The double continuity of algebra

Al Cuoco

William McCallum

pp. 49-63

in: Gabriele Kaiser, Helen Forgasz, Mellony Graven, Alain Kuzniak, Elaine Simmt, Binyan Xu (eds), Invited lectures from the 13th international congress on mathematical education, Berlin, Springer, 2018

Abstract

We consider Klein's double discontinuity between high school and university mathematics in relation to algebra as it is studied in both settings. We give examples of two kinds of continuities that might mend the break: (1) examples of how undergraduate courses in algebra and number theory can provide useful tools for prospective teachers in their professional work, as they design and sequence mathematical tasks, and (2) examples of how questions that arise in secondary pre-college mathematics can be extended and analyzed with methods from algebra and algebraic geometry, using both a careful analysis of algebraic calculations and the application of algebraic methods to geometric problems. We discuss useful sensibilities, for high school teachers and university faculty, that are suggested by these examples. We conclude with some recommendations about the content and structure of abstract algebra courses in university.