No calculation necessary: Accessing magnitude through decimals and fractions

Cognition 199 (2020) 104219

Contents lists available at ScienceDirect

Cognition journal homepage: www.elsevier.com/locate/cognit

No calculation necessary: Accessing magnitude through decimals and fractions

T

John V. Binzak , Edward M. Hubbard ⁎

University of Wisconsin–Madison, Dept. of Educational Psychology, Educational Sciences Bldg, 1025 W. Johnson Street, Madison, WI 53706-1796, USA

ARTICLE INFO

ABSTRACT

Keywords: Numerical cognition Magnitude processing Fractions Decimals Diffusion model

Research on how humans understand the relative magnitude of symbolic fractions presents a unique case of the symbol-grounding problem with numbers. Specifically, how do people access a holistic sense of rational number magnitude from decimal fractions (e.g. 0.125) and common fractions (e.g. 1/8)? Researchers have previously suggested that people cannot directly access magnitude information from common fraction notation, but instead must use a form of calculation to access this meaning. Questions remain regarding the nature of calculation and whether a division-like conversion to decimals is a necessary process that permits access to fraction magnitudes. To test whether calculation is necessary to access fractions magnitudes, we carried out a series of six parallel experiments in which we examined how adults access the magnitude of rational numbers (decimals and common fractions) under varying task demands. We asked adult participants to indicate which of two fractions was larger in three different conditions: decimal-decimal, fraction-fraction, and mixed decimal-fraction pairs. Across experiments, we manipulated two aspects of the task demands. 1) Response windows were limited to 1, 2 or 5 s, and 2) participants either did or did not have to identify when the two stimuli were the same magnitude (catch trials). Participants were able to successfully complete the task even at a response window of 1 s and showed evidence of holistic magnitude processing. These results indicate that calculation strategies with fractions are not necessary for accessing a sense of a fractions meaning but are strategic routes to magnitude that participants may use when granted sufficient time. We suggest that rapid magnitude processing with fractions and decimals may occur by mapping symbolic components onto common amodal mental representations of rational numbers.

1. Introduction A fundamental aspect of numerical cognition is the capacity to learn how numerical symbols, such as Arabic numerals, represent different forms of numerical meaning. Studying how people acquire an understanding of these symbols and their meaning, sometimes referred to as the symbol-grounding problem (Harnad, 1990; Leibovich & Ansari, 2016; Reynvoet & Sasanguie, 2016), is complex. Most investigations into the symbol-grounding problem with numbers have focused exclusively on the process of accessing whole number meaning (De Smedt, Noël, Gilmore, & Ansari, 2013; Leibovich & Ansari, 2016; Piazza, Pinel, Le Bihan, & Dehaene, 2007) and how this capacity with number symbols relates to an understanding of quantities grounded in perception. Here we extend this line of research to examine the processes involved in accessing rational number meaning from symbolic fractions, and test how these processes may differ between common fraction and decimal fraction notation. Addressing the symbol grounding problem with rational numbers is

⁎

complex because there are multiple ways to symbolically represent rational numbers and multiple interpretations of what rational numbers mean. The two main symbolic forms of fractions are decimal fractions (e.g. 0.125, referred to hereafter as decimals) and common fractions (e.g. 1/8, referred to hereafter as fractions). Learning the meaning of decimals and fractions involves learning how these number symbols can represent multiple numerical relationships including part-whole comparisons, ratios, quotient solutions to division, and rational number magnitudes (Behr, Lesh, Post, & Silver, 1983). These multiple meanings of rational numbers can be defined through formal mathematics and also understood through physical and perceptual experiences of these concepts in the world. Among these multiple interpretations of rational numbers, the current study focuses on how decimals and fractions represent specific rational number magnitudes and the cognitive processes necessary to access this meaning. Advancing research on how people understand symbolic fractions and decimals as representations of rational number meaning stands to further the empirical understanding of human numerical cognition and

Corresponding author at: 1075C Educational Sciences Building, 1025 Johnson Street, Madison, WI 53706, USA. E-mail addresses: [email protected] (J.V. Binzak), [email protected] (E.M. Hubbard).

https://doi.org/10.1016/j.cognition.2020.104219 Received 21 February 2018; Received in revised form 28 January 2020; Accepted 31 January 2020 0010-0277/ © 2020 Elsevier B.V. All rights reserved.

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

may have important educational implications. Although accurate proportional reasoning in physical space can be observed as early as infancy (Duffy, Huttenlocher, & Levine, 2005; McCrink & Wynn, 2007), multiple studies with children and adults have demonstrated the problems people experience reasoning with fractions as symbolic representations of magnitudes (e.g. 1/4 or 0.25) (Ni & Zhou, 2005; Siegler & Pyke, 2013, for a review, see Siegler, Fazio, Bailey, & Zhou, 2013). The prevalence of difficulties learning fractions is significant because developing rational number knowledge may play a significant role in the development of math knowledge more broadly (I. Resnick et al., 2016; Siegler, Thompson, & Schneider, 2011). For example, accurately mapping decimals and fractions onto visuo-spatial representations of magnitude (i.e., a number line) is a unique predictor of future mathematics achievement (Bailey, Hoard, Nugent, & Geary, 2012; Siegler et al., 2012). Furthermore, performance on fractions knowledge assessments is a significant predictor of early algebraic reasoning (DeWolf et al., 2015b), even when accounting for differences in whole-number knowledge (Booth & Newton, 2012). To better understand how fractions knowledge is related to math achievement and why fractions may be difficult to learn, it is important to study the psychological mechanisms underlying how adults access holistic magnitude meaning from different forms of fraction notation. Some studies have explored the psychological mechanisms of fraction magnitude processing by observing performance in magnitude comparison tasks with fractions and decimals. These studies have revealed multiple ways that magnitude processing with fractions and decimals is complex and influenced by task demands. For instance, when pairs of fractions share common numerator or denominator values (e.g. 1/4 vs 3/4 or 2/7 vs 2/9) educated adults routinely process only the whole number components that differ (e.g., numerators or denominators) rather than accessing the holistic magnitude of the fraction (Bonato, Fabbri, Umiltà, & Zorzi, 2007; Zhang, Fang, Gabriel, & Szücs, 2014). Likewise, researchers have identified ways that magnitude processing with multi-digit decimals can lead people to focus on whole-number components instead of or in parallel to the decimal's holistic rational number magnitude (Roell, Viarouge, & Houde, 2017; Varma & Karl, 2013). These findings of componential processing with fractions and decimals demonstrate ways that people exhibit wholenumber biases (Ni & Zhou, 2005) in tasks that do not require holistic magnitude processing. Researchers have also observed behavioral and neural evidence that adults can process the holistic magnitude of fractions even when componential features of fractions may also influence performance (Ischebeck, Schocke, & Delazer, 2009; Schneider & Siegler, 2010). For example, manipulations of magnitude comparison tasks, such as including comparisons between fractions that do not have common numerator or denominator components, can shift adults towards forms of magnitude processing that access a holistic sense of rational number magnitude (Meert, Grégoire, & Noël, 2009; Toomarian & Hubbard, 2018). Additionally, eye tracking studies have observed ways that adults explore fractions in comparison tasks to adaptively apply holistic or componential processing (Huber, Moeller, & Nuerk, 2014; Ischebeck, Weilharter, & Körner, 2016; Obersteiner & Tumpek, 2016). These findings indicate that holistic magnitude processing with fractions can occur, but descriptions of how people integrate fractions' and decimals' component parts to access this holistic sense of magnitude remain underspecified. One description of this process states that specific magnitude values of unique fractions are not encoded in long term memory (Kallai & Tzelgov, 2009), and that magnitudes of fractions (but not decimals) are accessed through online calculation (DeWolf, Grounds, Bassok, & Holyoak, 2014). The term calculation is often mentioned to describe the magnitude processing steps involved in understanding fractions. However, the term does not provide any insight into what is being calculated or how these processes differ when magnitudes are represented by decimal notation. For instance, calculating a fraction's magnitude may involve

syntactic processes akin to division where numerator and denominator components are mentally manipulated to compute or estimate a decimal equivalent (referred to hereafter as symbolic calculation). If people cannot directly access rational number meaning from a given fraction, but can more easily from decimal notation, then symbolic calculation may be a necessary intervening step during magnitude judgments. Alternatively, stages of so-called calculation may not involve symbolic mental representations at all but may instead involve mapping symbolic referents to nonsymbolic or amodal representations of rational number meaning. Specifically, if people can access a sense of a fraction's or decimal's magnitude without symbolic calculation, then more direct mappings between fractions and magnitudes may occur via forms of ratio processing similar to the perception of visual ratios and proportions (Jacob, Vallentin, & Nieder, 2012; Lewis, Matthews, & Hubbard, 2015). Nevertheless, magnitude processing with fractions and decimals in the form of symbolic calculation and perceptual-based ratio processing may not be mutually exclusive. Thus far previous studies have not been able to disentangle which processes are necessary and which processes individuals choose to apply contingent on task demands. 1.1. Symbolic calculation processing One process that may be necessary for understanding fraction magnitudes is a form of symbolic calculation, where fractions are converted to decimals via mental operations akin to division. Symbolic calculation may be necessary if the bi-partite form of fraction notation (numerator/denominator) is in some way incompatible with how the mind represents rational number magnitude meaning. Some researchers have proposed that decimal notation may be a more effective symbolic representation than common fractions for conveying rational number magnitude information because decimals are arranged in a base-10 notation consistent with whole numbers (DeWolf et al., 2015a). Others argue that the concept of rational numbers, and not the notation specifically, may be difficult to understand because humans' foundational number systems lack rational number intuitions to directly support fraction knowledge (Feigenson, Dehaene, & Spelke, 2004). Similarly, some argue that the semantic meaning of fraction magnitudes, unlike whole number magnitudes, are not encoded in long-term memory (DeWolf et al., 2014; Kallai & Tzelgov, 2009). Symbolic calculation may therefore be necessary for rearranging fractions into decimal forms more similar to whole numbers. Symbolic calculation processes may be learned through formal math education. Indeed some researchers argue that the ability to understand rational numbers may be limited to a subset of people in educated cultures who learn to formalize number representations beyond human core number intuitions (Feigenson et al., 2004). This may be particularly true if the basis of human numerical intuitions are whole number concepts established through experiences with countable and discrete numbers (Gallistel & Gelman, 1992). Fraction instruction often occurs in the math classroom years after children are introduced to whole number meaning(Common Core Standards Initiative, 2010). This convention may make learning about rational numbers difficult, if building an understanding of fractions and decimals requires reconceptualizing and reorganizing entrenched knowledge about whole numbers (DeWolf & Vosniadou, 2015; Gelman, 2015). If, as these theorists suggest, people lack a grounded sense of rational numbers, then accessing the meaning of decimals and fractions may indeed rely on generating an emergent sense of meaning through online strategies such as calculation with the integer components (Bonato et al., 2007; DeWolf et al., 2014). We suggest that symbolic calculation processes are likely to be cognitively effortful and slow. Previous studies have shown that adults can complete very rapid addition and subtraction operations, but not multiplication operations (Fayol & Thevenot, 2012). To our knowledge, no evidence has been observed to suggest that adults can similarly conduct rapid forms of 2

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

division, which are likely to support a precise translation from fractions to decimal equivalents. Furthermore, when adults completing magnitude comparison tasks report that converted fractions to decimals their performance was found to be relatively more accurate but also relatively slower than other strategies (Faulkenberry & Pierce, 2011). Empirical support for symbolic calculation accounts of magnitude processing has come from studies examining differences in magnitude comparison performances between fraction and decimal notations. DeWolf et al. (2014) examined differences in performance when adults compared magnitudes in decimal, fraction and multi-digit whole number notations. Distance effects were observed in all three notations: comparisons between numbers closer in magnitude led to more frequent errors and longer response times (Moyer & Landauer, 1967). The strongest distance effects were observed in fraction comparisons when participants were given unlimited time to respond (Experiment 2). Fraction comparisons took as much as seven seconds longer than decimal comparisons at equivalent distances. DeWolf et al. (2014) concluded from these findings that adults access a less precise magnitude representation from fractions through online strategies such as calculation. Similarly, in an experiment where adults made magnitude comparisons between mixed fraction-decimal pairs presented sequentially, accurate responses were faster when a fraction was presented prior to a decimal than vice versa (Zhang, Fang, Gabriel, & Szücs, 2015). The authors conclude that these differences reflect the processing time necessary to convert fractions into decimals. These conclusions are consistent with the suggestion that decimal notation offers more efficient access to unidimensional representations of magnitudes relative to fractions (DeWolf et al., 2015a) and thus support the argument that calculating a decimal quotient from a fraction may be an important route to accessing fraction magnitudes. If symbolic calculation is a necessary step in understanding the magnitude meaning of fractions, then differences in accuracy and response time between processing decimals and fractions may arise from the effortful steps necessary to compute fractions into decimals.

studies examining the relationship between magnitude processing with symbolic fractions and non-symbolic ratios (e.g. the relative number of dots in one array vs another). Matthews, Lewis, and Hubbard (2015) observed that individual differences in RPS acuity (magnitude judgments with integrated and separate line ratios and dot ratios) predict multiple measures of math competence, including algebra tests in college entrance exams, even when controlling for absolute magnitude processing and executive functions. These relationships between RPS acuity, fractions knowledge, and higher-level mathematical skills offer evidence that aspects of numerical cognition are grounded in lowerlevel perceptual abilities. Additionally, Matthews and Chesney (2015) found that adults exhibited distance effects during rapid discriminations of magnitude between symbolic fractions and non-symbolic representations of ratios and argued that perceiving the magnitude of fractions may involve direct mapping onto a common analog magnitude representation. Bonn and Cantlon (2017) observed that adults are sensitive to relative-magnitude information and that this acuity for recognizing ratio relationships is deployed automatically and ubiquitously to multiple sensory modalities (e.g., the perception of loudness, size, or number). They then suggest that ratio perception may therefore be a common code underlying a generalized magnitude system. Taken together, these studies suggest that magnitude processing of fractions and decimals may not be exclusively symbolic but may also be supported by forms of ratio processing grounded in perception (Jacob et al., 2012; Lewis et al., 2015). Symbolic calculation may therefore be a valuable mathematical strategy to reorganize or simplify fractions' bipartite form but may not be a necessary step in accessing the holistic magnitude meaning. Moreover, symbolic calculation may be one of several possible strategic alternatives that people might use to access the meaning of fractions and decimals. These rational number magnitude representations may be activated in parallel with explicit symbolic strategies. The extent to which responses on any given fraction comparison trial depend on perceptual-based ratio processing, symbolic calculation strategies or other possible shortcuts may therefore depend on a variety of task demands, such as the amount of time available for processing and the precision with which participants are required to compare fractions. In the current study, we manipulated these two sets of task demands to explore the necessary processes involved in magnitude processing with decimals and fractions.

1.2. Perceptual-based ratio processing Alternatively, some researchers argue that understanding rational number magnitudes from symbols may involve evoking a grounded sense of ratios and proportions built up from perceptual experiences (Jacob et al., 2012; Lewis et al., 2015). Throughout development, humans must perceive and interact with relative magnitudes, proportional quantities, and spatial relationships in our physical environment in ways that are not specifically enumerable or discrete. Perceiving these ratio relationships may even have cognitively primitive roots. In a review of studies describing how human and non-human primates perceive and discriminate nonsymbolic ratios, Jacob et al. (2012) argued that a magnitude understanding of symbolic fractions might be founded upon a perceptual understanding of ratios, rather than whole number knowledge or the perception of discrete quantities. Supporting this argument, Lewis et al. (2015) presented a review of findings across neuroscience, psychology and education research to argue for the existence of a ratio-processing system (RPS) sensitive to the perception of non-symbolic ratios in humans and non-human primates. They further argue that the RPS serves as a “neurocognitive start-up tool” capable of grounding a foundational understanding of symbolic decimals and fractions if properly leveraged during education. Thus, representations of rational number magnitude may be accessed through mechanisms analogous to the processing whole number magnitudes described in the triple-code model of number processing (Dehaene, 1992). Specifically, a common mental representation of rational number magnitude may support the understanding of fractions across varied external representations including symbolic common fractions, decimal fractions, nonsymbolic ratios (e.g. visual, temporal, or physical). Evidence that accessing the meaning of rational numbers involves processes shared with the perception of non-symbolic ratios comes from

1.3. Current study Previous findings that show more efficient magnitude processing with decimals than fractions have led some to suggest that accessing the magnitude meaning of fractions, but not decimals, involves calculation (DeWolf et al., 2014). However, details regarding how people differentially process fractions and decimals notation remain largely unexplored. For instance, does slower and less accurate magnitude comparison performance with fractions relative to decimals reflect processing stages necessary to convert fractions to an alternative form? If so, does this conversion involve calculating decimals from fractions via processes akin to division? How might the processes involved in accessing the magnitude meaning of decimals and fractions be contingent on task demands? Previous studies have largely observed magnitude comparison performance in the absence of a time limit (DeWolf et al., 2014; Faulkenberry & Pierce, 2011; Sprute & Temple, 2011), which may allow participants to apply slower strategies and forms of calculation to facilitate magnitude judgments with greater certainty. However, tasks without time limits do not directly test whether it is necessary to convert fractions to decimals to understand their meaning. In this study, we aimed to address this critical limitation and, in so doing, explore earlier stages of fraction and decimal processing that may occur before deliberate strategies can be applied. Specifically, we examined magnitude comparison performance under various task demands to test whether forms of symbolic calculation are necessary for accessing the magnitude meaning of decimals and 3

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

accurately as possible whether the larger fraction was presented on the left (press F key with left index finger) or on the right (press J key with right index finger). For experimental groups completing the task with catch trials mixed throughout the comparison trials, participants were also asked to identify if the two magnitudes were equivalent (press the spacebar with their thumbs). Participants were instructed to respond while the comparison and catch trial pairs were presented on the screen. Depending on group assignment, stimulus duration was restricted to 1000 ms, 2000 ms, or 5000 ms. Between each trial a fixation square was presented for 500 ms. Each block began with instructions, followed by 10 practice items not included in the experimental subset. Practice trials provided response feedback. Feedback included, “too slow” statements if responses were not entered within the response window and accuracy feedback to responses that were “correct” or “incorrect.” Subsequent experimental trials did not provide feedback.

Table 1 Factorial design of experimental groups. Response Window

1000 ms 2000 ms 5000 ms

Presence of catch trials Present

Absent

Experiment 2A Experiment 1A Experiment 3A

Experiment 2B Experiment 1B Experiment 3B

Note: Data from each experimental group was collected sequentially. We tested magnitude processing with 2000 ms response windows in Experiments 1A and 1B, further constrained responses to 1000 ms in Experiments 2A and 2B, and expanded response windows to 5000 ms in Experiments 3A and 3B.

fractions. 2. General methods

2.1.1. Fractions stimuli The experimental task included magnitude judgments between fraction pairs with an array of numerical distances from very dissimilar pairs (1/7 vs 8/9) to extremely close pairs (4/9 vs 3/7). Fraction pairs included in the experimental list were selected to minimize the possibility that participants could easily rely on componential strategies. First, we excluded any fraction comparison pairs that shared components (e.g., 1/3 vs. 1/7 or 2/7 vs. 4/7) (Bonato et al., 2007; Huber et al., 2014; Toomarian & Hubbard, 2018). Second, we excluded pairs where the numerators or denominators of the fractions were multiples of one another to rule out simple operations to reorganize pairs into forms with common components (e.g., 1/3 vs. 4/9) (Miller Singley & Bunge, 2018). Third, component-based strategies were further discouraged by including many numerator-incongruent trials, where judgments based on numerator value would lead to incorrect judgments (e.g. 1/2 vs. 3/8) (Meert et al., 2009; Sprute & Temple, 2011; Zhang et al., 2014). Collectively, these three stimulus selection factors were implemented to maximally reduce the possibility that participants could perform the task using fast component-based strategies and increase the possibility that participants' were engaging in holistic magnitude processing. To maximize our ability to observe the numerical distance effects on response times and accuracy, we carefully evaluated the impact that controlling the components of fraction pairs has on the distribution of comparison distances (see tables in Appendix A and B). This led us to retain pairs with unit fractions in order to maximize the overall range of pair distances and the number of pairs where the larger fraction had the smaller numerator (e.g. 1/3 vs. 2/7). Since every integer is a multiple of one, unit fractions were excluded from the criteria that pairs could not

2.1. Overall study design Over a series of six experiments, we observed how quickly and accurately adults were able to complete magnitude comparisons with fractions and decimals under varying task demands, which we manipulated to directly test the necessity of symbolic calculation processing. Task demands varied between the six experimental groups according to a 3 (response window) × 2 (catch trials) factorial design. First, we manipulated the length of response windows between groups to observe magnitude comparison performance when responses were limited to 5000, 2000, or 1000 milliseconds. Second, we observed the effect of asking participants to monitor for the presence of catch trials (pairs with equivalent magnitudes) among the comparison pairs (described in Section 2.1.1 below). This resulted in six experimental groups (see Table 1). All participants completed a magnitude comparison task with decimals and fractions using a paradigm adapted from Lyons, Ansari, and Beilock (2012) (see Fig. 1). Pairs of fractions were presented in blocks of common fraction pairs (FF; e.g., 1/4–2/9), decimal fraction pairs (DD; e.g., 0.250–0.222), and mixed fraction pairs (MX; e.g., 0.250–2/ 9). To avoid possible contamination in the within-notation conditions, MX blocks were always presented last. The order of DD and FF blocks was counterbalanced across participants. Stimuli were presented on a Windows 7 64-bit PC using E-prime 2.0.8.90a (Schneider, Eschman, & Zuccolotto, 2002). Each trial began with a fixation square followed by the stimulus pair presented simultaneously in white font in the middle of a black computer screen (21.5″ monitor). Participants were instructed to respond as quickly and

Fig. 1. Experimental paradigm depicting the presentation of fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) pairs. 4

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

share common multiple between their numerators or denominators (e.g., a pair such as 2/7 vs. 4/9 was excluded but not 1/7 vs 4/9). Using these criteria, we created a list of 108 unique comparison pairs using only the 27 irreducible proper fractions with single digit numerators and denominators. In this final list of 108 fraction-fraction (FF) pairs, each of the 27 single-digit fractions were presented 8 times. We included 108 unique fraction pairs , which was about three times as many unique pairs as previous magnitude comparison studies (DeWolf et al., 2014; Meert et al., 2009; Sprute & Temple, 2011; Zhang et al., 2014), to better observe how error rates and response times change as continuous functions of numerical distance between the pairs. The decimal equivalents of each fraction pair were then calculated to 3 decimal places to create the decimal-decimal (DD) comparison stimuli. Trailing zeros were added if the significant digits did not extend to the thousandths place. Participants completed two blocks of the 108 FF and DD pairs, which allowed the pairs to be presented in each orientation on the screen (e.g., Block 1 trial, 1/2–3/5 and Block 2 trial, 3/ 5–1/2). Since MX pairs yielded four possible combinations of orientation and notation, four lists of 108 MX pairs were assembled and matched into two sets. Half of the participants completed Set A (e.g., Block 1 trial, 1/2–0.600 and Block 2 trial, 3/5–0.500) and half completed Set B (e.g., Block 1 trial, 0.500–3/5 and Block 2 trial, 0.600–1/ 2). Stimuli within blocks were presented in random order. See Appendix C for a full list of stimuli pairs.

fractions will exhibit higher drift rates, corresponding to more efficient evidence accumulation to make a correct judgment. 2.2.2. Effect of limiting response windows on magnitude processing If symbolic calculation is a cognitively demanding and slow process that is necessary to access a fraction's magnitude, then limiting the time allowed to make a response should disrupt accurate holistic magnitude processing (H2). Specifically, markers of magnitude processing, in the form of negative distance effect slopes in error rates and reaction times, should be observed in all notation conditions when groups are given longer response windows (5000 ms). However, when response times are limited (1000 or 2000 ms response windows), these distance effects may not be observed in conditions that are thought to require calculation (FF and MX pairs). Decimal-decimal comparisons, requiring no calculation processes, should not be impacted by these shorter time limits. 2.2.3. Ability to rapidly recall fraction-decimal equivalence facts Besides effortful calculation, adults may convert fractions to decimals by recalling memorized fraction-decimal equivalence pairs. In the FF and DD conditions, identifying catch trials only requires recognizing two perceptually identical fractions or decimals. However, in the MX condition, recognizing catch trials involves identifying equivalent fraction-decimal pairs across perceptually dissimilar surface representations. Accuracy on these trials offers specific insight into whether adults are able to apply more rapid recall strategies and, if so, for which fractions. By analyzing the accuracy of MX catch trials for each of the 27 unique fractions (and corresponding decimal equivalents), we may identify and quantify any heterogeneity in adult knowledge and familiarity with specific fractions. If adults can rapidly recall memorized decimal-fraction facts and identify these pairs among their MX comparison judgments, then this should be seen in accurate (better than chance, see Section 2.3.3) catch trial performance even when response windows are restricted (e.g. 2000 ms) (H3). We predict that when response windows are limited (1000 ms or 2000 ms) accurate performance should be observed among a subset of well-known fractions (e.g. 1/2) (Schneider & Siegler, 2010), but not for the majority of the fractions included in the experiment. The longer response window of 5000 ms may provide sufficient time for the symbolic calculation processes described above. As such, accurate catch trial performance in these groups may correspond to either these more effortful calculation processes or fast recall-based processes.

2.1.2. Catch trials In half of the experimental groups, participants also were instructed to monitor for the presence of catch trials. Within each block, 27 catch trial pairs were randomly distributed among the comparison pairs. In the DD and FF condition, these catch trials appeared as two identical decimals (e.g. 0.125 vs 0.125) or fractions (e.g. 1/8 vs 1/8). In the MX condition, catch trials appeared as a single digit fraction and its decimal equivalent (e.g. 0.125 vs 1/8). Manipulating the presence of these of catch trials between groups allowed us to test the impact of adding additional task demands on the precision of magnitude comparison performance. More importantly, analyzing the accuracy of MX catch trials allowed us to evaluate how well participants could retrieve decimal-fraction equivalence facts. 2.2. Hypotheses 2.2.1. Effect of notation-specific symbolic calculation Previous findings with unrestricted response windows have shown that DD magnitude judgments require less time than FF trials. However, as the authors note, this difference may have been partially due to affordances of decimal notation (DeWolf et al., 2014). Specifically, two decimals may be compared more quickly by assessing the differences between the first digit (tenths place) in each decimal, rather than comparing the holistic rational number value that multi-digit decimals represent (Huber, Klein, et al., 2014). Therefore, MX comparisons in the current study offer specific insights into how adults access the holistic magnitude of decimals when comparing them against fractions. If symbolic calculation processing is necessary and specific to fractions but not decimals, then the time to make accurate judgments and error rates of these judgements should increase as comparisons contain more fractions (H1). Specifically, mean response times and error rates should be lowest in the DD condition, relatively higher in the MX condition, and highest in the FF condition. Furthermore, if response times are greater when comparisons involve more fraction calculations, then the presence of fractions in a pair may also correspond to greater error rates when response times are limited. Effects of notation on response times and error rates should also be reflected together in the drift rate parameters estimated by the diffusion model analysis. If decimal notation provides adults with a superior representation of magnitude meaning of rational numbers relative to fractions, then we would predict that responses to comparison pairs containing more decimals than

2.3. Analyses 2.3.1. Sample size and exclusion criteria We aimed to enroll at least 36 participants per experimental group in order to test our hypotheses with similar power to previous studies with fractions (Matthews & Chesney, 2015; Meert et al., 2009) and have several data points from each of our counterbalance orders (described in Section 2.1). After an initial round of data collection, we applied our exclusion criteria (see below). We then completed a second round of data collection to replace excluded participants and applied our exclusion criteria a second time before running our analyses. Variation across these aspects of participant recruitment and exclusion criteria resulted in slightly different sample sizes across the experiments. The following exclusion criteria were applied to data from comparison trials (excluding catch trials): Anticipation responses (RTs < 250 ms) and missed trials (no response entered) were removed from the data set prior to analysis. Our anticipation cutoff of 250 ms and exclusion of missed trials was based on previous research using diffusion models to examine performance on numerical tasks (Ratcliff, Thompson, & McKoon, 2015). Participants whose anticipation and missed response rates in any notation condition were far greater than the group mean (above 25%) were then excluded from analyses (For further details see Appendix D). After removing anticipation and missed 5

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

responses, participants were also excluded from our analysis if they had average error rates exceeding 40% in any notation condition. All participants were recruited from a subset of undergraduate students enrolled in educational psychology courses. Data for each of the six experiments were collected sequentially over multiple semesters. While this did not result in true random assignment between groups, we have no strong reasons to suspect that this method of group assignment contributed to large intergroup heterogeneity. Therefore, our primary hypotheses were tested within experimental groups, and further exploratory between-group analyses were also carried out to quantify and describe the effects of task demands manipulated across experimental groups.

2.3.5. Diffusion model analysis Response time and error rate results provide insight into the overall differences between magnitude comparison performance in the different notation conditions. However, as separate measures, response times and error rates do not offer specific insight into the subcomponents of the decision process. Although RT and error rate measures should be explained by a unified cognitive mechanism, RT and error rate results are often described separately and interpreted inconsistently across different cognitive tasks in the numerical cognition literature (for a review, see Ratcliff et al., 2015). To address the limitations of interpreting RTs and error rates separately, we also analyzed these two aspects of task performance together using a diffusion model analysis (Ratcliff, 1978; Ratcliff & McKoon, 2008). Using this theorybased model of decision making, we tested how correct and incorrect response times together explain differences in how quickly adults encode decimal and fractions notations (non-decision time), how efficiently they accumulate information from stimuli to made decisions (drift rate), and how carefully they approach their decision-making process (decision boundary). Specifically, Ratcliff's diffusion model (Ratcliff, 1978; Ratcliff & McKoon, 2008) depicted in Fig. 2, assumes that making a two-alternative forced choice decision involves extracting information from the stimulus through a noisy accumulation of information until one of two decision boundaries are reached. The Ratcliff diffusion model affords the opportunity to estimate and extract subprocesses of the magnitude comparison decision. Specifically, we used the Fast-DM program (Voss & Voss, 2007) to estimate three unique components of the magnitude comparison decision assumed by Ratcliff's diffusion model.

2.3.2. Analysis of notation effects on response time latency and error rate To analyze whether the number of fractions in a comparison pair corresponded to the amount of symbolic calculation processing (H1), we tested for group mean differences response times (for accurate responses) between the FF, DD, and MX conditions. Specifically, we calculated each participant's mean response time latency for each notation condition, and we tested for differences in mean response times for each condition using within-subjects ANOVAs and planned pairwise comparisons. Using the same approach, we also compared differences in error rates between conditions. 2.3.3. Analysis of numeric distance effect slopes To test for evidence of holistic magnitude processing, we examined whether response time and error rate patterns exhibited numerical distance effects (H2), where decreasing numerical distance between the two fractions was associated with increasing error rates and response times (Moyer & Landauer, 1967). We estimated distance effect slopes for each participant by modeling the distribution of participants' response times and error rate probabilities for each notation condition as a function of the numerical distance between the two magnitudes in the comparison pair. Using only correct responses, linear regression models were used to fit numerical distance effect slopes for response times. Logistic models were used to predict error rate probabilities. Following the repeated measures analysis approach described by Lorch and Myers (1990), each participant's estimated slope coefficients for the three notation conditions provided the units of analysis for single-sample ttests to determine if the group mean distance effect slopes were negative (non-zero) in each condition. 2.3.4. Catch trial analysis Catch trials were analyzed separately from comparison trials for the three experimental groups assigned to monitor for and respond to equivalent pairs. First, we compared group mean catch trial accuracy across notation conditions to test for differences in identifying catch trials composed of perceptually identical fractions (e.g., 1/2 vs 1/2) and decimals (e.g. 0.500 vs 0.500) from identifying catch trials composed of visually unique fractions and decimals that represent the same magnitude (e.g. 1/2 vs 0.500). To test the predictions of H3 we analyzed group mean accuracy in the mixed condition where participants needed to recognize a decimalfraction equivalence pair instead of making a magnitude comparison response. Specifically, we determined if adults tend to have these equivalence pairs memorized, and if so for which pairs, by testing whether the participants' accuracy in these trials exceeded chance performance. We estimated chance performance on catch trials for each decimal-fraction pair by calculating the binomial probability of a trial being a catch trial based on the proportion of catch trials to comparison trials (20%) and the number of opportunities each group had to identify each fraction-decimal pair (2 trials per pair per participant). Group mean accuracy on a given decimal-fraction pair above 28% (approximately, group specific values presented below) was therefore determined to be performance greater than chance.

Fig. 2. An illustration of the Ratcliff diffusion model. The top panel's jagged lines depict the diffusion model's assumed noisy accumulation of information when making a decision. Information is accumulated over time until reaching decision boundary threshold. The bottom panel indicates how the model breaks total response time into this decision process of accumulating information and non-decision components of stimulus encoding and response output. Figure originally published in Ratcliff et al., 2015.

First, the diffusion model estimates the drift rate, corresponding to how quickly participants accumulate evidence to make a correct decision. Second, the model estimates the length of non-decision time, quantifying the amount of time necessary to encode the stimulus into a decision related representation and the time to execute the physical response when a decision is made. In our interpretation of non-decision time parameters, we assumed that the time to generate a physical response was consistent across conditions, as responses always involved the same simple button presses. Therefore, we attributed variation in non-decision times to differences in the encoding process. Third, the model estimates participant's decision boundaries describing how 6

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Table 2 Group mean response times across experimental groups. DD

1000-Catch 1000-NoCatch 2000-Catch 2000-NoCatch 5000-Catch 5000-NoCatch

MX

FF

Mean

(SD)

Mean

(SD)

Mean

(SD)

717 688 894 846 918 921

109.2 122.0 220.9 210.4 280.6 343.8

720 680 1152 1047 1589 1433

132.9 139.9 335.0 305.9 808.6 723.5

717 679 1124 1077 1605 1635

136.2 150.0 341.6 319.8 863.5 844.7

Note: Group means for decimal-decimal (DD), mixed pair (MX), and fraction-fraction (FF) comparisons were calculated from each subject's mean RTs from correct responses in each condition.

carefully participants made decisions based on speed-accuracy tradeoffs. Wider decision boundaries represent more careful decisions such as emphasizing accuracy over speed. For example, if differences in speed accuracy tradeoffs are driving the error rate and response time effects, we would predict shorter response times to correspond with narrow decision boundaries and longer response times to correspond to wider decision boundaries.

3.2. Results 3.2.1. Response times As seen in Fig. 3a, mean response times differed between notations, F(2,68) = 83.5, p < .001. Means and standard deviations of response times for each notation condition are presented in Table 2. Participants made accurate magnitude judgments with two decimals (DD) about 230 ms faster than when comparing two fractions (FF), t(68) = −10.5, p < .001, and 258 ms faster than mixed pairs (MX), t(68) = −11.8, p < .001. Contrary to our prediction that FF comparisons would require more processing time to calculate two fractions than MX comparisons with one fraction (H1), no significant difference was observed between response times with FF and MX pairs, t(68) = 1.25, p = .422. These results indicate that reaching accurate magnitude judgments with two fractions or a mixed pair required the same amount of processing time. Evidence of holistic magnitude processing with fractions and decimals was observed in significant negative distance effect slopes for all three notation conditions within the time limit of 2000 ms (Table 4). Critically, as seen in Fig. 3a, the magnitude of these slopes did not differ between notations, F(2,68) = 0.313, p = .732. For all notations, mean response times decreased as the magnitude between fractions in the comparison pair increased, with parallel slopes between each notation. Therefore, we observed evidence of holistic magnitude processing in all conditions, including pairs with fractions, despite implementing a restrictive response window of 2000 ms. These results are inconsistent with the predictions that a 2000 ms response window would disrupt magnitude processing by preventing symbolic calculation strategies (H2).

3. Experiment 1A: Comparisons with 2000 ms response windows and catch trials 3.1. Methods 3.1.1. Participants Thirty-seven undergraduate students participated for partial course credit. Two participants were excluded because > 25% of their responses in a condition were missed responses. On average, only 3.5% of trials were missed responses. 35 adults were included in the final experimental group (30 female, 4 male, 1 not reporting; average age = 19.6, SD = 1.2). 3.1.2. Procedure All participants completed the magnitude comparison task as described in Section 2.1. Response windows limited participants to respond within 2000 ms of stimulus onset, and participants were instructed to monitor for catch trials which were randomly distributed among the comparison trials.

Fig. 3. Magnitude comparison performance in Experiment 1A with 2000 ms response windows and catch trials present. Thicker lines depict group averages of (a) individual linear models predicting mean response times and (b) logistic models predicting error rate probabilities across numerical distance for fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) comparisons. (c) Graphical renderings of diffusion model parameters depict the group averages of individuals' predicted non-decision time (length of the horizontal midline), drift rate (slope from the midline to the positive boundary), and decision boundaries (width between upper and lower bounds). Equivalent overlapping decision boundaries across notations are shown with dotted lines. 7

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Table 3 Group mean error rates across experimental groups. DD

1000-Catch 1000-NoCatch 2000-Catch 2000-NoCatch 5000-Catch 5000-NoCatch

MX

FF

Mean

(SD)

Mean

(SD)

Mean

(SD)

0.107 0.122 0.046 0.046 0.034 0.031

0.310 0.328 0.210 0.209 0.182 0.174

0.271 0.271 0.211 0.189 0.160 0.134

0.445 0.444 0.408 0.391 0.367 0.341

0.251 0.257 0.210 0.204 0.161 0.155

0.434 0.437 0.407 0.403 0.367 0.362

Note: Group means for decimal-decimal (DD), mixed pair (MX), and fraction-fraction (FF) comparisons were calculated from each subject's mean error rate in each condition.

3.2.2. Error rates As seen in Fig. 3b, we observed significant differences between notations in the frequency of errors participants made, F(2,68) = 324, p < .001. Means and standard deviations of error rates for each notation condition are presented in Table 3. Participants made errors less often when comparing DD pairs than when comparing FF, t (68) = −22.0; p < .001, or MX pairs, t(68) = −22.2, p < .001. However, no significant difference was observed in error rates between MX pairs with one fraction or FF pairs with two, t(68) = 0.190, p = .980. Consistent with response time results, the similarity in error rates between the MX and FF conditions does not suggest that accessing the magnitudes of two common fractions required more cognitively taxing forms of symbolic calculation than when comparisons were made with one fraction and one decimal (H1). Furthermore, applying a 2000 ms response window did not differentially affect the accuracy of pairs with two fractions (FF) relative to pairs with only one fraction (MX). Despite restrictions on response time, group mean error rates in both conditions with fractions were < 25%. As seen in Fig. 3b, numerical distance effects were observed in all notation conditions (Table 5) in the form of decreasing error rates as the difference between magnitudes in the pairs increased. These results are inconsistent with the prediction that restricting response times would inhibit holistic magnitude processing in comparisons containing fractions (H2). Significant differences were observed in the rate that error rates decreased over increasing distance, F(2,68) = 7.23, p = .001, but these findings may be influenced by a floor effect in the DD condition.

Adults reach almost errorless performance in all conditions as comparisons pairs approach the furthest numerical distances, however higher error rates at near distances for comparisons containing at least one fraction lead to steeper probability curves in the FF, t (68) = −3.01, p = .008, and MX comparisons, t(68) = 3.52, p = .001, than DD comparisons. FF and MX error rate probability curves showed no difference, t(68) = 0.52, p = .864.. 3.2.3. Diffusion model parameters 3.2.3.1. Drift rates. We observed significant differences in drift rates between the three notation conditions, indicating that the manipulation of fractions notation had a significant effect on how quickly participants were able to extract magnitude information from the comparison pair, F (2,68) = 485, p < .001. As seen in Fig. 3c, drift rates were greater for DD comparisons (steeper slope) than MX, t(68) = 27.1, p < .001, or FF comparisons, t(68) = 26.8, p < .001. However, drift rates of FF comparisons were no less efficient than MX comparisons, t (68) = −0.300, p = .951. These results indicate that when decimals are compared within notation, information to make accurate magnitude decisions is accumulated rapidly. This appears consistent with the hypothesis that decimals may be a more efficient symbolic representation of magnitude than fractions (H1); however, the similarity in drift rates between MX and FF comparisons suggests that this predicted efficiency with decimals does not translate to an advantage when making MX comparisons. 70 60

0.8 50 0.6

40 30

0.4

20 0.2 10 0.0

count of correct trials

proportion of correct trials

1.0

0

− 8/9 − 7/9 − 5/9 − 4/9 − 2/9 − 1/9 − 7/8 − 5/8 − 3/8 − 1/8 − 6/7 − 5/7 − 4/7 − 3/7 − 2/7 − 1/7 − 5/6 − 1/6 − 4/5 − 3/5 − 2/5 − 1/5 − 3/4 − 1/4 − 2/3 − 1/3 − 1/2 89 0.8 78 0.7 56 0.5 44 0.4 22 0.2 11 0.1 75 0.8 25 0.6 75 0.3 25 0.1 57 0.8 14 0.7 71 0.5 29 0.4 86 0.2 43 0.1 33 0.8 67 0.1 00 0.8 00 0.6 00 0.4 00 0.2 50 0.7 50 0.2 67 0.6 33 0.3 00 0.5

catch trial pair Fig. 4. Catch trial accuracies for mixed decimal-fraction pairs in Experiment 1A with 2000 ms response windows. Purple points indicate group mean accuracy for identifying each catch trial pair. Error bars indicate 95% confidence intervals around the group mean. The red horizontal line marks the proportion of trials the group would have to identify to say that performance was greater than chance. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 8

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

3.2.3.2. Non-decision times. We observed significant differences in the mean non-decision time between notations indicating that there were differences in how adults encoded the comparison pairs, F (2,68) = 24.2, p < .001. As seen in the Fig. 3c, this main effect is driven by significantly faster non-decision times when comparisons are made between two decimals than when comparisons are made between two fractions, t(68) = −4.84, p < .001, or MX pairs, t(68) = −6.74, p < .001. However, we did not observe a significant difference in nondecision times between FF and MX comparisons, t(68) = 1.90, p = .138.

4. Experiment 1B: Comparisons with 2000 ms response windows and no catch trials 4.1. Methods 4.1.1. Participants Forty undergraduate students participated for partial course credit. Data reduction (described in Section 2.1.3) was < 25% for all participants in all notation conditions. Two participants were excluded for error rates > 40% in a notation condition. The remaining 38 adults (33 F/5 M; average age = 19.8, SD = 1.3) were included in the analyses.

3.2.3.3. Decision boundaries. Comparisons of decision boundary parameters across notation conditions revealed no differences in how carefully participants made responses, F(2,68) = 0.006, p = .995. Therefore, these model estimates indicate that differences observed in group performance (RT or error rate) cannot be attributed to differences in how individuals set speed-accuracy tradeoffs.

4.1.2. Procedure The experimental procedure, materials, and fraction stimuli were identical to Experiment 1A except that catch trials were not interspersed among comparison pairs and the corresponding catch trial instructions were omitted. Identical to Experiment 1A, participants had 2000 ms to respond.

3.2.4. Catch trial analysis In the group given 2000 ms to respond, there were significant differences in catch trial accuracy between notation, F(2,68) = 372.0, p < .001. This main effect was driven by lower ability to accurately identify MX catch trials (M = 0.321, SD = 0.391) than FF (M = 0.932, SD = 0.186), t(68) = 23.2, p < .001, and DD (M = 0.954, SD = 0.156), t(68) = 24.0, p < .001. No differences were observed between the two conditions were participants only had to match identical decimal or fraction catch trials, t(68) = 0.678, p = .677. As seen in Fig. 4, participant's abilities to accurately identify fraction-decimal equivalence pairs (MX condition) varied across the fraction values. Fractions with denominators of 5 or smaller and unit fractions (numerator of 1) with larger denominators were identified at a rate greater than chance (28.6%). Furthermore, participants identified the fraction-decimal pairs of 1/2, thirds, and fourths more than half of the time. These results are consistent with our predictions that adults may have more rapid abilities to recall decimal values for a subset of fractions, even under time pressures that may limit symbolic calculation processing. However, this rapid recall processing does not appear to be accurately applied to the majority of single-digit fractions used in this study (H3).

4.2. Results 4.2.1. Response times As seen in Fig. 5a, differences were observed between notation conditions in the mean response times, F(2, 74) = 57.1, p < .001. On average, accurate magnitude judgments for DD pairs were made faster than FF, t(74) = −9.83, p < .001, or MX pairs, t(74) = −8.55, p < .001. Consistent with Experiment 1A, mean response times of FF comparisons were not significantly different from MX comparisons, t (74) = −1.28, p = .407. Across both experimental groups, we did not observe evidence that comparisons pairs with more fractions would require more processing time (H1). On average, linear models of individuals' response times across numerical distance show evidence for magnitude processing in the form of negative distance effect slopes in all notations (Table 4). As seen in Fig. 5a, the magnitudes of these slopes differed significantly across notations, F(2,74) = 5.10, p < .01. Specifically, as the numerical distance between fractions increased, response times decreased at a greater rate in MX comparisons than FF comparisons, t(74) = 3.18, p = .004. No differences in distance effects slopes were observed between DD and MX or FF comparisons (DD – MX: t(74) = 1.85, p = .154; DD – FF: t(74) = −1.33, p = .378).

Fig. 5. Magnitude comparison performance in Experiment 1B with 2000 ms response windows and without catch trials. Thicker lines depict group averages of (a) individual linear models predicting mean response times and (b) logistic models predicting error rate probabilities across numerical distance for fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) comparisons. (c) Graphical renderings of diffusion model parameters depict the group averages of individuals' predicted non-decision time (length of the horizontal midline), drift rate (slope from the midline to the positive boundary), and decision boundaries (width between upper and lower bounds). Equivalent overlapping decision boundaries across notations are shown with dotted lines. 9

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Table 4 One-group t-tests of response time slope coefficients. df

1000-Catch 1000-No Catch 2000-Catch 2000-No Catch 5000-Catch 5000-No Catch

31 36 34 37 39 35

DD

MX

FF

mean

t

p

mean

t

p

mean

t

p

−137.8 −140.6 −298.4 −289.3 −320.1 −331.3

−20.8 −19.3 −23.5 −23.4 −21.8 −19.8

< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

−43.8 −58.9 −324.2 −332.2 −1059.5 −1028.4

−5.7 −7.6 −10.3 −14.2 −11.8 −13

< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

−69.2 −70.9 −310.6 −259.5 −962.4 −1082.6

−6.7 −7.2 −10.7 −15.0 −9.6 −13.0

< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

Note: Slope coefficients were estimated for each subject using linear regression predicting their distribution of response times over changes in numerical distance and notation condition.

4.2.2. Error rates Analysis of mean error rates across notations showed significant differences between the notation conditions, F(2,74) = 262, p < .001. Means and standard deviations of error rates for each comparison notation are presented in Table 3. Similar to the results of Experiment 1A when catch trials were included, planned comparisons showed that participants made errors less often when comparing DD pairs than FF, t (68) = −20.8; p < .001, or MX pairs, t(68) = 18.7, p < .001. However, differences in error rates between MX pairs with one fraction or FF pairs with two fractions were not statistically significant, t (68) = 2.08, p = .094. Numerical distance effects on error rates were observed in all notation conditions (Table 5). As seen in Fig. 5b, significant differences were observed in the rate that error probability decreased over expanding distance, F(2,74) = 8.14, p < .001, and these differences were potentially influenced by a floor effect in the DD condition. The strongest numerical distance effects were observed in the FF and MX conditions (DD – FF: b = −8.07, t(74) = −3.73, p < .001; DD – MX: b = −6.93, t(74) = 3.20, p = .004) relative to DD. FF and MX slopes showed no difference, t(74) = 0.525, p = .859.

4.2.3.2. Non-decision times. Congruent with the non-decision time estimates in Experiment 1A when catch trials were absent, a significant main effect of notation was observed, F(2,74) = 15.9, p < .001. Likewise, this main effect is driven by significantly faster non-decision times with DD comparisons than FF, t(74) = −5.09, p < .001, or MX comparisons, t(74) = −4.63, p < .001. No difference in mean non-decision times was observed between FF and MX comparisons, t(74) = 0.465, p = .888. 4.2.3.3. Decision boundaries. Similar to the findings of Experiment 1A, within-subjects analysis of decision boundary parameters showed no significant differences, F(2,74) = 0.337, p = .715. Therefore, we assume that differences in RT and error rate observed between notations were not driven by varying speed accuracy tradeoffs. 5. Experiment 2A: Comparisons with 1000 ms response window and catch trials In Experiments 1A and 1B, we tested the hypothesis that magnitude processing with fractions, but not decimals, involves symbolic calculation by examining magnitude comparison performance judgments were restricted to a 2000 ms response window. Against our prediction that this 2000 ms limit on response time would inhibit accurate magnitude processing by disrupting symbolic calculation, we still observed evidence of holistic magnitude processing (distance effects) in comparisons with symbolic fractions (MX and FF). These results may suggest that accurate magnitude processing with decimals and fractions can occur without deliberate and cognitively effortful calculation processes. Alternatively, it remains possible that response time limit in Experiments 1A and 1B was not sufficient rule out symbolic calculation processing. Given our participants are highly educated college students, converting fractions to decimals may still occur within 2000 ms. Experiments 2A and 2B were therefore carried out to examine the same measures of magnitude comparison performance with the response window reduced to 1000 ms.

4.2.3. Diffusion model parameters 4.2.3.1. Drift rates. Differences were observed between drift rates estimated for each notation condition, F(2,74) = 401.9, p < .001. This effect of notation was driven by lower drift rates in the FF, t (74) = −25.4, p < .001, and MX conditions, t(74) = −23.6, p < .001, relative to the DD condition. No difference was observed between drift rates calculated from FF and MX comparisons, t (74) = 1.89, p = .141. This pattern of results is consistent with the results of Experiment 1A. Specifically, we observed that the predicted advantages of decimal notation for accessing rational number magnitudes (H1) were observed within DD comparisons but not did not contribute to more efficient evidence accumulation to make MX comparisons relative to FF.

Table 5 One-group t-tests of logit slope coefficients of error rate. df

1000-Catch 1000-No Catch 2000-Catch 2000-No Catch 5000-Catch 5000-No Catch

31 36 34 37 39 35

DD

MX

FF

mean

t

p

mean

t

p

mean

t

p

−7.34 −6.60 −11.73 −14.33 −15.0 −8.25⁎

−8.0 −11.4 −6.0 −5.5 −3.11 −5.458⁎

< 0.001 < 0.001 < 0.001 < 0.001 0.003 < 0.001⁎

−3.78 −4.28 −6.18 −7.40 −8.38 −9.85

−14.4 −14.9 −14.3 −14.0 −9.11 −11.3

< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

−4.35 −4.49 −7.00 −6.27 −8.22 −9.94

−14.4 −20.8 −16.3 −20.4 −11.2 −11.9

< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

Note: Slope coefficients were estimated for each subject using logistic regression models predicting the log odds of errors over numerical distance and notation condition. ⁎ One participant's extremely negative slope coefficient (−573) was excluded from the one-group t-test. 10

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Fig. 6. Magnitude comparison performance in Experiment 2A with 1000 ms response windows and catch trials present. Thicker lines depict group averages of (a) individual linear models predicting mean response times and (b) logistic models predicting error rate probabilities across numerical distance for fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) comparisons. (c) Graphical renderings of diffusion model parameters depict the group averages of individuals' predicted non-decision time (length of the horizontal midline), drift rate (slope from the midline to the positive boundary), and decision boundaries (width between upper and lower bounds). Equivalent overlapping decision boundaries across notations are shown with dotted lines.

5.1. Methods

presented in Table 3. Consistent with the results of Experiments 1A and 1B, participants given only 1000 ms to respond made errors the in the DD condition less often than in the FF, t(62) = −14.1, p < .001, and MX conditions, t(62) = 16.9, p < .001. Contrary to the prediction that having more factions in a pair to calculate would lead to higher error rates (H1), we observed the opposite effect with 2.7% higher mean errors rates in the MX condition than the FF condition, t(62) = 2.78, p = .015. As seen in Table 5, error rate probabilities significantly increased as the numerical distance between comparison pairs decreased for all notations. These findings of magnitude dependent processing under strict response windows is inconsistent with the prediction that time consuming calculation steps are necessary for processing fraction magnitudes (H2). These distance effect slopes, as seen in Fig. 6b, were not equivalent across conditions, F(2,62) = 13.2, p < .001. Distance effects were stronger (more negative) in the FF and MX conditions than in the DD condition (FF – DD: t(62) = −4.02, p < .001; MX – DD: t (62) = −4.78, p < .001), but slope coefficients were not different between the FF and MX conditions, t(62) = 0.762, p = .727.

5.1.1. Participants Forty-one undergraduate students participated for partial course credit. Three participants were excluded because data reduction from cleaning was > 25% in at least one notation condition. Six participants were excluded for error rates > 40% in a notation condition. The remaining 32 adults (26 F/6 M; average age = 19.8, SD = 1.4) were included in the analyses. 5.1.2. Procedure The experimental procedure, materials, and fraction stimuli were identical to the previous experiments (see Section 2.1). Participants in this experimental condition completed magnitude comparisons with additional catch trials and were given only 1000 ms to respond. 5.2. Results 5.2.1. Response times When given 1000 ms to respond, adults showed no differences in mean response times between notations, F(2, 62) = 0.612, p = .546. Under this stringent limit on response times, these results did not support the hypothesis that magnitude processing with decimals should be more efficient than fractions (H1). Inconsistent with response times observed in Experiments 1A and 1B, we did not observe the effects of efficient DD processing relative to MX or FF that were observed in Experiments 1A and 1B. Despite the restrictive 1000 ms response window, linear models of participant response times on average showed evidence of holistic magnitude processing (H2) in the form of significant negative distance effect slopes in all three conditions (Table 4). These distance effect slopes were not equivalent across all conditions, F(2,62) = 53.0, p < .001. Specifically, response times in the DD condition showed the strongest negative slope compared to FF, t(62) = −7.99, p < .001, and MX conditions, t(62) = −9.62, p < .001. No significant difference was observed between the slopes fit to FF and MX responses, t (62) = 1.63, p = .232.

5.2.3. Diffusion model parameters 5.2.3.1. Drift rates. The main effect of notation on drift rates, F (2,62) = 132, p < .001, was driven by faster rates of evidence accumulation when comparing DD pairs than when comparing FF, t (62) = 13.6, p < .001, or MX pairs, t(62) = 14.4, p < .001. Estimated drift rates for FF and MX comparisons did not differ, t (62) = 0.798, p = .704. Consistent with the findings of Experiments 1A and 1B when participants had 2000 ms to respond, these results do not support H1. The advantages of decimal notation for rapid evidence accumulation were evident in within notation comparisons, but not when magnitude judgments were made between decimal and fractions notations. 5.2.3.2. Non-decision times. Comparisons of non-decision time parameters across notation conditions revealed no significant differences, F(2,62) = 0.358, p = .701. Thus, when response windows were restricted to 1000 ms, we did not observe the differences in non-decision times between notation conditions present when participants had 2000 ms to respond in Experiments 1A and 1B.

5.2.2. Error rates As seen in Fig. 6b, we observed significant differences in error rates between the notation conditions, F(2,62) = 164.1, p < .001. Means and standard deviations of error rates for each comparison notation are

5.2.3.3. Decision boundaries. Within-subjects analysis of decision boundary parameters showed differences across notation, F 11

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

64 56

0.8 48 40

0.6

32 0.4

24 16

0.2

count of correct trials

proportion of correct trials

1.0

8 0.0

0 − 8/9 − 7/9 − 5/9 − 4/9 − 2/9 − 1/9 − 7/8 − 5/8 − 3/8 − 1/8 − 6/7 − 5/7 − 4/7 − 3/7 − 2/7 − 1/7 − 5/6 − 1/6 − 4/5 − 3/5 − 2/5 − 1/5 − 3/4 − 1/4 − 2/3 − 1/3 − 1/2

89 0.8 78 0.7 56 0.5 44 0.4 22 0.2 11 0.1 75 0.8 25 0.6 75 0.3 25 0.1 57 0.8 14 0.7 71 0.5 29 0.4 86 0.2 43 0.1 33 0.8 67 0.1 00 0.8 00 0.6 00 0.4 00 0.2 50 0.7 50 0.2 67 0.6 33 0.3 00 0.5

catch trial pair Fig. 7. Catch trial accuracies for mixed decimal-fraction pairs in Experiment 2A with 1000 ms response windows. Red points indicate group mean accuracy for identifying each catch trial pair. Error bars indicate 95% confidence intervals around the group mean. The red horizontal line marks the proportion of trials the group would have to identify to say that performance was greater than chance. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6.1.2. Procedure The experimental procedure, materials, and fraction stimuli were identical to the previous experiments, as described in Section 2.1 except catch trials were not interspersed among comparison pairs and the corresponding catch trial instructions were omitted. Participants again had 1000 ms to respond.

(2,62) = 6.77, p = .002. With only 1000 ms to make a response, decision boundaries estimated by the diffusion model were wider (more careful) when comparing two decimals than when comparing two fractions, t(62) = 3.27, p = .003, or mixed pairs, t(62) = 3.10, p = .006. However, mean decision boundary estimates fraction to fraction comparisons were not significantly different from mixed pair decision boundaries, t(62) = 0.181, p = 982.

6.2. Results

5.2.4. Catch trial analysis In the group given 1000 ms to respond, there were significant differences in catch trial accuracy between notations, F(2,62) = 349.1, p < .001. This main effect was driven by lower ability to accurately identify MX catch trials (M = 0.085, SD = 0.214) than FF (M = 0.732, SD = 0.342), t(62) = 22.2, p < .001, and DD (M = 0.769, SD = 0.324), t(62) = 23.5, p < .001. No differences were observed between the two conditions where participants only had to match identical decimal or fraction catch trials, t(68) = 1.3, p = .422. Accuracy across fraction values varied slightly but did not exceed chance performance (28.1%) for any of the fraction values (Fig. 7). Catch trials with ½ and 0.500 were the most accurate fraction value yet were only identified 26.6% of the time. These results do not offer any additional support of the prediction that adults have rapid recall abilities with certain decimal-fraction pairs (H3). Given that participants in Experiment 1A were able to identify catch trials within specific fraction-decimal pairs when given 2000 ms, these results may indicate that even when adults may have abilities to recall decimal-fraction equivalence facts, they are limited in their ability to express this knowledge accurately within one second.

6.2.1. Response times Comparison of mean response times between notation conditions did not identify any significant differences, F(2, 72) = 0.754, p = .474. Similar to results in Experiment 2a, when adults were restricted to a 1000 ms response window, processing times to make accurate magnitude judgments were similar regardless of how many decimals or fractions are in the comparison pair (H1). Despite the restrictive 1000 ms response window, the distribution of response times across numerical distance showed significant distance effects in all notation conditions (H2) (Table 4), and the magnitude of these negative slopes differed across conditions, F(2,72) = 42.5, p < .001. As seen in Fig. 8a, response times from DD comparisons showed the strongest negative slope compared to FF, t(72) = −8.04, p < .001, and MX comparisons, t(72) = −7.92, p < .001. However, no significant difference was observed between the FF and MX conditions, t(72) = 0.11, p = .993. 6.2.2. Error rates Significant differences were observed between mean error rates across notations, F(2,72) = 194, p < .001. Means and standard deviations of error rates for each comparison notation are presented in Table 3. Consistent with all previous experimental groups, magnitude comparisons with two decimals were associated with significantly fewer errors than comparing two fractions, t(72) = −16.1, p < .001, or mixed pairs, t(72) = −17.9, p < .001. Unlike when catch trials were present in the task (Experiment 2A), error rates were not higher among MX comparisons than FF comparisons, t(72) = 1.78, p = .176. Therefore, error rate results in this experiment, as well as the previous 3 experiments, do not indicate that limiting response times disrupted magnitude processing more when comparison pairs included two fractions relative to when one fraction was compared against a decimal (H1).

6. Experiment 2B: Comparisons with 1000 ms response windows and no catch trials 6.1. Methods 6.1.1. Participants Thirty-eight undergraduate students participated for partial course credit. One participant was excluded because data reduction from cleaning was > 25% in at least one notation condition. No additional participants were excluded for error rates > 40% in a notation condition. The remaining 37 adults (28 F/8 M/1 NR; average age = 20.2, SD = 2.9) were included in the analyses. 12

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Fig. 8. Magnitude comparison performance in Experiment 2B with 1000 ms response windows and without catch trials. Thicker lines depict group averages of (a) individual linear models predicting mean response times and (b) logistic models predicting error rate probabilities across numerical distance for fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) comparisons. (c) Graphical renderings of diffusion model parameters depict the group averages of individuals' predicted non-decision time (length of the horizontal midline), drift rate (slope from the midline to the positive boundary), and decision boundaries (width between upper and lower bounds).

strict response windows. Experiments 3A and 3B were carried out to see how patterns of performance compare to when response windows are relaxed to 5000 ms. If longer response windows allow for symbolic calculation processes to occur, Experiments 3A and 3B may show unique evidence of these processes relative to the previous experiments.

Furthermore, evidence of magnitude processing in the form of distance effects on error rates was observed in all notation conditions (Table 5), and the magnitudes of these distance effects differed between conditions, F(2,72) =12.3, p < .001. As seen in Fig. 8b, the slopes of error probability were stronger (more negative) in the FF, t (72) = −4.09, p < .001, and MX comparisons, t(72) = −4.48, p < .001, than the DD comparisons. However, between FF pairs with two fractions and MX pairs with a decimal and a fraction, distance effect slopes again showed no difference, t(72) = 0.394, p = .918.

7.1. Methods 7.1.1. Participants Forty-three undergraduate students participated for partial course credit. One participant was excluded because data reduction from cleaning was > 25% in at least one notation condition. Two additional participants were excluded for error rates > 40% in a notation condition. The remaining 40 adults (32 F/8 M; average age = 20.2, SD = 1.5) were included in the analyses.

6.2.3. Diffusion model parameters 6.2.3.1. Drift rates. Effects of notation on drift rates show the same pattern of results as when catch trials were present in Experiment 2A. Specifically, significant differences between estimated drift rates across the notation conditions, F(2,72) = 135, p < .001, were driven by faster rates of evidence accumulation in DD comparisons than in either FF, t(72) = 14.0, p < .001, or MX, t(72) = 14.4, p < .001, comparisons. Mean drift rates in the FF and MX conditions were not significantly different from each other, t(72) = 0.467, p = .887. These results, along with the findings of the previous three experiments, are inconsistent with the hypothesis that magnitude processing with fractions is less efficient than decimal processing. We also do not see evidence in these experiments that symbolic calculation is a necessary part of processing the magnitude of each fraction.

7.1.2. Procedure The experimental procedure, materials, and fraction stimuli were identical to the previous experiments, described in Section 2.1, except participants were given 5000 ms to respond. Catch trials were included among comparison pairs during the task. 7.2. Results 7.2.1. Response times Analysis of mean response times showed significant differences between notations, F(2,78) = 99.4, p < .001. With 5000 ms to respond, adults made accurate magnitude judgments with two decimals about 686 ms faster than judgments with FF, t(78) = −12.4, p < .001, and about 668 ms faster than MX pairs, t(78) = −12.1, p < .001. No significant difference was observed between mean response times in the FF and MX conditions, t(78) = 0.309, p = .949. This pattern of notation effects on response times was similar to Experiments 1A and 1B when response times were restricted to 2000 ms. Despite having more time to make magnitude judgments, these results do not support the hypothesis that FF comparisons would involve more symbolic calculation to access the magnitudes of both fractions than MX comparison with only one fraction (H1). Consistent with the results of the previous four experiments, (nonzero negative) distance effect slopes on response times were again observed in all notation conditions (Table 4). As seen in Fig. 9a, these slopes were not equivalent across all conditions, F(2, 78) = 43.6, p < .001, with DD comparisons showing the weakest negative slope

6.2.3.2. Non-decision times. Similar to results in Experiment 2A, nondecision time parameters were not significantly different across the notation conditions, F(2,72) = 1.65, p = .200. 6.2.3.3. Decision boundaries. Within-subjects analysis of decision boundary parameters showed differences across notation, F (2,72) = 10.3, p < .001. Specifically, estimates of decision boundaries from comparing DD pairs were wider (more careful) than when comparing FF, t(72) = −3.13, p = .005, or MX pairs, t (72) = −4.40, p < .001. Decisions boundaries estimated for comparing MX and FF pairs did not show significant differences, t (72) = 1.27, p = .410. 7. Experiment 3A: Comparisons with 5000 ms response windows and catch trials Experiments 1A, 1B, 2A, and 2B were designed to test the limits of magnitude processing with decimals and fractions by implementing 13

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Fig. 9. Magnitude comparison performance in Experiment 3A with 5000 ms response windows and catch trials present. Thicker lines depict group averages of (a) individual linear models predicting mean response times and (b) logistic models predicting error rate probabilities across numerical distance for fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) comparisons. (c) Graphical renderings of diffusion model parameters depict the group averages of individuals' predicted non-decision time (length of the horizontal midline), drift rate (slope from the midline to the positive boundary), and decision boundaries (width between upper and lower bounds).

80 70

0.8 60 50

0.6

40 0.4

30 20

0.2

count of correct trials

proportion of correct trials

1.0

10 0.0

0

− 8/9 − 7/9 − 5/9 − 4/9 − 2/9 − 1/9 − 7/8 − 5/8 − 3/8 − 1/8 − 6/7 − 5/7 − 4/7 − 3/7 − 2/7 − 1/7 − 5/6 − 1/6 − 4/5 − 3/5 − 2/5 − 1/5 − 3/4 − 1/4 − 2/3 − 1/3 − 1/2 89 0.8 78 0.7 56 0.5 44 0.4 22 0.2 11 0.1 75 0.8 25 0.6 75 0.3 25 0.1 57 0.8 14 0.7 71 0.5 29 0.4 86 0.2 43 0.1 33 0.8 67 0.1 00 0.8 00 0.6 00 0.4 00 0.2 50 0.7 50 0.2 67 0.6 33 0.3 00 0.5

catch trial pair Fig. 10. Catch trial accuracies for mixed decimal-fraction pairs in Experiment 3A with 5000 ms response windows. Blue points indicate group mean accuracy for identifying each catch trial pair. Error bars indicate 95% confidence intervals around the group mean. The red horizontal line marks the proportion of trials the group would have to identify to say that performance was greater than chance. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

compared to FF, t(78) = −7.46, p < .001, and MX, t(78) = −8.59, p < .001. No significant difference was observed between the FF and MX conditions, t(78) = −1.13, p = .497.

distance effects in the form of significant negative slopes in all notation conditions (Table 5). Differences in distance effect slopes between notation conditions did not reach statistical significance, F(2,77) = 0.978, p = .381. However, this null effect may be due to a floor effect in the DD condition, which created drastically higher variability in slope coefficients calculated in the individual logistic regression models, than the previous experiments. Specifically, coefficients were extremely negative (min = −181.4) when participants made very few errors and only at very close distances and coefficients approached zero when participants made very few errors but not distributed at very close distances. Therefore, although the slopes seen in the group means plotted in Fig. 7b are visually dissimilar, these differences are not statistically different.

7.2.2. Error rates Consistent with groups that completed the tasks with shorter response windows, responses showed similar patterns of error rate distance effects when participants were given 5000 ms response windows. As shown in Fig. 7b, we observed significant differences between mean error rates across notation, F(2,78) = 168, p < .001. Means and standard deviations of error rates for each comparison notation are presented in Table 3. Once again, magnitude comparisons with two decimals (DD) were associated with significantly fewer errors than comparing two fractions, t(78) = −15.9, p < .001, or mixed pairs, t (78) = −15.8, p < .001. Even with 5000 ms to respond, conditional error rates did not differ between MX and FF conditions, t (78) = −0.088, p = .996. Slope estimates of error rates over numerical distance showed

7.2.3. Diffusion model parameters 7.2.3.1. Drift rates. Consistent with the effects observed in the previous experiments, the significant differences in drift rates across the conditions, F(2,78) = 679, p < .001, were driven by more rapid 14

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

experiment 3A, participants had 5000 ms to respond.

evidence accumulation during DD comparisons than during FF, t (78) = 31.8, p < .001, and MX, t(78) = 32.1, p < .001, comparisons. Once again, no difference was observed between the FF and MX conditions, t(78) = 0.262, p = .963. These results add to the findings of the previous experiments, indicating again that evidence accumulation for magnitude judgments with two fractions is no less efficient that processing a fraction and a decimal (H1).

8.2. Results 8.2.1. Response times Analysis of mean response times showed significant differences in adult responses across conditions, F(2, 70) = 151, p < .001. As was seen in all other experimental groups DD comparisons were made father than either condition containing a fraction. Specifically, accurate DD comparisons were made on average about 713 ms faster than FF, t (70) = −16.9, p < .001, and 510 ms faster than MX comparisons, t (70) = −12.1, p < .001. Unique to this experimental condition and consistent with the prediction that fraction processing involves symbolic calculation (H1), FF comparisons were on average made about 203 ms faster than MX comparisons, t(70) = −4.81, p < .001. Negative slope estimates of response times across numerical distances were observed in all notation conditions (Table 4). As seen in Fig. 11a, these slopes were not equivalent across all conditions, F (2,70) = 65.5, p < .001, with DD comparisons showing the weakest negative slope compared to FF, t(70) = 10.2, p < .001, and MX comparisons, b = 697.1, t(70) = 9.5, p < .001. No significant difference was observed between the FF and MX conditions, t(70) = 0.7, p = .762.

7.2.3.2. Non-decision time. Comparisons of non-decision time parameters across notation conditions revealed no significant differences between conditions, F(2,78) = 2.96, p = .058. 7.2.3.3. Decision boundaries. Within-subjects analysis of how adults set decision boundaries when given 5000 ms to respond show significant differences between notations, F(2,78) = 85.8, p < .001, that are different from the effects seen in the groups with shorter response windows. With more time, estimated decision boundaries were narrower (less careful) when comparing two decimals than when comparing two fractions, t(78) = −11.0, p < .001, or mixed pairs, t (78) = −11.6, p < .001. Decision boundary parameters when comparing two fractions or a mixed pair, however, were not significantly different, t(78) = 0.615, p = .812. 7.2.4. Catch trial analysis In the group given 5000 ms to respond, there were significant differences in catch trial accuracy between notation, F(2,78) = 234.1, p < .001. This main effects was driven by lower ability to accurately identify MX catch trials (M = 0.447, SD = 0.420) than FF (M = 0.0.971, SD = 0.119), t(78) = 18.6, p < .001, and DD (M = 0.979, SD = 0.102), t(78) = 18.9, p < .001. No differences were observed between the two conditions were participants only had to match identical decimal or fraction catch trials, t(78) = 0.296, p = .953. This pattern of results is the same as for groups given less time to respond. Abilities to accurately identify fraction-decimal equivalence pairs varied across the fraction values, resembling the results when participants were given 2000 ms to respond more so than the group given only 1000 ms. Moreover, adults given 5000 ms to respond were able to identify the majority of fraction-equivalence pairs better than chance (Fig. 10). Participants accurately identified most fraction decimal-pairs above chance performance (27.5%). Below chance performance was only observed with the fractions 5/6, 3/7, 4/7, 5/7, & 6/7. These results further suggest that adults are more familiar with the decimal equivalents of some common fractions. However, this analysis does not provide a strong test of whether these equivalence facts can be rapidly applied, given participants had multiple seconds to respond. The higher proportion of catch trials identified among these participants relative to the 2000 and 1000 ms conditions may also reflect the application of symbolic calculation processing when time permits.

8.2.2. Error rates Analysis of mean error rates showed an effect of notation, F (2,70) = 143, p < .001. Means and standard deviations of error rates for each comparison notation are presented in Table 3. Similar to all other experimental groups, error rates were lowest in the DD condition. Specifically, error rates during DD comparisons were lower than during MX, t(70) = 13.2, p < .001, and FF comparisons, t(70) = 15.8, p < .001. Unlike all other experimental groups, error rates during FF comparisons were significantly higher than during MX comparisons, t (70) = −2.63, p = .024. Single-group t-tests performed on the conditional slope coefficient of error rate over numerical distance showed distance effects in the form of significant negative slopes in all three notation conditions (Table 5). We did not initially observe significant differences in distance effect slopes between conditions, F(2,70) = 0.84, p = .435, As observed in Experiment 3A, slope estimates in the DD condition were extremely variable due to limitations of modeling sparse errors at only very close distances. 8.2.3. Diffusion model parameters 8.2.3.1. Drift rates. We observed significant differences in drift rates between notation conditions, F(2,70) = 546, p < .001. Similar to the results of the previous experiments, drift rates were fastest in the DD condition relative to FF, t(70) = 30.1, p < .001, and MX comparisons, t(70) = 26.9, p < .001. Unlike the results of the previous experiments, drift rates in the MX condition were significantly higher than in the FF condition, t(70) = 3.16, p = .004. These results are consistent with the hypothesis that magnitude processing specific to fractions may involve symbolic calculation (H1), but only when participants were given significant time to respond.

8. Experiment 3B: Comparisons with 5000 ms response windows and no catch trials 8.1. Methods 8.1.1. Participants Thirty-six undergraduate students participated for partial course credit. No participants were excluded for excessive data reduction or problematic error rates. All 36 adults (31 F/4 M/1 NR; average age = 19.9, SD = 1.8), were included in the analyses.

8.2.3.2. Non-decision time. Non-decision time estimates also showed a main effect of notation, F(2,70) = 6.6, p = .002. Specifically, nondecision times parameters for FF comparisons were significantly longer than non-decision times for DD, t(70) = 3.30, p = .002, and MX comparisons, t(70) = 2.98, p = .006. Non-decision time estimates for MX and DD conditions were not significantly different, t(70) = 0.32, p = .945.

8.1.2. Procedure The experimental procedure, materials, and fraction stimuli were identical to the previous experiments, as described in Section 2.1 except catch trials were not interspersed among comparison pairs and the corresponding catch trial instructions were omitted. Identical to

8.2.3.3. Decision boundaries. Within-subjects analysis of notation effects on decision boundaries revealed significant differences between notation conditions, F(2,70) = 49.4, p < .001. As was seen 15

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Fig. 11. Magnitude comparison performance in Experiment 3B with 5000 ms response windows and without catch trials. Thicker lines depict group averages of (a) individual linear models predicting mean response times and (b) logistic models predicting error rate probabilities across numerical distance for fraction-fraction (FF), decimal-decimal (DD), and mixed (MX) comparisons. (c) Graphical renderings of diffusion model parameters depict the group averages of individuals' predicted non-decision time (length of the horizontal midline), drift rate (slope from the midline to the positive boundary), and decision boundaries (width between upper and lower bounds). Equivalent overlapping decision boundaries across notations are shown with dotted lines.

and the within-subjects effects of notation using mixed ANOVAs and planned pairwise comparisons.

3.5

Group 1000−Catch 1000−NoCatch 2000−Catch 2000−NoCatch 5000−Catch 5000−NoCatch

3.0

drift rate

2.5

9.1. Drift rates Drift rate parameters have been described as a way to measure the quality of information extracted from a stimulus to facilitate an accurate decision (Ratcliff, Love, Thompson, & Opfer, 2012). Across the six experiments in this study we observed a consistent pattern of notation effects on mean drift rates within five of the experimental groups. As shown in Fig. 12, the highest drift rates were always observed in DD comparisons with lower drift rates in FF and MX comparisons. FF and MX drift rates did not differ, except when participants were given the longest response windows and no catch trials. In this case, drift rates were higher in MX than FF judgements (see Experiment 3B). Consistent with these results within each experiment, the mixed ANOVA indicated that drift rate parameters differed between notations for all experimental groups, F(2,424) = 1777.2, p < .001. However, these effects of notation within groups were observed in the presence of two significant two-way interactions with effects of response window length, F (4,424) = 4.75, p < .001, and catch trial presence, F(2,424) = 3.52, p = .031. The three-way interaction was not significant, F(4, 424) = 0.85, p = .492. To identify the factors driving these interactions, three separate between-subjects 3 (response window) × 2 (catch) ANOVAs were run on the notation conditions separately. Within DD comparisons, drift rates did not differ between groups based on the presence of catch trials, F(1,214) = 0.002, p = .969, or the length of response windows, F (2,214) = 0.555, p = .575. Within MX comparisons, we observed significant effects of response window length, F(2,214) = 18.0, p < .001, and the presence of catch trials, F(1,214) = 16.8, p < .001. Drift rates in MX comparisons were lower when catch trials were present, t(216) = 4.1, p < .001, and higher when response windows were 1000 ms than when they were 2000 ms, t(140) = 4.61, p < .001, or 5000 ms, t(143) = 6.43, p < .001. Drift rates for MX comparisons did not differ between the 2000 ms and 5000 ms groups, t(147) = 0.79., p = 1.00. Within FF comparisons, we observed a similar pattern of significant differences in drift rates between response window conditions as seen among MX comparisons, F(2,216) = 66.7, p < .001, but unlike MX comparisons, no differences were observed based on the presence of catch trials, F(1,216) = 0.740 p = .391. Mean drift rates were faster for FF comparisons when response windows were 1000 ms than when they were 2000 ms, t(140) = 9.1, p < .001, or 5000 ms, t

2.0

1.5

1.0

0.5 DD

Mx

FF

notation Fig. 12. Group means of drift rates across notations. Error bars indicate 95% confidence intervals. Data points within each notation have been grouped horizontally by response window. Groups that did not complete catch trials are indicated in a lighter color and a dashed line connecting their data points.

when catch trials were included in the task (Experiment 3A), decision boundaries were narrower (less careful) when comparing DD pairs than when comparing FF, t(70) = −9.53, p < .001, and MX pairs, t (70) = −7.20, p < .001. Decision boundary estimates for FF and MX comparisons were not significantly different, t(70) = 2.33, p = .051. 9. Mixed-effects analyses across experimental groups The six experiments in this study offer separate examinations of magnitude processing performance with fractions and decimals under varying task demands. To facilitate the discussion of how varying response window length or adding catch trial demands influences magnitude processing, we conducted a set of exploratory analyses on data from all six experimental groups. Specifically, we analyzed how the experimental conditions of each study impacted sub-processes of magnitude comparison captured by the three diffusion model parameters: drift rate, non-decision time, and decision boundaries. For each of the three diffusion model parameters, we tested the between-subjects effects of manipulating response window length and catch trial presence 16

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

(143) = 10.8, p < .001, but no difference was observed between drift rates when response windows were 2000 ms and 5000 ms, t (147) = 1.61, p = .324. These between-groups results indicated ways that the task demands of response time limits and additional catch trial judgements affected how efficiently participants extracted magnitude meaning from decimals and fractions. First, adding catch trial judgements to the magnitude comparison task did not affect the efficiency of magnitude processing with decimals or fractions among DD and FF pairs. However, drift rates with MX comparisons were lower in the groups that made catch trial judgements. Catch trials may have specifically impacted MX notation comparisons due to the difficulty of identifying matching magnitudes represented by unique symbolic notations in the MX notation catch trials relative to identifying matching magnitudes with identical surface features in DD and FF catch trials. Second, we observed that requiring participants to respond within strict time limits (1000 ms) relative to longer amounts of time (2000 ms and 5000 ms) increased drift rates among MX and FF comparisons, but had no effect on DD comparisons. This may indicate that comparing two decimals regardless of time limits, may be made via the same or similar processes, but comparing two fractions or mixed decimal-fraction pairs under extreme time demands may involve different forms of magnitude processing than when time is abundant. The current findings further support the conclusion that decimal notation provides efficient access to evidence for making accurate magnitude judgments, but only in the within-notation case. In the MX comparisons, we did not observe evidence of a decimal-notation advantage in the drift rate estimates relative to FF comparisons in any of the experimental conditions, except when participants were given the most time to respond and did not have additional catch trial demands. Thus, the results from this analysis do not support the symbolic calculation hypothesis that accessing the holistic magnitude of fractions necessarily involves additional and less efficient processing stages than decimals (H1).

effects analysis of non-decision time showed that main effects of notation, F(2,424) = 22.6, p < .001, and response window, F (2,212) = 22.5, p < .001, were observed in the presence of a significant interaction, F(4,424) = 12.6, p < .001. This interaction was consistent with the varying notation effects observed in non-decision time results across the six experiments above. First, we did not observe differences in non-decision times across notations when comparisons were made with only 1000 ms to respond (see Experiments 2A Section 5.2.3.2 and 2B Section 6.2.3.2). In the groups given 2000 ms to respond non-decision times with DD comparisons were faster than when comparisons contained one or two fractions, but no differences were observed between FF and MX conditions (see Experiments 1A Section 3.2.3.2 and 1B Section 4.2.3.2). Participants receiving the longest times to respond showed no difference in non-decision times (see Experiments 3A Section 7.2.3.2. and 3B Section 8.2.3.2). Interestingly, the lack of a difference between the FF and MX conditions across groups with different response window lengths suggest that adults do not experience a decimal notation advantage when encoding the meaning of mixed pairs containing a decimal relative to FF pairs containing only fractions (H1). The lack of notation effects under the strictest response windows may indicate that limits of 1000 ms may have pushed adults near the limit of their abilities. With so little time to respond, fractions and decimals may have been encoded into fuzzier estimates of magnitude in order to initiate rapid judgments. Conversely, the lack of significant notation effects under the longest response windows may indicate that limits of 5000 ms may have allowed adults to demonstrate the widest variability in encoding processes (as seen in wider confidence intervals depicted in Fig. 13). As seen in Fig. 13, the interaction between response window length and notation on mean non-decision times did not follow a pattern that would suggest decreasing response windows directly decreased nondecision time. In all notations, main effects of response window length led to differences in non-decision times (DD: F(2,215) = 6.96, p = .001; MX: F(2,215) = 24.0, p < .001; FF: F(2,215) = 17.1, p < .001). In almost all cases non-decision times were shorter when responses were limited to 1000 ms than when limits were longer (DD 2000–1000: t(140) = 3.72, p < .001; DD 5000–1000: t(143) = 2.15, p = .098; MX 2000–1000: t(140) = 6.82, p < .001; MX 5000–1000: t (143) = 2.51, p = .039; FF 2000–1000: t(140) = 5.73, p < .001, FF 5000–1000: t(143) = 4.02, p < .001). However, non-decision times

9.2. Non-decision times In Ratcliff's diffusion model, non-decision time includes the time to generate the physical response (same button presses across all experiments) and the encoding time to convert the stimulus into a representation that permits decisions to be made (Ratcliff, 1978). Mixed-

3.0

Group 1000−Catch 1000−NoCatch 2000−Catch 2000−NoCatch 5000−Catch 5000−NoCatch

0.8

decision boundaries

non−decision time

0.9

0.7

0.6

0.5

Group 1000−Catch 1000−NoCatch 2000−Catch 2000−NoCatch 5000−Catch 5000−NoCatch

2.5

2.0

1.5

1.0

0.5

DD

Mx

FF

DD

notation

Mx

FF

notation

Fig. 13. Group means of non-decision times across notations. Error bars indicate 95% confidence intervals. Data points within each notation have been grouped horizontally by response window. Groups that did not complete catch trials are indicated in a lighter color and a dashed line connecting their data points.

Fig. 14. Group means of decision boundaries across notations. Error bars indicate 95% confidence intervals. Data points within each notation have been grouped horizontally by response window. Groups that did not complete catch trials are indicated in a lighter color and a dashed line connecting their data points. 17

18

N/A N/A N/A Partially Supported N/A H3

Notes: Measures used to test Hypotheses 1 and 2 (H1 and H2) include response time (RT), error rate (ER), and Drift Rate (DR). Catch trial accuracy (Acc) was used to test Hypothesis 3 (H3).

RT ER DR

If symbolic calculation is a cognitively demanding and slow process, and is necessary to access a fraction's magnitude, then limiting the time allowed to make a response should disrupt accurate holistic magnitude processing If adults can rapidly recall memorized decimal-fraction facts and identify these pairs among their MX comparison judgments, then this should be seen in accurate (better than chance, see Section 2.3.3) catch trial performance even when response windows are restricted (e.g. 2000 ms) H2

DR

Not Supported

Partially Supported Partially Supported Not Supported Not Supported N/A Partially Supported Partially Supported Not Supported Not Supported N/A ER

Acc

Not Supported Not Supported N/A

Supported

Supported

In all experiments, efficient decimal processing was observed within notation (DD), relative to pairs containing at least one fraction. Evidence of less efficient fraction processing in FF (two fractions) relative to MX (one fraction) was only observed with long response windows and no catch trials. Group mean distance effects were significant in response times and error rates for all experimental groups including groups where responses were restricted to 1000 ms and 2000 ms. Supported

Partially Supported Partially Supported Partially Supported Not Supported Not Supported N/A Partially Supported Partially Supported Partially Supported Not Supported Not Supported N/A Partially Supported Partially Supported Partially Supported Not Supported Not Supported N/A Not Supported Not Supported

Present Present

RT

Present

Catch Trial Catch Trial

Hypotheses

Table 6 Summary of results across experimental groups.

Measure

Absent

2000 ms 1000 ms

Absent

Catch Trial

5000 ms

Absent

Decision boundaries can be described as how carefully responses are made, how much accuracy is prioritized over speed, or how much evidence is necessary before a response is given (Ratcliff, 1978). Group mean decision boundaries observed across notations and experimental groups are presented in Fig. 14. Results of the mixed ANOVA revealed a significant interaction, F(4,424) = 96.7, p < .001, between the main effects of response window duration, F(2,212) = 462, p < .001, and notation, F(2,424) = 72.7, p < .001. Interestingly, results from our decision boundary analysis do not indicate that the manipulation of catch trials prompted participants to set larger decision boundaries and thus respond more carefully, F(1,212) = 0.091, p = .763. Furthermore, the three-way interaction between response window length, catch trial presence, and notation was not significant, F(4,424) = 1.39, p = .238. The significant interaction between notation and response window was consistent with the varying notation effects observed across the six experiments. Within groups given 1000 ms to respond, slightly wider decision boundaries were observed in DD comparisons relative to notations containing at least one fraction, but no differences were observed between FF and MX comparisons (Experiment 2A 5.2.3.3 and 2B 6.2.3.3). Groups given 2000 ms to respond showed no differences in decision boundaries across notations (Experiment 1A 3.2.3.3 and 1B 4.2.3.3). Groups who were given the most time to respond and potentially apply more complex strategies (Experiment 3A section 7.2.3.3 and 3B section 8.2.3.3) set wider decision boundaries when comparisons involve fractions than when the choice was between two decimals. As seen in Fig. 14, the effects of varying response window length on decision boundaries estimated for each notation condition were consistent with the idea that participants will wait to be more certain of their responses when they have the extra time to do so. In all notations significant differences were observed between decision boundaries across groups with varying response window limits (DD: F (2,215) = 129, p < .001; MX: F(2,215) = 494, p < .001; FF: F (2,215) = 433, p < .001). Specifically, decision boundaries were narrowest for all notations when response windows were limited to 1000 ms (DD 2000–1000: t(140) = 10.7, p < .001; DD 5000–1000: t (143) = 15.8, p < .001; MX 2000–1000: t(140) = 12.6, p < .001; MX 5000–1000: t(143) = 31.2, p < .001; FF 2000–1000: t (143) = 11.4, p < .001; FF 5000–1000: t(140) = 29.1, p < .001) and decision boundaries were wider when response windows were 5000 ms than when they were 2000 ms (DD: t(147) = 5.07, p < .001; MX: t (147) = 18.7, p < .001; FF: t(147) = 17.9, p < .001). Furthermore, confidence intervals of decision boundaries shown in Fig. 14, indicate that within group variation was greater when response windows were longer, which may further indicate that strict limitations (1000 ms) may have pushed participants to the limits of their abilities. Taken together, these results affirm that the manipulation of the response windows across experimental groups was effective in influencing how carefully participants chose to make responses.

If symbolic calculation processing is necessary and specific to fractions but not decimals, then the time to make accurate judgments and error rates of these judgements should increase as comparisons contain more fractions

9.3. Decision boundaries

H1

Summary of Results

with MX comparisons were higher in groups given 2000 ms to respond than in groups given 5000 ms, t(147) = 4.44, p < .001, and nondecision times across these groups did not differ among DD, t (147) = 1.63, p = .313, and FF comparisons, t(147) = 1.79, p = .225. The presence of catch trials had a significant main effect on participants' non-decision times, F(1,136) = 4.07, p = .046, as seen by longer non-decision times estimated for comparison decisions when catch trials were included in the block (see Fig. 13). No interaction effects were observed between the manipulation of catch trials and the other experimental factors (notation × catch trials: F(2,424) = 1.77, p = .171; response window × catch: F(2,424) = 0.116, p = .981).

Under the strictest response windows (1000 ms), participants were unable to recognize fraction-decimal equivalence pairs at rates higher than chance. When restrictive response windows were less stringent (2000 ms), participants were able to recognize a small subset of fraction-decimal equivalence pairs.

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

10. Discussion

fractions relative to decimals is not consistently observed. Perhaps similar to the ways that adults can adaptively apply holistic and componential processing during magnitude comparison tasks depending on fraction values and task demands (Huber, Klein, et al., 2014; Ischebeck et al., 2016), symbolic calculation may be a way to process fractions when conditions allow for the most careful responses.

Through a series of six experiments, we manipulated elements of the same magnitude comparison task with fractions, decimals, and mixednotation pairs to examine how adults access an understanding of rational magnitudes when constrained by time and additional task demands. Previous studies observing slower magnitude processing for fractions relative to decimals often suggest that this additional processing time may correspond to a process of calculating the fraction's magnitude specific to fractions and not decimals. Here we tested two predictions based on the hypothesis that accessing representations of magnitude from common fractions, but not decimals, requires forms of symbolic calculation to mentally manipulate the fraction into a decimal form. First, we predicted that symbolic calculation with fractions should result in increasing response time latencies as more fractions are involved in the comparison pair (H1). Second, we predicted that imposing short response windows should disrupt necessary symbolic calculation processes and thus inhibit access to the magnitude of fractions (H2). Furthermore, we tested for evidence that adults can convert fractions to decimals more rapidly by recalling equivalence pair facts. Evidence of fluent recall abilities should be seen in accurate catch trial performance with mixed pairs, where participants needed to recognize fraction-decimal equivalence pairs (H3). A summary of results is presented in Table 6. Across the six experiments we did not observe strong evidence to support the argument that symbolic calculation is necessary for accessing magnitude representations of fractions and not decimals. Alternatively, converting fractions to decimals may be better described as a strategy that people may utilize differently depending on task demands, even within the same magnitude comparison task. Alternative forms of magnitude processing must therefore explain how adults access holistic representations of magnitude from fractions and decimals when strict time limits or additional task demands restrict the use of slower strategies like symbolic calculation.

10.2. Effects of response time limits on magnitude processing Furthermore, by examining adult performance under the pressure of time and task demands, we observed that rapid and successful forms of holistic magnitude processing may occur before slow forms of calculation can be applied. Thus, the observed results did not support the hypothesis that restricting responses times would prevent access to magnitude representations of fractions by disrupting symbolic calculation processes (H2). All experimental groups exhibited evidence for successful magnitude processing in the form of significant distance effects on response times and accuracy. These distance effects were still observed when responses were restricted to 2000 ms or even 1000 ms. Furthermore, mean error rates in all experimental groups show that accuracies within all three notation conditions were higher than 70% (Table 3). However, results of the experimental group that received only 1000 ms to respond and instructions to monitor for catch trials offer an indication that magnitude processing abilities were affected by these task demands. Specifically, a larger percentage of participants in this group were excluded from the analysis due to problematic numbers of missed responses and poor task accuracy. It is difficult to determine whether the excluded participants' performance reflects a disruption of symbolic calculation specifically, or other capacities such as overall task confidence or even basic speed of processing. Looking between the two groups that were given 1000 ms to compete the task, it appears that the inclusion of catch trials may have been the critical manipulation that made the task too hard for some individuals to complete, but not too hard to disrupt forms of magnitude processing. Specifically, a much smaller proportion of participants were excluded from the 1000 ms group that did not have to monitor for catch trials, and the pattern of distance effect slopes were similar between these groups. Evidence for successful magnitude processing in the participants that were not excluded from the 1000 ms group with catch trials may suggests that some adults can still access a sense of rational numbers under extreme conditions that preclude slower calculation strategies.

10.1. Effects of notation-specific symbolic calculation First, the effects of notation on response times and error rates did not support the hypothesis that processing each fraction in a comparison pair requires calculating a decimal equivalent of the fraction (H1). The predicted pattern of mean response times based on this hypothesis (DD < MX < FF) was observed only when adults were given abundant time to respond (5000 ms) and participants did not have additional catch trial task demands (Experiment 3B). In all other experimental groups—with shorter response windows or the addition of catch trial monitoring—response times in the FF condition were not longer than the MX condition. This lack of a calculation effect, where RT and ER were predicted to correspond to the number of fractions, could also be described as a lack of a decimal processing advantage in MX pairs relative to FF pairs. Even though DD comparisons were made more quickly and accurately than either notation containing at least one symbolic fraction (except when response windows were 1000 ms), this did not correspond to faster or more accurate MX comparisons relative to FF. Based on these findings, we argue that symbolic calculation to convert fractions into decimals may not be necessary for processing each fraction's magnitude but it may be a useful strategy that adults can apply when they are given enough time to make careful responses. Consistent with this interpretation, Faulkenberry and Pierce (2011) observed that adults who reportedly compared fractions by converting fractions to decimals showed slower and more accurate performance relative to other strategies. Furthermore, our results are consistent with studies that did not impose time limits on magnitude judgments and concluded that large differences between fraction and decimal comparison response times were due to processing necessary to calculate a fraction's magnitude (DeWolf et al., 2014). However, our critical comparisons of MX and FF performance in the current study under varied task demands indicates that slower magnitude processing of

10.3. Abilities to recall fraction-decimal equivalence facts Third, results of catch trial performance in the MX comparison condition indicated that adults do not tend to have fraction-decimal equivalence facts memorized for all single digit fractions but do appear to have the ability to accurately recognize a subset of more familiar fractions. Specifically, results of our MX catch trial analysis confirm that equivalence facts are more likely to be recalled for unit fractions and fractions with small denominators, such as halves, thirds, and fourths. For the majority of the single digit fractions included in the experiment, adults did not appear to have quick and accurate recall of these crossnotation equivalence facts. Critical to the first hypothesis, which states symbolic calculation is a necessary step in magnitude processing with fractions, adults in the experimental groups who had the longest response windows (5000 ms), and who may have had time to calculate decimal equivalents, still failed to identify more than half of the equivalence pairs. Differing levels of accuracy on catch trials across different fraction values, may indicate that there is considerable heterogeneity at a trial-by-trial level with regards to how adults process each fraction in a comparison pair. Thus, the application of symbolic calculation strategies may depend both on task demands and varying familiarity with different fractions. 19

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

10.4. Efficiency of decimal notation processing within and across fraction notations

that humans' innate ability to perceive and discriminate ratios provides a basis upon which the meaning of symbolic fractions can be built (Lewis et al., 2015). Limitations of this study present interesting questions to resolve in future research. First, this study examined fraction and decimal understanding by observing performance with visually presented fractions composed of single-digit numerators and denominators. To fully understand the key processes underlying how individuals construct an understanding of rational number magnitudes, future research should consider how the results described here extend to understanding complex fractions (e.g. multi-digit or improper) and various forms of presentation (e.g. nonsymbolic ratios, auditory). Second, catch trial analysis in the current study reflected ways that fraction-decimal familiarity differed across different fractions. Thus, it is possible that there is considerable heterogeneity in fraction processing and strategy use not only across differing task demands but also between individual fractions. Third, results from adult participants in the present study provide insight into the mechanisms of fractions understanding when individuals have significant mathematical education. Research across child and adult development is necessary to examine how math education and specific experiences mapping symbolic fractions to nonsymbolic instantiations influences how adults rely on symbolic calculation and perceptual-based ratio processing across contexts. Lastly, more research is necessary to explore how models of decision making, such as Ratcliff's diffusion model, apply to studies of fraction and decimal understanding. Magnitude judgments modeled in the current study present a relatively complex decision-making process, especially in the case where including catch trials introduces a third response beyond the assumed two-alternative forced choice. In the current study, drift diffusion models performed similarly in experimental groups with and without catch trials, providing some evidence that the use of these models is appropriate and informative to understanding the subcomponents in the magnitude comparison decision. Investigations into how people access the meaning of fractions and decimals are complex, as the mechanisms underlying processing of these numerical symbols may largely depend on the strategies allowed by task demands. Calculating a fraction into its decimal equivalent may provide a valuable means to support precise magnitude judgments in conditions where time limits are not imposed. However, the results from this study suggest that adults can access fraction magnitudes even before forms of symbolic calculation can take place. Greater specificity should be applied in future research when using terms like calculation in order to better distinguish mechanisms of human cognition and learned strategies. Uncovering the mechanisms by which fractions, including decimals, are mapped to analog magnitudes without calculation requires further investigation, but our results are consistent with theories that analog magnitude meaning may be built up from perceptual experience. Similar to findings with non-symbolic ratios (Matthews, Lewis, & Hubbard, 2016) adults may have a sense of fraction magnitudes via an appreciation of the analog magnitudes from their symbolic components.

Researchers have suggested that the symbolic form of decimal fractions may be better suited for representing ratio magnitudes along a unidimensional scale than common fractions (DeWolf et al., 2015a). More efficient DD performance in the current experiments relative to FF is consistent with this idea that decimals are more efficient symbolic representation of magnitude. However, this theory does not explain why mixed fraction comparisons were neither faster nor less error prone than FF comparisons in five of the six experiments. Efficient magnitude judgment performance in DD comparisons relative to FF, in the current study and previous studies alike, may not reflect differences in how these notations support access to rational number magnitudes, because DD comparisons may not rely on accessing the rational number meaning of the stimuli. For example, magnitude judgments between two decimals can be accurately and quickly determined if the digits to the right of the decimal place are treated as whole numbers (Huber et al., 2014). Such a strategy may be specifically encouraged in the current study given that decimals in the DD condition were zero padded to have equal decimal lengths. Additionally, the use of componential strategies, such as comparing the first digit of two deicmals, may be used in conjunction with holistic magnitude processing, as has been seen with multi-digit whole number comparisons (Dehaene, Dupoux, & Mehler, 1990; Nuerk, Weger, & Willmes, 2001). Findings of efficient decimal processing in the DD condition that are not seen in MX condition may thus indicate that the processes involved in accessing an analog magnitude representation with decimal notation is similar to or the same as mapping to this meaning with fractions. Whereas the rational magnitude of a given fraction is defined by the value of the numerator relative to the value of the denominator, the rational magnitude of a given decimal is defined by the value of its digits relative to implicit denominator values given by the digits' positions to the right of the decimal place (e.g., 0.025 = 25/1000) (Resnick et al., 1989) Unlike fractions however, the base 10 nature of the decimal system is constant. As seen in studies of fraction comparison, when denominators are held constant, accurate magnitude comparisons can shift towards componential strategies that make accessing the holistic magnitude unnecessary (Bonato et al., 2007; Toomarian & Hubbard, 2018). In this experiment, fraction comparisons pairs with common denominators (or even common multiples) were excluded to eliminate the possibility of participants using these componential strategies. Thus, the implicit base-ten denominator in decimal fractions may provide one explanation for why mixed decimal-fraction comparisons show similar distance effects as fraction-fraction comparisons. Specifically, for all mixed comparison pairs in this study, the explicit denominator in the fraction (values 2-9) and the implicit denominator in the decimal fraction were unique (e.g. 0.200 can be read as 200 thousandths or reduced to two tenths). Numerical processing to make accurate magnitude judgments in these mixed comparison cases may therefore require that participants access sense of holistic magnitude for both numbers along a shared analog magnitude scale in a way that decimal to decimal comparisons may not.

Authors' contributions

10.5. Alternative mechanisms for accessing rational magnitude

JVB and EMH conceived the studies, JVB drafted the manuscript and JVB and EMH edited and revised the manuscript. All authors reviewed and approved the final manuscript.

If symbolic calculation is not a necessary process unique to understanding a fraction's magnitude, then what processes are necessary? One key component of magnitude processing with fractions and decimals is mapping the perceived symbolic stimuli to an internal grounded sense of rational number magnitudes. Akin to the triple code model of numerical processing with whole numbers (Dehaene, 1992; Skagenholt, Träff, Västfjäll, & Skagerlund, 2018), a shared amodal or nonsymbolic representation of rational number meaning may support an underlying sense of magnitudes regardless of a fraction or decimal's symbolic form (Jacob et al., 2012). Proponents of perceptual-based processing argue

Acknowledgements This research was supported by the National Science Foundation (NSF DRL 1420211). The opinions expressed are those of the authors and do not represent the views of the National Science Foundation. We would like to thank Percival Matthews, Elizabeth Toomarian and two anonymous reviewers who provided invaluable comments on previous drafts of this manuscript. 20

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard

Appendix A. Supplementary data

Experimental Psychology/Revue Canadienne de Psychologie Expérimentale, 70(1), 12–23. https://doi.org/10.1037/cep0000070. Lewis, M. R., Matthews, P. G., & Hubbard, E. M. (2015). Neurocognitive architectures and the nonsymbolic foundations of fractions understanding. Development of mathematical cognition (pp. 141–164). . https://doi.org/10.1016/b978-0-12-801871-2.00006-x. Lorch, R. F., & Myers, J. L. (1990). Regression analyses of repeated measures data in cognitive research. Journal of Experimental Psychology: Learning, Memory, and Cognition, 16(1), 149–157. https://doi.org/10.1037/0278-7393.16.1.149. Lyons, I. M., Ansari, D., & Beilock, S. L. (2012). Symbolic estrangement: Evidence against a strong association between numerical symbols and the quantities they represent. Journal of Experimental Psychology: General, 141(4), 635–641. https://doi.org/10. 1037/a0027248. Matthews, P. G., & Chesney, D. L. (2015). Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes. Cognitive Psychology, 78, 28–56. https:// doi.org/10.1016/j.cogpsych.2015.01.006. Matthews, P. G., Lewis, M. R., & Hubbard, E. M. (2015). Individual differences in nonsymbolic ratio processing predict symbolic math performance. Psychological Science, 27(2), 191–202. https://doi.org/10.1177/0956797615617799. Matthews, P. G., Lewis, M. R., & Hubbard, E. M. (2016). Individual differences in nonsymbolic ratio processing predict symbolic math performance. Psychological Science, 27(2), 191–202. https://doi.org/10.1177/0956797615617799. McCrink, K., & Wynn, K. (2007). Ratio abstraction by 6-month-old infants. Psychological Science, 18(8), 740–745. https://doi.org/10.1111/j.1467-9280.2007.01969.x. Meert, G., Grégoire, J., & Noël, M.-P. (2009). Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task. Quarterly Journal of Experimental Psychology (2006), 62(8), 1598–1616. https://doi.org/10.1080/ 17470210802511162. Miller Singley, A. T., & Bunge, S. A. (2018). Eye gaze patterns reveal how we reason about fractions. Thinking & Reasoning, 24(4), 445–468. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215(5109), 1519–1520. https://doi.org/10.1038/2151519a0. National Governors Association Center for Best Practices, & Council of Chief State School Officers (2010). Common Core State Standards for mathematics: Mathematics: Number & Operations-Fractions. Retrieved from http://www.corestandards.org/Math/Content/ NF/. Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. https://doi.org/10.1207/s15326985ep4001_3. Nuerk, H. C., Weger, U., & Willmes, K. (2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82(1), https://doi.org/10. 1016/S0010-0277(01)00142-1. Obersteiner, A., & Tumpek, C. (2016). Measuring fraction comparison strategies with eyetracking. ZDM - Mathematics Education, 48(3), 255–266. https://doi.org/10.1007/ s11858-015-0742-z. Piazza, M., Pinel, P., Le Bihan, D., & Dehaene, S. (2007). A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron, 53(2), 293–305. https://doi.org/10.1016/j.neuron.2006.11.022. Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review.. https://doi.org/ 10.1037/0033-295X.85.2.59. Ratcliff, R., Love, J., Thompson, C. A., & Opfer, J. E. (2012). Children are not like older adults: A diffusion model analysis of developmental changes in speeded responses. Child Development, 83(1), 367–381. https://doi.org/10.1111/j.1467-8624.2011. 01683.x. Ratcliff, R., & McKoon, G. (2008). The diffusion decision model: Theory and data for twochoice decision tasks. Neural Computation, 4(164), 1–44. https://doi.org/10.1126/ scisignal.2001449.Engineering. Ratcliff, R., Thompson, C. A., & McKoon, G. (2015). Modeling individual differences in response time and accuracy in numeracy. Cognition, 137, 115–136. https://doi.org/ 10.1016/j.cognition.2014.12.004. Resnick, I., Jordan, N. C., Hansen, N., Rajan, V., Rodrigues, J., Siegler, R. S., & Fuchs, L. S. (2016). Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Developmental Psychology, 52(5), 746–757. https:// doi.org/10.1037/dev0000102. Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8–27. https://doi.org/10.2307/749095. Reynvoet, B., & Sasanguie, D. (2016). The symbol grounding problem revisited: A thorough evaluation of the ans mapping account and the proposal of an alternative account based on symbol-symbol associations. Frontiers in Psychology. https://doi.org/ 10.3389/fpsyg.2016.01581. Roell, M., Viarouge, A., & Houde, O. (2017). Inhibitory control and decimal number comparison in school-aged children. 1–17. Schneider, M., & Siegler, R. S. (2010). Representations of the magnitudes of fractions. Journal of Experimental Psychology: Human Perception and Performance, 36(5), 1227–1238. https://doi.org/10.1037/a0018170. Schneider, W., Eschman, A., & Zuccolotto, A. (2002). E-Prime: User’s Guide.. Pittsburgh, PA: Psychology Software Tools. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., ... Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101. Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13–19. https://doi.org/10.1016/j.tics.2012.11.004. Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994–2004. https://doi.org/ 10.1037/a0031200.

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.cognition.2020.104219. References Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447–455. https://doi.org/10.1016/j.jecp.2012.06.004. Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational-number concepts. Acquisition of Mathematics Concepts and Processes, 91–126. Bonato, M., Fabbri, S., Umiltà, C., & Zorzi, M. (2007). The mental representation of numerical fractions: Real or integer? Journal of Experimental Psychology. Human Perception and Performance, 33(6), 1410–1419. https://doi.org/10.1037/0096-1523. 33.6.1410. Bonn, C. D., & Cantlon, J. F. (2017). Spontaneous, modality-general abstraction of a ratio scale. Cognition, 169, 36–45. https://doi.org/10.1016/j.cognition.2017.07.012. Booth, J. L., & Newton, K. J. (2012). Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37(4), 247–253. https://doi.org/10. 1016/j.cedpsych.2012.07.001. De Smedt, B., Noël, M. P., Gilmore, C., & Ansari, D. (2013). How do symbolic and nonsymbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2(2), 48–55. https://doi.org/10.1016/j.tine.2013.06. 001. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1–2), 1–42. https://doi. org/10.1016/0010-0277(92)90049-N. Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology. Human Perception and Performance, 16(3), 626–641. https://doi.org/10. 1037/0096-1523.16.3.626. DeWolf, M., Bassok, M., & Holyoak, K. J. (2015a). Conceptual structure and the procedural affordances of rational numbers: Relational reasoning with fractions and decimals. Journal of Experimental Psychology. General, 144(1), 127–150. https://doi.org/ 10.1037/xge0000034. DeWolf, M., Bassok, M., & Holyoak, K. J. (2015b). From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions. Journal of Experimental Child Psychology, 133, 72–84. https://doi.org/10.1016/ j.jecp.2015.01.013. DeWolf, M., Grounds, M. A., Bassok, M., & Holyoak, K. J. (2014). Magnitude comparison with different types of rational numbers. Journal of Experimental Psychology: Human Perception and Performance, 40(1), 71–82. https://doi.org/10.1037/a0032916. DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, 39–49. https://doi.org/ 10.1016/j.learninstruc.2014.07.002. Duffy, S., Huttenlocher, J., & Levine, S. (2005). It is all relative: How young children encode extent. Journal of Cognition and Development, 6(1), 51–63. https://doi.org/10. 1207/s15327647jcd0601_4. Faulkenberry, T. J., & Pierce, B. H. (2011). Mental representations in fraction comparison holistic versus component-based strategies. Experimental Psychology, 58(6), 480–489. https://doi.org/10.1027/1618-3169/a000116. Fayol, M., & Thevenot, C. (2012). The use of procedural knowledge in simple addition and subtraction problems. Cognition, 123(3), 392–403. https://doi.org/10.1016/j. cognition.2012.02.008. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. https://doi.org/10.1016/j.tics.2004.05.002. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44(1–2), 43–74. Gelman, R. (2015). Learning in core and non-core number domains. Developmental Review, 38, 185–200. https://doi.org/10.1016/j.dr.2015.07.010. Harnad, S. (1990). The symbol grounding problem. Physica D, 42, 335–346. https://doi. org/10.1016/0167-2789(90)90087-6. Huber, S., Moeller, K., & Nuerk, H. C. (2014). Adaptive processing of fractions - evidence from eye-tracking. Acta Psychologica, 148, 37–48. https://doi.org/10.1016/j.actpsy. 2013.12.010. Huber, S., Klein, E., Willmes, K., Nuerk, H. C., & Moeller, K. (2014). Decimal fraction representations are not distinct from natural number representations - evidence from a combined eye-tracking and computational modeling approach. Frontiers in Human Neuroscience, 8(1 APR), https://doi.org/10.3389/fnhum.2014.00172. Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing and representation of fractions within the brain. An fMRI investigation. NeuroImage, 47(1), 403–413. https://doi.org/10.1016/j.neuroimage.2009.03.041. Ischebeck, A., Weilharter, M., & Körner, C. (2016). Eye movements reflect and shape strategies in fraction comparison. Quarterly Journal of Experimental Psychology, 69(4), 713–727. https://doi.org/10.1080/17470218.2015.1046464. Jacob, S., Vallentin, D., & Nieder, A. (2012). Relating magnitudes: The brain’s code for proportions. Trends in Cognitive Sciences, 16(3), 157–166. https://doi.org/10.1016/j. tics.2012.02.002. Kallai, A. Y., & Tzelgov, J. (2009). A generalized fraction: An entity smaller than one on the mental number line. Journal of Experimental Psychology: Human Perception and Performance, 35(6), 1845–1864. https://doi.org/10.1037/a0016892. Leibovich, T., & Ansari, D. (2016). The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions. Canadian Journal of

21

Cognition 199 (2020) 104219

J.V. Binzak and E.M. Hubbard Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https:// doi.org/10.1016/j.cogpsych.2011.03.001. Skagenholt, M., Träff, U., Västfjäll, D., & Skagerlund, K. (2018). Examining the triple code model in numerical cognition: An fmri study. PLoS One, 13. https://doi.org/10.1371/ journal.pone.0199247. Sprute, L., & Temple, E. (2011). Representations of fractions: Evidence for accessing the whole magnitude in adults. Mind, Brain, and Education, 5(1), 42–47. https://doi.org/ 10.1111/j.1751-228X.2011.01109.x. Toomarian, E. Y., & Hubbard, E. M. (2018). The fractions SNARC revisited: Processing fractions on a consistent mental number line. Quarterly Journal of Experimental Psychology, 71(8), 1761–1770. https://doi.org/10.1080/17470218.2017.1350867.

Varma, S., & Karl, S. R. (2013). Understanding decimal proportions: Discrete representations, parallel access, and privileged processing of zero. Cognitive Psychology, 66(3), 283–301. https://doi.org/10.1016/j.cogpsych.2013.01.002. Voss, A., & Voss, J. (2007). Fast-dm: A free program for efficient diffusion model analysis. Behavior Research Methods, 39(4), 767–775. https://doi.org/10.3758/BF03192967. Zhang, L., Fang, Q., Gabriel, F. C., & Szücs, D. (2014). The componential processing of fractions in adults and children: Effects of stimuli variability and contextual interference. Frontiers in Psychology, 5(September), 1–8. https://doi.org/10.3389/fpsyg. 2014.00981. Zhang, L., Fang, Q., Gabriel, F. C., & Szücs, D. (2015). Common magnitude representation of fractions and decimals is task dependent. The Quarterly Journal of Experimental Psychology, (May 2015), 1–41. https://doi.org/10.1080/17470218.2015.1052525.

22