Learning from errors: effects of teachers training on studentsâ attitudes towards and their individual use of errors

Constructive error handling is considered an important factor for individual learning processes. In a quasi-experimental study with Grades 6 to 9 students, we investigate effects on students’ attitudes towards errors as learning opportunities in two conditions: an error-tolerant classroom culture, and the first condition along with additional teaching of strategies for analyzing errors. Our findings show positive effects of the error-tolerant classroom culture on the affective level, whereas students are not influenced by the cognitive support. There is no evidence for differential effects for student groups with different attitudes towards errors

The Role of General and Subject-specific Language Skills when Learning Mathematics in Elementary School

The role of skills in the language of instruction for mathematics learning is well established by longitudinal studies at the primary school level. Explanations for these relations lead to the question: are they mostly due to general, domain-overarching language skills, or does the command of subject-specific language registers play an important role? Integrating prior research threads, we propose two instruments to measure subject-specific language skills in mathematics: One measuring mathematical vocabulary, and one measuring mathematical text comprehension. We report on a longitudinal study with N = 237 German grade 3 students, which investigated the predictive value of these instruments beyond prior arithmetic skills, general language skills, and control variables such as general cognitive skills and socio-economic status on students’ later arithmetic skills. We applied a multidimensional assessment model to measure arithmetic skills. Apart from replicating the prominent role of general language skills found in earlier studies, our results indicate a substantial, additional role of subject-specific language skills for the development of mathematics skills. These relations could be identified for knowledge of mathematical vocabulary, as well as for mathematical text comprehension. The results indicate that fostering subject-specific language skills already at the primary school level is not only one of many goals of mathematics instruction, but is a core prerequisite to supporting mathematical skill acquisition

Contribution of flexibility in dealing with mathematical situations to word-problem solving beyond established predictors

To solve mathematical word problems, students need to build appropriate models of the described situations, which they can describe with mathematical operations. Various studies have confirmed the importance of general cognitive skills, basic arithmetic skills, and language skills for word-problem solving. Beyond these, we investigate flexibility in dealing with mathematical situations, a new construct that describes the skill to re-interpret everyday situations from various perspectives. In a study with N = 113 second graders, an instrument to measure this flexibility construct has been developed and investigated. We find that the construct explains word-problem solving skills beyond the established predictors. Being able to flexibly re-interpret everyday situations may be beneficial for word-problem solving

Supporting Mathematical Argumentation and Proof Skills: Comparing the Effectiveness of a Sequential and a Concurrent Instructional Approach to Support Resource-Based Cognitive Skills

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as 'resource-based,' as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students' work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students

Conceptualizing and measuring instructional quality in mathematics education: A systematic literature review

Conceptualizing and measuring instructional quality is important to understand what can be understood as “good teaching” and develop approaches to improve instruction. There is a consensus in teaching effectiveness research that instructional quality should be considered multidimensional with at least three basic dimensions rather than a unitary construct: student support, cognitive activation, and classroom management. Many studies have used this or similar frameworks as a foundation for empirical research. The purpose of this paper is to investigate the relation between the conceptual indicators underlying the conceptual definitions of the quality dimensions in the literature, and the various operational indicators used to operationalize these factors in empirical studies. We examined (a) which conceptual indicators are used to conceptualize the basic dimensions theoretically, (b) to which extent the operational indicator in the literature cover these conceptual indicators, and (c) if which additional indicators are addressed by the measurement instruments, which are not part of the theoretical conceptualization. We conducted a systematic literature review on the conceptualization and operationalization of Instructional Quality in Primary and Secondary Mathematics Education based on PRISMA procedures. We describe the span of conceptual indicators connected to the three basic dimensions over all articles (a) and analyze to which extent the measurement instruments are in line with these conceptual indicators (b, c). For each measurement dimension, the identified quality dimensions identified are, taken together, largely representative of the conceptual indicators connected to the core factor, but also a number of critical misconceptions occurred. Our review provides a comprehensive overview of the three basic dimensions of instructional quality in mathematics based on theoretical conceptualizations and measurement instruments in the literature. Beyond this, we observed that the descriptions of a substantial amount of quality dimensions and their conceptualizations did not clearly specify if the intended measurement referred to the learning opportunities orchestrated by the teacher, or the utilization of these opportunities by students. It remains a challenge to differentiate measures of instructional quality (as orchestrated by the teacher) from (perceived) teacher competencies/knowledge, and students’ reactions to the instruction. Recommendations are made for measurement practice, as well as directions for future research