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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.

Several studies have demonstrated that substantial learning gains are possible when teachers use formative assessment in their classroom practice. At the heart of most definitions of formative assessment lies the idea of collecting evidence of students’ thinking and learning, and based on this information modifying teaching to better meet students’ needs. Such regulation of learning processes would require skills to elicit the thinking underlying students’ oral and written responses, and the capacity to make suitable instructional decisions based on this thinking. When the continuation of the teaching is contingent on the information that appears in such assessments additional knowledge and skills are required compared with a more traditional approach to teaching.

Today, sufficient knowledge about how to help in-service teachers and pre-service teachers develop their formative classroom practice is lacking. In the pursuit of gathering research evidence about the specific content and design of professional development programs and teacher education courses in formative assessment, it is important that we know what kinds of skills and knowledge teachers need to successfully orchestrate a formative classroom practice.

The aim of this study is to identify activities and characterize the knowledge and skills that a teacher of mathematics uses in her formative assessment practice during whole-class lessons.

The study is a case study of a teacher’s formative assessment practice during mathematics lessons in year 5. The data were analysed by identifying a) the formative assessment practice b) the teacher’s activities during the formative assessment practice and c) the teacher knowledge and skills used during the activities.

The main result of the study shows that the formative assessment practice is a very complex, demanding and difficult task for the teacher in several ways. For example, during short term minute-by-minute formative assessment practice the teacher uses knowledge and skills to eliciting, interpreting and use the elicited information to modify instruction to better meet student learning needs. She also helps students’ to engage in common learning activities and take co-responsibility of their learning. In the minute-by-minute formative assessment practice the teacher also handles new mathematics (to her), unpredictable situations and makes decisions about teaching and learning situations in a matter of seconds. 

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.