Textbook accounts of the rules of indices with rational exponents

Abstract

The rules of indices, e.g. a n b n = (ab) n , are a particularly important part of elementary algebra. This paper reports results from a textbook analysis which examined how the shift from integer to rational exponents in the rules of indices is discussed in school textbooks. The analysis also considered related issues, such as notation and the introduction of complex numbers. A selection of popular textbooks from the period 1800-2000 was examined and the nature of the justification given for the extension of meaning to rational indices considered. In both the definition and computational rules, when extending the domain of n in a n to rational numbers the (potential) contraction of the domain of a to positive numbers was often quietly ignored. A wide variety of approaches are used in choosing what is to be a definition, what is to follow, and how this is justified. The difference between computational rules for practical algebraic manipulation and a formal definition was often blurred.

Opportunities to learn foundational number sense in three Swedish year one textbooks: implications for the importation of overseas-authored materials

“…Some years later, also from the specific perspective of school mathematics, Schutter and Spreckelmeyer (1959) compared US and European arithmetic textbooks, while Williams and Shuff (1963) compared textbooks from modern and traditional US series and Pinker (1981) a range of US textbooks with respect to their presentations of sets, a core topic of the new mathematics current at that time. In short, comparing textbooks, both with within and across educational systems, is a long-standing research tradition, a tradition well represented in the recent pages of IJMEST (Bütüner, 2018;Glasnovic Gracin, 2018;Kajander & Lovric, 2017;Sangwin, 2019). More recently, however, many studies have been motivated by a desire to understand how educational systems more successful than those of the writers, typically the US, present mathematical ideas (Ding, 2016;Li, Chen, & An, 2009;Yang, Reys, & Wu, 2010).…”