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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[subscript 3] + (a + b + c)x[subscript 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 (Dörfler 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.

As algebra teachers, the authors explore the following question in this article: "How can algebra 1, algebra 2, and precalculus teachers support students to develop algebraic reasoning and understanding of structure that can serve them in day-to-day algebraic computation?" The article shows how the algebraic identity "(a + b)[superscript 3] = a[superscript 3] + 3a[superscript 2]b + 3ab[superscript 2] + b[superscript 3]" can be generalized from quantitative reasoning in the context of combinatorics problems like the Card Problem (Tillema and Burch 2020). The article then illustrates how the combinatorial reasoning students develop in the Card Problem can support them to see combinatorial structure across algebraic computations. Throughout the article, the authors share work from students and teachers with whom the approach was developed.

Two-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11-12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented a balanced mix of students operating with each of three different multiplicative concepts that have been identified in prior research. This paper reports on the 11 students who were using the first or second multiplicative concept. Students operating with different multiplicative concepts all experienced similar perturbing elements in their solution of APs relative to their solution of CPPs, but they operated differently to resolve these perturbing elements. These differences are identified and their significance discussed in relation to other research findings on students' combinatorial and multiplicative reasoning.

Mr. Carter is about to start a two-day lesson on subtraction of integers with his sixth-grade prealgebra students. He plans to use contextualized problems that will allow his students to develop an interpretation of subtraction that involves the idea of "difference." This article outlines one way to teach students develop number line representations and equations for differences. In the lesson outline, sample work from sixth-grade urban middle school students is used to illustrate key ideas. (Contains 5 figures.)

This article describes how to multiply binomials so that middle school students produce the equivalence (a + b)[superscript 2] = a[superscript 2] + 2ab + b[superscript 2] after reasoning about and representing problems. First, a two-dimensional array representation is used so that students can become comfortable with multiplication embedded in their figures. Later, an area model is used. Such area models provide multiple representations of squaring a binomial; in so doing, these diagrams give students multiple ways to understand the concept. (Contains 7 figures.)

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students' construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students' whole number multiplicative concepts were found to be critical constructive resources for students' fraction composition schemes. Specifically, the "interiorization of two levels of units," a particular multiplicative concept, was found to be necessary for the construction of a "unit fraction composition scheme," while the "interiorization of three levels of units" was necessary for the construction of a "general fraction composition scheme." These findings contribute to previous research on students' construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered. (Contains 5 figures.)

Two-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11–12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented a balanced mix of students operating with each of three different multiplicative concepts that have been identified in prior research. This paper reports on the 11 students who were using the first or second multiplicative concept. Students operating with different multiplicative concepts all experienced similar perturbing elements in their solution of APs relative to their solution of CPPs, but they operated differently to resolve these perturbing elements. These differences are identified and their significance discussed in relation to other research findings on students' combinatorial and multiplicative reasoning. [ABSTRACT FROM AUTHOR]

A response from the author of the article "Interaction: A Core Hypothesis of Radical Constructivist Epistemology "in the 2014 issue is presented.

Context • This paper outlines how radical constructivist theory has led to a particular methodological technique, developing second-order models of student thinking, that has helped mathematics educators to be more effective teachers of their students. Problem • The paper addresses the problem of how radical constructivist theory has been used to explain and engender more viable adaptations to the complexities of teaching and learning. Method • The paper presents empirical data from teaching experiments that illustrate the process of second-order model building. Results • The result of the paper is an illustration of how second-order models are developed and how this process, as it progresses, supports teachers to be more effective. Implications • This paper has the implication that radical constructivism has the potential to impact practice. [ABSTRACT FROM AUTHOR]

The author discusses aspects of the radical constructivist approach in problem solving. The author states the identification of student's activity in problem solving which involves recognition, re-presentation and reflective abstraction. Also investigated are the methods of problem solving and its mathematical concept.

A quiz concerning the application of decimals and scientific notation through the use of the specks of dust is presented.

A quiz concerning the chance of having three girls in the family with the use of mathematical solutions is presented.