Empirical Research

Revisiting and Refining Relations Between Nonsymbolic Ratio Processing and Symbolic Math Achievement

Yunji Park*¹, Percival G. Matthews¹

[1] Department of Educational Psychology, University of Wisconsin-Madison, Madison, WI, USA.

Journal of Numerical Cognition, 2021, Vol. 7(3), 328–350, https://doi.org/10.5964/jnc.6927

Received: 2020-07-30. Accepted: 2021-03-05. Published (VoR): 2021-11-30.

Handling Editors: Mojtaba Soltanlou, University of Surrey, Guildford, UK; Krzysztof Cipora, Loughborough University, Loughborough, UK

*Corresponding author at: Department of Educational Psychology, University of Wisconsin-Madison, 1025 W. Johnson Street, Madison, WI 53706-1796, USA. Tel: +708 374 5461. E-mail: ypark246@wisc.edu

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In their 2016 Psych Science article, Matthews, Lewis and Hubbard (2016, https://doi.org/10.1177/0956797615617799) leveled a challenge against the prevailing theory that fractions—as opposed to whole numbers—are incompatible with humans’ primitive nonsymbolic number sense. Their ratio processing system (RPS) account holds that humans possess a primitive system that confers the ability to process nonysmbolic ratio magnitudes. Perhaps the most striking finding from Matthews et al. was that ratio processing ability predicted symbolic fractions knowledge and algebraic competence. The purpose of the current study was to replicate Matthews et al.’s novel results and to extend the study by including a control measure of fluid intelligence and an additional nonsymbolic magnitude format as predictors of multiple symbolic math outcomes. Ninety-nine college students completed three comparison tasks deciding which of two nonsymbolic ratios was numerically larger along with three simple magnitude comparison tasks in corresponding formats that served as controls. The formats included were lines, circles, and dots. We found that RPS acuity predicted fractions knowledge for three university math placement exam subtests when controlling for simple magnitude acuities and inhibitory control. However, this predictive power of the RPS measure appeared to stem primarily from acuity of the line-ratio format, and that predictive power was attenuated with the inclusion of fluid intelligence. These findings may help refine theories positing the RPS as a domain-specific foundation for building fractional knowledge and related higher mathematics.

Keywords: nonsymbolic ratio magnitudes, number sense, ratio processing system, fractions knowledge, higher mathematics, individual differences, replication

Numerous studies have demonstrated the importance of fractions knowledge for mastery of algebra and more advanced higher mathematics (e.g., Bailey, Hoard, Nugent, & Geary, 2012; Booth & Newton, 2012; Siegler et al., 2012; Siegler, Fazio, Bailey, & Zhou, 2013). Despite the importance of fractions knowledge, however, it has been reported as one of the most difficult mathematical topics for children to grasp (e.g., Lesh, Post, & Behr, 1987; Ni & Di Zhou, 2005; Vamvakoussi & Vosniadou, 2004, 2010). Indeed, fractions knowledge often proves problematic even for high school students (Lortie-Forgues, Tian, & Siegler, 2015) and adults (Stigler, Givvin, & Thompson, 2010).

One prevalent account explaining these widespread struggles with fractions suggests that fractions are difficult because they cannot be grounded in a primitive nonsymbolic number sense, the approximate number system (ANS) (Bonato, Fabbri, Umiltà, & Zorzi, 2007; Feigenson, Dehaene, & Spelke, 2004; Ni & Di Zhou, 2005; Wynn, 1995). On this argument, because the ANS is dedicated to processing numerosities (i.e., sets of discrete objects), it is well-suited to serve as foundation for natural conceptual analogs to numerosities—the whole numbers (Dehaene, 2011; Feigenson et al., 2004). In contrast, these researchers have argued that innate constraints of the ANS make it inappropriate for processing fractions, concluding that fractions learning (in contrast to whole number learning) cannot be supported by phylogenetically ancient protonumerical abilities. Ni and Di Zhou (2005) referred to such theories as innate constraints accounts, and Matthews, Lewis, and Hubbard (2016) was framed largely as a response to such accounts with regard to fractions. Here, we replicated a novel finding of Matthews et al. (2016) and extended the study to put this controversial account regarding symbolic fraction acquisition into the test.

Processing Nonsymbolic Ratio Magnitudes

However, growing evidence suggests that there may be a primitive nonsymbolic number sense that is different from the ANS (Jacob, Vallentin, & Nieder, 2012; Lewis, Matthews, & Hubbard, 2015; McCrink & Wynn, 2007; Sidney, Thompson, Matthews, & Hubbard, 2017). This primitive ability is dedicated to processing nonsymbolic ratios (e.g., ratios instantiated by juxtaposing two line segments; Figure 1). Multiple studies have shown that human adults (Bonn & Cantlon, 2017; Jacob & Nieder, 2009a; Matthews & Chesney, 2015; Meng, Matthews, & Toomarian, 2019), infants (McCrink & Wynn, 2007) and even some nonhuman species (Bastos & Taylor, 2020; Drucker, Rossa, & Brannon, 2016; Vallentin & Nieder, 2010) are capable of representing ratio magnitudes when presented nonsymbolically. Against this backdrop, Matthews et al. (2016) leveled a challenge against the innate constraints account that fraction learning is hard because it goes beyond the limits of our basic representational capacities (e.g., Feigenson et al., 2004). They proposed that a nonsymbolic ratio processing system (RPS) might help children acquire symbolic fractions knowledge efficiently (see also Jacob et al., 2012; Lewis et al., 2015; Matthews & Chesney, 2015). They suggested that the RPS might potentially serve as a neurocognitive foundation for learning symbolic fractions much as the ANS supports symbolic whole number learning. They further argued that the RPS may provide an additional perceptual route that can expand cognitive primitive accounts of numerical cognition to apply to fractions—and perhaps to all real numbers (Sidney et al., 2017; see also Gallistel & Gelman, 2000).

Click to enlarge

Figure 1

Example Stimuli for Simple Magnitude (Top) and Ratio Magnitude (Bottom) Comparison Tasks

Note. Line, circle, and dot formats were presented.

Several studies have provided evidence consistent with the hypothesis that the RPS and symbolic fraction representations are compatible. One line of studies has demonstrated that both adults and children are capable of rapidly translating magnitudes across nonsymbolic and symbolic formats (Binzak, Matthews, & Hubbard, 2019; Kalra, Binzak, Matthews, & Hubbard, 2020; Matthews & Chesney, 2015; Matthews & Lewis, 2017; Meert, Greǵoire, Seron, & Noël, 2013). In these studies, adults and children completed cross-format comparison tasks, whereby they determined the larger between a nonsymbolic and a symbolic ratio. The nonsymbolic ratios were presented in several formats, including lines (Kalra et al., 2020), circles or dots (Matthews & Chesney, 2015). Regardless of nonsymbolic format, participants’ responses were rapid and generally accurate. In fact, cross-format comparisons are typically completed faster than within-format comparisons of symbolic fractions. Furthermore, these comparisons showed a distance effect whereby performance improves as the distance between two magnitudes decreases (Moyer & Landauer, 1967), suggesting that both symbolic and nonsymbolic ratios were represented as analog magnitudes and that those analog magnitudes were compatible enough to facilitate rapid comparison.

A few neuroimaging studies have similarly demonstrated processing of nonsymbolic and symbolic ratios at the neural level (Ischebeck, Schocke, & Delazer, 2009; Jacob & Nieder, 2009a; Jacob et al., 2012; Mock et al., 2018). In these studies, adults performed symbolic fraction comparisons (Ischebeck et al., 2009; Jacob & Nieder, 2009a), nonsymbolic ratio comparisons (Jacob & Nieder, 2009b), or both (Mock et al., 2018). Researchers have found that the posterior parietal cortex, especially the intraparietal sulcus (IPS), is recruited for both nonsymbolic and symbolic fraction processing, which indicates a possible shared processing region.

The RPS and the Acquisition of Symbolic Fractions Knowledge

One important aspect of the RPS account is that it posits that RPS acuity can help support the acquisition of fractions knowledge and other downstream mathematics, such as algebra. Specifically, Matthews et al. (2016) hypothesized 1) that both formal and informal learning helps generate links among nonsymbolic ratios and their corresponding symbolic fractions, 2) that individual differences in RPS acuity might moderate the effects of instruction, and 3) that the RPS, which has been proven to operate even when it serves as a task irrelevant dimension (Jacob & Neider, 2009a; Matthews & Lewis, 2017), exerts its effects on learning even when it is not an explicit pedagogical focus.

On this hypothesis, RPS ability should be associated with fractions knowledge, and perhaps even higher mathematics such as algebra which require an understanding of relational magnitude. To test this hypothesis, Matthews et al. (2016) investigated possible relations between the RPS and symbolic math achievement scores. The authors used a series of ratio magnitude comparison tasks to construct a composite measure of RPS acuity and also used simple dot and line comparison tasks to measure acuities for number of dots and line-lengths as controls. In a novel finding, they observed that RPS acuity predicted symbolic numerical measures, including fractions knowledge and algebra assessment scores measured at college entry. These relations were significant even after controlling for inhibitory control, number acuity, and line-length acuity.

To date, however, this novel result has not been replicated. Although two studies with children also showed similar relations between nonsymbolic ratio comparison performance and symbolic fractions ability (Hansen et al., 2015; Möhring, Newcombe, Levine, & Frick, 2015), even these closest existing reports were quite different from Matthews et al. (2016) in at least three important ways: 1) Neither study measured RPS acuity explicitly, 2) they were conducted with child participants rather than adults, and 3) as a result, neither assessed relations between the RPS and advanced mathematics, such as algebra.

With the present study, we aimed to replicate Matthews et al.’s novel results using some identical tasks, a similar protocol, and a sample drawn from roughly the same population (i.e., students from the same introductory courses at the same university). At the same time, the current study aimed to refine and extend the results in three ways. First, we included the additional domain general control measure of fluid intelligence along with the inhibitory control measure from the original study. Fluid intelligence, the ability to solve novel and abstract problems, has been known to be related with mathematical attainment and higher order mathematics (Preusse, Elke, Deshpande, Krueger, & Wartenburger, 2011; Primi, Ferrão, & Almeida, 2010). Thus, including this measure imposes a tougher test of the extent to which RPS acuity can explain unique variance of higher order mathematics. Second, whereas Matthews et al. examined only algebra subtest scores from the university placement exam, we acquired additional subtests of higher order math achievement from the same exam. These were trigonometry and math fundamentals, which tested a combination of basic arithmetic, algebraic and geometry skills. Even though the math fundamentals featured some algebra items, they were confined to linear equations, whereas the algebra subtest consisted of more advanced materials such as non-linear equations and complex functions. Similar to the algebra subtest, both math fundamentals and trigonometry seem to be more distal outcomes when compared to fractions comparisons. Although there is no preexisting evidence to guide our hypotheses, our theory suggested the possibility that RPS acuity may be predictive of these two additional measures. Thus, we sought to explore how the RPS might predict these other tests of higher-order mathematics. Third, we added ratio and simple magnitude comparisons for circle stimuli (see Matthews & Chesney, 2015; Meng et al., 2019) as a new format of nonsymbolic comparison tasks along with the line and dot formats from the original study. We added circles for three reasons: 1) Unlike dot arrays, they have no obvious whole number analogs; 2) unlike lines, they are not easily partitioned such that count based strategies are plausible; and 3) despite their use in other RPS studies (e.g., Park, Viegut, & Matthews, 2020; Matthews & Chesney, 2015; Meng et al., 2019), their relations to symbolic mathematics performance have yet to be investigated.

In an analytical extension, we investigated the comparative predictive power of ratio processing ability for each separate format. Prior work has shown that the RPS acuities differ depending on format (Park et al., 2020). Moreover, other work suggests that the relations between acuity for magnitude in a specific format and math achievement may be specific to particular subdomains of mathematics (Lourenco, Bonny, Fernandez, & Rao, 2012; Odic et al., 2016; Park & Cho, 2017). However, Matthews et al. combined performance across different ratio formats to create a composite RPS measure that obscured potential difference by format. In the current study, we included acuity in each format as separate predictors to determine whether RPS acuity in each format differentially relates to mathematical achievement. Moreover, prior work has shown that perceptual acuity for magnitudes can vary substantially by format (i.e., Odic, 2017; Odic, Libertus, Feigenson, & Halberda, 2013; Starr & Brannon, 2015). For example, acuities for continuous magnitudes are typically found to be more accurate compared to acuity for number of dots (Odic, 2017; Odic et al., 2013; Park & Cho, 2017; Starr & Brannon, 2015). For instance, Odic (2017) found that acuity for discriminating line segments was the most accurate followed by acuities for discriminating area and numerosity. Based on this prior research, we hypothesized that our participants would demonstrate highest acuity for our line-based stimuli, followed by that for circles, and then dots. Finally, we expected that ratio comparisons would be more difficult compared to that of simple magnitude as previously found by Matthews et al. (2016).

Method

Participants

Ninety-nine undergraduate students from a large Midwestern university (85 Female; M_age = 20.12, SD = 1.14) participated for course credit.

Measures

Because this study was a conceptual replication of Matthews et al. (2016) we used several of the same measures from Matthews et al., and added a few more as well. For the chief predictors, we used three nonsymbolic ratio comparison tasks: separated formats of the dot and line ratio tasks from Matthews et al. and circle ratios, adapted from Meng, Matthews, and Toomarian (2019). We also included other cognitive tasks as covariate predictors. First we used simple magnitude comparison tasks in dot, line and circle formats to control for the ability to process the absolute magnitudes of the components of ratios (in contrast to their relative magnitudes). Each of these tasks in simple formats has been studied for decades (e.g., Krueger, 1984; Stevens, 1957; Teghtsoonian, 1965). We also included a flanker task to account for differences in inhibitory control and Raven’s Standard Progressive Matrices as a measure general intelligence.

Three of the outcome measures were identical to those in Matthews et al. (2016)—symbolic fractions comparisons, a fractions knowledge assessment (FKA), and Algebra scores from participants’ university placement examinations. We also included two additional placement examinations subtest scores—Trigonometry and Math fundamentals (all the measures are listed in Table 1). Each measure is described in more detail below. Task materials, data collected, and the R scripts are available online via the Open Science Framework (https://osf.io/c75xy).

Table 1

The List of Measures Used in the Current Study

Variable Type / Measures	Conducted in Matthews et al.	Computerized Task^a
Independent Variables
Line ratio comparison	Yes	Yes
Dot ratio comparison	Yes	Yes
Circle ratio comparison	No	Yes
Simple line comparison	Yes	Yes
Simple dot comparison	Yes	Yes
Simple circle comparison	No	Yes
Raven’s Progressive Matrices	No	No
Dependent Variables
Symbolic fractions comparison	Yes	Yes
Fractions Knowledge Assessment	Yes	No
Algebra (placement exam)	Yes	No
Math fundamentals (placement exam)	No	No
Trigonometry (placement exam)	No	No

^aAll computerized tasks were presented on 1,920 × 1,080 resolution screens using E-prime software (Schneider, Eschman, & Zuccolotto, 2002).

Nonsymbolic Comparisons (RPS Acuity and Simple Magnitude Control Tasks)

Nonsymbolic comparison tasks were blocked by type (i.e., ratio or simple magnitude) and format (i.e., dot, line or circle stimuli). For all nonsymbolic comparisons, participants were simultaneously presented with two stimuli and instructed to choose the larger one. Participants indicated their choices via key press—pressing “j” for right and “f” for left. Each trial began with a fixation cross for 200 ms, immediately followed by brief presentation of two comparison stimuli (Figure 1). Per Matthews et al. (2016), nonsymbolic ratio comparisons were presented for 1,500 ms before disappearing, and simple magnitude control trials were presented for only 750 ms. Trials did not advance until participants responded. Each ratio block started with 10 practice trials followed by 40 experimental trials, and each simple magnitude control block started with five practice trials followed by 40 experimental trials.

Task difficulty varied from trial to trial and was operationalized as the ratio distance, or the ratio between compared stimuli in a trial. Difficulty increased as ratio distance approached 1:1. Note that for ratio comparison trials, ratio distance was a ratio of ratios. The ranges of line and dot ratio distances were varied by format per Matthews et al. (Table 2). For circle stimuli, we adopted the distances for lines from Matthews et al., as our prior piloting demonstrated that participants had similar discrimination abilities in that format.

Table 2

Ratio Bins Indicating Difficulty in Each Task and Format

Task	Ratio Comparison			Magnitude Comparison
Format	Line	Circle	Dot	Line	Circle	Dot
Maximum	8:7	8:7	6:5	15:14	8:7	8:7
Minimum	1:2	1:2	1:3	12:11	2:1	2:1

Line-Ratio Comparisons

Line ratio stimuli were constructed by juxtaposing white and black line segments with jitter per Matthews et al. (2016). White segments ranged from approximately 24 to 128 pixels long, and black segments ranged from approximately 30 to 254 pixels. We followed Matthews et al.’s controls to minimize the likelihood that participants would choose based on the physical length of components as opposed to irrelevant dimension such as overall raw size. Hence, we controlled stimuli such that the larger ratio had longer summed lengths (numerator plus denominator) in half of all trials, and the larger ratio had shorter summed lengths in the other half of trials.

Simple Line Comparisons

Individual black line segments appeared on each side of the screen. Segments ranged from approximately 64 to 162 pixels in length. The two lines were always jittered relative to each other so that participants would be encouraged to consider the entire lengths of each line as opposed to merely focusing on the tops of the lines as would be possible if they were aligned at the bottom.

Circle-Ratio Comparisons

Stimuli were constructed of white circles in the numerator/top position and black circles in the denominator/bottom position. The size of white circles ranged from approximately 2,826 to 12,070 square pixels, and the size of black circles ranged from approximately 3,847 to 18,617 square pixels. We controlled summed areas such that the larger ratio had a larger summed area in half of all trials, and the larger ratio had smaller summed area in the other half of trials.

Simple Circle Comparisons

Two black circles were presented on each side of the screen. The size of circles ranged from approximately 1,661 to 5,539 squared pixels.

Dot-Ratio Comparisons

Ratio stimuli were constructed from juxtaposed pairs of white dot arrays against black backgrounds (numerators) and black dot arrays against white backgrounds (denominators). The number of dots in the numerators ranged from 11 to 67, and the number of dots in the denominators ranged from 30 to 118. We controlled the summed numerosities (i.e., the summed number of white and black dots) such that in half of all trials, the larger ratio featured a greater summed number of dots, and in the other half, the larger ratio had a smaller number of summed dots.

Simple Dot Comparisons

An array of black dots against a rectangular gray background appeared on each side of the screen. The number of dots in arrays ranged from 50 to 200 to preclude the possibility of counting given the rapid rate of response typical for such tasks (i.e., <1,000 ms). In half of the trials, the summed area of dots was constant across the two arrays, and in the other half, the dot size was constant across two arrays. Thus, in the first case, dot size was anticorrelated with numerosity, and in the other case, the cumulative area and density were correlated with numerosity.

Symbolic Fractions Comparisons

Participants selected the larger of two symbolic fractions via keypress. All fractions stimuli were irreducible and composed of single-digit numerators and denominators. We used the same 30 pairs used by Matthews et al. (2016), which excluded fractions pairs sharing common components (e.g., 3/5 vs. 3/6) to minimize dependency on componential strategies (i.e., judgments based solely on numerator or solely on denominator comparisons, rather than on overall fraction magnitude). On each trial, a fixation cross appeared for 200 ms followed by presentation of comparison stimuli until the participant responded or until the trial timed out at 5,000 ms. Symbolic comparison blocks started with five practice trials followed by 30 experimental trials. The side on which the larger fraction appeared (left/right) was counterbalanced across trials. Order of trial presentation was randomized.

Flanker Task

Our version of this measure of inhibitory control was identical to that from Matthews et al. (2016). Participants were asked to decide which direction the center arrow among five was pointing. On each trial, a fixation cross was presented for 500 ms followed by an array of five evenly spaced arrows, which appeared for up to 800 ms or until the participant’s response. Participants first received 12 practice trials followed by 80 test trials. Half of experimental trials were congruent, in which all stimuli pointed the same direction, and the other half of the trials were incongruent, in which the center stimulus pointed the opposite direction from the four flanking arrows. We used the congruity-based difference in RTs (RT_incongruent − RT_congruent) in our analyses.

Fractions Knowledge Assessment

The FKA was a 38-item pencil-and-paper test constructed by Matthews et al. (2016). Items were culled from key national and international assessments (e.g., National Assessment of Educational Progress and the Trends in International Mathematics, and Science Study), and from assessments developed by psychology and math education researchers (Carpenter, Corbitt, & National Council of Teachers of Mathematics, 1981; Hallett, Nunes, Bryant, & Thorpe, 2012). Items were intended to measure aspects of conceptual knowledge and of fraction arithmetic procedures. The conceptual knowledge items covered ordering of fractions, density, and how fractions operations affect magnitudes plotted on a number line (see sample items in Appendix). Participants had 20 minutes to complete the test. All participants were able to finish the test within the time limit.

College Mathematics Placement Exams

We obtained scores from three subtests of the math placement exam taken by all incoming freshman: Advanced Algebra (AALG), Math fundamentals (MFND), and Trigonometry & analytic geometry (TAG). The exams were taken by all freshman once admitted to the University for placement purposes and have been subject to years of validation work by the university testing services. As noted above, Math fundamentals tested a combination of basic arithmetic, algebraic and geometry skills. The subtests were composed of 30, 25, and 20 items respectively. The internal consistency reliability of each test (Cronbach’s α) was .89 for MFND, .88 for AALG, and .85 for TAG. The mean normalized assessment score for each test is 500 with a standard deviation of 100.

Raven’s Progressive Matrices

Raven’s is a widely-used standardized test measuring fluid intelligence (Raven, Raven, & Court, 1998). The test is composed of five sets of 12 items each. Each item requires analyzing a pattern of figures and reasoning about what figure would complete the pattern if placed in the blank. As the test progresses, the test set becomes more complex. Participants were given 20 minutes to finish the test. Each item of the test was worth 1 point, with a total possible raw score of 60. This test has been used across a wide range of population, with test-retest reliabilities of .83–.93 (Raven, 2000; Raven et al., 1998). We used the raw score in analysis.

Experimental Procedure

The experiment was divided into two sessions, each on a different day (M_gap = 5.25 days, SD = 2.8). In session 1, participants completed all comparison and flanker tasks on computers. First, participants completed the nonsymbolic comparison tasks. The order of format was always dots, lines, and then circles. In each format, participants completed the simple magnitude control tasks first followed by the ratio magnitude comparison task. Next, participants completed symbolic fractions comparison followed by the flanker test. In session 2, the participants completed the FKA followed by Raven’s Progressive Matrices.

Missing Cases and Outlier Removal

Data for simple line comparisons from one participant, simple circle comparisons from another, and the FKA for another were not collected due to experimenter error. Also, FKA and Raven’s scores were unavailable for six participants who failed return for session 2. Additionally, we were unable to secure placement exam scores for one participant. These specific data elements were missing for individual participants whose data remained otherwise intact. Remaining data for these participants were included in the analyses whenever possible. However, because regressions were run using listwise deletion, when an element was missing, participants’ with missing data elements were removed entirely from those regressions. We indicated the included sample size for each analysis in the corresponding tables for reference.

For all computerized tasks, trials with reaction times (RTs) shorter than 250 ms and trials with RTs more than 3 SD from a participant’s mean RT were trimmed. These steps resulted in the loss of 1.6 to 3.91% of the data for each computerized task. Additionally, for all comparison tasks, we excluded data from participants who scored beyond 3 standard deviations from the group mean for that task and from participants who scored below chance level. This step resulted in the exclusion of four participants’ ratio comparison task data (2 from circle-ratio, 1 from dot-ratio, 1 from line-ratio, 3 from symbolic fractions comparison). This resulted in a trimmed analytic sample of 80 participants. Note, we used this trimmed sample in the hierarchical regressions, except those predicting symbolic fractions comparisons, so that the sample would not vary depending on the number of predictors in each step. For models predicting symbolic fractions comparison, an additional three participants were excluded for below chance level performance, resulting in analytic sample of 77 participants.

Weber Fraction Analysis

We used task accuracy as the measure of acuity in most of our analyses. However, to compare acuity across different comparison tasks, we computed internal weber fractions (hereafter, w) as opposed to accuracy as measures of acuity. In comparison tasks, W represents the smallest discriminable difference expressed as a ratio between the two magnitudes (e.g., Halberda, Mazzocco, & Feigenson, 2008; Pica, Lemer, Izard, & Dehaene, 2004), with smaller w indicating better discrimination acuity. Because the weber fraction is invariant to differences in specific ratio difficulties used to measure it, w allows direct comparison across tasks. By contrast, it is inappropriate to compare discriminability across tasks using accuracy when the tasks use different ratio distances for comparisons.

The model for Weber’s Law assumes that the internal representation for magnitude can be represented as a Gaussian function. For instance, in the case of dot comparisons, if we use $n_{1}$ and $n_{2}$ to indicate the number of dots in each array, the Gaussian error function (erfc; Eq. 1) computes the degree of overlap between the two Gaussian functions. The overlap can be expressed as a new Gaussian with a mean of ( $n_{1}$ - $n_{2}$ ) and a standard deviation of $w \sqrt{n_{1}^{2} + n_{2}^{2}}$ . For each individual, the w can be calculated using the Levenberg-Marquardt algorithm for nonlinear least squares fit on mean accuracy as a function of magnitude ratio (Eq. 2).

Eq. 1. Gaussian Error Function

1

erfc (x) = 1 - \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t

Eq. 2. Expected Percentage Correct

2

Percentage correct = 1 - \frac{1}{2} erfc (\frac{n_{1} - n_{2}}{\sqrt{2} w \sqrt{n_{1}^{2} + n_{2}^{2}}})

We calculated w for each individual participant in each nonsymbolic comparison task. Prior to analysis, we excluded three participants’ ws from circle ratio comparisons and one participant’s w from dot ratio comparison due to extremely large values (>10, which would imply difficulty discriminating between 10 and 110 dots). We took such extreme values as indicators of noncompliance (see Odic [2017] for similar approach). We also excluded participants’ w data that fell beyond 3 standard deviations from the group mean for each task. This step resulted in the exclusion of several w data points distributed across tasks (3 from dot ratio, 5 from circle ratio, 1 from line ratio, 1 from dot and 1 from line comparisons).

Statistical Power

We conducted two different types of analysis of our results: the first analysis was to compare acuities across different tasks which using w as the dependent variable, the second type of analysis replicated Matthews et al.’s (2016) regressions, using acuity indexed by accuracy as independent variable. Because replication was our main goal, we powered the study for the analyses using accuracy as the chief independent variable. We used the “pwr” package in R to calculate statistical power based on the effect sizes of .18–.25 observed in the hierarchical models predicting symbolic fractions comparison (f² = .18), FKA (f² = .25), and Algebra (f² = .20) in Matthews et al. (2016) and α = .05 (Figure 2). Given these conditions, our initial recruitment of 99 participants would have resulted in power β = .96 for FKA, β = .91 for algebra, and β = .85 for symbolic fractions comparison. After cleaning, our regression framework with the trimmed samples using comparison task accuracy as predictors had the power to defect the above listed effects with β = .90 (n = 80) for FKA, β = .82 (n = 80) for Algebra, and β = .72 (n = 77) for symbolic fractions. For the supplementary analysis with models using ws as predictors, power was β = .87 (n = 74) for FKA, β = .78, (n = 74) for Algebra and β = .68 (n = 71) for symbolic fractions comparison (see Table S4, Supplementary Materials).

Click to enlarge

Figure 2

Expected Power (β) Changes Depending on the Sample Size

Results

RPS and Simple Magnitude Acuity

To compare acuity across magnitude types and formats, we conducted linear mixed effects regression models to account for within-subject correlation using “Imer” function of lme4 package in R software (Bates, Mächler, Bolker, & Walker, 2015). We regressed acuity (ws) against task (2 levels, ratio = 0, simple magnitude = 1) and format (3 levels, dot = 0, circle = 1, line = 2). We also included a task × format interaction term in the model to check whether observed format differences differed depending on whether participants were comparing ratios or making simple magnitude comparisons. To facilitate evaluation of our hypotheses for how ws would vary with format and task (i.e., the ratio > magnitude task and dot > circle > line), we used a backward difference coding scheme to compare adjacent levels of variables (i.e., coded so that the mean of a given level was compared with the mean of the immediately prior level). We estimated fixed effects for all predictors with random intercepts. Results from these regressions are presented in Table 3.

Table 3

Linear Mixed Effects Regressing ws Against Tasks and Formats

Regressor	β	t	p
Intercept	.18	31.00	< .001**
Simple magnitude	−.21	−27.03	< .001**
Circle—Dot	−.07	−7.31	< .001**
Line—Circle	−.02	−2.32	.021*
\|Simple—Ratio magnitude\|*\|Circle—Dot\|	.02	1.27	.209
\|Simple—Ratio magnitude\|*\|Line—Circle\|	.02	1.05	.296

*p < .05. **p < .01.

Participants showed significantly higher acuity (lower ws) for simple magnitude comparisons than for ratio comparisons (β = −.21, p < .001; Figure 3, Table 3). Moreover, there was a significant format effect. Consistent with our hypotheses, acuity was higher for lines than for circles (β = −.02, p = .021) and higher for circles than for dot arrays (β = −.07, p < .001). There were no significant interactions between tasks and formats, which suggests that main effects were additive (see Figure 3).

Click to enlarge

Figure 3

Task and Format Differences in ws

Note. Lower weber fractions indicate higher acuity. Simple magnitude acuity was sharper than that for ratio magnitudes, as all ws in the panel on the right are lower than those on the left. Acuity also varied by format with acuity increasing (i.e., w decreasing) from dot to circle to line format.

Relations Between Nonsymbolic Comparisons and Symbolic Math Outcomes

Revisiting Matthews et al. (2016)

We first replicated the analyses from Matthews et al. (2016). We calculated a composite RPS acuity by taking the mean of line ratio and dot ratio accuracies per Matthews et al. Zero-order correlations showed that RPS composite scores were significantly correlated with three of the five symbolic outcome measures (FKA: r = .26, p = .019, Math fundamentals: r = .28, p = .014, Trigonometry: r = .23, p = .038) but not with symbolic fraction comparison (r = .16, p = .16) and Algebra (r = .20, p = .069) (Table 4).

Table 4

Bivariate Correlation Among Accuracies From Comparison Tasks and Tests (n = 80, Except for FF: n = 77)

Measure	Line Ratio	Circle Ratio	Dot Ratio	Line	Circle	Dot	FF	Inhibition	FKA	ALG	MF	Trig	Raven’s
RPS Composite	.80**	.54**	.90**	.44**	.30**	.42**	.16	−.07	.26*	.20	.28*	.23*	.33**
Line Ratio		.40**	.46**	.23*	.03	.24*	.16	−.07	.32**	.28*	.35**	.29**	.32**
Circle Ratio			.51**	.18	.05	.21	−.02	−.07	.16	.00	.07	.06	.08
Dot Ratio				.48**	.42**	.44**	.12	−.05	.16	.10	.15	.13	.26*
Line					.47**	.34**	−.01	−.10	.03	.11	.08	.01	.17
Circle						.28*	.07	−.03	−.09	.03	−.07	−.01	−.07
Dot							−.17	−.01	.04	.25*	.15	.16	.22*
FF								−.02	.37**	.23*	.39**	.18	.16
Inhibition									−.05	−.04	−.12	−.07	−.18
FKA										.51**	.54**	.46**	.32**
ALG											.82**	.72**	.42**
MF												.68**	.43**
Trig													.30**

Note. FF = fractions comparison; FKA = fractional knowledge; ALG = algebra; MF = math fundamental; Trig = Trigonometry.

*p < .05. **p < .01.

Next, we conducted a series of two-stage hierarchical linear regressions (Table 5), one for each of the symbolic outcomes (i.e., symbolic fractions comparison, FKA, Algebra, Math fundamentals, Trigonometry). In the first stage, we entered the control variables simple dot acuity, simple line acuity, and flanker performance. In the second stage, we added composite RPS acuity. To facilitate the interpretation of effect size across variables, we reported standardized coefficients in all hierarchical regression models.

Table 5

Results From the Linear Regression Predicting From Each Format of Ratio Comparison Tasks

Regressor	Fractions Comparison			Fractions Knowledge			Algebra			Math Fundamentals			Trigonometry
Regressor	β	p	sr²	β	p	sr²	β	p	sr²	β	p	sr²	β	p	sr²
Step 1	R² = .03			R² = .004			R² = .07			R² = .04			R² = .03
Line	.057	.643	.003	.017	.890	.000	.019	.873	.000	.015	.904	.000	−.059	.629	.003
Dot	−.194	.118	.033	.030	.808	.001	.248	.039*	.054	.142	.241	.018	.179	.140	.028
Inhibition	−.017	.882	.000	−.048	.676	.002	−.038	.734	.001	−.120	.291	.014	−.076	.503	.006
Step 2	ΔR² = .07			ΔR² = .08			ΔR² = .01			ΔR² = .05			ΔR² = .04
Line	−.046	.723	.002	−.090	.476	.006	−.022	.864	.000	−.076	.544	.004	−.139	.276	.015
Dot	−.281	.028*	.063	−.069	.580	.004	.211	.094	.025	.058	.639	.003	.106	.399	.009
Inhibition	−.010	.933	.000	−.038	.735	.001	−.034	.761	.001	−.111	.317	.012	−.069	.541	.005
RPS composite	.298	.027*	.064	.327	.014*	.078	.124	.344	.011	.277	.035*	.056	.244	.064	.043

Note. Line = Line acuity; Dot = Dot acuity.

*p < .05.

Consistent with Matthews et al., we found that RPS composite acuity significantly predicted symbolic fractions comparison (β = .298, p = .027) and FKA performance (β = .327, p = .014) even after controlling for inhibition and simple magnitude acuity. We further found that RPS composite acuity significantly predicted math fundamental scores (β = .277, p = .035). However, unlike Matthews et al., we found RPS composite acuity failed to predict Algebra placement exam scores (β = .124, p = .344). It also did not predict trigonometry scores (β = .244, p = .064).

These results noted, use of an RPS composite does not allow insight into whether RPS acuity in different formats are differentially predictive of math outcomes. Thus, we expanded Matthews et al. (2016) and disaggregated RPS acuity by format and controlled for fluid intelligence using Raven’s Standard Progressive Matrices. We conducted analyses parallel to those above, but this time we included acuity in each format as separate predictors. Although we used accuracy as the indicator of acuity for the analyses reported below, we also conducted supplementary analyses using ws as an alternate measure of acuity. The results with ws were consistent with those reported using accuracy (see Supplementary Materials).

Bivariate correlations showed that line ratio acuity was significantly correlated with four of five symbolic math outcomes (FKA: r = .32, p = .004, Algebra: r = .28, p = .013, Math fundamentals: r = .35, p = .001, Trig: r = .29, p = .009) and with Raven’s scores (r = .32, p = .004). In contrast, dot and circle ratio acuities were not correlated with any of the five symbolic outcomes (Table 4). Regarding Raven’s, it was correlated with all math outcomes (FKA: r = .32, p = .004, Algebra: r = .42, p < .001, Math Fundamental: r = .43, p < .001, Trig: r = .30, p = .007) except for symbolic fractions comparison (r = .16, p = .16). Because our regressions used trimmed samples, we also conducted supplementary bivariate correlation analysis without using list-wise deletion (Table S2, Supplementary Materials), and the results showed similar correlation across variables.

Finally, we performed a new set of three-stage hierarchical linear regressions that extended Matthews et al. (2016) (Tables 6, 7, 8, 9, and 10). In Stage 1, we entered flanker scores and all simple magnitude acuities, in Stage 2, we entered ratio magnitude acuities, and in Stage 3, we added fluid intelligence. Below, we present results from regressions using comparison accuracies, but supplementary analyses conducted with weber fractions and yielded similar results (S4 Table a–e, Supplementary Materials).

Table 6

Results From the Hierarchical Regression Analyses Predicting Scores on Symbolic Fractions Comparison (n = 77)

Regressor	Step 1 (R² = .05)			Step 2 (ΔR² = .07)			Step 3 (ΔR² = .02)
Regressor	β	p	sr²	β	p	sr²	β	p	sr²
Line	−.004	.299	.000	−.091	.523	.005	−.108	.448	.007
Circle	.137	.894	.014	.107	.445	.007	.156	.279	.015
Dot	−.213	.091	.039	−.291	.026*	.066	−.321	.015*	.078
Inhibition	−.021	.900	.000	−.016	.887	.000	.011	.926	.000
Line Ratio				.204	.131	.030	.166	.225	.019
Circle Ratio				−.140	.310	.013	−.103	.460	.007
Dot Ratio				.225	.190	.022	.177	.309	.013
Raven’s							.178	.168	.024