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Abstract Algebra and Secondary School Mathematics Connections as Discussed by Mathematicians and Mathematics Educators

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Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers

Part of the book series: Research in Mathematics Education ((RME))

Abstract

Most certification programs in the USA for secondary mathematics require coursework in abstract algebra. Yet several researchers have shown that most undergraduate students struggle to understand even the most fundamental concepts of this course. Perhaps more troubling is that the participants in these studies were unable to articulate hardly any connections between abstract algebra and secondary school mathematics upon completion of the course. In this chapter, I elaborate on the results of a study involving interviews with 13 mathematicians and mathematics educators that research and teach abstract algebra. The aim of these interviews was to understand how field experts describe connections between abstract algebra and secondary mathematics. In my findings, I discuss the differences in the participants’ descriptions of connections as reflected by their experiences with the secondary curriculum and their individual conceptualizations of abstract algebra.

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Notes

  1. 1.

    Fortran is a programming language that was developed by IBM in the 1950s. This language has been especially useful for numeric computation and engineering applications.

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Author information

Authors and Affiliations

  1. Department of Liberal Arts, Savannah College of Art and Design, Atlanta, GA, USA

    Ashley L. Suominen

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  1. Ashley L. Suominen
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Correspondence to Ashley L. Suominen .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Science and Technology, Teachers College, Columbia University, New York, NY, USA

    Nicholas H. Wasserman

Appendix 1: Mathematical Connections List After Interviews

Appendix 1: Mathematical Connections List After Interviews

Abstract algebra concept

Secondary school mathematics concept

Algebraic structures (group, ring, integral domain, field) and their properties

Function and domain; identity; inverse; number systems and known operators; solving linear equations

Binary operator

Arithmetic operators and number systems; domain; function; function composition; function transformations

Commutative ring theory (localization)

Fractions

Compass/geometric constructions

Geometry concepts including: Points, lines, circles, regular n-gons, angles, intersection, and trisection

Congruence

Solving linear equations

Cyclic group

Division algorithm; greatest common divisor; imaginary unit; rotations and periodicity

Direct product

Cartesian plane and ordered pairs; matrices for area and volume

Equivalence

Equal sign; inequality; similarity; solving equations

Equivalence classes

Decimal expansions; equivalent fractions; linear functions

Equivalence relation

Congruence; inequality; similarity; symmetry

Extension field/splitting field

Complex numbers; domain; roots of a polynomial

Fundamental theorem of algebra

Roots of a polynomial

Galois theory

Radicals; roots of polynomial equations

Groups and specific types of groups

Function composition; geometric transformations and symmetries

Homomorphism/isomorphism

Equality; function; infinity and finitely infinite; invariance; mapping

Ideal

Number systems; subset

Inverse

Multiplicative reciprocal; negative numbers

Irreducible polynomial

Factoring polynomials

Kernel

Nullspace of a matrix

Lagrange’s theorem

Euclidean algorithm; greatest common factor; least common multiple

Nilpotent

Geometric series and convergence

Permutation group; product of cycle decomposition

Function and function composition; permutation; symmetry

Polynomial ring

Operations with polynomials and polynomial long division; polynomial vocabulary (degree, coefficients, roots, etc.); power series

Quotient group/quotient field

Equivalent fractions; fractions and operations with fractions;

Quaternions

Complex numbers

Sign rule in a ring

Product of two negative numbers is positive

Subgroup

Subsets

Unary operators

Negation; trigonometric functions

Unit

Invertible matrices

Zero divisors

Geometric reflections and rotations; solve quadratic equations by factoring

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Suominen, A.L. (2018). Abstract Algebra and Secondary School Mathematics Connections as Discussed by Mathematicians and Mathematics Educators. In: Wasserman, N. (eds) Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-99214-3_8

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  • DOI: https://doi.org/10.1007/978-3-319-99214-3_8

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