Umeå University's logo

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.

Educational design research (EDR) needs to further clarify how it leads to valid theoretical results. In this article, we aim to contribute to the development of EDR by suggesting an explication of what validity of design principles for teaching entails, and how the characteristics and phases of EDR can be employed to increase such validity. Our suggestions are based on our own adaptation of EDR within a research program focusing on teaching that promotes students’ mathematical reasoning, informed by discussions and elaborations of EDR and validity of causal inferences. Our results elaborates existing causal validity types in relation to design principles for teaching and adds two new validity types: relative benefit and guiding power. We also describe the importance of employing multi-coder level-raising analysis using an intermediate framework in collaboration with teachers to develop theory over iterations to increase not only explanatory power but also predictive and guiding power.

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.

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.

This paper reports the results of the initial cycles of a smaller part of an ongoing large-scale design research project, consisting of a five-week teaching sequence for the beginning of grade 1 focussing mathematical patters and structure and additive decomposition of number. The results include both emerging design and emerging insight, and how researcher and teacher knowledge and experiences contributed to those results.

Problem solving has been considered the gold standard of mathematical activity. It is a goal of mathematics education that students become problem solvers, and it is suggested that problem solving is a superior method for learning mathematics.  However, the arguments supporting the claim that problem solving leads to better learning are often vague. In specific studies, problem solving often constitutes mere one part of a compound design, making it difficult to determine the specific contribution of problem solving. The aim of this paper is to develop an initial theoretical model for problem solving as a learning activity, based on existing frameworks and previous research. Suggestions for how this model could be empirically tested are also discussed.