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Multi-digit subtraction is a central part of Grade 1–3 mathematics that many students struggle with and is challenging to teach. We used design research to investigate how and to what extent game-based activities could support Grade 1 students’ development of methods for two-digit subtraction. Our results show that a short sequence of activities based in the game led to 85 % of students being able to document at least one method for two-digit subtraction, and 42 % being able to document at least two different methods. However, the way the teacher picked up students’ methods for discussion and how the teacher documented them affected what methods students developed.

Several studies have shown that learning mathematics can be more effective if students solve problems by their own reasoning than if they follow given solution methods. However, when a student needs help with solving a task it is more challenging for the teacher to support the student’s own reasoning than to describe the method. In a collaborative research and development project, teachers and researchers developed a framework for diagnosing in- the-moment problem-solving difficulties. The present study compares the characteristics of experiment and control group teachers’ diagnoses of student difficulties. The study included 83 teachers making 298 diagnoses. The results show that the teachers in the experiment group, to a higher degree, focused on the problem-solving process, while teachers in the control group focused more on aspects of student competence. The teachers in the experiment group also anchored their diagnoses in the students’ statements and written notes to a greater extent than the control group teachers.

This teacher guide aims to provide guidance for teachers who want to support students’ own mathematical reasoning during mathematical problem solving. First, starting points, purpose, and the overall structure of the guide are formulated. The guide then describes what learning about and through problem solving means, what difficulties students encounter during problem solving, and how teaching can support students’ learning. The second half of the guide presents a framework with specific diagnoses and associated suggestions for feedback for students’ difficulties. The framework is summarized in an overview on the last page of the guide.

Denna lärarguide är en vägledning för lärare som vill stödja elevers egna matematiska resonemang i matematisk problemlösning. Inledningsvis formuleras utgångspunkter för problemlösning, lärarguidens syfte samt guidens övergripande struktur. Guiden beskriver därefter vad lärande om och genom problemlösning innebär, elevers svårigheter i problemlösning, samt hur undervisning kan stödja elevers lärande. Andra halvan av guiden presenterar ett ramverk med specifika diagnoser och tillhörande förslag på hjälp till elever, vilket sammanfattas i ett översiktsblad på sista sidan.

It is well-known that a key to promoting students’ mathematics learning is to provide opportunities for problem solving and reasoning, but also that maintaining such opportunities in student–teacher interaction is challenging for teachers. In particular, teachers need support for identifying students’ specific difficulties, in order to select appropriate feedback that supports students’ mathematically founded reasoning without reducing students’ responsibility for solving the task. The aim of this study was to develop a diagnostic framework that is functional for identifying, characterising, and communicating about the difficulties students encounter when trying to solve a problem and needing help from the teacher to continue the construction of mathematically founded reasoning. We describe how we reached this aim by devising iterations of design experiments, including 285 examples of students’ difficulties from grades 1–12, related to 110 tasks, successively increasing the empirical grounding and theoretical refinement of the framework. The resulting framework includes diagnostic questions, definitions, and indicators for each diagnosis and structures the diagnostic process in two simpler steps with guidelines for difficult cases. The framework therefore has the potential to support teachers both in eliciting evidence about students’ reasoning during problem solving and in interpreting this evidence.

Research has shown that formative assessment can enhance student learning. However, it is conceptualised and implemented in different ways, and its effects on student achievement vary. A need has been identified for experimental studies to carefully describe both the characteristics of implemented formative assessment practices and their impact on student achievement. We examined the effects on student achievement of changes in formative assessment of a random sample of 14 secondary school mathematics teachers after a professional development programme. This study describes practices implemented and students’ achievement as measured by pre-tests and post-tests. We found no significant differences in achievement on the post-test, after controlling for pre-test scores, between the intervention group and control group, and no significant correlation between the number of formative assessment activities implemented and the post-test scores (controlled for the pre-test scores). We discuss characteristics of formative assessment implementations that may be critical for enhancing student achievement.

It is long known that students’ learning in mathematics is facilitated by problem-solvingactivities, and school authorities all over the world have incorporated problem-solvingin their curricula. However, problem-solving does not have a clear definition, and itsmeaning risks being watered down in the process of implementation. In this study, weexamine how problem-solving is described and used in Swedish syllabi, commentarymaterials and national tests for school year 6–10, before and after the 2021 and 2022 revisions. Our results show that ‘problem-solving’ is increasingly conceptualised as agoal rather than a means for learning, and that as a goal problem-solving competencyis reduced. As a guidance for teachers the policy documents are often vague and evencontradictory. Implications for teaching practice and Swedish students are discussed.

Research has identified characteristics of professional development programmes (PDPs) that are important for accomplishing change in teachers’ practice and student achievement. However, these characteristics do not predict and guarantee the outcomes of a specific PDP. In the present study the authors investigated whether the expectancy-value theory of achievement motivation can be used to explain teachers’ change of practice. Through teacher interviews and questionnaires the authors investigated why randomly selected year-7 teachers, after participating in a PDP on formative assessment, changed their practice in ways identified in an earlier study, and why these changes differed from those of year-4 teachers participating in a similar PDP. The results show that expectancy-value theory can be used to explain both the schoolyear-7 teachers’ changes in practice, and why the PDP for the year-7 teachers produced different outcomes in teacher practice than the parallel PDP, with the same general design features, did for the year-4 teachers.

Formative assessment has been shown to have the power to positively affect student achievement, but professional-development initiatives for teachers to substantially develop their formative assessment practices have been frequently unsuccessful. In this study, a random selection of Year-7 teachers participated in a professional-development programme (PDP) in formative assessment. Classroom observations, questionnaires, and interviews were used to investigate the changes made in the teachers’ classroom practices after the PDP, and the conditions and barriers they perceived to have been important in their endeavours to develop a formative classroom practice. The results show how all teachers developed their practice, but to different degrees, and that a majority of the implemented activities focused on the teacher as the responsible agent for the formative assessment process. The teachers were satisfied with most of the conditions provided during the PDP, but experienced barriers for change from the teaching conditions at their schools after the PDP.

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.