Journal of Numerical Cognition (JNC - PsychOpen)
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EEG Decoding of Finger Numeral Configurations With Machine Learning
No full textIn this study, we used multivariate decoding methods to study processing differences between canonical (montring and count) and noncanonical finger numeral configurations (FNCs). While previous research investigated these processing differences using behavioral and event-related potentials (ERP) methods, conventional univariate ERP analyses focus on specific time intervals and electrode sites and fail to capture broader scalp distribution and EEG frequency patterns. To address this issue a supervised learning classifier—support vector machines (SVM)—was used to decode ERP scalp distributions and alpha-band power for montring, counting, and noncanonical FNCs (for integers 1 to 4). The SVM was used to test whether the numerical information presented in FNCs can be decoded from the EEG data. Differences in the magnitude and timing of accuracy rates were used to compare the three types of FNCs. Overall, the algorithm was able to predict numerical information presented in FNCs beyond the random chance level accuracy, with higher rates for ERP scalp distributions than alpha-power. Montring had lower peak accuracy compared to counting and noncanonical configurations, likely due to automaticity in processing montring configurations leading to less distinct scalp distributions for the four numerical magnitudes (1 to 4). Paralleling the response time data, the peak decoding accuracy time for montring was earlier for montring (472 ms), compared to counting (577 ms) and noncanonical FNCs (604 ms). The results provide support for montring configurations being processed automatically, somewhat similar to number symbols, and provide additional insights for processing differences across different forms of FNCs. This study also highlights the strengths of decoding methods in EEG/ERP research on numerical cognition
Potential Moderators of the Left Digit Effect in Numerical Estimation
No full textRecent work reveals a left digit effect in number line estimation such that adults' and children's estimates for three-digit numbers with different hundreds-place digits but nearly identical magnitudes are systematically different (e.g., 398 is placed too far to the left of 401 on a 0-1000 line, despite their almost indistinguishable magnitudes; Lai et al., 2018, https://doi.org/10.1111/desc.12657). In two preregistered studies (N = 218), we investigate the scope and malleability of the left digit effect. Experiment 1 used a typical forward-oriented 0-1000 number line estimation task and an atypical reverse-oriented 1000-0 number line estimation task. Experiment 2 used the same forward-oriented typical 0-1000 number line estimation task from Experiment 1, but with trial-by-trial corrective feedback. We observed a large left digit effect, regardless of the orientation of the line in Experiment 1 or the presence of corrective feedback in Experiment 2. Further, analyses using combined data showed that the pattern was present across most stimuli and participants. These findings demonstrate a left digit effect that is robust and widely observed, and that cannot be easily corrected with simple feedback. We discuss the implications of the findings for understanding sources of the effect and efforts to reduce it
A Direct Comparison of Two Measures of Ordinal Knowledge Among 8-Year-Olds
No full textChildren’s knowledge of the ordinal relations among number symbols is related to their mathematical learning. Ordinal knowledge has been measured using judgment (i.e., decide whether a sequence of three digits is in order) and ordering tasks (i.e., order three digits from smallest to largest). However, the question remains whether performance on these two ordinal tasks tap into similar cognitive processes. Canadian children (N = 87; Age M = 8.7 years, Grade 3) completed symbolic number tasks (i.e., number comparison, ordering, and order judgment) and measures of arithmetic fluency (i.e., addition and subtraction) and working memory (i.e., digit span backward). For both ordinal tasks, there was a reverse distance effect for ordered sequences such that children responded faster to adjacent than to non-adjacent sequences (e.g., 2 3 4 vs. 4 7 9) and a canonical distance effect for unordered sequences such that children responded faster to non-adjacent than to adjacent sequences (e.g., 4 2 3 vs. 4 9 7). Working memory and number comparison each predicted unique variance in the ordinal measures (ordering, order judgment, and a latent ordinal factor based on the two measures). Furthermore, ordinal skills superseded the role of number comparison as the key predictor of arithmetic, controlling for children’s gender and working memory skills. In summary, although both ordering and order judgment tasks index ordinal knowledge, a latent factor that excludes task-specific error may be a better index than either task separately
Non-Numerical Methods of Assessing Numerosity and the Existence of the Number Sense
No full textIn the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or non-numerical
The Effects of Operator Position and Superfluous Brackets on Student Performance in Simple Arithmetic
No full textFor students to advance beyond arithmetic, they must learn how to attend to the structure of math notation. This process can be challenging due to students' left-to-right computing tendencies. Brackets are used in mathematics to indicate precedence but can also be used as superfluous cues and perceptual grouping mechanisms in instructional materials to direct students’ attention and facilitate accurate and efficient problem solving. This online study examines the impact of operator position and superfluous brackets on students’ performance solving arithmetic problems. A total of 528 students completed a baseline assessment of math knowledge, then were randomly assigned to one of six conditions that varied in the placement of higher-order operator and the presence or absence of superfluous brackets: [a] brackets-left (e.g., (5 * 4) + 2 + 3), [b] no brackets-left (e.g., 5 * 4 + 2 + 3), [c] brackets-center (e.g., 2 + (5 * 4) + 3), [d] no brackets-center (e.g., 2 + 5 * 4 + 3), [e] brackets-right (e.g., 2 + 3 + (5 * 4)), and [f] no brackets-right (e.g., 2 + 3 + 5 * 4). Participants simplified expressions in an online learning platform with the goal to “master” the content by answering three questions correctly in a row. Results showed that, on average, students were more accurate in problem solving when the higher-order operator was on the left side and less accurate when it was on the right compared to in the center. There was also a main effect of the presence of brackets on mastery speed. However, interaction effects showed that these main effects were driven by the center position: superfluous brackets only improved accuracy when students solved expressions with brackets with the operator in the center. This study advances research on perceptual learning in math by revealing how operator position and presence of superfluous brackets impact students’ performance. Additionally, this research provides implications for instructors who can use perceptual cues to support students during problem solving
The Role of Basic Number Processing in High Mathematics Achievement in Primary School
No full textWhile symbolic number processing is an important correlate for typical and low mathematics achievement, it remains to be determined whether children with high mathematics achievement also have excellent symbolic number processing abilities. We investigated this question in 64 children (aged 8 to 10), i.e., 32 children with persistent high achievement in mathematics (above the 90th percentile) and 32 average-achieving peers (between the 25th and 75th percentile). Children completed measures of symbolic number processing (comparison and order). We additionally investigated the roles of spatial visualization and working memory. High mathematics achievers were faster and more accurate in order processing compared to average achievers, but no differences were found in magnitude comparison. High mathematics achievers demonstrated better spatial visualization ability, while group differences in working memory were less clear. Spatial visualization ability was the only significant predictor of group membership. Our results therefore highlight the role of high spatial visualization ability in high mathematics achievement
The Development and Assessment of Early Cardinal-Number Concepts
No full textNumber-recognition tasks, such as the how-many task, involve set-to-word mapping, and number-creation tasks, such as the give-n task, entail word-to-set mapping. The present study involved comparing sixty 3-year-olds’ performance on the two tasks with collections of one to three items over three time points about 3 weeks apart. Inconsistent with the sparse evidence indicating equivalent task performance, an omnibus test indicated that success differed significantly by task (and set size but not by time). A follow-up analysis indicated that the hypothesis that success emerges first on the how-many task was, in general, significantly superior to the hypothesis of simultaneous development. It further indicated the how-many-first hypothesis was superior to a give-n-first hypothesis for sets of three. A theoretical implication is that set-to-word mapping appears to develop before word-to-set mapping, especially in the case of three. A methodological implication is that the give-n task may underestimate a key aspect of children’s cardinal understanding of small numbers. Another is that the traditional give-n task, which requires checking an initial response by one-to-one counting, confounds pre-counting and counting competencies
Visual and Symbolic Representations as Components of Algebraic Reasoning
No full textSixty (35 girls) ninth graders were assessed on measures of algebraic reasoning and usage of visual and symbolic representations (with a prompt for visual use) to solve equations and inequalities. The study grouped visual representations into two categories: arithmetic-visual, which entailed the use of real-world objects to represent specific values of variables, and algebraic-visual, which involved formal representations like the number line and the coordinate plane. Symbolic representations, on the other hand, encompassed the use of standard algorithms to solve equations, such as changing the place of terms in an equation. The results reveal that the use of algebraic visuals, as opposed to arithmetic visuals, was associated with enhanced algebraic reasoning. Further, although the students initially relied on standard algorithms to explain equations and inequalities, they could produce accurate algebraic-visual representations when prompted. These findings suggest that students have multiple representations of equations and inequalities but only express visual representations when asked to do so. In keeping with the general relationship between visuospatial abilities and mathematics, self-generated algebraic-visual representations partially mediated the relation between overall mathematics achievement and algebraic reasoning
Correction of Batashvili, M., Cipora, K., & Hunt, T. E. (2022). Measurement of Mathematics Anxiety in an Israeli Adult Population
No full textCorrection to: Batashvili, M., Cipora, K., & Hunt, T. E. (2022). Measurement of mathematics anxiety in an Israeli adult population. Journal of Numerical Cognition, 8(1), 148-165. https://doi.org/10.5964/jnc.653
Number Line Estimation Patterns and Their Relationship With Mathematical Performance
No full textThere is ongoing debate regarding what performance on the number line estimation task represents and its role in mathematics learning. The patterns followed by children’s estimates on the number line task could provide insight into this. This study investigates children’s estimation patterns on the number line task and assesses whether mathematics achievement is associated with these estimation patterns. Singaporean children (n = 324, Age M = 6.2 years, Age SD = 0.3 years) in their second year of kindergarten were assessed on the number line task (0-100) and their mathematical performance (Numerical Operations and Mathematical Reasoning subtests from WIAT II). The results show that most children’s number line estimation patterns can be explained by at least one mathematical model (i.e., linear, logarithmic, unbounded power model, one-cycle power model, two-cycle power model). But the findings also highlight the high percentage of participants for which more than one model shows similar support. Children’s mathematical achievement differed based on the models that best explained children’s estimation patterns. Children whose estimation patterns corresponded to a more advanced model tended to show higher mathematical achievement. Limitations of drawing conclusions regarding what performance on the number line task represents based on models that best explain the estimation patterns are discussed