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Mastery of mathematics depends on the people’s ability to manipulate and abstract values such as negative numbers. Knowledge of arithmetic principles does not necessarily generalize from positive number arithmetic to arithmetic involving negative numbers (Prather & Alibali, 2008, https://doi.org/10.1080/03640210701864147). In this study, we evaluate the relationship between participant’s knowledge of the Relation to Operands arithmetic principle in both positive and negative numbers and their spontaneous on numerical relations. Additionally, we tested if the feedback that directs attention to relations affects participants’ attention to relation and their arithmetic principle knowledge. This study contributes to our understanding of the specific skills and cognitive processes that are associated with understanding high-level mathematics.

Principles are rules or regularities within a problem domain. For example, in arithmetic, when adding natural numbers, the sum is always larger than either addend. Arithmetic principles capture fundamental aspects of the number system and are part of people’s understanding of arithmetic, e.g., operation sense (

Knowledge of arithmetic principles is important not only for initial learning of natural numbers but also negative numbers. Negative numbers have a reputation within the education of being difficult for young children to learn. This may be in part due to the lack of a clear physical correspondence for negative values. Children’s learning of whole numbers in educational settings often relies on the use of physical manipulatives (

In the current study, we examine the relationship between people’s knowledge of the Relation to Operands principle and their spontaneous attention to relations (SAR). SAR is similar to spontaneous focus on number (SFON) and spontaneous attention to number (SAN). SFON involves differentiation of numbers from other aspects of the environment. SAN involves differentiation of small numbers for one another (

Individual variation in SFON and SAR may correlate with variation in other numerical skills. Children’s SFON is positively correlated with their performance on symbolic arithmetic tasks and standardized tests of arithmetic (

In the current study, we evaluate the relationship between participants’ arithmetic principle knowledge and their spontaneous attention to relations (SAR). We measure SAR using a non-symbolic shape-matching task, and arithmetic principle knowledge using a symbolic equation evaluation task, both based on prior work (

We also examine the effect of feedback on the shape-matching task. SAR is not only a fixed characteristic of an individual learner but may change with experience and vary across contexts. Recent work has shown that participant SFON task scores can change depending on the relative salience number has compared to other features in prior experience. For example, adults who had previously attended to number when it was paired with low salience features were more likely to continue to focus on number when later pair with high salience features (

Using a behavioral experiment, we evaluated the following hypotheses:

For Hypotheses 1 and 2, it is possible that confounds could create a correlation in the absence of the hypothesized mechanism. This issue is addressed by the experimental manipulation of providing feedback for Hypotheses 3, 4, and 5. We posit a direct causal mechanism between participants’ SAR and their knowledge of Relation to Operands.

In this experiment, we measured participants’ spontaneous attention to relational number. We also measure participants’ knowledge of the Relation to Operands principle. All participants completed the study tasks in the same order with the following task outline.

SAR measure task

Arithmetic Principle Knowledge Task

SAR measure Task with feedback

Arithmetic Principle Knowledge Task

The experimental session took approximately 30 minutes.

Participants were adults (

Both the matching task (e.g.,

We used a card-matching task based on the

Participants were given two different SAR scores. One based on how often they select additive relation as the best match. The other SAR score was based on how often they select multiplicative relation as the best match.

The second instance of the SAR task participants were told that prior responses from other participants had been collected and that the correct matches would be highlighted on some of the trials. Participants were in one of two conditions. For participants in the

Participants’ knowledge of the Relation to Operands principle was assessed using the equation evaluation task for subtraction (

The equation evaluation task was completed twice using different stimulus set (see

To characterize participants’ attention to relations, we calculated the participants’ score on the Matching task. We calculated the total number each participant selected for both additive relation and multiplicative relation. We calculated the total score given by the participant, ranging from 0 to 16. Participants selected the additive-relation match on average 5.81 (

We evaluated participants’ knowledge of the Relation to Operands principle both with positive numbers and including a negative number. Overall we find strong evidence for knowledge of the principle with positive numbers but not with negative numbers. We calculated how often each participant selected the principle-consistent student as understanding arithmetic better on the symbolic arithmetic principle task. Selecting the non-violation student as understanding arithmetic better corresponds with a higher score on this task. The analysis was the same as in prior work (

We evaluated if participants’ SAR score was significantly associated with their score on the principle knowledge task. We hypothesized those participants that show a strong attention to relations have higher principle knowledge scores. Overall we did not find strong relationships between participants’ SAR scores and their arithmetic principle knowledge. We used linear regression to predict arithmetic principle score using the mean ratio rating SAR score as a predictor: APK score = MultiplicativeScore + AdditiveScore + MultiplicativeScore* AdditiveScore. Additive score is the proportion of trials in which the participant selects the Additive Relation match as the best match. Ranging from 0 to 16. MultiplicativeScore is the proportion of trials in which the participant selects the Multiplicative Relation match as the best match. Arithmetic Principle Knowledge was not significantly predicted by AdditiveScore,

We calculated a regression predicting APK score for negative numbers separately: APK_{neg} = MultiplicativeScore + AdditiveScore. + MultiplicativeScore* AdditiveScore. All scores were arcsine transformed as they occur on a restricted scale and may not have normally distributed variation. Arithmetic Principle Knowledge for negative numbers was not significantly predicted by AdditiveScore,

We evaluated if there is a high correlation between participants MultiplicativeScore and AdditiveScore, which may affect the interpretation of the regression. We calculated the Variance Inflation Factor (VIF) to evaluate the multicollinearity in the positive and negative number models. A VIF above 5 would indicate unacceptable levels of colinearity. We calculated a VIF of 1.00 for the positive number APK model and a VIF of 1.02 for the negative number APK model. Thus additive and multiplicative relation scores did not have a high correlation.

We evaluated how participants complete the Matching task when given feedback. We evaluated the two feedback conditions, shape, and relation if participants' feature selection in the Matching task changed from initial to repeated measures. Overall we found evidence that feedback in the relational condition as associated with an increase in making relational matches. We used a 2(pre/post) x 2(Feedback condition) repeated measures ANOVA where SAR scores are the repeated measure and condition is across participants. We calculated separate ANOVAs for each of the SAR scores, additive relation, multiplicative relation and shape feature scores. We expected that participants in the shape condition would show a significant increase in the shape feature score. We expected that participants in the relation condition would show a significant increase in the additive relations score. Post-hoc contrasts were planned to compare additive relation, and multiplicative relation scores repeated measures for participants in the relation condition, and shape feature score repeated measures for participants in the shape condition. We used a 2(pre/post) x 2(Feedback condition) repeated measures ANOVA where SAR scores are the repeated measure and condition is across participants. For additive relation scores we found a significant effect for pre/post training,

Task | Pre training | Post training |
---|---|---|

5.35 | 13.41 | |

6.30 | 0.92 |

For multiplicative relation scores we found a significant effect for pre/post training,

Task | Pre training | Post training |
---|---|---|

8.19 | 1.92 | |

7.80 | 0.95 |

Additionally, to address potential ceiling effects, completed post-hoc comparisons for the subset of participants who select additive relations on 50% or less of trials on the initial Matching task. Post-hoc analysis for this set of participants showed a significant difference between pre and post training for participants in the relations condition,

We evaluated if SAR feedback condition is associated with a change in Arithmetic Principle knowledge score, for both positive and negative numbers. Overall we found mixed evidence of change in Arithmetic Principle knowledge scores across the two conditions. We used repeated measures ANOVA with 2 (feedback condition) between-subject conditions with 2 (pre/post) repeated measures, 2 (APK-positive and APK-negative) number types. The analysis also included planned contrasts to evaluate positive and negative number separately. We compared pre and post scores for participants in the two feedback conditions. We evaluated if there was an effect of feedback condition on either of the repeated measures. We used a 2 (pre/post) x 2 (Feedback condition) repeated measures ANOVA where APK scores are the repeated measure and condition is across participants For APK-positive scores we found a significant effect for pre/post training,

Task | Pre training |
Post training |
||
---|---|---|---|---|

Violation | Non-Violation | Violation | Non-Violation | |

2.63 | 2.36 | 2.46 | 2.46 | |

2.65 | 2.33 | 2.40 | 2.47 |

For APK-negative scores we found a significant effect for pre/post training,

Task | Pre training |
Post training |
||
---|---|---|---|---|

Violation | Non-Violation | Violation | Non-Violation | |

2.32 | 2.58 | 2.63 | 2.34 | |

2.30 | 2.54 | 2.62 | 2.35 |

As an exploratory analysis we evaluated if change on SAR task addition-relation predicted change in Arithmetic principle knowledge. We did not find a significant effect. For APK-negative the change in score is not significantly predicted by change in SAR task scores

We also include regression analysis with additive and multiplicative scores separated, as two scores are not independent. We used linear regression to predict arithmetic principle score using the mean ratio rating SAR score as a predictor: APK score = MultiplicativeScore and as a separate model APK score = AdditiveScore. Additive score is the proportion of trials in which the participant selects the Additive Relation match as the best match. Ranging from 0 to 16. MultiplicativeScore is the proportion of trials in which the participant selects the Multiplicative Relation match as the best match. Arithmetic Principle Knowledge was not significantly predicted by the model with AdditiveScore,

We repeated the same analysis to predict APK scores with negative numbers. Arithmetic Principle Knowledge with negative numbers was not significantly predicted by the model with AdditiveScore,

In this study we investigated the relationship between Spontaneous Attention to Relations and knowledge of the Relation to Operands principle with positive and negative numbers. We found support for the hypothesis that brief feedback can change participants’ Spontaneous Attention to Number. Participants showed significant changes in behavior on the SAR task after feedback. We were able to direct attention to relations on this task with brief feedback. As hypothesized only feedback focused on relations was associated with an increase in attending to relational matches on the SAR task. Participants in the Relations condition showed a market increase in their selection of additive relation matches, while those in the Shape condition did not. We conclude participants are able change their attention towards relations with only brief feedback.

We did not find strong evidence that feedback also affected participants’ arithmetic principle knowledge. For arithmetic with positive numbers there was not an increase in principle knowledge after feedback. In fact, participants appear to show a decrease in rating the violation equation as worse. However for arithmetic with negative numbers there was an increase in principle knowledge for both groups. This is consistent with prior work showing a lack of correlation between principle knowledge with positive and negative number. However, it is odd that participants might show knowledge gains with negative numbers while simultaneously failing to consistently show knowledge with positives. We found no significant relationship between participants’ initial SAR scores and their Arithmetic Principle knowledge for either positives or negatives.

This study should be viewed in the context of prior work on training of numerical and arithmetic skills across symbolic and non-symbolic formats. Overall there is evidence that training can improve performance, however it is unclear when training may transfer from non-symbolic to symbolic formats. There are several examples of successful training of symbolic arithmetic skills using non-symbolic training (

We conclude that adults are able to adjust their attention to relations with minimal feedback. The effect of feedback on the SAR task stands in contrast to studies completed with children, in which such feedback did not result in a change of behavior on the SAR task (

Why are the results different for negative numbers? Prior work suggests that adult participants’ principle knowledge with negative numbers may be lower (

We are left with the questions of what drives individual differences in SAR, and why is there not an observed relationship between SAR and Arithmetic Principle knowledge. One potential issue is the difference in format. The SAR task is non-symbolic, while the Arithmetic principle knowledge is symbolic. The lack of observed relationship between SAR and Arithmetic Principle knowledge could be due, in part, to the difference between non-symbolic and symbolic tasks. Knowledge of arithmetic principles can vary across contexts. Individuals that show knowledge of arithmetic principles in a non-symbolic context did not necessarily show knowledge in a symbolic context (

The Supplementary Materials contain the pre-registration protocol for this Registered Report (for access see

The author has no funding to report.

The author has declared that no competing interests exist.

The author has no additional (i.e., non-financial) support to report.