Generalizing with algebraic expressions and variables is a challenging area of algebra content that requires conceptual understanding. Instructional videos can contribute to conceptual understanding only if students deeply engage with the video content. This deep engagement can be scaffolded by activating design features (e.g., brief prompts). However, limited research attention has been given to specifying their content-specific focus. This paper reports on a design research approach concerning generalizing with expressions, conducted in two design-experiment cycles with 16 students. Through qualitative comparative scaffolding analysis, we specified which content-specific processes needed focus at what moment through which activating feature. The first design experiment cycle, using an unscaffolded video, revealed which processes occurred spontaneously and which required scaffolding through activating features. The second cycle indicated that the designed activating features could indeed scaffold the intended processes. We discuss how this design research specification approach for content-specifically focused scaffolds could be transferred to other subject-matter content.

For implementing instructional approaches in classrooms, material or personnel implementation strategies have been applied, yet rarely compared with respect to the separate effectiveness of each strategy. Our effectiveness study (75 teachers, 1466 students) compared different implementation strategies for an intervention on mathematical word problems, with students' self-efficacy beliefs and word problem performance as dependent variables. In four implementation conditions, teachers received weak or strong personnel and/or material support with an additional control condition. Multilevel models revealed that students of teachers with support had significantly higher post-test performance in both dependent variables than the control group. High personnel or material support led to significantly higher post-test word-problem performance than only weak support. Combined high personnel and material support led to higher outcomes in self-efficacy beliefs than either did separately. In conclusion, implementation strategies should provide both strong personnel and material support, as the combination can best enhance students' performances and self-efficacy beliefs.

Assessing conceptual understanding in mathematics education requires students to explain meanings and to make connections between concept elements explicit. However, many students struggle with written answers to open-ended questions in assessments, but are able to explain their thought processes orally. The growing availability of devices capable of automatic speech recognition (ASR) and the improved performance of ASR models calls for research on integrating speech input into digital formative assessments. This study explores how speech-input answers (n = 354) from 53 Grades 5 and 6 students differed from typed answers (n = 595) in a between-group comparison and analyses in detail which individual and item factors shaped the features of the speech-input answers. Results from multi-level models indicated that speech input answers were substantially longer, belonged more often to an everyday language register, but were richer, including more arguments and explanative discourse practices so that shallow understanding became more visible. Especially for low-achieving students, speech input seemed to be beneficial, independent from the chosen input format, for both dictation tools and voice messages.

Erklärvideos werden für fachliches Lernen im Fachunterricht dann lernwirksam, wenn sie fachdidaktisch treffsicher ausgewählt und aktivierend eingesetzt werden. Der Beitrag stellt ein Fortbildungskonzept vor, mit dem Lehrkräfte ihre Praktiken zum fachdidaktisch treffsicheren und (kognitiv und kommunikativ) aktivierenden Einsatz für verschiedene didaktische Anforderungssituationen ausbauen und dazu wichtige medienbezogene fachdidaktische Hinter gründe (ausgewählte Aspekte des Technological Pedagogical Content Knowledge, TPCK) lernen können. Der Beitrag stellt den konzeptionellen Rahmen und die Realisierung in Fortbildungsaktivitäten vor. Erste Evaluationseinblicke zeigen vielversprechende Entwicklungen in den adressierten Praktiken and Kategorien.

For many out-of-field teachers, limitations in mathematical and pedagogical content knowledge have been identified. However, the ways in which these limitations shape their instructional practices have not been systematically explored. Furthermore, little is known about how out-of-field teachers’ practices can grow through professional development. A qualitative single-case study contributes to reducing these research gaps by comparing teaching practices. These were video-recorded before and after completing a session that provided access to pedagogical content knowledge about key concept elements for place value understanding. The comparison reveals that the teacher’s practices shift from untargeted surface practices to deep conceptually targeted practices. It disentangles how targeting concept elements supports deeper, conceptually focused instruction. Unpacking the background of substantial professional growth can contribute to emphasizing the value of topic-specific professional development opportunities.

Many teachers seek to foster students’ conceptual understanding, e.g., for meanings of multiplication, but are only partially prepared to go beyond surface translations between multiple representations. This concerns out-of-field teachers as well as in-field teachers (i.e., with mathematics teaching certificate). The paper presents a professional development program in which teachers engage in PD inquiry activities (a) to specify what understanding multiplication entails and how deeper connections of representations are articulated, (b) to monitor students’ understanding and uncover surface translations, and (c) to enhance students’ understanding by engaging them in explicating unit structures in graphical and symbolic representations. The randomized controlled trial with n = 94 teachers investigates the extent to which out-of-field and in-field teachers’ approximated practices can change in a PD session, and whether their professional learning gains depend on an explicit summary in the PD systematization phase. The comparison of pre-vignettes and post-vignettes reveals significant professional learning gains for all three practices, slightly more strongly for out-of-field teachers. For out-of-field teachers, the explicit summary in the systematization phase has significant additional effects for specifying and enhancing, less for monitoring practices. The findings show that out-of-field teachers can undergo very promising changes, especially when including an explicit systematization phase on specifying the mathematical learning content.

Instructional videos play an increasingly important role in subject-matter education. They can enhance conceptual understanding by connecting representations and providing language to explain ideas. However, since videos are typically consumed in a receptive manner, eliciting sufficient cognitive engagement poses a particular design challenge. This controlled trial investigates design conditions and individual use factors that promote learning with videos using a two-phase instructional macro-design: a video was integrated into a knowledge organization phase following an exploration phase and combined with language and cognitive engagement support. In the controlled trial with N = 288 second-year undergraduate students (mathematics pre-service teachers) exploring part-whole structures in conditional probability information, we examined conditions for deepening conceptual learning by comparing four randomly assigned micro-design engagement support conditions for the knowledge organization phase: (1) video without engagement support, (2) video with segmentation, (3) video with prompts, and (4) video with prompts and interactive features. Beyond comparing these conditions, we examined students’ use of the provided learning opportunities. ANOVAs showed immediate use effects on students’ explanation correctness and academic language use but no significant differences between micro-design support conditions. A multiple linear regression model, controlling for pretest scores and pre-video language use, revealed that learning outcomes depended on students’ cognitive engagement during exploration and their academic language uptake during knowledge organization, rather than on specific micro-design support. The findings highlight implications for supporting engagement in two-phase learning environments: future designs may benefit from deliberate language input and targeted prompts that foster language uptake.

Self-learning environments with instructional videos are well-established for procedural skill remediation, but their effectiveness for conceptual understanding of complex concepts has been less examined. This study examines the impact of more or less structured prompts in instructional videos on students’ understanding of variables as generalizers while controlling for effects of students’ individual engagement. A randomized controlled trial with N = 239 students (grades 9–11) compared three self-learning environments covering the same algebraic content—understanding variables and algebraic expressions: a control group with written explanations and practice tasks, a video group with moderately structured prompts, and a video group with highly structured prompts. All groups significantly improved their understanding; however, when accounting for students’ individual engagement operationalized through prompt use, both video groups achieved significantly higher effects than the control group. No significant differences emerged between the two video groups. For students with low pretest scores, the control group proved less inclusive. The findings highlight the potential of video-supported self-learning for conceptual understanding, but also indicate that further design improvements are needed to engage all students.

Dynamically linked multi-representation tools are designed to promote students’ understanding of mathematical concepts. While qualitative studies have shown their potential for deepening students’ understanding for four decades, the quantitative evidence for their efficacy on measurable learning gains has remained very limited and has not yet been disentangled from other potential benefits of digital teaching–learning environments in which they are integrated. In this class-wise randomized controlled trial with 136 fifth graders from six classes, we aimed to disentangle the effects of a dynamically linked multi-representation tool from its digital environment, in this case, understanding multiplication as counting in units in dot arrays, by comparing it with a digital environment with flexible non-linked manipulatives. The statistical analysis in linear mixed models revealed that students in both digital environments significantly outperformed the students in the waiting control group. The dynamically linked dot array (supporting the unit structures in dot arrays) had a significant yet only small additional effect compared to flexible non-linked dots. While the evidence of a small additional effect of dynamically linked multi-representation tools can inform the design of future digital teaching–learning environments, its small effect size calls for further studies to disentangle conditions for increasing the effects.

Pandemic-related school disruptures (by closures or individual distance learning periods) appear to have social and immigration-related disparities in the mathematics achievement of students in Germany. Previous research has addressed the disparity in home and distance learning conditions. However, characteristics of origin, home and distance learning conditions have hardly been related to mathematics achievement. This cross-sectional study aims to gain insight into the relationships between background variables, home and distance learning conditions and mathematics achievement. To this end, the achievement data of n = 2023 children at the beginning of Grade 5 were analyzed descriptively and with multilevel models. As in other studies, it was found that children with a low socio-economic status or from first immigrant generation had below-average home learning conditions, but received slightly above-average intensive support from teachers compared to their classmates. For the first time for this age group, connections between learning conditions during school closures and mathematics achievements were examined: The home learning conditions such as technology equipment as well as the distance learning conditions such as regular written homework assignments proved to be significant predictors of mathematics achievement. Assumptions about social and immigration-related disadvantages during school disruptures can be differentiated with these findings.

While existing surveys on professional development (PD) programs for mathematics teachers revealed a substantial growth in PD research studies, there has been a call for more in-depth reports. Inspired by reviews on practice-based pre-service teacher education, this systematic review focuses on PD activities developed and investigated for enhancing teachers’ practices for handling relevant tasks of mathematical teaching, to provide an overview of the state of PD research for practicing teachers. To accomplish this, keywords were used to identify relevant research, complemented by hand-searched papers which are in the core of the topic yet do not use the keywords. Based on the full-text analysis of 32 peer-reviewed studies, this systematic review uncovered the emphasis of the most studies on the task of supporting and enhancing student learning followed by noticing and assessing student thinking, with variation in associated teaching practices. Differences in the level of reporting on PD activities were revealed concerning both the PD design and the research about PD. Tendencies in the types of frameworks for designing PD were also found, with almost half the studies situating teacher learning in practice. Overall, this review not only advances the field by making findings on teacher practices to handle tasks of teaching more comparable, but by gaining insights into the types of corresponding PD activities.

Facilitating whole-class discussions based on tasks that support inquiry and provide conceptual learning opportunities has been identified as a central, challenging task of teaching. We drew upon practice-based and interactionist research traditions to decompose how eight experienced Grades 5-7 mathematics teachers facilitated whole-class discussions after their students had determined volumes for rectangular prisms. The discussions highlighted ways that students worked with wooden cubes, imposed multiplicative structures, and developed formal calculations. Qualitative analysis of the video-recorded lessons revealed five practices that the teachers used to facilitate whole-class discussions. Three of the practices supported extensive, explicit connections between higher-order and lower-order knowledge elements related to volumes of rectangular prisms. We characterized such instruction as providing deep conceptual learning opportunities. The remaining two practices jumped quickly to calculations and did not make as many explicit connections to the organization of cubes in rows, columns, and layers. We characterized such instruction as providing shallow learning opportunities. Identifying the five facilitation practices contributes to theorizing about relations between teaching and learning that support conceptual understanding and underscores the importance of examining such relations at a micro-level and in the context of specific mathematical topics. Moreover, the results can inform professional development programs that seek to help teachers to enact productive conceptual learning opportunities during whole-class discussions.

For several decades, digital formative assessment tools and platforms have been promoted as promising for supporting adaptive teaching. For learning goals such as procedural fluency, this promise has been held across a range of subject-matter topics. For conceptual learning goals, however, the available digital formative assessment tools are much less widespread, and the research needed to develop them still seems to be underestimated. This paper reports on the design process of the Mastering Math Online-Check to illustrate approaches to a research-based design of targeted digital formative assessment tools that (a) are relevant, (b) are valid, and (c) provide in-depth, informative insights into students’ understanding of subject-matter concepts (in this case, basic arithmetic concepts such as the meaning of multiplication and division). The described research needs and the considerations about research-informed designs refer to improving the depth of insights into students’ understanding and their explanations of meanings. This conceptual (rather than empirical) paper discusses pivotal questions that must be addressed to fully leverage technological advances for the assessment of deep conceptual understanding, not only for mathematical concepts but also for other abstract concepts that rely on explanations and visual representations for the assessment of their understanding.

Although curriculum materials are essential for implementing instructional innovations, little is known about how exactly they support teachers’ enactment of innovative approaches and how explicitly they must integrate their key components. The paper contributes to reducing this research gap for one instructional innovation: explicit strategy instruction for word problem comprehension strategies. In our qualitative video study, teachers participated in a professional development on explicit strategy instruction and experimented with it in their classrooms. We compared the teachers’ enactment of strategy instruction in two implementation conditions: in the strong material support condition, teachers received curriculum materials in which components of strategy instruction were explicitly integrated. In the weak material support condition, the curriculum materials only provided the sequence of word problems, but the teachers had to implement explicit strategy instruction themselves. In the qualitative analysis of four cases of teachers’ videorecorded enactment, we unpacked how one teacher with weak support provided only limited learning opportunities by non-targeted enactment, with limited measurable student strategic learning gains. However, in the same condition, another teacher provided productive learning opportunities by explicit and targeted strategy instruction in spite of weak material support. In the strong material support condition, learning opportunities were assured even in a case of partially non-targeted enactment (yet high measurable student strategic learning gains). The findings bear consequences for combining curriculum materials with strong support and professional development.

Die pandemiebedingten Schulunterbrechungen (durch Schließungen und individuelle Fernzeiten) scheinen die sozialen und migrationsbedingten Disparitäten in den erzielten Mathematikleistungen von Lernenden in Deutschland verstärkt zu haben. Bisherige Forschung hat sich mit der Unterschiedlichkeit der häuslichen und fernunterrichtlichen Lernbedingungen befasst. Jedoch sind Herkunftsmerkmale, häusliche und fernunterrichtlichen Lernbedingungen bisher kaum mit Fachleistungen in Beziehung gesetzt worden. Die vorliegende querschnittlich angelegte Studie verfolgt das Ziel, Einblick in die Zusammenhänge zwischen Herkunftsmerkmalen, häuslichen und fernunterrichtlichen Lernbedingungen und Mathematikleistung zu gewinnen. Dazu wurden die Leistungsdaten von n = 2023 Kindern zu Beginn der fünften Klasse deskriptiv und mehrebenenanalytisch ausgewertet. Wie in anderen Studien zeigt sich, dass Kinder mit niedrigem sozioökonomischem Status oder der ersten Migrationsgeneration unterdurchschnittlich ungünstige häusliche Lernbedingungen aufweisen, aber im Durchschnitt etwas intensiver von Lehrkräften betreut wurden als ihre Klassen. Erstmalig ermittelt wurde für diese Altersgruppe, welche der Lernbedingungen prädiktiv für die Mathematikleistungen waren: Häuslichen Lernbedingungen wie Technik-Ausstattung sowie fernunterrichtlichen Lernbedingungen wie regelmäßige Abgaben erwiesen sich als signifikante Prädiktoren für die Mathematikleistung. Vermutungen zur sozialen und migrationsbedingten Benachteiligung während der Schulschließungen und individuellen Fernzeiten können mit diesen Befunden ausdifferenziert werden.

Low-achieving students need targeted interventions to remediate the critical foundational knowledge required for successful further learning trajectories. For Grade 5, critical basic concepts (e.g., place-value understanding and meaning of multiplication) and basic skills (e.g., counting in steps and multiplication facts) have been identified as equally important and highly intertwined. However, evidence for the effectiveness of foundation intervention programs has mainly been provided for basic skills and for younger children's understanding of basic concepts but has not yet been sufficiently provided for fifth graders' understanding of basic concepts. In this paper, we present a quasi-experimental implementation trial (n = 295) on the effectiveness of the Mastering Math foundation intervention program that aims at fostering understanding of basic concepts in arithmetic. Teachers in the intervention group were prepared in professional development courses and supported by detailed formative assessment and teaching materials to foster low-achieving students' understanding in small-group interventions supplemental to the regular mathematics classrooms. Teachers in the control classes received assessment feedback on students' achievement and the suggestion to work on basic skills and basic concepts, but no professional development or teaching materials for their regular mathematics classrooms. The intervention group was compared to a control group (selected by propensity score matching) by analyses of variances with repeated measures, applying gender, socioeconomic status, immigrant background, and reading proficiency as co-variates. The intervention group significantly outperformed the control group in the explicitly covered basic concepts. No transfer effect was found for basic skills. Further analyses showed that the intervention group learned significantly more than their higher-achieving classmates. Most interesting was a positive spillover effect that all students in the Mastering Math schools learned significantly more basic concepts of below-grade content, so the teachers' professional experiences in the small-group foundation intervention seem to also have diffused into their regular classrooms. The results show that, when teachers gain additional expertise on fostering understanding of basic concepts, conceptually focused foundation intervention programs targeting understanding of basic concepts can be effectively implemented and can even have an impact on the learning of those students who did not attend the small-group intervention.

The cascade model, in which facilitators provide professional development to teachers who then implement innovations, can result in diluted impact along the cascades. Effectiveness has rarely been shown for the level of student outcomes and supportive conditions, such as teacher collaboration and principal leadership, are needed. This study examines whether implementing a research-based, language-responsive mathematics innovation through a cascade model impacts student outcomes and whether teacher collaboration and principal leadership support implementation. Student outcomes were assessed before and after the teaching unit in view in an intervention group and a control group (n = 479), with school context conditions reported by teachers. Multilevel models revealed that implementation in the cascade model was effective in improving student outcomes. Principal leadership positively predicted outcomes, especially in the control group providing mathematics instruction as usual, while teacher collaboration was a negative predictor. Implications for future research and the design of professional development programs are discussed.

Enhancing understanding is crucial for all students. Instructional approaches have been developed to achieve this goal, yet little is known about the kind of support that teachers need for their effective implementation. We compare two support conditions designed to enable teachers to enhance students’ understanding of the mathematical topic percentages. 51 teachers and their grade 7 classes participated in a quasi-experimental pre-post-test control group design, comparing three groups. (1) Teachers of the material condition received the curriculum material of the MATILDA teaching unit on percentages, with a comprehensive teacher manual on the main principles of the approach. (2) Teachers in the workshop condition received the same materials and additionally participated in professional development (PD) workshops covering key ideas of the approach. (3) The waiting control group used their regular textbooks. Hierarchical regressions were used to compare the effects on students’ learning outcome. The analysis reveals that students in both support conditions achieved significantly higher learning outcomes than the control group, i.e., curriculum materials and teacher manuals seem to provide effective support. No additional effect on student outcomes was found for the workshop condition, but a substantially lower teacher dropout than in the material condition. This finding indicates that PD workshops may not be necessary for highly motivated volunteer teachers but can be completed by a wider range of teachers. The findings can inform implementation projects for educational innovations and inspire further research to determine for which teachers a written PD is sufficient and for which face-to-face workshops are more promising.

In individuellen und selbstgesteuerten Lernsettings, aber auch in gemeinsamen Klassengesprächen besteht die Gefahr, das Lernen auf Oberflächenwissen zu reduzieren. Welche Gesprächsstrategien helfen Lehrkräften und Lernenden, zu einem tieferen Verständnis von Unterrichtsinhalten zu gelangen?