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Examples play a critical role in mathematical practice, in general, and in proving-related activities (e.g., developing conjectures, exploring conjectures, justifying conjectures), in particular. Yet, despite the critical role examples play in proving-related activity, we contend that students typically receive very little, if any, explicit instruction on how to become more deliberate and strategic in their use of examples. The goal of the study reported here was to explore teachers’ beliefs about the role examples play in proving-related activities, and the instructional practices they implement to foster the development of students’ abilities to strategically think about and productively use examples. Fifty-four middle school mathematics teachers responded to a series of on-line survey questions that focused on the role and use of examples during proving-related classroom activities. We found that many teachers have limited views of what it means to use examples strategically during proving-related activities, and that they tended not to provide explicit instruction designed to help students learn to strategically think about and productively use examples during their engagement in proving-related activities. The findings suggest the need for both professional development and curricular resources to support teacher efforts to help their students learn to strategically think about and productively use examples during proving-related activities.

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.

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.

Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K-1 students' thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K-1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students' practice of argumentation from the start of formal schooling.