Journal of Numerical Cognition

Routine and Adaptive Experts: Individual Characteristics and Their Impact on Multidigit Arithmetic Strategy Flexibility and Mathematics Achievement

Lóa Björk Jóelsdóttir — 2024-12-20

Motivated by a curriculum privileging number-based strategies but national tests highlighting students’ reliance on standard algorithms, this study analyses 2,216 Danish Grade 3, 6 and 8 students’ solutions to various multidigit arithmetic tasks, each designed to elicit shortcut strategies, against background variables including sex, ethnicity and familial socio-economic status (SES), and outcomes including strategy flexibility, and national tests for both mathematics and reading. Students offering multiple solutions to a task were defined as flexible, while arithmetic experts (defined by accuracy) were distinguished by their use of shortcut strategies; routine experts never used them, while adaptive experts used them in at least one third of all tasks. With respect to mathematics achievement, experts scored 0.86 SD-units higher than non-experts, and within the former, adaptive experts scored 0.49 SD-units higher than routine experts. With respect to reading, experts achieved 0.57 SD-units higher than non-experts, while adaptive experts achieved 0.19 SD-units higher than routine experts. Boys were significantly more adaptive and flexible than girls. The proportion of experts increased from Grade 3 to Grade 8, whereas the proportion of adaptive experts increased from Grade 3 to 6 but then remained constant. Familial SES was significantly higher for experts than for non-experts but not for adaptive experts in relation to routine. Neither quarter of birth nor the existence of older siblings influenced any outcomes, although the proportion of experts was higher for children with Western backgrounds than for children with non-western background. The results suggest a relationship between adaptive expertise, strategy flexibility, and achievement.

From One-Half to 12th: Fraction Writing in Children and Adult Education Students

Isabella Starling-Alves — 2024-12-20

Learning fractions is essential for academic and daily life success. A critical first step in acquiring fractions is learning to transcode them (e.g., writing ½ when hearing “one half”). However, little is known about how students master fraction transcoding. We addressed this gap by assessing fraction writing in two groups of Brazilian students with limited education: adults in the first year of an adult education program (AEP-1) and 2nd graders. Both groups made frequent transcoding errors. Errors were classified into three categories, Syntactic: correct numerator/denominator values with an incorrect notation (12th for “one half”); Lexical: incorrect numerals with the correct notation (⅓ for “one half”); Combined: incorrect numerals and notation (15th for “one-half”). AEP-1 students’ performance was strongly bimodal: those with weak fraction writing skills made predominantly syntactic errors, whereas those with strong fraction writing skills made mostly lexical errors. Second graders did not transcode any fractions correctly making exclusively syntactic or combined errors. Approximately half the AEP-1 students with the lowest levels of schooling (< 3 years) succeeded in writing fractions, suggesting an important role of informal experiences for this group.

Perceiving Precedence: Order of Operations Errors Are Predicted by Perception of Equivalent Expressions

Jeffrey Kramer Bye — 2024-12-04

Students often perform arithmetic using rigid problem-solving strategies that involve left-to-right-calculations. However, as students progress from arithmetic to algebra, entrenchment in rigid problem-solving strategies can negatively impact performance as students experience varied problem representations that sometimes conflict with the order of precedence (the order of operations). Research has shown that the syntactic structure of problems, and students’ perceptual processes, are involved in mathematics performance and developing fluency with precedence. We examined 837 U.S. middle schoolers’ propensity for precedence errors on six problems in an online mathematics game. We included an algebra knowledge assessment, math anxiety measure, and a perceptual math equivalence task measuring quick detection of equivalent expressions as predictors of students’ precedence errors. We found that students made more precedence errors when the leftmost operation was invalid (addition followed by multiplication). Individual difference analyses revealed that students varied in propensity for precedence errors, which was better predicted by students’ performance on the perceptual math equivalence task than by their algebra knowledge or math anxiety. Students’ performance on the perceptual task and interactive game provide rich insights into their real-time understanding of precedence and the role of perceptual processes in equation solving.

Examining the Role of Spatial and Mathematical Processes and Gender in Postsecondary Precalculus

Robert C. Wilbur — 2024-12-04

Passing the introductory calculus sequence is critical to undergraduate students’ retention in STEM programs. This study examines the relations between three interrelated processes found to influence mathematics learning and achievement: spatial skills, spatial anxiety, and math anxiety. Additionally, it examines the role of gender on these relations and if and how they help explain precalculus achievement. Findings revealed that spatial skills, spatial anxiety, and gender were linked to math anxiety. Furthermore, spatial anxiety and math anxiety were related to strong final exam performance, but spatial skills and gender were not related to achievement. The presented evidence is in accordance with prior research and corroborates the existence of these relational patterns in a postsecondary academic context in addition to the laboratory context. These findings have broad implications for the development and implementation of efforts aimed at improving postsecondary mathematics outcomes, and subsequent persistence, retention, and representation in STEM programs.

Do Errors on Classic Decision Biases Happen Fast or Slow? Numeracy and Decision Time Predict Probability Matching, Sample Size Neglect, and Ratio Bias

Ryan Corser — 2024-11-04

Higher numeracy is associated with better comprehension and use of numeric information as well as reduced susceptibility to some decision biases. We extended this line of work by showing that increased numeracy predicted probability maximizing (versus matching) as well as a better appreciation of large sample sizes. At the same time, we replicated the findings that the more numerate were less susceptible to the ratio bias and base rate neglect phenomena. Decision time predicted accuracy for the ratio bias, probability matching, and sample size scenarios, but not the base rate scenarios. Interestingly, this relationship between decision time and accuracy was positive for the ratio bias problems, but negative for the probability matching and sample size scenarios. Implications for research on cognitive ability and decision biases are discussed.

Parental Math Anxiety Is Associated With Negative Emotional Activation During Hypothetical Health Decision Making

Alyvia Meanor — 2024-09-10

Dealing with numbers is an inherent aspect of interpreting health statistics, and negative emotions may interfere with medical decision making. One emotionally charged decision-making context is parents making medical decisions for their children. Knowing which factors–such as anxiety specific to math contexts–are associated with parents’ negative emotions during the decision-making process may inform ways to better support families as they make critical medical decisions. The current study involved secondary data analyses of an experiment with 249 parents. Participants were randomly assigned to make hypothetical health decisions for themselves, their child, or a stranger. We examined which domain-specific math (e.g., math anxiety), domain general (i.e., need for cognition), and demographic variables (e.g., parents’ health-care coverage) were associated with ratings of negative emotional activation immediately after making the decisions. Results indicated that two factors were significantly associated with parents’ ratings of negative emotional activation: (1) the person they were making decisions about (i.e., higher negative emotion activation if they were randomly assigned to make hypothetical health decisions about their child versus themselves or a stranger), and (2) parents’ ratings of their own math anxiety (i.e., parents with higher self-reported math anxiety also reported higher negative emotional activation). Future research may further consider the joint roles of emotional activation and math anxiety in how parents make health decisions for their children. Further, understanding how much math anxiety causally contributes to people’s overall negative emotional activation could lead to a more nuanced understanding of negative emotional activation in health decision making.

Capturing Math Language Use During Block Play: Creation of the Spatial and Quantitative Mathematical Language Coding System

Lindsey M. Bryant — 2024-07-16

The goals of the current study were: 1) to modify and expand an existing spatial mathematical language coding system to include quantitative mathematical language terms and 2) to examine the extent to which preschool-aged children used spatial and quantitative mathematical language during a block play intervention. Participants included 24 preschool-aged children (Age M = 57.35 months) who were assigned to a block play intervention. Children participated in up to 14 sessions of 15-to-20-minute block play across seven weeks. Results demonstrated that spatial mathematical language terms were used with a higher raw frequency than quantitative mathematical language terms during the intervention sessions. However, once weighted frequencies were calculated to account for the number of codes in each category, spatial language was only used slightly more than quantitative language during block play. Similar patterns emerged between domains within the spatial and quantitative language categories. These findings suggest that both quantitative and spatial mathematical language usage should be evaluated when considering whether child activities can improve mathematical learning and spatial performance. Further, accounting for the number of codes within categories provided a more representative presentation of how mathematical language was used versus solely utilizing raw word counts. Implications for future research are discussed.

Ninth-Grade Students’ Conceptual Understanding of the Number Line

Zehra E. Ünal — 2024-07-05

Sixty (35 girls and 25 boys) 9th-grade students’ conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students’ number line descriptions and the explanations coalesced around six features: sequential ordering (i.e., numbers are sequentially represented), positivity-negativity of numbers (i.e., the number line contains positive and negative numbers), non-centrality (i.e., zero does not have to be in the center), infinity, increment flexibility (i.e., number line increments can vary), and continuity (i.e., numbers can be placed anywhere between minus infinity and plus infinity without breaks). The students’ explanations show that these six features emerge in five successive stages in the conceptual understanding of the number line. These stages are (1) no knowledge, (2) sequential ordering and positivity-negativity, (3) infinity and non-centrality, (4) incremental flexibility, and (5) continuity. The last two stages were not found in most descriptions. The results suggest that students’ understanding of the number line is incomplete and may be overestimated by commonly used quantitative assessments of number line knowledge.

Assessment of Computation Competence and Non-Count Strategy Use in Addition and Subtraction in Grade 1

Delia Leuenberger — 2024-04-17

Computation competence (CC) in simple addition and subtraction using non-counting (NC) strategies is an important learning objective in Grade 1 mathematics but many children, especially low achievers in mathematics, struggle to acquire these skills. To provide these students with the support they need, it is important to have valid and reliable tools for assessing progress in CC and NC strategy use. Developing an assessment instrument for use in Grade 1, when some children start the year unable to solve any problems, is challenging, as is ensuring measurement invariance over a school year when children generally make large achievement gains. This paper presents a new assessment tool for CC and NC strategy use in Grade 1 that was tested in a longitudinal study with N = 1,017 children. Analyses using the Rasch model revealed acceptable mean square scores (MNSQ 0.83 – 1.20). Warm’s Weighted Likelihood Estimate (WLE) reliability scores were acceptable (pre-test .77; post-test .87). Measurement invariance over time was given. The instrument is promising for assessing CC and NC strategy use efficiently and accurately in Grade 1.

Understanding the Role of Working Memory and Phonological Memory in Mathematics and Response to Intervention for Emergent Bilingual Kindergartners

Madison A. Cook — 2024-04-17

This study explores how kindergarten students from a multilingual sample (n = 131) representing 23 different languages differ in response to intervention, based on their skill in mathematics and domain general cognitive skills. Analyses for this study indicate significant correlations between initial math skill, phonological memory, working memory, and language proficiency. There was no statistically significant relationship demonstrated between gains in mathematics and phonological memory, working memory, and language proficiency. No moderation effect was found between domain general skills and response to math intervention. Implications of this work will inform development and delivery of math interventions for multilingual students in kindergarten.