Extraneous Solutions and Students’ Opportunities for Reasoning with Equations

When solving radical or rational equations, students make reasonable decisions about what to use or choose (i.e., true solutions) and what not to (i.e., extraneous solutions). This study investigates mathematics textbook problems regarding extraneous solutions that are related to students’ reasoning competencies. This study examines the following research questions: (1) How do mathematics textbook problems regarding extraneous solutions provide opportunities to help students develop their skills in reasoning with equations? (2) What is the relationship between how textbook problems regarding extraneous solutions are worded and the opportunities for students’ development of skills in reasoning with equations? The textbook problems are collected from high school mathematics textbooks, Big Ideas Math Algebra 1 and Algebra 2. By using Anderson and Krathwohl’s (2001) framework, sentences of the textbook problems regarding extraneous solutions are classified into different codes according to how the sentences are worded. In doing so, the framework helps understand what opportunities the textbook problems provide students to improve their reasoning skills. As a result, out of 228 sentences analyzed, the most frequent code is Applying Procedural Knowledge, which constitutes 47.81% of the total sentences. This indicates that the textbook problems disproportionately provide students with opportunities for reasoning with equations because one out of 24 codes accounts for almost half, and some codes only appear once or twice. The findings show that the textbook problems regarding extraneous solutions focus mainly on applying conceptual knowledge and that the textbooks provide insufficient opportunities to develop students’ skills in reasoning with equations.