Journal of Numerical Cognition (JNC - PsychOpen)
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Knowledge of Mathematical Symbols Goes Beyond Numbers
No full textThe written language of mathematics is dense with symbols and with conventions for combining those symbols to express mathematical ideas. For example, reading a factored polynomial function such as f(x) = x²(2x + 15) requires the knowledge that parenthesis can be used to signify function notation in one context and multiplication in another. Mathematical orthography is defined as orthographic knowledge of symbolic mathematics. It entails both knowledge of discrete mathematical symbols and the conventions for combining those symbols into expressions and equations. The ability to read text written in the base-ten system, comprised of digits and conventions for combining digits to express whole and rational quantities, is an important aspect of mathematical orthography. However, success in secondary and post-secondary programs requires more advanced mathematical orthography. The goal of this research was to determine if a simple and novel measure of mathematical orthography captures individual differences in adults’ mathematical skills. Mathematical orthography was measured with a timed dichotomous symbol decision task. Adults (N = 58) discriminated between conventional and non-conventional combinations of mathematical symbols (e.g., x² vs. ²x; |y| vs. ||y). The mathematical symbol decision task uniquely predicted individual differences in whole-number arithmetic, fraction/algebra procedures, and word problem solving. These findings suggest that the symbol decision task is a useful index of symbol associations in mathematical development and, thus, provides a tool for understanding the role of mathematical orthography in individual differences in adults’ mathematical skills
The Effects of Auditory Numerosity and Magnitude on Visual Numerosity Representation: An ERP Study
No full textNumerical representation is not restricted to sensory modalities. It remains unclear how numerosity processing in different modalities interacts within the brain. Moreover, the effect of continuous magnitudes presented in one modality on the representation of numerosity in another modality has not been well studied. By using event-related potential (ERP) and source localization analyses, the present study examined whether there was an interaction between auditory numerosity and continuous magnitude on visual numerosity representation. A visual dot array (visual standard stimulus) was preceded by sound in which numerosity (Multiple-tone vs. One-tone conditions) and magnitude (Loud-tone vs. Soft-tone conditions) information were manipulated. Then, another visual dot array (visual comparison stimulus) was presented, and participants were required to compare the numerosities of the visual dots. Behavioural results revealed that participants showed smaller just-noticeable differences (JNDs) when visual stimuli were preceded by multiple tones than those when visual stimuli were preceded by one tone. The subsequent ERP analysis of visual standard stimuli revealed that the peak amplitude of N1 was more negative under the Loud-tone condition than that under the Soft-tone condition, which could be related to better preparatory attention. Moreover, a significant interaction between auditory numerosity and magnitude was found within the P2p time window for the standard stimuli. Further source localization analysis identified the effect of N1 and P2p to be in the right middle frontal gyrus (MFG) and left inferior parietal lobule (IPL). The present study suggests that numerosity information presented in one sensory modality could spontaneously affect the numerical representation in another modality
Hand Position Affects Performance on Multiplication Tasks
No full textWe investigated whether or not hand placement affects people’s ability to apply learned mathematical information in new and familiar contexts. Participants learned a set of arithmetic facts presented one way (i.e., in a × b = c format) and then were tested on those same facts shown in either a novel format (b × a = __) in Experiment 1 or in the previously-learned format (a × b = __) in Experiment 2. Throughout study and test, participants’ hands were either near to or far from the stimuli. Performance on the novel format was worse when the hands were near compared to far, but performance on the previously-learned format did not depend on hand placement. Together, results indicate that hand proximity impairs mathematical performance when performance depends on the abstracting of conceptual information from sensory information. We conclude that hand placement may be involved in the application of knowledge
Eye Gaze Patterns Reflect How Young Fraction Learners Approach Numerical Comparisons
No full textLearning fractions is notoriously difficult, yet critically important to mathematical and general academic achievement. Eye-tracking studies are beginning to characterize the strategies that adults use when comparing fractions, but we know relatively little about the strategies used by children. We used eye-tracking to analyze how novice children and mathematically-proficient adults approached a well-studied fraction comparison paradigm. Specifically, eye-tracking can provide insights into the nature of differences: whether they are quantitative—reflecting differences in efficiency—or qualitative—reflecting a fundamentally different approach. We found that children who had acquired the basic fraction rules made more eye movements than did either adults or less proficient children, suggesting a thorough but inefficient problem solving approach. Additionally, correct responses were associated with normative gaze patterns, regardless of age or proficiency levels. However, children paid more attention to irrelevant numerical relationships on conditions that were conceptually difficult. An exploratory analysis points to the possibility that children on the verge of making a conceptual leap attend to the relevant relationships even when they respond incorrectly. These findings indicate the potential of eye-tracking methodology to better characterize the behavior associated with different levels of fraction proficiency, as well as to provide insights for educators regarding how to best support novices at different levels of conceptual development
Fraction Magnitude: Mapping Between Symbolic and Spatial Representations of Proportion
No full textFraction notation conveys both part-whole (3/4 is 3 out of 4) and magnitude (3/4 = 0.75) information, yet evidence suggests that both children and adults find accessing magnitude information from fractions particularly difficult. Recent research suggests that using number lines to teach children about fractions can help emphasize fraction magnitude. In three experiments with adults and 9-12-year-old children, we compare the benefits of number lines and pie charts for thinking about rational numbers. In Experiment 1, we first investigate how adults spontaneously visualize symbolic fractions. Then, in two further experiments, we explore whether priming children to use pie charts vs. number lines impacts performance on a subsequent symbolic magnitude task and whether children differentially rely on a partitioning strategy to map rational numbers to number lines vs. pie charts. Our data reveal that adults very infrequently spontaneously visualize fractions along a number line and, contrary to other findings, that practice mapping rational numbers to number lines did not improve performance on a subsequent symbolic magnitude comparison task relative to practice mapping the same magnitudes to pie charts. However, children were more likely to use overt partitioning strategies when working with pie charts compared to number lines, suggesting these representations did lend themselves to different working strategies. We discuss the interpretations and implications of these findings for future research and education. All materials and data are provided as Supplementary Materials
Refuting Misconceptions: Computer Tutors for Fraction Arithmetic
No full textFractions, known to be difficult for both children and adults, are especially prone to misconceptions and erroneous strategy selection. The present study investigated whether a computer tutor improves fraction arithmetic performance in adults and if supplementing problem solving with erroneous examples is more beneficial than problem solving alone. Seventy-five undergraduates solved fraction arithmetic problems using a computer tutoring system we designed. In a between-subjects design, 39 participants worked with a problem-solving tutor that was supplemented with erroneous examples and 36 participants worked with a traditional problem-solving tutor. Both tutors provided hints and feedback. Overall, participants improved after the tutoring interventions, but there were no significant differences in gains made by the two conditions. For students with low prior knowledge about fraction arithmetic, the numerical gains were higher in the erroneous-example group than the problem-solving group, but this effect was not significant. Thus, computer tutors are useful tools for improving fraction knowledge. While erroneous examples may be particularly beneficial for students with low prior knowledge who may hold more misconceptions, more research is needed to make this conclusion
Iconicity in Mathematical Notation: Commutativity and Symmetry
No full textMathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects – those which visually resemble in some way the concepts they represent – offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd’s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance
Approximate Number System Discrimination Training for 7-8 Year Olds Improves Approximate, but not Exact, Arithmetics, and Only in Children With Low Pre-Training Arithmetic Scores
No full textWe investigated whether training the Approximate Number System (ANS) would transfer to improved arithmetic performance in 7-8 year olds compared to a control group. All children participated in Pre- and Post-Training assessments of exact symbolic arithmetic (additions and subtractions) and approximate symbolic arithmetic abilities (a novel test). During 3 weeks of training (approximately 25 minutes per day, two days per week), we found that children in the ANS Training group had stable individual differences in ANS efficiency and increased in ANS efficiency, both within and across the training days. We also found that individual differences in ANS efficiency were related to symbolic arithmetic performance. Regarding arithmetic performance, both the ANS training group and the control group improved in all tests (exact and approximate arithmetics tests). Thus, the ANS training did not show a specific effect on arithmetic performance. However, considering the initial arithmetic level of children, we found that the trained children showed a higher improvement on the novel approximate arithmetic test compared to the control group, but only for those children with a low pre-training arithmetic score. Nevertheless, this difference within the low pre-training arithmetic score level was not observed in the exact arithmetic test. The limited benefits observed in these results suggest that this type of ANS discrimination training, through quantity comparison tasks, may not have an impact on symbolic arithmetics overall, although we cautiously propose that it could help with approximate arithmetic abilities for children at this age with below-average arithmetic performance
Can You Trust Your Number Sense: Distinct Processing of Numbers and Quantities in Elementary School Children
No full textTheories of number development have traditionally argued that the acquisition and discrimination of symbolic numbers (i.e., number words and digits) are grounded in and are continuously supported by the Approximate Number System (ANS)—an evolutionarily ancient system for number. In the current study, we challenge this claim by investigating whether the ANS continues to support the symbolic number processing throughout development. To this end, we tested 87 first- (Age M = 6.54 years, SD = 0.58), third- (Age M = 8.55 years, SD = 0.60) and fifth-graders (Age M = 10.63 years, SD = 0.67) on four audio-visual comparison tasks (1) Number words–Digits, (2) Tones–Dots, (3) Number words–Dots, (4) Tones–Digits, while varying the Number Range (Small and Large), and the Numerical Ratio (Easy, Medium, and Hard). Results showed that larger and faster developmental growth in the performance was observed in the Number Words–Digits task, while the tasks containing at least one non-symbolic quantity showed smaller and slower developmental change. In addition, the Ratio effect (i.e., the signature of ANS being addressed) was present in the Tones–Dots, Tones–Digits, and Number Words–Dots tasks, but was absent in the Number Words–Digits task. These findings suggest that it is unlikely that the ANS continuously underlines the acquisition and the discrimination of the symbolic numbers. Rather, our results indicate that non-symbolic quantities and symbolic numbers follow qualitatively distinct developmental paths, and argue that the latter ones are processed in a semantic network which starts to emerge from an early age
Natural Number Bias in Arithmetic Operations With Missing Numbers – A Reaction Time Study
No full textWhen reasoning about numbers, students are susceptible to a natural number bias (NNB): When reasoning about non-natural numbers they use properties of natural numbers that do not apply. The present study examined the NNB when students are asked to evaluate the validity of algebraic equations involving multiplication and division, with an unknown, a given operand, and a given result; numbers were either small or large natural numbers, or decimal numbers (e.g., 3 × _ = 12, 6 × _ = 498, 6.1 × _ = 17.2). Equations varied on number congruency (unknown operands were either natural or rational numbers), and operation congruency (operations were either consistent – e.g., a product is larger than its operand – or inconsistent with natural number arithmetic). In a response-time paradigm, 77 adults viewed equations and determined whether a number could be found that would make the equation true. The results showed that the NNB affects evaluations in two main ways: a) the tendency to think that missing numbers are natural numbers; and b) the tendency to associate each operation with specific size of result, i.e., that multiplication makes bigger and division makes smaller. The effect was larger for items with small numbers, which is likely because these number combinations appear in the multiplication table, which is automatized through primary education. This suggests that students may count on the strategy of direct fact retrieval from memory when possible. Overall the findings suggest that the NNB led to decreased student performance on problems requiring rational number reasoning