Mathematics is not only relevant in formal situations such as in a classroom, but students use mathematical knowledge and pay attention to mathematical features even in situations that are not explicitly mathematical. Adaptive expertise in mathematics is required to apply mathematical skills to novel and everyday problems. Situations in everyday life may also provide opportunities to develop adaptive expertise in mathematics. We suggest that Spontaneous Focusing on Quantitative Relations (SFOR), Spontaneous Focusing on Numerosity (SFON), and Adaptive rational Number Knowledge (ArNK) are different manifestations of adaptive expertise in mathematics. Thus, we investigated how adaptive expertise in mathematics manifests in general upper secondary school students ( n = 106). We also examined whether these skills are related and differ if a student chooses an advanced or basic mathematics syllabus track. The results showed individual differences in how adaptive expertise in mathematics manifests in general upper secondary school students. Furthermore, ArNK was found to have a positive correlation with SFOR and SFON. However, there were task-based differences. In addition, students in the advanced mathematics syllabus track had a higher ArNK than those in the basic mathematics track, but there were no consistent differences in SFOR or SFON. This research furthers our understanding of individual differences in mathematics learning outcomes and adaptive expertise in mathematics at the start of upper secondary education.
Spatial skills are robustly associated with mathematical performance, yet the cognitive mechanisms underlying this relationship in adults remain poorly understood. The present research examined whether unbounded number line estimation mediates the association between spatial skills and mathematical performance in adults. Ninety (90) undergraduate students completed measures of spatial visualization, mental rotation, algebra, fractions, arithmetic, and unbounded number line estimation. Results indicated that spatial skills significantly predicted mathematical performance and that unbounded number line estimation indirectly mediated this relationship across all three mathematical domains. These findings extend spatial representation theory to adulthood, suggesting that stronger spatial skills may support the construction of more precise internal representations of numerical magnitude, which in turn facilitate mathematical proficiency.
Understanding the sources of individual differences in mathematical abilities is considered essential by both researchers and society alike. Almost 50 years ago, the Russian psychologist V.A. Krutetskii proposed the concept of a mathematical cast of mind (MCOM), which he described as a predisposition to perceive the world through ‘mathematical eyes’. This MCOM should be particularly evident in gifted individuals, enabling them to acquire mathematical knowledge more rapidly and to achieve higher levels of mathematical performance. Although Krutetskii's concept is often referred to in models of mathematical giftedness, empirical attempts to measure it have been limited. The aim of this study is to identify specific tasks that capture aspects of the MCOM and to determine whether these tasks contribute to an MCOM factor that is associated with mathematical achievement above and beyond verbal and figural intelligence. To this end, 82 university students worked on tasks that measure the MCOM components of obtaining mathematical information, processing of mathematical information, and motivational tendencies. The results of an exploratory factor analysis showed a three-factor structure in line with theoretical components of MCOM but not a single factor of MCOM. From the three components, only motivational tendencies was a significant predictor of mathematical achievement. Obtaining mathematical information, in contrast, showed negative correlations with the other two factors and did not significantly predict individual differences in mathematical competence. These findings provide further insights into the concept of MCOM and if it can be measured reliably to eventually predict individual differences in mathematics.
We investigated the development of mathematical vocabulary understanding in young French children aged 3 to 6 years old as a function of the socioeconomic environment. A total of 306 French children were administered two tasks, respectively assessing their understanding and production of French mathematical vocabulary. The results showed differences in understanding and production of mathematical vocabulary between children in preschool/kindergarten Grade Levels 1 (3-4 yo), 2 (4-5 yo), and 3 (5-6 yo). Children from REP+ (“Réseau d’Education Prioritaire”) schools (that concerned neighborhoods or isolated sectors with the greatest concentrations of social difficulties) understood and produced less numerous mathematical terms than those from regular schools, whatever the grade level. Theoretical and practical contributions for teaching are discussed. Despite the measures as a priority education policy organized by the government in REP+ schools, the school fails to reduce the impact of social inequalities on mathematical vocabulary. Efficient intervention is therefore needed.
Using lab-based eye-trackers, previous studies have demonstrated that visual presentation, such as positions of high-order operators (HOOs) and presence of superfluous brackets, influences gaze behaviors during arithmetic problem solving (e.g., Egorova et al., 2024). The current study explored the feasibility of moving eye-tracking arithmetic problem solving studies online by replicating Egorova et al.’s (2024) study using a webcam-based eye-tracker (i.e., WebGazer). Furthermore, we examined whether the proportion of gazes on the HOO over time, a gaze measure that was not analyzed in Egorova et al. (2024) but commonly used in WebGazer studies, can suggest participants’ online problem solving strategies. We analyzed gaze data from 119 college students who mentally evaluated simple arithmetic expressions where the HOO appeared in left, center, or right positions with or without superfluous brackets. Replicating Egorova et al.’s (2024) findings, participants registered their first gaze on the HOO faster when the HOO was on the left compared to on the right (a left-to-right tendency), and slower when brackets were present compared to absent around the center HOO (center brackets effects). Nevertheless, the online gaze data did not replicate the gaze difference in left versus center HOO positions. Furthermore, results of the proportion of gazes on the HOO over time confirmed the left-to-right tendency and further indicated that superfluous brackets guide gaze behaviors in both early and late stages of evaluations. Based on these findings, we discuss the feasibility and cautions of using WebGazer in online eye-tracking arithmetic problem solving studies.
