical distance between them (e.g., 5 6 7, Distance = 1), than when judging between ordered number sequences of a larger numerical distance (e.g., 1 4 7, Distance = 3). This is termed the reversed distance effect (RDE).
The RDE is in contrast to the numerical distance effect (NDE), which is the finding in tasks of magnitude processing that participants tend to respond more quickly and more accurately when discriminating between the quantities of target numbers that have a larger numerical distance between them (e.g., 2 and 9, Distance = 7) than when judging between target numbers with a smaller numerical distance (e.g., 5 and 6, Distance = 1) (e.g., Goldfarb et al., 2011; Moyer & Landauer, 1967; Sekuler & Mierkiewicz, 1977). While the NDE is taken to be an indicator for the processing of numerical magnitude, the RDE is commonly employed as an indicator for the processing of numerical order (e.g., Goffin & Ansari, 2016; Lyons & Beilock, 2013; Rubinsten & Sury, 2011; Turconi et al., 2006). In addition, in line with dominant interpretations of the size of the NDE (cf. De Smedt et al., 2013; De Smed, & Gilmore, 2011), it is possible that the size of the RDE may index the precision of an individual’s representations of numerical order. This interpretation is supported by findings that the size of the RDE is negatively correlated with mathematical achievement (Goffin & Ansari, 2016).
The existing body of literature has suggested three possible mechanisms for order processing: serial scanning, strength of associations, and numerical counting. According to the serial scanning perspective, when participants are presented with ordered number sequences, they access the mental number line and scan through the items one by one to locate the numbers that match what is presented in the task (Franklin et al., 2009; Turconi et al., 2006; Vos et al., 2017). Ordered number sequences with a smaller numerical distance between them have fewer items for participants to scan through mentally than those with a larger numerical distance. For example, the sequence 2 3 4 (Distance = 1) has three items whereas the sequence 2 5 8 (Distance = 3) has seven items. This is said to give rise to faster reaction time (RT) and higher accuracy for ordered number sequences with a smaller numerical distance between them than those with a larger distance, which is the RDE (Caplan, 2015).
The strength of associations perspective is a frequency-based account in which it is proposed that there are stronger bidirectional associations between adjacent items in a well-learnt ordered list of items, and weaker associations between non-adjacent items (Lewandowsky & Murdock, 1989; Lovelace & Snodgrass, 1971; Vos et al., 2017 also see LeFevre & Bisanz, 1986). It is argued that ordered number sequences with a smaller numerical distance between them co-occur more often in the environment, have stronger inter-item associations, and are more easily retrieved than ordered number sequences with a larger numerical distance (Franklin et al., 2009; Vos et al., 2017). Hence, ordered number sequences with a smaller numerical distance are said to yield faster RT & higher accuracy than those with a larger numerical distance, giving rise to the RDE (Lyons & Beilock, 2013).
According to the numerical counting account, ordered number sequences with adjacent numbers (i.e., Numerical Distance = 1) are part of the counting sequence and are more frequently rehearsed than ordered number sequences with non-adjacent numbers (i.e., Numerical Distance > 1) (e.g., Lyons et al., 2016). Similar to the strength of associations perspective, it is said that the greater familiarity and rehearsal associated with ordered number sequences with adjacent numbers leads to faster RT & higher accuracy than ordered number sequences with non-adjacent numbers, thereby giving rise to the RDE (Lyons et al., 2016). All three perspectives above predict that the RDE will occur in tasks of numerical ordinality. Therefore, the RDE continues to be widely used as an indicator of numerical order processing (e.g., Goffin & Ansari, 2016; Lyons & Beilock, 2013; Orrantia et al., 2019).
Influence of direction on order processing for number symbols
To what extent does the direction of the number sequence influence order processing? Typically, three types of directional conditions are employed: ascending trials (e.g., 2 3 4), descending trials (e.g., 4 3 2), and mixed direction trials (e.g., 3 2 4). When investigating the specific directional conditions under which order processing occurs, studies do not yield consistent findings. For example, using ascending trials, some studies report the RDE (e.g., Franklin et al., 2009; Goffin & Ansari, 2016), whereas others report the NDE (e.g., Sasanguie, De Smedt & Reynvoet, 2017). Using descending trials, Franklin et al. (2009) reported the RDE.
It is possible to further examine whether there are systematic differences in the RDE depending on the direction in which the numerical stimuli are presented. Statistically, this is referred to as an interaction between Direction and Numerical Distance. This issue has implications on the extent to which the mental representations and processes associated with numerical order are dependent on or independent of the direction of the number sequence. Some behavioural studies report a significant interaction between Direction and Numerical Distance. For example, Lyons and Beilock (2013) and Vos et al. (2017) presented participants with three Arabic numerals at a time and instructed them to indicate whether the numbers were in order or not. They found a significant two-way interaction between Direction and Distance, with the RDE for ascending and descending trials, and the NDE for mixed direction trials. Morsanyi et al. (2017) found a significant two-way interaction between Direction and Distance, with no distance effect for ascending trials, and the NDE for mixed direction trials. Turconi et al. (2006) found a significant two-way interaction between Direction and Distance, with the RDE for ascending trials, and the NDE for descending trials. Together, these studies consistently report a significant interaction between Direction and Numerical Distance, which suggests that the processing of numerical order is dependent on the direction in which the stimuli are presented. However, these studies do not concur on the type of distance effect that is associated with each directional condition; hence this issue requires further investigation.
Vos et al. (2017) further proposed that investigating the influence of direction on order processing can be used to disentangle the different theoretical mechanisms that are said to underlie numerical order processing. Specifically, the strength of associations perspective would predict that ascending number sequences co-occur more often in our daily environments than descending sequences, resulting in stronger inter-item associations and hence, larger RDEs and faster RTs for ascending trials than for descending trials (Vos et al., 2017). However, the serial scanning and numerical counting perspectives would predict that the size of the RDE for descending trials would be the same or larger than in ascending trials. Specifically, if the time taken to scan or count the items is the same for ascending and descending sequences, there should not be a significant difference in the size of the RDE or average RTs between both trial types. However, if a longer time is taken to scan or count descending than ascending number sequences, then the size of the RDE should be significantly larger for descending than ascending trials and RTs should be significantly longer for descending than ascending trials (Vos et al., 2017).
Therefore, investigating the influence of direction on the processing of numerical order is an important issue that will help to clarify (1) the directional conditions under which numerical order processing occurs, (2) the extent to which the representations and processes associated with numerical order are dependent on or independent of the direction in which the numerical stimuli are presented, and to (3) disentangle the different theoretical mechanisms that are said to underlie numerical order processing.
Influence of format for order processing for number symbols
When judging whether a sequence of numbers are in order or not, are the representations and processes that are associated with order judgements for “one three five” similar or different to those for “1 3 5”? To address the influence of format on order processing for number symbols, it is possible to examine whether there are systematic differences in the RDE depending on the notational format in which the numerical stimuli are presented. Statistically, this can be referred to the presence or absence of an interaction between Format and Numerical Distance in tasks that assess numerical order. This issue has implications on whether the mental representations and processes associated with numerical order are identical, regardless of whether the number symbols are represented as Arabic numerals, number words, or any other notational format (i.e., format independent), or whether the mental representations and processes associated with numerical order differ according to the notational format in which the numerical stimuli are presented in (i.e., format dependent) (e.g., Barth et al., 2003; Campbell & Epp, 2004; Cohen et al., 2019; Cohen Kadosh et al., 2007; Dowker & Nuerk, 2016).
The broader debate on the extent to which numerical cognition is format dependent or format independent has been a long-standing issue (Cohen Kadosh & Walsh, 2009). Researchers have developed theoretical models to better understand the role of notational format in numerical cognition. Some models support a format independent view of numerical cognition, where the representations and processes associated with numerical magnitude are said to be abstract and shared across notational formats. These include the Abstract Code Model (McCloskey, 1992), the Multiroute Model of Number Processing (Cipolotti & Butterworth, 1995), and the Triple Code Model (Dehaene & Cohen, 1995; Dehaene et al., 1998, 2003). Other models support a format dependent view of numerical cognition, where the representations and processes associated with numerical magnitude are said to be non-abstract and linked to specific classes of expressions of quantity (e.g., Cohen, 2010; Cohen et al., 2019). These include the Encoding Complex Model of Number Processing (Campbell & Epp, 2004), the Preferred Entry Code Model (Noel & Seron, 1993), and the Multiple Quantity Representation Model (Cohen et al., 2002, 2019). Finally, the Dual Code Model (Cohen Kadosh & Walsh, 2009) advocates for the mixed perspective, where the representations and processes associated with numerical cognition are said to be both format independent and format dependent, depending on the type of stimuli employed and/or the task conditions used. These theories on the role of notational format on numerical cognition generally address magnitude processing, but have not directly addressed the role of format on numerical ordinality. Therefore, when participants are asked to make judgements about whether a string of numerals are in numerical order or not, it is unclear whether the representations and processes that are used in order judgements are abstract and linked across different notational formats, or whether there are multiple, non-abstract representations and processes for the different formats in which the stimuli are presented. Thus, it is important to directly examine the role of format on numerical order judgements so as to substantiate the above models.
Existing behavioural studies have examined the influence of notational format on the processing of numerical magnitude by investigating the interaction between Format and Numerical Distance. It is argued that, when two factors are independent of each other, manipulating one of them will result in an additive effect, or no overall effect; however, when two factors depend on each other, manipulating one of them would result in an interaction (Cohen Kadosh & Walsh, 2009; Sternberg, 1969). Thus, a significant interaction between Format and Numerical Distance in tasks of numerical magnitude processing is taken as support for the idea that numerical magnitude processing is dependent on format (e.g., Campbell et al., 1999; Cao et al., 2010; Cohen Kadosh, 2008; Dehaene & Akhavein, 1995; Lukas et al., 2014 for reviews, see Cohen Kadosh & Walsh, 2009; Göbel, 2018; Wong et al., 2018), while the lack of an interaction between Format and Numerical Distance is often taken as evidence for the format independent account (e.g., Ganor-Stern & Tzelgov, 2008; Ito & Hatta, 2003, Experiment 2; Pinel et al., 1999, 2001).
While existing studies present a mixed background of findings on the influence of notational format on the processing of numerical magnitude, it is not possible to assume that the influence of format on magnitude processing is the same as that for order processing. This is because the representations and processes associated with magnitude processing have been found to be distinct from those of order processing (e.g., Goffin & Ansari, 2016; Orrantia et al., 2019). Therefore, it cannot be assumed that the factors which influence magnitude processing will influence also order processing in the same way. For example, since magnitude and order processing have been shown to be different, they may have different susceptibilities with respect to the format in which the stimuli is presented. Thus, to obtain a holistic understanding of the influence of format on symbolic numerical cognition, it is important to assess the influence of format in tasks designed to measure the processing of numerical order.
To date, only one study has investigated the influence of notational format on the processing of numerical order. Using a Numerical Ordinality Task, Lyons and Beilock (2013) presented participants with Arabic numerals or dot arrays. Among ordered trials, the RDE was found for Arabic numerals while the NDE was found for dot arrays, suggesting that the processing of numerical order is format dependent. However, there have not yet been any studies or existing literature that explores the similarities or differences in order processing among different formats of number symbols (e.g., languages that are based on alphabetic vs. logographic scripts). Investigating this issue will clarify the extent to which the processing of numerical order is dependent on or independent of the format of the number symbols.
In addition, existing studies have not directly investigated whether Direction and Format interact to influence order processing for number symbols. Although Lyons and Beilock (2013) tested both Direction and Format in their study, they did not report whether or not the factors interacted with each other. This is an important gap to address because it could mean that there is an additional factor (Direction × Format) that influences the processing of numerical order, but that remains unaccounted for. For example, since numbers are commonly presented in an ascending direction, but less commonly in a descending or mixed direction, it is possible that the influence of format on order processing of number symbols may differ depending on the direction in which the numbers are presented. Alternatively, it is possible that the influence of format could be consistent regardless of the direction of the trials. Therefore, examining whether or not Direction and Format interact to influence order processing will clarify the cognitive mechanisms that underlie the processing of numerical order for number symbols.
Furthermore, it is noteworthy that existing behavioural studies have relied solely on frequentist analyses to infer format independence based on a non-significant interaction between Format and Numerical Distance. However, it may not be appropriate to do so. In recent years, statisticians have emphasised that frequentist statistics can only be used to reject the null hypothesis on the basis of statistically significant results, and that non-significant results cannot be used to infer support for the null hypothesis (Harms & Lakens, 2018; Lakens et al., 2020). To draw inferences based on null findings, it is argued that Bayesian hypothesis testing may be more appropriate (Harms & Lakens, 2018; Lakens et al., 2020). This is because a Bayesian hypothesis testing approach allows researchers to quantify evidence in favour of the null and alternative hypotheses by comparing the probability of the data given one hypothesis against the probability of the data given a second hypothesis (e.g., Hawes et al., 2019; Hutchison et al., 2019; Ortega & Navarrete, 2017; Wagenmakers et al., 2008). Such an approach would allow for an assessment of the strength of support for the format independent and format dependent accounts of numerical processing.
In summary, investigating the influence of format on the processing of numerical order will help to clarify the extent to which the representations and processes associated with numerical order are dependent on or independent of the notational format in which the numerical stimuli are presented, and this will have theoretical implications for existing models of numerical cognition. It may also be useful to examine whether the factors of Direction and Format interact to influence the processing of numerical order. Methodologically, it is important that experimental studies rely not just on frequentist statistics, but also to directly weigh the strength of evidence for the format independent/format dependent hypotheses, such as using a Bayesian hypothesis testing approach.
Relationship between numerical order processing and mathematical achievement
Studies have reported a significant relationship between order processing for ascending Arabic number sequences and mathematics achievement for adults (e.g., Goffin & Ansari, 2016; Lyons & Beilock, 2011; Morsanyi et al., 2017; Sasanguie, Lyons, et al., 2017; Vogel et al., 2017; Vos et al., 2017) and typically developing children (e.g., Attout & Majerus, 2018; Lyons et al., 2014). These findings suggest that order processing of number symbols may be one of the core numerical-specific competencies that are related to individual differences in mathematical achievement.
For example, Goffin and Ansari (2016) reported that smaller RDEs based on participants’ RTs in the Numerical Ordinality Task were significantly associated with higher mathematical achievement. The authors chose not to speculate on the processes and representations that may have influenced the size of the RDE. Vos et al. (2017) reported that faster median RTs (Experiment 2), but not the size of the RDE (Experiment 1), was significantly associated with higher mathematical achievement. Together, their results suggest that different measures (e.g., median RT, RDE) could index different aspects of numerical ordinality, and point to the need for further research to fully characterise the nature of the relationship between the processing of numerical order and complex mathematical skills.
In addition, existing research has not yet investigated whether the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals, or generalisable to other types of number symbols, such as number words in various languages. While the relationship between numerical order processing and mathematical achievement is well-established for Arabic numerals, it seems possible that this relationship could extend to other notational formats, given that numerical ordinality is argued to be an ability that is present very early on in human development, as early as in infancy (e.g., Brannon, 2002; Suanda et al., 2008; de Hevia et al., 2017; Picozzi et al., 2010).
Impact of language profile on numerical order processing
A significant body of literature has examined the impact of language and culture on numerical processing. Some commentaries have observed that the conceptual, lexical, and orthographic properties of an individual’s first language influences how that individual processes information about number, space, and mathematics in that language and overall (Daroczy et al., 2015; Dowker & Nuerk, 2016 Göbel, 2018; Göbel et al., 2011). For example, the inversion property of Dutch and German number words (i.e., where the rightmost digit of a two-digit number word is named before the leftmost digit) has been found to yield differences between speakers of German and/or Dutch compared to speakers of other non-inverted languages in numerical tasks such as estimation (Savelkouls et al., 2020), transcoding (Poncin et al., 2020), arithmetic processing (Lonnemann & Yan, 2015), as well as in markers of numerical processing such as the compatibility effect for number words (Nuerk et al., 2005, also see Bahnmueller et al., 2019).
While the above studies and commentaries explore the differences between speakers of different languages (e.g., German-speakers vs. English-speakers), few studies have investigated for possible differences in numerical processing within bilingual speakers of the same languages with different patterns of language dominance (e.g., among English-Chinese bilinguals, those who are English-dominant vs. Chinese-dominant vs. balanced bilinguals) (Campbell & Epp, 2004; Santos & Finger, 2020). Furthermore, existing research has not investigated the influence of participants’ language dominance profiles on the processing of numerical order. It is important to address this research gap because a disparity in participants’ fluency and/or frequency of use of the notational formats could influence numerical processing in at least two ways. First, a main effect of Format, consisting of faster RTs to the more fluent or frequently used format, could occur simply as a result of participants having more experience with that notational format. This would constitute a source of error, as opposed to a true effect from format-related factors. Second, a smaller distance effect to the more fluent or frequently used format may occur because participants have more precise semantic (in this case, order) representations with less distributional overlap for that notational format. Hence, to strengthen the methodological rigour of existing research, it is important for studies involving bilingual participants to explore the influence of participants’ language dominance profiles on numerical processing.
The present study
The aims, research questions, experimental design, proposed methods, and analysis plan were pre-registered on the Open Science Framework prior to data collection (https://osf.io/62tq7). The broad aim of the study was to probe the cognitive mechanisms that underlie order processing for number symbols. Specifically, we investigated the extent to which the direction and notational format that number symbols were presented in influenced order processing. We also examined the extent to which the relationship between numerical order processing and mathematical achievement was specific to Arabic numerals or generalisable to other notational formats of number symbols. Addressing these issues would help to disentangle the theoretical mechanisms that are said to underlie numerical order processing, and to provide empirical evidence to substantiate or refute existing models of numerical cognition. Our study recruited participants who were bilingual in English and Chinese to examine the processing of numerical order among multiple notational formats (Arabic numerals, English number words, and Chinese number words) within a single group of individuals.
Here, we examined the interaction between Direction and Numerical Distance. Based on the findings of studies reviewed above (e.g., Franklin et al., 2009; Goffin & Ansari, 2016; Lyons & Beilock, 2013 but see Morsanyi et al., 2017; Sasanguie, Lyons, et al., 2017), it was predicted that order processing would be direction dependent such that ascending and descending trials, but not mixed direction trials, would yield the RDE. Taking the RDE as an indicator for the processing of numerical order (e.g., Goffin & Ansari, 2016; Lyons & Beilock, 2013; Rubinsten & Sury, 2011; Turconi et al., 2006), this would mean that order processing occurs for ordered, but not unordered, number sequences.
Here, we examined the interaction between Format and Numerical Distance. Based on the findings of Lyons and Beilock (2013) who reported a significant interaction between Format and Numerical Distance, it was predicted that order processing for number symbols would be format dependent. However, Lyons and Beilock (2013) compared between symbolic and non-symbolic formats, and the representations and processes that have been implicated in symbolic numbers are said to be different from those for non-symbolic numbers (Bulthé et al., 2014; De Smedt et al., 2013; Gilmore et al., 2018; Sasanguie, De Smedt, & Reynvoet, 2017; Schneider et al., 2016). Thus, it was uncertain whether the finding of format dependence would still hold when comparing among different notational formats of number symbols in the present study.
To address this research question, the interaction between Format, Direction, and Numerical Distance was examined. To date, studies have not directly examined this issue. Thus, it was hoped that investigating this research question would yield a better understanding of the factors that influence order processing for number symbols.
Here, two indices of numerical order processing were employed. First, bivariate correlations were examined between the processing speed for Arabic numerals, English number words, and Chinese number words in the Numerical Ordinality Task and mathematical achievement, split by direction and distance conditions. Based on the results of existing studies, it was predicted that faster RT to correct trials on the Numerical Ordinality Task would be associated with higher mathematical achievement (e.g., Attout & Majerus, 2018; Goffin & Ansari, 2016; Lyons & Beilock, 2011; Morsanyi et al., 2017; Sasanguie, Lyons, et al., 2017).
Second, bivariate correlations were examined between the distance effects of Arabic numerals, English number words, and Chinese number words in the Numerical Ordinality Task and mathematical achievement, split by direction and distance conditions. In light of mixed findings in the literature, it was uncertain as to what findings this analysis would yield. For example, Goffin and Ansari (2016) reported a significant relationship between the size of the RDE and mathematical achievement, but not Vos et al. (2017).
In addition to the confirmatory analyses, three sets of exploratory analyses were conducted. The purpose of the exploratory analyses were to (1) disentangle the different theoretical mechanisms that are said to underlie numerical order processing, (2) investigate the influence of participants’ language dominance profiles on numerical order processing, and (3) test for possible associations between the size of the distance effects across notational formats.
Method
Participants
Seventy participants aged 21–39 years who were bilingual in English and Chinese completed the study. The sample size (minimum: 65 participants) was determined based on a statistical power analysis performed with G*Power 3.1.9, with α = .05, a minimum β of .80, and a minimum partial η2 of .14 (i.e., small–medium effect size, based on reported effect sizes in related studies, e.g., Morsanyi et al., 2017; Sasanguie et al., 2017; Vos et al., 2017). Outlier participants were defined as those individuals whose average RTs or accuracies differed by at least ±3 standard deviations (SD) from the average RT or accuracy of all participants. There were no outlier participants. Thus, all 70 participants were included in the analyses (49 females, M = 28.85 years, SD = 4.68 years). Participants were recruited via word of mouth, online newsletters to university students and staff, and posters around the University. All participants provided written informed consent and declared normal or corrected-to-normal vision. The study was approved by the Institutional Review Board of Nanyang Technological University, Singapore.
Although all participants declared being bilingual in English and Chinese, they comprised a heterogeneous variety of nationalities, including Singapore (63%), People’s Republic of China (28%), Malaysia (5%), Taiwan (3%), and Hong Kong (1%). Participants provided a subjective judgement as to what they perceived their first language to be: English (60%), Chinese (33%), or equally balanced in both languages (7%). To obtain detailed information on their language backgrounds, all participants completed the Language Background Questionnaire.
Procedure
Each participant completed the behavioural tasks in an individual testing session, along with the Language Background Questionnaire. The Numerical Ordinality Task was computerised and presented on a 14-inch laptop computer screen (Acer TravelMate P238, Model: N15 W8, refresh rate: 60 Hz), with participants sitting approximately 50 cm away from the screen. E-Prime 2.0 (Psychology Software Tools, Pittsburgh, PA, USA) was used to programme and present the Numerical Ordinality Task. The Arithmetic Task was a paper-and-pencil measure. All participants completed the Numerical Ordinality Task first, followed by the Arithmetic Task.
Measures and stimuli
Numerical Ordinality Task
Participants viewed three numerical stimuli that were simultaneously presented in the centre of the computer screen. They were instructed to place their left and right index fingers on two yellow stickers on a keyboard to indicate a left or right button press response respectively. Participants were asked to make a button press response to indicate whether the stimuli were in numerical order (i.e., ascending/descending direction) or not (i.e., mixed direction). They were told to respond as quickly and as accurately as possible. The left/right button assignment used to indicate order/not in order was counterbalanced within subjects.
The numerical stimuli displayed on the screen ranged from 1 to 9 and were presented in black on a white background. The stimuli were presented in three levels of Format: Arabic numerals, English number words, and Chinese number words (Table 1). In addition, three levels of Direction were employed: Ascending (e.g., 2 3 4), Descending (e.g., 4 3 2), and Mixed (e.g., 2 4 3). The font used for the Arabic and English formats was Arial (size 72), while MS Gothic was used for the Chinese format. The two levels of numerical distance were: Small (Distance = 1) and Large (Distance = 2, 3).
Stimulus list for Arabic, English, and Chinese formats.
This task consisted of six blocks in total: two blocks for each notational format (Arabic, Chinese, and English). Each block only contained one type of notational format. The order of the formats and the order of the button assignment were counterbalanced between subjects with the restriction that the order of the button assignment could only switch after the first three blocks.
Each block began with a practise block of five trials, where participants were given feedback for their accuracy and RT at the end of each trial. After the practise block, participants could ask any questions they had about the task. This was followed by the experimental blocks, where participants were not given any feedback. Fifty percent of the total trials in each experimental block were in order (25% ascending, 25% descending) and the rest were not in order. Within each directional condition, there were an equal number of small distance (Distance = 1) and large distance (Distance = 2, 3) trials. Each experimental block consisted of 112 experimental trials, giving rise to a total of 672 trials.
The experimental schematic was identical for all task conditions. As seen in Figure 1, each block of trials began with a fixation star lasting 300 ms. The experimental trials, comprising three number symbols at a time, were presented on the screen for an unlimited duration until a response was made. The experimental trials were interspersed by interstimulus intervals, comprising of a blank screen that lasted 800 ms.
Example of the experimental schematic for the Arabic notation condition in the Numerical Ordinality Task.
Arithmetic task
This task was used as a measure of mathematical achievement. Participants completed as many single-digit addition, subtraction, and multiplication questions as quickly and as accurately as possible within a 2-min period. This was an adapted version of the Mathematical Fluency subtest from the Woodcock Johnson III Tests of Achievement (Woodcock et al., 2001) as pilot testing of the original test revealed ceiling effects. Most of our pilot participants completed all the items under 3 min, and the pilot data lacked variability. Thus, for this adapted version, the number of items were increased and the duration was shortened.
Language Background Questionnaire
The full questionnaire is available on our OSF project page (https://osf.io/fsab4/), and was previously developed by Wong et al. (2018) to assess the language dominance profiles of bilingual respondents. Here, the questionnaire was used to assess participants’ fluency and experience with their first and second languages, which were the stimulus formats in the Numerical Ordinality Task. The measures were: Language History (i.e., age at which participant learnt English and Chinese in both formal and informal settings), Subjective Language Proficiency (i.e., participants’ ratings for how proficient they were in understanding, speaking, reading, and writing in English and Chinese on a scale of 0 (very few words) to 6 (native proficiency), Frequency of Use in various modalities and contexts (i.e., frequency of using English and Chinese in different communicative modalities such as speaking, reading, and writing in different daily life settings), and Objective Language Proficiency (i.e., previous academic grades for English and Chinese).
Results
Descriptive statistics
The full dataset is available at https://osf.io/fsab4/. For the Arithmetic Task, scores ranged from 60 to 151 correctly answered items (M = 102, SD = 21). A Kolmogorov–Smirnov test indicated that the scores were normally distributed, D(70) = 0.04, p = .20. For the Numerical Ordinality Task, overall percentage accuracy was close to ceiling (M = 93%, SD = 3%). Trials which differed at least ±3 SD from each participant’s mean RT to correct trials were deemed outliers. Outlier trials (1.56%) and incorrect trials (6.46%) were excluded from the data analyses. Given the ceiling effect for the accuracy data, participants’ accuracy was not analysed directly. Because the data showed no significant speed-accuracy tradeoffs, participants’ mean RT to correct trials were used as a dependent variable. Mean average RT to correct trials ranged from 507 to 3,003 ms (M = 1,320 ms, SD = 444 ms).
Main analyses for the Numerical Ordinality Task
For the main analyses, a two-fold statistical approach was adopted. Bayesian and frequentist statistics were conducted on a Direction (3) × Format (3) × Numerical Distance (2) repeated-measures ANOVA on participants’ RT data. Bayesian and frequentist statistics comprise different approaches that offer unique insights to the data. Bayesian statistics quantifies the evidence for the null and alternative hypotheses by providing information about the probability of each of the hypotheses being true, given the data at hand (Chambers, 2019; Dienes, 2011; Dienes & McLatchie, 2018; Gelman & Loken, 2014; Ortega & Navarrete, 2017; Simonsohn, 2015; Wagenmakers et al., 2008), whereas frequentist statistics provides information about the long run probability of the data under the null hypothesis (Harms & Lakens, 2018). Furthermore, Bayesian statistics provide information on which combination of terms best fit the overall data (e.g., Etz & Vandekerckhove, 2018; Harms & Lakens, 2018; Lakens et al., 2020; Wagenmakers, Love, et al., 2018), where frequentist statistics offer a more fine-grained perspective on the measure of significance for each term, as well as a useful comparison between different trial types. Given the limitations of the frequentist approach as explained in the Introduction, we prioritised the use of Bayesian statistics, as we believe that they provide a more holistic perspective to the data. Although the frequentist and Bayesian results are presented separately for methodological clarity, they ultimately address the same research questions. Hence, we holistically examine the interpretation of both sets of results in the Discussion section.
Bayesian statistics
A Bayesian repeated-measures ANOVA was conducted using JASP (Version 0.10.0; JASP Team, 2018). Given the lack of prior knowledge about the influence of direction and format on numerical order processing, we employed the default zero-centred Cauchy distribution prior with a scale of 0.707, which is the default setting in JASP. Different statistical models were specified and were compared between on the basis of the size of their Bayes Factors (BFs) and posterior probabilities (Marsman & Wagenmakers, 2017; Wagenmakers et al., 2017; Wagenmakers, Love, et al., 2018; Wagenmakers, Marsman, et al., 2018). To interpret the size of the BFs, we employed the recommendation of Jeffreys (1961), Andraszewicz et al. (2015), and Lee and Wagenmakers (2014), where a BF of 0–3 offers anecdotal support for the model, 3–30 moderate support for the model, 30–100 strong support for the model, and values over 100 indicating decisive evidence for the model.
Full details of all models are available in the Online Supplementary Material. The null model did not contain any terms and represented the null hypothesis that participants’ RT data did not depend on any factors. The candidate models contained three main effects (Direction, Format, Distance) plus various combinations of two-way interactions (Direction × Distance, Format × Distance, Format × Direction) and the three-way interaction (Format × Direction × Distance). The BFs for each of these models were examined. Overall, the model that best predicted the data contained three main effects (Direction, Format, Distance) and a two-way interaction (Direction × Distance) (i.e., Model 2, BF = 9.15 × 10283). Thus, with regard to the influence of direction on order processing (RQ1), the inclusion of the Direction × Distance term in the best-fitting model supports the idea that the processing of numerical order is dependent on the direction in which the numerical stimuli are presented.
To address the influence of notational format on order processing (RQ2), the term Format × Distance was of interest. The inclusion BF for this term, generated by JASP, was 0.049. We computed BF01 for this term, which takes into account the posterior probability of the best-fitting model (i.e., which did not include the Format × Distance term) relative to the posterior probability of the equivalent model that included Format × Distance as a predictor (van den Bergh et al., 2020), where BF01 = 0.723/0.016 = 45.19. Thus, the data were 45.19 times more likely to occur under the best-fitting model (Direction + Format + Distance + Direction × Distance) than the equivalent model that included the Format × Distance term. 1
In addition, to address the extent to which Format and Direction interact to influence the processing of numerical order (RQ3), the term Format × Direction × Distance was of interest. The inclusion BF for this term was 0.006. Again, we computed BF01 for this term, 0.723/3.09 × 10−4 = 2,339.81. Thus, the data were 2,339.81 times more likely to occur under the best-fitting model (Direction + Format + Distance + Direction × Distance) than the equivalent model that included the Format × Direction × Distance term.
Frequentist statistics
Mean RT to correct trials for all Direction, Format, and Distance conditions are displayed in Figure 2. The paragraphs below present a summary of the frequentist statistics for the terms in the best-fitting Bayesian model (i.e., three main effects and a two-way interaction between Direction and Numerical Distance, Model 2). Full frequentist results are available at https://osf.io/fsab4/. In this set of frequentist analyses, the size of the RDE for ascending and descending trials was calculated by subtracting mean RTs for small distance trials from mean RTs for large distance trials. Conversely, the size of the NDE for mixed direction trials was calculated by by subtracting mean RTs for large distance trials from mean RTs for small distance trials.
Mean RT to correct trials for the Numerical Ordinality Task for all direction, format, and distance conditions. Error bars represent within-participant standard errors of the means.
All main effects were significant. There was a significant main effect of Direction, F(2, 138) = 131.28, p < .001, partial η2 = .66, where RT was fastest for ascending trials (M = 1,206 ms, SE = 31 ms), followed by descending (M = 1,354 ms, SE = 39 ms) and mixed direction trials (M = 1,402 ms, SE = 36 ms). Follow-up Bonferroni pairwise comparisons indicated that RT was significantly faster for ascending trials than descending trials, which were in turn significantly faster than mixed direction trials (p < .001 for all). Next, there was a significant main effect of Format 2 , F(1.67, 115.12) = 188.75, p < .001, partial η2 = .73, where RT was fastest for Arabic numerals (M = 975 ms, SE = 27 ms), followed by Chinese (M = 1,383 ms, SE = 42 ms), and finally English (M = 1,604 ms, SE = 46 ms) number words. Follow-up Bonferroni-corrected pairwise comparisons revealed that RT was significantly faster for Arabic numerals than Chinese number words, which were in turn significantly faster than English number words (p < .001 for all). There was also a significant main effect of Numerical Distance, F(1, 69) = 11.44, p = .001, partial η2 = .14, where RT was significantly faster for small distance trials (M = 1,306 ms, SE = 34 ms) than large distance trials (M = 1,335 ms, SE = 35 ms), yielding an overall RDE.
There was also a significant interaction between Direction and Numerical Distance, F(2, 138) = 134.06, p < .001, partial η2 = .66, with an RDE for ascending (M = 118 ms, SE = 14 ms) and descending (M = 83 ms, SE = 13 ms) trials, and an NDE for mixed direction trials (M = 113 ms, SE = 9 ms). Simple effects analyses revealed that all distance effects were significant (p < .001 for all).
Exploratory analyses for the Numerical Ordinality Task
Three sets of exploratory analyses were conducted for the Numerical Ordinality Task. First, to disentangle the different theoretical mechanisms that are said to underlie numerical order processing, we conducted a follow-up analysis to compare the size of the RDEs for ascending vs. descending trials (Vos et al., 2017). We calculated RDEs by subtracting mean RT to small distance trials from mean RT to large distance trials (Vos et al., 2017) and ran a Direction (2) × Format (3) follow-up repeated-measures ANOVA on participants’ RT data. There was a significant main effect of Direction, F(1, 69) = 5.73, p = .02, partial η2 = .08, where RDEs were significantly larger for ascending trials (M = 118 ms, SE = 14 ms) than descending trials (M = 83 ms, SE = 13 ms).
Second, we investigated the extent to which participants’ language dominance profiles influenced RTs in the Numerical Ordinality Task. Based on participants’ responses to the Language Background Questionnaire, they were categorised into three language dominance groups following Wong et al. (2018). In total, 35 participants were categorised as English-dominant, 14 as Chinese-dominant, and 21 as balanced bilinguals. Frequentist statistics were conducted using a Direction (3) × Format (3) × Numerical Distance (2) × Language Dominance (3) mixed ANOVA on participants’ mean RT to correct trials. There was no significant main effect of Language Dominance (p = .42). However, there was a significant interaction between Format and Language Dominance, F(4, 134) = 18.05, p < .001.
To follow-up, Bonferroni-corrected pairwise comparisons were conducted to compare RT across the notational formats for each language dominance group. For English-dominant participants, RT was significantly faster for Arabic numerals (M = 947 ms, SE = 40 ms) than Chinese (M = 1,443 ms, SE = 59 ms) and English (M = 1,445 ms, SE = 60 ms) number words (both p < .001), although RT for the latter two formats did not differ significantly (p > .99). For Chinese-dominant participants, RT was significantly faster for Arabic numerals (M = 1,005 ms, SE = 63 ms) than Chinese number words (M = 1,232 ms, SE = 94 ms), which were in turn significantly faster than English number words (M = 1,761 ms, SE = 95 ms) (p ⩽ .001 for all). For participants who showed a balanced profile of language dominance, RT was significantly faster for Arabic numerals (M = 1,002 ms, SE = 51 ms) than Chinese number words (M = 1,384 ms, SE = 76 ms), which were in turn significantly faster than English number words (M = 1,765 ms, SE = 77 ms) (p < .001 for all). In other words, all participants showed fastest RT to correct trials for Arabic numerals, but only the Chinese-dominant and balanced bilinguals showed a significant RT advantage for Chinese number words compared to English number words. Aside from this, there were no other systematic differences between the language dominance groups.
Third, as requested by a Reviewer, we assessed the correlation among the size of the distance effects for Arabic, English, and Chinese notational formats, according to the direction in which the stimuli were presented. For ascending and descending trials, we calculated the size of the RDE by subtracting mean RTs for small distance trials from mean RTs for large distance trials, divided by mean RTs of small and large distance trials (Goffin & Ansari, 2016; Orrantia et al., 2019). For mixed direction trials, we calculated the size of the NDE by subtracting mean RTs for large distance trials from mean RTs for small distance trials, divided by mean RTs of small and large distance trials. For ascending trials, there was moderate to strong evidence that the RDEs for the different notational formats were correlated (2.98 ⩽ BF10 ⩽ 41.11). For descending trials, there was anecdotal evidence that the RDEs for the different formats were correlated (1.94 ⩽ BF10 ⩽ 2.86). For mixed direction trials, there was no evidence that the NDEs for the different formats were correlated (0.24 ⩽ BF10 ⩽ 0.52). Full results are presented in Table 2. 3
Bivariate correlations between distance effects a for all format and direction conditions.
BF: Bayes factor; ρ: Spearman’s rho.
As specified in the main text, reverse distance effects (RDEs) were calculated for ascending and descending trials, while numerical distance effects (NDEs) were calculated for mixed direction trials.
p < .05; **p < .01.
Numerical order processing and mathematical achievement
Previous studies have examined the relationship between the processing of numerical order and mathematical achievement for Arabic numerals (e.g., Attout & Majerus, 2018; Goffin & Ansari, 2016; Lyons & Beilock, 2011; Morsanyi et al., 2017; Sasanguie, Lyons, et al., 2017). In line with this, and since tests of mathematical achievement are typically administered with Arabic numerals, we investigated the relationship between numerical order processing and mathematical achievement for Arabic numerals in the Numerical Ordinality Task, then extended this analysis to English and Chinese number words. Bayesian and frequentist statistics were performed for all analyses.
The procedure of Vos et al. (2017) was adopted because their version of the Numerical Ordinality Task was most similar to that employed in the present study. First, bivariate correlations were conducted between mean RT to correct trials for Arabic number sequences in the Numerical Ordinality Task and scores on the Arithmetic Task, split by the different conditions for Numerical Distance and Direction (Table 3). The Bayesian statistics yielded strong to decisive evidence that the speed of processing for the different directional conditions of Arabic number sequences in the Numerical Ordinality Task was associated with arithmetic scores (41.15 ⩽ BF10 ⩽ 1,034.89). Similarly, the frequentist statistics showed that faster mean RT to correct trials for Arabic numerals for all distance and direction conditions in the Numerical Ordinality Task was positively and significantly associated with better arithmetic performance (p < .002 for all).
Bivariate correlations (Bayes factor, Pearson’s r, and Spearman’s ρ) between RT for each notational format in the Numerical Ordinality Task and arithmetic scores.
Both Pearson’s r and Spearman’s ρ are reported because several trial types did not fulfil the assumption of normality. Smaller RT values indicate faster responses.
p < .05; **p < .01; ***p < .001.
The same procedure was followed for Chinese and English number words in the Numerical Ordinality Task (Table 3). For Chinese number words, the Bayesian statistics yielded decisive evidence that the speed of processing for the different directional and distance conditions in the Numerical Ordinality Task were associated with arithmetic scores (817.22 ⩽ BF10 ⩽ 4,477.80). Similarly, the frequentist statistics showed that faster RT to correct trials for Chinese number words for all conditions in the Numerical Ordinality Task was positively and significantly associated with better arithmetic performance (p < .001 for all). For English number words, the Bayesian statistics yielded anecdotal to strong evidence that the speed of processing for the different directional and distance conditions in the Numerical Ordinality Task were associated with arithmetic scores (3.38 ⩽ BF10 ⩽ 92.34). Similarly, the frequentist statistics showed that faster RT to correct trials for English number words for all conditions in the Numerical Ordinality Task was positively and significantly associated with better arithmetic performance (p < .001 for all).
Next, bivariate correlations were conducted between distance effects for the different notational formats in the Numerical Ordinality Task and scores on the Arithmetic Task, split by the different format and direction conditions (Table 4). For consistency with the frequentist analyses and to directly compare the results of this analysis with previous work by Vos et al. (2017), the size of the RDE was calculated by subtracting mean RTs for small distance trials from mean RTs for large distance trials for ascending and descending trials. For mixed direction trials, the size of the NDE was calculated by subtracting mean RTs for large distance trials from mean RTs for small distance trials. The Bayesian and frequentist statistics did not generally yield any evidence that the distance effects for the different format and directional conditions were related to arithmetic scores, apart from anecdotal evidence for Arabic numerals presented in the mixed direction (BF10 = 2.12; r = −.28, p = .02) and English number words presented in the descending direction (BF10 = 1.71; r = −.27, p = .03).
Bivariate correlations (Bayes factor, Pearson’s r, and Spearman’s ρ) between distance effects a for each notational format in the Numerical Ordinality Task and arithmetic scores.
As specified in the main text, reverse distance effects (RDEs) were calculated for ascending and descending trials, while numerical distance effects (NDEs) were calculated for mixed direction trials.
p < .05; **p < .01. For Chinese number words, all ps were not significant (p > .05 for all).
Discussion
The broad aim of this study was to probe the cognitive mechanisms that underlie order processing for number symbols. First, we investigated the influence of the direction in which the number symbols were presented, which would help to clarify the directional conditions under which numerical order processing occurs, identify the extent to which the representations and processes associated with numerical order are dependent on or independent of direction, and disentangle existing theoretical mechanisms that are said to underlie numerical order processing. Second, we investigated the influence of notational format on order processing for number symbols, which would clarify the extent to which the representations and processes associated with numerical order are dependent on or independent of the format in which the stimuli are presented, and provide empirical evidence to substantiate or refute contemporary models of numerical cognition. Third, the study investigated the extent to which the relationship between numerical order processing and mathematical achievement was specific to Arabic numerals or generalisable to other notational formats of number symbols.
Order processing of number symbols is dependent on the direction in which the stimuli are presented
The processing of numerical order was found to be dependent on the direction in which the numerical stimuli were presented (RQ1). The Bayesian model that best predicted the data contained three main effects, and critically, a two-way interaction between Direction and Numerical Distance. In parallel, the frequentist statistics yielded a significant Direction × Distance interaction (partial η2 = .66), with the RDE for ordered trials and the NDE for non-ordered trials. These results suggest that when participants are presented with a numerical sequence and are instructed to engage in order processing, they process numerical order directly for ascending and descending (i.e., ordered) trials, but not for mixed direction (i.e., non-ordered) trials. These findings are in line with previous studies that reported the RDE for ascending (Franklin et al., 2009; Goffin & Ansari, 2016; Lyons & Beilock, 2013; Turconi et al., 2006; Vos et al., 2017) and descending trials (Lyons & Beilock, 2013; Vos et al., 2017), and studies that report the NDE for mixed direction trials in the Numerical Ordinality Task (e.g., Lyons & Beilock, 2013; Morsanyi et al., 2017; also see Lyons et al., 2016). However, in contrast to these studies which tested Arabic numerals only, the present study confirms and extends their findings to that for ordered number sequences presented in Arabic, Chinese, and English formats.
Follow-up analyses further revealed that RT to correct trials were significantly faster for ascending than descending trials, with a significantly larger RDE for ascending trials than descending trials. Following Vos et al. (2017), these results support the strength of associations account of numerical ordinality, which predicts that ascending trials would be associated with larger RDEs and faster RTs than descending trials, on the basis that ascending number sequences occur more frequently than descending number sequences. Therefore, the present results are in line with the idea that the strength and closeness of associations between items in the number sequence could underlie the processing of numerical order for number symbols presented in ascending/descending directions.
That the NDE was found for mixed direction sequences could indicate that participants employ a strategy based on magnitude comparison to process non-ordered trials, such as sequentially and iteratively comparing the quantity of the two numbers at a time, to assist in making a global judgement about the order of the numerical sequence (Lyons & Beilock, 2013; Vos et al., 2017). 4 For example, in the mixed sequence 4 2 3, participants might compare that 4 is larger than 2, and 2 is smaller than 3. Since the numbers do not consistently increase or decrease in quantity, participants then judge that the numbers are not in numerical order. Thus, when viewing a non-ordered numerical sequence, participants may make multiple, iterative comparisons about the magnitude of two numbers at a time before formulating an overall judgement about numerical order for the entire number sequence. Vos et al. (2017) further argue that a strategy based on comparing the quantities of two numbers at a time in the entire sequence is likely to occur for mixed direction trials because non-ordered numerals rarely occur in daily life, resulting in weak associations between the number sequence. Hence, it is possible that the direction in which the stimuli are presented may cue participants to use particular processing strategies to ascertain whether the numbers are in order or not, that is, a strategy involving magnitude comparison for non-ordered trials (i.e., mixed direction), and a strategy involving the strength and closeness of associations between items in the number sequence for ordered (i.e., ascending/ descending) trials.
Order processing of number symbols is independent of the format in which the stimuli are presented
The processing of numerical order was found to be independent of the notational format in which the stimuli were presented (RQ2). The Bayesian results showed that the odds of including Format × Distance strongly decreased after observing the data; moreover, the model that best predicted the data did not contain the Format × Distance interaction. In line with this, the Format × Distance interaction was not significant in the frequentist analyses. Together, these results suggest that the processing of numerical order is independent of the notational format in which the stimuli are presented. In other words, the psychological representations and processes associated with numerical order appear to be abstract, regardless of whether the stimuli are represented as Arabic numerals, Chinese number words, or English number words. This suggests that our understanding of numerical order is unaffected by the different possible notational formats that a number symbol is represented in (Cohen et al., 2019).
The present results are broadly in line with the format independent view of numerical cognition advocated by models such as the Abstract Code Model (McCloskey, 1992), the Multiroute Model of Number Processing (Cipolotti & Butterworth, 1995), the Triple Code Model (Dehaene & Cohen, 1995), and do not exclude a mixed view of numerical cognition (Cohen Kadosh & Walsh, 2009). To date, these models of numerical cognition have not directly addressed the processing of numerical order. Thus, our results extend the perspectives of the above models by providing experimental evidence, for the first time, that the processing of numerical order (and not just numerical magnitude) is abstract across different formats of number symbols. Taking the Abstract Code Model (McCloskey, 1992), the Multiroute Model of Number Processing (Cipolotti & Butterworth, 1995), and the Triple Code Model (Dehaene & Cohen, 1995) as an example, our results are consistent with the idea that the system that contains internal analogue representations of number is indeed format-independent.
In the exploratory analyses, we found moderate to strong evidence that the RDEs for the different formats of ascending number sequences were correlated, and anecdotal evidence that the RDEs for the different formats of descending sequences were correlated. While there was moderate evidence from the Bayesian and frequentist statistics that the behavioural signature for order processing was similar across formats, we are cautious as to whether format independence necessitates equally sized distance effects across notational formats. It is possible that such correlations may be influenced by the size and reliability of distance effects across individuals. Furthermore, it is important to note that the present study was set up as an experimental study and not an individual differences one, so future work may consider the question of format-dependence/independence from an individual differences perspective.
Format and direction do not interact to influence order processing for number symbols
The factors of Format and Direction were not found to interact to influence the processing of numerical order (RQ3). The Bayesian results showed that the odds of including Format × Direction × Distance decreased very strongly after observing the data; moreover, the model that best predicted the data did not contain the Format × Direction × Distance interaction. This indicates that the notational format and the direction in which numerical stimuli are presented are separate factors that do not interact to influence how participants make order judgements for number symbols.
Main effects of direction and format
While the main effects of Direction and Format in the Numerical Ordinality Task do not directly relate to the processes or representations associated with numerical order, they offer useful information about the similarities and differences in the speed of order judgements depending on the directional conditions and formats/orthographies employed.
First, the results indicated that the direction in which the stimuli are presented influences the speed of order judgements. Previous studies have suggested that cultural and linguistic factors may underlie this main effect of Direction. For example, participants from cultures with a left-to-right reading/writing direction have been found to respond to ascending trials (i.e., smallest to largest) significantly faster than other trial types (e.g., Turconi et al., 2006; Vos et al., 2017), whereas participants from cultures with a right-to-left reading/writing direction tend to respond to descending trials (i.e., largest to smallest) significantly faster than other trial types (e.g., Rubinsten et al., 2013; Rubinsten & Sury, 2011). In our study, the dominant languages of participants were English and Chinese, both of which have a left-to-right reading/writing direction. Indeed, the Bayesian and frequentist statistics yielded a main effect of Direction where RT was significantly faster for ascending trials than other trial types (i.e., descending and mixed directions). Whereas previous studies investigating languages of a left-to-right direction tested speakers of French and Dutch (Turconi et al., 2006; Vos et al., 2017), the present study extends these findings to include bilingual speakers of English and Chinese. Furthermore, the current results are in line with the mental number line theory, where number representations are said to be oriented from left to right (Dehaene et al., 1993; Restle, 1970) or right to left, depending on the directionality of the reading/writing system (e.g., Fischer et al., 2009; Zebian, 2005).
Second, the results indicated that the format of the number symbols influences processing time. Both Bayesian and frequentist analyses yielded a main effect of Format, where RT was significantly faster for Arabic numerals than Chinese number words, which were in turn significantly faster than English number words. The finding of faster RTs for Arabic numerals than number words in the Numerical Ordinality Task parallels similar findings in tasks of numerical magnitude, where Arabic numerals have been found to yield better performance than number words in alphabetic (Dehaene & Akhavein, 1995; Lukas et al., 2014) or logographic scripts (Campbell et al., 1999; Cao et al., 2010). In the current research, it seems likely that the greater orthographic economy and higher prevalence of Arabic numerals to convey numerical information compared to English and Chinese number words contributed to significantly faster processing time. Visually, written Arabic numerals are less complex than written number words in English and Chinese (Table 1), and Arabic numerals are the most commonly used notational format to convey numerical information in daily life, whether among bilingual speakers of English and Chinese, or among native monolingual speakers of English or Chinese. It is likely that the high prevalence and familiarity of Arabic numerals in daily life contributed to participants’ faster responses to Arabic numerals than to English and Chinese number words.
Chinese-dominant and balanced bilinguals showed an overall speed advantage for Chinese trials compared to English trials
The language dominance profiles of the bilingual participants influenced their speed of responding to the different notational formats, but did not impact the cognitive processes associated with order processing. This can be observed from the exploratory analyses, which yielded a significant interaction between Language Dominance and Format; however, there were no significant interactions between Language Dominance and Numerical Distance. Follow-up analyses to the Language Dominance × Format interaction showed that all language dominance groups responded significantly faster to Arabic numerals than to other notational formats; furthermore, Chinese-dominant and balanced bilinguals responded significantly faster to Chinese than English number words, whereas the English-dominant bilinguals did not have such an advantage.
It seems possible that these findings could be attributed to the differences in writing systems for English and Chinese and participants’ levels of fluency and experiences with these languages. In Chinese, the written form maps directly to meaning (spelling-to-semantic correspondences), with each character/radical concisely and directly conveying a unit of meaning (Yang et al., 2013). In English, the written form maps to sound, and then to meaning (spelling-to-sound correspondences), where a group of letters correspond to phonemes, which then combine to convey a unit of meaning (Yang et al., 2013). As speakers develop fluency and experience in the Chinese language, it seems possible that they can draw on the more concise orthography, along with the direct and strong links between the written form and meaning, to support processing and reading in Chinese (Gottardo et al., 2017; Shu et al., 2006). Taking the psycholinguistic literature as a background from which to frame the findings in this study, it is possible that bilinguals who rated themselves to be at least as fluent and experienced with Chinese as they are with English, or more fluent and experienced with Chinese than English, drew on the orthographic conciseness and strong spelling-to-semantic links of the Chinese language to reap the benefits of an overall speed advantage for Chinese than English trials in the Numerical Ordinality Task. This result parallels related findings in tasks of numerical magnitude where logographic scripts have been found to yield better performance than alphabetic scripts (Wong et al., 2018), but extends and nuances this finding to order processing for Chinese-dominant and balanced bilinguals.
In summary, this is the first study in the literature to systematically investigate the influence of bilingual participants’ language dominance profiles on the processing of numerical order, 5 and yielded unique insights on how Chinese-dominant and balanced bilinguals, but not English-dominant bilinguals, make faster order judgements for Chinese than English number words. The finding that participants’ language dominance profiles did not significantly influence order processing further suggests that it may not be necessary for future research in numerical cognition to carefully control for the language dominance profiles of bilingual participants in related tasks.
Processing speed of numerical order judgements is related to mathematical achievement
In line with previous work (e.g., Lyons & Beilock, 2011; Morsanyi et al., 2017; Sasanguie et al., 2017; Vogel et al., 2017; Vos et al., 2017), the present study found a relationship between the processing speed of numerical order judgements and mathematical achievement. The evidence for this relationship was strongest for Arabic and Chinese numerals, and weaker for English number words. In other words, faster RT to correct trials of different directional and distance conditions in the Numerical Ordinality Task were associated with higher scores on the Arithmetic Task, particularly for numbers presented in Arabic or Chinese formats. That the relationship between the processing speed of order judgements and mathematical achievement cuts across different notational formats may be in line with the idea that the numerical ordinality is present as early as in infancy, and may be an innate numerical ability (e.g., Brannon, 2002; Suanda et al., 2008; de Hevia et al., 2017; Picozzi et al., 2010).
However, we found little to no evidence for a relationship between the size of participants’ distance effects and mathematical achievement, regardless of the format that the number words were presented in. These findings echo those of Vos et al. (2017) who reported a significant relationship between processing speed for numerical order judgements and mathematical achievement, but not between the size of RDEs and mathematical achievement.
Together, our findings and those of Vos et al. (2017) suggest that the speed of making accurate judgements about numerical order, rather than the specific mental representations associated with numerical ordinality, is associated with mathematical achievement. However, taking our findings in the wider context of related studies (e.g., Goffin & Ansari, 2016; Lyons & Beilock, 2011; Morsanyi et al., 2017; Sasanguie et al., 2017; Vogel et al., 2017; Vos et al., 2017), the wider body of literature suggests that further research is needed to clarify the nature of the relationship between numerical order processing (whether indexed by differences in the size of participants’ RDE or overall RT measures) and mathematical achievement.
Conclusion
The present study offers behavioural evidence that the processing of numerical order for number symbols depends on the direction in which the numerical stimuli are presented, but not their notational format. Order processing was specific to number sequences that were presented in an ascending or descending direction. Furthermore, ascending sequences yielded faster RTs and larger RDEs than descending sequences, suggesting that the strength and closeness of bidirectional associations between items in a number sequence could account for numerical order processing. The results also indicated that the cognitive representations and processes associated with numerical order were abstract across notational formats – whether the stimuli were presented as Arabic numerals, Chinese number words, or English number words. This supports theoretical models that advocate for a format independent view of numerical cognition (e.g., Cipolotti & Butterworth, 1992; Cohen Kadosh & Walsh, 2009; Dehaene & Cohen, 1995; McCloskey, 1992), but does not exclude a mixed view of numerical cognition (Cohen Kadosh & Walsh, 2009). With regard to processing speed, the main effect of Direction supports the mental number line theory, where the direction of internal number representations is thought to parallel the directionality of one’s reading/writing system (Dehaene et al., 1993; Fischer et al., 2009). In addition, participants were generally fastest to respond to Arabic numerals, with those rating themselves to be at least as fluent and experienced in Chinese as they were in English having an additional speed advantage when responding to Chinese over English number words. This suggests that these individuals could draw upon the orthographic conciseness and strong spelling-to-semantic links of the Chinese language to enhance speed of processing for Chinese over English number words. The results also indicated that the speed of making accurate judgements about numerical order (for Arabic numerals, Chinese number words and, to a weaker extent, English number words) is associated with mathematical achievement.
While the present study has started to unpack the factors that influence the processing of numerical order for number symbols, as well as the relationship between numerical order processing and mathematical achievement, much is yet to be learnt about the processing of numerical order for fractions/decimals and multi-digit numbers, which are said to employ additional representations, manipulations, and processes beyond those of single-digit numbers (Nuerk et al., 2015). There also remains room for brain imaging research to further explore the neural and cognitive mechanisms that underlie the mental representations of numerical order (e.g., Attout et al., 2014; Fias et al., 2007; Kaufmannet al., 2009; Vogel et al., 2013). As forthcoming research continues to examine the representation and processing of numerical order, it is hoped that the field will be able to achieve a more holistic understanding of the factors that influence how humans represent and process numerical information.
Supplemental Material
sj-docx-1-qjp-10.1177_17470218211026800 – Supplemental material for Order processing of number symbols is influenced by direction, but not format
Supplemental material, sj-docx-1-qjp-10.1177_17470218211026800 for Order processing of number symbols is influenced by direction, but not format by Becky Wong, Rebecca Bull, Daniel Ansari, David M Watson and Gregory Arief D Liem in Quarterly Journal of Experimental Psychology
Footnotes
Acknowledgements
The authors would like to thank all participants who volunteered for this research. They also thank Prof. Kevin Patterson and four anonymous reviewers for their helpful comments.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: B.W. was supported by the Nanyang President’s Graduate Scholarship under Nanyang Technological University, Singapore.
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Notes
References
Supplementary Material
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