Misconceptions and resulting errors displayed by in service teachers in the learning of linear independence

Abstract

The aim of this paper is to identify errors and misconceptions that student demonstrated when learning linear independence and linear dependence concepts. A case study is presented involving 73 in-service mathematics teachers at a university in Zimbabwe who were studying for a Bachelor of Science Education Honors Degree in mathematics. Data was generated from a content analysis of the written responses of the participants to two items from a structured activity sheet. Follow up interviews with five participants were used to gain a better understanding of their misconceptions. The study found that the participants had different kinds of misconceptions leading to errors which could be described as procedural, conceptual, and foundational respectively and the distribution of the errors differed across the two problems. For question 1 which was set within the vector space M2×2, students found it harder to move past the first few two steps of formulating the general vector equation and doing scalar multiplication; those who passed those two steps were mostly able to get to a correct solution. For question 2 which was set within ℝ 3 most students went past the first two steps formulating the general vector equation and converting that to an augmented matrix but then made many foundational errors, most of which were related to misinterpretations of the solutions to the system of equations.

Undergraduate students’ conceptualization of elementary row operations in solving systems of linear equations

“…The results revealed many types of errors made by students as they learn linear algebra (Mutambara & Bansilal, 2022). Some errors were foundational in nature, meaning manipulation and baseline knowledge errors.…”

“…3 / 15 2019). Hence, literature reports that students grapple to come to terms with the key concept of linear algebra courses (Harel, 2017;Kazunga & Bansilal, 2017;Mutambara & Bansilal, 2018, 2022Stewart & Thomas, 2010;Tai, 2020). Tai (2020) posits that part of the difficulties may be attributed to the transition from elementary to advanced mathematics.…”

“…In the same vein, students' lack of foundational knowledge might trifle full understanding of a given concept, similar to lack of pre-requisite concepts. On a closer inspection of the misconceptions and errors committed by 73 in-service teachers registered for mathematics teaching degree in the concept of linear independence, Mutambara and Bansilal (2022) discovered that students' errors were mostly foundational. The students erred in processes of row reductions and interpretation of solutions to SLEs.…”

Undergraduate Students’ Cognitive Obstacles in the Learning Power Series Concepts Using APOS

“…This will help to address students' cognitive obstacles. The knowledge of these obstacles is important because it can provide guidance to the instructor in structuring the learning materials so that they can take these unexpected obstacles into account and help improve students' learning (Mutambara & Bansilal, 2022;Oktaç, Trigueros & Romo, 2019). The APOS-ACE teaching cycles in mathematics are useful in order to refine, detect and explain students' ways of thinking, their difficulties and connections among different constructions they can or cannot make (Trigueros & Oktaç, 2019).…”

Misconceptions of mathematical concepts vis-à-vis how they pose as barriers to developing students’ conceptual understanding

“…Developing students' conceptual understanding of mathematical concepts is paramount in attaining mathematical proficiency [3,5,4]. However, assumptions, preconceptions and misconceptions of mathematical theories and concepts obstruct students' understanding [24]. These develop into errors in students' work, leading to students experiencing severe learning difficulties; this is displayed in their written responses in examinations making them under-achieve in the subject.…”