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For teacher feedback to be most beneficial for student learning, students would need to proactively seek and actively engage with it. Embedding feedback within a formative assessment practice has been proposed as a means to support such student engagement. In this study, we investigate a mathematics teacher’s development of a formative assessment practice and the associated changes in students’ engagement with feedback during individual task solving. Inductive thematic coding of classroom audio-recordings was used to identify characteristics of changes in student-teacher interactions, and Orthogonal Projections to Latent Structures Discriminant Analysis was employed to examine multivariate differences between lessons. Results indicate that changes in the teacher’s practice were accompanied by increased student engagement with feedback.

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.

Teachers’ formative feedback provides motivational and cognitive support for student engagement in learning and raises their achievement. Because feedback arises from assessment, teacher conceptions of both assessment and feedback matter to how feedback is practised. This study determines how teacher conceptions of assessment and feedback are linked to each other and to self-reported practices in a repeated measures survey 18 months apart. All factors in the Swedish Teachers Conceptions of Assessment inventory predicted all six Conceptions of Feedback, which fully mediated influence of assessment conceptions on teachers’ self-reported formative feedback practices. Two conceptions of feedback (i.e., feedback improves learning, and students ignore feedback) were important predictors of teachers’ practice, and these feedback conceptions were predicted by the corresponding conceptions of assessment (i.e., it improves learning and it is ignored by students). The results imply that teacher formative feedback practices depend on an improvement-oriented conception of assessment as a prior belief. 

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.

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.

Student self-report data for intrinsic motivation and behavioural engagement in mathematics was collected differently for Grade 1-3 students compared to Grade 4-9 students. For the younger students the teacher read the items aloud and students responded using a three-point smiley face. The older students read the items themselves and used a five-point agreement response scale. The challenge was how to link scores across the two different measures of psychological constructs so that the potential impact of a teacher professional development project in formative assessment could be evaluated. Confirmatory factor analysis established three-item scales for both constructs in both instruments. Equating used the small sample circle-arc method because only 517 students from Grades 2-3 did the second measure when they were in Grades 4-5. The difference in self-reported scores over the two-year period was calculated using three scores; that is, raw average scores, z-scores based on in grade and data wave, and circle-arc equated scores. MANOVA revealed a grade by condition interaction that was statistically significant only for intrinsic motivation. The result was the same for each score type. This suggests that once robust scale scores are created using latent trait theory, scores for the same constructs can be compared regardless of differences in administration or response format. Equating is not necessary in these circumstances. 

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.