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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 outlines the research program for the formative assessment group at Umeå Mathematics Education Research Centre. The program was presented in a symposium at the conference, and focuses on the study of the development and impact of formative assessment. The main purpose of the research carried out by the research group is to provide research results that will be used outside the research community for educational decisions on systemic level, or as support for improved teaching and learning at classroom level. The paper outlines the fundamental ideas of the program, current studies, and examples of completed studies.

Research reviews show that formative assessment has great potential for raising student achievement in general, but there is a need for reviews of formative assessment in individual subjects. This review examines its impact on student achievement in mathematics through an assessment of scientific journal articles published between 2005 and 2014 and indexed in Web of science. Through the use of search terms such as ”formative assessment”, ”assessment for learning” and ”self-regulated learning”, different approaches to formative assessment were included in the review. While varying in approach, they all share the defining characteristic of formative assessment: agents in the classroom collect evidence of student learning and, based on this information, adjust their teaching and/or learning. The results show positive relations between student achievement in mathematics and the ways of doing formative assessment included in the review.

The current project focuses on mathematics education, and is partitioned into three subprojects mapping research on formative assessment, classroom teaching, and curriculum programs in mathematics. The rationale for focusing on these three areas is that they are all highly relevant for understanding and improving Swedish mathematics education and students’ knowing of mathematics. Therefore, the aim of the project is to map research on formative assessment, classroom teaching, and curriculum programs in mathematics education.

The methodology of the literature review has been inspired by Gough, Oliver, and Thomas (2013), and we have focused on the mapping on journal articles published on Web of Science (WoS).

The results from the sample of articles on formative assessment show that strategies of formative assessment in mathematics are positively correlated to students’ performance in mathematics with medium and large effect sizes. However, based on the current mapping it is difficult to specify aspects of how the formative strategies are to be implemented in order to promote students’ knowing of mathematics.

Despite the change in perspective of what constitutes knowledge in mathematics to also include reasoning, problem-solving and communication, the map shows that research is mainly focused on examining teaching methods and their effects on students’ skills in mathematics. A closer examination of the studies that do focus on teaching for supporting students in developing competencies like reasoning and problem-solving shows that connections between and comparison of students’ solutions, as well as teachers’ ways of asking questions to support students in explaining their solutions clearly and in detail, are important for students’ learning of these competencies.

A central finding stemming from this review of curriculum programs is the complexity involved in how the programs can support teachers in establishing classroom practices. Curriculum resources and teacher resources, as well as other influencing factors, impact the quality of instruction, and studies have begun to point out how curriculum resources and teacher resources uniquely and jointly impact classroom practices. Multiple research articles have expressed the need for teacher support in implementing curriculum programs, by means of professional development, teacher education and support provided by the curriculum programs themselves. Interesting in this regard is the state of the research field concerning the design of educative curriculum programs, and how teachers make use of such support. Studies have proposed design approaches, regarding both the actual development of educative curriculum programs as well as how to use them in teacher education to support prospective teachers’ development of knowledge. Further, although research has revealed that it is important to prepare for teaching in certain ways, we found very little research that explicitly analyzed how teachers actually prepare for teaching a mathematics lesson.

Limitations of the project include: (1) the lack of searching in potentially relevant databases, (2) the fact that a relatively small proportion of articles found in the search have been coded, (3) that we have not engaged in deep considerations as to whether and in what ways results from international research are relevant in the Swedish context, and (4) that we therefore have not been able to synthesize the results of the study. In relation to the Swedish context (Hemmi & Ryve, 2014; Boesen et al., 2014), international research (Hattie, 2009; Smith & Stein, 2011), and the current project’s findings, we recommend that Skolforskningsinstitutet focus on two aspects of great importance for developing students’ knowing of mathematics. First, Skolforskningsinstitutet should synthesize research that supports actors, such as teachers and principals, in acting within school practices. In the case of teachers, support is needed to engage them in actively anticipating students’ thinking, using curriculum programs effectively, introducing mathematical content, acting in group work, formatively assessing students’ learning, and orchestrating whole-class mathematical discussions. Secondly, actors within school practices need support not only in initiating and implementing developments but also in institutionalizing such developments. Skolforskningsinstitutet should specify the kind of support needed in order to ensure that material, routines, competences, and organizations become integral and permanent features of Swedish school practice. 

Previous research has shown that substantial learning gains are possible when formative assessment and support for students’ development of self-regulated learning skills are implemented in classroom practice. Such implementation is not straightforward and there is a need for both further understanding of the knowledge and skills teachers require to practice formative assessment, and further insights into how different characteristics of ordinary teaching practices support students’ in becoming proficient self-regulated learners. This doctoral thesis includes a licentiate thesis and two articles. In the licentiate thesis, classroom observations are used to investigate the knowledge and skills used by a teacher engaged in a comprehensive formative classroom practice. The results show that the teacher's practice is complex and requires advanced knowledge and skills that are often used simultaneously and under time pressure. For example, the teacher, sometimes in a matter of seconds, handles new (to her) mathematics, makes inferences from students’ responses to their understanding, and based on these inferences makes decisions about her teaching. The first article is a literature review focusing on the effects of formative assessment on student achievement in mathematics since there is a lack of knowledge of the effects of formative assessment on student achievement, in particular for subject areas such as mathematics. In the review, a systematic literature search is made for articles studying the effects of both teacher-centered approaches and approaches emphasizing student involvement in the formative assessment processes. The latter type of approaches includes teacher practices that support students’ development of aspects of self-regulated learning competence. The results show that all approaches included in the review have significant positive effects on student achievement in mathematics. The second article examines in what ways learning situations in authentic classroom practices provide opportunities for the students to develop self-regulated learning skills, and how students experience these opportunities. The analysis is based on data from classroom observations of three teachers’ mathematics lessons, and on interviews with their students. The results show that instruction in self-regulated learning skills mostly occurred implicitly, and the opportunities to develop the skills were mainly provided and experienced at the observational level.

Ett flertal studier har visat att implementering av formativ bedömning ökar elevers lärande i matematik. I hjärtat av de flesta definitioner av formativ bedömning finns iden om att samla belägg om elevers tänkande och lärande, och utifrån beläggen anpassa undervisningen så att den möter elevernas behov. En sådan reglering av lärandeprocessen kräver kunskap om hur man framkallar den typ av svar som visar hur elever tänker. Det krävs också förmåga att välja ett lämpligt sätt anpassa undervisningen. När undervisningen kontinuerligt anpassas efter den typ av bedömning som beskrivits, så behövs ytterligare kunskap jämfört med den kunskap som behövs för att genomföra en mer traditionell undervisning.

Idag saknas tillräcklig kunskap om hur man kan utbilda och stödja lärare och blivande lärare att utveckla sin formativ bedömningspraktik. För att veta vilket innehåll och vilken design kompetensutbildningar i formativ bedömning bör ha så är det viktigt att vi vet vilka specifika kunskaper och förmågor en lärare behöver för att kunna genomföra en framgångsrik formativ bedömning.

Studiens syfte är att identifiera aktiviteter och att karaktärisera de kunskaper och förmågor en matematiklärare använder i sin formativa bedömningspraktik under lektioner i helklass.

Fallstudien följer en lärares formativa bedömningspraktik under matematiklektioner i årskurs 5. Data analyserades med avseende på a) lärarens formativa bedömningspraktik b) lärarens aktiviteter under denna praktik och c) de kunskaper och förmågor som läraren använde för att genomföra dessa aktiviteter.  Studiens huvudresultat visar att den formativa bedömningspraktiken är mycket komplex, krävande och svår på flera sätt för läraren att göra. Till exempel, under en formativ bedömningspraktik som sker minut-för-minut så använder läraren kunskaper och förmågor för att framkalla, tolka och använda den informationen för att bättre kunna möta elevens behov. Läraren hjälper också eleverna att engagera sig i och ta ansvar för sitt lärande under lärandeaktiviteter. Under denna praktik så hanterar läraren även för henne ny matematik, oförutsägbara situationer och fattar beslut om undervisning inom loppet av sekunder.

This case study investigates a teacher ́s use and need of knowledge and skills during interaction in whole-class is using formative assessment. This practice includes eliciting information about student learning, interpreting the responses, and modifying teaching and learning activities based on the given information. The teacher has participated in a professional development program and teaches grade 5 in mathematics. The results of the study will be presented at the conference.

The ability to self-regulate one’s learning has been associated with academic success, and recognized as an important component in successful life-long learning. But studies on how characteristics of ordinary classroom practices provide or impair opportunities for students to develop self-regulate learning skills are rare. This study examines in what ways learning situations in authentic classroom practices provide opportunities for the students to develop self-regulated learning skills, and how students experience these opportunities. The analysis is based on data from classroom observations of three teachers’ mathematics lessons, and on interviews with their students. The results show that the opportunities for students to develop self-regulated learning skills were mainly provided by learning situations in which the teacher modeled the skills, and most often the students did not recognize these situations as opportunities to develop the skills. However, they did so more often when they had an active role in the learning situations.

The study investigates a teacher's use of activities, knowledge and skills when conducting formative assessment during interaction in whole-class. This formative assessment practice includes eliciting information about student learning, interpreting the responses, and modifying teaching and learning activities based on elicited information. Results show that the teacher used activities that help students to engage in common learning activities and take co-responsibility for their learning. Furthermore, while orchestrating the activities the teacher used knowledge and skills that are complex, demanding and difficult.

A reader of mathematical text must often switch between reading mathematical symbols and reading words. In this study, five different categories of structural connections between symbols and language, which invite such switches, are presented in a framework. The framework was applied in a study of Swedish mathematics textbooks, where 180 randomly selected pages from different educational stages were analyzed. The results showed a significant change in communication patterns as students progress through school. From a predomination of connections based on proximity found in year two, there is a gradual change to a predomination of symbols interwoven in sentences in year eight. Furthermore, more qualitative investigations of the different connections complemented the quantification, both through further explanations of the quantitative results, and through more examples of differences in communication patterns. The implications for readers of mathematics texts are discussed.