This paper reports on an evaluation of the professional development program Boost for Mathematics in Sweden. 200 mathematics lessons were visited, and the teachers were interviewed after each lesson.The analysis used an analytical framework based on Lithner et al. (2010).The findings indicate that the PD-program has had a significant impact on the teachers’ knowledge about the mathematical competencies as they are presented in the national curriculum documents, and that the teaching practice has improved and give the students better possibilities todevelop the competencies.The results also show that these improvements are still present one year after the program had ended.
STEAM (Science, Technology, Engineering, Arts, and Mathematics) research emphasises interdisciplinary approaches to developing skills needed for the 21st century. In this paper, we explore three aesthetically rich mathematical activities involving dance and bodily performance (e.g., creative body postures) with 6-year-old students. Supported by an embodied perspective of task design and a Deweyan view of aesthetics, we argue that a high degree of bodily engagement and a high degree of task integration allow space for creativity and imagination that may involve aesthetic dimensions. Specifically, we argue that body postures, bodily rotation, rhythm, and fluency in composite sequences of movement may connect art and mathematics in a way that treats the students as active creators of aesthetic experiences, in contrast to other STEAM approaches that do not consider bodily performance as a way of connecting art and mathematics.
This presentation reports on an ongoing study, which aims to create more knowledgeon the relationship between different types of curriculum resources when identified inthe practice of teachers planning collaboratively. These resources are describedthrough the Design Capacity for Enactment framework, augmented with domains of theMathematical Knowledge for Teaching framework. The aim is to identify and examinethe connection, rather than to claim to explain the relation. Preliminary results showthat there are many different types of resources used, both digital and analogue, andthat teachers’ Knowledge of Content and Students and Knowledge of Content andTeaching guide the reasons for what types of resources are used.
This study investigates resource use by upper-secondary mathematics teachers in the context of collaborative planning. A thematic content analysis is conducted on audio-recorded teacher discussions in order to find out what resources are used by the teachers, how they are used, and for what reasons. The findings show that although teachers use a variety of resources to support their instructional enactment as well as instructional design, there is a difference in how they use different resources to support different planning practices. For instructional design, curriculum resources provide support for the mathematical content, while social resources, self-generated documents, and cognitive resources provide support for the design of instructional activities. Authority is given by teachers to curriculum resources, but conflicts of authority emerge in discussions, when teachers’ abilities to exert their agency are not supported by curriculum resources. We discuss the findings in relation to authority and resource use, as well as for established conceptualizations of resources. The findings bear practical implications for the design of curriculum resources.
If students are to build knowledge, it is important to connect disciplinary knowledge to students’ everyday knowledge when teaching. In this study, the relationship between disciplinary- and everyday knowledge in subject-teaching is analysed, based on the semantic dimensions used in Legitimation Code Theory. Un-packing disciplinary concepts through concrete examples implies a shift towards a context-dependent everyday knowledge, while re-packing them entails a shift towards disciplinary abstractions and context-independent generalizations. Over time these shifts constitute so-called semantic waves argued to facilitate students´ knowledge building. Earlier research suggest that the form of these semantic waves can differ between school-subjects. Therefore, this article examines semantic waves in two contrasting subjects – Mathematics and Social Science Education, aiming at a better understanding of how and why semantic waves differ. The results reveal that the forms of semantic waves differ between the two subjects. In Mathematics, when teaching different geometrical concepts, the semantic shifts through un-packing and re-packing were frequent and evenly distributed. When teaching about pricing and household in Social Science, un-packing activities dominated, making shifts biased towards everyday knowledge. These differences are discussed in terms of semantic waves constituting a different pulse when making knowledge-building possible in Mathematics and Social Science.
In the last decade research in mathematics education has grown and consolidated in the Nordic countries. New and younger researchers are entering the field as a result of the organization of solid doctoral programs and the growth of teaching and research environments in higher education institutions. The social, cultural and economic contexts of our countries have also changed in the last decade. The question of which are the main challenges for research in the years to come is worth exploring for a community starting a new decade of work. The intention of the panel is to generate a discussion in the community by having young researchers present their views on: (1) which topics of research the Nordic region has contributed to the world and that deserve being strengthened in the future; (2) which new topics of research that may become relevant as social changes pose new demands on our knowledge and understanding of the practices of mathematics education in the educational system; and (3) which institutional affordances and constraints young researchers experience for the development of research
As teachers’ informal professional development is visible in social media, this study probes teachers’ participation in self-organized Facebook groups in mathematics or Swedish-language education. In total, 553 posts from six Facebook groups were categorized using Shulman’s knowledge-base framework, and analysed using systemic functional grammar. Teachers use “questions” and “offers” most frequently (88%). Within these speech functions, pedagogical content knowledge dominates (63%), indicating that these groups constitute professional learning communities that teachers use as a professional development resource, focusing the interaction on pedagogical content knowledge. This study finds a largely similar practice in Facebook groups across the two subjects.
The new technologies, in particular social media in Web 2.0, enable rapid change in people’s behaviour, which needs to be considered in research on teacher empowerment and teacher professional development and growth. In this chapter we discuss how teachers in an informal, yet structured, way use social media to expand their professional learning communities beyond the local school context in Sweden. This is an example of how a new behaviour is emerging among teachers that changes the opportunities and the frames for professional development and growth. Through teachers’ engagement in social media, such as Facebook, extended professional learning communities arise and teachers’ professional development and growth become evident. Global levels influence local levels: teachers from different schools engage in structured discussions related to everyday practice, such as issues of learning goals in pre-school or topics related to a specific course in upper secondary school. The teachers’ arena for professional development and growth has changed, which means that the context of teacher empowerment is rapidly changing too. Consequently, the chapter includes theoretical reflections on professional learning communities in a Web 2.0 world and how this phenomenon may affect our approach to enhancing teachers’ professional development.
Teachers worldwide are using social media as a professional development resource. In studying social media as ‘a place’ for teachers’ professional development, we investigated large Facebook groups with themes connected to teaching and learning in compulsory schools. The interaction in these groups was analysed within the framework of systemic functional grammar. In order to reveal knowledge known and shared by teachers as a community, we have also used Shulman’s (1987) framework. Most posts received responses and this response is in line with the expected response pattern. The speech functions ‘Questions’ and ‘Offers’ were most common. Further, most posts addressed subject specific knowledge. The multi-theoretical approach used when researching mathematics teachers’ professional development in self-organized online groups showed that these large Facebook groups facilitated professional learning.
This study explores how assessment is presented in Swedish early years’ steering documents and considers risks for young gifted students in relation to assessment (or lack thereof). Document analysis was undertaken on, firstly, Swedish curriculum documents for the preschool and for the compulsory school, and secondly, mapping materials used in the preschool class with six-year-old children. Results show that assessment is not a term used in Swedish early years curricula. Instead, preschool teachers are asked to evaluate their own practice; preschool class teachers are asked to engage with mapping and only to consider working toward later assessment goals in year 3 of school. A plethora of alternative assessment terms are used in the curriculum without definition. Giftedness is also invisible in the curriculum. However, the mapping materials used with six-year-old students in the subject areas of mathematics and Swedish do encourage teachers to consider children who achieve mastery early. Further, these materials provide supportive questions and activities for teachers to use in exploring further. The specific examples of assessment discourses and the need to consider gifted children are combined in this article to highlight aspects of teacher work that are important for the educational rights of an often-forgotten group of learners.
Giftedness is a special education need that can receive more attention in Nordic early years’ education. Here, we draw upon examples provided in early years’ research literature and on a new narrative example of practice to illustrate inclusive and responsive teaching possibilities where gifted children’s special learning needs are acknowledged. Gifted children have a need for specific cognitive support and a need for social and emotional support. Specific strategies to account for such support include curriculum differentiation, higher-order questioning, partnerships, and opportunities to work with like-minded peers. Through a framework for engaging with gifted education in the early years titled SPARK, with the elements support, policy, awareness, relationships, and knowledge, giftedness in inclusive education is illustrated and discussed. The elements were inclusively integrated in the everyday context of a Swedish preschool that caters for children with various abilities and needs. There is a policy mandate for teachers to engage with gifted education in Sweden; however, it is largely implicit, and more work is required before the special needs of gifted children are explicitly recognised.
Studies on high ability in mathematics rarely take a teacher’s perspective. The purpose of our study is to add such a perspective, which we will do by using positioning theory to analyze elementary teachers’ discussions on mathematical tasks, aiming to challenge all pupils, including the highly able. The study is conducted in the context of a two-year long teacher professional development program on high ability. Teachers expressed both their teaching and the mathematical tasks as helpful in orchestrating teaching suitable for highly able pupils. They highlight the opportunities given by some tasks as well as the importance of guiding highly able pupils to go further in such tasks. However, they expressed their own limited mathematical knowledge and time needed for pupils with learning difficulties as obstacles to orchestrate teaching for the highly able. The results show that it is important to, in close cooperation with teachers, further explore how to orchestrate teaching that challenges highly able pupils.
Förutom en introduktion till ett problemlösande arbetssätt presenteras genomförda problemuppgifter med tydliga kopplingar till styrdokumenten. Varje problemuppgift har testats och bearbetats, och i boken finns förslag till fördjupning eller förenkling av problemuppgifterna. Författarna ger även exempel på hur eleverna kan utvärdera både problemuppgifterna och det problemlösande arbetssättet. Boken vänder sig till blivande och verksamma lärare i förskola, förskoleklass och de tidiga skolåren.
The focus of this article is methodological, on how teachers’ participation in practice-based research coacts with research quality. Educational design research is an example of a practice-based research approach often used in mathematics education with the goal of developing both the theories and the practice of teaching and learning mathematics. In this article, one such educational design research study on problem solving in Swedish preschool class is used as an example of how teachers’ participation in practice-based research can develop and of how different kinds of collaboration between researchers and teachers coact with research quality. One conclusion of the methodological meta-analysis is that there is a challenging tension between ensuring external validity of a study versus enabling internal validity and improvement of practice.
This article is about the systematization and representation young children spontaneously use when they are working on a combinatorial task. In this article, documentations from 123 children working on the same task are analysed. The question asked is if there are any connections between the systematizations and representations used in the documentations and how the children solve the task. The results indicate that there are some connections between systematization and representations and that both prepossess children’s solutions. In this paper, we provide some possible reasons; however, we also state that more studies are needed to give deeper insights on these issues.
This paper is about the representation young students use when they are working on a problem-solving task on combinatorics. Results from previous studies on young students and combinatoricshave shown connections between the representations used and to what extent students solved the task. Based on these previous results, young students in this study were interviewed about their choice ofrepresentation. In this paper, the rationales expressed by the students are connected to therepresentation and stage of systematization shown in their documentation. The results indicate thatdifficulties in representing the context of the problem-solving task, may force some of the students towork with a representation on a level of abstraction not suitable for them. Working with representations on a non-suitable level of abstraction may in turn influence how the students manageto solve the problem-solving task.
The empirical data in this study are from a series of two lessons on measurement implemented in seven classes with 119 six-year-old students in Sweden. Both problem solving and problem posing were shown to be important in early mathematics when students in this study worked on one problem-solving task and one problem-posing task on measurement. As there are few studies specifically on problem posing in early mathematics and on young children’s understanding of measurement, this study adds knowledge of value for both teachers and researchers. In the study, paper-and-pen work from the students was analysed together with interviews conducted after the students had worked on the two tasks. When solving the task on measurement, the students discerned shape, size, distance, and number as mathematical aspects of measurement. When asked to pose a similar task, only size and number reoccurred as mathematical aspects of measurement. However, other features from the problem-solving task reoccurred in the posed tasks: similar drawings were used in combination with questions on measurement as the mathematical content.
According to the Swedish curriculum, problem-solving is to be part of mathematics teaching from preschool continuing throughout all grades in school. However, little is known about young students’ feelings towards problem-solving tasks. This paper reports on an educational design research study investigating the potential in teaching problem-solving in preschool classes (6-year-olds). Two examples are presented showing how the students evaluate their feelings towards the problem-solving tasks they have been working on. The results show that understanding a task from the beginning or being able to solve it quickly are not necessary prerequisites for young students to experience enjoyment when working with the tasks. Quite the opposite, the majority of the students evaluated the tasks as fun and accessible, even though their initial solutions were often incorrect and they had to struggle a lot to solve the problems.
This paper focuses on problem solving and problem posing in mathematics education with 6-year-olds. After working ona problem-solving activity, the young students were asked to pose a similar task to a friend. This article explores how thestudents interpret the notion of similar. To be able to pose a problem-solving task themselves the students had to changeperspective, from searching for information to providing information, and from searching for a solution to searching for aquestion. Also, to create a similar task the students had to reflect on the original problem-solving task. Thus, their posedtasks shed light on their interpretation of what the original problem-solving task was really about. The results show that thelarge majority of the students included some three-dimensional aspects from the original problem-solving task in their posedtasks. However, the questions they posed varied in terms of whether or not they included mathematical elements.
A considerable amount of research shows that it is both possible and plausible to teach young children mathematics. However, there is less agreement regarding the content and the framing of such teaching. In this symposium we will present and discuss possibilities with teaching young children mathematics through challenging problem solving. Based on empirical studies of children 3-6 years old we will imply that it is both possible and plausible to teach young children mathematics through challenging problem solving. This since the children in the presented studies both learn a lot of mathematics and enjoy the activities they are involved in.
Med utgångspunkt i det praktiknära forskningsprojektet Problemlösning i Förskoleklass, fokuserar vi i den här presentationen på hur lärarnas olika deltagande i en intervention samverkar med forsknings-kvalitet. Studien genomförs enligt modell för designforskning med målet att utveckla både teori och praktik avseende lärande och undervisning i matematik. Genom de snart tio år som studien har pågått har de medverkande förskoleklasslärarnas deltagande i interventionen varierat, från passiva observatörer och därmed konsumenter av forskning till att ha ansvar för såväl genomförande av undervisning som datainsamling och därmed producenter av forskning. Dessa olika samarbeten har resulterat i data där frågan om autenticitet är aktuell i förhållande till om intern eller extern validitet är utgångspunkten. Dessa olika autenticitet diskuteras på presentationen i relation till forskningskvalitet.
The understanding of mathematical concepts has been described in terms of conceptdefinition and concept image. We suggest an elaboration of these constructs, theconcept element, to find a way to theoretically describe students’ understanding.The concept element construct was tested in a setting with students working withlinear functions at the secondary school level. Our empirical findings reveal tracesof students’ concept elements regarding linear functions. Some concept elementsappeared early in the process while others appeared after a cognitive conflict (e.g.evoked by the task construction and setting). The detailed grid on which concept elementsare defined was a useful tool, yielding new insights into students’ knowledgeand understanding.
This article reports a study on members in self-organised Facebook groups for teachers. The aim is to investigate how teachers’ needs and actions form their roles in the extended staffroom. 26 teachers from six different Facebook groups within two different school subjects, mathematics and Swedish, are interviewed. The results reveal a trajectory from a lurking behaviour as authorised visitor to a key person within the community of the group or an influencer, where teachers can function as authorities in their field. In relation to this trajectory, the interviewed teachers describe the choice to become public as a crucial step, emphasising the courage needed. A next important step can be identified when teachers leave the role as consumers of content that others provide, asking questions and commenting on others’ posts, in favour of a more contributing role, publishing their own posts and expressing an agenda. Further, these kinds of large Facebook groups can be seen as a node in teachers’ social media networks.
Which dimensions of instruction can be reliably captured using student perception surveys, is subject for debate. The aim of this study is to empirically explore the validity and limitations of two different measures of cognitive activation: systematic classroom observations and student perceptions. 34 video-recorded lessons from ten lower secondary mathematics teachers in Iceland were analysed using an observation system and compared to 217 responses to the Tripod student perception survey. The results indicate that for the cognitive activation dimension, the connection between observer ratings and student perceptions is weak, raising questions about the validity of different measures of instructional quality.
Instructional quality is a research topic that has received increased attention over the past decades. However, despite evidence for its importance to student learning, few studies are designed to examine patterns of prevalent instruction. The present study aimed to enhance the understanding of instructional quality in Swedish lower secondary school by examining patterns of instruction in 7th grade language arts and mathematics. 274 lessons from 73 separate classrooms were video-recorded and analyzed using the Protocol for Language Arts Teaching Observation (PLATO). Findings of the study show that lessons were largely organized through either whole-class instruction or individual seatwork. Mathematics included significantly more explicit teacher scaffolding and mathematics teachers scored higher than language arts teachers on nearly all instructional dimensions. However, in both subjects, there were substantial differences between teachers, meaning that students in different classrooms received systematically different kinds of instruction. Implications for instructional development and future research are discussed.
Att formulera problemuppgifter är en del av problemlösning, en del som dock skiljer sig från att lösa problemuppgifter. Denna text inriktas på hur undervisning med fokus på problemformulering kan utformas, varför det är viktigt att elever i samband med problemlösning ges möjlighet att formulera egna uppgifter samt vad eleverna då kan lära
Erfarna lärare har kunskaper om elevers vanliga missuppfattningar i matematik. Genom att på ett klokt sätt ta hänsyn till det då diagnostiska frågor konstrueras, kan elevers svar avslöja en del om hur de tänker. I artikeln finns exempel på sådana svar och missuppfattningar.
Erfarna lärare har kunskaper om elevers vanliga missuppfattningar i matematik. Genom att på ett klokt sätt ta hänsyn till det då diagnostiska frågor konstrueras, kan elevers svar avslöja en del om hur de tänker. I artikeln finns exempel på sådana svar och missuppfattningar.
Didactiek van de rekendidactiek!
Waar moet ik aan denken als lerarenopleider bij het geven van rekenen-wiskunde in een cursus didactiek? Welke stappen moeten studenten maken om de didactiek tot zich te kunnen nemen en wat zijn belangrijke factoren om mee te nemen in de voorbereiding van didactiek lessen?
Antwoorden op deze vragen staan beschreven in mijn proefschrift Improving teaching, improving learning, improving as a teacher – Mathematical knowledge for teaching as an object of learning. De studie beschreven in het proefschrift werd uitgevoerd door lerarenopleiders rekenen-wiskunde in een cursus didactiek op een Zweedse lerarenopleiding (vergelijkbaar met de Pabo). De lerarenopleiders wilden hun lessen in didactiek verbeteren om zo de wiskundedidactische kennis van studenten te verbeteren. De methode die gebruikt werd was een zogenaamde learning study, een mix van design research en de in Japan gebruikte lesson study. Specifiek voor een learning study is dat een leerteorie, meestal variatietheorie, wordt gebruikt zowel in het plannen als in het analyseren van lessen en testen. Door te focusseren op het leerobject wordt er gezocht naar kritische factoren voor het begrijpen van dat leerobject. Vier zulke aspecten kwamen in de studie naar voren om in acht te nemen bij het onderwijzen van reken- en wiskundedidactiek.
In de presentatie wordt er kort aandacht besteed aan het design van de studie, waarna het resultaat wordt gepresenteerd, ondersteund met voorbeelden uit de verzamelde data. Om de studie te kunnen plaatsen binnen de context van de Zweedse lerarenopleiding zal een kort overzicht van het Zweedse schoolsysteem en de bijbehorende structuur van de lerarenopleiding worden gegeven. Afsluitend wil ik graag kijken naar in welke mate de resultaten herkenbaar en toepasbaar zijn voor lerarenopleiders in Nederland.
The aim of the study described in this article was to improve the teaching of the didactics of mathematics. A Learning Study was conducted during a course in the didactics of mathematics at a teacher training college, and three lessons were planned and analysed with the help of variation theory. The object of learning consisted of five elements of Mathematical Knowledge for Teaching (MKT ) and for this object of learning, four critical features were found. To be able to formulate goals for a lesson, to be able to give detailed descriptions, to have adequate mathematical knowledge and to be able to take a role as a mathematics teacher appeared to be necessary for understanding the object of learning, MKT .