Journal of Numerical Cognition (JNC - PsychOpen)
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Biased Problem Distributions in Assignments Parallel Those in Textbooks: Evidence From Fraction and Decimal Arithmetic
No full textImbalances in problem distributions in math textbooks have been hypothesized to influence students’ performance. This hypothesis, however, rests on the assumption that textbook problems are representative of the problems that students encounter in classroom assignments. This assumption might not be true, because teachers do not present all problems in textbooks and because teachers present problems from sources other than textbooks. To test whether distributions of problems that students encounter parallel distributions of textbook problems, we analyzed fraction and decimal arithmetic problems assigned by 14 teachers over an entire school year. Five of the six documented biases in textbook problem distributions were also present in the classroom assignments. Moreover, the same biases were present in 16 of the 18 combinations of bias and grade level (4th, 5th, and 6th grade) that were examined in assignments and textbooks. Theoretical and educational implications of these findings are discussed
Linking Quantities and Symbols in Early Numeracy Learning
No full textWhat is the foundational knowledge that children rely on to provide meaning as they construct an exact symbolic number system? People and animals can quickly and accurately distinguish small exact quantities (i.e., 1 to 3). One possibility is that children’s ability to map small quantities to spoken number words supports their developing exact number system. To test this hypothesis, it is important to have valid and reliable measures of the efficiency of quantity-number word mapping. In the present study, we explored the reliability and validity of a measure for assessing the efficiency of mapping between small quantities and number words – speeded naming of quantity. Study 1 (N = 128) with 5- and 6-year-old children and Study 2 (N = 182) with 3- and 4-year-old children show that the speeded naming of quantities is a simple and reliable measure that is correlated with individual differences in children’s developing numeracy knowledge. This measure could provide a useful tool for testing comprehensive theories of how children develop their symbolic number representations
An Input Lexicon for Familiar Numbers
No full textNeuropsychological case-studies suggested that dates and encyclopedic numbers may be processed differently than unknown numbers. However, this issue was seldom investigated in healthy participants. Therefore, it is unclear whether known dates are read like words (as lexical items), or like numbers (each position strictly defines digits’ values in a base-10 system). Here, we compared dates to unknown numbers in an experiment using a paradigm from the word recognition literature. We assessed the word-superiority effect by testing experts (students/ teachers in History) with dates. A 4-characters stimulus (xxxx; letters or numbers, half known/unknown) was presented centrally, masked, and followed by 2 characters above and below the mask, at position 2 (xXxx) or 3 (xxXx) in an alternative-forced-choice recognition task. Both accuracy and reaction times were better for dates than unknown numbers, similarly to the results obtained with words by comparison to non-words. However, this effect was modulated by position in the string. These results show a “date-superiority effect” revealing that dates are processed differently than unknown numbers, and suggest that similar orthographical mechanisms might be used to process dates and words
A Systematic Review of Secondary Students’ Attitudes Towards Mathematics and its Relations With Mathematics Achievement
No full textFor a significant number of students, attitudes towards mathematics decrease notably during secondary education. Thus, there is an urgent need to improve students’ mathematics attitudes because attitudes may negatively affect conceptual understanding of mathematics or mathematics performance. However, without a clear unified construct of mathematics attitudes, the ambiguity surrounding this construct prevents researchers from drawing broad conclusions about how to improve students’ overall mathematics attitudes. Therefore, we conducted a systematic review of 95 studies focused on mathematics attitudes to clarify the construct and measurement of mathematics attitudes, and to provide a holistic picture of the relations between mathematics attitudes and math achievement. The review suggested the adoption of a multidimensional definition that regards mathematics attitudes as a combination of specific mathematical cognitions (value, gender roles/beliefs, confidence, self-concept), affects (enjoyment, anxiety), and behavioural intentions (i.e., willingness and tendency to spend more time learning mathematics subjects). The review then explored the relations between each subdimension of attitudes and mathematics performance. In general, anxiety and gender roles were negatively correlated with mathematics performance (r = -.27 to -.48; -.21) whereas enjoyment, self-concept, confidence, perceived value, and behavioural intentions were positively related to achievement (r = .27 to .68; .21 to .76; .34 to .42; .11 to .30; .21 to .34, respectively). Thus, mathematics attitudes appear to comprise three components with several subdimensions that each uniquely contribute to mathematics achievement. Going forward, researchers of mathematics attitudes should a) specify the components of mathematics attitudes used to guide their investigation b) adopt measures in line with their chosen components, and c) investigate how each subdimension of mathematics attitudes uniquely and cumulatively contribute to mathematics ability
No Clear Support for Differential Influences of Visuospatial and Phonological Resources on Mental Arithmetic: A Registered Report
No full textSeveral working memory processes have been hypothesized to influence different arithmetic operations. Working memory has been compartmentalized into a number of different sub-processes, such as phonological memory and visuospatial memory that are believed to have unique contributions to the performance of two distinct arithmetic operations: multiplication and subtraction. A previous dual task experiment produced these effects, but subsequent experiments have yielded inconsistent results. Because the reasons for these inconsistencies are not immediately apparent, the current study systematically reviewed these subsequent attempts and attempted to replicate this effect in a within-subjects dual task experiment using tasks developed from prior work across a number of different subsamples. In contrast to the original finding, we observed no differential impact of specific working memory secondary tasks by arithmetic operation in any of our analyses. However, our analyses do not entirely rule out the possibility of differential effects of working memory tasks. Our findings suggest that the working memory facet by arithmetic operation interactions observed in previous work may be idiosyncratic in nature and difficult to predict a priori in subsequent experiments
Guiding Students’ Attention Towards Multiplicative Relations Around Them: A Classroom Intervention
No full textLearning fractions poses a challenge for many elementary school students, including applying fraction knowledge in novel contexts. For instance, there are substantial individual differences in students’ tendency of spontaneous focusing on quantitative relations (SFOR), which is related to the development of rational number knowledge. In this study, 4th grade students (N = 129) took part in a quasi-experimental study comparing an intervention condition (n = 71) aimed at improving students’ multiplicative relational reasoning and fraction knowledge with a control condition (n = 58) of business as usual fraction instruction. Five lessons of intervention activities were designed to promote students ability to recognize and describe multiplicative relations in their everyday surroundings. There was an overall positive effect on the students’ mathematical knowledge. Students who participated in the intervention improved their ability to recognize and describe multiplicative relations embedded in pictures representing everyday situations. There were no significant differences in the development of fraction knowledge despite replacing five traditional fraction lessons. These findings provide further evidence that researchers and educators should continue to pay attention to issues surrounding students’ spontaneous mathematical focusing tendencies
Depression Induced Quantity Estimation Bias Manifests Only Under Time Constraints
No full textHere, we assess whether quantity representations are influenced by the perceptual biases hypothesized to manifest in depressive individuals. In contrast to this clinical model, several prominent models of numerical cognition assume that quantity representations are abstract, and therefore are independent of the items that are being quantified. If this is the case, then the depression induced perceptual biases should not manifest with respect to the perception of quantity. We tested these predictions in two experiments in which we presented participants with a number-line with a tick mark that indicated the time until a positive, neutral, or negative event. The participant’s task was to estimate the quantity of time indicated by the tick mark. In both experiments, we assessed participants’ BDI-II score. To assess the role of controlled, strategic processing on the manifestation of these biases, we manipulated the amount of time that participants were able to study the number-line prior to responding. In Experiment 1, we attempted to motivate participants to respond quickly voluntarily. The results revealed no influence of time pressure on participants’ RTs, nor any relation between quantity bias and depression. In Experiment 2, we restricted the amount of time participants could spend viewing the number-line. The results revealed estimation biases consistent with the perceptual biases predicted by Beck’s cognitive theory of depression for the short presentation times. These findings (1) confirm that level of depression is linked to the predicted perceptual biases of quantity and (2) implicate controlled processing in the masking of perceptual bias
Decoding Fact Fluency and Strategy Flexibility in Solving One-Step Algebra Problems: An Individual Differences Analysis
No full textAlgebraic thinking and strategy flexibility are essential to advanced mathematical thinking. Early algebra instruction uses ‘missing-operand’ problems (e.g., x – 7 = 2) solvable via two typical strategies: 1) direct retrieval of arithmetic facts (e.g., 9 – 7 = 2) and 2) performance of the inverse operation (e.g., 2 + 7 = 9). The current study investigated the strategies people choose when solving these problems, and whether some people are more flexible in their choices than others. U.S. undergraduates (n = 59) solved missing-operand problems and made speeded verifications of arithmetic sentences corresponding to the direct- and inverse-matched facts. To ‘decode’ their strategy as direct or inverse, each participant’s response times (RTs) for missing-operand problems were regressed on their RTs for the corresponding direct and inverse facts. Our findings replicated the problem size effect for the arithmetic verification task and extended this effect to missing-operand (i.e., one-step) algebra problems, suggesting that the two tasks draw on common representations and processes in the addition (but not subtraction) context. We found individual differences in strategy choice and flexibility such that participants varied both in terms of fluency for retrieving the direct fact and sensitivity to the potential benefit of switching to the inverse fact, which was validated by self-report. We did not find a predicted relation between strategy flexibility and standardized mathematical achievement. These findings inform our understanding of the cognitive processes involved in strategy flexibility in algebra and establish an RT-decoding paradigm for future examination of individual differences in students’ learning of early algebra concepts
The Power of One: The Importance of Flexible Understanding of an Identity Element
No full textThe number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of students indicated that 36/36 = 1), most students did not recognize and apply knowledge of fraction forms of one to estimate numerical magnitudes, solve arithmetic problems, and evaluate arithmetic operations. Specifically, students were less accurate in locating fraction forms of one on number lines than integer forms of the same number; they also were slower and less accurate on fraction arithmetic problems that included one as a fraction (e.g., 6/6 + 1/3) than one as an integer (e.g., 1 + 1/3); and they were less accurate evaluating statements involving fraction forms of one than the integer one (e.g., lower accuracy on true or false statements such as 5/6 × 2/2 = 5/6 than 4/9 × 1 = 4/9). Analyses of three widely used textbook series revealed almost no text linking fractions in the form n/n to the integer one. Greater emphasis on flexible understanding of fractions equivalent to one in textbooks and instruction might promote greater understanding of rational number mathematics more generally
The Unique Role of Spatial Working Memory for Mathematics Performance
No full textWe explored the multi-dimensionality of mathematics and working memory (WM) by examining the differential relationships between different areas of mathematics with visual, spatial, and verbal WM. Previous research proposed that visuospatial WM is a unique predictor of mathematics, but neuroimaging and cognitive research suggest divisions within visuospatial WM. We created a new WM task to isolate visuospatial WM’s visual and spatial components and maintained consistent design across tasks and found that spatial WM predicted mathematics and visual WM did not. We also found that verbal WM predicted all mathematics areas included, while spatial WM was a unique predictor of numerical understanding and geometry, not arithmetic and estimation. These findings integrate previous neuroimaging, cognitive and educational psychology research and further our understanding of the relationship between WM and mathematics