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The potential of using formative assessment is well demonstrated, but studies about the use of formative assessment from a special education perspective are lacking. This study adds to this gap by investigating the view of formative assessment in a group of 39 special education teachers in mathematics (SETMs) who had learned about formative assessment within the SETM-program 2–6 years earlier. Five respondent interviews were used to design a questionnaire answered by the rest of the group. The SETMs had perceived formative assessment beneficial and useful in all their common sub-responsibilities and reported experiences of benefits as well as challenges. The article discusses the importance of reaching an inclusive formative assessment practice in mathematics education.

Research shows that substantial learning gains are possible through the use of formative assessment. However, little is known about Swedish mathematics teachers’ use of formative assessment, and thus about the possible value of professional development programmes. This study uses teacher interviews and classroom observations to examine the classroom practice of 38 randomly selected primary and secondary school teachers in a mid-sized Swedish municipality. A framework of formative assessment comprising one big idea and five Key strategies structured the analysis. The study identifies characteristics of current formative assessment practices. The results show that the teachers do use a variety of formative assessment activities, but also that there is much room for development towards a more effective formative classroom practice. 

Formative assessment has been shown to have the potential to significantly enhance student achievement, but a strong research base about how to support teachers to implement such a practice is lacking. This is particularly so for a conceptualisation of formative assessment as a unity of integrated formative assessment strategies. The aim of this study is to investigate why the mathematics teachers who participated in a successful professional development programme in formative assessment developed their formative classroom practice to such an extent that it had a significant impact on student achievement. An analysis of data from teacher questionnaires and interviews shows that the teachers' actions can be explained by expectancy-value theory of achievement motivation. Characteristics of the professional development programme that the teachers experienced as important for their development of a formative classroom practice are identified, and the characteristics' affordances for the development of the teachers' expectancy and value beliefs are discussed.

A random sample of 22 Year 4 teachers in mathematics from a middle-sized Swedish municipality participated in a teacher professional development programme in formative assessment. The content ofthe programme was formative assessment conceptualised as a unity of different, integrated strategies.The study examines the effects on student achievement of the changes in the teachers’ formative classroom practice that followed the professional development input. Results show that, after controlling for pretest scores, the classes in the intervention group significantly out performed the classes in thecontrol group in a posttest administered one school year after the end of the programme (p = 0.036, d = 0.66). The study contributes to the understanding of under-studied areas of the impact of professional development in formative assessment, and the impact of teacher practice based on formative assessment conceptualised as a unity of different formative assessment strategies.

Multilingual students’ participation in national tests in mathematics – discursive prerequisites. This article explores discursive prerequisites in test-taking for second language learners with other mother tongues than Swedish. Four students were interviewed in 2016 during their final year of compulsory school. The results imply that multilingual students are positioned as disadvantaged within testing. This phenomenon is mainly situated in a competitive discourse with several subordinated discourses that further position the students: A discourse of justice positioned the students as being sorted or left behind, a discourse of handling the assessment positioned the students as caretakers and a discourse of future challenges positioned the students as struggling while learning, being capable to learn or facing positive challenges. The results imply that national testing is a personal and relational experience and gives rise to issues of legitimacy and equality. These issues should be considered in policy-making, the construction and the carrying out of tests as well as in the conclusions which are based on the results on individual, group and organisational levels.

This paper presents and discusses parts of a large-scale [1] ethnographical and longitudinal study that has followed the process of implementing the National Test in Mathematics for third graders (Ntm3) in Sweden during its first three years (2010, 2011, and 2012). Pupil talk from 2011 about pressure and what is at stake was used to construe three positions of need that might characterize pupils during the National Test in Mathematics in their third school year: the position of shame, the position of unfamiliarity, and the position of stress. How these might be handled in educational practices is discussed briefly.

This thesis presents the contribution to research that my doctoral education led to. My starting point was a large scale qualitative research project (here after called the VR-project) which reviewed the implementation of national tests in the third grade on the subject of mathematics. The VR-project investigated how the test affected the pupils with a special focus on pupils in need of special support. An urge to look further into issues concerning the support, the pupil in need and the test was revealed in he initial VR-project. These issues therefore constitutes the problem area of this thesis. The VR-project studied a total of 22 classrooms in two different municipalities' during 2010- 2012. The methodology used for this project was inspired by ethnography and discourse analysis. The raw data consisted of test instructions, video observations of the actual test subjects, interviews from teachers and pupils about the test, the support that was given throughout the testing as well as the observations and interviews of the pupils requiring special assistance. Activated discourses and positions of the participants were demarcated. The results revealed that a traditional testing discourse, a caring discourse and a competitive discourse are activated during the tests. The testing discourse is stable and traditional. Much of what was shown and said in classrooms, routines and rules regarding the test were repeated in all the schools and in all the classrooms. The discourse on support is affected by ambiguity, which is revealed especially when issues of pupils’ equity is put against the tests equality. This is connected to the teachers restricted agency to give support due to the teacher position as a test taker. The positions in need that are available to students are not the same in pupils, teachers and steering documents. The situation is especially troublesome for pupils that do not manage Swedish good enough to take the test and for pupils in need of special support. Some of the conclusions from this thesis is that the national test format: Disciplines not only the pupil, but also the teacher, the classroom and the school at large. Results indicate that the test:

Activates a focus on achievementLeads attention away from learning Activates issues of accountability Influences pupils and teachers with stakes involved

Besides evaluating knowledge, the test disciplines not only the pupil, but also the teacher, the classroom and the school at large. Discussing the national test as an arena for equity might be a way towards attaining equality in education for all pupils.

This chapter considers quality and equity in mathematics education and addresses how discursive circumstances on national tests position students, particularly those in need of special support, at both the individual and group levels. It discusses that the processes and systemic patterns prompting inequalities need to be evaluated at the local, national, and international levels. National tests in mathematics in students' third grade in school were reinstated in Spring 2010, and to date, third graders have taken national tests for seven years, something that might change given suggestions in a social democratic governmental investigation on national education assessment. In short, it is suggested that tests in third grade be eliminated, since the several purposes of national tests have been deemed incompatible. Throughout national testing in mathematics in the third grade, a secondary purpose was cited as an important argument for tests and connected to the decreased levels of knowledge on the subject: to identify pupils in need of early support.

Denna artikel studerar förutsättningarna för det stöd som kan ges till elever i samband med det nationella provet i matematik i det tredje skolåret (härefter kallat Ntm3), i Sverige under 2010 och 2011. Den stöddiskurs som aktiveras i samband med proven har avgränsats genom att provets instruktioner samt åtta intervjuer med lärare har analyserats. Resultaten indikerar att provet leder till att provdeltagarnas fokus förskjuts från lärande till kontroll och att provets dubbla syften positionerar läraren som både testtagare och testgivare. Detta visar sig som en ambivalens i stöddiskursen och leder till ett pedagogiskt dilemma i situationer där provets likvärdighet sätts mot elevens.

This paper discusses parts of the discourse on multilingual pupils in need of support in the national test in mathematics in the third grade. A content analysis was done on the test-instructions from the years 2010–2014. A shift in thediscourse was seen, from being about students in need to being about students with disabilities. The results show that instructions have moved from a relational towards a more categorical perspective on the student. One implication following this is that teachers receive less guidance in their mission to help pupils who need language support.

In present pilot study the relation between mathematical performance, test anxiety and mathematical self image is being examined in two Swedish grade 3 pupils. A difference between the high achieving and the low achieving pupils is possible to discern. A negative or uncertain mathematical self image seems to be connected to test anxiety, but not to performance when we look at these two young pupils. The experience of taking the test is affecting the high and low achiever differently. There are indications that the context and the kind of information given about the tests and the pupil's abilities affect both the level of test anxiety and the pupil’s evolving mathematical self image. Educational implications are being discussed.

The focus of this paper is the conceptualisation of students in special educationalneeds in mathematics (SEM students) in the research fields of mathematics andspecial education. A difference between fields regarding the perspectives takenon the SEM student is obvious in the reviewed articles. Those in the specialeducational field were individual oriented in their view of the difficulties, whilstreviewed articles from the field of mathematics education more often discusssocio-cultural settings. The content in the selected 28 articles reveals that theoverall conceptualisation of SEM student has to do with the social construct ofthe SEM student, as well as with students’ experiences, affects, andprerequisites; with the specific training methods or interventions applied; withspecial areas in the subject of mathematics; with special groups of students; andwith teachers’ knowledge about all these factors.

One of the latest reforms to increase the equity and quality in mathematics education is mandatory national assessment in Preschool-class. It is supposed to govern the action, focus and professional language of the pedagogues. In an earlier study of preparatory work, policy decisions and assessment material, we discovered that there is a risk for schoolification of the Preschool-class, through this reform. A second risk is that students could be locked into "levels" of achievement and a third risk lies in a potential narrowing down of the curriculum (Bagger, Björklund, Vennberg, Accepted). At the same time, earlier research show that early assessment can lead to positive development for students in need of support (Vennberg & Norqvist, 2018). Deriving from our initial study of national assessment in preschool class, we have planned for a follow up project with the aim to contribute to knowledge of and developing the practice about the preschool class teacher’s work with national assessment in mathematics. During the conference we will present and discuss ethical and methodological challenges in this project. More specifically, we want to discuss what is required of practice close approaches in these ethically demanding situations of vulnerability and assessment in mathematics with young students. A key issue is the opportunities for teachers and researchers to manage the information collected during the  national assessment and handling of sensitive data, how they jointly can contribute to the development and management of the knowledge that generated about students' opportunities to learn and to demonstrate their knowledge. And finally, in what way this might contribute to assessment and teaching in mathmatics continues to have its starting points in the unique assignment of the preschool-class.

References

Bagger, A., Björklund, L. & Vennberg, H. (accepted). The politics of early assessment in mathematics education. CERME11 in Utrecht, January 2019.

Vennberg, H. & Norqvist, M. (2018). Counting on – long term effects of an early intervention program. In Bergqvist, E., Österholm, M., G, C & Sumpter, L. (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, (pp. 355-362). Umeå, Sweden: PME.

One of the latest reforms in Sweden in order to increase equity and quality in education is making national assessment compulsory in preschool-class (age 6). The claimed political volition is all students’ best possible mathematical development. In this paper, we examine the preparatory work, the assignment to the National Agency of Education, and the assessment material itself, searching for what meaning is inscribed regarding the student, mathematics and assessment. The results imply that the politics of the assessment might exaggerate a search for flaws and control instead of promoting all students learning and in addition contributing to the schoolification of preschool-class.

This chapter presents nine case studies in which school students engage in challenging mathematics outside their immediate classroom environment. In each case, students are encouraged to collaborate in investigations that go beyond the standard curriculum and creatively use the ingredients of the particular context. In Italy, students visit a mathematical laboratory to understand and utilize mathematical machines. Morning assembly at an Indian school brings students from many classes together in the solution of mathematical problems. Four of the projects are from France: students analyze the configuration of a heap of sand, pursue astronomical investigations with software, obtain a flavor of research by having secondary school teams investigate interesting problems, and are presented at all levels with open-ended research problems. There are three programs from the United States, the first, an advanced geometry sequence for secondary students completing the regular syllabus early, the second, activities arising from exhibits in an art museum, and the third, using the school lawn to deepen student understanding of geometric constructions. All such activities need to be evaluated for their effectiveness, so that they move from just being initiatives of dynamic individuals to serve as the foundation for systemic improvements in the way in which students learn, understand and use mathematics. In the early part of this chapter, we briefly mention how research into such activities might be approached.

This study uses a statistical method to identify verbal items among mathematical items from PISA 2003. The verbal items are preliminary analysed and compared to the non-verbal items concerning number of text lines, response types, cognitive level, and competences measured. The results show that the verbal items, to a higher percentage than the non-verbal items, measures the reproduction competency, are straigh­t­forward, and of open constructed-response type. These results and proposed further analyses are discussed.

Students often use imitative reasoning, i.e. copy algorithms or recall facts, when solving mathematical tasks. Research show that this type of imitative reasoning might weaken the students' understanding of the underlying mathematical concepts. In a previous study, the author classified tasks from 16 final exams from introductory calculus courses at Swedish universities. The results showed that it was possible to pass 15 of the exams, and solve most of the tasks, using imitative reasoning. This study examines the teachers' views on the reasoning that students are expected to perform during their own and others mathematics exams. The results indicate that the exams demand mostly imitative reasoning since the teachers think that the exams otherwise would be too difficult and lead to too low passing rates.

Over the last decades, there has been an on-going international reform for school mathematics, which has, not surprisingly, been difficult to implement. This study focuses on teachers’ interpretation of formal written curriculum documents, especially whether their interpretations align with how a concept (the concept of problem) is conveyed in the documents (in Sweden). The results show that the formal written documents are vague, but that it to some extent conveys the concept of problem as “a task for which the solution method is not known in advance to the solver.” The interviews show that about 53 % of the teachers interpreted problem as “any task,” and that teachers’ interpretationstherefore are not aligned with how the concept is (albeit vaguely) conveyed in the documents.

More than 15 years after the introduction of a standards-based curriculum reform, the mathematics teachers are positive towards the reform message but have not changed their classroom practice accordingly. To improve the impact of future reforms, it is important to learn from this situation and to better understand the role of the national policy documents. The purpose of this study is therefore to examine how the standards-based reform in mathematics in Sweden was conveyed in the formal written curriculum. The research questions focus on to what extent and how clearly the national policy documents convey the message. The results show that the message is present to a large extent in the policy documents, but that it is vague and formulated with complex wording. The study gives concrete examples and shows in detail in what ways the documents are vague and complex.

In mathematical text and talk, natural language is a constant companion to mathematical symbols. The purpose of this study is to identify different types of relations between natural language and symbolic language in mathematics textbooks. Here we focus on the level of integration. We have identified examples of high integration (e.g., when symbols are part of a sentence), medium integration (e.g., when the shifts between natural and symbolic language occurs when switching to a new line), and low integration (e.g., when symbols and written words are connected by the layout).

In this study, we examine how the use of natural language varies, considering the symbolic language in procedural and conceptual aspects of mathematics.

This paper focuses on relationships between vocabulary in mathematical tasks and aspects of reading and solving these tasks. The paper contains a framework that highlights a number of different aspects of word difficulty as well as many issues to consider when planning and implementing empirical studies concerning vocabulary in tasks, where the aspect of common/uncommon words is one important part. The paper also presents an empirical method where corpora are used to investigate issues of vocabulary in mathematical tasks. The results from the empirical study show that there are connections between different types of vocabulary and task difficulty, but that they seem to be mainly an effect of the total number of words in a task.

This empirical study examines how different types of symbols, familiar and unfamiliar, are processed in working memory; phonologically and/or visuo-spatially.

Våren 2004 infördes på försök två olika versioner av de nationella kursproven i matematik på kurserna B, C och D. På varje kurs skilde sig de båda versionerna åt i några av de ingående uppgifterna, medan övriga uppgifter var identiska. Syftet med denna studie är att undersöka om, hur och varför dessa förändringar i matematikuppgifterna påverkar uppgifternas svårighetsgrad.

Resultaten visar att en förändring av de i uppgifterna ingående talen endast i ett fåtal fall påverkat uppgifternas svårighetsgrad i någon större utsträckning. Dessa fåtal fall studeras vart och ett för sig. När två uppgifters sammanhang och formulering skiljer sig åt, även om det matematiska innehållet är i stort sett identiskt, visar ett exempel på att skillnaden i svårighetsgrad kan vara mycket stor.

The mathematics PISA tasks are primarily supposed to measure mathematical ability and not reading ability, so it is important to avoid unnecessary demands of reading ability in the tasks. Many readability formulas are using both word length and sentence length as indicators of text difficulty. In this study, we examine differences and similarities between English, German, and Swedish mathematics PISA tasks regarding word length and sentence length. We analyze 146 mathematics PISA tasks from 2000–2013, in English, German, and Swedish. For each task we create measures of mean word and sentence length. To analyze if there are any differences between the three language versions of the tasks, we use t-tests to compare the three languages pairwise. We found that in average, the German versions have the longest words, followed by Swedish and then English. Average sentence length was highest for English, followed by German and then Swedish.

The purpose of this study is to deepen the understanding of the relation between the language used in mathematics tasks and the difficulty in reading and solving the tasks. We examine issues of language both through linguistic features of tasks (word length, sentence length, task length, and information density) and through different natural languages used to formulate the tasks (English, German, and Swedish). Analyses of 83 PISA mathematics tasks reveal that tasks in German, when compared with English and Swedish, show stronger connections between the examined linguistic features of tasks and difficulty in reading and solving the tasks. We discuss if and how this result can be explained by general differences between the three languages.

In this paper we suggest a theoretical model of the connection between the process of reading and the process of solving mathematical tasks. The model takes into consideration different types of previous research about the relationship between reading and solving mathematical tasks, including research about traits of mathematical tasks (a linguistic perspective), about the reading process (a psychological perspective), and about behavior and reasoning when solving tasks (a mathematics education perspective). In contrast to other models, our model is not linear but cyclic, and considers behavior such as re-reading the task.

In this study we analyse the communication competence included in two different frameworks of mathematical knowledge. The main purpose is to find out if mathematical communication is primarily described as communication of or about mathematics or if it is (also) described as a special type of communication. The results show that aspects of mathematics are mostly included as the content of communication in the frameworks but the use of different forms of representation is highlighted both in the frameworks and also in prior research as a potential cause for characterising mathematical communication differently than "ordinary" communication.

Det språk vi använder oss av i matematikklassrummet kan fokuseras på många olika sätt. Språket är också nödvändigt att förhålla sig till vid utvecklingen av sitt matematiska tänkande. Författarna diskuterar här relationer mellan språk och lärande.

There is much research on the role of theory in mathematics education research, at least from more overarching or theoretical perspectives. Micro analyses of the role of theory in particular research studies are rarer. We contribute by analysing one empirical study to allow for in-depth analyses and discussions around the role of theory in a specific case, concerning relationships between mathematics and reading. Our results show that studies that do not use an explicit theoretical model can still be strongly influenced by implicit theoretical assumptions. We conclude that it is important to identify existing theoretical assumptions in an empirical research study and try to convey them as clearly as possible, and we discuss specific issues concerning research on relationships between mathematics and reading.

Eight teachers were interviewed concerning how students verify conjectures. The study is a sequel to a previous study, “How Students Verify Conjectures” [Bergqvist, T. (2000). How students verify conjectures. Research reports in Mathematics Education 3]. Teachers’ expectations of students’ reasoning and performance are examined, and also how they wish students would work. The results indicate that the teachers tend to underestimate the students’ reasoning levels and that they believe that only a small group of students in each class can use higher level reasoning in mathematics.

In this study podcasts, i.e. short video clips, were analysed to ascertain how mathematical competencies were addressed. We analysed how the mathematical content was presented in the podcasts, and then how these characterizations related to eight students’ interest to learn mathematics. The mathematical presentations was analysed using the Mathematical Competencies Research Framework (Lithner et al., 2010). The results indicate that podcasts that address several mathematical competencies receive overall higher grades by the students in the study.

Proceedings from the 13^th ProMath conference, September 2 – 4, 2011, in Umeå, Sweden.

Med utgångspunkt i ett stort antal artiklar om problemlösning i de 15 senaste årens Nämnaren diskuteras i artikeln några intressanta problem och problemlösningens roll i skolans matematikundervisning.