Understanding School Mathematics in Terms of Linear Measure and Discrete Real Additive Groups

  • Hyman BassEmail author
Part of the Research in Mathematics Education book series (RME)


In this chapter, I describe a capstone course developed for secondary teachers. Its content represents my mathematical perspective on school mathematics, aimed at conceptual coherence. It centrally features the real number line, with its geometric and arithmetic structures. It starts with linear measurement, expressed through division with remainder (DwR), which leads directly to place value and modular congruence. Abstract algebra enters through the study of discrete additive groups of real numbers, from which multiplicative arithmetic and commensurability (irrationality) naturally emerge; DwR is the foundation of this development. Brief treatments of polynomial algebra and combinatorics then culminate in Discrete Calculus, the natural generalization of the “pattern generalization” activities in school mathematics. Finally, I present and discuss some problem-solving designs, which are intended to cultivate important mathematical practices in the course.


Abstract algebra Group theory Division with remainder Commensurability Discrete calculus 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and School of EducationUniversity of MichiganAnn ArborUSA

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