Journal of Numerical Cognition (JNC - PsychOpen)
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Bayesian Inference in Numerical Cognition: A Tutorial Using JASP
No full textResearchers in numerical cognition rely on hypothesis testing and parameter estimation to evaluate the evidential value of data. Though there has been increased interest in Bayesian statistics as an alternative to the classical, frequentist approach to hypothesis testing, many researchers remain hesitant to change their methods of inference. In this tutorial, we provide a concise introduction to Bayesian hypothesis testing and parameter estimation in the context of numerical cognition. Here, we focus on three examples of Bayesian inference: the t-test, linear regression, and analysis of variance. Using the free software package JASP, we provide the reader with a basic understanding of how Bayesian inference works “under the hood” as well as instructions detailing how to perform and interpret each Bayesian analysis
NASCO: A New Method and Program to Generate Dot Arrays for Non-Symbolic Number Comparison Tasks
No full textBasic numerical abilities are generally assumed to influence more complex cognitive processes involving numbers, such as mathematics. Yet measuring non-symbolic number abilities remains challenging due to the intrinsic correlation between numerical and non-numerical dimensions of any visual scene. Several methods have been developed to generate non-symbolic stimuli controlling for the latter aspects but they tend to be difficult to replicate or implement. In this study, we describe the NASCO method, which is an extension to the method popularized by Dehaene, Izard, and Piazza (2005). Their procedure originally controlled for two visual dimensions that are mediated by Number: Total Area and Item Size (i.e., N = TA/IS). Here, we additionally propose to control for another twofold dimension related to the array extent, which is also mediated by Number: Convex Hull Area and Mean Occupancy (i.e., N = CH/MO). We illustrate the NASCO method with a MATLAB app—NASCO app—that allows easy generation of dot arrays for a visually controlled assessment of non-symbolic numerical abilities. Results from a numerical comparison task revealed that the introduction of this twofold dimension manipulation substantially affected young adults’ performance. In particular, we did not replicate the relation between non-symbolic number abilities and arithmetic skills. Our findings open the debate about the reliability of previous results that did not take into account visual features related to the array extent. We then discuss the strengths of NASCO method to assess numerical ability, as well as the benefits of its straightforward implementation in NASCO app for researchers
The Complexity of Mental Integer Addition
No full textAn important paradigm in modeling the complexity of mathematical tasks relies on computational complexity theory, in which complexity is measured through the resources (time, space) taken by a Turing machine to carry out the task. These complexity measures, however, are asymptotic and as such potentially a problematic fit when descriptively modeling mathematical tasks that involve small inputs. In this paper, we argue that empirical data on human arithmetical cognition implies that a more fine-grained complexity measure is needed to accurately study mental arithmetic tasks. We propose a computational model of mental integer addition that is sensitive to the relevant aspects of human arithmetical ability. We show that this model necessitates a two-part complexity measure, since the addition tasks consists of two qualitatively different stages: retrieval of addition facts and the (de)composition of multidigit numbers. Finally, we argue that the two-part complexity measure can be developed into a single response-time measure with the help of empirical study of the two stages
Book Review of “An Introduction to Mathematical Cognition” by C. Gilmore, S. M. Göbel, and M. Inglis
No full textNo abstract available
More Problems After Difficult Problems? Behavioral and Electrophysiological Evidence for Sequential Difficulty Effects in Mental Arithmetic
No full textThis study investigated whether sequential difficulty effects emerge during processing of a mixed set of small, easy and large, more difficult arithmetic problems. Furthermore, we assessed if these sequential difficulty effects are reflected in event-related (de-)synchronization (ERS/ERD) patterns. To this end, we analyzed data of 65 participants, who solved two separate blocks (additions and subtractions) of arithmetic problems while their EEG was recorded. In each block, half of the problems were difficult problems (two-digit/two-digit with carry/borrow), and the other half were easy problems (one-digit/one-digit). Half of the problems were preceded by a problem of the same difficulty (repeat trials), and half were preceded by problems of the other difficulty (switch trials). In subtractions a sequential difficulty effects pattern emerged. Participants solved easy repeat trials faster than easy switch trials, while difficult repeat trials were solved slower and less accurately than difficult switch trials. In the EEG, we found the strongest effects in left hemispheric beta band (13–30 Hz) ERD. Specifically, participants showed a stronger beta band ERD in easy switch trials than in easy repeat trials. Furthermore, beta band ERD was stronger in difficult problems than in easy problems within repeat trials, but stronger in easy problems than in difficult problems within switch trials. In summary, our results are in line with the presence of sequential difficulty effects, as processing of easy and difficult problems was impaired if they were preceded by a difficult problem. Furthermore, these sequential difficulty effects are reflected in ERD patterns
The Hierarchical Symbol Integration Model of Individual Differences in Mathematical Skill
No full textSymbolic number knowledge is strongly related to mathematical performance for both children and adults. We present a model of symbolic number relations in which increasing skill is a function of hierarchical integration of symbolic associations. We tested the model by contrasting the performance of two groups of adults. One group was educated in China (n = 71) and had substantially higher levels of mathematical skill compared to the other group who was educated in Canada (n = 68). Both groups completed a variety of symbolic number tasks, including measures of cardinal number knowledge (number comparisons), ordinal number knowledge (ordinal judgments) and arithmetic fluency, as well as other mathematical measures, including number line estimation, fraction/algebra arithmetic and word problem solving. We hypothesized that Chinese-educated individuals, whose mathematical experiences include a strong emphasis on acquiring fluent access to symbolic associations among numbers, would show more integrated number symbol knowledge compared to Canadian-educated individuals. Multi-group path analysis supported the hierarchical symbol integration hypothesis. We discuss the implications of these results for understanding why performance on simple number processing tasks is persistently related to measures of mathematical performance that also involve more complex and varied numerical skills
Fraction Errors in a Digital Mathematics Environment: Latent Class and Transition Analysis
No full textStudent struggles with fractions are well documented, and due to fractions’ importance to later mathematics achievement, identification of the errors students make when solving fraction problems is an area of interest for both researchers and teachers. Within this study, we examine data on student fraction problem errors in pre- and post-quizzes in a digital mathematics environment. Students (n = 1,431) were grouped by prevalence of error types using latent class analysis. Three different classes of error profiles were identified in the pre-quiz data. A latent transition analysis was then used to determine if class membership and class structure changed from pre- to post-quiz. In both pre- and post-quiz, there was a class of students who appeared to be guessing and a class of students who performed well. One class structure was consistent with the idea that early fraction learners rely heavily on whole number principles. Identification of co-occurrence of and changes to fraction errors has implications for curricular design and pedagogical decisions, especially in light of movements toward personalized learning systems
Same-Different Judgments With Alphabetic Characters: The Case of Literal Symbol Processing
No full textLearning mathematics requires fluency with symbols that convey numerical magnitude. Algebra and higher-level mathematics involve literal symbols, such as "x", that often represent numerical magnitude. Compared to other symbols, such as Arabic numerals, literal symbols may require more complex processing because they have strong pre-existing associations in literacy. The present study tested this notion using same-different tasks that produce less efficient judgments for different magnitudes that are closer together compared to farther apart (i.e., same-different distance effects). Twenty-four adolescents completed three same-different tasks using Arabic numerals, literal symbols, and artificial symbols. All three symbolic formats produced same-different distance effects, showing literal and artificial symbol processing of numerical magnitude. Importantly, judgments took longer for literal symbols than artificial symbols on average, suggesting a cost specific to literal symbol processing. Taken together, results suggest that literal symbol processing differs from processing of other symbols that represent numerical magnitude
Adolescents and Adults Need Inhibitory Control to Compare Fractions
No full textFor children, adolescents and educated adults, comparing fractions with common numerators (e.g., 4/5 vs. 4/9) is more challenging than comparing fractions with common denominators (e.g., 3/4 vs. 6/4) or fractions with no common components (e.g., 5/7 vs. 6/2). Errors are related to the tendency to rely on the “greater the whole number, the greater the fraction” strategy, according to which 4/9 seems larger than 4/5 because 9 is larger than 5. We aimed to determine whether the ability of adolescents and educated adults to compare fractions with common numerators was rooted in part in their ability to inhibit the use of this misleading strategy by adapting the negative priming paradigm. We found that participants were slower to compare the magnitude of two fractions with common denominators after they compared the magnitude of two fractions with common numerators than after they decided which of two fractions possessed a denominator larger than the numerator. The negative priming effects reported suggest that inhibitory control is needed at all ages to avoid errors when comparing fractions with common numerators
Mathematics Students Demonstrate Superior Visuo-Spatial Working Memory to Humanities Students Under Conditions of Low Central Executive Processing Load
No full textPrevious research has demonstrated that working memory performance is linked to mathematics achievement. Most previous studies have involved children and arithmetic rather than more advanced forms of mathematics. This study compared the performance of groups of adult mathematics and humanities students. Experiment 1 employed verbal and visuo-spatial working memory span tasks using a novel face-matching processing element. Results showed that mathematics students had greater working memory capacity in the visuo-spatial domain only. Experiment 2 replicated this and demonstrated that neither visuo-spatial short-term memory nor endogenous spatial attention explained the visuo-spatial working memory differences. Experiment 3 used working memory span tasks with more traditional verbal or visuo-spatial processing elements to explore the effect of processing type. In this study mathematics students showed superior visuo-spatial working memory capacity only when the processing involved had a comparatively low level of central executive involvement. Both visuo-spatial working memory capacity and general visuo-spatial skills predicted mathematics achievement