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This study investigated the role of tools in supporting students to reason about even and odd numbers. Participants included Kindergarten, Grade 1, and Grade 2 students (ages 5–8) at two schools in the USA. Students took part in a cross-sectional early algebra intervention in which they were asked to generalize, represent, justify, and reason with mathematical structure and relationships. Interviews were conducted with students before, during, and after the year-long intervention to explore the ways in which tools mediated their engagement in these practices in the context of questions about parity. In addition to cubes and "dot cards," which were introduced during the interviews, we found that students also used their fingers and hands as tools. We outline the affordances and constraints of these tools in supporting students' reasoning about the properties of even and odd numbers. [ABSTRACT FROM AUTHOR]

Gestures are one of the ways in which mathematical cognition is embodied and have been elevated as a potentially important semiotic device in the teaching of mathematics. As such, a better understanding of gestures used during mathematics instruction (including frequency of use, types of gestures, how they are used, and the possible relationship between gestures and student performance) would inform mathematics education. We aim to understand teachers' gestures in the context of early algebra, particularly in the teaching of the equal sign. Our findings suggest that the equal sign is a relatively rich environment for gestures, which are used in a variety of ways. Participating teachers used gestures frequently to support their teaching about the equal sign. Furthermore, the use of gestures varied depending on the particular conception of the equal sign the instruction aimed to promote. Finally, teacher gesture use in this context is correlated with students' high performance on an early algebra assessment. [ABSTRACT FROM AUTHOR]

We present a study that explores Kindergarten and first-grade students' understandings and representations of arithmetic properties. The 16 students participated in a classroom teaching experiment designed to explore children's algebraic understandings, including their understandings and symbolic representations of three arithmetic properties: additive identity, additive inverse, and commutativity. We characterized students' understandings in terms of Skemp's framework of instrumental (rules without reason) and relational (knowing what to do and why) understandings. Following Vergnaud, we analyzed the types of additive relationships (transformation, comparison, or combination) and representations used by students. Our findings show that students' understandings developed in sophistication over time. We observed the least sophisticated understandings for the commutative property, particularly among Kindergarten students who exhibited instrumental understandings even after instruction. [ABSTRACT FROM AUTHOR]

We share here results from a quasi-experimental study that examines growth in students' algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3–5. Analyses showed that, while there were no significant differences between experimental and control students on a grade 3 pre-assessment measuring students' capacity for generalizing and representing generalizations, experimental students significantly outperformed control students on post-assessments at each of grades 3–5. Moreover, experimental students were able to more flexibly interpret variable in different roles and were better able to use variable notation in meaningful ways to represent arithmetic properties, expressions and equations, and functional relationships. This study provides important evidence that young children can learn to think algebraically in powerful ways and suggests that the earlier introduction of algebraic concepts and practices is beneficial to students. [ABSTRACT FROM AUTHOR]

Recent research suggests that children in elementary grades have some facility with variable and variable notation in ways that warrant closer attention. We report here on an empirically developed progression in first-grade children's thinking about these concepts in functional relationships. Using learning trajectories research as a framework for the study, we developed and implemented an instructional sequence designed to foster children's understanding of functional relationships. Findings suggest that young children can learn to think in sophisticated ways about variable quantities and variable notation. This challenges assumptions that young children are not 'ready' for a study of such concepts and raises the question of whether difficulties adolescents exhibit might be ameliorated by an earlier introduction to these ideas. [ABSTRACT FROM AUTHOR]

Through this study we explored a community of practice framework applied to faculty professional development at a mid-size state university in order to examine the issues unique to discipline-specific professional development in higher education. Through content-focused professional development activities conducted by the authors, several key areas were identified that point to challenges in building a faculty community of teaching practice: (a) the need for a culture of professional development, (b) developing old-timers and recruiting newcomers, (c) the need for teaching scholars to coordinate professional development, (d) challenging the “culture of service”, and (e) the need for a language to mediate thinking about practice. [ABSTRACT FROM AUTHOR]

The original version of this article unfortunately contained a typo in co-author name. [ABSTRACT FROM AUTHOR]

This study explores the use of Valsiner’s zone theory as a way to interpret the zones of proximal development of three secondary teachers in mathematics and science. Specifically, we used classroom discourse to identify what the participating teachers promoted (zone of promoted action) or allowed (zone of free movement) in the classroom as a way to understand better their potential for development. We found that teachers who promoted actions or events that they ultimately did not allow students to experience seemed to be at a point in their development analogous to Vygotsky’s pseudo-conceptual stage, prior to full concept formation. Moreover, we found that, at thisillusionary zone, the teacher’s capacity to listen actively to students’ thinking seemed to affect the teacher’s transition toward more inquiry-based forms of practice. We concluded that understanding the zone of promoted action and the zone of free movement the teacher organizes in the classroom can indicate the existence or absence of an illusionary zone and thereby provide insight into a teacher’s potential for development. [ABSTRACT FROM AUTHOR]

Many of the morphological defects associated with embryonic alcohol exposure are a result of cell death. During limb development, ethanol administration produces cell death in the limb and digital defects, including postaxial ectrodactyly. Because an accumulation of reactive oxygen species (ROS) is produced in adult and embryonic tissues by ethanol exposure, this investigation examines the possibility that ethanol-induced cell death in the limb is a result of ROS. Using an in vitro primary culture of limb mesenchyme, the effects of hydrogen peroxide (H2O2) and ethanol on cell death and differentiation were examined. In addition, a dichlorofluorescein diacetate assay was performed to determine the relative intracellular ROS levels after exposure to several concentrations of ethanol and H2O2. Exposure of 1 to 100 μ M H2O2 resulted in a 1.08–1.21 times control increase in cartilage matrix accumulation. Cell death was increased 1.69–2.76 times the untreated control value. Production of ROS ranged from 1.25–1.51 times untreated controls. Ethanol exposure of 0.25 to 1.00% (v/v) did not affect cartilage matrix accumulation but resulted in an increase of cell death (1.45–2.31 lower than that produced by 1 μ M H2O2. On the basis of the correlation between ROS level produced by H2O2, it was concluded that ethanol-induced cell death in limb mesenchyme is a result of a non-ROS-mediated mechanism. Therefore, in addition to ethanol-induced cell death mediated by ROS reported in the literature, ethanol-induced cell death can be induced in limb mesenchyme by mechanisms that are not dependent upon ROS. [ABSTRACT FROM AUTHOR]

This investigation uses classroom discourse in an undergraduate mathematics course to challenge pre-service secondary mathematics teachers' notions about mathematical discourse, what it might resemble in the classroom, and how its various forms can be cultivated by classroom teachers. The research setting was a required geometry course taught by the author. Eleven pre-service mathematics teachers in their junior or senior year of an undergraduate program participated. Results indicated participants made a transition toward an image of discourse as an active process by which students use the collective knowledge of their peers to build mathematical understanding and developed in their ability to participate in such discourse. This awareness, along with participants' analyses of their own habits of discourse as classroom teachers, prompted shifts in their projected image of the role of discourse in their future practices of teaching. Results suggested further that the undergraduate mathematics classroom (as opposed to the methods classroom) offers a powerful and unique forum in which pre-service secondary teachers can practice, articulate, and collectively reflect on reform-minded ways of teaching. [ABSTRACT FROM AUTHOR]

Our investigation explored a pedagogy for supervision through a case study of one prospective middle school mathematics teacher during her student teaching semester. Classroom observations by the university supervisor, teaching episode interviews between the supervisor and student teacher, and focused journal reflections by the student teacher were coordinated to challenge the student teacher's existing models of teaching. The emerging pedagogy of the teaching episodes, a central focus of this study, was characterized by (a) the use of open-ended questions that centered the student teacher in the process of sense making; (b) a shift away from the supervisor's direct, authoritative evaluations of the student teacher's practice; (c) a sustained focus throughout supervision derived from the student teacher's classroom experiences; and (d) an effort to maintain sensitivity to the student teacher's zone of proximal development. We found our approach to be coincident with the notion of instructional conversation (IC) advanced by Gallimore and Goldenberg (1992). The nature of the teaching episodes seemed to open the student teacher's zone of proximal development so that her practice of teaching could be mediated with the assistance of a more knowing other. [ABSTRACT FROM AUTHOR]