Journal of Numerical Cognition (JNC - PsychOpen)
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The Numeric Ebbinghaus Effect: Evidence for a Density-Area Mechanism of Numeric Estimation?
No full textOne model of numeric perception is a density-area mechanism: a process that estimates both density and area of an array, then multiplies them to create an estimate of number. One line of evidence that supports this is the surprising numeric Ebbinghaus illusion: smaller context circles lead to greater perceived number than larger context circles, potentially via larger perceived area. This registered report re-tested this effect with a number of simple but potentially important improvements in the method and analysis. Participants were asked to indicate the number of blue dots in arrays that were surrounded by grey context circles of three different sizes. Both experiments confirmed that larger context circles lead to a proportional increase in perceived number. Experiment 1 (N = 50) did so with denser, more texture-like arrays (50-100 dots filling 35% of the area). Experiment 2 (N = 50) did so with sparser, more scatter-like arrays (10-30 dots filling 5% of the area). These findings confirm the existence of the numeric Ebbinghaus effect. This in turn confirms a specific prediction derived from a density-area mechanism and rules out alternatives that begin by stripping away context to non-verbally count discrete entities. No further significant evidence was found to suggest that this depends on the array being particularly dense or texture-like, nor to suggest that anything moderates the impact of increasing perceived area as a direct proportional effect on increasing perceived number. This further builds the case that this kind of numeric perception relies on a density-area mechanism
The Correlates of Statistics Anxiety: Relationships With Spatial Anxiety, Mathematics Anxiety and Gender
No full textThis study investigates the correlates of statistics anxiety. Considering that statistics anxiety and spatial anxiety have been separately correlated with related constructs (e.g., mathematics anxiety, academic performance, etc.), the possibility that spatial anxiety plays a role in statistics anxiety is explored. When facing statistics or mathematics operations, people may imagine or visualize the task operations they must do to obtain the result. To examine this hypothesis, 778 students in a Social or Health Sciences program, enrolled in a –often mandatory– statistics course from Canadian, French and Belgian universities completed an online survey. The results show moderate to strong positive correlations between all three types of anxiety (spatial, mathematics, and statistics). In addition, a mediation analysis reveals the intermediate role played by mathematics anxiety in the relationship between spatial and statistics anxieties. Nonetheless, the direct link from spatial anxiety to statistics anxiety is non-negligible in the model. Finally, the results also indicate that women report higher levels of statistics anxiety, which may be partly explained by their higher level of spatial anxiety
Spatial Biases in Approximate Arithmetic Are Subject to Sequential Dependency Effects and Dissociate From Attentional Biases
No full textThe notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes that operate across trials
The Role of Domain-General and Domain-Specific Skills in the Identification of Arithmetic Strategies
No full textIndividuals solve arithmetic problems in different ways and the strategies they choose are indicators of advanced competencies such as adaptivity and flexibility, and predict mathematical achievement. Understanding the factors that encourage or hinder the selection of different strategies is therefore important for helping individuals to succeed in mathematics. Our research contributed to this goal by investigating the skills required for selecting the associativity shortcut-strategy, where problems such as ‘16 + 38 – 35’ are solved by performing the subtraction (38 – 35 = 3) before the addition (3 + 16 = 19). In a well-powered, pre-registered study, adults completed two tasks that involved ‘a + b – c’ problems, and we recorded a) whether and b) when, they identified the shortcut. They also completed tasks that measured domain-specific skills (calculation skill and understanding of the order of operations) and domain-general skills (working memory, inhibition and switching). Of all the measures, inhibition was the most reliable predictor of whether individuals identified the shortcut, and we discuss the roles it may play in selecting efficient arithmetic strategies
Characterizing Mathematics Anxiety and Its Relation to Performance in Routine and Adaptive Tasks
No full textMathematics anxiety hinders students' mathematical achievement already in primary school, but research on its effects beyond whole number knowledge is limited. The main aim of the current study is to examine how state and trait mathematics anxiety relate to performance across five tasks that are relevant for the development of mathematics in primary school, including a measure of adaptive expertise with school mathematics. These include mathematical tasks with non-symbolic quantities, whole numbers, and rational numbers. The participants were 406 primary school students attending the 5th grade (N = 188) and 6th grade (N = 218). Our results showed that state anxiety varies across task type. Furthermore, students' self-evaluated state and trait mathematics anxiety had varying negative relations with performance depending on the task type. In particular, we found that mathematics anxiety may limit students' adaptive expertise with rational numbers, even after controlling for other relevant mathematical skills. Overall, our results indicate that existing accounts on the role mathematics anxiety plays in school mathematics should expand to consider differences across task type and measures of anxiety
The Effects of Mental Abacus Expertise on Working Memory, Mental Representations and Calculation Strategies Used for Two-Digit Hindu-Arabic Numbers
No full textIn Asia, some children are taught a calculation technique known as the ‘mental abacus’. Previous research indicated that mental abacus experts can perform extraordinary feats of mental arithmetic, but it disagrees as to whether the technique improves working memory. The present study extended and clarified these findings by contrasting performance from several numerical and working memory tasks across three groups of participants: Japanese mental abacus experts, abacus-naïve Australian undergraduates, and abacus-naïve Japanese undergraduates. It also investigated whether the mental representations and strategies used to process two-digit numbers differed across the three groups. First, the results showed that the Japanese mental abacus experts only performed better when the numerical and working memory tasks involved arithmetic problems, suggesting domain-specific transfer rather than domain-general improvements to numerical processing or working memory. Second, the results suggest that the Japanese mental abacus experts were less reliant on decomposed magnitude representations, and used a processing strategy that is less sensitive to the perceptual overlap between numbers. Finally, performance was less discrepant between the Australian and Japanese abacus-naïve undergraduates than either group with the Japanese mental abacus experts, indicating that mental abacus training, rather than socio-cultural differences, was responsible for the observed group differences
No Influence of Masked Priming on the Multiplication Fact Retrieval in a Result Verification Task
No full textIn three experiments, we used a masked prime in a verification task to investigate the processing stages occurring during multiplication fact retrieval. We aimed to investigate the retrieval process by overlapping its execution with the processing of a masked prime consisting of a number. Participants evaluated the correctness of multiplication equations, where the result was preceded by a masked prime (presented for 30 ms, with different stimulus onset asynchrony between the operands and the prime). Decade consistency and relatedness of the prime were manipulated. For example, given the equation 4 x 7 = 28, the prime could be: a neighbor either decade consistent (24) or inconsistent (32), or an unrelated number either decade consistent (23) or inconsistent (31). We expected that the feature of the prime (relatedness or decade consistency) that generates interference depends on the processing stage reached when the prime is processed. Although Experiment 1 showed promising results, Experiments 2 and 3 suggest that the pattern found in Experiment 1 was a false positive. Overall, the paradigm used in this study (i.e., masked prime with a verification ask) does not seem to produce a stable interference during the retrieval process
A Complicated Relationship: Examining the Relationship Between Flexible Strategy Use and Accuracy
No full textThis study explores student flexibility in mathematics by examining the relationship between accuracy and strategy use for solving arithmetic and algebra problems. Core to procedural flexibility is the ability to select and accurately execute the most appropriate strategy for a given problem. Yet the relationship between strategy selection and accurate execution is nuanced and poorly understood. In this paper, this relationship was examined in the context of an assessment where students were asked to complete the same problem twice using different approaches. In particular, we explored (a) the extent to which students were more accurate when selecting standard or better-than-standard strategies, (b) whether this accuracy-strategy use relationship differed depending on whether the student solved a problem for the first time or the second time, and (c) the extent to which students were more accurate when solving algebraic versus arithmetic problems. Our results indicate significant associations between accuracy and all of these aspects— we found differences in accuracy based on strategy, problem type, and a significant interaction effect between strategy and assessment part. These findings have important implications both for researchers investigating procedural flexibility as well as secondary mathematics educators who seek to promote this capacity among their students
Comparing Fraction Magnitudes: Adults’ Verbal Reports Reveal Strategy Flexibility and Adaptivity, but Also Bias
No full textMany studies have used fraction magnitude comparison tasks to assess people’s abilities to quickly assess fraction magnitudes. However, since there are multiple ways to compare fractions, it is not clear whether people actually reason about the holistic magnitudes of the fractions in this task and whether they use multiple strategies in a flexible and adaptive way. We asked 72 adults to solve challenging fraction comparisons (e.g., 31/71 vs. 13/23) on a computer. In some of these comparisons, using benchmarks (i.e., reference numbers such as 1/2) was potentially beneficial. After each trial, participants provided verbal reports of their strategies. We found that participants used a large variety of strategies. The majority of strategies were holistic and relied on fraction magnitudes, and most of these strategies were based on benchmarks. Participants sometimes used gap comparison (i.e., comparing the differences between each fraction’s numerator and denominator), a heuristic that is not always valid and that does not rely on fraction magnitudes. Participants used strategies flexibly: they used many different strategies, they used highly efficient strategies most often, and they adapted their strategy use to features of the items. However, participants sometimes used gap comparison on items for which it did not yield the correct response, and this lack of adaptivity partly explained the “natural number bias” observed in this study
Enhancing Cognitive Flexibility Through a Training Based on Multiple Categorization: Developing Proportional Reasoning in Primary School
No full textProportional reasoning is a key topic both at school and in everyday life. However, students are often misled by their preconceptions regarding proportions. Our hypothesis is that these limitations can be mitigated by working on alternative ways of categorizing situations that enable more adequate inferences. Multiple categorization triggers flexibility, which enables reinterpreting a problem statement and adopting a more relevant point of view. The present study aims to show the improvements in proportional reasoning after an intervention focusing on such a multiple categorization. Twenty-eight 4th and 5th grade classes participated in the study during one school year. Schools were classified by the SES of their neighborhood. The experimental group received 12 math lessons focusing on flexibly envisioning a situation involving proportional reasoning from different points of view. At the end of the school year, compared to a control group, the experimental group had better results on the posttest when solving proportion word problems and proposed more diverse solving strategies. The analyses also show that the performance gap linked to the school’s SES classification was reduced. This offers promising perspectives regarding multiple categorization as a path to overtake preconceptions and develop cognitive flexibility at school