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The topic of how symbolic and non-symbolic number systems relate to exact calculation skill has received great discussion for a number of years now. However, little research has been done to examine how these systems relate to approximate calculation skill. To address this question, performance on symbolic and non-symbolic numeric ordering tasks was examined as predictors of Woodcock Johnson calculation (exact) and computation estimation (approximate) scores among university adults (N = 85, 61 female, Mean age = 21.3, range = 18-49 years). For Woodcock Johnson calculation scores, only the symbolic task uniquely predicted performance outcomes in a multiple regression. For the computational estimation task, only the non-symbolic task uniquely predicted performance outcomes. Symbolic system performance mediated the relation between non-symbolic system performance and exact calculation skill. Non-symbolic system performance mediated the relation between symbolic system performance and approximate calculation skill. These findings suggest that symbolic and non-symbolic number system acuity uniquely relate to exact and approximate calculation ability respectively.

Participants consisted of 85 undergraduate students (Mean age = 21.3 years,

Participants completed an iPad version, Test Runner, of the numeric ordering task previously used by

Example of a symbolic and a non-symbolic trial of the numeric ordering task.

Participants completed a computational estimation task using a custom-designed iPad program, modelled on a pencil-and-paper test by

Participants completed the Math Calculation subtest of the Woodcock-Johnson III Battery (

Participants were seated in a quiet room in front of an iPad. Once comfortable, participants completed the symbolic and non-symbolic variants of the numeric ordering task. Presentation order (symbolic then non-symbolic or the reverse) was counterbalanced across participants. Following the numeric ordering tasks, participants completed the computational estimation task. Following the iPad tasks, the iPad was removed and participants completed the Math Calculation subtest of the Woodcock-Johnson III Battery. These tasks were completed in one session as part of a larger study, which lasted approximately one hour and 45 minutes.

Descriptive statistics for all measures are reported in

Variable | ||
---|---|---|

Approximate calculation (max = 60) | 15.53 | 7.91 |

Exact calculation (max = 58) | 30.08 | 7.97 |

Symbolic composite | 1425 | 469.02 |

Symbolic RT (ms) | 1209 | 358.46 |

Symbolic error (% error) | 8.80 | 6.50 |

Non-symbolic composite | 1895 | 559.24 |

Non-symbolic RT (ms) | 1449 | 306.37 |

Non-symbolic error (% error) | 15.2 | 11.7 |

Variable | 1 | 2 | 3 |
---|---|---|---|

1. Approximate calculation | |||

2. Exact calculation | .520** | ||

3. Symbolic composite | -.385** | -.403** | |

4. Non-symbolic composite | -.403** | -.217* | .577** |

*

There was a significant correlation between the exact calculation measure and the approximate calculation measure. This correlation indicates that higher exact calculation scores were associated with higher approximate calculation scores. There was a significant correlation between the symbolic composite measure and the exact calculation measure. This correlation indicates that lower symbolic composite scores (indicative of better performance) were associated with higher levels of exact calculation skill. There was a significant correlation between the non-symbolic composite measure and exact calculation measure. This correlation indicates that lower non-symbolic composite scores (indicative of better performance) were associated with higher levels of exact calculation skill. There was a significant correlation between the symbolic composite measure and the approximate calculation measure. This correlation indicates that lower symbolic composite scores (indicative of better performance) were associated with higher levels of approximate calculation skill. There was a significant correlation between the non-symbolic composite measure and the approximate calculation measure. This correlation indicates that lower non-symbolic composite scores (indicative of better performance) were associated with higher levels of approximate calculation skill. Using Fisher’s r to z transformation to test for differences between two dependant correlation coefficients (i.e., based on the same sample;

Data were analyzed using multiple regression to determine whether performance on symbolic and non-symbolic numeric ordering trials predicted exact calculation skill. Both the symbolic and non-symbolic composite scores were entered in a single step. As shown in ^{2} = .16,

Calculation skill measure |
||
---|---|---|

Predictor | Exact Skill | Approximate Skill |

Symbolic composite (β) | -.42** | -.23 |

Non-symbolic composite (β) | .02 | -.27* |

Total ^{2} |
.16 | .20 |

85 | 85 |

*

It was found that symbolic composite scores were a significant predictor of exact calculation scores,

To examine the nature of the relation of the two number representation systems to exact calculation skill, a series of four regression analyses was employed to test the mediation model presented in

*

Path A was tested using a bivariate regression in which performance on the non-symbolic measure was the predictor variable and performance on the symbolic measure was the criterion variable. Non-symbolic system performance was found to be a significant predictor of symbolic system performance, β = .58,

As both A and B paths were significant, mediation analyses (path C’) were tested using the bootstrapping method, with bias-corrected confidence intervals (_{change} = .24, 95% CI [-.40, -.11]. When controlling for symbolic system performance (path C’), non-symbolic system performance was not a significant predictor of exact calculation skill, β = .024,

Data were analyzed using multiple regression to determine whether performance on symbolic and non-symbolic numeric ordering trials predicted approximate calculation skill. Both the symbolic and non-symbolic composite scores were entered in a single step. As shown in ^{2} = .20,

To examine the nature of the relation of the two number representation systems to approximate calculation skill, a series of four regression analyses was employed to test the mediation model presented in

*

For path A, performance on the symbolic measure was the predictor variable and performance on the non-symbolic measure was the criterion variable. Symbolic system performance was found to be a significant predictor of non-symbolic system performance, β = .58,

As both A and B paths were significant, mediation analyses were tested using the bootstrapping method, with bias-corrected confidence intervals (_{change} = .15, 95% CI [-.30, -.04]. When controlling for non-symbolic system performance (path C’), symbolic system performance was not a significant predictor of approximate calculation skill, β = -.23,

As our measures for exact and approximate calculation skill both required some basic mathematical knowledge (i.e., how to multiply, divide, add and subtract with whole numbers, decimals, and fractions), we examined whether controlling for exact calculation skill changed the relation between non-symbolic number system acuity and approximate calculation. Data were analyzed using multiple regression to determine whether performance on the exact calculation measure, and the symbolic and non-symbolic numeric ordering trials predicted approximate calculation skill. The exact calculation measure was entered into the first step of the regression, and both the symbolic and non-symbolic composite scores were entered into the second step.

As shown in ^{2} = .36,

Predictor | Approximate Skill |
---|---|

Exact calculation skill (β) | .44 |

Symbolic composite (β) | -.04 |

Non-symbolic composite (β) | -.28 |

Total R^{2} |
.36 |

85 |

*

In the present study, how both symbolic and non-symbolic number system acuity relate to measures of exact and approximate calculation was examined. Previous research has yielded a variety of findings with regards to how symbolic and non-symbolic number representation systems relate to exact calculation ability. Findings from this study were consistent with previous research indicating that both non-symbolic system acuity (

The novel contribution of this study is the examination of how both symbolic and non-symbolic number representation systems relate to a measure of approximate calculation. Similar to exact calculation findings, both symbolic and non-symbolic number system acuity were correlated with a measure of approximate calculation. However, after entering both symbolic and non-symbolic measures into a single-step regression predicting approximate calculation, only the non-symbolic measure remained a significant predictor. It was found that non-symbolic system performance mediates the relation between symbolic system performance and approximate calculation skill. These results provide the first evidence in adults that only non-symbolic, and not symbolic, number system acuity is uniquely predictive of performance on a measure of approximate calculation.

Each of the above findings fit well within the current understanding of how both (1) symbolic and non-symbolic systems differ and (2) exact and approximate calculation differ. For example, it has been suggested that at the neural level, symbolic and non-symbolic number systems represent numerosity in different ways; symbolic in a digital manner and non-symbolic in an analogue manner (

One limitation of the present study is that while the approximate calculation measure was designed to limit efforts to exactly calculate correct answers, it is possible that some participants could have exactly calculated some answers in the time allotted. However, spontaneous self-reports provided during the debriefing indicate this was unlikely to be the case for many, if any, participants. Another thing to be considered is that our requirement of specific estimates, rather than a choice of response options, may have resulted in some participants feeling compelled to calculate a more exact answer, despite clear instructions to provide an estimate. In future studies, we look to control for such confounds by adding another estimation measure in which participants are forced to choose from a set of possible answers, as such a task would allow us to more easily limit the answer time frame, preventing exact calculation while possibly being less demanding on participants. Additionally, we will collect RT data from this forced-choice estimation task.

Another point of note is our decision to employ a numeric ordering task as a measure of number system acuity, rather than more conventional tasks such as number comparison. As mentioned previously, numeric ordering tasks typically demonstrate a difference in distance effect patterns between symbolic and non-symbolic task variants that are not typically demonstrated in magnitude comparison tasks. This robust finding suggests that symbolic and non-symbolic number representation systems perform the task of determining numeric order in different ways (

One final thing to consider is that in the numeric ordering task, some of the trials involved stimuli with a numeric value less than or equal to four. Non-symbolic stimuli are enumerated differently when four or fewer items are present than when five or more items are present (

Regarding the practical implications of the present study’s findings, a next logical step for this research is training studies, which we are currently designing, to assess whether or not training individual’s non-symbolic number representation system can improve their approximate calculation ability in absence of general math training. If such a training program improves approximate calculation ability, it could reduce mistakes – small and large – resulting from inaccurate day-to-day estimations. Improved approximation abilities could also benefit exact mathematic performance, as approximation skills can play an important role in double-checking answers and the development of general number sense (

In summary, the present study supports existing research suggesting that only symbolic number representation system acuity is uniquely predictive of exact calculation performance, while also presenting the novel finding that only non-symbolic number system acuity is uniquely predictive of approximate calculation performance. These findings further emphasize differences between symbolic and non-symbolic number representation systems and the mathematical processes that each underlie. Overall, this study points to the importance of continued study on both number representation systems and the areas where they uniquely and/or jointly contribute to mathematics performance.

(1, 2, 3), (3, 2, 1), (1, 3, 2), (3, 1, 2), (2, 3, 4), (4, 3, 2), (2, 4, 3), (4, 2, 3), (1, 3, 5), (5, 3, 1),

(1, 5, 3), (5, 1, 3), (3, 5, 7), (7, 5, 3), (3, 7, 5), (7, 3, 5), (3, 4, 5), (5, 4, 3), (3, 5, 4), (5, 3, 4),

(2, 4, 6), (6, 4, 2), (2, 6, 4), (6, 2, 4), (4, 6, 8), (8, 6, 4), (4, 8, 6), (8, 4, 6), (4, 5, 6), (6, 5, 4),

(4, 6, 5), (6, 4, 5), (5, 7, 9), (9, 7, 5), (5, 9, 7), (9, 5, 7)

This project was funded by a grant from King’s University College at Western University. The funding source did not have a role in the research design, execution, analysis, interpretation or reporting of this research.

The authors have declared that no competing interests exist.

The authors wish to thank Lynda Hutchinson for assistance with the mediation analyses and Chris Roney and Lynda Hutchinson for feedback on an earlier draft of this paper. The authors also wish to thank members of the Cognitive Science and Numeracy Lab, Cindel White, Adam Newton, Robert Nowosielski, Olivia Wassing, Sarah Hunt, Leo Bigras, and John Nowak, for assistance with data collection and data entry. Finally, the authors wish to thank Douglas Wilger (