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Division is regarded as the most difficult of the four basic operations for students to learn. Yet, there is little research on how teaching can support students learning of methods for multi-digit division. Supporting students’ own construction of methods, rather than letting students imitate given methods, has been shown to result in more sustainable learning for other mathematical topics. However, designing teaching that is contingent to students’ own construction requires knowledge of students’ construction processes for the method in question. In this case study, we provide a detailed example of such a process regarding a method for multi-digit division. The results show that students can construct increasingly sophisticated methods for multi-digit division based on additive partitions of the dividend into multiples of the divisor.

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.

The purpose of this intervention study was to develop and evaluate a support model for teachers, designed to assist them in diagnosing students’ (age 16–19 years) difficulties and providing feedback to support students’ mathematical problem solving. Reporting on an iteration in a design research project, the results showed that the support helped the teachers to provide less procedural information and instead help students construct solutions for themselves. Constraints in achieving this included the nature of some tasks, difficulties in making reasonable diagnoses, and students’ inability to communicate their difficulties.

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.

Implementing teaching through mathematical problem-solving entails substantial challenges and calls for sustained teacher-researcher collaboration. The joint research and development project ”Teaching that supports students’ creative mathematical problem-solving” has a fundamental ambition to be symmetric in that both teachers’ and researchers’ needs and conditions are attended to and complementary in that their different areas of expertise are utilised and valued. In this paper we show how the interplay and development of symmetry and complementarity can function as a means for studying teacher-researcher collaborations.

Ett centralt skäl till elevers svårigheter med matematik är att undervisningendomineras av utantillinlärning och arbete med rutinuppgifter trots att vi vet atten undervisning som lyfter fram problemlösning är mer effektiv för att utvecklamatematisk förståelse. Genom ett aktivt val av uppgifter kan lärandet förbättras.

A selection of secondary school mathematics textbooks from twelve countries on five continents was analysed to better understand the support they might be in teaching and learning mathematical problem solving. Over 5700 tasks were compared to the information provided earlier in each textbook to determine whether each task could be solved by mimicking available templates or whether a solution had to be constructed without guidance from the textbook. There were similarities between the twelve textbooks in the sense that most tasks could be solved using a template as guidance. A significantly lower proportion of the tasks required a solution to be constructed. This was especially striking in the initial sets of tasks. Textbook descriptions indicating problem solving did not guarantee that a task solution had to be constructed without the support of an available template.