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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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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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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