Errors and learning in the study of the domain of a function of two variables

Abstract

This work aims to report and reflect on how a mathematical error made by a student when determining the domain of a function of two variables can be explored as a learning tool. To this end, we considered explorations carried out in a class of Multivariable Calculus, part of the Mathematics Teaching degree at the State University of Paraná – Paranavaí Campus, in which the objective was to discuss the mistakes made by students in a written exam. We understand that mathematical errors committed by students can be explored in the classroom as sources of new discoveries. In this sense, according to Borasi (1996) and Cury (2008), it is possible for an error to become a problem so that students, guided by the teacher, work in search of solutions that promote learning. With this intent, the instructor asked one of the students (the first author of this paper) to present on the board his solution to a question that asked for the domain of the two-variable function defined by . The answer presented was the set  ,in which the student explained that it was necessary to verify that the radicand should be greater than or equal to zero, and to do so, he considered that since 16 is fixed and positive, the remaining expression  should result in 16 or less so that the square root value would be a real number. From this, the instructor invited the students to investigate whether this strategy was correct, as it seemingly made sense. First, one of the students went to the board and justified: if we let , then the function can be written as , and in this case, to determine the domain, we must consider that , that is,   or . Thus, he concluded that the strategy presented by the student was correct. However, the instructor questioned whether the function   could indeed be written in that way, and on the board, together with the students, she invited them to analyze by writing . From these equalities, they observed a sign difference in the initial expression that defined the function   , leading them to identify that , that is, parentheses cannot be inserted in the algebraic expression without performing the proper sign changes. Thus, the student, along with the class, understood the error while also valuing the strategy used, which could have led to the correct answer. For that, it was necessary to consider , resulting in the domain being the set . In this brief report, we highlight the importance of valuing students’ productions in the classroom through interventions such as this one, especially in Calculus classes, where it is common for students to feel incapable of justifying their answers due to the abstract nature of the concepts addressed in the course. Instead of simply correcting the question on the board, the instructor chose to explore the student’s response, validating his reasoning and guiding him to understand where and why he was wrong. In this approach, various types of knowledge were mobilized: the concept of the domain of multivariable functions, algebraic manipulation, logical reasoning, among others.