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An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - Implications for the Modern Teacher

Nicole M. Wessman-Enzinger (Illinois State University)
Implications for the Modern Teacher

The teaching and learning of subtraction is just as important today as it was in the past. Innovations in technology and mathematics curriculum have certainly occurred since the 1700s and 1800s, but the need for the teaching and learning of subtraction has not changed. Today, in many classrooms, subtraction is often taught through student-invented algorithms. Looking to the past may give teachers insight into invented algorithms or other algorithms students may use. Additionally, many teachers who do not encourage students to invent strategies teach only the “standard subtraction algorithm” presented in nearly every textbook across the United States, the decomposition algorithm. This research and analysis provides the modern teacher with an opportunity to reflect on the algorithms being taught in his or her classroom and allows the teacher to begin to think about why decomposition became the dominant algorithm in the United States. Teachers can ask their students to reflect on whether they agree with this historical turn of events. Incorporating the history of subtraction algorithms into modern elementary school mathematics invites robust mathematical discussion of subtraction and also of how, for many mathematical operations, there isn’t just one algorithm, but rather many algorithms from which to choose.

Exploring the history of subtraction in past school mathematics may provide us with insight into students’ mathematical struggles as they attempt to conceptualize not only subtraction, but also negative numbers and other notoriously challenging mathematical concepts. As educators and researchers, we need to devote more attention to issues in mathematics education such as the development of specific algorithms in elementary mathematics.


The images of pages of cyphering books in this article have been reproduced, with permission, from cyphering books belonging to Nerida F. Ellerton and M. A. (Ken) Clements, who, as of January 2014, own a collection of 350 North American cyphering books, dating from 1701 to 1861. This is the largest collection, public or private, of North American cyphering books. For details relating to any of these cyphering books, consult Rewriting the History of School Mathematics in North America 1607–1861, by N. F. Ellerton and M. A. Clements (Springer, 2012). Ellerton and Clements have indicated that photographs of excerpts from the cyphering books in their collection can be reproduced in scholarly papers or presentations or in curriculum materials, provided it is indicated that the cyphering books from which the photographs were taken are in the Ellerton-Clements collection and provided reference is made to the above-mentioned book by Ellerton and Clements (2012).

Nerida F. Ellerton and M. A. (Ken) Clements also own a collection of arithmetic textbooks that I have consulted and from which I have used images. I would also like to thank Dr. Ellerton and Dr. Clements not only for the use of their arithmetic and cyphering books, but also for their feedback on this project. They have a passion for the history of mathematics education and have been an inspiration to me.

About the Author

Nicole Wessman-Enzinger is a Ph.D. candidate in mathematics education within the Department of Mathematics at Illinois State University. She is working on her dissertation about developing conceptual models of student thinking about negative integers. Her dissertation is focused on fifth graders’ thinking about negative integers and operations with them. She is broadly interested in the teaching and learning of number, from both historical and psychological perspectives.

Nicole M. Wessman-Enzinger (Illinois State University), "An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - Implications for the Modern Teacher," Loci (January 2014)