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.

Bridging cognitive neuroscience and education: Insights from EEG recording during mathematical proof evaluation.

BACKGROUND: Much of modern mathematics education prioritizes symbolic formalism even at the expense of non-symbolic intuition, we contextualize our study in the ongoing debates on the balance between symbolic and non-symbolic reasoning. We explore the dissociation of oscillatory dynamics between algebraic (symbolic) and geometric (non-symbolic) processing in advanced mathematical reasoning during a naturalistic design. METHOD: Employing mobile EEG technology, we investigated students' beta and gamma wave patterns over frontal and parietal regions while they engaged with mathematical demonstrations in symbolic and non-symbolic formats within a tutor-student framework. We used extended, naturalistic stimuli to approximate an authentic educational setting. CONCLUSION: Our findings reveal nuanced distinctions in neural processing, particularly in terms of gamma waves and activity in parietal regions. Furthermore, no clear overall format preference emerged from the neuroscientific perspective despite students rating symbolic demonstrations higher for understanding and familiarity.

Fostering diversity in mathematics cognition.

Integrating diverse perspectives in psychological science can enhance innovation in research and allow research teams to better study diverse populations of individuals through an authentic lens. Despite recent efforts to better address issues of race and ethnicity in research samples, the field of psychology broadly-and the area of mathematics cognition specifically-has largely failed to support scientists from diverse racial and ethnic backgrounds. In this essay, we consider the unique contributions that scholars of color can make to psychological research in mathematics cognition. Next, we reveal common challenges faced by scholars of color and challenges to recruiting and maintaining scholars of color in our community with a focus on Black scholars. Finally, we propose actions for diversifying the "pipeline" of promising scholars.

Developing a cognitive model of solid geometry based on Interpretive Structural Modeling method.

With the advancement of science and artificial intelligence, education is experiencing significant innovation. The adaptive learning system is emerging as a promising approach to achieving personalized learning. The cognitive model plays a crucial role as the fundamental rationale behind the adaptive learning system. Currently, there is no uniform and highly operational method for constructing cognitive models. This study adopts Interpretive Structural Modeling (ISM) as the foundational approach for constructing a cognitive model of solid geometry. Based on literature and expert opinions, 17 cognitive attributes of high school solid geometry were identified. Subsequently, a questionnaire survey involving 40 experts was conducted to establish the contextual relationships among these attributes. Applying the ISM method resulted in a seven-level model. This model was then revised based on expert opinions to create the final cognitive model, revealing three primary paths within the domain of high school solid geometry. This paper contends that the use of the ISM method for constructing cognitive models is effective and objective. The resulting cognitive model unveils the content structure of high school solid geometry, and provides an innovative perspective on the construction of cognitive models.

A review of learning analytics opportunities and challenges for K-12 education.

Use of learning analytics to improve educational outcomes is a growing area of research. While learning analytics research has been more prevalent in higher education contexts, this study presents findings from a qualitative metasynthesis of 47 research publications related to opportunities and challenges relevant to learning analytics design, implementation, and research at the PK-12 level. Our findings indicate that, while many see the educational benefits of learning analytics (e.g., more equitable instruction, individualized learning, enhanced assessment for learning); others remain unconvinced by a lack of evidence of improved outcomes and concerned about persistent challenges and potentially harmful impacts (e.g., infringement on users' privacy, misuse or misinterpretation of data by educators). We conclude by considering implications for mathematics education stemming from our analysis of learning analytics and posing questions to shape future research and key developments in mathematics education as learning analytics become more prevalent.

O Homem Que Calculava em Libras: Tradução, Contação de História e Matemática / The Man Who Counted in Libras: Translation, Storytelling and Mathematics

RESUMO: Esta pesquisa teve por objetivo compreender os desafios e as possibilidades de tradução e contação em Língua Brasileira de Sinais (Libras) das histórias presentes no livro O homem que Calculava, do escritor modernista brasileiro Malba Tahan. Tratase de uma investigação qualitativa que se debruçou sobre o processo de tradução audiovisual das histórias presentes no livro, as adaptações feitas em suas narrativas, os recursos de preparação para a produção e edição dos vídeos, a implementação de recursos gráficos e visuais e, por fim, sua divulgação como produto da pesquisa. As análises indicam o potencial didático que emerge na interação entre os envolvidos no processo de tradução das histórias: professores, intérpretes e discentes (surdo e ouvinte), possibilitando formas de expressão artísticas que exploram a visualidade inerente da língua de sinais aliada a recursos gráficos, tomando as narrativas como elemento central para o desenvolvimento criativo ancorado no ensino da Matemática por meio da leitura e da interpretação de histórias.

ABSTRACT: This research aimed to understand the challenges and possibilities of translation and storytelling in the Brazilian Sign Language (known by the acronym LIBRAS) of the stories present in the book O homem que calculava [The man who counted] by the Brazilian modernist writer Malba Tahan. This is a qualitative investigation that focused on the audiovisual translation process of the stories present in the book, on the adaptations made in its narratives, the preparation resources for the production and editing of the videos, the implementation of graphic and visual resources and, finally, its dissemination as a research product. The analyzes indicate the didactic potential that emerges in the interaction between those involved in the process of translating the stories: teachers, interpreters and students (deaf and hearing), enabling forms of artistic expression that explore the inherent visuality of sign language combined with graphic resources, taking the narratives as a central element for the creative development anchored in the teaching of Mathematics through the reading and interpretation of stories.

John and Eliza Ware Rotch Farrar: A dual-career marriage in sickness and in health-but mostly sickness.

The story of John Farrar (1779-1853) and Eliza Ware Rotch (1791-1870) is neither a tale about a female mathematician nor one of the "making" of a mathematical career. Rather, the productivity of John, who is known to historians of American mathematics for the Harvard College series of mathematics and natural philosophy textbooks that bears his name, had already begun to decline by the time he married Eliza in 1828. She then embarked on her own career of adapting novels and biographies for children and writing self-improvement manuals. Despite their joint efforts to continue writing and to preserve their social and professional connections, the activities of both John and Eliza were effectively brought to an end by the decades John spent fully incapacitated by illness. An exploration of how their marriage was interwoven with their careers adds details to their biographies and suggests connections to the historical analysis of dual-career marriages and to examinations of how such unions can be impacted by the poor health of one or both partners.

International comparative study of learning trajectories based on TIMSS 2019 G4 data on cognitive diagnostic models.

Learning trajectory describes the student's learning progress or steps in one area through which teaching and learning can be linked. The cognitive diagnostic model (CDM) is an emerging evaluation theory in education measures. Researchers can determine students' mastery of fine-grained knowledge points by describing the learning trajectory based on CDM. The present study is based on the Trends in International Mathematics and Science Study (TIMSS) 2019, particularly, 21 mathematical items in the fourth grade. It analyzes the response data of 2,854 students from 17 countries from 10 attributes based on CDM. This study explores students' different learning trajectories in the content field by analyzing the relationships between knowledge states and attribute patterns. The study found that the 17 countries differ in learning trajectories but have commonalities. The learning starting points of 17 countries can be roughly divided into two categories and geometry attribute and fractions and decimals attribute are the last two attributes to master.

Analyzing teacher-student interactions through graph theory applied to hyperscanning fNIRS data.

Teacher-student relationships have been found consistently important for student school effectiveness in mathematics in the last three decades. Although this observation is generally made from the teacher's perspective, neuroscience can provide new insights by establishing the neurobiological underpinning of social interactions. This paper further develops this line of research by utilizing graph theory to represent interactions between teachers and students at the neural level. Through hyperscanning with functional near-infrared spectroscopy (fNIRS), we collected data from the prefrontal cortex and the temporoparietal junction of 24 dyads composed of a teacher and a student. Each dyad used a board game to perform a programming logic class that consisted of three steps: independent activities (control), presentation of concepts, and interactive exercises. Graph theory provides results regarding the strength of teacher-student interaction and the main channels involved in these interactions. We combined graph modularity and bootstrap to measure pair coactivation, thus establishing the strength of teacher-student interaction. Also, graph centrality detects the main brain channels involved during this interaction. In general, the teacher's most relevant nodes rely on the regions related to language and number processing, spatial cognition, and attention. Also, the students' most relevant nodes rely on the regions related to task management.

Exploring the potential of eye tracking on personalized learning and real-time feedback in modern education.

Eye tracking is one of the techniques used to investigate cognitive mechanisms involved in the school context, such as joint attention and visual perception. Eye tracker has portability, straightforward application, cost-effectiveness, and infant-friendly neuroimaging measures of cognitive processes such as attention, engagement, and learning. Furthermore, the ongoing software enhancements coupled with the implementation of artificial intelligence algorithms have improved the precision of collecting eye movement data and simplified the calibration process. These characteristics make it plausible to consider eye-tracking technology a promising tool to assist the teaching-learning process in school routines. However, eye tracking needs to be explored more as an educational instrument for real-time classroom activities and teachers' feedback. This perspective article briefly presents the fundamentals of the eye-tracking technique and four illustrative examples of employing this method in everyday school life. The first application shows how eye tracker information may contribute to teacher assessment of students' computational thinking in coding classes. In the second and third illustrations, we discuss the additional information provided by the eye-tracker to the teacher assessing the student's strategies to solve fraction problems and chart interpretation. The last illustration demonstrates the potential of eye tracking to provide Real-time feedback on learning difficulties/disabilities. Thus, we highlight the potential of the eye tracker as a complementary tool to promote personalized education and discuss future perspectives. In conclusion, we suggest that an eye-tracking system could be helpful by providing real-time student gaze leading to immediate teacher interventions and metacognition strategies.

Decoding of spatial proportions using somatosensory feedback in sighted and visually impaired children.

BACKGROUND AND PURPOSE: Humans can naturally operate with ratios of continuous magnitudes (proportions). We asked if sighted children (S) and visually impaired children (VI) can discriminate proportions via somatosensory feedback. PROCEDURES: Children formed a proportion by tracing a pair of straight lines with their finger, and compared this proportion with a second proportion resulting from the tracing of another pair of lines. MAIN FINDINGS: Performance was 68% in S, thus significantly lower (p < 0.001) compared to VI (75%). Tracing velocity (p < 0.01) and trial-to-trial variability of tracing velocity (p < 0.05) was higher in S compared to VI. CONCLUSIONS: Operating with proportions solely from somatosensory feedback is possible, thus tracing lines might support learning in mathematics education. Kinematic variables point to the reason for the difference between S and VI, in that higher trial-to-trial variability in velocity in S leads to biased estimation of absolute line lengths.

Disruptiveness of COVID-19: Differences in Course Engagement, Self-Appraisal, and Learning.

We investigated how the transition to remote instruction during the COVID-19 pandemic affected students' engagement, self-appraisals, and learning in advanced placement (AP) Statistics courses. Participants included 681 (Mage=16.7 years, SDage=.90; %female=55.4) students enrolled in the course during 2017-2018 (N=266), 2018-2019 (N=200), and the pandemic-affected 2019-2020 (N=215) school years. Students enrolled during the pandemic-affected year reported a greater improvement in affective engagement but a decrease in cognitive engagement in the spring semester relative to a previous year. Females enrolled in the pandemic-affected year experienced a greater negative change in affective and behavioral engagement. Students enrolled during the pandemic-affected year reported a greater decrease in their anticipated AP exam scores and received lower scores on a practice exam aligned with the AP exam compared to a prior year. Although students were resilient in some respects, their self-appraisal and learning appear to have been negatively affected by pandemic circumstances.

A Network Model for Connecting Mathematics Faculty in Communities of Practice: Where is the Value?

Creating sustained, transformative change within and across organizations is challenging, particularly when those undertaking change act as individuals. COMmunities of Practice (CoPs) are organically created collaborations among like-minded participants, working toward a common set of goals (Lave & Wenger, 1991; Wenger-Trayner & Wenger-Trayner, 2014). CoPs offer an avenue for members to connect individuals across various boundaries. In this paper, we investigate the ways in which regional CoP leaders experience value participating in their community, using the Communities for Mathematics Inquiry in Teaching (COMMIT) Network as our unit of study. The COMMIT Network is a grant funded project aimed at engaging mathematics faculty at institutions of higher education in regional CoPs around teaching with inquiry. In this study we examine the experiences of CoP leaders nested within this network setting. We interviewed 19 leaders from eight United States regions to understand their perceptions of individual and collective value participating in the regional CoP and COMMIT Network structures. We framed our study on Wenger et al. (2011). Promoting and assessing value creation in communities and networks: A conceptual framework. Open University of the Netherlands.) Value Framework. Our findings show that leaders found Immediate Value as individuals participating in a collaborative, supportive CoP environment and they found Realized Value in terms of the impact their CoP could make on instructional practices, both in their region and the network. An unexpected finding examines how future opportunities for value creation may influence long-term sustainability and transformation of college mathematics instruction. We provide implications for the ways that regional CoPs, along with CoP networks, can provide value for members through such communities.

Trayectorias de la enseñanza de las matemáticas en ingeniería: entornos colombiano y norteamericano

RE S U M E N Este texto presenta una revisión sistemática de artículos científicos, tipo scoping review, sobre educación matemática en programas de ingeniería, en los entornos colombiano y norteamericano, durante el periodo 2000 - 2020. La exploración, surgida del proyecto de investigación "Fundamentos teóricos y metodológicos de la enseñanza de la matemática en ingeniería", siguió los protocolos del Instituto Joanna Briggs y consideró literatura especializada incluida en Scopus y Publindex. La información, clasificada como "experiencia pedagógica", "reflexión" o "revisión", permitió describir tendencias de investigación y acción respecto a la enseñanza, el aprendizaje, el currículo y la evaluación de las matemáticas en ingeniería. Los resultados permitieron concluir que en el contexto norteamericano existe un interés significativo por la investigación sobre el aprendizaje, el currículo y la evaluación de las matemáticas en ingeniería, desde enfoques sistêmicos, interdisciplinarios y socioculturales. En contraste, la literatura sobre el tema en Colombia presenta una tendencia al estudio de procesos de enseñanza, desde perspectivas disciplinares y cognitivas de la educación matemática.

This paper presents a systematic review of scientific articles, scoping review type, on mathematics education in engineering programs, in the Colombian and North American environments, during the period 2000 - 2020. The exploration, which arose from the research project "Theoretical and methodological foundations of mathematics education in engineering", followed the protocols of the Joanna Briggs Institute and considered specialized literature included in Scopus and Publindex. The information, classified as "pedagogical experience", "reflection" or "review", allowed describing research and action trends regarding teaching, learning, curriculum and assessment of engineering mathematics. The results allowed concluding that in the North American context there is a significant interest in research on learning, curriculum and assessment of engineering mathematics, from systemic, interdisciplinary and sociocultural approaches. In contrast, the literature on the subject in Colombia presents a tendency to study teaching processes from disciplinary and cognitive perspectives of mathematics education.

Este texto apresenta uma revisão sistemática de artigos científicos, tipo revisão de escopo, sobre educação matemática em programas de engenharia, nos ambientes colombiano e norte-americano, durante o período de 2000 a 2020. A exploração, decorrente do projeto de pesquisa "Fundamentos teóricos e metodológicos da educação matemática em engenharia", seguiu os protocolos do Instituto Joanna Briggs e considerou literatura especializada incluída no Scopus e no Publindex. As informações, classificadas como "experiência pedagógica", "reflexão" ou "revisão", permitiram descrever as tendências de pesquisa e ação em relação ao ensino, aprendizagem, currículo e avaliação da matemática de engenharia. Os resultados levaram à conclusão de que no contexto norte-americano há um interesse significativo na pesquisa sobre aprendizagem, currículo e avaliação de matemática de engenharia a partir de abordagens sistêmicas, interdisciplinares e socioculturais. Em contraste, a literatura sobre o assunto na Colômbia mostra uma tendência a estudar os processos de ensino a partir de perspectivas disciplinares e cognitivas da educação matemática.

Development, testing, and application of a mathematics learning scale of self-direction.

Many countries' curriculum reforms focus on developing the next generations' competencies of self-directed learning (SDL) to address rapid social changes and sustainable environmental development. Taiwan's curriculum reform corresponds with the global trend in education. The latest curriculum reform, which proposed a 12-year basic education, was implemented in 2018 and included SDL explicitly in its guidelines. The reformed curriculum guidelines have been followed for over 3 years. Thus, it is necessary to conduct a large-scale survey to examine its impact on Taiwanese students. However, existing research instruments help provide a generalized analysis of SDL and have yet to be designed specifically for SDL of mathematics. Therefore, we developed a mathematics SDL scale (MSDLS) and examined its reliability and validity in this study. Subsequently, MSDLS was utilized to investigate Taiwanese students' SDL of mathematics. The MSDLS consists of four sub-scales with 50 items. It has acceptable reliability, validity, and measurement invariance across gender and grade groups. The MSDLS was administered online to 5,575 junior high school students, and 5,456 valid responses were collected. The findings highlight the gender and grade differences in SDL of mathematics. Male students are higher than female students in many factors. It is noted that the SDL in mathematics does not increase with grade. In sum, the MSDLS is a helpful instrument for examining secondary school students' SDL of mathematics.

Conceptualizing teachers' interactions with resources in crossing languages and cultures.

This paper introduces this special issue aiming to deepen and extend research on mathematics teachers' work, from a resource perspective, by taking language and culture into account, and exploring two questions: How are teachers' interactions with resources interpreted and modeled across contexts? And, What challenges and insights emerge through recent efforts to engage these models in cross-cultural (and linguistic) research? The fields of resources, language and culture in mathematics education are each extensive, and we do not attempt to survey comprehensively across them. We have chosen instead to propose three approaches on resources in mathematics teachers' work that developed somewhat contemporaneously from three different countries with differing linguistic, curricular, and social contexts, corresponding to the work of the three guest editors. The models developed through these approaches are driven by the educational, and so cultural and material conditions of the time and the location of each author, and allow us to propose preliminary answers to our two guiding questions. We then move to pull the threads from these models together, and discuss the contributions to this Special Issue. This results in more robust and nuanced responses to our questions, and in identifying two themes that emerge from research that sit at the convergence of studies of teachers' interactions with resources, languages, and cultures: an invisibility-visibility dialectic and a local-global tension. Finally, this study leads us to consider a new region of mathematics education research.

The Impacts of Three Educational Technologies on Algebraic Understanding in the Context of COVID-19.

The current study investigated the effectiveness of three distinct educational technologies-two game-based applications (From Here to There and DragonBox 12+) and two modes of online problem sets in ASSISTments (an Immediate Feedback condition and an Active Control condition with no immediate feedback) on Grade 7 students' algebraic knowledge. More than 3,600 Grade 7 students across nine in-person and one virtual schools within the same district were randomly assigned to one of the four conditions. Students received nine 30-minute intervention sessions from September 2020 to March 2021. Hierarchical linear modeling analyses of the final analytic sample (N = 1,850) showed significantly higher posttest scores for students who used From Here to There and DragonBox 12+ compared to the Active Control condition. No significant difference was found for the Immediate Feedback condition. The findings have implications for understanding how game-based applications can affect algebraic understanding, even within pandemic pressures on learning.

Influence of primary students' self-regulated learning profiles on their rating of a technology-enhanced learning environment for mathematics.

The importance of learning technologies for mathematics education is increasing as new opportunities arise for mathematics education for all students, in school and at home. These so-called technology-enhanced learning environments (TELEs) incorporating technology with mathematical content are useful for developing mathematical knowledge and can simultaneously foster self-regulated learning (SRL) and motivational learning in mathematics. However, how do primary students' differences in their SRL and motivation affect their rating of the quality of mathematical TELEs? To answer this research question, we asked third and fourth-grade primary students (n = 115) to evaluate both their SRL, including metacognition and motivation, and the quality characteristics of the ANTON application, a frequently and intensively used TELE in Germany. Using a person-centered research approach by conducting a cluster analysis, we identified three SRL profiles of primary students-motivated self-learners, non-motivated self-learners, and average motivated non-self-learners-who differ in their ratings of the quality characteristics of the TELE (output variables). Our results highlight that motivated self-learners and non-motivated self-learners vary significantly in their rating of the adequacy of the TELE to their mathematical learning and highly but not significantly concerning the TELE's reward system. Moreover, differences existed between the motivated self-learners and the average motivated non-self-learners regarding their rating of the characteristic differentiation. Based on these findings, we assume that technical elements associated with adequacy, differentiation, and rewards of mathematical TELEs should be tailorable to the needs of individuals and groups of primary schoolchildren.

Understanding the complexities of mathematical cognition: A multi-level framework.

Mathematics skills are associated with future employment, well-being, and quality of life. However, many adults and children fail to learn the mathematics skills they require. To improve this situation, we need to have a better understanding of the processes of learning and performing mathematics. Over the past two decades, there has been a substantial growth in psychological research focusing on mathematics. However, to make further progress, we need to pay greater attention to the nature of, and multiple elements involved in, mathematical cognition. Mathematics is not a single construct; rather, overall mathematics achievement is comprised of proficiency with specific components of mathematics (e.g., number fact knowledge, algebraic thinking), which in turn recruit basic mathematical processes (e.g., magnitude comparison, pattern recognition). General cognitive skills and different learning experiences influence the development of each component of mathematics as well as the links between them. Here, I propose and provide evidence for a framework that structures how these components of mathematics fit together. This framework allows us to make sense of the proliferation of empirical findings concerning influences on mathematical cognition and can guide the questions we ask, identifying where we are missing both research evidence and models of specific mechanisms.