Quantitative and Qualitative Differences in the Canonical and the Reverse Distance Effect and Their Selective Association With Arithmetic and Mathematical Competencies

Abstract

Understanding the relationship between symbolic numerical abilities and individual differences in mathematical competencies has become a central research endeavor in the last years. Evidence on this foundational relationship is often based on two behavioral signatures of numerical magnitude and numerical order processing: the canonical and the reverse distance effect. The former indicates faster reaction times for the comparison of numerals that are far in distance (e.g., 2 8) compared to numerals that are close in distance (e.g., 2 3). The latter indicates faster reaction times for the ordinal judgment of numerals (i.e., are numerals in ascending/descending order) that are close in distance (e.g., 2 3 4) compared to numerals that are far in distance (e.g., 2 4 6). While a substantial body of literature has reported consistent associations between the canonical distance effect and arithmetic abilities, rather inconsistent findings have been found for the reverse distance effect. Here, we tested the hypothesis that estimates of the reverse distance effect show qualitative differences (i.e., not all participants show a reverse distance effect in the expected direction) rather than quantitative differences (i.e., all individuals show a reverse distance effect, but to a different degree), and that inconsistent findings might be a consequence of this variation. We analyzed data from 397 adults who performed a computerized numerical comparison task, a computerized numerical order verification task (i.e., are three numerals presented in order or not), a paper pencil test of arithmetic fluency, as well as a standardized test to assess more complex forms of mathematical competencies. We found discriminatory evidence for the two distance effects. While estimates of the canonical distance effect showed quantitative differences, estimates of the reverse distance effect showed qualitative differences. Comparisons between individuals who demonstrated an effect and individuals who demonstrated no reverse distance effect confirmed a significant moderation on the correlation with mathematical abilities. Significantly larger effects were found in the group who showed an effect. These findings confirm that estimates of the reverse distance effect are subject to qualitative differences and that we need to better characterize the underlying mechanisms/strategies that might lead to these qualitative differences.

The role of basic number processing in high mathematics achievement in primary school

“…It is plausible to assume that group differences in symbolic number processing ability might be observed when more fine-grained indices of the symbolic magnitude processing and order processing are considered, i.e., the canonical numerical distance effect and the reversed distance effect, respectively. Performance on a numerical magnitude compari son task is poorer (i.e., slower reaction times) for numbers that are closer together (e.g., 5 and 6) compared to numbers further apart (e.g., 2 and 7), which is the classic numerical distance effect (Vogel et al, 2021). It has been posited that smaller numerical distance effects are reflective of more precise representations of the number representation system (e.g., Schwenk et al, 2017).…”

“…It has been posited that smaller numerical distance effects are reflective of more precise representations of the number representation system (e.g., Schwenk et al, 2017). A reverse distance effect is often reported for the numerical order task (Vogel et al, 2021), where participants are often faster when the distance between the three in-order numbers is small (3 4 5) compared to when the distance is large (2 4 6). Interestingly, a very recent study by Hohol et al (2020) examined whether professional mathematicians showed a smaller canonical numerical distance effect as compared to engineers, social scientists, and a reference group from the general population.…”

Distance in depth: A comparison of explicit and implicit numerical distances in the horizontal and radial dimensions.

“…Patterns similar to the priming DEs (i.e., faster performance with small compared to large numerical distances) arise also in tasks that focus on the ordinal property of numbers, such as the relative order judgment task (requiring participants to judge whether the order in number pairs is ascending or descending; e.g., Turconi et al, 2006) or the number order verification task (requiring participants to verify the correctness of order in number pairs or triplets; e.g., Franklin et al, 2009;Lyons & Beilock, 2011). The emerging effect, known as reverse DE, although present both in children (Lyons & Ansari, 2015;Vogel et al, 2015) and in adults (Vogel et al, 2017(Vogel et al, , 2019, is more variable across individuals when compared to the comparison DE, possibly due to the involvement of multiple individual strategies, such as long-term memory retrieval and sequential-procedural comparisons (Vogel et al, 2021). Notably, all the above-mentioned methods except Holyoak's (1978) study assessed the DE implicitly, that is, without explicit processing of the numerical distance between the presented numbers.…”

The hierarchical relations among mathematical competencies: From fundamental numeracy to complex mathematical skills.

“…In contrast, Orrantia et al (2019) highlight the extensive overlap between cardinal (number comparisons) and ordinal processing (order judgements) in adults’ arithmetic, suggesting that both processes draw on underlying symbolic magnitude associations. Other researchers have suggested that both familiarity and magnitude associations may influence performance on the order judgement task, depending on the task demands (e.g., stimulus characteristics; Vos et al, 2021) and individuals’ strategy use (Dubinkina et al, 2021; Muñez et al, 2021; Vogel et al, 2021; Vos et al, 2021). Extending the latter views, strong correlations between order judgements and arithmetic may reflect the extent to which individuals have developed an integrated knowledge system (Xu et al, 2019).…”