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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.

However, growing evidence suggests that there may be a primitive nonsymbolic number sense that is different from the ANS (

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 (

A few neuroimaging studies have similarly demonstrated processing of nonsymbolic and symbolic ratios at the neural level (

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,

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,

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 (

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 (

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 (

Ninety-nine undergraduate students from a large Midwestern university (85 Female; _{age}

Because this study was a conceptual replication of

Three of the outcome measures were identical to those in

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 |

^{a}All computerized tasks were presented on 1,920 × 1,080 resolution screens using E-prime software (

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 (

Task difficulty varied from trial to trial and was operationalized as the ratio

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 stimuli were constructed by juxtaposing white and black line segments with jitter per

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.

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.

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.

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.

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.

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

Our version of this measure of inhibitory control was identical to that from _{incongruent} − RT_{congruent}) in our analyses.

The FKA was a 38-item pencil-and-paper test constructed by

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 is a widely-used standardized test measuring fluid intelligence (

The experiment was divided into two sessions, each on a different day (_{gap} = 5.25 days,

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

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,

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

We calculated

We conducted two different types of analysis of our results: the first analysis was to compare acuities across different tasks which using ^{2}^{2}^{2}

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 (

Regressor | β | ||
---|---|---|---|

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 |

*

Participants showed significantly higher acuity (lower

We first replicated the analyses from

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** |

*

Next, we conducted a series of two-stage hierarchical linear regressions (

Regressor | Fractions Comparison |
Fractions Knowledge |
Algebra |
Math Fundamentals |
Trigonometry |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
||||||

Step 1 | ^{2} = .03 |
^{2} = .004 |
^{2} = .07 |
^{2} = .04 |
^{2} = .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 | Δ^{2} = .07 |
Δ^{2} = .08 |
Δ^{2} = .01 |
Δ^{2} = .05 |
Δ^{2} = .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 |

*

Consistent with Matthews et al., we found that RPS composite acuity significantly predicted symbolic fractions comparison (β = .298,

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

Bivariate correlations showed that line ratio acuity was significantly correlated with four of five symbolic math outcomes (FKA:

Finally, we performed a new set of three-stage hierarchical linear regressions that extended

Regressor | Step 1 (^{2} = .05) |
Step 2 (Δ^{2} = .07) |
Step 3 (Δ^{2} = .02) |
||||||
---|---|---|---|---|---|---|---|---|---|

β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
||||

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 |

*

Regressor | Step 1 (^{2} = .02) |
Step 2 (Δ^{2} = .10) |
Step 3 (Δ^{2} = .05) |
||||||
---|---|---|---|---|---|---|---|---|---|

β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
||||

Line | .078 | .563 | .004 | −.009 | .945 | .000 | −.032 | .813 | .001 |

Circle | −.145 | .269 | .016 | −.128 | .344 | .011 | −.060 | .658 | .002 |

Dot | .049 | .689 | .002 | −.036 | .777 | .001 | −.072 | .565 | .004 |

Inhibition | −.047 | .686 | .002 | −.030 | .790 | .001 | .014 | .900 | .000 |

Line Ratio | .282 | .034* | .057 | .226 | .086 | .035 | |||

Circle Ratio | .018 | .894 | .000 | .066 | .623 | .003 | |||

Dot Ratio | .090 | .590 | .004 | .023 | .888 | .000 | |||

Raven’s | .258 | .039* | .052 |

*

Regressor | Step 1 (^{2} = .07) |
Step 2 (Δ^{2}= .07) |
Step 3 (Δ^{2} = .11) |
||||||
---|---|---|---|---|---|---|---|---|---|

β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
||||

Line | .045 | .734 | .001 | .025 | .851 | .000 | −.007 | .956 | .000 |

Circle | −.061 | .633 | .003 | −.016 | .906 | .000 | .083 | .520 | .004 |

Dot | .257 | .036* | .057 | .243 | .053 | .046 | .192 | .109 | .028 |

Inhibition | −.037 | .740 | .001 | −.030 | .785 | .001 | .034 | .750 | .001 |

Line Ratio | .305 | .020* | .067 | .223 | .075 | .035 | |||

Circle Ratio | −.132 | .325 | .012 | −.062 | .626 | .002 | |||

Dot Ratio | −.087 | .595 | .003 | −.184 | .245 | .015 | |||

Raven’s | .375 | .002** | .110 |

*

Regressor | Step 1 (^{2} = .06) |
Step 2 (Δ^{2} = .10) |
Step 3 (Δ^{2} = .08) |
||||||
---|---|---|---|---|---|---|---|---|---|

β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
||||

Line | .083 | .529 | .005 | .012 | .931 | .000 | −.017 | .897 | .000 |

Circle | −.163 | .206 | .020 | −.143 | .275 | .014 | −.058 | .653 | .002 |

Dot | .164 | .178 | .023 | .099 | .420 | .008 | .054 | .652 | .002 |

Inhibition | −.119 | .297 | .014 | −.108 | .326 | .011 | −.052 | .625 | .003 |

Line Ratio | .344 | .008** | .086 | .273 | .031* | .052 | |||

Circle Ratio | −.129 | .327 | .011 | −.069 | .599 | .003 | |||

Dot Ratio | .065 | .687 | .002 | −.018 | .906 | .000 | |||

Raven’s | .327 | .007** | .084 |

*

Regressor | Step 1 (^{2} = .04) |
Step 2 (Δ^{2} = .07) |
Step 3 (Δ^{2} = .04) |
||||||
---|---|---|---|---|---|---|---|---|---|

β | sr^{2} |
β | sr^{2} |
β | sr^{2} |
||||

Line | −.037 | .782 | .000 | −.097 | .478 | .006 | −.012 | .391 | .009 |

Circle | −.052 | .691 | .002 | −.032 | .815 | .001 | .027 | .844 | .000 |

Dot | .186 | .131 | .030 | .131 | .302 | .013 | .100 | .428 | .008 |

Inhibition | −.076 | .509 | .006 | −.066 | .561 | .004 | −.028 | .807 | .001 |

Line Ratio | .293 | .028* | .062 | .245 | .067 | .042 | |||

Circle Ratio | −.094 | .491 | .006 | −.052 | .700 | .002 | |||

Dot Ratio | .045 | .786 | .001 | −.012 | .943 | .000 | |||

Raven’s | .223 | .076 | .039 |

*

Dot acuity was the only significant predictor from among the controls entered in stage 1. It significantly predicted Algebra performance (β = .257

When all ratio acuities were entered in stage 2 it explained an additional 7–10% of the variance in the models. Dot acuity was no longer a significant predictor of symbolic math outcomes. However, line ratio acuity emerged as a significant predictor for all symbolic math outcomes except for symbolic fractions comparison: line ratio acuity significantly predicted FKA (β = .282,

When Raven’s scores were added in the final step, it explained additional 2–11% of the variance in the models. Fluid intelligence significantly predicted math achievement in three subdomains: FKA (β = .258,

Finally, we note the unexpected finding that symbolic fractions comparison was inversely correlated with acuity for simple dot comparisons. Upon further analysis, this correlation appears to be coincidental; indeed, we found that it was only present for the sample after it was trimmed for listwise deletion in the regressions. When supplemental bivariate correlations were conducted without list-wise deletion, the correlation disappeared (

The current research was a partial replication and extension of

Consistent with Matthews et al., when we operationalized RPS acuity as a composite of line and dot ratio performance, we found that composite acuity predicted symbolic fractions comparison and general fractions knowledge. This was true even when controlling for simple magnitude acuities and inhibitory control. On the other hand, the relations between composite RPS acuity and Algebra failed to replicate. However, the RPS composite was predictive of Math fundamentals which also tested some basic algebra concepts.

When we disaggregated the composite to check the predictive power of each format, we found that effects of the RPS composite were largely driven by performance in the line ratio format. Prior to the addition of general intelligence in the third round of our hierarchical regression, the line ratio format predicted performance on 4 of 5 outcome measures—the FKA, Algebra, Math fundamentals, and Trigonometry. Indeed, a standard deviation improvement on line ratio comparisons was associated with anywhere from one-fourth to one-third of a standard deviation improvement on these outcomes. The current findings both corroborate and refine Matthews et al.’s prior results showing that nonysmbolic ratio processing ability was predictive of symbolic math performance, with predictive power confined to the line ratio format.

It is unclear why the line format was the most predictive. Although it was reasonable to expect that acuity would be higher for line ratios than for circles or dots based on prior research (e.g.,

It is striking that the low-level perceptual ability to discriminate line ratios—an ability which has been found even among rhesus macaques (

Why would ratio processing ability predict higher order mathematics? RPS theorists (

One possibility is that the most effective aspect of nonsymbolic ratio processing lies less in the ability to accurately map from a given nonsymbolic to a specific symbolic and resides more in the ability to focus on the relations between ratio components. That is, it may be that performance on RPS tasks effectively measures participants’ abilities to attend to the fact that there is a multiplicative relation between components. If this is the case, then this sort of nonsymbolic relational reasoning may fuel the development of more general relational reasoning, even in the case that some perceptual bias results in an inaccurate map between fractions symbols and their nonsymbolic analogs.

Two pieces of evidence are consistent with this account. First,

This potential individual difference in attending to multiplicative relation between ratio components may parallel the construct of spontaneous focusing on relational information (SFOR) which has been described in recent work (

Second, there appears to be important shared variance between line ratio processing ability and performance on our measure of fluid intelligence—Raven’s Standard Progressive Matrices. As detailed above, we observed a significant effect of fluid intelligence on FKA, Math Fundamentals, and Algebra scores. Moreover, adding Raven’s scores to our models rendered line ratio nonsignificant for all but Math Fundamentals. Here it is important to note both 1) that Raven’s is often characterized as a test of relational reasoning (e.g.,

We would like to note two important limitations of our study. First, there were the failures to find the expected relations between multiple predictors and symbolic fraction comparisons: line, circle and dot ratio performance failed to predict symbolic fractions performance, and performance on dot comparisons was actually negatively correlated with symbolic fractions comparisons. We have no strong explanations for these results. It is well known that there can be considerable variability in the strategies that people attempt to use when comparing symbolic fractions, and these vary depending upon attributes of the fractions compared (

Second, although the current study confirms an association between perceptually-based ratio processing abilities and symbolic math outcomes, our design was not adequate for testing proposed mechanisms connecting the two. One of the most interesting predictions of RPS-based theories is that ratio processing ability might be effectively leveraged to improve intuitions about symbolic fractions, thereby improving math performance. Our study cannot speak to this issue empirically, and neither can other existing RPS studies to our knowledge beyond nods to the idea that number line estimation may in some way leverage RPS ability (e.g.,

The current study replicated

For this article, a dataset is freely available (

The Supplementary Materials contain the following additional information (for access see

Analysis of the relations between ratio acuities measured by Weber fractions (

Bivariate correlation tables relating performance among various comparison tasks and math tasks, as well as inhibitory control. Separate tables were constructed, alternatively using accuracy and

Results from the hierarchical regression analyses predicting math abilities from

Results from the hierarchical regression analyses predicting symbolic fraction comparison without including Ravens as a covariate.

Task materials, data collected, and the R scripts for analysis are available via the Open Science Framework (

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.