232 | wileyonlinelibrary.com/journal/ssm School Science and Mathematics. 2020;120:232–243. © 2020 School Science and Mathematics Association 1 | INTRODUCTION Mathematical foundations for algebra success begin in the early grades. The National Mathematics Advisory Panel em- phasized the importance of understanding fraction concepts for success in algebra and real-world problem solving. The Panel stated that proficiency with fractions is “foundational for algebra and, at the present time, seems to be severely un- derdeveloped” (U.S. Department of Education, 2008, p. xvii). Wu (2001) argued that “the proper study of fractions provides a ramp that leads students gently from arithmetic to algebra. But when the approach to fractions is defective, that ramp collapses” (p. 1). Success in algebra is important because it can help students attain the skills needed to succeed in post- secondary education and society (Weissert, 2014). As noted by Usiskin (1995), while people can live without algebra, they will not appreciate what is occurring in their world and “will not be able to participate fully in our technological soci- ety” (p. 37). Not knowing algebra is like a person in a foreign country and, not knowing the local language, does not know what they are missing. The present study extends the work of Zientek, Younes, Nimon, Mittag, and Taylor (2013) who investigated the bias and accuracy of prospective teachers' confidence in perform- ing fraction operations and the extent to which fraction op- erations predicted prospective teachers' ability to solve three algebra equations in one variable. Using the same data set, the current study reports on the solution pathways chosen by respondents to solve an arithmetical equation that contains fractions and the errors that were made along those pathways. Received: 29 July 2019 | Revised: 29 January 2020 | Accepted: 18 February 2020 DOI: 10.1111/ssm.12402 RESEARCH PAPER – MATHEMATICS EDUCATION Solving equations with fractions: An analysis of prospective teachers’ solution pathways and errors Dustin L. Jones 1 | Linda Reichwein Zientek 1 | Valerie V. Sharon 2 | Mary B. Swarthout 1 1 Department of Mathematics & Statistics, Sam Houston State University, Huntsville, TX, USA 2 Independent consultant, Spring, TX, USA Correspondence Dustin L. Jones, Department of Mathematics & Statistics, Sam Houston State University, Box 2206, Huntsville, TX 77341, USA. Email: DLJones@shsu.edu Abstract We analyzed the solution pathways and errors found in the written responses of 469 prospective teachers solving an equation containing fractions. The majority (332, or 70%) used an algebraic method; 141 of the 332 (42%) were correct, and 22% of the algebraic methods were abandoned before a solution was obtained. We identified the steps in the written solutions, determined which solution pathways led to the correct solution, and identified common errors in the solution pathways of respond- ents who incorrectly solved the equation. Respondents initially attempted different methods. The most common method was solving by using fractions, but the majority of respondents who solved by using mixed numbers were able to correctly solve the problem. Common errors related to fraction arithmetic and the distributive property. Nearly all of the abandoned pathways contained no errors, but ended with a step that likely would precede an operation with fractions. Our findings suggest that the ability to solve an arithmetic equation with no fractions was necessary, but not sufficient, to solve an arithmetic equation involving fractions, and that the problem of solving equations with fractions was more closely tied to one's difficulties with rational num- ber arithmetic and less with one's understanding of algebra. KEYWORDS algebra, fractions, prospective teachers, solving equations