Chapter 5 Knowing Ratio and Proportion for Teaching James J. Madden Abstract Ratio and proportion have been part of school mathematics since the earliest manifestations of anything like school math in the Middle Ages. In this paper, I compile and comment on statements from primary sources of the last 2300 years to exhibit ideas that appear to have influenced the treatment of these topics in schoolbooks today. Historical sources clarify many points about the contemporary curriculum, supporting the contention that an understanding of history of ideas concerning ratio and proportion is an important component of knowledge of mathematics for teaching. 5.1 Introduction Some of the occasionally puzzling things that we read in school mathematics textbooks, or find in discussions about standards or in commentaries about school math, can best be explained by reference to the long, complicated history of the curriculum. When we read that the quantities used in forming a ratio must be of the same kind, we are catching an echo of Definition 3 of Book V of Euclid’s Elements: “A ratio is a sort of relation in respect of size between two magnitudes of the same kind.” Similarly, the statement that a proportion is an equality between two ratios refers back to Definition 6 of the same book: “Let magnitudes which have the same ratio be called proportional.” Euclid, and two millennia of scholarly writings on Euclid, have influenced the way we speak about proportion today. Another powerful influence, largely independent of the classical tradition, de- veloped with the emergence of mercantilism in Europe in the Middle Ages. The rule of three is a method for solving the proportions that arise in trade, such as deducing the cost of one amount of a commodity from the cost of another amount, assuming that the conditions of the sale remain the same. The rule was known in antiquity and was described in texts such as al-Khw¯ arizm¯ ı’s Algebra (c. 820 CE) and Fibonacci’s Liber Abaci (1202 CE). It was always closely associated with numerical computations and the use of units. From the thirteenth to the sixteenth century, the J.J. Madden () Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA e-mail: mmmad@lsu.edu © Springer International Publishing AG 2018 Y. Li et al. (eds.), Mathematics Matters in Education, Advances in STEM Education, DOI 10.1007/978-3-319-61434-2_5 93