Not toeing the number line for simple arithmetic: Two large-n conceptual replications of Mathieu et al. (Cognition, 2016, Experiment 1)

We conducted two conceptual replications of Experiment 1 in Mathieu, Gourjon, Couderc, Thevenot, and Prado (2016, https://doi.org/10.1016/j.cognition.2015.10.002). They tested a sample of 34 French adults on mixed-operation blocks of single-digit addition (4 + 3) and subtraction (4 – 3) with the three problem elements (O1, +/-, O2) presented sequentially. Addition was 34 ms faster if O2 appeared 300 ms after the operation sign and displaced 5° to the right of central fixation, whereas subtraction was 19 ms faster when O2 was displaced to the left. Replication Experiment 1 (n = 74 recruited at the University of Saskatchewan) used the same non-zero addition and subtraction problems and trial event sequence as Mathieu et al., but participants completed blocks of pure addition and pure subtraction followed by the mixed-operation condition used by Mathieu et al. Addition RT showed a 32 ms advantage with O2 shifted rightward relative to leftward but only in mixed-operation blocks. There was no effect of O2 position on subtraction RT. Experiment 2 (n = 74) was the same except mixed-operation blocks occurred before the pure-operation blocks. There was an overall 13 ms advantage with O2 shifted right relative to leftward but no interaction with operation or with mixture (i.e., pure vs mixed operations). Nonetheless, the rightward RT advantage was statistically significant for both addition and subtraction only in mixed-operation blocks. Taken together with the robust effects of mixture in Experiment 1, the results suggest that O2 position effects in this paradigm might reflect task specific demands associated with mixed operations.

Language effects in early development of number writing and reading

Reading and writing multidigit numbers requires accurate switching between Arabic numbers and spoken number words. This is particularly challenging in languages with number-word inversion such as German (24 is pronounced as four-and-twenty), as reported by Zuber, Pixner, Moeller, and Nuerk (2009, https://doi.org/10.1016/j.jecp.2008.04.003). The current study aimed to replicate the qualitative error analysis by Zuber et al. and further extended their study: 1) A cross-linguistic (German, English) analysis enabled us to differentiate between language-dependent and more general transcoding challenges. 2) We investigated whether specific number structures influence accuracy rates. 3) To consider both transcoding directions (from Arabic numbers to number words and vice versa), we assessed performance for number reading in addition to number writing. 4) Our longitudinal design allowed us to investigate transcoding development between Grades 1 and 2. We assessed 170 German- and 264 English-speaking children. Children wrote and read the same set of 44 one-, two- and three-digit numbers, including the same number structures as Zuber et al. For German, we confirmed that a high amount of errors in number writing was inversion-related. For English, the percentage of inversion-related errors was very low. Accuracy rates were strongly related to number syntax. The impact of number structures was independent of transcoding direction or grade level and revealed cross-linguistic challenges of transcoding multidigit numbers. For instance, transcoding of three-digit numbers containing syntactic zeros (e.g., 109) was significantly more accurate than transcoding of items with lexical zeros (e.g., 190). Based on our findings, we suggest adaptations of current transcoding models.

Acquiring the cardinal knowledge of number words: A conceptual replication

Understanding the way in which counting represent numerosities was shown to be a long-lasting process. As shown in the Give-a-number task, acquiring the meanings of verbal number words goes through successive developmental stages in which children first learn the cardinal meanings of small number words one at a time before generalizing the cardinal principle they have induced from the first three number words to all number words within their counting range. This acquisition would take about a year, and would be completed by the age of 3 ½ years. The aim of the present study was to provide a conceptual replication of the developmental sequence described in Wynn’s study nearly 30 years ago using the Give-a-number task. A first cross-sectional study was conducted on 213 Belgian children aged between 39 and 74 months using the Give-a-number task to examine the developmental pattern and the influence of age on this acquisition. The time span of acquisition was examined in a second study in which 34 children were tested five times every months between the age of 36 to 52 months. Results showed that acquiring the cardinal meanings of number words spread out over a protracted period, far more extended than assumed by Wynn. Furthermore, children do not generalize all-at-once to large number words, the cardinal knowledge they learned on small number words. Rather, number words were found to be learned one at a time in a really progressive manner. Results were discussed with regard to their implications for the existing theories and in relation with other tasks assessing the acquisition of verbal number symbols.

Effectiveness of a numeracy intelligent tutoring system in kindergarten: A conceptual replication

Intelligent Tutoring Systems are a genre of highly adaptive software providing individualized instruction. The current study was a conceptual replication of a previous randomized control trial that incorporated the intelligent tutoring system Native Numbers, a program designed for early numeracy instruction. As a conceptual replication, we kept the method of instruction, the demographics, the number of kindergarten classrooms (n = 3), and the same numeracy and intrinsic motivation screeners as the original study. We changed the time of year of instruction, changed the control group to a wait-control group, added a maintenance assessment for the first group of participants, and included a mathematical language assessment. Analysis of within- and between-group differences using repeated measures ANOVA indicated gains of numeracy were significant only after using Native Numbers (Partial Eta Square = 0.147). Results of intrinsic motivation and mathematical language were not significant. The effect size of numeracy achievement did not reach that of the original study (Partial Eta Square = 0.622). Here, we compared the two studies, discussed plausible reasons for differences in the magnitude of effect sizes, and provided suggestions for future research.

Pick the smaller number: No influence of linguistic markedness on three-digit number processing

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.

When does the story matter? No evidence for the foregrounding hypothesis in math story problems

Math story problems are difficult for many solvers because comprehension of mathematical and linguistic content must occur simultaneously. Across two studies, we attempted to conceptually replicate and extend findings reported by Mattarella-Micke and Beilock (2010, https://doi.org/10.3758/PBR.17.1.106) and Jarosz and Jaeger (2019, https://doi.org/10.1002/acp.3471). Mattarella-Micke and Beilock found that multiplication word problems in which an irrelevant number was associated with the protagonist of the problem (i.e., foregrounded in the text) were solved less accurately than problems in other conditions. Jarosz and Jaeger used similar materials but tested the more general inconsistent-operations hypothesis that association with the protagonist would interfere with multiplication whereas dissociation would interfere with division. They found partial support: When division problems were primed with dissociative scenarios, solvers made more errors, but they failed to replicate the associative findings for multiplication. In the present research, we conducted two studies (Ns = 205 and 359), in which we similarly manipulated whether irrelevant content was associated with or dissociated from the story protagonist. In these studies, we did not find support for either the foregrounding or inconsistent-operations hypotheses. Exploratory error analyses suggested that solvers’ errors were most often the result of calculation difficulties or inappropriate operation choices and were unrelated to the presence of associative or dissociative story elements. Our careful implementation of this manipulation and much greater power to detect effects suggests that the association manipulation in irrelevant text does not influence adults’ performance on simple math story problems.

The causal impact of objective numeracy on judgments: Improving numeracy via symbolic and non-symbolic approximate arithmetic training yields more consistent risk judgments

Park and Brannon (2013, https://doi.org/10.1177/0956797613482944) found that practicing non-symbolic approximate arithmetic increased performance on an objective numeracy task, specifically symbolic arithmetic. Manipulating objective numeracy would be useful for many researchers, particularly those who wish to investigate causal effects of objective numeracy on performance. Objective numeracy has been linked to performance in multiple areas, such as judgment-and-decision-making (JDM) competence, but most existing studies are correlational. Here, we expanded upon Park and Brannon’s method to experimentally manipulate objective numeracy and we investigated whether numeracy’s link with JDM performance was causal. Experimental participants drawn from a diverse internet sample trained on approximate-arithmetic tasks whereas active control participants trained on a spatial working-memory task. Numeracy training followed a 2 × 2 design: Experimental participants quickly estimated the sum of OR difference between presented numeric stimuli, using symbolic numbers (i.e., Arabic numbers) OR non-symbolic numeric stimuli (i.e., dot arrays). We partially replicated Park and Brannon’s findings: The numeracy training improved objective-numeracy performance more than control training, but this improvement was evidenced by performance on the Objective Numeracy Scale, not the symbolic arithmetic task. Subsequently, we found that experimental participants also perceived risks more consistently than active control participants, and this risk-consistency benefit was mediated by objective numeracy. These results provide the first known experimental evidence of a causal link between objective numeracy and the consistency of risk judgments.