Generalizing is a critical aspect of mathematics learning, with researchers and policy documents highlighting generalizing as a core mathematical practice. It can also be challenging to foster in class settings, and teachers need access to better resources to teach generalizing, including an understanding of effective forms of instruction. This article proposes Classroom Supports for Generalizing (CSGs), investigating how multiple elements—such as tasks, teacher moves, student interactions, and representations—interact to meaningfully foster student generalizing. Drawing on class video data from a middle school teacher and two high school teachers, we present the CSG Framework, which identifies three categories of supports: Interactions for Generalizing, Structures for Generalizing, and Routines for Generalizing. © 2023 by The National Council of Teachers of Mathematics, Inc.

Over the past few decades, Piaget's forms of abstraction have proved productive for developing explanatory models of student and teacher knowledge, yet the broader applicability of his abstraction forms to mathematics education remains an open question. In this paper, we adopt the Piagetian forms of abstraction to accomplish two interrelated goals. Firstly, we analyze instructional tasks to develop hypothetical accounts of the abstractions that might occur during students' engagement with them. Secondly, we draw on middle- and secondary-grades classroom data to discuss the abstractions that occurred during the implementation of those instructional tasks. Because this paper represents an initial attempt at extending the applicability of Piagetian forms of abstraction, we close with potential implications of such use and possible avenues for future research. Most notably, we highlight the complexities involved in supporting abstraction through curriculum and instruction.

Combinatorial proofs of binomial identities involve establishing an identity by arguing that each side enumerates a certain set of outcomes. In this paper, we share results from interviews with experienced provers (mathematicians and upper-division undergraduate mathematics students) and examine one particular aspect of combinatorial proof, namely the kinds of contexts that experienced provers used to establish combinatorial proofs of binomial identities. Our findings show that overall, our participants used a variety of contexts in their work; we also demonstrate ways in which previous experiences influenced the contexts they chose, and we offer some instances in which features of a context supported their combinatorial proof production. We offer some theoretical implications of our work, and we conclude with a discussion of limitations and avenues for future work.

Teachers' knowledge of students' mathematical thinking is a growing area of research in mathematics education. The literature has reported plentiful evidence of the interplays between teachers' mathematical knowledge and their knowledge of students' mathematical thinking. The present study builds on this body of work to explain how such interplays occur. Over a semester, I worked in partnership with a secondary school mathematics teacher on cycles of task design, interactions with a student, and in-depth reflection on the student's thinking about linear programming. Adopting and extending Piagetian constructs of assimilation and accommodation, I describe several key mental processes that illuminate the teacher's learning of mathematics and of the student's mathematical thinking. I conclude with a discussion of the study's empirical and theoretical contributions to understanding teachers' mathematical learning in relation to student thinking.

This paper reports on the results of a four-day teaching experiment that supported two algebra teachers to develop a combinatorial meaning for algebraic structure. The purpose of the teaching episodes was to support the teachers (a) to establish a combinatorial understanding for algebraic structure (Tillema & Burch, 2022) by generalizing the cubic identity, (a + b)3 = a3 + 3a2b + 3ab2 + b3, as a symbolization of quantitative and combinatorial relationships out of a contex-tualized problem (Tillema & Gatza, 2016) and (b) to develop a combinatorial meaning as a mobilization of their understanding through a series of algebraic tasks (cf. Thompson et al., 2014). The findings from this study contribute to research literature on teachers' mathematical meanings within secondary algebra by investigating how teachers' combinatorial meanings developed and how differences in their combinatorial meanings impacted their algebraic reasoning. The findings demonstrate a combinatorial pathway for supporting the development of expanding and factoring as reversible polynomial operations (cf. Sangwin & Jones, 2017).

This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers' combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of polynomials. The data in this paper examines the learning that occurred as one teacher transitioned from making a generalization from a sequence of contextualized combinatorics problems to applying her combinatorial reasoning to symbolic problems common in algebra curricula. The findings from the study include the identification of three planes of learning that can be used to differentiate among ways that combinatorial reasoning can be used to engage in binomial expansion. The highest plane involved constructing a combinatorial scheme for binomial expansion, a scheme that supported the teacher to produce the equivalence, (x + a)(x + b)(x + c) = x(3) + (a + b + c)x(2) + (ab + ac + bc)x + abc), and to see important algebraic structure in it. The contributions of the study include: (a) expanding earlier arguments about the ways that combinatorics can be integrated into goals of extant curricula (e.g., Maher et al. in Combinatorics and reasoning: Representing, justifying and building isomorphisms. Springer, 2011); and (b) proposing how reflecting abstraction can be used to study the transition between generalizations learners make from contextualized problem situations to operating with and on generalizations expressed with conventional algebraic symbols. This second contribution is an under-researched area in the algebra literature (Dorfler in ZDM - Int J Math Educ 40(1):143-160, 2008), and points to an important role that combinatorial reasoning can play in algebra learning.