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The exploration of children's drawings as mathematical representations is a current focus in early years mathematics education research. This paper presents a qualitative analysis of 72 kindergarten to Grade 3 (5 to 8 years old) children's drawings produced during problem-solving tasks centred on multiplicative strategies. Existing frameworks for the developmental sequence of mathematical drawings and the progression of children's strategies for multiplicative situations were an interpretive lens through which to analyse the drawings. Children used pictographic and iconic drawing types to represent the "story" in the problem and the multiplicative strategies employed to solve the tasks. Exploration of the children's drawings suggested that as children's drawings become more structural, schematic in nature, it may be easier for children to show their understanding of the structural elements of multiplicative relationships. Results revealed that structural elements of multiplicative relationships were more easily seen in iconic representations; however, both pictographic and iconic drawings were useful to observe counting, additive, and multiplicative strategies when mathematical elements of the problem were visible. Additional representations attached to the drawings (e.g. numerical) were needed to confirm children's strategies when their drawings lacked structure. These findings have implications for how young children's drawings are interpreted by classroom teachers. The interpretation of these drawings suggested that some children may not yet realise how their drawings in mathematics need to shift from illustrations of the problem's story context to representing mathematical ideas and processes — which requires intentional teaching of the purpose of drawings for mathematical contexts. [ABSTRACT FROM AUTHOR]

In recent years, there has been an increasing amount of literature on online teaching and learning. Teaching mathematics in the fourth industrial revolution offers entry-level mathematics teachers formidable challenges. Despite using various teaching aids in explaining the learning area, entry-level mathematics teachers, those with zero to three years of teaching experience, have experienced challenges in blended teaching during the COVID-19 pandemic. Such challenges could support the preconceived notion identified by others that the field of learning (mathematics) is difficult. This paper hinged on constructivist epistemology and investigated the challenges that entry-level mathematics teachers face in conducting blended teaching in the Sedibeng West District of Gauteng Province in the Republic of South Africa. Eight entrylevel mathematics teachers were purposely sampled from four selected schools that participated in the article. This article adopted a case study design and responses were analysed thematically. The focus group interviews were used as tools to collect data in this study. The study revealed a variety of perceptions that entry-level mathematics teachers shared about the impact of blended teaching in mathematics learning. These included, but were not limited to, the use of teaching programs or software, the effects of load shedding and blended teaching challenges related to learners' performance and behaviour. The results of this study could provide program developers, subject advisors, school principals with other members of the management team, and mathematics teachers to support the entry-level mathematics teacher's confidence, sense of future and communication skills, as well as foster multigenerational connections in blended teaching. [ABSTRACT FROM AUTHOR]

It is the intention of this special issue to provide an opportunity for mathematics education researchers from all around the world to offer their complementary accounts and to engage in reflective scholarly exchange regarding research into the complexity of classroom practice. Other major international projects informed by the LPS, and led by Clarke, cohered around a desire to understand, model and ultimately optimise the practices of the mathematics classroom. For most of his career Clarke's research activity centred on capturing the complexity of classroom practice through a programme of international video-based classroom research in more than 20 countries. The work of David Clarke The work of David Clarke profoundly influenced the field of mathematics education. [Extracted from the article]

This paper draws from a broad study that explored mathematics teachers’ engagement with textbooks within the Lesson Study context. In the current paper, we report on teachers’ use of mathematics textbook activities to inform their instructional decisions during collaborative lesson planning of numeric and geometric patterns. The study took place within the Lesson Study (LS) context – a teacher development practice that emphasises, inter alia, collaborative lesson planning. Data was collected by observing teachers during collaborative lesson planning and through interviews. Data analysis was informed by the Mathematics Knowledge for Teaching (MKT) framework and Modes of teacher engagement with the textbook. Findings suggest that teachers’ instructional decisions are mainly stimulated by the textbooks, teachers use textbook activities with fidelity, and they seldom adapt the activities drawn from the textbooks. However, where adaptation occurs teachers do it superficially. We contend that the capacity to conduct in-depth interrogation of the text is a derivative of mathematics subject matter knowledge (SMK) as well as pedagogical content knowledge (PCK). Therefore, the widespread adoption of text without interrogation could be indicative of the lack of requisite content knowledge and pedagogical skills. [ABSTRACT FROM AUTHOR]

Mathematical modelling competencies have become a prominent construct in research on the teaching and learning of mathematical modelling and its applications in recent decades; however, current research is diverse, proposing different theoretical frameworks and a variety of research designs for the measurement and fostering of modelling competencies. The study described in this paper was a systematic literature review of the literature on modelling competencies published over the past two decades. Based on a full-text analysis of 75 peer-reviewed studies indexed in renowned databases and published in English, the study revealed the dominance of an analytical, bottom-up approach for conceptualizing modelling competencies and distinguishing a variety of sub-competencies. Furthermore, the analysis showed the great richness of methods for measuring modelling competencies, although a focus on (non-standardized) tests prevailed. Concerning design and offering for fostering modelling competencies, the majority of the papers reported training strategies for modelling courses. Overall, the current literature review pointed out the necessity for further theoretical work on conceptualizing mathematical modelling competencies while highlighting the richness of developed empirical approaches and their implementation at various educational levels. [ABSTRACT FROM AUTHOR]

The paper addresses the potential relationships between shared authority in mathematics classrooms and students' mathematical reasoning. Even though tensions and challenges related to shared authority are explicated in the literature, there are few examples of how these issues play out in mathematics teaching. We investigate the case of a mathematics teacher attempting to share authority as well as applying several moves recognized as supporting meaningful student learning. Data has been collected in a fourth-grade Norwegian classroom and is analyzed by means of open coding, inspired by literature. We identify the moves used by the teacher, and we rank these moves along two dimensions: (1) their potential to support mathematical reasoning and (2) their potential for sharing authority. From this, we uncover how a teacher's work of orchestrating mathematical discussions involves moves in all four quadrants, and we discuss how the interplay of moves affects the authority structures and the collaborative reasoning in the classroom. [ABSTRACT FROM AUTHOR]

Student motivation has emerged as a major issue in mathematics education in recent years. Utilizing the capabilities of contemporary technology and the accessibility of a STEM environment is crucial for the creation and application of educational materials. Integration of the created resources will increase the interest of students to pursue in-depth studies of science and math. In this paper are presented the results of the conducted study on the effectiveness of using problem-based learning (PBL) modules to learn mathematics. The study was carried out in Israel with the participation of teachers and ninth graders. As part of our research, we created a novel program and conducted an experimental study, evaluated its impacts on the designated target groups, and analyzed how the program affected student subgroups. Pre- and post-program questionnaires measured ability feeling, relevance perception, and motivation. The results of the analysis of the students indicate a considerable rise in the motivating factors associated with self-esteem and the utility of mathematics training. [ABSTRACT FROM AUTHOR]

In the research reported in this paper, using a modified version of the interconnected model of teacher professional growth (IMTPG) proposed by Clarke and Hollingsworth (Teaching Teacher Educ 18:947–967, 2002), we aimed to understand the learning dynamics in a lesson study of a group of five teachers of grades 5–6, during their work around mathematical tasks and in the elaboration and analysis of a diagnostic test of students' knowledge. We draw on examples from a lesson study that took place in a school in Lisbon, involving five teachers that constituted the whole mathematics group of the school. The main feature of our adaptation is the consideration of a Group Domain instead of the Personal Domain of the original model. In the original model, the Domain of Practice had a broad definition but usually it was restricted to classroom experimentation. In the research reported in this paper, the lesson study sessions were taken as part of the Domain of Practice. For data collection, there was audio and video recording of the lesson study sessions and of the research lesson and the writing of a research journal. Data analysis was done using the modified IMTPG. The results suggest that this model is helpful in understanding the processes that make lesson study a valuable professional development setting, contributing to responding to the question of 'how lesson study works' (Lewis et al., Theory and practices of lesson study in mathematics: an international perspective, Springer, New York, 2019). In addition, the use of this model showed that the lesson study promoted the teachers' development regarding knowledge of didactics, especially knowledge of students and their learning processes and of teaching practice. [ABSTRACT FROM AUTHOR]

Teacher quality in primary schools in South Africa is a serious concern, and how best to prepare teachers in initial teacher education—with regard to mathematics—remains an open and vexing question. There is a confluence of challenges facing the mathematics education community in South Africa: poor conceptual understanding of mathematics amongst teachers; poor attainment in Primary Teacher Education mathematics tests at both first and fourth year levels; and few initial teacher education programmes having evidence of improving knowledge for teaching mathematics. In addition, few lecturers in initial teacher education make explicit the theoretical tenets informing their course design for mathematics. In response to the latter challenge, this paper makes explicit the design choices made in the Emergent Number Sense first year mathematics module in Bachelor of Education degrees, used across six universities in South Africa. It then theorises the choice to commence the B.Ed mathematics programmes with two pathways into number knowledge: (1) a counting pathway, making use of discrete objects or events which can be quantified by counting in ones; and (2) a measurement pathway, making use of continuous extents which can be quantified by introducing a unit of measurement. Each has a distinct metaphor, strategy for the reification of the 10 and supporting representation. This offers the lecturers using the course in their own design trials, and initial teacher educators beyond the Maths4Primary Teachers collective, a conceptual framework for this design choice, which can now be critiqued and further improved. [ABSTRACT FROM AUTHOR]

Informal Australian discussion suggests that calculators are still widely regarded as devices for calculation, consistent with their naming, and are rarely recognised as potentially helpful to support students learning mathematics. In this paper, Barry Kissane explores some of the ways in which modern calculators might support students learning about irrational numbers, of key significance to the study of the real number system, characterising lower secondary school mathematics. [ABSTRACT FROM AUTHOR]

Engaging students in processes of mathematizing and active knowledge organization (instead of telling and rehearsing ready-made mathematics) is a key demand for high quality mathematics instruction. Although many mathematically rich context problems have been designed and published in textbooks, their impact on regular mathematics classrooms remains limited, as teachers face challenges in shaping and enacting the important phase of knowledge organization. The 15-year project KOSIMA presented in this paper was aimed at supporting teachers in really enhancing these processes by means of a textbook for Grades 5–10. The paper provides an overview of different studies conducted during the project in order to show how the project developed a research base for the support of teachers. The paper also shows that different research approaches need to be combined to establish an appropriate research base: In iterative design research cycles of topic-specific didactical design research, learning environments were designed and investigated with the aim of understanding the obstacles encountered in implementing the desired processes. In this paper, we focus on an example of one particular obstacle, namely, teachers' need of support in engaging students in processes of active knowledge organization. This obstacle was overcome by developing new types of tasks for active knowledge organization. The theoretical outcome of the design research resulted in two hypotheses for improving relevant features of the design of these tasks, which were tested in two controlled trials. Finally, a longitudinal field study of 312 students gave first indications that the classes that worked with the textbook had significantly higher learning gains than classes using other textbooks. In sum, we can show that textbooks can support changes in classroom practices when the obstacles that occur for both teachers and students are carefully investigated and treated. [ABSTRACT FROM AUTHOR]

Creativity, attitudes, and conceptions of teachers regarding creativity in teaching mathematics have been examined in this study. The instrument used is a questionnaire consisting of three groups of questions. The first group of questions includes questions to gather general information about the teachers, while the second and third groups of questions include questions to gather data about teachers' attitudes and conceptions of creativity in teaching mathematics. The questionnaire was completed by 115 class and Maths teachers. After data analysis, the findings indicate that the majority of teachers have knowledge about creativity and believe that teachers are responsible for developing students' creativity. From the proven hypotheses of this paper, we can conclude that if the teachers know creativity, then they will teach the students to be creative, which they will further develop creativity, as a requirement of the 21st century. [ABSTRACT FROM AUTHOR]

Classroom management is an emotionally demanding task for mathematics teachers, especially if students exhibit frequent discipline problems. Intense classroom conflicts can result in persistent latent emotional dispositions, such as fear or anger, that teachers are not directly aware of but that have a strong influence on their classroom management. However, little is known about the relationship between mathematics teachers' latent emotions and their classroom management strategies. This paper reports on findings from an exploratory, video-based single case study in an urban secondary school in Germany. One mathematics double lesson (95 minutes in total) was videotaped to conduct a depth-hermeneutical analysis of identified classroom management strategies and the corresponding latent teacher emotions. The results suggest that mathematics teachers use classroom management strategies not only to establish orderly lessons but also to regulate intense latent emotions that arise during classroom teaching. [ABSTRACT FROM AUTHOR]

Deep learning is an important concept introduced into modern learning science. It is different from the surface learning of mechanically and passively acquiring knowledge and storing individual information but emphasizes learners' active and critical learning. It wants them to understand the full meaning of what they have learned. By establishing a link between existing knowledge and new knowledge, it transfers existing knowledge to a new environment, makes decisions, and solves problems. Deep learning plays an important role in students' learning. Deep learning ability is the key factor affecting the quality of learning and the development of students' academic ability. The quality of in-depth teaching is difficult to guarantee, which requires a complete, comprehensive, and evaluation system to evaluate it. This paper introduces the analytic hierarchy process to weight the indexes in mathematics deep learning and puts forward some suggestions on creating an environment for deep learning. The experimental results show that teachers' teaching accounts for the highest proportion of primary indicators, reaching 67%. Multimedia resources account for the highest proportion of secondary indicators, reaching 73.01%. This paper then puts forward some suggestions for indicators with large weights. [ABSTRACT FROM AUTHOR]

This is a methodological critique of research by the Best Practice in Grouping Students (BPGS) project claiming teacher bias in allocating students to first‐year secondary school mathematics teaching sets (British Educational Research Journal, 45(4), 873–879). The research assumes that bias could be shown by non‐random relationships between 'misallocations' to sets and memberships of gender, ethnic and/or socio‐economic subgroups. This paper questions the authors' prescriptions for correct set placements and demonstrates two ways in which non‐random relationships between subgroup memberships and (alleged) misplacements could be generated despite all students with similar scores within a school being given the same chance of a set position deemed correct by the researchers. It suggests that the BPGS project results are largely artefacts of their evaluation approach. No claim is made herein that teacher bias plays no part in set allocation, merely that, if it does, the approach by Connolly et al. could not evidence it. A lesson from the critique is that, before drawing inferences from data aggregated from diverse clusters, it is wise for researchers to investigate the way the data are generated at source and patterned by aggregation. [ABSTRACT FROM AUTHOR]

This paper determined the effectiveness of Science Teachers Academy for the Regions (STAR) trainings on the teaching practice of participants in Region 6 - Western Visayas. The research participants of this study were STAR trainees who attended the STAR Trainings from 2014-2017. The researchers employed an explanatory sequential mixed methods design. Quantitative data were collected through an online survey with regards to the effectiveness of their teaching practice. The extent and quality of adaptation of STAR training content and skills in their practice through focus group discussions consisting of 8 participants each in Science and Mathematics were determined. The methods of data collection were online survey, Focus Group Discussion (FGD), classroom observation, and interview guide questions for In-depth interview. The findings of the study showed that Science and Mathematics teachers showed a great change in their teaching experiences after undergoing Project STAR training. Moreover, the participants insights about the STAR trainings were the different inquiry-based activities that they applied in their classroom, application of the 7Es in the lesson plan, improvement of the learning pedagogy, concepts and skills were easily learn through interdisciplinary contextualization of teachers in preparing a lesson plan was effective in ensuring the appropriate strategies in the lesson. Furthermore, students developed and enhanced their science process skills from the participant's learnings in the STAR trainings. The quality of the adaptation of STAR trainings revealed that trainees had a great transformation from the traditional method of teaching to the inquiry-based method of teaching. The STAR trainings were unique in such a way that participants did not feel that they are bored, it is interactive and less stressful, it is specific and lots of hands-on activities, and it focus more in the content on how to apply it in the classroom. The STAR trainings made a great impact to the STAR trainees in their professional growth by giving them an opportunity to become speakers in their respective institutions and by living the expectations of other people and enabling them to study more. [ABSTRACT FROM AUTHOR]

Over the past decade, teaching mathematics ambitiously has received increased attention. In this paper, we argue that to materialize this vision in contemporary classes we need to understand how practicing teachers experiment with certain aspects of this teaching and what challenges they encounter. Toward this end, we focus on an aspect of teaching ambitiously—designing and using enablers and extenders—and examine how four elementary schoolteachers experimented with it in their practice while also participating in a video-club setting. Drawing on a corpus of data including lesson plans, videotaped lessons, pre- and post-lesson interviews, end-of-program interviews, and videotaped video-club sessions, and looking across the four cases, we sketch how these teachers worked with enablers and extenders and the challenges they faced. Our analysis helped identify certain components entailed in working with enablers and extenders during the phases of lesson planning and enactment; it also yielded a classification of observed and reported challenges encountered as teachers engage with this work. This mapping of work associated with designing and using enablers and extenders, along with the classification of challenges generated, can inform professional learning development attempts aiming to support teachers enact ambitious teaching by identifying and naming separate components of practice that merit consideration and by providing insights into the types of scaffolds needed to support teachers in teaching ambitiously. [ABSTRACT FROM AUTHOR]

In the Hungarian "Guided Discovery" approach to teaching mathematics, teachers' planning work plays a crucial role. Teachers following the approach develop teaching trajectories based on "series of problems". This work includes the choice, creation, transformation, organization and networking of problems with regard to various teaching objectives. However, most of this planning work remains implicit; the structure and rationale of the teaching trajectories are often inaccessible for an external observer (including, for example, student teachers). An ongoing project based on teacher–researcher collaboration aims to reveal the hidden characteristics and principles of teachers' work with series of problems through the creation of innovative resources for teachers. In this paper, we present the design of this collaborative work, a process we designate "reverse engineering", and analyse the development process of resources, focusing particularly on the emergence of new vocabulary and representation tools serving as mediators in the communication between teachers and researchers. [ABSTRACT FROM AUTHOR]

In this paper, we report an enquiry into elementary preservice teachers' learning, as they engage in doing mathematics for themselves. As a group of researchers working in elementary Initial Teacher Education in English universities, we co-planned and taught sessions on growing pattern generalisation. Following the sessions, interviews of fifteen preservice teachers at two universities focused on their expressed awareness of their approach to the mathematical activity. Preservice teachers' prospective planning and post-teaching evaluations of similar activities in their classrooms were also examined. We draw on aspects of enactivism and the notion of reflective "spection" in the context of teacher learning, tracing threads between preservice teachers' retro-spection of learning and pro-spection of teaching. Our analysis indicates that increasing sensitivity to their own embodied processes of generalisation offers opportunities for novice teachers to respond deliberately, rather than to react impulsively, to different pedagogical possibilities. The paper contributes a new dimension to the discussion about the focus of novice elementary school teachers' retrospective reflection by examining how deliberate retrospective analysis of doing mathematics, and not only of teaching actions, can develop awarenesses that underlie the growth of expertise in mathematics teaching. We argue that engaging preservice teachers in mathematics to support deliberate retrospective analysis of their mathematics learning and prospective consideration of the implications for teaching can enable more critical pedagogical choices. [ABSTRACT FROM AUTHOR]

Another significant research area relevant to teacher awareness and its development is teacher reflection on teaching. In this process the role of the researcher/mathematics teacher educator involves development of awareness in mathematics, teaching, teacher education and research. [Extracted from the article]

Teacher expectations not only relate positively to student achievement, but also to student beliefs such as their self-concept. Nevertheless, most studies focus on the relations with student achievement, followed by studies on beliefs. Beliefs are a significant determinant of academic success and can include student self-concept or emotions, such as anxiety. The extent to which anxiety can be influenced by teacher expectations has been investigated in very few studies. This paper examined how teacher expectations related to changes in student achievement, self-concept, and anxiety in mathematics within a school year. The data were from a longitudinal study "Outcomes of teacher education", funded by the Swiss National Science Foundation and based on 28 teachers and 509 primary school students. Teacher expectations were operationalized using the residual approach. Student mathematics achievement was assessed via a standardized mathematics test and their self-concept as well as their mathematics anxiety via a questionnaire. The multi-level structure was considered in the analyses, because the interclass-correlation of student mathematics achievement exceeded the critical value of 10%. The results showed that teacher expectations were positively related to student achievement as well as self-concept and negatively related to anxiety towards mathematics. The change in the explained variance was small (self-concept and anxiety) to large (achievement). The results extend findings on the expectation effect in the classroom, as they focus not only on student achievement but also on student beliefs and are discussed regarding their significance for academic success. [ABSTRACT FROM AUTHOR]

This paper focuses on the importance of identifying students' Prior Knowledge (PK) for the teaching and learning of mathematics. Specifically, we present a framework that exemplifies the core elements teachers need to consider in relation to the purpose, design and implementation of learning experiences that enable PK to be activated and assessed. The paper concludes by offering practical insights for the design and implementation of a PK learning experience using this framework, with a geometry-based example. [ABSTRACT FROM AUTHOR]

In this paper we report on an activity developed in the context of the Erasmus+ PUNTE Project, using the Poly-Universe material, which led to the learning of logic and mathematics among the 10th grade students of a school in the central region of Portugal. Three teachers, two of mathematics and one of philosophy, and 21 students of secondary education participated in the study. The data obtained, following the observation and application of questionnaires and pre- and post-tests, reveal the usefulness of Poly-Universe for the acquisition of notions about logical operations, for analysing arguments and their validity and solving mathematical problems, motivating the students for carrying out the activity. [ABSTRACT FROM AUTHOR]

It is wildly held belief that one way to interest students in mathematics is to use real-world examples. Mathematics teachers are aware of the lack of interest in mathematics by many students and seek ways to improve their motivation. It seems reasonable to suggest that demonstrating to students the ubiquitous nature of mathematics may help overcome this lack of interest. This paper examines the literature pertaining to the use of real-world examples and suggests, that, unless carefully chosen, their use may not be successful in improving students' motivation. [ABSTRACT FROM AUTHOR]

Improving mathematical skills is a priority in England, and a series of policy levers and government change projects have focused on improving mathematical outcomes in further education (FE) in recent years. Yet little is known about the mathematics teacher workforce that supports these students on vocational and technical programmes. This paper addresses this knowledge gap by examining important features of the mathematics teacher workforce in colleges and the implications for recruitment, initial training and ongoing professional development. We report findings from a national survey targeted at all mathematics teachers in around one-sixth (N = 31) of England's general FE colleges. Teachers have transitioned into FE from three main areas: another career, curriculum area, or educational context. We discuss the varied assets and training needs of these three subgroups and argue that the mathematics teacher cohort in FE should be seen as distinct from that in either primary or secondary education. We highlight the bimodal distribution of mathematics qualifications amongst this workforce; those having level 2 mathematics qualifications being more likely to be teaching functional skills mathematics courses than GCSEs. The rapid expansion of the mathematics teacher workforce that followed the changes to the condition of funding in 2014 has come disproportionately from teachers who were previously working in schools. This, together with a trend of funnelling increasing numbers of FE students into academic mathematics rather than functional skills courses, raises important questions about the nature of mathematical education for learners on vocational and technical programmes, and the teachers thereof. [ABSTRACT FROM AUTHOR]

Background: Eye movement modelling examples (EMME) are demonstrations in which learners' not only see a model's (e.g., a teacher's) task performance on a computer screen (as in regular video examples) but also the model's eye movements (represented as moving coloured dots overlaid on the screen). Thereby EMME help guide learners' attention towards the relevant information and can model cognitive strategies which are otherwise unobservable for learners. Objectives: This study investigated whether EMME can help to learn deductive reasoning strategies and how the presence/absence of a teacher's verbal explanation affects learning from EMME. Methods: Secondary education students (N = 137) were randomly assigned to study video examples under one of four conditions in a 2 (EMME: yes/no) x 2 (verbal explanations: yes/no) between‐subjects design. Results and Conclusions: Results revealed only a beneficial effect of the presence of verbal explanations on performance on the practice problems, but no pretest‐to‐posttest learning gains. Implications: Seeing the teacher's eye movements does not appear to enhance learning of deductive reasoning. The presence/absence of the teacher's verbal explanation does not seem to affect learning deductive reasoning. Lay Description: What is currently known about the subject matter?: Eye movement modelling examples (EMME) are demonstrations in which learners not only see a teacher's task execution on a computer screen (as in video examples) but also the teacher's eye movements.The eye movements in EMME were visualized as moving coloured dots overlaid on the screen recording.EMME help direct attention to relevant information and enhance learning; it is not yet known if this also applies to reasoning problems.It is unknown if and how presence/absence of a teacher's verbal explanation affects learning from EMME. What our paper adds to this?: Secondary education students watched EMME or regular video examples with/without a teacher's verbal explanation.The examples demonstrated how to solve deductive reasoning problems (adapted version of the Mastermind game).Performance and learning gains were examined and revealed no differences between conditions. The implications of study findings for practitioners: Seeing the teacher's eye movements does not appear to enhance learning of deductive reasoning.The presence/absence of the teacher's verbal explanation does not seem to affect learning deductive reasoning. [ABSTRACT FROM AUTHOR]

Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers' knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers' backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed. [ABSTRACT FROM AUTHOR]

This paper reports on nine elementary, middle, and high school in-service teachers who participated in a series of workshops aimed at exploring the wonder, joy, and beauty of mathematics through the creation and application of digitally fabricated tools (i.e., laser-cut and 3D printed). Using the Technological Pedagogical and Content Knowledge (TPACK) framework to investigate technological, pedagogical, contextual, and content knowledge, researchers applied qualitative methods to uncover the affordances and constraints of teaching and learning math concepts with digitally fabricated tools and examined how the workshops supported broadening participation in mathematics by focusing on the connections between mathematical inquiry, nature, and the arts. Affordances include opportunities for hands-on learning, visual support at the secondary level, and real-world connections that go beyond the state standards. Barriers include purchasing a laser-cutter, ventilation and noise issues, time constraints, misalignment with school and district priorities, and a lack of administrative support. All participants indicated that they were interested in additional workshops focused on designing their own digitally fabricated mathematics tools that better align with their grade level(s) and standards. [ABSTRACT FROM AUTHOR]

Mathematical structure is central to the interconnectedness of numerical, spatial or logical relationships, but it is not known how well teachers understand this concept or implement it in pedagogical practice. In this paper, we examine three junior secondary mathematics teachers' understanding of mathematical structure and observe whether they apply it in their teaching. Case study data comprised teacher interviews and mathematics lesson observations. Analysis of interviews and teacher-directed communication (utterances) in lessons utilised an emergent framework for categorising mathematical structure: connections to other learning (C), recognising patterns (R), identifying similarities and differences (I) and generalising and reasoning (G). Findings from the interviews indicated that teachers supported the development of mathematical structure, but the interview responses were not generally reinforced by their utterances in mathematics lesson observations. Analysis of teacher utterances revealed superficial understanding and use of mathematical structure although there was some evidence of CRIG components in the teacher-directed communication for individual cases. [ABSTRACT FROM AUTHOR]

Parents' and teachers' beliefs and evaluations of young people are important. Using a feminist institutionalist perspective, and drawing on rich data from one in seven nine-year-old children in Ireland, this paper examines mothers' (who make up the overwhelming majority of primary care-givers) and teachers' perceptions of boys' and girls' mathematics performance. The evidence shows that girls' mathematics performance is underestimated by both relative to boys'. Mother's gender bias was evident among high performing children, at all levels of children's academic self-concept, and among mothers with at least third level education. While the judgements reflect children's actual performance and engagement, a notable gender gap remains. It is suggested that the results reflect gender stereotypes: overestimating boys' and underestimating girls' mathematics achievements. The article indicates the importance of the informal dimension of institutions and the part played by women in the effective devaluation of girls by endorsing gendered stereotypes. Women teachers are less likely to rate children highly in mathematics, taking account of performance: arguably reflecting their own lack of confidence in mathematics assessment. The findings raise concerns for girls' futures since mathematics is seen as an indicator of intelligence. Given the move towards teacher-assessed grading during COVID-19, understanding, and challenging, gender-stereotyping is pressing. [ABSTRACT FROM AUTHOR]

Mathematics teaching efficacy is an important construct as confidence in one's ability to teach influences teaching practices. This paper explores pre-service primary teachers' mathematics teaching efficacy on entry to initial teacher education and the extent that pre-tertiary mathematics experiences and resultant beliefs affected their mathematics teaching efficacy. A mixedmethods approach combined the Mathematics Teaching Efficacy Beliefs Instrument (N=420) and qualitative interviews (N=30). The findings suggest medium personal mathematics teaching efficacy among participants with limited conceptions of what mathematics teaching involves. While uncertain regarding their immediate teaching ability, participants reported confidence regarding their potential. Mathematics teaching outcome expectancy was high; however, an undercurrent of conviction exists that external factors, most notably learners' natural mathematical ability, are critical to student learning. [ABSTRACT FROM AUTHOR]

In the research reported in this paper we investigated teachers' changes when adopting and adapting to emergency online teaching during the COVID-19 pandemic, from the perspective of the Interconnected Model of Professional Growth (IMPG). By adapting complementary accounts methodology to research into teachers' changes when addressing the unexpected migration to online teaching, an integrated data set, including online teaching videos, teacher daily reflections, and teacher interviews from two purposefully selected teachers over two weeks of online teaching, was collected and analyzed qualitatively. Both teachers encountered different difficulties and thus had different knowledge changes displayed in different change routes. For the experienced teacher, students' mistakes in homework and her online teaching practice triggered her knowledge changes. For the young teacher, the online video lessons, relevant resources on the Internet and students' performance were her primary sources that triggered the changes of her knowledge for teaching. These differences between the experienced teacher and young teacher provide evidence of the complexity of teacher's professional growth, which is related to a variety of external and internal factors. This study demonstrates how the IMPG model helps uncover teachers' changes in such an unprecedented virtual-teaching environment. Finally, the implications of this study for teacher professional development in general are discussed. [ABSTRACT FROM AUTHOR]

This cross-sectional study looks into how students' perceived teacher support is linked to achievement emotions in mathematics via academic control and value. The data was gathered from 602 secondary students (13-16-year-olds) in 16 mathematics classrooms from three public schools in Jiangsu province, China. Participants were asked to respond to the paper-pencil questionnaire assessing students' perceived teacher support, cognitive appraisals, and achievement emotions in mathematics. Findings showed that cognitive appraisals played complete mediation roles in teacher support and achievement emotions. Intrinsic value and mathematics self-concept were shown to serve a positive role in teacher support and achievement emotions, whereas extrinsic value was found to play a negative one. Besides, teacher support showed the greatest indirect effect on achievement emotions through mathematics self-concept. Lastly, findings indicated the great relevance of teacher support to students' anger, boredom, enjoyment and relaxation in mathematics classrooms. [ABSTRACT FROM AUTHOR]

Early number sense requires teachers to have explicit knowledge and understanding to develop learners' skills and knowledge of early number sense. Early grade teachers play a critical role in ensuring learners develop appropriate number sense. The South African National Curriculum Statement allocated 65% of time towards teaching Numbers Operations and Relationship which forms the basis of number sense. Most South African teachers in the early grades are not mathematics specialists and are dependent on support from their school's head of department and the subject curriculum specialist from the district offices. This paper explored teachers’ experience of the support they received from their head of department in the early grades and the curriculum specialist for the Foundation Phase. This study used the qualitative approach and Fuller's Concerned Based Model for Teacher Development as a theoretical lens to investigate the phenomena. This was a single case study research conducted at one school in the Gauteng Province with three Foundation Phase teachers, one head of department and one subject advisor. Findings revealed teachers understanding and knowledge of number sense and their ambivalent views regarding the support they received. All teachers concluded that the support they received from their department head and subject advisor were partially beneficial to them. They were very concerned with level and depth of the training provided to them. They were unhappy with the once-off short training. Teachers recommended they needed more intensive training on pedagogical content knowledge and content knowledge. Professional development training should be ongoing and that the network learning communities should be established in each district. [ABSTRACT FROM AUTHOR]

Teaching mathematical modeling is a demanding task. Thus, fostering teachers' competencies in this regard is an essential component of teacher education. Recent conceptualizations of teachers' competencies include situation-specific skills based on the concept of noticing, which is of particular interest for the spontaneous reactions needed when teaching mathematical modeling. The study described in this paper aims to analyze the development of a video-based instrument for measuring teachers' noticing competencies within a mathematical modeling context and obtain evidence for the validity of the instrument. Three kinds of validity are examined in three different studies: content validity, elemental validity and construct validity. Indicators for content validity could be found through different expert ratings and implementation with the target group, where participants were able to perceive all relevant aspects. The qualitative analysis of participants' reasoning, which is consistent with the coded level, indicates elemental validity. Moreover, the results of the confirmatory factor analysis suggest construct validity with one overall factor of noticing competence within a mathematical modeling context. Taken together, these studies imply a satisfactory validity of the video-based instrument. [ABSTRACT FROM AUTHOR]

Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers' professional mathematical modeling competencies in three different regions—Germany, Mainland China, and Hong Kong—where educational and cultural traditions differ, including the role of mathematical modeling. In total, 232 pre-service mathematics teachers from the three regions completed a modeling task covering mathematics content knowledge (MCK) of modeling and mathematical pedagogical content knowledge (MPCK) of modeling. The results show that pre-service teachers from Germany demonstrated the strongest MCK and MPCK of mathematical modeling; by contrast, pre-service mathematics teachers from Mainland China and Hong Kong demonstrated relatively weaker MCK and MPCK of mathematical modeling. MCK and MPCK of mathematical modeling were also found to be unevenly developed at different competence levels for the three regions. These differences may be attributed to the history of mathematical modeling in mathematics curricula, teacher education, and teaching culture in these three regions. [ABSTRACT FROM AUTHOR]

The redesign of national curricula across the Anglophone world since the 1990s is demonstrably shaped by common policy trends. Focusing on the profound and uncritiqued changes that have been implemented in New Zealand education, this paper provides a critical commentary on the characterising features of the current New Zealand mathematics curriculum, describing a context within which mathematics education at schools is severely compromised. Drawing on the evidence available from large-scale international indicators, such as PISA and TIMSS, to benchmark associated curriculum changes implemented by the New Zealand government, we hypothesise that the ongoing decline of student mathematical achievement is the result of four main interdependent features which characterise the New Zealand curriculum. The features are (1) its highly generic nonprescriptive nature, (2) a commitment to teacher autonomy in curriculum knowledge selection, (3) competency-based outcomes approach, and (4) a commitment to localisation in curriculum selection. Recognising socio-political forces and ideological and intellectual ideas associated with those forces, we discuss each characterising feature, in turn, to show how they contribute to and draw from the others to create a 'curriculum without content'. We conclude with explicit recommendations and a call for future studies to establish the extent to which each of these four features contributes to the decline of student achievement. [ABSTRACT FROM AUTHOR]

For the past forty years, United States school districts have increasingly hired mathematics specialists to support the teaching and learning of mathematics. Despite the prevalence of this professional development structure, this is a relatively new research topic for the mathematics education field. In this paper, we report findings from an exploration of literature between 1981 and 2018 to examine the role of mathematics specialists (MSs). In particular, we examine: (a) MS positioning across research; (b) historical trends of school‐based MS research; and (c) orientations of school‐based MSs within research. Using the McGatha and Rigelman framework as an analytic lens, we found that beyond the positions identified in their framework (coach, intervention specialist, teacher), there were four additional MS positionings within the research (learner, other p‐12 stakeholder, university stakeholder, unknown). Ultimately, we forward an expansion to the McGatha and Rigelman framework to include these new categories, as well as contextual descriptions and working definitions to support future research in more accurately and robustly capturing the ways in which MSs are investigated and reported on in research. [ABSTRACT FROM AUTHOR]

Interim assessments have been defined variously in different contexts and can be used for predictive purposes or instructional purposes. In this paper, we present results from a study where we evaluated reporting needs for interim assessments designed for instructional purposes and intended to be used at the end of defined curriculum units. Results from such unit assessments should help teachers determine gaps in student understanding and inform ongoing instructional decision‐making. Our goal was to determine if learning progressions (LPs) could serve as the cognitive lens through which teachers can evaluate how their students' understanding of key constructs improves through periodic unit assessments. Therefore, we used the LP framework in mathematics and the key practices (KP) framework for English language arts (ELA) to design preliminary teacher report mock‐ups for these unit assessments. Within a utilization‐oriented evaluation framework, we conducted six needs‐assessment focus groups with elementary and middle school mathematics (n = 12) and ELA (n = 11) teachers to specifically evaluate the extent to which they find results presented within the LP and KP frameworks understandable and useful for their instructional practice. Results from the focus groups show teachers' overall needs for types of information sought from unit assessment reports, the extent to which teachers are familiar with the LP and KP frameworks, their interpretations (including confusions) of the information presented in the preliminary mock‐ups, and their additional needs for reports from unit assessments to be instructionally useful. Report Number: RR‐21‐08 [ABSTRACT FROM AUTHOR]

This paper reports an action research study I conducted into the mathematical productive disposition of 29 tenth grade learners in the Philippines by analyzing their metaphorical conceptualizations of mathematics in a series of lessons following principles of Realistic Mathematics Education. Analysis of participants' pre-intervention metaphors of mathematics revealed four conceptualizations that capture learners' perceptions of mathematics reflective of their productive disposition: mathematics as a process, mathematics as a puzzle, mathematics as a problem, and mathematics as a difficulty. Insightful changes in the learners' metaphors of mathematics support the argument that making mathematics matter to students can be achieved if teachers make considerable and devoted attempts at making what matters to students mathematical. Several challenges in the implementation of realistic mathematics lessons on Permutation included developing and enabling students' awareness of their social and cultural environment and encouraging critical thinking. Reflecting on my experience, I argue that constant exposure to lessons that present mathematics as contextually and socially relevant can potentially improve students' mathematical productive disposition together with other strands of their mathematical proficiency. [ABSTRACT FROM AUTHOR]

Professional noticing of children's mathematical thinking (Jacobs et al. in J Res Math Educ 41(2):169–202, 2010) is a construct that is used frequently to understand the ways in which teachers attend to, interpret, and respond to children's mathematical thinking. In this paper, we present our analysis of prospective elementary mathematics teachers' (PTs') responses to a pedagogical activity designed to engage them in an approximation (Grossman et al. in Teach Coll Rec 111(9): 2055–2100, 2009) of professional noticing of children's mathematical thinking in the moment of teaching, with a focus on developing responses which extend the student's thinking. PTs were provided two different examples of student work and a series of questions designed to elicit information about their ability to professionally notice student work. We present data and analysis of 73 PTs' abilities to engage in the individual skills associated with noticing as well as their abilities to use the skills cohesively in concert. Our results demonstrate the type of representation used determined PTs' success in all three facets of noticing, with a particular effect on interpreting. Further, our focus on a specific type of response move, along with the introduction of a scaffold for that response move, made a difference in PTs' ability to successfully respond to student work when compared to a prior study with the same PTs (Tyminski et al. 2014). We unpack these results along with suggestions for pedagogical practice as well as directions for future research. [ABSTRACT FROM AUTHOR]

In this paper, we approach modelling mathematics dispositions from a different methodological perspective in order to shed more light into the complex interplay between teaching practices and students' learning outcomes. We draw on survey data from around 5000 students from Year 7–11 (age 11–16) from 40 Secondary schools in England. Our methodological approach includes Rasch modelling to produce measures of attitudinal outcomes as well as students' perceptions of pedagogic practices. We then employ fuzzy-set Qualitative Comparative Analysis (fs/QCA) to explore the relationships between students' characteristics and the perceived type of teaching they receive in mathematics. We use two measures of 'transmissionist teaching' which aim to quantify the degree to which teaching practices are perceived as 'teacher-centred'. One measure gives the students' perceptions and the other gives the teacher's perspective. We find that different configurations of student and teacher perceptions of transmissionist teaching are associated with high and low mathematical dispositions for different year groups and for boys and girls. We discuss the methodological merits of this approach along with the substantive educational implications of these findings. [ABSTRACT FROM AUTHOR]

Problem posing, the process of formulating problems based on a given situation, is an essential practice in mathematics and other disciplines. Although this is acknowledged in policy documents, problem posing is neither substantively nor consistently included in the school mathematics curriculum. In this paper, we first comment on the state of problem posing in school mathematics and discuss three recommendations to improve its integration into curriculum materials and classroom practice. These recommendations present a low barrier to entry for teachers and students and require only minor changes to common mathematics classroom activities and curriculum materials. Based on the three recommendations, as well as the features of effective teacher professional development, a program was created to investigate longitudinally the impact of problem-posing professional development on teachers' conceptions of problem posing and their design of lessons to teach mathematics using problem posing, as well as on students' learning. Initial findings are presented, including the significant changes in teachers' conceptions of problem posing and teachers' design of lessons to teach mathematics using problem posing, as well as the impact on students' learning. [ABSTRACT FROM AUTHOR]

Inquiry into the teaching and learning of mathematics is a central aspect of the work of both the mathematics teacher and the mathematics education researcher, yet there are profound differences between the practices and processes underlying educational inquiries within each community. These differences are known to hinder implementation of research, yet they can also become a valuable resource. This paper explores a particular model of implementation of research: mathematics teachers adopt and adapt practices and processes of disciplined educational inquiry in a co-learning partnership with mathematics education researchers. We present a narrative inquiry of a teacher–researcher community that designed and studied classroom activities aimed at encouraging students to ask meaningful mathematical questions. The data analysis, informed by the literature on boundary objects and boundary crossing, highlights how the teachers and researchers leveraged their different processes and practices of educational inquiry as a resource in their collaborative inquiry. We conclude by suggesting that informed and thoughtful attention to the differences between teacher inquiry and disciplined inquiry may support and enhance mathematics education research implementation. [ABSTRACT FROM AUTHOR]

Implementation support strategies such as professional development (PD) are pivotal in guaranteeing the assistance that teachers need to enact mathematics teaching innovations and to secure lasting change. Within large-scale efforts an important issue for researchers is to understand how to design and put into effect these implementation support strategies. Although some work has been done to examine the challenges of implementing PD to advance the enactment of mathematics teaching innovations in new school settings, little is known about the multiple challenges novice facilitators face when providing PD for fellow teachers in their local districts and communities. In this paper we report on a 3-year long study that examined how a novice facilitator learned to deal with these challenges and become confident in his new role. Drawing on Graven's (Educational Studies in Mathematics 57(2), 177–211, 2004) expanded model of learning, a case study and an interpretative approach are used to investigate a novice facilitator's process of becoming confident. The findings provide evidence that becoming confident is part of an ongoing process of learning through (1) experiencing or constructing new meanings for problem solving as an instructional approach; (2) doing or enacting the instructional approach; and (3) engaging in the core PD practices. Recommendations for the education of PD facilitators in large-scale initiatives are provided. [ABSTRACT FROM AUTHOR]

Background: Building on the Realistic Accuracy Model, this paper explores whether it is easier for teachers to assess the achievement of some students than others. Accordingly, we suggest that certain individual characteristics of students, such as extraversion, academic self‐efficacy, and conscientiousness, may guide teachers' evaluations of student achievement, resulting in more appropriate judgements and a stronger alignment of assigned grades with students' actual achievement level (as measured using standardized tests). Aims: We examine whether extraversion, academic self‐efficacy, and conscientiousness moderate the relations between teacher‐assigned grades and students' standardized test scores in mathematics. Sample: This study uses a representative sample of N = 5,919 seventh‐grade students in Germany (48.8% girls; mean age: M = 12.5, SD = 0.62) who participated in a national, large‐scale assessment focusing on students' academic development. Methods: We specified structural equation models to examine the inter‐relations of teacher‐assigned grades with students' standardized test scores in mathematics, Big Five personality traits, and academic self‐efficacy, while controlling for students' socioeconomic status, gender, and age. Results: The correlation between teacher‐assigned grades and standardized test scores in mathematics was r =.40. Teacher‐assigned grades more closely related to standardized test scores when students reported higher levels of conscientiousness (β =.05, p =.002). Students' extraversion and academic self‐efficacy did not moderate the relationship between teacher‐assigned grades and standardized test scores. Conclusions: Our findings indicate that students' conscientiousness is a personality trait that seems to be important when it comes to how closely mathematics teachers align their grades to standardized test scores. [ABSTRACT FROM AUTHOR]

Learning to name and notice students' mathematical strengths is a challenging process requiring time and multiple iterations of practice for prospective teachers (PTs) to adopt. Mathematics teacher educators (MTEs) can approximate and decompose the complex practice of naming and noticing students' mathematical strengths so PTs learn to teach mathematics while emphasizing what students know and can do. This study uses two tools MTEs can use to support PTs as they learn to name and notice students' mathematical strengths: A LessonSketch experience, a digital platform with comic-based storyboards showing children engaged in a mathematics task, and a strengths-based sentence frame. Our study presents the findings from the 111 noticing statements from 18 PTs as they engaged in the LessonSketch digital experience and practiced making noticing statements about what children know about mathematics. The study found that after a sentence-frame intervention, the PTs are more likely to use strengths-based language and more likely to identify mathematical evidence in their noticing statements. Uncommitted language (statements that do not align with a strength- or deficit-based coding scheme), suggests a fruitful, yet complex space for supporting more PTs as they learn to name and notice students' mathematical strengths. The paper concludes with implications for future research in teacher education. [ABSTRACT FROM AUTHOR]

The study investigated the effects of condonation on the promotion of senior phase mathematics learners into the FET phase in a high school in KwaZulu-Natal. Condoned learners were assisted to progress, despite not meeting the pass requirements. The study suggests learning support strategies teachers could use to improve condoned FET mathematics learners' performance. An exploratory case study investigated the condonation policy and its effects thereof. Interviews were conducted with two mathematics teachers, and open-ended questions were posed. The data analysis entailed collating the data, followed by a thematic data analysis approach. The literature study was done on pertinent theories, prior studies, and pragmatic research. The findings suggest that condoned learners did not master the foundational knowledge and accumulated knowledge gaps. Consequently, learners struggle to achieve good performance in FET mathematics. This may lead to condoned learners failing FET mathematics, dropping mathematics in favour of more uncomplicated mathematical literacy, or dropping out of school rather than completing Grade 12. The study recommends that condoned senior phase mathematics learners struggling to perform well in FET mathematics need to undergo an individualised learning support programme. Teachers will then employ various academic teaching and learning support strategies to help improve learners' FET mathematics performance. The success of learning support depends on a collaborative relationship between nurturing teachers and parents and positive discipline in group teaching within supplemental classes. [ABSTRACT FROM AUTHOR]