8th Grade Students’ Algebraic Thinking Processes based on Different Meanings of Variable
Author(s):
Fadime Ulusoy (presenting / submitting)
Conference:
ECER 2012
Network:
21. Emerging Researchers' Group (for presentation at Emerging Researchers' Conference)
Format:
Paper

Session Information

ERG SES B 11, Mathematics Education

Parallel Paper Session

Time:
2012-09-17
11:00-12:30
Room:
FCEE - Aula 4.3
Chair:

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

Case study design was found appropriate for this study to gather descriptive information about what actually occurs during the students’ thinking processes (Merriam 1998). Participant selection procedure was implemented in two stages via purposeful sampling method. In the first stage, an elementary school to be contacted easily was chosen. After that, Chelsea Diagnostic Algebra Test (CDAT) which designed by Kuchemann (1978) was applied to twenty 8th grade students in a class. In the second stage, three participants were determined according to the results of Chelsea Diagnostic Algebra Test, students’ mathematics scores of first semester, mathematics teacher’s review about students’ mathematics achievement. To gather and triangulate information, following data collection tools were used: i) Chelsea Diagnostic Algebra Test (CDAT) including 50 gap filling questions related to six ways in which students interpreted and used variables: the variable evaluated; the variable ignored; the variable as an object; the variable as a specific unknown; the variable as a generalized number; and the variable as a varying quantity; ii) semi-structured interviews. Data was analyzed with respect to three main aspects of variable as unknown, as a general number, as a functional relationship to determine the nature of 8th grade students’ understanding about variable.

Expected Outcomes

The results of the interviews indicated that there were three initial themes about the perception of the concept of variable: variable as unknown, variable as a general number, and variable as a functional relationship. According to results of the study, students found easy to struggle with the unknown, but it was procedurally-oriented as stated Skemp’s study (1978). In addition, findings revealed that students have a lot misconceptions and inability of dual nature of a concept in the process of their algebraic thinking. Common misconceptions which interrupted students’ algebraic thinking processes can be ordered as confusion with rank of letter in alphabet and its value, the inclination on singular operation by perceiving x + y as same with xy (Kuchemann, 1981), and the wrong label-object relationship (Booth, 1988; MacGregor & Stacey, 1997). These results are consistent with the result of research conducted in past decades. It is believed that the results of this study will give valuable implications for the teachers, teacher educators and policy makers because instructional materials and curricula may be designed to provide opportunities to students for transition from arithmetic to algebra through the construction of an arithmetic/algebra bridge.

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.

Author Information

Fadime Ulusoy (presenting / submitting)
Middle East Technical University
Elementary Education
Ankara

Update Modus of this Database

The current conference programme can be browsed in the conference management system (conftool) and, closer to the conference, in the conference app.
This database will be updated with the conference data after ECER. 

Search the ECER Programme

  • Search for keywords and phrases in "Text Search"
  • Restrict in which part of the abstracts to search in "Where to search"
  • Search for authors and in the respective field.
  • For planning your conference attendance, please use the conference app, which will be issued some weeks before the conference and the conference agenda provided in conftool.
  • If you are a session chair, best look up your chairing duties in the conference system (Conftool) or the app.