Session Information
ERG SES B 11, Mathematics Education
Parallel Paper Session
Contribution
The analysis of the historical development of algebra indicates that the construction of the symbolic language was very slow and difficult because mathematics arose with the necessities of civilizations (Gallardo, 2000; Sfard, 1995). For instance, numerical cuneiform signs belonging to Mesopotamia show their commercial counting activities. In spite of slow development of algebraic language, variables are indispensable concept in algebra because scientists and mathematicians developed a lot of theory and abstraction by means of the concept of variable (Katz, 2007).
The concept of variable which leads to transition process of arithmetic to algebra is a downturn for students because it requires formal and abstract thinking. In contrast to the importance of variables and their heavily usages in algebra, many of studies present that students are generally confused while passing from arithmetic to algebra in their learning processes due to the concept of variable having different meanings in different context (MacGregor & Stacey, 1997; Schoenfeld & Arcavi, 1988). At this point, researchers have wondered students’ misconceptions and difficulties related to the concept of variable in any algebraic context (Davis, 1988; Kuchemann, 1978;1981; Philipp, 1992, Schoenfeld & Arcavi, 1988; Wagner, 1999). Although there are a lot of studies expressing students’ difficulties, little attention has been given to students’ reasoning processes. At this circumstance, Sfard’s theory (1991) sheds some light on understanding of students’ algebraic reasoning by proposing an inherent process-object duality in most mathematical concepts. According to Sfard (1991), mathematical concept often has two faces: an operational process side and a structural object side such as ``two sides of the same coin.''
Why do I pay attention to research transition from arithmetic to algebra and relationship between structural and operational understanding by examining students algebraic thinking processes, especially in terms of 8th grade students? Not only math subject but also scientific subjects in physic or chemistry involve many of letters (variable) in order to make a generalization or identify a parameter in any formula. Due to the importance of transition from elementary education to secondary education and 8th grade students’ cognitive development, studying with eight grade students was found meaningful and useful in order to find solutions for their misconceptions and the nature of their algebraic thinking processes according to different meanings of variables in algebra. As a result, the nature of students’ algebraic thinking process related to different meaning of variables is examined by proposing a research question in this study:
What is the nature of 8th grade students’ algebraic thinking processes based on different meanings of the concept of variable?
Method
Expected Outcomes
References
Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 20-32). Reston, VA: National Council of Teachers of Mathematics. Davis, R. B. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex. Gallardo, A. (2000). Historical-epistemological analysis in mathematics education: Two works in didactics of algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspective on School Algebra (pp. 121-139). Dordrecht, The Netherlands: Kluwer Academic Publishers. Katz, V. J. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66, 185-201.doi:10.1007/s10649-006-9023-7. Kuchmann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7(4). Kuchmann, D. (1981). Algebra. In K. M. Hart (Ed.), Children’s understanding of mathematics: 11-16 (pp.102-119). London: John Murray. MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation. Educational Studies in Mathematics, 33, 1-19. Merriam, S. B. (1998). Qualitative research and case study applications in education (Rev. ed.). Jossey-Bass, Inc., San Francisco. Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher, 81, 420-427. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15-39. Skemp, R. R. (1978). Relational Understanding and Instrumental Understanding, Mathematics Teaching, 77: 20-26. Wagner, S. (1999). What are these things called variables. In B. Moses (Ed.), Algebraic thinking, Grades K-12: Readings from the NCTM’s school-based journals and other publications (pp. 316-320). Reston, VA: National Council of Teachers of Mathematics. Philipp, R. A. (1992). The many uses of algebraic variables. Mathematics Teacher, 85(7), 560.
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