Common Errors in Algebra

In the late 1970s, Hendrik Radatz issued a call for action models for teachers to integrate diagnostic teaching and findings from educational and social psychology, claiming that an “analysis of individual differences in the absence of a consideration of the content of mathematics instruction can seldom give the teacher practical help for individualizing instruction or providing therapy for difficulties in learning a specific task” (Radatz, 1979, p. 164). Societal and curricular differences make this connection difficult and, thus, instructors should consider other factors such as the teacher, the curriculum, the environment, and interactions. Given these multiple forces involved in the learning of mathematics, analyzing errors “in the learning of mathematics are the result of very complex processes. A sharp separation of the possible causes of a given error is often quite difficult because there is such a close interaction among causes” (Radatz, 1979, p. 164).

In order to simplify this set of complex causes, mathematical errors have been classified into five areas: language errors; difficulty with spatial information; deficient mastery of prerequisite skills, facts, and concepts; incorrect associations and rigidity of thinking; and incorrect application of rules and strategies (Radatz, 1979). Common mistakes and misconceptions in algebra can be rooted in the meaning of symbols (letters), the shift from numerical data or language representation to variables or parameters with functional rules or patterns, and the recognition and use of structure (Kieran, 1989).

Language Errors

Language errors can have multiple sources including gaps in knowledge for English Language Learners (ELL) and English as a Second Language (ESL) students, as well as gaps in academic language knowledge. This is particularly true for all students working on word problems. Students may lack reading comprehension skills that are required to interpret the information needed to solve a problem. Students may also have difficulty understanding academic language required to solve a problem. Prompts, word banks, and fill-in-the-blank questions may be used to help students solve open-ended questions. For example, a prompt and fill-in-the-blank could be used when asking a student to distinguish similarities and differences between polygons: “Squares and rectangles both have ___ sides but are different because _______________.” Word banks can be used when defining properties of angles. For example, the words “acute,” “obtuse,” “vertical,” “equal” and “not equal” can be included with other terms in a word bank to help students fill-in the following sentences:

An angle that is less than 90 degrees is ________. (acute)

An angle that is greater than 90 degrees is ________. (obtuse)

________ angles are formed when two lines intersect and have ______ measurements. (vertical, equal)

Spatial Information Errors

Difficulties in obtaining spatial information can also cause errors. A strong correlation was found between spatial ability and algebraic ability (Poon & Leung, 2009). When problems are represented using icons and visuals, mathematics assessments assume students can think spatially. For example, students may make errors on questions about Venn diagrams due to difficulties in understanding that lines represent boundaries and may ignore the lines. “Perceptual analysis and synthesis often make greater demands on the pupil than does the mathematical problem itself” (Radatz, 1979, p. 165). Without considering this lack of spatial ability as a possible cause of the incorrect responses, teachers may invest a lot of time and energy in presenting new materials that would not address the root cause of the problem.

Poor Prerequisite Skills/Flexibility/Procedural Errors

When a student does not possess the necessary prerequisite skills, facts, and concepts to solve a problem, he or she will not be able to solve the problem correctly. For example, if a student does not know how to combine like-terms, he or she may face difficulty solving multistep equations involving combining like-terms. Difficulties due to incorrect associations or rigidity of thinking are also common areas of error in mathematics. “Inadequate flexibility in decoding and encoding new information often means that experience with similar problems will lead to habitual rigidity of thinking” (Radatz, 1979, p. 167). Further, students make procedural

errors when they incorrectly apply mathematical rules and strategies. Rushed solutions and carelessness can also cause errors. Interviews revealed that errors in simplifying expressions were caused by carelessness and could be fixed with improved working habits (Poon & Leung, 2009). In addition, many students do not have linear problem solving skills. In fact, for many students, when reaching a point of difficulty in a problem, they go back and change their translation of the problem to avoid the difficulty (VanLehn, 1988, as cited by Sebrechts, Enright, Bennet, Martin, 1996).

The ability to use assessments in order to reduce common algebra errors may, in turn, increase understanding and build prerequisite skills that may lead to a stronger understanding of more advanced topics for both students and teachers alike. The use of open-ended quiz or test items allows teachers to see all of a student’s work rather than just an answer, such as in the case of multiple-choice questions. However, teachers who use quizzes or tests with multiple-choice questions provided in textbooks and other curricula could discuss in professional learning communities (PLC) what potential errors could have led a student to choose that multiple-choice answer, whether it be procedural, conceptual, spatial, language, or random. From there, teachers may be able to see a pattern that arises among classes and share ideas on how to approach reteaching. Teachers in a PLC setting could share and discuss common errors that have surfaced in their classrooms and what strategies have helped to address the errors.