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The ability to efficiently compare number symbols, such as digits, is associated with mathematics competence across the lifespan. Performance on symbolic number comparison tasks differ across age groups; young students who are developing fluency with digits improve on symbolic number comparison, and performance is better in adults than children. However, whether this improvement continues for older students who are fluent with number symbols, and what cognitive factors relate to this improvement, is unknown. This study used a longitudinal sample of U.S. middle school students (n = 394) to examine whether symbolic number comparison performance changes over middle school (i.e., students aged 11-14), whether there are individual differences in students’ rate of change, and potential predictors of that change. Students completed measures of single-digit symbolic number comparison, nonsymbolic number comparison, executive function (EF), and mathematics competence in Grade 5 (M = 11.02 years; SD = 0.32), and double-digit symbolic number comparison in Grades 6-8. Results showed that, on average, students’ symbolic number comparison performance improved from Grades 6-8. Grade 5 Symbolic number comparison performance predicted Grade 8 symbolic number comparison and rate of change over Grades 6-8. Grade 5 nonsymbolic number comparison, EF, and mathematics competence predicted Grade 8 symbolic number comparison performance. Results suggest that numerical magnitude processing, executive functions, and mathematics competence are related to symbolic number processing well into middle school, and that students continue to refine their ability to process number symbols into adolescence.

The symbolic number comparison task has been widely used to investigate the cognitive representation and underlying processes of multi-digit number processing. The standard procedure to establish numerical distance and compatibility effects in such number comparison paradigms usually entails asking participants to indicate the larger of two presented multi-digit Arabic numbers rather than to indicate the smaller number. In terms of linguistic markedness, this procedure includes the unmarked/base form in the task instruction (i.e., large). Here we evaluate distance and compatibility effects in a three-digit number comparison task observed in Bahnmueller et al. (2015, https://doi.org/10.3389/fpsyg.2015.01216) using a marked task instruction (i.e., ‘pick the smaller number’). Moreover, we aimed at clarifying whether the markedness of task instruction influences common numerical effects and especially componential processing as indexed by compatibility effects. We instructed German- and English-speaking adults (N = 52) to indicate the smaller number in a three-digit number comparison task as opposed to indicating the larger number in Bahnmueller et al. (2015). We replicated standard effects of distance and compatibility in the new pick the smaller number experiment. Moreover, when comparing our findings to Bahnmueller et al. (2015), numerical effects did not differ significantly between the two studies as indicated by both frequentist and Bayesian analysis. Taken together our data suggest that distance and compatibility effects alongside componential processing of multi-digit numbers are rather robust against variations of linguistic markedness of task instructions.

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.