Journal of Numerical Cognition (JNC - PsychOpen)
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Time as a Measure: Elementary Students Positioning the Hands of an Analog Clock
No full textElementary students have difficulty with the topic of time. The present study investigated students’ actions to position hour and minute hands on an analog clock to indicate particular times of the day. Using one-on-one interviews with students in Grades 2 and 4 (n = 48), we analyzed whether students were more accurate for one hand indicator (hour or minute) versus the other as well as their solution approaches as they positioned each hand. We first present a quantitative analysis of student performance to document whether hour and minute hands posed differential challenges for students as they positioned hands to indicate particular times. Results indicate the hour hand is significantly more challenging to position accurately than the minute hand. Students’ solutions reflected varied approaches, including consideration of the quantitative hour-minute multiplicative relationship, attention to part-whole relations, and matching numbers from the provided time to numerals on the clock. We discuss implications for theory and instruction, including the relationship of time to length measure learning trajectories and the current treatment of time in K-12 mathematics standards for the United States
A New Method for Calculating Individual Subitizing Ranges
No full textA large body of research has shown that human adults are fast and accurate at enumerating arrays of ~1-4 items. This phenomenon has been called subitizing. Above this range, enumeration is slower and less accurate. The subitizing range has been related to individual differences in variables such as mathematical abilities, working memory, etc. The two most common methods for calculating subitizing range today – bilinear fit and sigmoid fit – have their strengths and weaknesses. By combining these two methods, we overcome their biggest limitations and come up with a novel way for calculating Individual Subitizing Range (ISR). This paper introduces this new method as well as empirical studies designed to test the new method. We replicated classic effects from the literature and obtain a high correlation with the sigmoid fit method. This paper includes a Matlab code for easy calculation of ISR as well as a ready-to-use experimental file for testing ISR. We hope that these tools would be of use to researchers studying individual differences in the subitizing range
Reactive and Proactive Control in Arithmetical Strategy Selection
No full textIndividual differences in arithmetic have been explained by differences in cognitive processes and by arithmetic strategy use and selection. In the present study, we investigated the involvement of reactive and proactive control processes. We explored how variation in proactive and reactive control was related to individual differences in strategy selection. We correlated proactive and reactive measures obtained from the AX-CPT and an adjusted N-back task with a measure of strategy adaptiveness during a numerosity judgment task. The results showed that both measures of reactive control (of the AX-CPT and N-back task) correlated positively with strategy adaptiveness, while proactive control was not. This suggests that both cognitive control modes might have a different effect on adaptive strategy selection, where adaptive strategy selection seems to benefit from a transient (late) control mode, reactive control. We discuss these results in the light of the Dual Mechanisms Framework
Natural Alternatives to Natural Number: The Case of Ratio
No full textThe overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students’ non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students’ perceptually based intuitions about quantity and b) students’ reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies
Cognitive Predictors of Counting Skills
No full textRote counting skills have found to be a strong predictor of later arithmetic and reading fluency. However, knowledge of the underlying cognitive factors influencing counting skill is very limited. Present study examined to what extent language skills (phonology, vocabulary, and morphology), nonverbal reasoning skills, and memory at the age of five could explain counting skill at the beginning of first grade. Gender, parents’ education level and child’s persistence were included as control variables. The question was examined in a longitudinal sample (N = 101) with a structural equation model. Results showed that language skills together with memory, nonverbal reasoning skills and parent’s education explained only 22% of the variance in counting at the beginning of the first grade. Vocabulary, morphology, and verbal short-term memory were found to be interchangeable predictors, each explaining approximately 7%–9%, of counting skill. These findings challenge the interpretation of counting as a strongly language-based number skill. However, additional analysis among children with dyslexia revealed that memory and language skills, together with a child’s persistence and gender, had a rather strong predictive value, explaining 34%–46% of counting skill. Together these results suggest that verbal short-term memory and language skills at the age of five have not the same predictive value on counting skill at the beginning of school among a population-based sample as found in subjects with language impairment or learning difficulties, and thus, other cognitive factors should be taken into account in further research related to typical development of counting skill
Retrieval Priming in Product Verification: Evidence From Retrieval-Induced Forgetting
No full textThe conditions under which multiplication verification (3 × 6 = 12, true or false?) involves product retrieval and comparison or familiarity-based recognition judgements has not been clearly established. In two experiments examining verification of single-digit multiplication problems, we used Retrieval-Induced Forgetting (RIF), a signature of retrieval use, as an index of product retrieval in multiplication verification. In Experiment 1, 72 adults practiced multiplication either in a production format or in a verification format and then were tested on corresponding addition and control problems. The results showed RIF (i.e., slower answer production for addition problems whose multiplication counterparts had been practiced) in both the production-practice and the verification-practice groups, but RIF was stronger following true than false verification. Experiment 2 tested verification with related-false and unrelated-false products. Related-false equations produced longer RTs than unrelated false equations. Practice of true, related-false and unrelated-false multiplication equations all produced RIF of the addition counterparts but, overall, related-false multiplication equations produced relatively weak RIF. The results indicated that product retrieval mediates multiplication verification even when false answers are weak associative lures and suggest that a retrieve-and-compare process is the default strategy when false answers are at least plausible. We conclude that the presented answer in verification equations act as retrieval-priming stimuli with true equations priming correct answer retrieval and related-false answers interfering with correct answer retrieval
The Differential Relationship Between Finger Gnosis, and Addition and Subtraction: An fMRI Study
No full textThe impact of fingers on numerical cognition has received a great deal of attention recently. One sub-set of these studies focus on the relation between finger gnosis (also called finger sense or finger gnosia), the ability to identify and individuate fingers, and mathematical development. Studies in this subdomain have reported mixed findings so far. While some studies reported that finger gnosis correlates with or predicts mathematics abilities in younger children, others failed to replicate these results. The current study explores the relationship between finger gnosis and two arithmetic operations—addition and subtraction. Twenty-four second to third graders participated in this fMRI study. Finger sense scores were negatively correlated with brain activation measured during both addition and subtraction. Three clusters, in the left fusiform, and left and right precuneus were found to negatively correlate with finger gnosis both during addition and subtraction. Activation in a cluster in the left inferior parietal lobule (IPL) was found to negatively correlate with finger gnosis only for addition, even though this cluster was active both during addition and subtraction. These results suggest that the arithmetic fact retrieval may be linked to finger gnosis at the neural level, both for addition and subtraction, even when behavioral correlations are not observed. However, the nature of this link may be different for addition compared to subtraction, given that left IPL activation correlated with finger gnosis only for addition. Together the results reported appear to support the hypothesis that fingers provide a scaffold for arithmetic competency for both arithmetic operations
Approximate Number System Task Performance: Associations With Domain-General and Domain-Specific Cognitive Skills in Young Children
No full textWe investigated the associations between young children’s domain-general executive functioning (EF) skills and domain-specific spontaneous focusing on number (SFON) tendencies and their performance on an approximate number system (ANS) task, paying particular attention to variations in associations across different trial types with either congruent or incongruent non-numerical continuous visual cues. We found that children’s EF skills were strongly related to their performance on ANS task trials in which continuous visual cues were incongruent with numerosity. Novel to the current study, we found that children’s SFON tendencies were specifically related to their performance on ANS task trials in which continuous visual cues were congruent with numerosity. Children’s performance on ANS task trials in which children can use both congruent numerical and non-numerical continuous visual cues to approximate large quantities may be related to their unprompted tendency to focus on number in their early environment when there are not salient distractors present. On the other hand, children’s performance on incongruent ANS trials may be less a function of number-specific knowledge but more of children’s domain-general ability to inhibit salient but conflicting or irrelevant stimuli. Importantly, these effects held even when accounting for global math achievement and children’s cardinality knowledge. Overall, results support the consideration of both domain-specific and domain-general cognitive factors in developmental models of children’s early ability to attend to numerosity and provide a possible means for reconciling previous conflicting research findings
Impairment of Arabic- and Spoken-Number Processing in Children With Mathematical Learning Disability
No full textThe performance of 24 French-Quebec 8‒9-year-old children with Mathematical Learning Disability (MLD) in Arabic and spoken number recognition, comprehension and production tasks designed to assess symbolic number processing was compared to that of 37 typically developing children (TD). Children with MLD were less successful than TD children in every symbolic numerical task, including recognition of Arabic and spoken numbers. These results thus suggested that this deficit of symbolic number recognition could compromise symbolic number comprehension and production. Children with MLD also presented with general cognitive difficulties as reading difficulties. Taken together, our results clearly showed that children with MLD presented with a symbolic numerical processing deficit that could be largely attributed to their poorer written language skills
