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Accountability measures have quickly entered into formal teacher-preparation programs. As a response, we introduce the use of structural equation modeling vis-à-vis path analysis in secondary-grade mathematics teacher preparation as a methodology to test models to understand the strength of relationships to recommendations of prominent professional organizations and standards for entering the teaching profession. This longitudinal, 6-year, five-cohort study examines the relationship of program design sequencing and core components (internal measures) to an externally scored high-stakes teacher licensing examination portfolio intended to measure pedagogical content knowledge and first-year teacher readiness. The internal measures and program sequencing model explains 49.2% of the variance in relation to the standardized outcome teaching portfolio examination with high-power and medium- to large-effect statistics. We provide implications for teacher preparation with respect to recommendations of professional organizations, governments, and accreditation standards. Results should stimulate discussions and fuel future research efforts.

Mathematics growth trajectory plays a significant role to predict future outcomes and careers. We examined the existence of different growth trajectories in mathematics during the late elementary-school period (from Grades 4 to 6) and the relationship between the trajectory group classification and later mathematics achievement (Grade 8). Then, this study examined the moderating effects of mathematics teachers' teaching styles. Two thousand six hundred eighty students participated in a 5-year longitudinal study in South Korea. Growth mixture modeling provided four distinctive classes of growth trajectories in elementary mathematics achievement, and all classes significantly predicted later achievement in Grade 8. Furthermore, the student's perception of teaching styles revealed different associations with mathematics achievement depending on growth trajectory groups. While direct teaching is positively associated with the high and the average performing groups, dialogic teaching is positively associated with the low performing group. The findings highlighted the growth trajectory is a critical component in predicting later academic achievement and students need appropriate instructional supports.

Over the last three decades, educational researchers and policymakers have increasingly promoted instructional strategies that centralize group work in mathematics. One difficulty teachers face in implementing group-based instruction in mathematics involves facilitating meaningful group interaction amongst students. In this paper, we explore student noticing as a novel strategy for supporting college students to collaborate effectively during group work in mathematics. First, we construct a noticing framework named "student noticing of collaborative practices" which provides a lens for "seeing" how students notice their collaborative practices. Then, we use the framework to explore how 25 college students noticed their collaborative practices in mathematics. After working on a novel mathematics task in groups, the college students listened to audiorecordings of their group interactions and responded to reflection questions about the effectiveness of their collaboration. We identified themes regarding how and what students noticed related to their collaborative practices. The findings reveal that students attended to many aspects of their collaboration, including their talking turns and propensity to listen to others. Students demonstrated a desire to change their collaborative practices in the future. The findings imply that teachers and researchers might leverage student noticing as a tool for improvement in mathematics group-based classrooms.

The purpose of this study was to identify and categorize in-the-moment teaching moves which support productive general social norms and sociomathematical norms during dialogic instruction by drawing on transcripts of classroom episodes. To understand how teachers construct in-the-moment teaching moves, this paper is grounded in a revised version of the Advancing Children's Thinking framework. In particular, the framework reveals three interconnected dimensions of teaching moves: (1) eliciting group strategies, (2) supporting conceptual understanding "and" group coordination, and (3) extending group mathematical thinking "and" coordination. Drawing on this framework, we use a novel self-study methodology to construct a sequence of in-the-moment teaching moves that support the development of productive social and sociomathematical norms in dialogic classrooms. These teaching moves include (1) inviting strategies and encouraging all members to actively listen; (2) exploring, clarifying, and questioning the mathematical details of all strategies as a group; (3) promoting understanding of differing strategies; (4) comparing and evaluating differing strategies as a group; and (5) connecting the group's thinking for making progress on the task/advancing current strategy to become viable. Implications for research and practice are offered.

Understanding how students develop learning opportunities through peer-to-peer interaction is vital for advancing research and practice on collaborative learning environments. This study investigated the discourse practices middle grade students utilized to resolve conflict in ways that promoted or inhibited learning opportunities. Seventy-seven middle grade students were audio/video recorded as they worked in groups of three on cognitively demanding mathematical tasks over three days. Analysis centered on the nature of learners' discourse practices within instances of small group conflict. The findings revealed 17 specific discourse practices which promote learning opportunities within conflict and six discourse practices which limit learning opportunities. Practitioners might leverage these findings by explicitly teaching and modeling the discourse practices which promote learning opportunities. Researchers might build upon the theoretical and methodological contributions of this study to examine promotive discourse practices under different constraints than were examined in the present analysis.

Teachers' attitudes and beliefs play an important role in their integration of digital games in the classroom thus the transformative potential of game-based learning. In this study, we adapt the Technology Acceptance Model to examine elementary teachers' acceptance of a digital mathematics game and investigate antecedents to their intention to use the game to teach mathematics. The hypothesized extension of Technology Acceptance Model includes redefined factors in the context of game use during mathematics instructions (perceived ease of use, attitude towards game use, perceived usefulness for mathematical learning), a social factor (environmental support), and specified outcome factors depending on orientations (game-driven intention, mathematics-driven intention). Using survey data from 304 elementary teachers in the USA, the findings confirm associations within the adapted model, including direct links from attitudes and environmental support towards intentions to use the digital game and indirect links from perceived ease of use and usefulness. Implications for acceptance theories and teaching practices to use digital games are discussed.

Over the past three decades, research and policy in many geographic regions has promoted a shift from direct, lecture-oriented mathematics instruction to inquiry-based, "dialogic" forms of instruction. While theory and research support dialogic instructional approaches, some have noted that the complexities of dialogic teaching make it difficult for teachers to implement. One mechanism by which teachers can improve their decision-making practices in dialogic classrooms is learning to notice (i.e. becoming aware of learners' processes). While research has contributed frameworks for understanding how teachers notice individual learners' mathematical thinking, there is little conceptualization regarding how teachers notice group processes in mathematics classrooms, which is integral to dialogic instruction. We offer a noticing framework termed "professional noticing of coordinated mathematical thinking" that describes how teachers notice group activity in mathematics classrooms. Professional noticing of coordinated mathematical thinking is conceptualized as a bi-dimensional process: noticing groups' mathematical activity and noticing groups' coordinated activity. Teachers must become aware of how groups approach the mathematical and collaborative nature of a task, since both of these aspects inform whether learners develop opportunities to learn in groups. The framework describes noticing practices integral to dialogic instruction and promotes inquiry for future research related to teaching moves in dialogic classrooms.

In recent decades, mathematics education scholarship has promoted a shift from traditional teaching and learning approaches to novel approaches that privilege peer-to-peer collaboration. To mobilize researchers towards advancing scholarship on collaborative learning, scholars need to understand the methodologies available to them for examining group work. In this study, I synthesize and critically examine discourse analytic methodologies for analyzing group work in mathematics. Using a systematic review methodology, an initial 576 articles were considered, and 31 articles met the inclusion criteria for analysis. Articles were separated according to their research foci and methodologies were synthesized and critically evaluated. The review provides researchers with methodological options for analyzing group discourse in mathematics.

Previous research making claims about students' proving capabilities usually only explored one modality (e.g. oral or written arguments). This study compares the written and oral representations of eighth-grade students' mathematical arguments to determine whether the two modalities portray similar understandings. We replicate Stylianides' (2019) analysis using a different temporal sequencing of oral and written arguments and a larger sample size. Using an existing proof scoring scheme, we found a statistically significant difference between students' written and oral representations of proof. Students often struggled to write arguments they could otherwise represent orally, and we found no evidence that students' written arguments were more sophisticated than their oral arguments. We also share two illustrative cases representing the disparity between students' written and oral arguments. The findings of this study suggest that students' proving capabilities might not be as dire as previous research implies.

This study explored how approximating the practice of mathematicians by communally negotiating the standards for proof mediates middle grade students' abilities to collaboratively construct mathematical arguments. Forty-seven eighth-grade students engaged in an instructional sequence wherein they, along with the instructors, negotiated communal criteria for proof and subsequently worked in small groups to collaboratively construct arguments which attempted to meet the communally-negotiated criteria. The findings revealed there was no significant correlation between the number of times a group appealed to communal criteria and the quality of their argument. However, the qualitative analysis revealed groups who created valid arguments utilized communal criteria in productive ways, while groups who created invalid arguments superficially engaged with communal criteria or exhibited fundamental misunderstandings of some criteria. These findings imply developing and utilizing communal criteria is a promising support for improving school-aged students' proving capacities, but further theoretical and empirical research is needed to determine how to develop communal criteria in ways that all students within a classroom community can meaningfully utilize the criteria to mediate their abilities to create arguments.

Proof and argumentation are essential components of learning mathematics, and technology can mediate students' abilities to learn. This systematic literature review synthesizes empirical literature which examines technology as a support for proof and argumentation across all content domains. The themes of this review are revealed through analyzing articles related to Geometry and mathematical content domains different from Geometry. Within the Geometry literature, five subthemes are discussed: (1) empirical and theoretical interplay in dynamic geometry environments (DGEs), (2) justifying constructions using DGEs, (3) comparing technological and non-technological environments, (4) student processing in a DGE, and (5) intelligent tutor systems. Within the articles related to content different from Geometry, two subthemes are discussed: technological supports for number systems/algebra and technological supports for calculus/real analysis. The technological supports for proof revealed in this review could aid future research and practice in developing new strategies to mediate students' understandings of proof.

Current international curriculum frameworks and research advocate for K-12 mathematics to emphasize proof and argumentation. This review synthesizes literature related to conceptions, content, and supportive actions for proof and argumentation across four grade-level bands: K-2, 3-5, 6-8, and 9-12. Our systematic search and subsequent screening process yielded 76 articles for coding and analysis. The findings suggest (a) a variety of conceptualizations that are imbalanced amongst the grade bands, (b) there is more research related to proof and argumentation in middle (6-8) and high school (9-12) compared to elementary school (K-5), (c) the type of content used to solicit argumentation is not well-balanced, and (d) supportive actions include aspects related to classroom culture, individual accountability, and conjecturing before justifying.

While a growing body of research examines teachers' facilitation of discourse-rich classrooms, surprisingly little research is devoted to learners' responsibilities in such classrooms. In this paper, we share a theoretical model for explaining students' responsibilities in yielding mathematical learning in discourse-rich classrooms. These responsibilities consist of the following: (1) determined listening and striving to understand others' contributions; (2) proactive contribution; (3) maintaining equal positioning; (4) willingness to resolve incommensurability; and (5) on-task talk. Each of the responsibilities is interdependent, suggesting that failure to meet one responsibility decreases the likelihood that another responsibility will be met. The model suggests important implications for supporting learners in discourse-rich classrooms.

Accountability measures have quickly entered into formal teacher-preparation programs. As a response, we introduce the use of structural equation modeling vis-à-vis path analysis in secondary-grade mathematics teacher preparation as a methodology to test models to understand the strength of relationships to recommendations of prominent professional organizations and standards for entering the teaching profession. This longitudinal, 6-year, five-cohort study examines the relationship of program design sequencing and core components (internal measures) to an externally scored high-stakes teacher licensing examination portfolio intended to measure pedagogical content knowledge and first-year teacher readiness. The internal measures and program sequencing model explains 49.2% of the variance in relation to the standardized outcome teaching portfolio examination with high-power and medium- to large-effect statistics. We provide implications for teacher preparation with respect to recommendations of professional organizations, governments, and accreditation standards. Results should stimulate discussions and fuel future research efforts. [ABSTRACT FROM AUTHOR]

Understanding how students develop learning opportunities through peer-to-peer interaction is vital for advancing research and practice on collaborative learning environments. This study investigated the discourse practices middle grade students utilized to resolve conflict in ways that promoted or inhibited learning opportunities. Seventy-seven middle grade students were audio/video recorded as they worked in groups of three on cognitively demanding mathematical tasks over three days. Analysis centered on the nature of learners' discourse practices within instances of small group conflict. The findings revealed 17 specific discourse practices which promote learning opportunities within conflict and six discourse practices which limit learning opportunities. Practitioners might leverage these findings by explicitly teaching and modeling the discourse practices which promote learning opportunities. Researchers might build upon the theoretical and methodological contributions of this study to examine promotive discourse practices under different constraints than were examined in the present analysis. [ABSTRACT FROM AUTHOR]

In recent decades, mathematics education scholarship has promoted a shift from traditional teaching and learning approaches to novel approaches that privilege peer-to-peer collaboration. To mobilize researchers towards advancing scholarship on collaborative learning, scholars need to understand the methodologies available to them for examining group work. In this study, I synthesize and critically examine discourse analytic methodologies for analyzing group work in mathematics. Using a systematic review methodology, an initial 576 articles were considered, and 31 articles met the inclusion criteria for analysis. Articles were separated according to their research foci and methodologies were synthesized and critically evaluated. The review provides researchers with methodological options for analyzing group discourse in mathematics. [ABSTRACT FROM AUTHOR]

Previous research making claims about students' proving capabilities usually only explored one modality (e.g. oral or written arguments). This study compares the written and oral representations of eighth-grade students' mathematical arguments to determine whether the two modalities portray similar understandings. We replicate Stylianides' (2019) analysis using a different temporal sequencing of oral and written arguments and a larger sample size. Using an existing proof scoring scheme, we found a statistically significant difference between students' written and oral representations of proof. Students often struggled to write arguments they could otherwise represent orally, and we found no evidence that students' written arguments were more sophisticated than their oral arguments. We also share two illustrative cases representing the disparity between students' written and oral arguments. The findings of this study suggest that students' proving capabilities might not be as dire as previous research implies. [ABSTRACT FROM AUTHOR]

While a growing body of research examines teachers' facilitation of discourse-rich classrooms, surprisingly little research is devoted to learners' responsibilities in such classrooms. In this paper, we share a theoretical model for explaining students' responsibilities in yielding mathematical learning in discourse-rich classrooms. These responsibilities consist of the following: (1) determined listening and striving to understand others' contributions; (2) proactive contri - bution; (3) maintaining equal positioning; (4) willingness to resolve incommensurability; and (5) on-task talk. Each of the responsibilities is interdependent, suggesting that failure to meet one responsibility decreases the likelihood that another responsibility will be met. The model suggests important implications for supporting learners in discourse-rich classrooms. [ABSTRACT FROM AUTHOR]

Current international curriculum frameworks and research advocate for K-12 mathematics to emphasize proof and argumentation. This review synthesizes literature related to conceptions, content, and supportive actions for proof and argumentation across four grade-level bands: K-2, 3–5, 6–8, and 9–12. Our systematic search and subsequent screening process yielded 76 articles for coding and analysis. The findings suggest (a) a variety of conceptualizations that are imbalanced amongst the grade bands, (b) there is more research related to proof and argumentation in middle (6–8) and high school (9–12) compared to elementary school (K-5), (c) the type of content used to solicit argumentation is not well-balanced, and (d) supportive actions include aspects related to classroom culture, individual accountability, and conjecturing before justifying. [ABSTRACT FROM AUTHOR]