Introduction

A question central to the field of numerical cognition is “What representations underlie the ability to think and reason about number?” (Feigenson et al., 2004). One prominent theory holds that our sense of the value of number is grounded in a sense of whole number magnitudes (numerosities) provided by a perceptually based Approximate Number System (ANS) that can rapidly enumerate sets without counting (Dehaene, 2011; Piazza, 2010). However, Sidney et al. (2017) noted this theory fails a simple test. Research demonstrates that adults’ understanding of the relative values of symbolic numbers is both very accurate and very linear (e.g., Opfer & Siegler, 2007; Siegler et al., 2009; Siegler & Opfer, 2003). However, the map between ANS estimates and symbolic numbers is demonstrably inaccurate and non-linear. When participants evaluate non-symbolic numerosities (e.g., dot arrays), they typically underestimate with responses well fit by logarithmic or compressive power functions, rather than by linear functions (e.g., Crollen et al., 2011; Izard & Dehaene, 2008; Taves, 1941). It follows that inaccurate, non-linear ANS estimates alone do not provide stable ground for an accurate, linear sense of symbolic numbers.

There is an emerging hypothesis that number concepts might be grounded in part by using non-symbolic ratios rather than individual non-symbolic numerosities (Chesney & Matthews, 2018; Kalra et al., 2020; Lewis et al., 2016; Matthews & Chesney, 2015; see also Bonn & Cantlon, 2017; Jacob et al., 2012). Chesney and Matthews (2018) recently demonstrated that participants’ symbolic numerical estimates of ratios composed of pairs of non-symbolic numerosities are both linear and accurate. Note that these patterns are to be distinguished from estimates that are quite linear, but nevertheless inaccurate. For instance, when placing exponential stimuli (e.g., .500 × 104.5) on a number line from 0 – 31623, regressions of participants’ responses on the true stimuli values were well fit by a linear function, but the slope of that function was .35: Responses, while linear, were inaccurate, consistently underestimating the stimulus values (Chesney & Matthews, 2013). The finding of accurate results with non-symbolic ratios have led to hypotheses that a non-symbolic ratio sense could help ground symbolic numbers – addressing the inaccuracy and non-linearity of the map to traditional ANS estimates (e.g., Lewis et al., 2016; Matthews & Chesney, 2015; Sidney et al., 2017; compare with Clarke & Beck, 2021)

Theories hypothesizing that ratio sense may help ground symbolic numbers are relatively new. As such, there are several gaps in the supporting research. For instance, Chesney and Matthews (2018) focused on ratios composed of non-symbolic numerosities (specifically dot arrays) that arguably still implicate ANS involvement (see also Clarke & Beck, 2021). However, non-symbolic ratios can be instantiated by a wide variety of stimuli, and existing theory suggests that the map between non-symbolic and symbolic ratios might remain stable regardless of stimulus format (e.g., Matthews & Chesney, 2015). Here, we extend the reporting of Chesney and Matthews (2018) and present previously unpublished data consistent with this format irrelevance in the context of estimation. We investigated whether ratios made from continuous non-numerical stimuli – circle areas – would yield results similar to ratios made from dot arrays. If ratios made from continuous stimuli can support accurate numerical estimation, it would stand as strong evidence that a general ratio-processing ability, wholly separate from ANS-based whole number estimation, can provide a stable map to their analogous numerical symbols.

Although several studies have yielded findings consistent with this ratio-as-potential-ground hypothesis, most have employed alternative forced-choice comparison paradigms (e.g., Kalra et al., 2020; Matthews et al., 2016; Matthews & Chesney, 2015; Matthews & Lewis, 2017). Although well suited to test discrimination, forced-choice comparisons are not a good means for assessing participants’ conscious abilities to map between non-symbolic and symbolic analogues. Rather, this ability is perhaps best explored using estimation paradigms (e.g., Chesney & Matthews, 2018; Stevens & Galanter, 1957). Chesney and Matthews (2018) found that participants were quite accurate at assigning symbolic fractions values to ratios composed of pairs of dot arrays. By contrast, participants were much less accurate at mapping the numerosities of those individual dot arrays to symbolic integers, yielding linear estimates, but with an average slope of .327, as opposed to one. Currently, it is an open question whether similar patterns would be found with alternative non-symbolic stimuli.

Here, we present data from additional tasks collected in parallel with those reported in Chesney and Matthews (2018). In these tasks, participants estimated values derived from non-numeric visual stimuli (circle areas) rather than stimuli with a numeric component (i.e., dot arrays). Circles were chosen because they have no obvious whole number analogs and are not easily partitioned into regular pieces that facilitate algorithmic or procedural mapping to specific numbers. Our predictions for these circle-based tasks mirrored Chesney and Matthews’ predictions for dot-based tasks:

  • On Free estimation tasks (Fig. 1a), where participants are asked to assign symbolic numerical values to individual circles, estimates should be characterized by compressive power functions as has been found in past work on perceived circle area (Teghtsoonian, 1965). Moreover, estimates should be marked by scalar variability (i.e., variability will increase linearly with stimulus size) as typically found in free estimation tasks (Cordes et al., 2001; Izard & Dehaene, 2008; Thurley, 2016).

  • On Line estimation tasks, where participants are asked to place circles on a line anchored by a very small circle on one end and a large circle on the other (Fig. 1b), we predicted that responses would be approximately linear by way of a cyclical power model. Barth and Paladino (2011) cogently argued that analogous number line estimation tasks can generally be characterized as proportion judgment tasks that employ strategic subdivision of the line to create an additional anchor point at the center (see also Hollands & Dyre, 2000; Hollands et al., 2002; Spence, 1990). Thus, median estimates should be close to the correct value of the stimulus, standard deviations should decrease near end-point anchors and near the midpoint of the line, and responses should exhibit a cyclical pattern of over- and then under-estimation. However, we note that if these subdivision-based strategies were not employed, and line estimation were to proceed from a direct perceptual mechanism, one might instead expect a performance pattern similar to the Ratio task (see Jacob & Nieder, 2009; Lewis et al., 2016; Matthews & Chesney, 2015).

  • On Ratio estimation tasks, where participants are to assign symbolic fractions values to paired circles (Fig. 1c), we predicted an accurate linear relation (i.e., a slope near 1 and an intercept near 0) between the circle ratio stimuli presented and participants’ estimates. This is because prior work has indicated that ratio estimation may proceed from a more direct perceptual mechanism (Jacob & Nieder, 2009; Lewis et al., 2016; Matthews & Chesney, 2015; Vallentin & Nieder, 2008) as opposed to the strategy-bound method that results in cyclical performance on line-based proportion judgment (Barth & Paladino, 2011). Moreover, we predicted there would be reduced variability near the natural anchors of 0 and 1, as the bounds should naturally constrain the estimates (see Chesney & Matthews, 2018).

Fig. 1
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Diagrams of trials in the three estimation conditions: a Free estimation, b Line estimation, c Ratio estimation

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Method

Participants

Participants were 31 undergraduates at the University of Notre Dame who participated for course credit (16 female; age 18–22 years). Participants had normal or corrected-to-normal vision.

Materials and design

All training and testing stimuli were presented using Superlab 4 software (Cedrus Corporation, 2007) on Apple® iMac 5.1 computers running OS10.6. Each computer had a 17-in. LCD display with a resolution of 1,440 × 900 pixels and a refresh rate of 60 Hz. These screen dimensions subtended approximately 34 × 22° of visual angle with participants seated ~60 cm from the screen. No restraints were used to restrict head motion, so degrees of visual angle are only approximate. We created images of black circles on white backgrounds to be used as stimuli in three distinct blocks of trials. Circles ranged from 1.6 cm2 to 49.5 cm2 in area (54–300 pixels in diameter, see Appendix Table 4).

Free estimation stimuli

Free estimation trials presented 26 circles ranging from 1.6 cm2 to 49.5 cm2 in area (see Appendix Table 4). Circles were presented alone on a white screen, one at a time, over 27 trials. The first circle (24.0 cm2) served as a modulus and was repeated later in the set.

Ratio estimation stimuli

Each ratio estimation trial presented a pair of circles arranged above and below a bar to form a fraction, paralleling the non-symbolic numerosity stimuli used in Chesney and Matthews (2018). These pairs instantiated ratios between 0 and 1, yielding 41 total trials (Fig. 1c and Appendix Table 4). We varied the areas of circle denominators to ensure responses were not reliant on a particular denominator.

Line estimation stimuli

For each trial, participants were shown a “number-line” anchored by a small circle on the left and a larger circle on the right. To ensure answers were not dependent on a single set of anchors, half the trials featured circles with areas of 3.0 and 49.5 cm2 as the small and large anchors, and half the trials featured circles with 1.6 and 24.7 cm2 anchors. Stimuli to be estimated appeared 2.5 cm above the center of the line. For each anchor, there were 17 to-be-estimated stimulus circles, corresponding to the numerators for the ratio estimates with corresponding denominators of 24.7 and 49.5 cm2 (Fig. 1b and Appendix Table 4). Thus, there were 34 trials in total. All circles and the line were rendered in black against a white background.

Procedure

Participants first completed the Ratio estimation block, followed by the Line estimation block, and finally the Free estimation block. This order was set to reduce the likelihood that participants would deliberately attempt to use recalled responses from one block when assigning estimates in subsequent blocks. Each block began with a set of instructions, using example stimuli that were different from the experimental stimuli. Trial order was randomized within each block. As described in Chesney and Matthews (2018), participants also completed similar tasks involving dot arrays, a symbolic number line task, and several mathematics assessments not discussed in this article. All stimulus presentations were speeded to preclude the possibility that participants might measure the stimuli and subsequently use algorithms to generate more accurate estimates.

Ratio estimation

In ratio estimation trials, participants were instructed “to estimate the ratio between the sizes of two circles” composing each stimulus. For clarity, they were told that “by size here we mean area or how much room a circle takes up on the screen.” Trials began by presenting a fixation cross for 500 ms, followed by stimulus presentation for 1,500 ms. Once the stimulus disappeared, a textbox appeared asking, “What was the proportion?” Participants then typed their answers into a text box via the keyboard. After responding, they clicked a <Next> button to initiate the subsequent trial (see Fig. 1c).

Line estimation

A line bounded by circle anchors appeared at the beginning of each trial for 500 ms before a stimulus circle was presented 2.5 cm above the center of the line for 1,500 ms. Once the stimulus disappeared, participants used a mouse to indicate the position on the line corresponding to the stimulus magnitude. The line and anchors remained on the screen throughout the duration of each trial. After responding, participants clicked a <Next> button to initiate the subsequent trial (see Fig. 1b).

Free estimation

Instructions for the free estimation trials were based loosely on those from Teghtsoonian (1965). However, participants were free to provide their own numbers for every stimulus rather than being given a numeric value for a standard figure. This has been demonstrated to be a good way to elicit an unconstrained quantitative judgment of magnitude (e.g., Marks & Florentine, 2011; Stevens, 1957). Participants were told: “When you see the first circle, give it a number – any number that seems appropriate to you corresponding to how much room it takes up on the screen. Then you will be shown more circles, and you will give each a number as well.” Participants were further told that they should try to make the ratios of the numbers they assigned to different circles proportional to the relative sizes of those circles’ areas. Trials began when participants indicated that they understood the instructions (see Fig. 1a). Using this method to elicit unconstrained absolute magnitude judgments frequently yields underestimates that are related to actual magnitudes by a power function, in accordance with the Weber-Fechner law (Stevens, 1957; Zwislocki, 1983).

For each trial, a stimulus circle was presented in the center of the screen for 1,500 ms. Once the stimulus disappeared, a textbox appeared asking participants to “Please give the circle a number.” Participants then typed their answers into a text box. After responding, they clicked a <Next> button to initiate the subsequent trial. The first trial always presented a 24.0 cm2 circle, and this circle served as the standard for the task. The subsequent trials presented 26 different circles ranging from 1.6 cm2 to 49.5 cm2 in area (see Appendix Table 4). Order was randomized for each participant.

Coding

Participants’ spatial position responses on the Line estimation trials were converted to numerical form corresponding to each response’s relative location on a 0–1 line representing circle area. For example, a click on the midpoint of the line was coded as a response of .5. Circles with areas half that of the upper anchor should be placed at that location. We believe our treatment of the lower bound as 0 is justified, given the conventional understanding of number line tasks; however, we note results were similar when line positions were coded to reflect the relative size of the upper and lower anchors. Responses for ratio estimation trials were accepted in fractional, decimal, and mixed number formats, but all were converted to decimal format for analysis. A total of 10 of 1,271 trials (0.8%) were excluded due to clear participant error, including five not in a valid format (i.e., ‘1/’,‘1/’, ‘1/’, ‘7/’, & ‘/4’) and five outliers (i.e., 78/7, 8, 5, 1.9, & 1.9).

Free estimation items were scaled individually based on the responses provided to the first item (i.e., the modulus), which was always a 24 cm2 circle. Participants’ free responses were divided by their idiosyncratic modulus estimates and then multiplied by .485, as the 24 cm2 modulus was 48.5/100th the area of the largest stimulus (49.5 cm2). Stimulus values were scaled by area such that the value of the 49.5 cm2 circle was 1. This allowed us to chart participants’ response patterns on the same scale.

Results

Analysis

Blocks were analyzed separately. For each block, we found the median and standard deviation for responses for each stimulus value. To be conservative, values from trials where the stimuli equaled the modulus, denominator, or upper anchor were excluded, as finding the ratio between two different circles might be considered a different task to identifying that two circles are identical. Following the procedure of Chesney and Matthews (2018), we fit the median responses to four different models:

$${\displaystyle \begin{array}{c} Linear:\mathrm{median}=\mathrm{B}\^ast \mathrm{stimulus}+\mathrm{C}\\ {} Logarithmic:\mathrm{median}=\mathrm{B}\^ast \ln \left(\mathrm{stimulus}\right)+\mathrm{C}\\ {}\begin{array}{c} Power:\mathrm{median}=\mathrm{C}\^ast \mathrm{stimulus}\mathrm{^B}\\ {}1- cycle\ cyclical\ power\ model=\left(\left(\left(\mathrm{stimulus}\mathrm{^B}\right)/\left(\left(\ \mathrm{stimulus^B}\right)+\left(\mathrm{Range}\hbox{-} \mathrm{stimulus}\right)\mathrm{^B}\right)\right)\right)\^ast \mathrm{Range}\end{array}\end{array}}$$

Also per Chesney and Matthews (2018), all models were fit by minimizing the sum of squares distance to the predicted value, and all R2s were calculated as 1 - (Residual Sum of Squares) / (Corrected Sum of Squares). We allowed parameters B and C to vary freely in all models, with the 1-cycle cyclical power model including a C parameter, rather than a Range parameter. We ran the 1-cycle model both with Range fixed at 1, and with Range allowed to vary, but constrained to be ≥ 1. We also analyzed variability patterns by regressing standard deviations against the stimulus values. We conducted these analyses using the non-linear regression function of SPSS version 24. Results are presented in Table 1 and Fig. 2.

Table 1 Various regressions on median estimates and linear regressions on standard deviations for the Free estimation, Line estimation, and Ratio estimation tasks
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Fig. 2
figure 2

Median responses and SDs in the Free estimation (top), Line estimation (middle) and Ratio estimation (bottom) blocks

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Regression Results

Median estimates for line and ratio estimation tasks were largely accurate. That is, linear regressions yielded slopes that were very near 1 for both line (1.02) and ratio (1.14) estimation blocks, with intercepts indistinguishable from zero. Only the free estimation block showed any evidence of compression, with a slope of .77 and an intercept that was significantly different from 0. Regression results’ proximity to the canonical slope of 1 and intercept of 0 indicated that line and ratio estimation accurately reflected intervals between stimulus magnitudes, whereas free estimation did not.

The individual data reflected aggregate trends in the median data. As with the aggregate data, line and ratio estimates for most individuals were well-fit by both linear functions (i.e., slopes near 1 and intercepts near 0) and by power functions with exponents near 1, corroborating linear fits (see Table 2). Indeed, all 31 participants’ linear slopes for both the Line estimation block and the Ratio estimation block fell between .7 and 1.25. Free estimation responses were much more variable, with only 16 of 31 participants slopes falling in this range. As can be seen in Table 2, both linear and power models typically explained about 86% of the variability within individuals for the Free estimation trials, 88% for the Line estimation trials, and 90% for the Ratio estimation trials. The power model was the best predictor for most participants on all three tasks, but was typically only able to explain less than 1% more of the variance than the linear model. Both the linear and power models outperformed the logarithmic model for a large majority of the participants (see Table 2). Matched pairs t-tests on participants’ R2s indicated these differences were significant (see Table 3).

Table 2 Mean parameter estimates for various regressions on individual participants for the Free estimation, Line estimation, and Ratio estimation tasks
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Table 3 Comparison of model fits (paired samples t-tests on R2s) for various regressions on individual participants for the Free estimation, Line estimation, and Ratio estimation tasks
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Discussion

The goal of this study was to test whether Chesney and Matthews’ (2018) predictions regarding the effects of task differences on estimation patterns would hold for non-numeric perceptual stimuli just as they did for ANS-processed stimuli. Such a finding would support emerging research that perceived ratio magnitudes constructed from non-numeric stimuli might provide an accurate and stable map to their symbolic numerical analogs. These predictions were largely confirmed. Median estimates for the Free estimation task were somewhat compressive, whereas those for the Ratio estimation task were quite accurate, with linear slopes near 1 and intercepts near 0. Also as predicted, variability increased linearly with stimulus size on Free estimation tasks, whereas Ratio estimation showed a rising and falling variability pattern, with standard deviations decreasing as stimulus ratios approached the natural anchors of 0 and 1. The one divergence from our predictions was that line estimation did not yield a cyclical power function. Rather, performance was very similar to that seen for the Ratio estimation task - high accuracy with a slope near 1 and an intercept near 0 and with reduced variability as stimuli approached the anchors. As noted in the Introduction, this performance pattern would be predicted if participants based their line responses on direct ratio perception but did not use subdivision strategies to construct midline anchors. These data are consistent with the propositions that (a) people can perceive ratios from paired non-symbolic stimuli even when individual components of the ratios are clearly non-numeric, and (b) these ratios can in turn provide a stable map to symbolic number. This has implications for theorizing about cognitive primitives that may exist for grounding numerical reasoning.

From non-symbolic magnitude to symbolic number

Both our Free estimation and Ratio estimation tasks asked participants to assign symbolic numbers to non-symbolic circle stimuli. As in Chesney and Matthews (2018), participants provided accurate estimates for the ratio task, but were less accurate with free estimations. In fact, participant estimates of ratio magnitudes with circles were no less accurate than those Matthews and Chesney found with ratios made from dot arrays, even though stimuli only flashed for a brief 1,500 ms. Given that symbolic number estimates offered for non-symbolic ratios are (a) demonstrably more accurate than free estimation even for numerosity arrays, and (b) that this accuracy is maintained even for ratios constructed from continuous circle stimuli, there is ample evidence that the human perceptual apparatus provides analog representations of non-symbolic ratios (see also Bonn & Cantlon, 2017; Clarke & Beck, 2021; Jacob et al., 2012; Park et al., 2020). Further, it is clear that humans can map between those analog representations and the appropriate corresponding number symbols (Chesney & Matthews, 2018; Matthews & Chesney, 2015). Finally, it is also clear that maps between symbolic numbers and non-symbolic ratios – even those made from circles – are less variable and more linear (i.e., more canonically accurate) than is typically observed in mappings between symbolic numbers and the numerosity of individual dot arrays.

We suggest that the accuracy of human analog representations of non-symbolic ratio magnitudes makes them a good candidate for grounding number symbols. Young children typically have poor understanding of symbolic fraction magnitudes (e.g., Fazio et al., 2014; Siegler et al., 2013), but seem to have robust abilities to accurately compare non-symbolic ratio magnitudes (Park et al., 2020). Thus, whereas the mapping among adults may be bi-directional in nature, it appears that sensitivity to non-symbolic ratio is developmentally prior. This raises the question of how an initial map from non-symbolic magnitudes to number symbols might be established. After all, the symbolic numbers we first learn as children are integers and not fractions.

Chesney and Matthews (2013, 2018) offered a solution to this problem, suggesting that non-symbolic ratios might be mapped to symbolic numbers via cross-format analogy. Because humans are good at matching proportions across symbolic and non-symbolic formats, such analogies may help imbue unknown number symbols with meaning. Specifically, pairing an initial numerical symbol X1 for one non-symbolic modulus S1 (e.g., a circle, a dot array, or line segment length) can create a size reference point for the number symbol. From this point, given a second non-symbolic stimulus S2 and its corresponding symbolic marker X2, the non-symbolic ratio of S1/S2 must be equivalent to X1/X2. This in essence sets up a cross-notation proportion-matching task. This may explain much of the power that number line estimation tasks have shown in assessing understanding of numerical magnitudes (Matthews & Hubbard, 2017; Sidney et al., 2017). Indeed, this mechanism is arguably only a minor corrective to that offered by (Opfer & Siegler, 2007) to explain their number-line findings.

Number-line estimation, ratio, and the importance of task demands

It is widely accepted that linearity in number line estimation is indicative of mature numerical competence (e.g., Rouder & Geary, 2014; Schneider et al., 2018; Siegler et al., 2009). However, there is still much disagreement about the nature of the number line estimation task. Whereas one prominent theory posits that it is an example of pure numerical estimation that reflects participants’ underlying basal representations of numerical magnitude (Booth & Siegler, 2006), another holds that it is essentially a proportion-matching task (Barth & Paladino, 2011; Slusser et al., 2013). However, Chesney and Matthews (2018) highlighted historical differences in patterns observed from line estimation versus free estimation tasks, tasks that have also been considered “pure,” or indicative of peoples’ underlying subjective representations of magnitudes in psychophysics (Kaufman et al., 1949; Krueger, 1984; Stevens, 1957; Taves, 1941).

Even though we used identical ranges of circle stimuli across tasks, we found substantially different patterns of estimation and errors among our unbounded free estimation task and our bounded line estimation and ratio estimation tasks. This performance pattern closely matched what Chesney and Matthews (2018) reported on parallel tasks with dot arrays, which are often considered to be numerical stimuli. Thus, the differences between patterns with free unbounded estimation and bounded line estimation across both numerical and non-numerical stimuli suggest we should be cautious about regarding the general class of line estimation tasks as indicative of underlying representations of magnitude.

Limitations and future directions

There is more work to be done to delimit this ability. First, we need to address more systematically how people translate from ratios to traditional symbols. Ratios (in small/large format) can instantiate an uncountable infinity of real numbers bound by 0 and 1. This arguably seems somewhat different – prima facie – from the unbounded, countable infinity of natural numbers. Thus, there are natural questions about the limits of the applicability of leveraging non-symbolic ratio analogies, especially as numbers become large (Landy et al., 2013, 2017). Second, we need to investigate the divergence of performance on the current line estimation task from performance on line estimation using symbolic numbers and non-symbolic numerosities (e.g., Barth & Paladino, 2011; Chesney & Matthews, 2018). Might the employment of strategies yielding the cyclical power functions be dependent on stimulus format? Third, we need to do modeling work to theoretically unpack similarities and differences between ratio estimation (which maps non-symbolic ratio to symbolic numbers) and line estimation (which maps non-symbolic ratios to other non-symbolic ratios). Finally, the specialized nature of our (relatively small) sample of high achieving undergraduates limits the generalizability of our findings. Although our results indicated that this subpopulation can do an excellent job of mapping between symbolic numbers and ratios, replications with more diverse samples are needed before we can speak to the generality of our findings.

Conclusion

The current results are consistent with theories speculating that ratios are abstract quantities that might undergird a generalized magnitude system (Balci & Gallistel, 2006; Bonn & Cantlon, 2017; cf., Gallistel & Gelman, 2000). They are additionally consistent with recent theories suggesting that non-symbolic ratio magnitude processing might serve as a cognitive primitive for grounding numerical cognition (e.g., Kalra et al., 2020; Lewis et al., 2016; Park et al., 2020). This growing body of work highlights that humans represent non-symbolic analogs to rational numbers both accurately and “abstractly,” in an amodal fashion unbounded to particular units. There is considerable evidence that this human capacity to represent non-symbolic ratios may stand as a much more flexible sort of number sense than that afforded by the ANS as typically conceived (cf. Clarke & Beck, 2021). This ratio processing ability does not end with the ability to compare ratios across symbolic and non-symbolic formats (Matthews & Chesney, 2015) or with the ability to extract and symbolically represent accurate ratios from non-symbolic numerical stimuli (Chesney & Matthews, 2018). It is also the case that people can similarly extract and symbolically represent accurate ratios from non-symbolic stimuli that are non-numeric. These results confirm that non-symbolic ratios are a good candidate for facilitating the grounding of numerical symbols. If this indeed proves to be the case, then a large swath of the human psychophysical apparatus, not merely the ANS, could be recruited to aid in the grounding of symbolic numbers.