Abstract:

According to Dorier (2002), "(…), linear algebra represents, with calculus, the two main mathematical subjects taught in science universities. However, this teaching has always been difficult." The main criticisms pointed out by the students of the Linear Algebra course concern the excessive use of formalism, the excessive number of new definitions and results involved in learning a concept, and the lack of connection with the mathematical knowledge already acquired (Dorier et al., 2000). According to Costa and Catarino (2007), these aspects constitute real obstacles to Linear Algebra learning. Recently Lohgheswary et al. (2018) wrote that "(…) the underachievement of engineering students in engineering mathematics is due to poor prior knowledge and anxiety towards mathematics." To Barros et al. study (2021), "as a result, the lack of prior knowledge can compromise the chances of success in the course and lead students to abandon it." Barros et al. study (2021) also emphasized that "(…) it is up to teachers to help students to overcome their difficulties, developing and implementing strategies that promote reflection and debate on concepts, representations, and procedures related to them."

In this context, to identify the errors and difficulties and understand the students' reasoning in solving questions considered preparatory to the learning of linear algebra and analytical geometry, a questionnaire was applied to students who attended the course of Linear Algebra from Communication and Multimedia degree of a university from northern Portugal.

Algebraic manipulations are not the main aspect of teaching algebra but contribute to the development of specific mathematics learning, that is, algebraic thinking. Using Kaput's (1998) framework, we consider four elements of algebraic thinking: Generalized arithmetic, functional thinking, modeling languages, and abstract algebra and algebraic proof. In this study, we will also use the four levels of algebraization suggested by Godino (2015): Level 0, absence of algebraic reasoning; level, which identifies general entities but uses a common language; level 2, which uses alphanumeric language but cannot perform operations; and level 3, consolidated algebraic level.
To establish the errors and difficulties in algebraic thinking and the Godino levels of algebraization in a questionnaire given to the first-year students in a linear algebra course from Multimedia Communication Degree at a university in the north of Portugal. The questionnaire consisted of two parts. The questionnaire's first part comprises six questions to identify the student's profile and their affinity with Mathematics. The second part, also consisting of six questions aimed at exploring the aspects of algebraic thinking, was elaborated from a questionnaire validated by Pitta-Pantazi et al. (2020).

The qualitative research has a descriptive goal and was carried out in a class of 54 students through the application of a questionnaire during a linear algebra class that lasted 90 minutes.

This work aimed to identify which of the four elements of algebraic thinking the students had achieved. Finally, the analysis of the results indicated the presence of some aspects of algebraic thinking in some students, but not in a generalized way. That level 4, established as ideal for a good performance in learning the subject of linear algebra, was rarely observed in students.

Keywords:

Algebraic thinking, algebraization levels, questionnaire.