Development of Mathematical Cognition
Chapter 9 - Individual Differences in Arithmetic Fact Retrieval
Abstract
In this chapter I summarize research on individual differences in arithmetic fact retrieval through the lens of educational neuroscience. By generating predictions about underlying cognitive processes based on neuroimaging data, developmental behavioral studies have revealed that symbolic numerical magnitude processing plays a unique role in children’s early arithmetic facts acquisition. Such studies also suggest that phonological processing might play a role in fact retrieval. Other studies in educational neuroscience point to the recruitment of a widespread brain network during fact retrieval, including the prefrontal cortex, inferior parietal cortex, and the medial temporal lobe. The fact that the organization of this network changes over time highlights the continuing need for developmental imaging studies.
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Learning and education in numerical cognition: We do need education
2021, Heterogeneous Contributions to Numerical Cognition: Learning and Education in Mathematical CognitionThis chapter reflects on the impact of education on numerical cognition. Mathematics is a symbolic activity, which must be learned via education, and this learning process will impact on how we process number. There are huge differences in the educational contexts around the globe in which this learning occurs, but these contexts are relatively underappreciated in studies on numerical cognition. I will illustrate the impact of the educational context on numerical cognition by showing that the processing of natural Arabic numerals might be changed by learning other symbolic representations (e.g., fractions), which children learn early in primary school, and by highlighting that the association between number processing and arithmetic depends on the methods children learn to calculate (e.g., mental arithmetic vs. algorithmic computation). I further demonstrate that the educational experience of learning to calculate by itself affects children’s numerical processing. There is a need for experimental studies that manipulate learning to test causal associations between numerical cognition and mathematics learning. The educational context can be used as a natural experiment, via the school cutoff design and cross-national comparisons, to obtain more direct evidence on the broad impact of education on numerical cognition. I end with the suggestion to take the educational context more seriously in future studies on numerical cognition, which will increase both their external and internal validity.
Developmental changes in size effects for simple tie and non-tie addition problems in 6- to 12-year-old children and adults
2021, Journal of Experimental Child PsychologyIn the domain of cognitive arithmetic, the size effect corresponds to an increase in solution times as a function of the size of the operands involved in the problems. In this study, we tracked the evolution of size effects associated with tie and non-tie addition problems across development. We scrutinized the progression of solution times for very small problems involving operands from 2 to 4, larger problems, and 1-problems (problems involving 1 as one of the operands) in children from Grade 1 to Grade 5 and adults. For the first time, we document the presence of a size effect for tie problems with a sum up to 8 in Grade 1 children. In contrast, from Grade 3 until adulthood, this size effect could not be evidenced. Crucially, for non-tie problems, whereas a general size effect is observed when contrasting small one-digit additions with large additions, we show that, from Grade 1 until adulthood, a continuous size effect as a function of the sum of the problems is not observed. In fact, for all age groups, medium problems with sums of 8, 9, and 10 do not present a size effect at all. Given that the problem size effect is sometimes referred to as one of the most robust and reliable effects in the numerical cognition literature, our results necessarily challenge its theoretical interpretation.
Temporo-frontal activation during phonological processing predicts gains in arithmetic facts in young children
2019, Developmental Cognitive NeuroscienceBehavioral studies have shown discrepant results regarding the role of phonology in predicting math gains. The objective of this study was to use fMRI to study the role of activation during a rhyming judgment task in predicting behavioral gains on math fluency, multiplication, and subtraction skill. We focused within the left middle/superior temporal gyrus and left inferior frontal gyrus, brain areas associated with the storage of phonological representations and with their access, respectively. We ran multiple regression analyses to determine whether activation predicted gains in the three math measures, separately for younger (i.e. 10 years old) and older (i.e 12 years old) children. Results showed that activation in both temporal and frontal cortex only predicted gains in fluency and multiplication skill, and only for younger children. This study suggests that both temporal and frontal cortex activation during phonological processing are important in predicting gains in math tasks that involve the retrieval of facts that are stored as phonological codes in memory. Moreover, these results were specific to younger children, suggesting that phonology is most important in the early stages of math development. When the math task involved subtractions, which relies on quantity representations, phonological processes were not important in driving gains.
More than number sense: The additional role of executive functions and metacognition in arithmetic
2019, Journal of Experimental Child PsychologyArithmetic is a major building block for children’s development of more complex mathematical abilities. Knowing which cognitive factors underlie individual differences in arithmetic is key to gaining further insight into children’s mathematical development. The current study investigated the role of executive functions and metacognition (domain-general cognitive factors) as well as symbolic numerical magnitude processing (domain-specific cognitive factor) in arithmetic, enabling the investigation of their unique contribution in addition to each other. Participants were 127 typically developing second graders (7- and 8-year-olds). Our within-participant design took into account different components of executive functions (i.e., inhibition, shifting, and updating), different aspects of metacognitive skills (i.e., task-specific and general metacognition), and different levels of experience in arithmetic, namely addition (where second graders had extensive experience) and multiplication (where second graders were still in the learning phase). This study reveals that both updating and metacognitive monitoring are important unique predictors of arithmetic in addition to each other and to symbolic numerical magnitude processing. Our results point to a strong and unique role of task-specific metacognitive monitoring skills. These individual differences in noticing one’s own errors might help one to learn from his or her mistakes.
Patterning counts: Individual differences in children's calculation are uniquely predicted by sequence patterning
2019, Journal of Experimental Child PsychologyMany studies have examined the cognitive determinants of children’s calculation, yet the specific contribution of children’s patterning abilities to calculation remains relatively unexplored. This study investigated whether children’s ability to complete sequence patterns (i.e., add the missing element into 2–4–?–8) uniquely predicted individual differences in calculation and whether these associations differed depending on the type of stimuli in these sequence patterns (i.e., number, letter, time, or rotation). Participants were 65 children in first and second grade (Mage = 7.40 years, SD = 0.44). All children completed four tasks of sequence patterning: number, letter, time, and rotation. Calculation was measured via addition and subtraction tasks. We also measured cognitive determinants of individual differences in calculation—namely symbolic number comparison, motor processing speed, visuospatial working memory, and nonverbal IQ—to verify whether patterning predicted calculation when controlling for these additional measures. We observed significant relationships between the patterning dimensions and calculation, except for the rotation dimension. Follow-up regressions, controlling for the aforementioned cognitive determinants of calculation, revealed that the number and time dimensions were strong predictors of calculation, whereas the evidence for the letter dimension was only anecdotal and the evidence for the rotation dimension was nonexistent, suggesting some degree of specificity of different types of sequence patterning in predicting calculation. Symbolic magnitude processing remained a powerful unique correlate of calculation performance. These findings add to our understanding of individual differences in calculation ability, such that sequence patterning could begin to be considered as one of the cognitive skills underlying calculation ability in young children.
Arithmetic in the developing brain: A review of brain imaging studies
2018, Developmental Cognitive NeuroscienceBrain imaging studies on academic achievement offer an exciting window on experience-dependent cortical plasticity, as they allow us to understand how developing brains change when children acquire culturally transmitted skills. This contribution focuses on the learning of arithmetic, which is quintessential to mathematical development. The nascent body of brain imaging studies reveals that arithmetic recruits a large set of interconnected areas, including prefrontal, posterior parietal, occipito-temporal and hippocampal areas. This network undergoes developmental changes in its function, connectivity and structure, which are not yet fully understood. This network only partially overlaps with what has been found in adults, and clear differences are observed in the recruitment of the hippocampus, which are related to the development of arithmetic fact retrieval. Despite these emerging trends, the literature remains scattered, particularly in the context of atypical development. Acknowledging the distributed nature of the arithmetic network, future studies should focus on connectivity and analytic approaches that investigate patterns of brain activity, coupled with a careful design of the arithmetic tasks and assessments of arithmetic strategies. Such studies will produce a more comprehensive understanding of how the arithmetical brain unfolds, how it changes over time, and how it is impaired in atypical development.