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Over the past two decades, the landscape of research on mathematics teacher educators (MTEs) has grown considerably. One particular area of interest has focused on MTE knowledge and the ways in which it is developed and used in teaching practice. However, studies have conceptualized MTE knowledge in different ways and have employed considerably different methodologies and approaches to its study. In an effort to understand this varied landscape, we conducted an extensive review of research on the nature and development of MTE knowledge. This review provides a broad descriptive analysis of the existing theoretical and empirical research on MTEs' knowledge, explores the theoretical underpinnings of the existing frameworks for and studies on MTE knowledge, and considers implications for future research. [ABSTRACT FROM AUTHOR]

There exists a substantial body of research evidence that embodied learning modes can enhance children's mathematics learning, yet implementation of the research findings in classrooms has so far been limited. This paper argues the need for translational research to bridge the gap between psychological, neuroscientific, and narrow-focused educational research and teaching practice in schools. As a foundation for such research, and with a focus on early year mathematics education (4 to 8 years), we derive a set of 12 principles of embodied learning from the research literature by applying an integrative literature review method. Based on the findings from the literature, we offer some advice on how these principles might be put into practice with young students. [ABSTRACT FROM AUTHOR]

Foundational thinking for later use of technology, particularly coding, is necessary for an inclusive and sustainable future. Inclusive practices beginning in early childhood recognize children's innate development of computational thinking—sequencing, repetition and looping, debugging, decomposing and composing, representation, and causality. This qualitative research describes processes of developing and evaluating hypothesized developmental progressions. Inclusive engagement of children with and without disabilities is described in examples for each level of each developmental progression. Implications for teaching and learning with inclusive practices are described for children with and without disabilities. [ABSTRACT FROM AUTHOR]

This manuscript explores the integration of spatial reasoning into K-12 education through the lens of Joyce VanTassel-Baska's Integrated Curriculum Model (ICM) and a talent development approach. It emphasizes the significance of nurturing spatial reasoning components—visualization, rotation, orientation, and perception—for academic success, problem-solving, and career prospects, especially for academically talented students. Through collaboration with talent development strategies, including mentorship and differentiated instruction, this approach provides a dynamic, real-world-focused learning experience. Inclusive and designed to engage the educational community, it aims to prepare students for complex global challenges, highlighting the transformative power of tailored education in developing versatile societal contributors. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Research in Education Sciences is the property of National Taiwan Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This study was to investigate the relationship between metacognition and the mathematical modeling skills of high school students, as well as the mediating role of computational thinking. A cluster sampling method was adopted to investigate 661 high school students, using the metacognition scale, computational thinking scale, and mathematical modeling skill test questions. The results showed that metacognitive knowledge and metacognitive monitoring had a direct and positive correlation with high school students' mathematical modeling skills. Additionally, the critical thinking dimension of computational thinking mediated the relationship between metacognitive knowledge, experience, monitoring, and mathematical modeling skills. These findings indicated that sufficient metacognition could improve the critical thinking of high school students' computational thinking and enhance their mathematical modeling skills. [ABSTRACT FROM AUTHOR]

In this study, 176 Turkish seventh-grade students were asked to pose three problems for each of the swimming and purchasing tasks to examine how the solvability and complexity of the problems changed depending on the order of responses. The purchasing task required to pose a problem to derive a specific answer before posing new problems, whereas the swimming task required to pose problems based on existing data. The findings indicated that students performed poorly in terms of posing solvable problems in both tasks. The majority of solvable responses in the purchasing task were at the high complexity level, but most in the swimming task were at the low complexity level. No evidence of a relationship between the order of the responses and any of the analysis units (solvability and complexity) was found. Furthermore, when comparing the groups that posed and did not pose solvable problems for the specific answer in the purchasing task, the former group had a stronger tendency to maintain the complexity points in their other responses compared to their first response. [ABSTRACT FROM AUTHOR]

Using an in-depth case study approach, this study explored the mathematics educational values held by two in-service teachers at the primary and secondary levels in Hong Kong through lesson observations and deep structured interviews. The findings revealed that the primary teacher emphasized fun, flexibility, critical thinking, and coherence, while the secondary teacher prioritized achievement, control, higher-order thinking, and progression. Using Seah and Bishop's five complementary pairs framework, the primary teacher aligned with an activist, relational, relevance, and accessibility perspective, contrasting sharply with the secondary teacher. This reflected a shift in values in response to students' educational stages, revealing the challenges of balancing social expectations and nurturing future science, technology, engineering, and mathematics talent. We discussed the implications for local education reforms, including modernizing assessment methods and curricula, reshaping the social climate, and updating professional development to address these tensions. [ABSTRACT FROM AUTHOR]

Starting early in life, children, especially girls, experience obstacles when it comes to developing interest in STEM. Although teachers face an important task in promoting girls (and boys) in STEM, they often encounter hurdles in doing so. A three-month-long training for pre-and in-service teachers in elementary education was developed to counter this phenomenon. An important training feature was teaching ideas for STEM classrooms. Teachers' evaluation of the training and teaching ideas, changes in their self-concept, and elementary students' assessment of the teaching ideas were investigated. Students rated the teaching ideas favorably, with no gender differences. Even though each idea incorporated relevant didactic features for gender-sensitive STEM instruction, the teachers rated certain ideas and contents more critically than others. Nevertheless, their assessments speak in favor of the training intervention, while also indicating gaps in teachers' professional knowledge regarding gender-sensitive didactics. Implications for the design of STEM teacher education are outlined. [ABSTRACT FROM AUTHOR]

Ninth grade on-track is predictive of high school graduation, more than race, socio-economic status, and prior achievement combined. Although initiatives characterized by an intense focus on the ninth-grade year are being increasingly implemented, research has not fully documented and tested mechanisms linked to improved outcomes. Using survey and transcript-level data and causal mediation analysis, this study tests the effects of students attending high teacher efficacy (self and collective—TSE and CTE) schools on ninth grade on-track in an urban school district in a northeast state in the United States. It further examines the extent to which ambitious instructional practices, defined as culturally relevant and transformative pedagogy, mediate the effects of TSE on ninth grade on-track and how levels of supportive school culture moderate these relationships. The findings indicate that urban ninth-graders attending schools with high TSE and CTE are more likely to be on track at the end of ninth grade. Additionally, the direct effect of students attending a school with high TSE was mediated by the level of ambitious instruction. We discuss implications for teacher education (TE) and professional development. [ABSTRACT FROM AUTHOR]

In early mathematics education, the beliefs of the teacher are essential for facilitating the integration of technology into teaching mathematics. This study explores the influence of physical and digital interactive learning environments on the development of early childhood teachers' beliefs about integrating technology into early mathematics classrooms. To understand the development of teachers' beliefs, a half-year, content-based professional development program that incorporates interactive digital and physical environments was designed for this research on early childhood mathematics teachers (N = 58). We administered a questionnaire to measure teachers' beliefs regarding employing an interactive technological environment in mathematics classrooms before and after engaging in an intervention program. In addition, a reflective writing strategy was implemented with the participants (N = 10) to understand the development of their beliefs about technology via an evaluation of their levels of reflection. In general, the research findings indicate that an interactive learning environment supports an improvement in teachers' beliefs about technology. Furthermore, positive qualitative findings were drawn from the reflective writing essays of early childhood teachers. The qualitative findings indicate that an interactive environment enables teachers to be conscious of selecting effective math-specific technologies that facilitate children's exploration and discovery of mathematics concepts. [ABSTRACT FROM AUTHOR]

Little is discussed regarding the meaning of "definition" in primary school, where children often memorize definitions. In geometry, students frequently confuse "definition" with the "description" of a geometric object. The aim of this study is to verify whether a new hybrid methodology between storytelling and game-based learning called Geometrikoland exercises skills that fall within what we understand as "definitory thinking", which the set of competencies and meta-competencies is associated with acquiring a true mastery of definitions in geometry. This goal can be achieved because the dynamics of Geometrikoland alter the approach to definitions. The methodology applied is a hybrid of storytelling and adventure game-based learning applied to a "engine" of quadrilateral theory. The experiment took place in a fifth-grade class in an Italian school. The analysis of some data based on the accuracy of geometry actions during workshops and qualitative analysis based on the analysis of oral arguments produced during the labs lead to the conclusion that, on average, children have gained a greater awareness of what it means to define a geometric object and a better understanding of the relationships between various subsets of the set of quadrilaterals, previously seen as separate sets. Further confirmation has been provided by comparing the results of two mini-tests (a placement test and an exit test), which are useful for assessing the skills associated with the criterion of arbitrariness, the criterion of uniqueness, and classificatory thinking. [ABSTRACT FROM AUTHOR]

According to contemporary research, there exists an imbalance within the disciplines of Science, Technology, Engineering, and Mathematics (STEM), wherein certain subjects are lacking representation due to the neglect or omission of mathematical elements. The purpose of this study is to address this issue through the analysis of an established learning sequence that has been well-tested to promote mathematical skills. We adapted the selected biology-based learning sequence to foster mathematical problem solving and conducted it with a school class. The qualitative analysis of the recorded video footage of this adapted learning sequence revealed that the modified task effectively stimulated mathematical problem-solving skills. This successful adaptation demonstrates one approach through which mathematics can be strengthened and effectively utilized in STEM subjects. [ABSTRACT FROM AUTHOR]

Mathematical modelling involves a set of interconnected skills that allow students to translate real-world situations into mathematical representations and vice versa. This research aimed to develop the instrument consisting of sub-competency; simplifying, mathematising, computing, interpreting and validating. The study utilized a cross sectional survey research design. A total of 135 pre-service teachers of mathematics voluntarily selected using convenience sampling methods, participated in this study. The data was analysed by item-CVI (I-CVI), S-CVI and exploratory factor analysis (EFA). This tool's preliminary versions demonstrated good content validity for both the individual items and overall, for STEM-based material (S-CVI/UA = 0.88; S-CVI/Ave = 0.93). The Kappa value was K = 0.88, indicating an excellent value of validity. The development of STEM-Based modelling instrument revealed great item-content validity and scale-validity for assessing sub competency in pre-service teachers. At the same time, EFA revealed that STEM-based modelling instrument had five sub-components; simplifying, mathematising, computing, interpreting and validating. The results showed that the STEM-based modelling instrument's reliability was good. Creating and verifying the STEM-focused modelling tool designed for preservice teachers is essential in mathematics education and research. It offers a valuable means to evaluate and improve the essential skillset related to mathematical modelling, benefiting educators and STEM students alike. Moreover, the research's approach and discoveries have the potential to influence forthcoming investigations and educational methodologies across diverse fields. [ABSTRACT FROM AUTHOR]

This study sought to examine the kinds of mindsets that the participating teachers demonstrated in their teaching and the impact of teachers' mindsets on their ways of teaching mathematics and science. Seven teachers who were teaching science or mathematics in high-needs schools in the urban context of the Bronx, New York were the participants of the study, with interview data and analysis provided for three participants representing the low, medium, and high mindset groupings. Qualitative research method was used to analyze the data. The data were collected through three different questionnaires, lesson observations, and interviews. Data analysis revealed that six participating teachers demonstrated a growth mindset with different degrees and one teacher seemed to have a fixed mindset. Teachers with a growth mindset adopted instructional practices that are more aligned with those that are recommended in the growth mindset literature. [ABSTRACT FROM AUTHOR]

In the large body of literature that is available about the role of motivation in mathematics education, the focus is mostly on students' attitudes and feelings toward mathematics in general. In the current small-scale explorative study, we zoomed in on what students think about the tasks they can come across in mathematics education. To investigate students' appraisement of mathematics tasks, that is, their judgment of how much they like mathematics tasks, we used a mixed method approach with an online questionnaire including closed and open questions. We asked 67 Norwegian eight graders to give an appraisal score for 24 tasks and to express in their own words their reasons for liking/disliking them. In addition, the students also had to indicate for each task whether they think they can solve the tasks. The analysis of the data indicated that the students may prefer bare number problems over context problems. Similarly, students inclined to like puzzle-like tasks more than estimation or straightforward tasks. The reasons for liking/disliking a task most often refer to the difficulty or easiness of the task. The data about the perceived solvability revealed that the more the students believed that they can solve the task, the more they liked the task. Given the different appraisals that the students granted to the different task types and the comments they came with, there is much reason to afford students a stronger voice in mathematics education research, particularly when it is about the tasks used in mathematics education. [ABSTRACT FROM AUTHOR]

Lesson study is recognised as a cultural activity and its implementation in a context different from Japan is a complex process. Researchers' role in this process is assumed to be critical, although this has rarely been investigated. In this paper, we analyse a teaching experiment to introduce lesson study into a professional development course for prospective teachers at an Italian university, focusing our investigation on a group of researchers acting as teacher educators (didacticians). Using the anthropological theory of the didactic and meta-didactical transposition frameworks, we investigate their dual position as researchers and as teacher educators. We observe the evolution of the didacticians' teacher-education praxeology (a model of practice and knowledge) during their interactions with prospective teachers. The results indicate that the didacticians' teacher-education praxeology is distinguished, shaped, and actively influenced by their research praxeology. The results also imply that coordinating the two theoretical frameworks may guide the design and analysis of teachers' professional development courses. [ABSTRACT FROM AUTHOR]

Previous research has shown that language is an essential part of the development of mathematical skills and, specifically, in solving verbal problems. We know that using a different language from one's mother tongue is highly beneficial for students, and that language plurality and bilingualism is more and more habitual and present in current educational environments. However, what is still not clear is how it influences certain tasks, especially the most naturalistic ones, such as the formation of the concept of numbers and in tasks with a greater verbal component, such as solving problems in the early ages. The present research examined the problem-solving performance of first- and second-grade elementary education students in bilingual environments, comparing the problem solving of students whose language of instruction (LI) is the same as their mother tongue (MT) and those whose MT differs from the LI. Through an analysis of variance, the results showed that there exist differences in change and combination problems. Discrepancies in performance were also found, depending on the evolutionary moment. These results suggest that it is necessary to study how the LI can shape mathematical skills in the early years. [ABSTRACT FROM AUTHOR]

This systematic review examines 31 articles published from 2000 to 2021, comprehensively addressing mathematical resilience. These articles originate from English-language journals indexed in Google Scholar, Web of Science, ERIC, and Scopus databases. The analysis considered specific aspects including years, countries, research methods, designs, sample characteristics, data collection instruments, analysis techniques, findings, and recommendations. The majority of studies were conducted in 2021, mainly in Indonesia and the United Kingdom, with a focus on high school students. Qualitative approaches predominated, particularly qualitative descriptive research, along with quantitative approaches featuring semi-experimental designs and correlational studies. Quantitative studies commonly employed mean and frequency tables, while surveys and interviews were frequently utilized data collection tools. These investigations revealed diverse dimensions of mathematical resilience, spanning cognitive, affective, pedagogical, demographic, and social domains. The studies offer valuable recommendations for educators, researchers, teacher education programs, learning environments, students, school administration, psychological counsellors, parents, and society at large. [ABSTRACT FROM AUTHOR]

The teachers' beliefs in ethnomathematics-based numeracy learning are a crucial element to be able to develop students' numeracy skills. The teachers' beliefs in developing students' numeracy skills through ethnomathematics-based numeracy learning is a teacher's personal view about certain practices that are considered correct, which include four dimensions. The first dimension is the teachers' beliefs about numeracy. The second dimension is teachers' beliefs about the role of ethnomathematics in learning numeracy. The third dimension is teachers' beliefs about ethnomathematics-based numeracy teaching. The fourth dimension is teachers' beliefs about ethnomathematics-based numeracy learning. This study aimed to investigate the characteristics of teachers' beliefs in developing students' numeracy skills through ethnomathematics-based numeracy learning. The teacher's response was acquired through the completion of a questionnaire. The findings indicated that most teachers believed fostering students' numeracy skills could be achieved by implementing ethnomathematics-based numeracy learning inside the semi-realisticmechanistic (SRM) categories. Further analysis reveals that more extended teaching experience may mean something other than that a teacher has a realistically oriented outlook. Teachers with teaching experience of fewer than five years (new teachers) and 5 to 10 years (junior teachers') have a semi-realisticmechanistic (SRM) view. Most teachers with more than 10 to 15 years (semi-senior-teachers) of teaching experience have a dominant realistic (DR) idea. In contrast, teachers with more than 15 years (senior teachers) of teaching experience have a semi-realistic-mechanistic (SRM) view and a dominant realistic (DR) view. Teachers' positive beliefs will likely improve students' numeracy skills in general and based on teaching experience in the dominant realistic (DR) and realistic (R) categories regarding ethnomathematics-based numeracy learning. [ABSTRACT FROM AUTHOR]

With increasing attention on the potential overlap between computational thinking (CT) and mathematical reasoning, STEM education researchers seek to understand how integrating CT and mathematics can deepen student learning across disciplines. Although there are various professional development programs that introduce teachers to CT concepts and strategies for curriculum integration, limited research exists on how teachers might apply this knowledge to create math + CT activities for use with their students. Additionally, the majority of research on CT integration through programming has focused on upper elementary grades, leaving the early grades (K-2) relatively unexplored. This qualitative exploratory study aims to examine how teachers in a graduate STEM education program collaborated with university STEM faculty to explore and critique a set of integrated math + CT block-based programming activities designed to build place value conceptual understanding. In-service elementary teachers enrolled in an online graduate CT course for educators (n = 13) explored these activities as learners and drew on their experiences as classroom teachers to offer feedback for program redesign. A sequence of deductive pattern coding and inductive holistic coding of course transcripts, collaborative problem-solving slides, and individual teacher reflections provided insights into how teachers were able to establish connections between their mathematical knowledge related to teaching place value and their emerging understanding of CT concepts, such as abstraction, algorithms, decomposition, and debugging. Implications for the design of professional development for elementary teachers on integrating CT and mathematics are offered. [ABSTRACT FROM AUTHOR]

We articulate the principles behind our practices as teacher educators for an Initial Teacher Education (ITE) provider in South-West England working in partnership with local schools. We position them as 'braces' that enable us not only to withstand but 'embrace' the challenges of the current policy environment, which imposes braces of its own. All ITE providers in England must present their curriculum and teaching plans for approval by the Department for Education, one of many significant changes to ITE required in England over the last decade. Furthermore, for such plans to be approved they must assume a model of schooling based on knowledge acquisition in tension with not only our own motivations but also a broader (global) conversation among teacher educators critical of policy priorities for schooling and professional teacher formation that prioritise success in high-stakes tests and international league tables. The braces/principles emerge through articulating our collective understandings of and concerns with the policy braces. Our principles connect to the processes we feel confident are necessary aspects of 'becoming' teachers and can hold our Post Graduate Certificate in Education (PGCE) together, going forwards. The braces—becoming an informed educator; becoming an ethical actor; becoming a reflective practitioner—have a distinctive form of reflexivity at their heart and should apply to everyone across our partnership: us, as university-based teacher educators; school-based colleagues; and pre-service teachers. We offer our approach and the braces/principles we have developed to others also struggling to reconcile their own notions of good teacher education with national policy imperatives. [ABSTRACT FROM AUTHOR]

The study was to assess the algebraic thinking levels of prospective basic education teachers using SOLO taxonomy. Sequential explanatory mixed methods design was used for the study. 250 Pre-service mathematics teachers from four selected colleges of education in the Eastern region of Ghana were purposely sampled for the study. The data were collected with the aid of super item test and unstructured interviews as instruments. Descriptive statistics (means and standard deviations) and analysis of variance were used to analyse the quantitative data whereas thematic analysis was employed to analyse the qualitative data. The study revealed that the pre-service teachers showed good algebraic thinking at uni-structural and multi-structural levels of the SOLO taxonomy, but performed abysmally at relational and extended abstract levels. The study recommended that Mathematics tutors at the Colleges of Education in Ghana should use the SOLO model for teaching and evaluation of their students. This would help improve pre-service teacher algebraic thinking levels early enough for preservice teachers' overall progress in teaching mathematics after school. [ABSTRACT FROM AUTHOR]

Since the practice turn, the contemporary education landscape has been shifting from mere knowledge dissemination to empowering students to solve problems. Special emphasis is given to problems on which students work for an extended period (at least several hours; frequently multiple school days). While working on such problems, it is essential to employ a variety of activities. Two of these are working with models and evaluating models and their results. One topic that has received little attention up to now is the question of to what extent educators are able to apply these skills. This study, fundamentally exploratory in nature, seeks to delve into such an assessment by evaluating the competence of n = 20 educators in estimating and evaluating building evacuation duration using digital simulations. Our results show that the participants self-assessed as being able to solve such exercises. However, this was contrasted by our external assessment of the solutions provided by the participants, which showed that the solutions lacked in quality. [ABSTRACT FROM AUTHOR]

Problem posing and problem solving, involving significant situations for students, encourage effective learning, mathematical reasoning, communication, and connections in/with mathematics, enhancing the emergence of different literacies. Although problem solving is more frequently present in teaching practices, the ability to formulate them must also be developed, since it requires other skills, such as writing a problem statement, establishing connections between reality and mathematical knowledge, and creating and idealizing problematic situations, among others. Financial Literacy and Consumer Literacy promote contexts related to the student's daily life and are promising in the development of problem posing. These literacies are also relevant to the development of a healthy, balanced, and responsible relationship with money and to raising awareness of the student regarding the importance of their actions in society, the economy, and the environment. This text presents part of a qualitative, descriptive, and interpretive research, developed by a teacher of children in the third year of a Portuguese primary school, involving problem posing and problem solving in Financial and Consumer Education contexts, with the following research question: How does posing and solving problems, in contexts close to the students' reality, influence the development of different literacies? The results point out the mobilization of different mathematical concepts, in articulation with other areas of knowledge, and the development of skills and knowledge innate to different literacies, namely Mathematical Literacy, Financial Literacy and Consumer Literacy. [ABSTRACT FROM AUTHOR]

Copyright of Humanities & Educational Sciences Journal is the property of Humanities & Educational Sciences Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper describes the results of a study whose main aim was to find the interrelationship between pupils’ school grades in Czech language (native), mathematics, and physics and pupils’ cognitive predispositions to problem-solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. A total of 180 pupils from the Czech Republic aged 14-15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2 and between those assessed by grades 3 or 4. Pupils’ performance in the two tests might help to strengthen the objectivization of grading at school. They might also help identify the indicators important for developing problem-solving skills. The research specifically points to the need to develop algebraic thinking, the conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning, and the ability to control variables. [ABSTRACT FROM AUTHOR]

Mental computation strategies in addition on two-digit natural numbers used by preservice elementary teachers are assessed and analyzed in order to examine if in the research literature the mental computation strategies used by the students are adequately categorized. A written test is used to gather mental computation strategies used by preservice elementary teachers. The results indicate a few new strategies used by preservice elementary teachers not already covered by the research literature. On the other hand, the results indicate that there is a theoretical need to clarify some of the definitions of mental computation strategies in research literature. [ABSTRACT FROM AUTHOR]

In this study we first attempt to present relevant research and analyze the types of number lines, then present two studies examining components of number line, namely the direction and negative numbers based on historical evolution. Then a historical analysis of the evolution of number line is presented. After that, we present the difficulties encountered by students in using the number line as a representation tool, and through the historical evolution of the number line we try to contrast these difficulties with critical points in its historical evolution. 1st period. Mathematics up to Euclid: Separation of numbers and lines. In Greek mathematics there was a clear separation between number and magnitude. Numbers (natural numbers) were simply collections of discrete units that measured a multitude. Magnitude Size, on the other hand, was usually described as a continuous quantity divided into parts and is infinitely divisible. This distinction between number and magnitude also resulted in the distinction between arithmetic and geometry. Arithmetic dealt with a discrete or non-extended quantity, while geometry dealt with a continuous or extended quantity. It also imposed a different way of solving and handling many problems. 2nd period. Until the 16th Century: Foundations of Integers - Rational Numbers - Empirical Geometry From the 12th to the 16th century an orientation in empirical Geometry is observed, as well as its relation to computational methods and the use of measuring tools. Michael Stifel (1487-1567) was the first to define negative numbers as numbers less than zero and positive numbers as greater than zero. He was the first to describe zero, fractional and irrational numbers as numbers. In the 16th century a transformation began in the classical conception of number and magnitude. Francois Viete (1540-1603) introduced a new form of symbolism to denote unknown magnitudes and numbers, stating that numbers and magnitudes can be interchanged. This relationship between numbers and magnitudes encouraged the idea that numbers could also be treated as if they were continuous, in the Aristotelian sense of continuity. However, even during this period there is no concept of the number line and the identification of each point with a number. Nevertheless, the elements that make it possible to display it are beginning to emerge. 3rd period. From the 17th century to the beginning of the 19th century: The first connection of numbers and geometric lines - Algebraization of geometry Mapping lines of numbers was not a common idea among mathematicians until the end of the 16th century. However, the concept of the number line begins to emerge in the 17th century by some pioneering mathematicians. John Wallis (1685) was the first to use a number line in his book, Treatise of Algebra, in order to interpret addition and subtraction with negative numbers. The development of algebraic symbolism and the connection of curves to their equations led Descartes to the algebraization of geometry with the help of the coordinate system. Descartes does not use the terms abscissa, ordinate or axis in his work. Descartes did not introduce the number line through the discovery of the coordinate system in 1637 in his work La Géometrie, as he never mentions the concept of axis, and none of his illustrations depicts an axis or a numerical system of coordinates, coordinates, even when values for specific magnitudes are specified. The first recording of number lines appears in the first half of the nineteenth century in the work of Ernst Gottfried Fischer (1754-1831). Fischer works with negative and positive quantities without limitation. He explicitly associates each point on the axis of the abscissa with the values of x and corresponds to one point on the axis of the ordinate. 4th period. From the beginning of the 19th century to the present: The foundation of the number line in its present form. The first attempt to develop a theory of real numbers was made in the early 1830s by Bolzano, who saw real numbers as limits of progressions of rational numbers. Around the same time, Rowan Hamilton (1805-1865) made an attempt to define real numbers, but could not escape the logic of ordinary geometric tradition. Karl Weierstrass (1855-1897), wanting to base infinite calculus solely on the concept of number, believed that he had to define irrational numbers irrespective of the concept of limit. He therefore considered the convergent sequence itself as the number or the limit. Thus, irrational numbers are defined as sets of rational numbers, rather than ordered sequences of rational numbers. In 1871 Georg Cantor (1845-1918), launches a new numbering program, similar to the Meray and Weierstrass programs. At the same time Heine (1821-1881) proposed some simplifications that led to the so-called Cantor- Heine development, which resembles that of Meray, in which convergent sequences that do not converge to rational numbers are considered to define irrational numbers. Dedekind tried to give a clear definition of continuity, first for the points of a straight line and then for a set of numbers starting from the set of rational numbers, after observing that the ordering properties of rational numbers apply just as the relations between the points of a straight line. Dedekind considered that the set of rational numbers could be extended to a continuous set of real numbers if the Cantor - Dedekind principle accepted that the points of a straight line could be mapped one by one with the real numbers. Therefore, we have the foundation of the number line in its current form. First of all, we should note that the mathematical integration and constitution of the notion of the number line, as we know it today, took place very slowly in the history of mathematics. As we have seen, it was only with the foundations of Dedekind and Cantor in the late 19th and the early 20th century that we can now consider that there is a one-to-one correspondence between the points of a straight line and the numbers of the set of real numbers. This in itself shows that while the concept of the number line appears to be simple, its composition to its present form has taken a long time. As critical points in this mathematical constitution of the notion of the number line over the course of mathematical history, we can in principle consider the separation between the numbers and the magnitude or the separation between the numbers and the straight line. We can see this separation, as we indicated above, in the mistakes of students who often manage numbers separately from the measures on the straight lines of numbers. The second critical point that appears in the historical evolution, but also as a difficulty for students, is the negative numbers and the orientation on the number line in the positive or negative direction. The third critical point is the density of rational numbers and the extra unit intervals needed to place them on the line of numbers. This manifests as a problem for students, when they have to place fractions, and generally rational numbers, on the number line where it is necessary to determine the extra unit space such as, for example, specifying the unit of ¼ to place ¾ on the number line. The fourth critical point focuses on the density of the irrational numbers, the separation of the rational from the irrational numbers and the representation of the irrational numbers on the number line. In this work we examined the difficulties of students and teachers of mathematics when dealing with the density and separation of rational and irrational numbers and the representation of irrational numbers on the number line. As it was pointed out earlier and drawing on historical information, the identification of real numbers and their correspondence with points on the number line reaches its conclusion very late, actually between the late 19th and the early 20th centuries. [ABSTRACT FROM AUTHOR]

Developing algebraic thinking in elementary school has gained consensus among mathematics educators. The objective of this study is to understand the developmental trajectory of early algebraic thinking in elementary school students so as to assist teachers and curriculum developers in implementing instruction that aligns with students' cognitive development. This study adopted a cross-sectional survey approach, involving 526 students from grades three to five in Shanghai, who were tested using a 12-item assessment that measured three aspects: "generalized arithmetic", "functional thinking", and "quantitative reasoning". Latent class analysis was used to analyze students' response strategies, and, in conjunction with individual interviews, this study identified potential developmental pathways in students' early algebraic thinking, progressing from "arithmetic thinking" to "concrete algebraic thinking", "generalized algebraic thinking", and finally to "symbolic algebraic thinking". As thinking levels advanced, significant differences in students' response strategies emerged, with notable improvements in "generalization abilities" and "symbolization abilities". This study suggests that educational practices should encompass content in elementary arithmetic curricula that fosters generalization abilities. Additionally, providing students with opportunities for diverse representations can effectively stimulate the development of early algebraic thinking. [ABSTRACT FROM AUTHOR]

Many systematic reviews have examined flipped learning models in various fields. However, there is little research on the use of the flipped learning model in mathematics education, which could help researchers and practitioners use and develop a model to implement mathematics learning processes. To fill this gap, this study aimed to analyze and synthesize current knowledge and practices in the implementation of flipped learning in mathematics education. Systematic literature review was adopted as the research method following an article selection and screening process using the preferred reporting items for systematic review and meta-analysis (PRISMA) protocol. Articles published from 2012 to 2021 in some reputable databases (Web of Science, Scopus, and ERIC) were reviewed, and 17 of 137 articles were included for detailed analysis and synthesis. The findings of this study showed that research in the implementation of the flipped learning model in mathematics education focused on documenting the affectivity of the implementation of the flipped learning model, developing learning processes, and sharing preliminary findings and student feedback. Future research is highly recommended to examine different aspects of flipped learning implementation, promote longitudinal data based on multiyear research for implementing flipped learning, and review various learning media to strengthen students’ understanding of mathematics [ABSTRACT FROM AUTHOR]

To promote optimal learning in their students, mathematics teachers must be proficient in problem posing, making this skill a cornerstone in teacher training programs. This study presents a formative action in which pre-service teachers are required to create and analyze a problem involving proportional reasoning within a probabilistic context. For this problem, they must identify the objects and processes involved in its resolution, recognize the degree of algebraic reasoning implied and identify potential difficulties for students. Subsequently, they need to formulate and analyze a new problem with variation, which mobilizes higher-level algebraic activity. Results indicate that prospective teachers struggle to pose problems that engage proportional reasoning, as well as to identify in their analysis which elements of proportional and algebraic reasoning are present in their solutions. Despite this fact, a significant percentage of participants adequately modify the original problem to address higher levels of algebraic reasoning, identifying in these cases the new algebraic objects and potential difficulties that might arise as the degree of generalization required in the solution increases. The study concludes by underscoring the importance of training in problem posing to enhance the knowledge and competences of prospective teachers concerning proportional and algebraic reasoning. [ABSTRACT FROM AUTHOR]

This study investigates the effects of teaching mathematics supported by problemposing strategies on fourth-grade students' problem-posing skills. This study also seeks to determine students' views about the process. To this end, the study employed an explanatory sequential design, a mixed method that incorporates collecting quantitative and qualitative data. The study group consisted of fourth-grade students studying in two different classrooms of a public school in the west of Türkiye in the 2021-2022 academic year. Data were collected through a "Problem-Posing Skills Test" and a "Semi-Structured Interview Form." The research concludes that teaching mathematics supported by problem-posing strategies improves students' problem-posing skills. In addition, this method was more effective than the one used in the control group in developing students' structured, semi-structured, and free problem-posing skills. At the end of the interviews, it was determined that teaching mathematics supported by problem-posing strategies was an innovative, studentcentered, and emotionally stimulating technique. It was also found that the students had more difficulty in the semi-structured and free problem-posing tasks. [ABSTRACT FROM AUTHOR]

This study investigates sixth-grade Turkish students' pattern-generalization approaches among arithmetical generalization, algebraic generalization, and naïve induction. A qualitative case study design was employed. The data was collected from four sixth-grade students through the Pattern Questionnaire (PQ) and individual interviews based on the questionnaire. The findings revealed that all students generalized near terms using arithmetical generalization as the first step and then they mostly looked for a general rule through memorized procedures by skipping far term generalization. When they found the general rule, far terms were calculated by rote. In other words, students did not generalize the pattern to far terms using an algebraic generalization. The current study's findings would give valuable information to the mathematics educators regarding the necessity of avoiding creating a procedural instructional environment by focusing on the rote procedure of finding the general rule of a pattern. These findings would also expand the horizons of curriculum developers regarding the importance of objectives about both near terms and far term generalization by progressing from arithmetical generalization to algebraic generalization. [ABSTRACT FROM AUTHOR]

Since teachers have the greatest impact on student learning, it is crucial to consider how professional development programs (PDP) for teachers can enhance their contribution, especially in designing mathematical tasks for teaching. This paper focuses on identifying patterns of practices of mathematics teacher educators related to crucial aspects of two teacher PDPs: one conducted face-to-face and the other using a Massive Open Online Course (MOOC). The Meta-Didactical Transposition is employed as the theoretical framework for comparing the two PDPs and for identifying patterns of practices. The findings suggest that educators, both in face-to-face and online settings, consider certain practices to guide teachers in designing mathematical tasks. This paper aims to share experiences of good practices that can be implemented by other researchers seeking to guide teachers in task design, either alone or in small groups. [ABSTRACT FROM AUTHOR]

This research article examines the widespread practice of ability grouping in educational systems across different countries, exploring its impact on academic achievement and its implications for equity among mathematics students. The findings underscore the urgent need for educational ecosystems to move away from ability grouping and embrace alternative pedagogical approaches that foster growth mindsets, equitable opportunities, and inclusive education for all students. The implications of this article extend beyond the immediate concerns of schools, school organizations, and mathematics education enthusiasts. Particularly, the insights gleaned from this research bear significant relevance to the development of adolescents and youths on a broader scale. The insights provided by this study are relevant to parents, educators, policymakers, and anyone interested in fostering an educational landscape that equips adolescents and young people with the skills, attitudes, and opportunities needed for success in a diverse and complex world. [ABSTRACT FROM AUTHOR]

This study explores the effect of mobile-assisted task-based learning (M-TBL) on vocabulary achievement and student attitude among English as a in content and language integrated learning (CLIL) science lessons, the researcher focused on foreign language (FL) students. To investigate this, a quasi-experimental research design was utilized. In the first semester of the 2022–2023 academic year, 34 fourth-grade students from a private school in Istanbul took part in the study. During the course of 6 weeks, the experimental group engaged in six different M-TBL activities and were assigned a one-page worksheet as homework, while the control group completed only the one-page worksheet. Quantitative data for the study were collected through vocabulary achievement tests and a student attitude questionnaire. The data underwent analysis through paired sample t-tests and descriptive analyses. The results indicated a significant difference in vocabulary achievement between the experimental and control groups, and they also revealed a positive impact on student attitudes, igniting their interest in learning the target language. In light of these findings, it can be concluded that mobile-assisted task-based activities yield positive effects on both vocabulary achievement and student attitude in primary FL learners participating in CLIL science lessons. [ABSTRACT FROM AUTHOR]

Copyright of Anadolu University Journal of Education Faculty (AUJEF) is the property of Anadolu Universitesi Egitim Fakultesi Dergisi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Anadolu University Journal of Education Faculty (AUJEF) is the property of Anadolu Universitesi Egitim Fakultesi Dergisi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Anadolu University Journal of Education Faculty (AUJEF) is the property of Anadolu Universitesi Egitim Fakultesi Dergisi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Quantitative literacy and critical thinking skills contribute to the skill of pre-service biology teachers to teach 21st-century learning. This study aims to describe the construction of structural correlation between quantitative literacy and critical thinking skills, and factors affected them among South Kalimantan pre-service biology teachers. The researchers employed a quantitative correlational research method. The samples were 245 biology education students in South Kalimantan province. The instrument used for this study was a test instrument and a questionnaire. The data analysis techniques used were partial least square, structural equation modeling, and Spearman correlation test to evaluate structural models and hypothesis testing. The results of this study indicated that there is a significant relationship between quantitative literacy ability and critical thinking skills of students of pre-service biology teachers in South Kalimantan. There is a significant influence from the factors that affected students' critical thinking skills and quantitative literacy, except age. [ABSTRACT FROM AUTHOR]

This commentary paper is a pedagogical reflection on the interplay between variation and invariant. It begins with a brief discussion on the concept of Unity of Opposites as an ancient philosophical theme. Ancient thinking systems regarded the variation and invariant pair as a Unity of Opposites. Next, the use of variation as a pedagogical approach in mathematics education is briefly examined under Marton's variational theory of learning, Gu's bianshi jiaoxue, and the related research done by the author in the context of Dynamic Geometry Environment (DGE). These lead to the formation of the concept of variational thinking, the main contribution of this paper, which is presented and explained. A DGE task design sequence example is presented to illustrate how variational thinking can be used to frame a process of geometrical reasoning and argumentation. [ABSTRACT FROM AUTHOR]

During the COVID-19 pandemic, researchers faced challenges conducting data collection. However, emerging communication technologies, such as web-conferencing platforms, provided an alternative mode of data collection. Among these platforms, Zoom stands out for its functional affordances and unique features that support data collection methods, including screen-capturing, digital photography, drawing, polling, and artifact-based interviews, which attempt to reveal a child's viewpoint and ability. In this article, we explore the feasibility of using Zoom to conduct educational research by investigating the mathematical learning experience among 26 primary 1 and 2 students in Hong Kong on two topics—understanding time and addition and subtraction. Five mathematics teachers were recruited to participate in 2-month online workshops with assessments among three groups of students to understand the suitability for data collection. Based on teachers' observations and students' responses, our findings suggest that Zoom is a viable and effective tool for qualitative data collection. It offers a cost-effective method for researchers and educators to examine students' mathematical performance in an online environment. We also investigate students' perceptions of online assessment strategies, perceived usefulness of various Zoom functions, and explore the benefits and challenges teachers and students perceived throughout mathematical education using Zoom. However, several technical difficulties were encountered, such as the digital literacy gap and privacy concerns. Therefore, improving students' digital literacy for remote learning and triangulation is important for online mathematical data collection and assessments. [ABSTRACT FROM AUTHOR]

The present study examines the role of everyday play-based activities, planned and implemented by parents, on the development of children's mathematical thinking at the age of up to 3 years old. Parents' knowledge and beliefs were measured by using an online questionnaire, while a nonformal training program was developed and implemented with the participation of a group of parents using technological tools. The aim was to improve their knowledge on how to use relevant play-based activities and the understanding of their role. Results indicated that parents asked for the contribution of experts in order to acquire the relevant knowledge and skills and develop positive self-efficacy beliefs about their role. They found the training practical, through the use of digital tools, as well as useful for collaboration and communication. The shared examples by them indicated that the training has to concentrate further on the use of daily activities outside of the school-based context, which is more known to the parents. [ABSTRACT FROM AUTHOR]

Over the past 30 years, teacher education has changed to incorporate a larger emphasis on understanding students' sociocultural backgrounds, knowing that these influence their learning. However, in terms of mathematics and mathematics education in teacher education, less has been done to recognise the sociocultural mathematics backgrounds of students. An example is provided to show how entrenched colonial attitudes to mathematics have developed into neocolonial policies that influence mathematics education. This example is based on a large historic research project in Papua New Guinea (PNG) that aimed to document and analyse the nature of mathematics education from tens of thousands of years ago to the present. Data sources varied from records of first contact and later records, archaeology, oral histories, language analyses, lived experiences, memoirs, government documents, field studies, and previous research especially doctoral studies. The impacts of colonisation, post-colonial aid and globalisation on mathematics education have been analysed, establishing an understanding of the current status of mathematics education as neocolonial. Neocolonial education policies diminish cultural ways of thinking. Thus, teacher education has an important role in sensitizing preservice and inservice teachers to the impact of neocolonial approaches as well as in developing with students some ways of reducing this impact and encouraging more holistic, culturally relevant mathematics education. [ABSTRACT FROM AUTHOR]

Copyright of Erzincan University Journal of Education Faculty / Erzincan Üniversitesi Egitim Fakültesi Dergisi is the property of Erzincan University Faculty of Education Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Academic researches about growth-mindset interventions show that growth-mindset strategies can be effective on academic achievement. However, it is not clear yet how this effect occurs and what is the background of this. The aim of this research is to examine the effect of the growth-mindset intervention including growth-mindset strategies on the academic achievement and mindset scores of 8th grade students who are continuing secondary education in Turkey by using mixed method. The sample of the research, which was designed in accordance with the embedded mixed design, consists of 42 8th grade students (Experimental group: 21, control group: 21). Within the scope of the research, an extracurricular 8-week training was applied. Mindset scale and academic achievement scale were applied before and after the intervention. At the same time, qualitative data were collected through semi-structured interviews and observations before, during and after the intervention. The findings obtained at the end of the research show that growth-mindset strategies make a significant change on students' academic success and mindset scores. [ABSTRACT FROM AUTHOR]