Evaluation of teacher candidates' knowledge about vectors

Basic science concepts have been considered as prerequisite for the understanding and explanation of subsequent science topics related to these concepts and they also take the responsibility for making sense of the associated concepts (Mann and Treagust, 2010). The some subjects of physics and mathematics also overlap and enrich one another with complementary perspectives. One of them is also vectors. Vectors are essential component of the mathematic language of the physics, even at the introductory level (Knight, 1995). Students require a good grasp of basic vector concepts to succeed in a physics course (Sheets, 1998).

The concept of vector can be associated to almost any topic in physics, but the shortcomings in the process of learning can lead to serious problems (Aguirre, 1988; Aguirre and Rankin, 1989; Barniol and Zavala, 2015; Zavala and Barniol, 2013). Especially while adding, subtracting, and identifying unit vector, identifying the magnitude as well as the direction of the vector tends to contribute towards the difficulty of the problem (Barniol and Zavala, 2010; Barniol and Zavala, 2012; Barniol and Zavala, 2014; D'Angelo, 2010; Flores, Kanim and Kautz, 2004; Hawkins, Thompson and Wittmann, 2009; Knight, 1995; Nguyen and Meltzer, 2003; Schaffer and McDermott, 2005; Van Deventer and Wittmann, 2007).

Redish (2005) with university physics students in classes from algebra-based introductory physics indicates that the gap between what students think they are supposed to be doing and what their instructors expect them to do can cause severe problems. Despite the fact that vector illustration is the easiest way for scientists of representing some concepts, it can be confusing or even inextricable for students. The most of students seem incapable of reasoning with vectors as abstract elements of a linear space (Hestenes, 2002). It can be that the learning problems of students in mathematics are transferred to the learning environment in physics (Basson, 2002). Mestre (2001) indicates from his own experience and from research findings that transfer is not easy to accomplish. As new knowledge is learned, students should be assisted in considering multiple contexts and in linking that knowledge to previously learned knowledge. The ways in which the students perceive the world in their past experiences influence the learning of the concept. It is known from the literature that students have some preconceptions from their experiences and a lot of them do not match with the scientific conceptions, as named misconceptions, alternative conceptions or alternative framework (Halloun and Hestenes, 1985; McDermott, 1984). Misconceptions are difficult to change and may affect how learners process new information and data (Beydoğan, 1998; Gilbert, Osborne and Fensham, 1982; Helm and Novak, 1983; Watts and Pope, 1989). It can be observed from studies conducted related to physics education that students have many misconceptions while learning concepts about vectors (Aguirre and Erickson, 1984; Flores et al. 2004; Heckler and Scaife, 2015; Knight, 1995; Nguyen & Meltzer, 2003; Schaffer & McDermott, 2005). Hence these misconceptions should be diagnosed and teaching should be designed to take students' conceptions into account (Dekkers and Thijs, 1998; Duit and Treagust, 1995; Hewson and Hewson 1984; Osborne and Wittrock, 1983).

Some researches carried out studies about how vectors scalar and vector product is performed (Knight, 1995; Van Deventer, 2008; Van Deventer & Wittmann, 2007; Zavala and Barniol, 2010) and its geometric interpretation (Van Deventer, 2008; Zavala and Barniol, 2010). While Van Deventer prepared multiple-choice questions regarding scalar structure of scalar product's results, other scientists conducted researches on identifying challenges faced by students as they tried to differentiate torque from force as well as torque magnitude (Ortiz, Heron and Shaffer, 2005; Rimoldini and Singh, 2005; Van Deventer, 2008).

This study aims to determine the level of knowledge and misconceptions of mathematics teacher-candidates with open-ended questions about vector properties and operation. The questions we hope to answer with this investigation are: 1) whether it is written the correct representation of vector magnitude and vector; 2) whether it can be determined the direction of a third vector is perpendicular to the plane that contains two vectors multiplied vector product by using the right-hand side; 3) whether mathematics teacher-candidates correctly distinguish the cosine or sine of the angle in dot or vector product of two vectors; 4) whether mathematics teacher-candidates achieve to transfer form algebra classes while the questions with scalar and vector product are answered.