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This paper set out to contribute to mathematics education research by elaborating on a methodology developed during a study, trying to understand, view and follow shifts in prospective teachers' discursive patterns. The methodology aims to illustrate and describe how prospective teachers adapt to the context of teaching through a flexible process. This flexible process is then described in the result as a narrative. It is argued that the methodology can be used in relation to different theoretical directions, such as research about beliefs, knowledge, or identity. Another contribution is that the methodology presented gives insights into bridging the gap between different analytical levels, micro and macro. With a theoretical foundation in 'Cultural Worlds' [Holland, D., Skinner, D., Lachicotte, W., & Cain, C. (1998). Identity and agency in cultural worlds. Harvard University Press.] the Social Semiotic approach of Systemic Functional Linguistics, SFL [Halliday, M., & Hasan, R. (1989). Language, context, and text: Aspects of language in a social-semiotic perspective (2nd ed.). Oxford University Press] is used as a methodological tool. SFL offers a toolkit that allows the analysis of meaning at the clause level to uncover how and why a speaker produces a particular wording rather than any other in a specific social practice. The paper aims to illustrate and describe how to go beyond findings in the micro-analysis and then present the result as a narrative case. [ABSTRACT FROM AUTHOR]

Improving the mathematical skills of undergraduate Engineering and Information Technology (EIT) students has been an enduring challenge for educators. Lecturers continually implement new strategies that try to overcome the challenges encountered in teaching university students with varying levels of success being encountered. This paper provides insight into how educators have handled the challenges students face in their mathematics learning journey. In addition, the paper highlights some of the methods used to improve the performance of these students as problem solvers, as well as the development and implementation of creative techniques that may have made the situation more successful and less challenging. The literature was searched through using a structured literature review approach based on the primary databases: Scopus; the Education Research Complete (ERC); and the Education Resources Information Centre (ERIC). This study followed the three main steps of a systematic literature review: retrieval, selection, and analysis of the literature. Analysis of the literature showed that there is still a need to further develop and deliver innovative pedagogical strategies for teaching mathematics to EIT students. It is recommended that an interdisciplinary approach, inclusive of technology, will lead to increased understanding, improve performance and, ultimately, enhance student satisfaction. [ABSTRACT FROM AUTHOR]

An editorial is presented on INDRUM being the acronym for the International Network for Didactic Research in University Mathematics. Topics include contributing to the development of research in didactics of mathematics at all levels of tertiary education; and focusing on building research capacity in the field showing concern for the development of early-carreer researchers and dialogue with the mathematics community.

In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this. [ABSTRACT FROM AUTHOR]

‘Group work’ is a vague description of an instructional activity, because many factors shape its character and effect on students. One important factor is group formation, that is, how groups are formed by the instructor. In this paper, we sought to better understand the variation of group work with respect to group formation by addressing: How do university mathematics instructors form groups over the course of a semester? To this end, we examined eight instructors' methods of forming groups in one multi-section introduction to proof course. Our findings include a classification of group formation methods and descriptions of how instructors varied their formation methods across the semester. Further, we sought to understand: How do students experience different group formation methods? We analysed interviews with 29 students from the eight instructors' classes and identified central themes among students' experiences of different group formation methods. Finally, we discuss the sometimes conflicting research on (when) which group formation method is most appropriate and offer our thoughts on how the differences between typical undergraduate and K-12 mathematics classrooms may contribute to different recommendations. [ABSTRACT FROM AUTHOR]

International literature is increasingly disclosing the relevance of cultural aspects within the processes of teaching and learning mathematics. Knowledge is inextricably linked to the activities in which the subjects engage, and must be considered in relation to the socio-cultural context wherein the activity takes place. Literature reveals the relationship between the background culture (e.g. language, nationality, etc.) of prospective elementary teachers and their beliefs about mathematics and its teaching. In this paper, we define culture with reference to the wider discussion from cross-cultural psychology literature about cultural values, and we investigate if and how differences in individuals' cultural values are related to prospective teachers' beliefs about being successful in mathematics and in its teaching. We adopt a questionnaire from the studies by Schwartz to measure participants' values. We assign each prospective teacher of our sample to a cluster of beliefs and we analyse how the beliefs of prospective teachers are related to their values. Results show that cultural values and beliefs about mathematics are related, while this is not the case for beliefs about the teaching of this subject. [ABSTRACT FROM AUTHOR]

This paper reports on the development of the Teaching Mathematics for Social Justice beliefs (TMfSJ-B) scale and its psychometric properties. Preservice elementary education teachers enrolled in teacher education programmes at seven universities completed multiple surveys designed to understand the evolution of attitudes and beliefs about teaching mathematics. To our knowledge, the TMfSJ-B scale is the only survey specifically designed to measure pre- and in-service teachers' beliefs about teaching mathematics for social justice. Confirmatory factor analyses (n = 655) provided here indicate that the TMfSJ-B scale is a valid and reliable assessment of beliefs about social justice in mathematics teacher education, with three factors: (1) value of mathematics for all students; (2) integration of social justice into mathematics instruction; and (3) mathematics and its relation to inclusiveness. The purpose of this study was to validate measures produced by the TMfSJ-beliefs scale and discuss its potential use in teacher education programmes and research. [ABSTRACT FROM AUTHOR]

Teachers' and students' values influenced the teaching and learning process in the mathematics classroom. Few studies were conducted to compare teachers' and students' value in mathematics learning. This paper explores Malaysian mathematics teachers' and students' values in mathematics learning. Fifty-three mathematics teachers and 354 secondary school students participated in the online survey. This study reports ten semantic differential items and an open-ended question related to mathematics learning. An independent t-test was conducted on the semantic differential items. Thematic coding analysis was performed to code the open-ended questions based on the framework of three intersecting values: mathematical values (the discipline), mathematics educational values (the pedagogy), and general educational values (the ethical and moral principles). The findings showed significant mean differences in mathematics educational values between teachers and students. No significant difference was found in mathematical values between teachers and students. Moreover, the qualitative data confirmed the similarities in mathematical values and differences in mathematics educational values between teachers and students. The current study also discusses possible reasons for the differences in teachers' and students' values. The findings imply that teachers' and students' values are crucial in enhancing teaching and learning in the mathematics classroom. [ABSTRACT FROM AUTHOR]

The article offers information on the increasing interest in affect-related research in mathematics education, particularly within the MAVI community. The topics include the growing recognition of the role of affect in teaching and learning, the collaborative spirit of MAVI in promoting research on mathematical beliefs and education, and the contributions from different communities in a special issue of the International Journal of Mathematical Education in Science and Technology (IJMEST).

Teacher practice is rife with tensions. Tensions around what and how best to teach, how to manage situations with students, and how to manage situations with colleagues, administrators, and parents. These tensions are often seen as pairs of opposing internal and external forces: this assessment is better, but it takes a lot of time. These forces are inescapable, and teachers have to learn to either manage them or resolve them. In this paper we look closely at tensions through the lens of opposing forces and, more interestingly, how teachers either learn to live with them or work to resolve them. And, in particular, we look at how this work differs if the tension exists between internal forces, external forces, or a tension between an internal and external force. Drawing on case studies of seven different participants we dive deep into the murky world of the lived experiences of teachers to understand better the way that tensions contribute to their beliefs, decisions, and actions. [ABSTRACT FROM AUTHOR]

This paper targets patterns of expressed emotional directions towards mathematics. By using TIMSS data and recoding two of the instruments, we compared responses from the same student cohort in Grade 4 (11 years old) and Grade 8 (15 years old) in Sweden. Two hypotheses were tested: negative emotions including negative motivations increase with age, and there will be gender differences. By using statistical analyses, including Cramer's V, the two hypotheses were confirmed. The results also illuminated patterns within the responses, such that boys and girls in Grade 4 differ in expressed intrinsic motivation but not extrinsic motivation. There are also indications that girls might be more likely to form negative emotional directions between grades 4 and 8 than boys, and some implications of this is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the learning experiences, beliefs and motivations of students in classes where the mathematics teachers have received support for using inquiry-based learning activities. Data were collected from 248 students in the age-range 11–16 using electronic questionnaires. Our results show that key features of inquiry-based mathematics were only moderately reflected in these students' beliefs about the subject, their dispositions towards mathematics were less positive across the transition from primary to secondary school, and with respect to motivation this decline was stronger for girls than for boys. Furthermore, medium to strong correlations between belief- and motivation subdomains were found, for instance, students who view mathematics as a creative subject and/or have a growth mindset of mathematics also tend to find this subject enjoyable and perceive it as useful. Finally, our results indicate that inquiry-based teaching has a potential for fostering positive dispositions towards mathematics, as students who often experience inquiry-related activities in class also tend to see mathematics as a creative and interesting subject that will be useful for them in the future. [ABSTRACT FROM AUTHOR]

Although research in mathematics education has yielded a great number of results on mathematics teachers' beliefs in recent decades, it is still an open question as to whether these teachers' beliefs are stable or not. However, since mathematics teachers' beliefs are perceived to be the default of their classroom practices, it is important to gain insight into these beliefs and into their development. In this paper, we contribute to this field of research by providing an approach designed to analyse a change in teachers' belief systems. We study a sample of 20 prospective teachers during their first practical experience. We analyse the structure of teachers' belief systems with interviews and a questionnaire set twice within one year. To analyse development of these teachers' belief systems, we assigned the young teachers of the first and second survey to clusters that were developed in a former study and re-analysed for our purposes here. Our results show considerable changes in mathematics teachers' central beliefs during their first practical experience. However, a general pattern of belief changes is not found. [ABSTRACT FROM AUTHOR]

This paper aims to examine the extent to which pre-service teachers' mathematical connection skills were promoted through the course designed within the scope of pedagogies of practice (Grossman et al. [2009]. Teaching practice: A cross-professional perspective. Teachers College Record, 111(9), 2055–2100. https://doi.org/10.1177/016146810911100905) in the teacher education programme. Data were collected in a course that was designed to develop pre-service mathematics teachers' (PSTs) mathematical connections skills through a classroom teaching experiment with individual contributions to the whole class learning process. Thirty-six PSTs who were enrolled in the course participated in the present study. Initial and revised lesson plans were used as primary data collection instruments. In the most general sense, the findings demonstrated that most of the PSTs improved their mathematical connection skills during the course. The results of the current study provide insight into how the courses in undergraduate education should be implemented to give PSTs opportunities to develop their mathematical connection skills. [ABSTRACT FROM AUTHOR]

Mathematical memes are an Internet phenomenon with an epistemic potential noteworthy for the teaching and learning of mathematics. The aim of this paper is to conceptualize this phenomenon on an empirical base, elucidating its educational potential. To pursue this goal, a two-year ethnographic research study has been conducted in a selection of social media communities that exchange mathematical memes. In the paper, a twofold theoretical lens is used to investigate the phenomenon from a memetic perspective focused on the general characteristics of Internet memes, and from a cultural perspective considering the Web 2.0 environment. Analysis tools derived from the memetic and cultural theoretical background are applied to an instrumental case study, to reconstruct how mathematical memes activate the interaction among members of an online community. The investigation elicits key characteristics of the phenomenon, provides grounds to conceptualize mathematical memes as hybrid representations of mathematical statements, and sheds light on the traits of the epistemic culture they establish in the community. [ABSTRACT FROM AUTHOR]

Numerous studies affirm the importance of students' attitudes for mathematical education. This study uses mathematical autobiographical essays to examine the mathematical experiences and attitudes of undergraduate students enrolled in mathematics classes at a Midwestern University in the United States. The essays of 41 students (20 men, 21 women) out of 170 enrolled in College Algebra, Precalculus, and Calculus were analyzed using the categories of (1) confidence in mathematics ability, (2) value of mathematics, and (3) liking of mathematics. This paper provides context and theoretical depth to student reports of their attitudes in these categories. Statements at the intersections of both confidence and liking, and value and liking provide additional insights. First, students who perceive mathematics as valuable are more likely to like it. Second, the vast majority of students talk about confidence and liking together, and with a positive correlation. Further, these students tend to talk about liking or not liking mathematics because of their perceived ability in mathematics. Our study suggests the importance of proactive teaching strategies to improve students' perceptions of the value, liking of, and their confidence in, mathematics, and also suggests self-efficacy as a potential theoretical basis for this work. [ABSTRACT FROM AUTHOR]

In this paper, we will show a characterization for pairs of commensurable diagonals of regular n-gons. This result gives a positive answer to a recent question. [ABSTRACT FROM AUTHOR]

Introducing the distributivity property of multiplication over addition is a well-known challenge in mathematics education, especially in primary school. As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation and evaluation of a modelling activity in which 2nd-grade students are introduced to the key concept of distributivity of multiplication over addition. The results show that modelling activities designed through the heuristics of didactical phenomenology, guided reinvention and emergent modelling may support the accessibility of distributivity, as these students were able to reinvent this concept solving an experientially significant problem. This result can be attributed to a combination of several factors: the choice of a realistic and rich problem, that stimulated students to elaborate formal mathematical concepts mathematizing their informal solving strategies, rooting new understandings in experientially real phenomena; the use of a suitable artifact, that presented mathematics as a means of interpreting and understanding reality and increasing the opportunities for observing mathematics outside of the school context; the role of the teacher, who guided students in reinventing mathematics in an active way. [ABSTRACT FROM AUTHOR]

Capitalizing on cognitive load theory and learning by analogy, we propose two instructional methods to learn a complex linear equation (e.g. two-step equation) by building on prior knowledge of a simpler linear equation (e.g. one-step equation). We will examine the proposal theoretically in this paper. In line with the design principles of cognitive load theory, we propose to strengthen students' prior knowledge of simpler linear equations before they learn complex linear equations with the aid of worked examples. Because a subset of the complex linear equation shares the same schema as the simpler linear equation, students can draw on their schema for the simpler linear equation to understand the complex linear equation, thus alleviating the limitation on working memory load. Based on the principles of learning by analogy, we place a simpler linear equation and a complex linear equation side-by-side and label the solution procedure of both linear equations to encourage active analogical comparison between these two equations. Making both the simpler linear equation and the complex linear equation visible to learners may help to reduce cognitive load demands in retrieving the simpler linear equation in order to facilitate the learning of the complex linear equation. [ABSTRACT FROM AUTHOR]

The paper is the result of extensive research carried out among Czech students and focuses on a conception of infinity. The questionnaire survey was taken by 861 students ranging from grades 7 to 13. The aim of the research was to describe the development of students' conceptions of infinity. These conceptions are built on the intuitive phenomenon of the horizon. We monitor the proportional representation of these conceptions in four combinations of views (into the distance and in depth) and contexts (arithmetical and geometrical). It can safely be maintained that the development of the proportional representation of the earliest conception comprehension of the concept of infinity, the so-called natural infinity, is not concurrent with the students' age. The development of the proportional representation of the conception of actual infinity is non-decreasing, at least in the view into the distance, in both contexts with respect to age. In general, it can be stated that the proportional representation of individual conceptions of infinity is strongly dependent on both context and view. [ABSTRACT FROM AUTHOR]

In this paper a class of regular surfaces, generated by rotation and twist of a rigid line segment, is considered. Firstly a complete description of those surfaces by means of local charts is obtained and then its orientability is characterized in terms of the number of half-twists, taking into account their direction (clockwise or counterclockwise) since changes in the sense of twist are allowed. Indeed a procedure to construct nonorientable surfaces is provided. Illustrative examples and computer-generated plots using the free software Maxima are included. [ABSTRACT FROM AUTHOR]

LEGO® stop-motion animations were developed to engage first-year university students in the topics of matrices, vectors, linear geometry and linear transformations. LEGO® provided a versatile medium through which a wide range of concrete and abstract concepts could be physically demonstrated in both two and three dimensions. It was also familiar to most students, making the mathematics videos accessible and fun. When designed according to key principles around reducing cognitive load in multimedia learning, animations in education have been shown to increase learning compared to static images for concepts involving dynamic systems and processes. Multimodal approaches including video and narration have also been shown to be beneficial to students in terms of demonstrating mathematical problem solving. The videos created in this project were a mix of conceptual explanations and problem-solving examples, where the word problems were physically modelled before being converted to formal mathematical notation and then solved. Some animations were short clips that were integrated into online lecture recordings, others were stand-alone videos designed either as short introductions to topics or examples of common questions. This paper describes the theory underpinning the design of the animations. [ABSTRACT FROM AUTHOR]

In Australian universities, mathematics is often an essential core subject for disciplines such as engineering, information technology and science. These first-year mathematics subjects may have large enrolments of which a significant number fail the subject and repeat. While interventions to retain students are common, those who have failed before and are repeating the subject are often overlooked. In this paper, we undertook a statistical analysis based on demographic and performance data of six cohorts of students undertaking a first-year university-level mathematics subject. We found that pass rates for students on their second attempt were significantly lower than pass rates for students enrolled on their first attempt, and that age, degree enrolled in and pathway into university were predictors for who was more likely to repeat. Students in their second attempt were less likely to complete all assessment items and were more than twice as likely not to attend the final exam. We argue for an increased focus on and efforts targeting repeating students; that demographic data are used to predict students likely to repeat before they have failed a subject, and that this group of students, as well as those repeating, are supported to increase their chances to pass. [ABSTRACT FROM AUTHOR]

Despite research demonstrating positive effects of writing on student achievement, writing-to-learn remains under-utilised in mathematics and mathematics teacher education. In this paper, we explore how writing-to-learn tasks in a mathematics methods course influenced elementary preservice teachers’ beliefs about teaching in two domains: mathematics and writing. Preservice teachers engaged in weekly writing-to-learn tasks designed to support their understanding of mathematics content and mathematics pedagogy over the course of a semester. Using survey data, we found no significant difference in their beliefs from pre to post. Such null findings led us to ask new qualitative questions, resulting in a concurrent mixed methods design. Through qualitative analyses of student writing, we found several meaningful and non-meaningful practices for engaging in writing-to-learn tasks. This research supports teacher educators, teachers, and researchers to understand how to implement writing-to-learn tasks and to support students towards meaningful engagement in the tasks. [ABSTRACT FROM AUTHOR]

Even prior to COVID-19, there had been an increase in the use of technology in undergraduate mathematics education to support the development of mathematical understanding and skill, and to enhance the student learning environment. Technology-enhanced resources can be used to provide formative assessment which can help students identify gaps in their knowledge and point towards appropriate action. In this paper, we report on the evaluations of a number of technology-enhanced resources that used formative assessment techniques and were provided for students attending non-specialist first-year undergraduate mathematics modules in Ireland. Analysis of the evaluations and consultation with literature identified 12 decisive factors that encourage students’ successful engagement with the resources, covering aspects such as the educational context, affordances and types of technology, technological communication and formative assessment. These factors constitute important knowledge for practitioners as they can use them in the planning, design, and development of technology-enhanced resources for use in their modules. The measures observed during the evaluations indicated both positive and negative engagement providing a link between observations and resultant student engagement, which was found lacking in previous studies on student engagement. [ABSTRACT FROM AUTHOR]

This article presents a networking of two theories, the Extended Theory of Mathematical Connections (ETC) and the Onto-semiotic Approach (OSA). In particular, concordances and complementarities are identified in the respective conceptions of mathematical connections, as a result of applying these theoretical frameworks to a student's response protocol when solving a task on the derivative. First the mathematical connections were identified from the perspective of the ETC and, second, an onto-semiotic analysis was carried out where the mathematical practices, the primary mathematical objects activated in them and the semiotic functions established between these objects were analysed. The main conclusion is that both theories complement each other to make a more detailed analysis of the mathematical connections. In particular, the more detailed analysis carried out with the OSA tools, visualizes a mathematical connection like the tip of an iceberg of a conglomerate of practices, processes, primary objects activated in these practices and semiotic functions that relate them, which enables a thorough analysis that details the structure and function of the connection. [ABSTRACT FROM AUTHOR]

This study used a modified mathematical knowledge for teaching (MKT) framework to compare the common content knowledge (CCK) of two groups of purposively selected pre-service primary mathematics teachers (n = 86) in relation to spatial visualization. Specifically, thirty-five pre-service teachers (n1 = 35) with a pure school mathematics background and fifty-one pre-service teachers (n2 = 51) with a high school mathematical literacy background worked on problems that involved spatial objects, their properties and relationships. The responses of pre-service teachers with a mathematical literacy content background to the Surface Development Test (SDT) were significantly different from their counterparts' which suggested that their CCK in relation to spatial visualization was thin. Analysis of the Probes questionnaire which inquired about the pre-service teachers' views on the concept of spatial visualization showed that misconceptions hindered their performance. Future research directions are discussed. [ABSTRACT FROM AUTHOR]

In Peru, national assessments repeatedly rank Indigenous mathematics students as the lowest performing across the entirety of Latin America and South America. Whereas lack of financial resources often predicts low measures, the history of educational practice teaches us that students' poverty need not predict their educational outcomes – creative instructional approaches may turn the tables. Here we report on an innovative, body-based arithmetic technique, the Botetano Method, that has been enabling poverty rural children from remote mountainous regions of Peru to match and even greatly surpass their urban peers on comparable test items. The article explains the method's guiding humanistic and cognitive principles and then reports on findings from explorative action research that implemented and evaluated the method. Using observational methodologies, we argue that the students developed in their conceptual understanding of the content as well as in their attraction to the discipline, their professional identity, their personal pride in their achievement, and their general epistemic capacity for concentration and self-regulation. Throughout, we emphasize the methodological limitations of this grassroots proof-of-concept action research, which threaten the validity of the assertions. We speculate on early extensions of the method to literacy studies. [ABSTRACT FROM AUTHOR]

This replicated research project [see Hine, G. S. C. (2019). Reasons why I didn't enrol in a higher-level mathematics course: Listening to the voice of Australian senior secondary students. Research in Mathematics Education, 21(3), 295–313. https://doi.org/10.1080/14794802.2019.15999982019] explored the reasons why senior secondary (i.e. Year 11 and Year 12) students elected not to enrol in a higher-level mathematics course. For this project, all senior secondary Australian Tertiary Admissions Ranking (ATAR) students within Western Australian schools (aged 17–18 years) were invited to participate in an anonymous, online survey comprised predominantly of qualitative items. The researcher analysed 1633 ATAR students' perspectives of enrolments in mathematics courses, according to a symbolic interactionist paradigm. Reasons included the extent to which students feel higher-level courses are too challenging, time-consuming, not required for university nor future life, and requiring more effort and stress to complete than lower-level mathematics courses. Especially outlined is the extent to which participants feel a 10% bonus is a sufficient incentive to increase higher-level course enrolments, an extension of the original project. Participants indicated a range of responses either supporting or opposing this incentive. While approximately 47% of participants agree that the incentive is sufficient, a small proportion of these supporters offer their agreement conditionally. Within the proportion asserting the incentive is insufficient, a majority of participants suggested that a higher percentage bonus should be offered to students enrolled in Mathematics Specialist. [ABSTRACT FROM AUTHOR]

This study investigates teachers' beliefs about the inquiry-based learning approach in mathematics. In particular, as the first research problem, it addresses teachers' beliefs about the appropriateness of inquiry activities for all students, after three years of attendance in a professional development programme, focused on inquiry. As a second research problem, it studies the evolution of teachers' beliefs and practices, during their fourth year of attendance of the programme. The results show that, at the beginning of the fourth year, the teachers, despite thinking that inquiry activities in mathematics have several valuable aspects, held the belief that they are not appropriate for all the students, but only for the high-achieving ones. The two case studies, analysed to address the second problem, refer to two teachers with different outcomes of their developmental paths. During the fourth year, in which the teachers have been invited to experiment with inquiry activities in their whole classes, one of them accepted the challenge and, as a consequence, had the opportunity to change both her practices and her beliefs. The second teacher, instead, continued to propose inquiry activities only to her high-achieving students and, consequently, showed no signs of change, either in her practices, nor in her beliefs. [ABSTRACT FROM AUTHOR]

In undergraduate mathematics education, students' collaborations have gained a reputation as a 'good' learning practice. A more complex image emerges once collaborations are construed as an arena where cognitive, social, and affective matters intertwine in ways that can fuel and impede learning. In this study, I take a close look at a twenty-minute-long interaction between two first-year students who engaged in a sequence of routine problems about dot products presented in a computer-based learning-support system. The commognitive analysis of the interaction revealed an asymmetrical structure: one student led the mathematizing, while his partner followed and invested considerable efforts to avoid making a mathematical contribution. The leader's attempts to switch positions and inquiry into the follower's recurrent refusals to mathematize entailed loaded exchanges that took a toll on both students in terms of identity and affect. I compare these findings to previous studies with school students to discuss possible sources of ineffective communication in situations where students are expected to learn through collaborations. [ABSTRACT FROM AUTHOR]

The study focused on the similarities and differences students and mathematics teachers have regarding values in mathematics classrooms in the same school culture. The study aimed to determine students' and mathematics teachers' values at Imam-Hatip high schools, a type of vocational school in Turkey. The study was carried out with students (n = 2,912) and mathematics teachers (n = 861) from Imam-Hatip high schools in seven regions of Turkey who voluntarily responded to the online data collection tools. The collected data were analysed using a latent profile analysis. As a result, students and mathematics teachers were divided into three latent profiles; however, these profiles differed according to value components. Some of the study's key results revealed practice as the essential value component. Information and Communication Technology (ICT) was the least for all teachers' and students' profiles, with relevance being the second least important value. [ABSTRACT FROM AUTHOR]

Teachers' beliefs about mathematics play an important role in the teaching of mathematics. Future teachers' views of mathematics are the basis of the quality of instruction they will deliver, as well as the teaching materials that they will present to their students and the means necessary to learn and participate in mathematical tasks. This article studies Colombian future teachers' beliefs about mathematical knowledge, taking into account two different training programmes for pre-service mathematics teachers from the same country, both of them qualify students to teach mathematics at elementary levels. Specifically, 213 mathematics graduate students and 405 normal school students from the Colombian Caribbean coast in their last term of their studies participated. Data were collected regarding four dimensions of beliefs: (i) on the nature of mathematics, (ii) about learning mathematics, (iii) about factors affecting students' achievement in mathematics, and (iv) about the quality of their training programme. Results show remarkable differences in the future teachers' beliefs depending on their training programme. [ABSTRACT FROM AUTHOR]

The transition from school to university is a challenging process for many students which is reflected in high dropout and low examination success rates during the first year at university. Qualitative studies have found hints that students' beliefs play an important role during the transition. Due to their dialectic nature with cognitive as well as affective valences, it is sensible to assume that beliefs influence rather cognitive and objective criteria of a successful transition such as exam achievement as well as subjective criteria like satisfaction that draw on affective sources. However, quantitative studies have been mostly focusing on the role of students' beliefs for study achievement, neglecting possible effects on students' satisfaction and actual dropout. In this contribution, I draw a more holistic picture of the effects of beliefs during the transition by analysing effects on students' achievement, students' satisfaction and students' actual dropout behaviour. Data from over 600 first-year students in a longitudinal quantitative study show that dynamic beliefs (that emphasize the process character of mathematics and the usability of mathematics in everyday life and other disciplines) have a positive impact during the transition, while static beliefs (focusing on schematic aspects of mathematics) are problematic. [ABSTRACT FROM AUTHOR]

This study explored school principals' and teacher educators' perceptions of STEM education based on how they described STEM as a discipline, their understanding of the nature of teaching and learning of STEM, and the capabilities of a STEM-educated person. Data were generated through the Draw a STEM Learning Environment (D-STEM) instrument comprising drawn and written descriptions where participants drew a picture of a STEM learning environment and completed five prompt statements about what STEM is and how an individual develops personal STEM capability. The Legitimation Code Theory (LCT) specialization codes were used for data analysis (198 individual response items in total) to understand how the participants perceive STEM education. Almost half the participant responses indicated knowledge-code perceptions with a smaller but significant number (approximately a third of responses) indicating knower-code perceptions. The remaining responses showed élite-code perceptions, indicating a small proportion of participants valued the development of both disciplinary knowledge/practices and generic skills/attributes in STEM education. We posit that curriculum structure and reporting requirements influence these perceptions. Further research in relation to the influence of such understandings on enacted curriculum is warranted. [ABSTRACT FROM AUTHOR]

There is evidence that student attitudes towards mathematics become more negative as students move through their schooling. However, less is known about the different factors that shape student attitudes at various stages. This qualitative study sought to shed light on this issue and consider how stage of schooling influenced self-perceived supports for what helped underperforming students to learn mathematics. One hundred and seventy-nine students in Grades 2/3 and 5/6 completed a questionnaire before they participated in a mathematics intervention program, targeting students who had fallen behind in mathematics. Participating students were asked to complete the sentence stem. The best thing about learning mathematics in my classroom is ... and answered the question: What helps you learn mathematics in the classroom? Thematic analysis revealed that the majority of younger students emphasized specific tasks (e.g. games) and mathematical content areas (e.g. counting) when describing the best thing about learning mathematics at school and were twice as likely to highlight such aspects compared with older students. By contrast, older students were far more likely to indicate the importance of being supported directly by their teacher or peers when learning mathematics, compared with younger students. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic not only impacted people's lives globally, but also pushed faculty to quickly adapt to an online teaching environment and continue it until the end of the school or academic year. This study is urgently needed to gain an understanding of the challenges STEM faculty members face during the COVID-19 era as they make the transition to teaching online, while many of them engage in this shift for the first time. The initial results of an online survey of 101 International STEM faculty members showed that online evaluation and pedagogy are the most disrupted dimensions of e-learning when instructors struggled to re-orchestrate their teaching during such an unprecedented event. In addition, while the affective domain of teaching is identified as the missing dimension of an e-learning framework, adoption of the new technology is rated as the area of least concern for teaching STEM online. [ABSTRACT FROM AUTHOR]

Flipped classrooms serve as a new pedagogical approach for teaching and learning, which involves switching from the traditional teacher-centred academic agenda. Unlike the traditional classroom model, the flipped classroom enables students to understand the lesson better while learning at their own pace and enhancing their communication skills among peers. At the same time, lecturers are provided with an opportunity to extend the discussion time in class through a more interactive session between students. While research works regarding the success of implementing flipped classrooms have been widely undertaken in the literature, only a few of these works considered exposing such models to secondary students. Engaging secondary students in the flipped classroom and investigating its impact on students paves the way for preparing an academic environment for students' advanced courses in the future. In this work, a case study was conducted at Barstow High School (Barstow, California) to evaluate the implementation of flipped classrooms in Mathematics education from an interplay of factors such as gender, ethnicity, and students' attitude. The students showed positive usability of the model from the end-activity evaluation and further noted an improved communication channel inside the classroom, allowing them to pace their own learning. [ABSTRACT FROM AUTHOR]

In light of the recent interest in mathematical competencies and the ways in which students communicate their ideas, this study aims to explore how undergraduate students communicate their ideas about mathematical tasks through written texts. Forty-three first-year undergraduate students participated in this study. They were given a graph of a function and tangent line and were asked to estimate the value of the derivative at the tangent point justifying their answer. We analyzed the justification of students to the given task by referring to the cooperative principle and to the Anthropological Theory of the Didactic. For the former, we employed the four Grice maxims: Quantity, Quality, Relation and Manner. For the latter, we employed the main components of praxeologies: the type of task, the technique needed to solve the task, the technology as knowledge needed to explain the technique, and a theory as basis to justify the technology. A model of the students' communication about mathematical ideas emerges and is described with reference to data analysis. The model sheds light on the relationship between the discourse of a student solving a mathematical task and how she performs the related problem-solving activity. [ABSTRACT FROM AUTHOR]

This article aims at examining the relationship between students' expressed notions of infinity and strategies used to compare the cardinalities of infinite sets. The notions of infinity were classified into two categorizations of potential and actual infinity, and the strategies were classified as part-whole, single infinity, incomparability and one-to-one correspondence. The participants were 104 senior high-school students and the data were collected through a questionnaire consisting of two parts. Analysis of the data showed that, despite the existence of inconsistent responses, there was a significant relationship between the notions of infinity and the strategies used to compare enumerable infinite sets, although the relation was not strong. [ABSTRACT FROM AUTHOR]

This study explores how the goals and enactment of three tailored calculus courses, referred to as course variations (standard calculus, life sciences calculus, and physics calculus), at the same university impact students' mathematical identity. In this mixed-methods study, we draw on the framework of figured worlds to interpret how students' mathematical identities can be refigured by the enactment of three calculus course variations. Results demonstrate that students in life sciences calculus had opportunities to refigure their identity as capable learners and doers of mathematics yet still viewed the subject as ancillary to their disciplinary interests. Students in physics calculus sustained their mathematical identity through strong peer collaborations. Finally, students in standard calculus shifted their identity as survivors of mathematics, conveying a sense of received knowing from their instructor. The ability of any particular course variation enactment to provide opportunities for refiguring mathematical identities resides within the goals, structures, students, and instructors that help to form these figured worlds. The results presented here provide insights into course variation instructional design and instructional practices that can positively affect students' mathematical identities, and in turn, their persistence in mathematics and science. [ABSTRACT FROM AUTHOR]

When students encounter challenging problems, difficulty or errors in the problem solving process, they must have the ability to adapt or change their problem solving methods; in other words, they must possess strategic flexibility. The aim of this study was to determine the strategic flexibility indicators displayed by gifted children in solving non-routine problems. A case study method was used. The study group consisted of 20 eighth, ninth, and tenth grade students attending the Science and Art Centre in Bursa, Turkey. A test consisting of seven non-routine problems was used. The problems in this test were asked to each student one by one by conducting individual interviews. The content analysis results revealed three themes. Under the first theme, strategy adaptivity, there were two indicators: Strategy knowledge and selection of the appropriate strategy. Under the second theme, intra-task strategic flexibility, four indicators were defined: changing the strategy when it did not work, solving the problem again with a different strategy, the ability to use several strategies simultaneously for solving a problem, checking the correctness of the solution with a different strategy. The last theme, inter-task strategic flexibility, consisted of only one indicator named changing strategies when encountering different problems. [ABSTRACT FROM AUTHOR]

In 2005, Osler et al. showed that algebraic irrational numbers of the form k n for n , k ∈ N can be approximated using specific Greek ladders (Osler, T. J., Wright, M., & Orchard, M. (2005). Theon's ladder for any root. International Journal of Mathematical Education in Science and Technology, 36(4), 389–398. https://doi.org/10.1080/00207390512331325969). In 2018, Herzinger et al., showed that every algebraic number of degree 2 or less can be approximated using rung ratios generated by a specific Greek ladder (Herzinger, K., Kunselman, C., & Pierce, I. (2018). Greek ladders via linear algebra. International Journal of Mathematical Education in Science and Technology, 49(7), 1119–1132. https://doi.org/10.1080/0020739X.2018.1440326). Using techniques from linear algebra, in particular the companion matrix of a polynomial, we will prove that every real algebraic number can be approximated by a Greek ladder, as well as show how to construct such a ladder. [ABSTRACT FROM AUTHOR]

This article reviews and proves nine necessary and sufficient properties of orthogonal circles. The first six properties are known to some degree, while the last three are unknown, to the best of our knowledge. [ABSTRACT FROM AUTHOR]

We investigate aspects of reform enacted in 66 class observations from 18 institutions collected as part of the National Study of University Calculus in the US. We examined four aspects related to how problems were solved in class: the extent of student involvement, the representations and technology used, and specific features of the problems solved. As a whole, we observed low enactment of reform-oriented practices in these lessons, with more variation within rather than across institutions. We describe lessons that illustrate different ways in which reform aspects were enacted and suggest some implications for further studies. [ABSTRACT FROM AUTHOR]

This work presents an evaluation of a unified Calculus 1 course in a large Brazilian federal university. By course unification we mean having all the sections following the same weekly schedule, under the same syllabus and common assessment. We analyse two sets of data: pass rates relative to the 9 semesters prior to the unification and to the following 12 semesters; results of objective tests based on the item response theory, taken by the students during the period following unification. Data analysis gives us the average students' performances grouped by programme for which Calculus 1 is compulsory and shows us no difference in pass rates over the two periods. These two results help us to point out the problems in the implementation of our unification. We also explain traditional low pass rates for some sections and why our Calculus 1 may not be adequate to some of the programmes for which it is compulsory. More importantly, we show that among the students' abilities assessed by the objective tests, two seem to be necessary for the learning of Calculus and proficiency in Calculus 1 appears to be related to seven of these abilities. [ABSTRACT FROM AUTHOR]

This study addresses high school students' conceptions of mathematical definitions of congruent and similar triangles. The findings indicate that many of the participants differentiated between definitions and theorems and did not always accept the congruent and similar triangles theorems as formal definitions of congruency and similarity. Based on the participants' explanations of their responses and from the interviews performed, it appears that two issues prevented some participants from accepting or preferring these theorems as definitions. The first was a concern for uniformity: there is only one known and accepted definition of each concept. The second was a focus on the essence of the concepts: the essence of the concepts of similarity and congruency lies primarily in the lengths of the sides of a triangle. The students who accepted these theorems as formal definitions explained their reaction as arising from the equivalence and from the theorems including necessary and sufficient attributes. Students' difficulties in understanding the characteristics and roles of mathematical definitions of geometric concepts affected their understandings of mathematical and geometric definitions. This behaviour indicates a tendency to interpret the content of theorems incorrectly and an inability to unpack the logical structure of the theorem. [ABSTRACT FROM AUTHOR]

We discuss a practical method for assessing mathematical proof online. We examine the use of faded worked examples and reading comprehension questions to understand proof. By breaking down a given proof, we formulate a checklist that can be used to generate comprehension questions which can be assessed automatically online. We then provide some preliminary results of deploying such questions. [ABSTRACT FROM AUTHOR]

We argue that progress in the area of research on mathematics teacher responses to student thinking could be enhanced were the field to attend more explicitly to important facets of those responses, as well as to related units of analysis. We describe the Teacher Response Coding scheme (TRC) to illustrate how such attention might play out, and then apply the TRC to an excerpt of classroom mathematics discourse to demonstrate the affordances of this approach. We conclude by making several further observations about the potential versatility and power in articulating units of analysis and developing and applying tools that attend to these facets when conducting research on teacher responses. [ABSTRACT FROM AUTHOR]

Combining mathematical and physical understanding in reasoning is difficult, and a growing body of research shows that students experience problems with the combination of physics and mathematics in reasoning beyond the introductory level. We investigated students' reasoning about boundary conditions (BCs) for the diffusion equation by conducting exploratory task-based, think-aloud interviews with twelve undergraduate students majoring in physics or mathematics. We identified several difficulties students experienced while solving the interview task and categorized them using the conceptual blending framework. This framework states that in reasoning, people draw from separate input spaces, in this case the mathematics and the physics input space, to form a blended space, where they make connections between elements from these spaces. To identify difficulties, we used open coding techniques. We observed few difficulties in the physics space. In the mathematics space, we identified several difficulties that we clustered in two main groups: findings about the mathematical meaning of BCs, and findings about reasoning with functions of two variables. Lastly, we identified four ways in which blending failed. Starting from our findings, we formulate recommendations for teaching and future research. [ABSTRACT FROM AUTHOR]