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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.

Formative assessment has been suggested as a means of supporting studentmotivation. However, empirical studies have shown mixed effects of formativeassessment interventions on students’ motivation, making it necessary to understandthe mechanisms underlying these effects. We analyzed a formative classroompractice implemented by a 10th-grade first-language teacher during 7 months.Teacher logs, classroom observations and a teacher interview were used to collectdata for characterizing the formative assessment practice. Changes in students’satisfaction regarding the basic psychological needs of perceived autonomy,competence and relatedness, as well as changes in student motivation manifestingas engagement in learning activities and autonomous types of motivation, weremeasured by pre- and post-questionnaires in the intervention class and fourcomparison classes. Since the intraclass correlation values ICC(1) and ICC(2)were low, we treated the comparison classes as one group and t-tests were usedin the significance testing of the differences in changes in psychological needssatisfaction and motivation between the intervention class and the comparisonclasses. Path analysis was conducted to investigate whether a possible influenceof the intervention on autonomous motivation and behavioral engagementwould be mediated by basic psychological needs satisfaction. The analysis ofthe classroom practice in the intervention class identifies that both teacher andstudents were proactive agents in formative assessment processes. The analysisof the quantitative data shows that students’ psychological needs satisfactionincreased more in the intervention class than in the comparison classes, and thatthis needs satisfaction mediated an effect on students’ behavioral engagementand autonomous motivation.

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.

In the present study we investigate one 7th-grade mathematics teacher’s eight-month long implementation of a comprehensive multi-approach formative assessment practice. Based on an analysis of classroom observations, interviews and written teacher logbooks, this classroom practice is described in detail to illustrate how multi-approach formative assessments may be enacted in practice. Furthermore, based on the same data, we identify characteristics of formative assessment practices that have been argued to influence student motivational characteristics that affect motivation manifested as behavioural engagement and autonomous types of motivation. We also demonstrate that while the students in the two control classes (n = 9 in both classes) decreased their engagement in learning activities and became less autonomous in their motivation during this period, the students in the intervention class (n = 14) became more engaged and did not become less autonomous in their motivation.

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.

Studies have shown that learning mathematics through programming can be complex and that the programming itself might even hamper students’ learning. However, few studies have focused on the role of the teacher and the teacher-student interaction that aims to support students’ learning when using programming. The present study examines a didactic design in which a teacher uses well-prepared questions and suggestions that focus on stu- dents’ reasoning to solve mathematical problems using Scratch, a block-based programming environment. Forty students, 10–11 years old, solved a geometry problem using Scratch supported by their teacher. The students’ screen activities and teacher-student interactions were recorded. The results indicate that well-prepared general and task-specific questions targeting students’ creative reasoning can help to overcome some of the complexities of learning mathematics while programming. The study furthermore shows that when students have sufficient programming skills, Scratch could even have the potential to support their reasoning and at the same time guide the teacher to give timely feedback. 

Research has shown that students’ learning gains in mathematics are greater when they work with problems rather than routine tasks. These learning gains from problem-solving activities may be enhanced by providing feedback that does not give away the solutions to the problems, but helps students construct their solution methods themselves and anchor their reasoning in intrinsic properties of the mathematical components involved in the reasoning. However, in order to use feedback, students would need to perceive it as useful, and not all students may find such feedback useful. In this study, we investigate how students’ ability and motivational beliefs affect how useful they perceive feedback aimed at supporting mathematical reasoning to be. In the study, students worked with mathematical problems and received metacognitive and heuristic feedback when they needed help. We used structural equation modeling (SEM) to analyze the effects. The results show that students’ mastery goals had a direct effect on the perceived usefulness of the feedback, but no such effects were found for students’ national test grades, self-efficacy beliefs, performance goals, or intrinsic or extrinsic forms of motivation. The proportion of successful use of feedback did not mediate the effects.

Background: Many learning methods of mathematical reasoning encourage imitative procedures (algorithmic reasoning, AR) instead of more constructive reasoning processes (creative mathematical reasoning, CMR). Recent research suggest that learning with CMR compared to AR leads to better performance and differential brain activity during a subsequent test. Here, we considered the role of individual differences in cognitive ability in relation to effects of CMR.

Methods: We employed a within-subject intervention (N=72, MAge=18.0) followed by a brain-imaging session (fMRI) one week later. A battery of cognitive tests preceded the intervention. Participants were divided into three cognitive ability groups based on their cognitive score (low, intermediate and high).

Results: On mathematical tasks previously practiced with CMR compared to AR we observed better performance, and higher brain activity in key regions for mathematical cognition such as left angular gyrus and left inferior/middle frontal gyrus. The CMR-effects did not interact with cognitive ability, albeit the effects on performance were driven by the intermediate and high cognitive ability groups.

Conclusions: Encouraging pupils to engage in constructive processes when learning mathematical reasoning confers lasting learning effects on brain activation, independent of cognitive ability. However, the lack of a CMR-effect on performance for the low cognitive ability group suggest future studies should focus on individualized learning interventions, allowing more opportunities for effortful struggle with CMR.