Journal of Numerical Cognition (JNC - PsychOpen)
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Predictors of Middle School Students’ Growth in Symbolic Number Comparison Performance
No full textThe ability to efficiently compare number symbols, such as digits, is associated with mathematics competence across the lifespan. Performance on symbolic number comparison tasks differ across age groups; young students who are developing fluency with digits improve on symbolic number comparison, and performance is better in adults than children. However, whether this improvement continues for older students who are fluent with number symbols, and what cognitive factors relate to this improvement, is unknown. This study used a longitudinal sample of U.S. middle school students (n = 394) to examine whether symbolic number comparison performance changes over middle school (i.e., students aged 11-14), whether there are individual differences in students’ rate of change, and potential predictors of that change. Students completed measures of single-digit symbolic number comparison, nonsymbolic number comparison, executive function (EF), and mathematics competence in Grade 5 (M = 11.02 years; SD = 0.32), and double-digit symbolic number comparison in Grades 6-8. Results showed that, on average, students’ symbolic number comparison performance improved from Grades 6-8. Grade 5 Symbolic number comparison performance predicted Grade 8 symbolic number comparison and rate of change over Grades 6-8. Grade 5 nonsymbolic number comparison, EF, and mathematics competence predicted Grade 8 symbolic number comparison performance. Results suggest that numerical magnitude processing, executive functions, and mathematics competence are related to symbolic number processing well into middle school, and that students continue to refine their ability to process number symbols into adolescence
Perceptual and Number Effects on Students’ Initial Solution Strategies in an Interactive Online Mathematics Game
No full textThis study investigated the effects of 1) proximal grouping of numbers, 2) problem-solving goals to make 100, and 3) prior knowledge on students’ initial solution strategies in an interactive online mathematics game. In this game, students transformed an initial expression into a perceptually different but mathematically equivalent goal state. We recorded students’ solution strategies and focused on the productivity of their first steps—whether their initial action led them closer to the goal. We analyzed log data within the game from 227 middle-school students solving four addition problems and four multiplication problems consisting of a total of 1,816 problem-level data points. Logistic regression modeling showed that students were more likely to use productive initial solution strategies to solve addition and multiplication problems when 1) proximity supported number grouping, 2) 100 was the problem-solving goal, and 3) students had higher prior knowledge in mathematics. Furthermore, when problem-solving goals were non-100s, students with lower prior knowledge were less likely to use productive initial solution strategies than students with higher prior knowledge. The findings of the study demonstrated that perceptual and number features influenced students’ initial solution strategies, and the effect of number features on initial solution strategies varied by students’ prior knowledge. Results yield important implications for designing instructional activities that support mathematics learning and problem-solving
Flexibility and Adaptivity in Arithmetic Strategy Use: What Children Know and What They Show
No full textCentral elements of adaptive expertise in arithmetic problem solving are flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from the Netherlands completed a comprehensive assessment of adaptive expertise in the domain of multidigit subtraction, designed to measure, first, the strategies students know and use to solve subtraction problems (potential and practical flexibility). Second, it measured to what extent students know which strategy is optimal and to what extent they use the optimal strategy (potential and practical adaptivity). Findings for flexibility showed that most students consistently used the same strategy across all problems: practical flexibility was low. When prompted, students knew more strategies than they used spontaneously, suggesting hidden potential in flexibility. Findings for adaptivity showed that students hardly ever spontaneously used the task-specific strategy that is efficient for specific problems since it has the fewest and easiest steps. However, almost half of the students could select this strategy from a set of given strategies at least once. Furthermore, an innovative, personalized version of the choice/no-choice method showed that the task-specific strategy was usually not the optimal strategy (fastest strategy leading to a correct answer) for individual students. Finally, students used the strategy with which they performed best more often than the other strategies, but there is hidden potential for the adaptive use of task-specific strategies
Measurement of Mathematics Anxiety in an Israeli Adult Population
No full textMaths anxiety is common and refers to feelings of anxiety, fear and other negative emotions and thoughts in individuals when confronted with mathematical tasks or numerical information. Self-report measures of maths anxiety have been created, but the majority are in English and are not culturally relevant to all countries. This study aimed to translate and validate existing measures for future use in Hebrew-speaking adult populations. The Mathematics Anxiety Scale – UK (MAS-UK) was translated to Hebrew and adult participants completed it alongside the Mathematics Anxiety Rating Scale – Revised (MARS-R) and a general anxiety measure. Factor structures were explored for both the translated MAS-UK and a Hebrew version of the MARS-R, as well as being checked for reliability and convergent and discriminant validity. Results from a final sample of 213 participants, indicated the shortened, Hebrew version of the MAS-UK and the MARS-R are internally consistent and suitable for use in future maths anxiety research in adult Israeli populations. Findings regarding sex differences in maths anxiety are also discussed
Children’s Mixed-Rounding Strategy Use in Computational Estimation
No full textBeing able to perform computational estimations efficiently is an important skill. Furthermore, computational estimation experiments are used to study general principles in strategy development. Rounding strategies are common in computational estimation. However, little is known about whether and when children use a mixed-rounding strategy (i.e., both rounding up and down in one estimation) and how demanding this is in comparison to only rounding-down or only rounding-up. Therefore, we systematically varied the size of unit digits (i.e., the rightmost digit in a whole number) in 72 addition problems. These estimation problems were presented to fourth graders. Most children preferred to use mixed-rounding on mixed-unit problems and therefore adjusted their strategy choice to the individual unit digits in a calculation. Additionally, the sum of units barely influenced children’s strategy choice. On mixed-rounding calculations, the proportion of best strategy use was comparable to that of rounding-up and the latencies to produce an estimate with mixed-rounding were between those for rounding-down and rounding-up. Therefore, the mixed-rounding strategy was in the difficulty range of the two more frequently studied rounding strategies; it was also the preferred strategy for mixed-unit problems by children who adapted their estimation strategies. Based on these findings we argue that research into strategy development with estimation tasks should also include mixed-rounding to improve ecological validity
Adults’ Use of Subtraction by Addition and its Association With Executive Functions
No full textThis study examined adults’ frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) in mental multi-digit subtraction with the choice/no-choice method. Participants were offered subtractions in one choice condition (choice between DS and SBA) and two no-choice conditions (mandatory use of either DS or SBA). SBA was used as frequently as DS in the choice condition. DS was most accurate on subtractions with a large difference (e.g., 502 – 18), while SBA was fastest on subtractions with a small difference (e.g., 903 – 886). In general, participants were adaptive for task characteristics and their personal speed characteristics. We further analyzed task-based adaptivity on an individual level via a Latent Class Analysis. Results showed that two-thirds of the participants were adaptive to task characteristics, and that these adaptive participants were the most proficient in accuracy and speed in the choice condition. We further examined whether executive functions (updating, inhibition, shifting) were related to individual differences in strategy efficiency and task-based adaptivity. In line with our hypothesis, updating was related to strategy efficiency, such that participants with higher updating skills were more accurate. In contrast to our expectations, inhibition and shifting were not related to task-based strategy adaptivity. This study highlights adults’ efficient and adaptive use of arithmetic strategies, and its association with their proficiency and executive functions
Mathematical Flexibility: Theoretical, Methodological, and Educational Considerations
No full textThe current paper presents an introduction to a special issue focusing on mathematical flexibility, which is an important aspect of mathematical thinking and a cherished, but capricious, outcome of mathematics education. Mathematical flexibility involves the flexible, creative, meaningful, and innovative use of mathematical concepts, relations, representations, and strategies. In this introduction we discuss the most relevant theoretical, methodological, and educational considerations related to mathematical flexibility, which form the background of the empirical studies presented in the special issue. Collectively, these studies provide a broader understanding of the mathematical flexibility, its subcomponents, influences, and malleability
Improving Mathematics Performance in 7-Year-Old Children: Training the Mapping From Estimated Quantities to Arabic Digits
No full textExact arithmetic abilities require symbolic numerals, which constitute a precise representation of quantities, such as the Arabic digits. Numerical thinking, however, also engages an intuitive non-linguistic number sense, the Approximate Number System (ANS). The ANS allows us to discriminate quantities, approximate arithmetic transformations, and estimate quantities, all without counting individual items. Although the ANS does not require language, estimations made by means of the ANS can be expressed with number words or Arabic digits. A connection between the ANS and school math performance has been established. A child’s accuracy in mapping from approximate quantities to Arabic digits is associated with children’s symbolic math abilities and can also predict their success at learning new arithmetic skills. Here, we explore whether directly training the mapping between estimated quantities and Arabic digits transfers to better math proficiency. The control training was based on discriminating quantity representations, without involving digits. Each of these three-week computer-based trainings were added to the school schedule. We measured improvements in approximate and exact arithmetic after training. Both the experimental and the control group improved in approximate arithmetic performance. However, in exact arithmetic, results show that strengthening the digit-quantity relation improved the 7-year-olds’ competence in symbolic additions and subtractions over and above the improvement measured in the control group. Our results speak to the complexity of the factors involved in developing mathematical abilities, making the case that training the mapping from estimated quantities to digits can be particularly effective in improving children’s mathematical performance
The Effect of Brief Anxiety Interventions on Reported Anxiety and Math Test Performance
No full textResearch suggests that math and test anxiety have detrimental impacts on performance in math. To prevent these effects, a number of interventions have been developed, but these interventions have not been extensively tested. In the current study, we examine whether four brief anxiety interventions reduce state anxiety and/or increase math performance. We also examine whether any of the interventions weaken the relation between math or test anxiety and math performance. Participants were 300 college students varying in math and test anxiety levels. Participants were randomly assigned to one of four single-session interventions, which each took 5 minutes or less (reappraisal as challenge, reappraisal as excitement, expressive writing, and look ahead), or a no intervention control group. Results generally show that none of the interventions had an effect on reports of state anxiety or performance on a difficult math assessment, with the exception that students in the expressive writing condition reported higher levels of state anxiety. None of the interventions served to attenuate the relation between math or test anxiety and math performance. These findings were not consistent with results of previous work, and suggest that interventions may need to be more extensive in order to have an effect on state anxiety and math performance
Pick the Smaller Number: No Influence of Linguistic Markedness on Three-Digit Number Processing
No full textThe symbolic number comparison task has been widely used to investigate the cognitive representation and underlying processes of multi-digit number processing. The standard procedure to establish numerical distance and compatibility effects in such number comparison paradigms usually entails asking participants to indicate the larger of two presented multi-digit Arabic numbers rather than to indicate the smaller number. In terms of linguistic markedness, this procedure includes the unmarked/base form in the task instruction (i.e., large). Here we evaluate distance and compatibility effects in a three-digit number comparison task observed in Bahnmueller et al. (2015, https://doi.org/10.3389/fpsyg.2015.01216) using a marked task instruction (i.e., ‘pick the smaller number’). Moreover, we aimed at clarifying whether the markedness of task instruction influences common numerical effects and especially componential processing as indexed by compatibility effects. We instructed German- and English-speaking adults (N = 52) to indicate the smaller number in a three-digit number comparison task as opposed to indicating the larger number in Bahnmueller et al. (2015). We replicated standard effects of distance and compatibility in the new pick the smaller number experiment. Moreover, when comparing our findings to Bahnmueller et al. (2015), numerical effects did not differ significantly between the two studies as indicated by both frequentist and Bayesian analysis. Taken together our data suggest that distance and compatibility effects alongside componential processing of multi-digit numbers are rather robust against variations of linguistic markedness of task instructions