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Combination problems are a cornerstone of combinatorics courses, but little research has been done examining the ways that students perceive and differentiate among different combination problems. In this article, we investigate how mathematics education students (n = 18) in a discrete mathematics course view two categorically different combination problems (Category I and II combination problems). In particular, we look at how participants conceptualized each problem's sets of outcomes, counting processes, and formulas, while also exploring the means by which students justified their relationships. Review of the data collected suggests that students tend to be less consistent and have more trouble utilizing and justifying combinations with a collection of ordered indistinguishable objects (Category II) than they do with a collection of unordered distinguishable objects (Category I). Based on these findings, we provide recommendations for the teaching and learning of combinations in combinatorics education. [ABSTRACT FROM AUTHOR]

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator – not just a descriptive statistic for the variation (spread) in data – is related to teachers’ instruction regarding standard deviation, particularly around the issue of division byn-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role in altering teachers’ planned instructional approaches in terms of student activity and cognitive demand in their instruction. [ABSTRACT FROM AUTHOR]

This article draws on semi-structured, task-based interviews to explore secondary teachers’ (N = 7) understandings of inverse functions in relation to abstract algebra. In particular, a concept map task is used to understand the degree to which participants, having recently taken an abstract algebra course, situated inverse functions within its group structure (i.e., the set of invertible functions under composition). In addition, their particular conceptions of functions and function composition throughout the interviews were then also considered as a means to explore further their responses during the interviews. Findings indicate that only two participants showed evidence of the desirable mathematically powerful understandings from abstract algebra in relation to inverse functions, and further analysis suggests a variety of challenges in terms of developing meaningful connections, which were more related to conceptions about secondary content than to the abstract algebra content. Implications for the mathematical preparation of secondary teachers are discussed. [ABSTRACT FROM AUTHOR]

Future teachers often claim that advanced undergraduate courses, even those that attempt to connect to school mathematics, are not useful for their teaching. This paper proposes a new way of designing advanced undergraduate content courses for secondary teachers. The model involves beginning with an analysis of the curriculum and practices of school mathematics and its teaching, and then using those to build up to the advanced mathematics – in this case, real analysis. After developing definitions, examples, theorems, and proofs, the model then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. To illustrate the model, we provide and discuss two example tasks. [ABSTRACT FROM AUTHOR]

Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic thinking is discussed by way of an example of an abstract group. Results of this approach in a mathematics for teachers course are presented, with K-12 teachers (n=12) having increased their content knowledge both about arithmetic properties and abstract algebra, as well as having their teaching beliefs and intended practices influenced. Implications for classroom teaching about a vertical slice of the K-12 curriculum related to algebraic structure are examined. [ABSTRACT FROM PUBLISHER]