Nicholas H. Wasserman

Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction

This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics—and their progression across elementary, middle, and secondary mathematics— where teaching may be transformed by teachers’ knowledge of abstract algebra are developed. In each of the four content areas (arithmetic properties, inverses, structure of sets, and solving equations), descriptions and examples of the transformational influence on teaching these topics are used to depict and support ways that study of more advanced mathematics can influence teachers’ practice. Implications for the mathematical preparation and professional development of teachers are considered.RésuméCet article se penche sur l’influence potentielle de certains aspects de l’algèbre abstraite sur l’enseignement de l’algèbre scolaire (et l’algèbre élémentaire). En utilisant les normes nationales d’analyse, on développe quatre domaines primaires communs dans les mathématiques scolaires, ainsi que leur évolution au travers des classes de mathématiques élémentaires, intermédiaires et secondaires, lorsque l’enseignement peut être modifié par les connaissances de l’enseignant en algèbre abstraite. Dans chacun des quatre domaines (propriétés arithmétiques, inverses, structure des ensembles et résolution d’équations), des descriptions et des exemples de l’influence transformationnelle sur l’enseignement de ces sujets sont utilisés pour décrire et soutenir l’idée que l’étude de mathématiques plus avancées peut influencer la pratique de l’enseignant. Les conséquences pour la préparation mathématique et le perfectionnement professionnel des enseignants sont examinées.

Introducing Algebraic Structures through Solving Equations: Vertical Content Knowledge for K-12 Mathematics Teachers

Abstract 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.

Making Real Analysis Relevant to Secondary Teachers: Building Up from and Stepping Down to Practice

Abstract 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.

Making Sense of Abstract Algebra: Exploring Secondary Teachers’ Understandings of Inverse Functions in Relation to Its Group Structure

ABSTRACT 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.

Quadrilaterals and Bretschneider's Formula

We talk about a formula deduced by Bretschneider in the mid-nineteenth century that not only allows the computation of areas of planar quadrilaterals, regular and irregular, but also provides a more thorough understanding of the geometry of such figures. We present one version of Bretschneiders formula and talk about its scope of application and what it can say about the possible shapes of quadrilaterals.

Shoelace formula: Connecting the area of a polygon and the vector cross product

Understanding how one representation connects to another and how the essential ideas in that relationship are generalized can result in a mathematical theorem or a formula. In this article, we demonstrate this process by connecting a vector cross product in algebraic form to a geometric representation and applying a key mathematical idea from the relationship to prove the Shoelace theorem.

A new look at an old triangle counting problem

The correct answer to the problem in figure 1 is 15. There are nine triangles congruent to ABE, three triangles congruent to ACH, two triangles congruent to BGH, plus ADJ, for a total of 15 equilateral triangles. Co-author Sutcliffe recently encountered this problem on a MATHCOUNTS® poster titled “What number do the following have in common” in her high school classroom while student teaching, and the problem generated a great deal of excitement among her students. Of fifty-four students in three different classes, none was able to identify BGH and CEI as equilateral, but they all tried very hard to find them. The second source of excitement came from the handful of students who were eager to know how the solution generalizes to larger arrays.

Solving and graphing quadratics with symmetry and transformations

One of the central components of high school algebra is the study of quadratic functions and equations. The Common Core State Standards (CCSSI 2010) for Mathematics states that students should learn to solve quadratic equations through a variety of methods (CCSSM A-REI.4b) and use the information learned from those methods to sketch the graphs of quadratic (and other polynomial) functions (CCSSM A-APR.3)

Statistics as unbiased estimators: exploring the teaching of standard deviation

ABSTRACT 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 by n-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.

Reflecting Upon Different Perspectives on Specialized Advanced Mathematical Knowledge for Teaching

Teachers’ knowledge assumes a major role in practice and in the students learning and achievement. In particular, the construct of horizon knowledge or, what can be termed specialized advanced mathematical knowledge for teaching (in order to capture the overall perspectives we are dealing with within this proposal) has been the focus of attention from some researchers with different foci of attack (e.g., Carrillo, Climent, Contreras, & Muñoz-Catalán, 2013; Jakobsen, Thames, Ribeiro, & Delaney, 2012; Wasserman & Stockton, 2013; Zazkis & Mamolo, 2011). In that sense, and aiming to deepen our understanding of such a construct, the aim of this working group is to discuss and reflect upon, different theoretical perspectives, methodological approaches and analytic methods used when focusing on such specialized advanced mathematical knowledge for teaching. In particular, we consider the activities of analysing and conceptualizing situations where access and development of such teachers’ knowledge is of primary importance.