# A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis

For example, they can use mental representa- tions of their environment and, on the basis of the representation, model relationships between objects in their environment. Later, children can compare lengths by measuring objects with manipulable units, such as centimeter cubes.

Number is particularly important to later success in school mathemat- ics, as number and related concepts make up the majority of mathemat- ics content covered in later grades. However, it is important to point out that concepts related to number and relations and operations can also be explored through geometry and measurement. In addition, geometry. Conclusion 7: Two broad mathematical content areas are particularly important as a focus for mathematics instruction in the early years: 1 number which includes whole number, operations, and relations and 2 geometry, spatial thinking, and measurement.

In the context of these core content areas, young children should engage in both general and specific thinking processes that underpin all levels of mathematics. These include the general processes of representing, problem solving, reasoning, connecting, and communicating, as well as the more specific processes of unitizing, decomposing and composing, relating and ordering, looking for patterns and structures, and organizing and clas- sifying information. In other words, children should learn to mathematize their world: focusing on the mathematical aspects of an everyday situation, learning to represent and elaborate the quantitative and spatial aspects of a situation to create a mathematical model of the situation, and using that model to solve problems.

Conclusion 8: In the context of each of these content areas, young chil- dren should engage in both general and specific mathematical thinking processes as described above and in Chapter 2. Program types range from friends and relatives who care for children in the home through informal arrangements, to large centers staffed by teachers offering a structured curriculum. This diversity in the early childhood education system characterizes the education and care arrangements of young children in the United States today.

About 40 percent of young children spend their day in a home-based setting, either with a parent or some other caregiving adult this percentage includes children in home-based relative and nonrelative care as well as children who do not have any regular early education and care arrange- ments , and about 60 percent are in some kind of center-based care this includes children in center-based non-Head Start and Head Start settings.

Depending on the type of setting, different regulations regarding edu-. Increasingly, policy makers are focused on how to provide high- quality preschool education for more children, especially to those whose families cannot afford to pay for it. A number of states are moving toward state-funded preschool education to provide early education and care for these children. Across all settings, there is a need to increase the amount and qual- ity of time devoted to mathematics.

Formal settings with an educational agenda represent the greatest opportunity for implementing a coherent, sequenced set of learning experiences in mathematics. For this reason, the committee focused attention on the kind of curriculum and instruction that can be implemented in centers and preschools.

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## Robert Edmund Edwards

The committee gave more limited attention to how to increase support for mathematics in informal settings. The committee first examined the extent to which the content and learning experiences embodied in the teaching-learning paths are represented in current curricula and preschool classrooms. Next, the committee explored what is known about effective mathematics instruction for young children and what might need to be done to improve existing practice.

The committee looked for evidence to address two sets of questions: What is known about how much mathematics in- struction is available currently to children in preschool settings and of what quality? What is known about the best methods of instruction and effective curriculum to teach mathematics to young children?

Although few system- atic data exist, the committee was able to identify some useful sources. We conducted original analyses of the standards documents pertaining to early childhood for 49 states and those pertaining to kindergarten for the 10 states with the largest student populations.

On the basis of these analyses, the committee concludes: Conclusion 9: Current state standards for early childhood do not, on average, include much mathematics. When mathematics is included, there is a pattern of wide variation among states in the content that is covered. Although standards represent broad guidance from the states regard- ing appropriate content for early childhood settings, they do not provide a. For the latter, the commit- tee examined data from a large-scale study of instruction in state-funded preschools drawn from 11 states as well as several, small-scale studies of curriculum.

The results show that when mathematics activities are incor- porated into early childhood classrooms, they are often presented as part of an integrated or embedded curriculum, in which the teaching of math- ematics is secondary to other learning goals. This kind of integration occurs when, for example, a storybook has some mathematical content but is not designed to bring mathematics to the forefront, a teacher counts or does simple arithmetic during snack time, or points out the mathematical ideas children might encounter during play with blocks.

However, data suggest that heavy reliance on integrated or embedded mathematics activities may contribute to too little time being spent on mathematics in early childhood classrooms. Furthermore, the time that is spent may be on activities in which the integrity and depth of the mathematics is questionable.

Few of the existing comprehensive early childhood curriculum approaches provide enough focused mathematics instruction for children to progress along the teaching-learning paths recommended by the committee. Conclusion Most early childhood programs spend little focused time on mathematics, and most of it is of low instructional quality. Many opportunities are therefore missed for learning mathematics over the course of the preschool day.

Evidence examined by the committee suggests that instructional time focused on mathematics is potentially more effective than embedded math- ematics. Emerging evidence from a few studies of rigorous mathematics curricula show that children who experience focused mathematics activi- ties in which mathematics teaching is the major goal have higher gains in mathematics and report enjoying mathematics more than those who do not.

Supplemental opportunities to use mathematics during mathematical play, sociodramatic play, and with concrete materials e. Further- more, in all these contexts, intentional teaching enhances the mathematics learning of young children. It should be noted that the committee does not endorse any specific model or curricu- lum; rather we hope to convey that the research-based principles described in this report should guide choices about development of early childhood mathematics curriculum and instruction.

Conclusion Effective early mathematics curricula use a variety of instructional approaches and incorporate intentional teaching. Such responsive instruction can be accomplished when teachers know how to use formative assessment to guide instruction. Formative assessment is an important component of what teachers need to know to effectively guide children along the mathematics teaching-learning paths. Evidence from studies of early childhood education indicates that any approach to curriculum and pedagogy is more effective if undertaken in the context of a positive learning environment.

Positive relationships between children and their teachers are a key aspect of high-quality early childhood education. In this kind of classroom, children are provided with a safe and nurturing environment that promotes learning and positive interactions between teachers and peers. Conclusion Successful mathematics learning requires a positive learning environment that fully engages children and promotes their enthusiasm for learning. This workforce consists of people who serve in a variety of roles, are located in a variety of settings, and have a wide range of education and training backgrounds.

About 24 percent of early childhood workers are in center-based settings, 28 percent are in regulated home-based settings, and about 48 percent work in informal care arrangements outside both of these systems. Although the majority of early childhood professionals work in informal care settings, the majority of children are in center-based settings.

Even in a single setting, individuals fill different roles, such as lead teacher, assistant teacher, classroom aide, or program administrator. Level and type of training can vary by both role and setting. This diversity of roles and educational backgrounds creates challenges for addressing the workforce needs related to supporting early childhood mathematics. Indeed, academic activities, such as mathematics learn- ing, can be a context in which social-emotional development flourishes. In large part, the heavy emphasis on social-emotional development in early childhood is based on misinterpretations of cognitive development theories; that is, the notion of young children engaging in more abstract thinking, such as mathematics, was believed to be at odds with the development and learning of preschool-age children.

Research on early childhood mathemat- ics has disproved this notion, but the idea is still pervasive in the field and continues to be a challenge in moving from research to practice. Conclusion Many in the early childhood workforce are not aware of what young children are capable of in mathematics and may not recognize their potential to learn mathematics. Based on studies at the K level, effective approaches to in-service professional development in mathematics are ongoing, grounded in theory, tied to a cur- riculum, job-embedded, and delivered at least partially onsite by a knowl- edgeable trainer who allows teachers time for reflection.

The committee reviewed emerging data from studies conducted in early childhood settings that support these findings. An effort to provide profes- sional development to teachers is one important component of successfully improving instruction, but sustainable change will also require collabora- tion from administrators, teachers, and parents. Conclusion In-service education of teachers and other staff to support mathematics teaching and learning is essential to effective implementation of early childhood mathematics education.

Useful pro- fessional development will require a sustained effort that involves help- ing teachers to a understand the necessary mathematics, the crucial teaching-learning paths, and principles of intentional teaching and curriculum and b learn how to implement a curriculum. Furthermore, licensure and credentialing systems exert a great deal of influence over the content and experience of pre-service education pro- grams in early childhood, and few incorporate mathematics requirements. Licensure and certifica- tion requirements for credentialing teachers and programs are both potential leverage points for increasing the amount of attention given to supporting mathematics.

Conclusion Improving the training and knowledge requirements for early childhood teachers will present significant challenges unless exist- ing issues of recruitment, compensation, benefits, and high turnover are also addressed. Math talk has been shown to be a particu- larly effective way for adults to support the development of mathematical ideas.

In addition, informal learning environments, such as libraries, museums, and community centers, have the potential to be resources that parents and caregivers can use to engage children in mathematics activities. Conclusion Families can enhance the development of mathematical knowledge and skills as they set expectations and provide stimulating environments. Evidence indicates, however, that low-SES families are less likely than families from higher socioeconomic groups to engage in the kind of prac- tices that promote language and mathematics competence.

Although many types of educational programs have been designed to promote the use of these practices with low-SES parents, there is little evidence about the qualities that make such efforts successful. Educational programs for par- ents based on models that place parents in the traditional role of students. Conclusion Educational programs for parents have the potential to enhance the mathematical experiences provided by parents; however, there is little evidence about how to design such programs to make them effective. The resources available to parents and other caregivers as well as those available through informal educational environments e.

Educational television programming and software, for example, can teach children about mathematics.

The com- mittee reviewed research on software and educational programs, as well as models of community-based programs that promote mathematics, and concludes: Conclusion Given appropriate mathematical content and adult support, the media e. Such resources can provide additional mathematics learning opportunities for young children, especially those who may not have access to high-quality early education programs.

Thus, the committee thinks it is critically important to begin an intensive national effort to enhance opportunities to learn mathematics in early childhood settings to ensure that all children enter school with the mathematical foundations they need for academic success. The research-based principles and mathematics teaching-learning paths de- scribed in this report can also reduce the disparity in educational outcomes between children from low-SES backgrounds and their higher SES peers.

The research to date about how young children learn key concepts in mathematics has clear implications for practice, yet these findings are not widely known or implemented by early childhood educators or even those who teach early childhood educators. This report has focused on synthesiz- ing and translating this evidence base into a usable form that can be used to guide a national effort.

Thus the committee recommends:. A number of specific recommendations for action follow from this overarching recommendation. The specific steps and the individuals or or- ganizations that must be involved in enacting them are outlined below. Recommendation 2: Mathematics experiences in early childhood set- tings should concentrate on 1 number which includes whole num- ber, operations, and relations and 2 geometry, spatial relations, and measurement, with more mathematics learning time devoted to num- ber than to the other topics.

The mathematical process goals should be integrated in these content areas. Children should understand the concepts and learn the skills exemplified in the teaching-learning paths described in this report. In both content areas, sufficient time should be devoted to instruction to allow children to become proficient with the concepts and skills outlined in the teaching-learning paths. In addition, the general and specific math- ematical process goals see Chapter 2 must be integrated with the content in order to allow children to make connections between mathematical ideas and deepen their mathematical reasoning abilities.

This new content focus will require that everyone involved rethink how they view and understand the mathematics that is learned in early childhood. Early childhood learn- ing goals, programs, curricula, and professional development will need to be informed by and adapted to the research-based teaching-learning paths laid out in this report.

The committee therefore recommends: Recommendation 3: All early childhood programs should provide high-quality mathematics curricula and instruction as described in this report. Early childhood programs will each need to implement a thoughtfully planned curriculum that includes a sequence of teacher-guided mathemat- ics activities as well as child-focused, teacher-supported experiences.

Such curricula must be based on models of instruction that are appropriate for young children and support their emotional and social development as well as their cognitive development. As noted previously, effective mathematics curricula use a variety of instructional approaches and should incorporate opportunities for children to extend their mathematical thinking through play, exploration, creative activities, and practice. Programs will need to review, revise, and align their existing stan-.

It is especially important that children living in pov- erty receive such high-quality experiences so that they start first grade on a par with children from more advantaged backgrounds. This means that implementation of our recommendations by programs serving economically disadvantaged children, such as Head Start and publicly funded early child- hood programs, is particularly urgent. To make the recommended changes, early childhood programs will need explicit policy directives to do so. To encourage this, the committee recommends: Recommendation 4: States should develop or revise their early child- hood learning standards or guidelines to reflect the teaching-learning paths described in this report.

Given the fresh knowledge and perspectives this report affords, it is important that states review their early learning and development stan- dards and guidelines to ensure that they reflect an appropriate emphasis on early mathematics. Recommendation 5: Curriculum developers and publishers should base their materials on the principles and teaching-learning paths described in this report.

Cur- riculum developers and publishers who produce materials for curriculum, instruction, and assessment should revise and update them so that they reflect the principles articulated in this report. The success of this overall effort will need to focus on reaching both the existing early childhood workforce and pre-service educators to provide them with skills and knowledge they need to teach mathematics.

Thus, we make several recommendations related to teachers and the workforce. Recommendation 6: An essential component of a coordinated national early childhood mathematics initiative is the provision of professional development to early childhood in-service teachers that helps them. All of these important areas should be included in professional development delivered by a highly qualified teacher educator. Abstract This article describes the results of an analysis of the mathematical register in Greenland concerning the characteristics for developing terms for mathematical concepts in a polysynthetic language.

The theoretical frame is registers, register continuum with respect to systemic functional linguistic. The analysis indicates that three techniques gerunding, circumlocutions, and metaphors are used simultaneously when generating new terms in Greenlandic. The mathematical concepts in Greenlandic thereby have a direct link to everyday meanings of similar concepts. Abstract This article discusses mathematical and cultural task design to support minority and endangered languages and cultures. More precisely, we propose a theoretical framework to design mathematical tasks for language immersion in mathematics for Kven students.

Drawing on previous studies, we suggest that traditional tools have the potential to support the learning of mathematics, language, and culture. One challenge for endangered languages and cultures is that the younger generations may have lost connections with their traditional language and culture. We argue that the older generations can mediate authentic aspects of Kven culture to students, which then become historical-cultural authentic HiCuA aspects. Huru Hilja L. Her background is in pure mathematics with a PhD in non-commutative algebra connected to mathematical physics.

Her research interest also includes multicultural classrooms with a focus on pressured minorities and indegenous mathematics. She is involved with language education programs for Kven language and leads Kven language nests projects in northern Norway. Her main research interests are sociolinguistics and language policy as well as language revitalization and language vitality.

Abstract The present study investigated how Swedish, grade 9 students, of whom 90 had an immigrant background, achieved on twelve written test items in the content area of number. Four of the twelve test items required good knowledge of arithmetic syntax, such as when it was appropriate to apply order-of-operation rules and the associative and distributive laws of arithmetic operations. On these four test items, the most-recently arrived students showed on average signi cantly more knowledge than the students who had immigrated when they were younger and had participated in Swedish schools for longer periods of time.

The outcome suggests that these two groups of immigrant students in later school years should be considered as separate sub-categories of second-language students when it comes to teaching, assessment and research. He wrote his PhD-dissertation on test achievement of second language students in the last year of compulsory school. Presently he is doing postdoctoral research on how foundational number sense appears in textbooks and homework in the rst year of compulsory school.

Abstract A lack of knowledge of the language of instruction is often believed to be the main reason for low achievement among students with an immigrant background. We regard language as a tripartite unit comprising aspects of concept formation, pragmatic language usage and the linguistic form. In a bilingual context, the labelling of concepts and meaning-making through argumentation are simultaneously processed in two languages.

Ahlholm is interested in the multilingual socialisation process of emerging second language. In their discussions students used Swedish, which was their second language and also the language of instruction. Key research interests are interaction in multilingual mathematics classrooms and educational design research. Abstract This paper explores the relation between discourses and identity formations as mathematical learners in a context of transition. The data consists of an interview with two 16 year-old immigrant girls, who were relocated when their school, in a multicultural and socio-economically disadvantaged area in Sweden, was closed.

The girls showed dynamic and unstable identities by drawing on di erent discourses. Social relational discourses, more than mathematical pedagogical discourses, governed their actions as learners of mathematics; enabling identities as noisy, un-engaged, but able students in the old school, and as engaged and accepted, but also as strangers, in the new school. Her main research interests concern socio-political issues in mathematics education and are related to multilingual and multicultural issues in mathematics education. However, simultaneously the newly-arrived student is thought of, in a more excluding discourse, as being in need of rescue and as lacking the most valuable asset, the Swedish language.

Her main research interest is multilingual students in mathematics classrooms. She has also researched gender issues related to mathematics teaching and learning. Since she is involved in a development and research project on programming in subject didactics. Her main research interest are critical mathematics education, teacher professional development and language diversity in mathematics classrooms.

Abstract This study adds to research on volume and spatial reasoning by investigating teacher-learner interactions in the context of Lesson study. Our analysis illustrates how the mathematical object of volume is realized, and what metarules of discourse that can be observed over two iterations of a research lesson. The study unpacks the mathematical work of teaching volume in terms of discourse, and shows how an undesirable and unexpected result from the first research lesson can be attributed to the communicational work of teaching rather than to lack of skills among students.

His research interests are related to mathematics teaching and developing mathematics teachers. Abstract Concrete applications of neuroscience to the classroom are yet to be confirmed. Hans forskningsinteresser omfatter anvendelse av funn fra kognitiv psykologi i matematikkundervisning, hjernens befatning med matematikk, samt bruk av matematikkhistorie og historie-fortelling i undervisning.

Abstract The article addresses the need for competence descriptions of disciplines as a means for fostering more productive communication between different disciplines and between the disciplines and their surroundings. It is argued that the usual competence descriptions devised for use within a discipline itself, e. The same is true for the general, non-disciplinary competence descriptions. Instead, specially devised disciplinary competence descriptions for external use are called for.

This description for external use is counterposed with one for internal use i. It is also counterposed with a competence description for external use for physics, taking into account the different justification problem of physics education. Together these two descriptions showcase how competence descriptions of disciplines for external use may support interdisciplinary collaboration and division of labor in the educational system.

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For many years he was involved in the governing of Roskilde University in various positions. Besides publications with more specific physics content he has published on university politics, general didactics, science didactics, and interdisciplinarity. Abstract In Finland, both prospective and in-service mathematics teachers report a discontinuity between university-level mathematics and mathematics taught at comprehensive and secondary school. In this study, ten prospective mathematics teachers PMTs were interviewed to examine their conceptions of the nature of this gap as well as their mathematical thinking.

The findings offer guidelines for further studies that could help the development of mathematics teacher education. He is involved with mathematics teacher education at the Department of Mathematics and Statistics. He has a background as a lecturer of mathematics and information technology at Helsinki Metropolia University of Applied Sciences. His main research interests are teacher knowledge and beliefs as well as cognitive aspects of mathematical thinking. Abstract The purpose of this study is to understand the school algebra offered in Swedish mathematic textbooks for grade 8.

Using a social semiotic perspective, textbook tasks are analysed with a method inspired by Systemic Functional Linguistics. Five school algebra discourses are identified: symbolic discourse, geometrical discourse, arithmetical discourse, un realistic discourse and the scientific discourse. It is argued that these offer different views on the nature of algebra and the positioning of students.

## CTET Syllabus Paper I & II: Section-Wise Topics

The main research interests are mathematics and language, especially algebra and social semiotics. To form such links, one needs knowledge of how children use and express these ideas. This is especially true in the intersection of arithmetic and geometry, where the intermingling of numerical and spatial concepts and skills is not yet fully understood. The results show that children can use spatial representations when reasoning about numbers, and that they are able to connect spatial and numerical structures.

Furthermore, it is shown that children not only use and invent effective procedures, but also are able to explain, justify and evaluate such procedures. Abstract In this paper, we develop an analytical tool for the role of the physical environment in mathematics education. We do this by extending the didactical triangle with the physical environment as a fourth actor and test it in a review of literature concerning the physical environment and mathematics education.

We find that one role played by the physical environment, in relation to mathematical content, is to portray the content in focus, such as geometry and scale. When focusing on teachers, students, and the interaction between them, the role of the physical environment appears to be a precondition, either positive enabling or negative hindering. Many of the findings are valid for education in general as well, such as the importance of building status.

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Key research interests are physical school environment and mathematics education. Key research interests are mathe-matical reasoning, affect and gender. Abstract This paper reports a study of the views held by Finnish students at the start of their university studies concerning their understanding of the knowledge and characteristics of a good mathematics teacher.

A total of 97 students following a basic university course responded to a questionnaire. The results showed that a knowledge of teaching mathematics was more often used to describe the good mathematics teacher than a knowledge of mathematics. Good mathematics teachers were also considered to be skilled in explaining, simplifying and transforming mathematical contents for their students.

A good mathematics teacher was often described by the respondents as a patient, clear, inspiring and consistent person. Those respondents who planned to become teachers demonstrated a more learner-centred concept of a good mathematics teacher than did those who were aiming at some other subject specialist profession than that of teaching. Mervi A. Asikainen Docent Mervi A. Asikainen directs the UEF physics and mathematics education research group.

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Her current field of interest include teacher knowledge of mathematics and physics teachers, teaching and learning of physics in higher and secondary education, and research-based development of STEM education. His research areas are mathematical beliefs, mathematics teacher education, learning materials textbooks etc.

He has used Mathematical Knowledge for Teaching MKT framework for evaluating and improving mathematics teacher education. In his dissertation study, he presented a novel approach for investigating teacher knowledge and its interconnections by making use of network analysis methods. His postdoctoral research continues from this work by focusing on how the components of teacher knowledge are interconnected. Pekka E. Hirvonen Docent Pekka E. Hirvonen has published more than 30 peer-reviewed articles in international journals, books, and proceedings. Abstract It is relatively straightforward to assess procedural knowledge and difficult to assess conceptual understanding in mathematics.

One reason is that conceptual understanding is better assessed using open-ended test questions that invite an unpredictable variety of responses that are difficult to mark. Recently a technique, called comparative judgement, has been developed that enables the reliable and valid scoring of open-ended tests. We applied this technique to the peer assessment of calculus on a first-year mathematics module. We explored the reliability and criterion validity of the outcomes using psychometric methods and a survey of participants. We report evidence that the assessment activity was reliable and valid, and discuss the strengths and limitations, as well as the practical implications, of our findings.

Prior to this he was a Royal Society Shuttleworth Education Research Fellow and taught in primary and secondary schools for ten years. He is enjoying spending some time working with education researchers to explore new ways of improving teaching and learning. Abstract The paper reports on the views and use of mathematical modelling MM in university mathematics courses in Norway from the perspective of lecturers. Our analysis includes a characterisation of MM views based on the modelling perspectives developed by Kaiser and Sriraman Through an online survey we aimed to identify the main perspectives held in higher education by mathematics lecturers and the underlying rationale for integrating or not MM in university courses.

The results indicated that most respondents displayed a realistic perspective on MM when it came to their professional practice. There was a more varied response when it came to their views on MM in teaching. Regarding conditions influencing the use or non-use of MM in teaching, these mainly concerned the mathematical content and the institutional practices.

Stephanie Treffert-Thomas Stephanie Treffert-Thomas is a lecturer at Loughborough University UK with experience of teaching mathematics at school level, tertiary college level and at university, mainly to engineering students. Her research interests are in university level mathematics teaching and learning using socio-cultural educational theories. She has a particular interest in the mathematical teaching practices of lecturers, including the use of mathematical modelling in teaching. His research is in university mathematics education, mainly focusing on the discursive practices of lecturers and students, and on the teaching and learning of mathematics, for instance mathematical modelling, in other academic disciplines.

His research is in post-compulsory Mathematics Education, where he uses socio-cultural educational theories to investigate teaching-learning practices e. Mathematical Modelling that have the potential to develop in students rich mathematical meanings while at the same time create in them positive dispositions towards the subject. His research interests include qualitative theory of ordinary, functional and impulsive differential equations, mathematical modelling, and mathematics education related to teaching and learning of differential equations and mathematical modelling at university level.

Abstract The main purpose of our research project is to gain insight into, and develop teaching on indices and their applications in society. The data analysed were collected in a numeracy across the curriculum class with practising teachers. The findings show that the practising teachers engaged in meaning making of the index formula, and they critically discussed how BMI is used in society and the role the BMI index can have in our lives. She has a background as mathematics teacher educator from Albania and Norway. Rangnes has a background as primary school teacher, textbook author and editor for Tangenten.

Her main research interests are critical mathematics education, teacher professional development and language diversity in mathematics classrooms. His research focuses on connections between communication qualities and learning in mathematics with a particular focus on argumentation and agency in real-life contexts and when students use digital learning tools. Herheim is the Editor in chief for Tangenten , a Norwegian journal on mathematics teaching. Meinrad Pohl Meinrad Pohl is associate professor in history.

His main research interests are early modern economic theory and economic policy, trade history and mining history. She has previous experience as mathematics teacher at the upper secondary school level. Her main research interests are critical mathematics education and mathematics teacher professional development. She received her master and PhD degrees from the University of Bergen within applied mathematics.

Hansen has a background in as a researcher in different modelling projects. Her main research interests are critical mathematics education and teacher professional development. Volume 22, No 4, December Developing practice through research into university mathematics education. Characterising undergraduate mathematics teaching across settings and countries: an analytical framework. Stimulating critical mathematical discussions in teacher education: use of indices such as the BMI as entry points. Oral presentations as a tool for promoting metacognitive regulation in real analysis.

Peer assessment of mathematical understanding using comparative judgement. Abstract Real Analysis is for many students their first proof-based mathematics course, and many find it challenging. The paper discusses several aspects tied to preparing for, and carrying out, oral presentations, that seem to spur important sub-components of metacognitive regulation such as planning, monitoring, and evaluating.