The forgotten p-versine and p-coversine family of functions revisited.

This study delves into special class of generalized p-trigonometric functions, examining their connection to the established counterparts like p-cosine and p-sine functions. We here explore the new class of functions called the p-versine, p-coversine, p-haversine, and p-hacovercosine, by providing comprehensive definitions and properties. Grounded in the characteristics of p-cosine and p-sine functions, the newly proposed functions offer unique mathematical insights. Our work contributes towards a thorough understanding of these new special functions, showcasing their potential applications in diverse scientific domains, from mathematical analysis to physics and engineering. This paper contributes as a valuable resource for future applied mathematics researchers, engaging with these new mathematical functions, enhancing the ability to model complex patterns from diverse real-world applications.

Gestures can help children learn mathematics: how researchers can work with teachers to make gesture studies applicable to classrooms.

The gestures we produce serve a variety of functions-they affect our communication, guide our attention and help us think and change the way we think. Gestures can consequently also help us learn, generalize what we learn and retain that knowledge over time. The effects of gesture-based instruction in mathematics have been well studied. However, few of these studies are directly applicable to classroom environments. Here, we review literature that highlights the benefits of producing and observing gestures when teaching and learning mathematics, and we provide suggestions for designing research studies with an eye towards how gestures can feasibly be applied to classroom learning. This article is part of the theme issue 'Minds in movement: embodied cognition in the age of artificial intelligence'.

Eye tracking study in children to assess mental calculation and eye movements.

Eye tracking technology is a high-potential tool for different mathematic cognition research areas. Moreover, there is a dire need for more studies that provide detailed information on the quality of registered eye data. This study aimed to illustrate the applicability of eye tracking in the examination of mathematical cognition, focusing specifically on primary school students completing a computerized mental arithmetic task. Results suggested that the eye tracking device effectively captured high-quality eye movement data when primary school children engaged in this specific task. Furthermore, significant negative correlations have been found between task performance and number of eye fixations. Finally, eye movements distinctions between "Areas of Interest" have been found, indicating different visual tracking associated with different components of arithmetic calculations. This study underscores the extensive possibilities for future research employing eye tracking devices during computerized calculation tasks as assessment tools to explore the complex visual and cognitive processes.

Effect of short-term intensive design-based STEM learning on executive function: an fNIRS study of the left-behind children.

Design-based STEM learning is believed to be an effective cross-disciplinary strategy for promoting children's cognitive development. Yet, its impact on executive functions, particularly for disadvantaged children, still need to be explored. This study investigated the effects of short-term intensive design-based STEM learning on executive function among left-behind children. Sixty-one Grade 4 students from a school dedicated to the left-behind children in China were sampled and randomly assigned to an experimental group (10.70 ± 0.47 years old, n = 30) or a control group (10.77 ± 0.43 years old, n = 31). The experimental group underwent a two-week design-based STEM training program, while the control group participated in a 2-week STEM-related reading program. Both groups were assessed with the brain activation from 4 brain regions of interest using functional near-infrared spectroscopy (fNIRS) and behavioral measures during a Stroop task before and after the training. Analysis disclosed: (i) a significant within-group time effect in the experimental group, with posttest brain activation in Brodmann Area 10 and 46 being notably lower during neutral and word conditions; (ii) a significant between-group difference at posttest, with the experimental group showing considerably lower brain activation in Brodmann Area 10 and Brodmann Area 46 than the control group; and (iii) a significant task effect in brain activity among the three conditions of the Stroop task. These findings indicated that this STEM learning effectively enhanced executive function in left-behind children. The discrepancy between the non-significant differences in behavioral performance and the significant ones in brain activation implies a compensatory mechanism in brain activation. This study enriches current theories about the impact of Science, Technology, Engineering, and Mathematics (STEM) learning on children's executive function development, providing biological evidence and valuable insights for educational curriculum design and assessment.

The impact of cognitive emotion regulation strategies on math and science anxieties with or without controlling general anxiety.

It is well-established that general anxiety associates with the lower use of adaptive emotion regulation and the higher use of maladaptive emotion regulation. However, no study has previously investigated the impact of cognitive emotion regulation on academic anxieties. Using a sample of secondary school students (N = 391), this study examined the impact of cognitive emotion regulation on math and science anxieties. Math anxiety showed stronger correlations with adaptive than maladaptive emotion regulation, whereas general anxiety showed stronger correlations with maladaptive than adaptive emotion regulation. Hierarchical regression analyses showed that math anxiety was associated with the high uses of acceptance, rumination and other-blame and the low uses of positive reappraisal and putting into perspective. However, with controlling science and general anxieties, math anxiety was associated with the high use of rumination and the low use of positive reappraisal. In contrast, science anxiety was associated with the high uses of acceptance and other-blame and the low use of positive reappraisal. Importantly, however, with controlling math and general anxieties, those science anxiety associations did not remain. Accordingly, these results might provide important insights for the specificity, etiology, and intervention of math anxiety.

Gender and circadian preferences influence emotions and motivation in secondary mathematics classrooms.

Many studies have reported poor school achievement in evening persons and general circadian fluctuations in cognition. The aim of this study was to analyze circadian fluctuations in a cross-sectional design and examine the effects of chronotype on situational emotions and intrinsic motivation. A cross-sectional survey study was conducted in three Turkish secondary schools with a total sample of 599 students (283 females and 316 males). Data were collected at the end of specific math lessons of the same grade level and content, using a form combining three scales. We found no gender-related differences in intrinsic motivation, while there were some differences in situational motivation. In math classes, female students exhibited higher level of interest, while boys scored higher on boredom. In addition, students who scored high on morning affect reported higher levels of interest, well-being, and less boredom. Students with higher stability (and lower fluctuations in mood and cognition during the day) reported a higher degree of enjoyment, perceived competence, perceived choice, and less pressure/tension in their math lessons. A positive association was observed between distinctness, interest, and well-being, while negative correlations existed between distinctness and boredom. This suggests that students with higher diurnal stability reported a higher level of interest, well-being, and a lower level of boredom. Additionally, the results of the analyses showed that morningness, distinctness, and eveningness were significant predictors of intrinsic motivation. Conversely, gender, time of application, morningness, and distinctness emerged as predictors for situational emotions in mathematics classes.

Instructional Influencers: Teaching Professors as Potential Departmental Change Agents in Diversity, Equity, and Inclusion.

At many research-intensive universities in North America, there is a disproportionate loss of minoritized undergraduate students from Science, Technology, Engineering, and Mathematics (STEM) majors. Efforts to confront this diversity, equity, and inclusion (DEI) challenge, such as faculty adoption of evidenced-based instructional approaches that promote student success, have been slow. Instructional and pedagogical change efforts at the academic department level have been demonstrated to be effective at enacting reform. One potential strategy is to embed change agent individuals within STEM departments that can drive change efforts. This study seeks to assess whether tenure-track, teaching-focused faculty housed in STEM departments are perceived as influential on the instructional and pedagogical domains of their colleagues. To answer this, individuals across five STEM departments at large, research-intensive campuses identified faculty who were influential upon six domains of their instruction and pedagogy. Social network analysis of individuals in these departments revealed heterogeneity across the instructional domains. Some, like the teaching strategies network, are highly connected and involve the majority of the department; while others, like the DEI influence network, comprise a significantly smaller population of faculty. Importantly, we demonstrate that tenure-track, teaching-focused faculty are influential across all domains of instruction, but are disproportionately so in the sparsely populated DEI influence networks.

Exploring the linguistic complexity of third-grade numerical literacy.

Reading numbers aloud, a central aspect of numerical literacy, is a challenging skill to acquire, but the origins of this difficulty remain poorly understood. To investigate this matter, we examined the performance of 127 third- and fourth-grade children who read aloud, in Hebrew, numbers with 2-5 digits. We found several key observations. First, we observed a substantial variation among the 3rd graders-7% and 59% errors in the top and bottom deciles, respectively. Second, the task difficulty stemmed from syntactic processing: Most errors were distortions of the number's syntax, as opposed to digit substitutions or transpositions, and the main factor affecting a specific number's difficulty was not its magnitude, as is commonly assumed, but rather its syntactic structure. Third, number reading performance was not predicted by a school-like task that assessed syntactic-conceptual knowledge of the decimal system structure, but rather by knowledge of specific syntactic-verbal rules, suggesting that the syntactic-verbal knowledge is separate from the syntactic-conceptual knowledge. Last, there was a double dissociation between 4-digit numbers and 5-digit numbers, which in Hebrew have completely different syntactic structures: Half of the children showed a significant advantage in one number length compared to the other, with equal numbers of children preferring either length. This indicates that the different syntactic-verbal rules are learned relatively independently of each other, with little or no generalization from one rule to another. In light of these findings, we propose that schools should specifically teach number reading, with focus on specific syntactic-verbal rules.

Children's confidence on mathematical equivalence and fraction problems.

Metacognition is how people think about their own thinking, and it includes children's confidence in their problem-solving solutions. We assessed children's metacognition in two areas of mathematics that are often plagued by misconceptions and often studied separately-mathematical equivalence (e.g., 3 + 4 = 5 +__) and fraction magnitude (e.g., 1/3 = 2/__). Specifically, we quantified children's metacognitive skills across these topics, examined whether these skills are topic-specific, and determined how these skills covary with domain-general executive functioning. A total of 80 elementary school children (Mage = 7.70 years, SD = 0.69) provided trial-by-trial performance data and confidence judgments on equivalence and fraction problems as well as assessments of executive function. Children's metacognitive skills were especially impaired when using incorrect strategies based on misconceptions, and we did not find robust evidence for cross-topic associations. We found modest associations between children's metacognitive skills and their working memory and inhibitory control. Findings have theoretical and practical implications for understanding which children struggle with these important mathematics concepts and why.

A 100-day mentoring program leads to positive shifts in girls' perceptions and attitudes towards biomechanics and related STEM disciplines.

The gender gap in STEM (Science, Technology, Engineering, and Mathematics) is among the widest across education and professional fields, with an underrepresentation of girls and women, particularly in engineering and biomechanics. This issue begins early in education and worsens as females progress into more senior roles. To address this gap, we designed and implemented the Biomechanics Research and Innovation Challenge (BRInC), a 100-day STEM program focused on mentoring and role modelling to engage high school girls and early-career biomechanists at key phases where they most commonly disengage in STEM. We evaluated the influence of the program on (i) identity and perceptions towards science, engineering, and biomechanics; (ii) attitudes towards biomechanics, maths and science; and (iii) attitudes towards gender bias, education and career aspirations in STEM, within high school girls following participation in the BRInC program. We observed significant and positive shifts in girls' perceptions of both biomechanics and engineering. Participation in the program appeared to lead to favourable shifts in attitudes towards biomechanics, maths, and science and fostered a positive influence on girls' education and career aspirations, igniting an interest in future research opportunities. Innovative STEM engagement programs, such as BRInC, highlight the promising potential of targeted and bespoke approaches to address the underrepresentation of females in biomechanics and STEM-related education and careers. Future programs should strive to enhance socioeconomic and cultural diversity, employ whole of life-cycle approaches by offering programs for girls and women at various phases of the STEM pathway, and prioritize impact assessments to effectively monitor progress.

The intersection of parent questions, child skills, and activity context in informal science, technology, engineering, and math learning.

Adult verbal input occurs frequently during parent-child interactions. However, few studies have considered how parent language varies across informal STEM (science, technology, engineering, and math) activities. In this study, we examined how open and closed parent questions (a) differed across three STEM activities and (b) related to math, science, and vocabulary knowledge in their preschool-aged children. A total of 173 parents and their preschool children (Mage = 4 years) from lower socioeconomic households were video-recorded participating in three STEM-related activities: (a) a pretend grocery store activity, (b) a bridge-building challenge, and (c) a book read about a science topic. Parent questions were categorized as open or closed according to the presence of key question terms. Results indicate that the three activities elicited different frequencies of parent open and closed questions, with the grocery store activity containing the most open and closed questions. Children's science knowledge was predicted by the frequency and proportion of parent open questions during the book read. These results enhance our understanding of the role of parent questions in young children's language environments in different informal learning contexts.

Knowing what they know: Preschool teachers' knowledge of math skills and its relation to instruction.

Math experiences during the preschool years play an important role in children's later math learning. Preschool teachers exhibit considerable variability in the amount and types of mathematics activities they engage in with their students; one potentially important source of these individual differences is adults' knowledge of early math development. The current study aimed to describe preschool teachers' knowledge of numeracy, patterning, and spatial/geometric skills developed in preschool and its relation to their reported mathematics instruction. Participants (N = 83) completed a survey in which they judged whether particular early math skills could be observed in typically developing 4-year-olds in the United States and reported their frequency of engaging in different math instructional activities. Pre- and in-service preschool teachers' knowledge varied across the different domains (i.e., numeracy, patterning, and spatial/geometric) of mathematical thinking, but their reported frequency of instruction did not. Teachers who were found to be more accurate in their knowledge of early math development were more likely to report higher frequency of math instruction; looking specifically at the domains, the strength of association between knowledge and instruction was the strongest for numeracy. Such findings highlight the possibility that supporting preschool teachers' knowledge of the range of math skills their students can be developing may be one component of improving early math teaching and learning.

Math talk by mothers, fathers, and toddlers: Differences across materials and associations with children's math understanding.

Learning words for numbers, shapes, spatial relations, and magnitudes-"math talk"-relies on input from caregivers. Language interactions between caregivers and children are situated in activity contexts and likely affected by available materials. Here, we examined how play materials influence the math talk directed to and produced by young children. We video-recorded parents (mothers and fathers; English- and/or Spanish-speaking) and their 24- to 36-month-olds during play with four sets of materials, transcribed and coded types of parent and toddler math words/phrases, and assessed toddlers' understanding of number, shape, and spatial relations terms. Categories of math words varied by materials. Numeracy talk (e.g., "one," "two," "first," "second") was more frequent during interactions with a picture book and toy grocery shopping set than with a shape sorter or magnet board; the reverse held for spatial talk (e.g., "out," "bottom," "up," "circle"). Parent math talk predicted toddler math talk, and both parent and toddler math talk predicted toddlers' understanding of spatial and number words. Different materials provide unique opportunities for toddlers to learn abstract math words during interactions with caregivers, and such interactions support early math cognition.

On the relativity of magnitudes: Delboeuf's forgotten contribution to the 19th century problem of space.

Faced with the mathematical possibility of non-Euclidean geometries, 19th Century geometers were tasked with the problem of determining which among the possible geometries corresponds to that of our space. In this context, the contribution of the Belgian philosopher-mathematician, Joseph Delboeuf, has been unduly neglected. The aim of this essay is to situate Delboeuf's ideas within the context of the philosophies of geometry of his contemporaries, such as Helmholtz, Russell and Poincaré. We elucidate the central thesis, according to which Euclidean geometry is given special status on the basis of the relativity of magnitudes, we uncover its hidden history and show that it is defensible within the context of the philosophies of geometry of the epoch. Through this discussion, we also develop various ideas that have some relevance to present-day methods in gravitational physics and cosmology.

Building integrated number sense in adults and children: Comparing fractions-only training with cross-notation number line training.

Mounting evidence points to the predictive power of cross-notation rational number understanding (e.g., 2/5 vs. 0.25) relative to within-notation understanding (e.g., 2/5 vs. 1/4) in predicting math outcomes. Although correlational in nature, these studies suggest that number sense training emphasizing integrating across notations may have more positive outcomes than a within-notation focus. However, this idea has not been empirically tested. Thus, across two studies with undergraduate students (N = 183 and N = 181), we investigated the effects of a number line training program using a cross-notation approach (one that focused on connections among fractions, decimals, and percentages) and a within-notation approach (one that focused on fraction magnitude representation only). Both number line approaches produced positive effects, but those of the cross-notation approach were larger for fraction magnitude estimation and cross-notation comparison accuracy. In a third study (N = 63), we adapted the cross-notation number line training for use in place of typical classroom warm-up activities for middle school students. Similar to the results with undergraduate students, the cross-notation training program yielded positive benefits for middle school students over a typical warm-up activity (fraction arithmetic practice). Together, these results suggest the importance of an integrated approach to teaching rational number notations, an approach that appears to be uncommon in current curricula.

How teachers make connections among ideas in mathematics instruction.

Conceptual understanding involves understanding connections among ideas within a domain. In this chapter, we consider how teachers support students in learning about connections among ideas in mathematics. We review research focusing on teachers' connection making in mathematics classrooms, and we consider several dimensions of variability in that connection making. Across three corpora of lessons that varied in students' grade levels (first grade to college), cultural settings (United States and China), and mathematics content, we found that all teachers produced linking episodes, but the frequency with which they did so varied substantially, raising new questions about the sources and consequences of that variability. Teachers of first-grade students in China routinely engaged their students in co-constructing links; teachers of middle schoolers and college students in the United States typically explained links to students. Linking episodes targeted many different types of connections, including connections between representations, connections between principles and exemplars, connections between procedures and concepts, and connections between concepts and real-world instantiations. Across all three corpora, teachers expressed linked ideas multimodally in a majority of linking episodes. Based on the findings, we present several hypotheses about how teacher behaviors may support students' understanding of connections among ideas, and we suggest directions for future work.

Adapting STEMM in Hawai'i: Necessary actions for one of the most diverse places in the United States.

Hawai'i's diverse population prime it to be an exemplary environment to study representation in science, technology, engineering, mathematics, and medicine (STEMM). In actuality, Hawai'i has low STEMM enrollment and therefore, low representation in STEMM. What primarily inhibits Hawai'i from having a strong STEMM workforce is the lack of education in STEMM, resources allocated to STEMM, and mentorship to succeed in STEMM. Other factors such as cultural values, high costs of living, and geographical barriers also contribute to Hawai'i's low STEMM enrollment. To combat these issues, I offer suggestions to encourage STEMM enrollment, such as directing funds toward after-school education. I also suggest combatting the lack of resources by providing more online opportunities for students and workers. As for Hawai'i's low mentorship, I suggest that more programs be created within communities and universities to create a platform for mentors and mentees to network. This manuscript seeks to highlight these areas of improvement and recognize lessons to be learned from Hawai'i, thus serving as a resource for individuals internationally.

Emotion regulation skills as a mediator of STEM teachers' stress, well-being, and burnout.

The teaching profession highly stressful, and teachers are often faced with challenging situations. This is particularly the case in STEM (science, technology, engineering, and math) education, which is a uniquely demanding and challenging field. This study examined the role of emotional regulation (ER) skills in STEM teachers' stress, well-being, and burnout. The sample included 165 STEM teachers in middle and high schools who completed standard online questionnaires on ER, stress, well-being, and burnout. They were also asked to comment on three videos depicting authentic mathematical and pedagogical situations. The results indicated that contrary to popular belief, seniority was not linked with levels of stress, difficulties in ER, lower levels of well-being, or higher levels of burnout. A structural equation model and bootstrapping analysis showed teachers' levels of stress predicted their well-being, and this link between stress and well-being was mediated by teachers' level of difficulty in ER. The study highlights the importance of STEM teachers' well-being and suggests the need to reduce stress and burnout by providing tools for teachers to regulate their emotions in the classroom.

Extent of magnitude representation deficit and relationship with arithmetic skills in children with 22q11.2DS.

BACKGROUND: Previous studies have produced conflicting results concerning the extent of magnitude representation deficit and its relationship with arithmetic achievement in children with 22q11.2 deletion syndrome. More specifically, it remains unclear whether deficits are restricted to visuospatial content or are more general and whether they could explain arithmetical impairment. METHODS: Fifteen 5- to 12-year-old children with 22q11.2 deletion syndrome and 23 age-matched healthy controls performed a non-symbolic magnitude comparison task. Depending on the trial, participants had to compare stimuli with high or low visuospatial load (visuospatial stimuli or temporal sequence of visual stimuli). The participants also completed a battery of arithmetic skills (ZAREKI-R) and a battery of global cognitive functioning (WISC-V or WPPSI-IV), from which working memory and visuospatial indices were derived. RESULTS: Children with 22q11.2DS responded as fast as healthy controls did but received fewer correct responses, irrespective of visuospatial load. In addition, their performance in the non-symbolic magnitude comparison task did not correlate with the ZAREKI total score, while the working memory index did. CONCLUSION: Children with 22q11.2DS might suffer from a global magnitude representation deficit rather than a specific deficit due to visuospatial load. However, this deficit alone does not seem to be related to arithmetic achievement. Working memory might be a better concern of interest in favoring arithmetic skills in patients with 22q11.2 deletion syndrome. TRIAL REGISTRATION: Clinicaltrials, NCT04373226 . Registered 16 September 2020.

Declarative memory supports children's math skills: A longitudinal study.

Substantial progress has been made in understanding the neurocognitive underpinnings of learning math. Building on this work, it has been hypothesized that declarative and procedural memory, two domain-general learning and memory systems, play important roles in acquiring math skills. In a longitudinal study, we tested whether in fact declarative and procedural memory predict children's math skills during elementary school years. A sample of 109 children was tested across grades 2, 3 and 4. Linear mixed-effects regression and structural equation modeling revealed the following. First, learning in declarative but not procedural memory was associated with math skills within each grade. Second, declarative but not procedural memory in each grade was related to math skills in all later grades (e.g., declarative memory in grade 2 was related to math skills in grade 4). Sensitivity analyses showed that the pattern of results was robust, except for the longitudinal prediction of later math skills when accounting for stable inter-individual differences via the inclusion of random intercepts. Our findings highlight the foundational role of early domain-general learning and memory in children's acquisition of math.