Abstract:

Numerous researchers in Mathematics education field agree that mathematics teachers need a high level of mathematical knowledge in order to be good teachers (Ball, Thames and Phelps, 2008). An issue of particular importance is the coherence of student knowledge (Taber, 2008, diSessa, 2006). Moreover, familiarity with different representations of mathematical concepts in different contexts and linkages between them is necessary for a better, deeper and more complete understanding of mathematics (Even, 1990, Goldin & Janvier 1998).

In this study we analyzed the level of understanding of mathematical concepts by students-teachers. We focused on whether mathematics student-teachers can transfer concept knowledge from one course to another, and whether they are able to recognize and use the same concept in different contexts.

The current study was conducted at a College of Education. All the participants were student-teachers in the educational program for secondary-school mathematics teachers. We concentrated on the Vector concept. The students met the concept during their first study year in the Linear Algebra course and then used it during the following year in the Physics course. The data consisted both of the students' assignment and their answers to a questioner about their understanding and utilization of the Vector concept. The collected data were analyzed both quantitatively and qualitatively.

We found from the students' answers to the questioner that all the students from both courses gave the correct definition of the concept. At the same time, the obtained results indicated that the students of the Physics course were familiar with additional representations with respect to those of the Linear Algebra course. In addition, the students of the Physics course reported more linkages between the Vector concept and other mathematical concepts than the students of the Linear Algebra course. At the same time, the results of the assignment given in both courses showed that the students did not acquire sufficient skills in using the Vector concept.

The results of the study indicate that the students have difficulties in identifying and using a mathematical concept when they meet it in different contexts. We believe that collaboration between the lecturers who teach the same concepts in different disciplines might improve learning and contribute to a deeper students' understanding of the material.

References:
[1] Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special?. Journal of teacher education, 59(5), 389-407.
[2] DiSessa, A. A. (2006). A History of Conceptual Change Research: Threads and Fault Lines. Cambridge University Press.
[3] Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational studies in mathematics, 21(6), 521-544.
[4] Goldin, G. A., & Janvier, C. (1998). Representations and the psychology of mathematics education. The Journal of Mathematical Behavior, 17(1), 1-4.
[5] Taber, K. S. (2008). Conceptual resources for learning science: Issues of transience and grain‐size in cognition and cognitive structure. International Journal of Science Education, 30(8), 1027-1053.

Keywords:

Subject matter knowledge, content knowledge, conceptual understanding.