ERG SES E 10, Evaluation in Education
Algebra has been a central argument for mathematics curriculum in middle school (Blanton, Stephens, Knuth, Gardiner, Isler & Kim, 2015) and is considered to be a gatekeeper in mathematics teaching and learning (Knuth, Alibali, McNeil, Weingberg & Stephens, 2005). Algebraic reasoning is defined as recognition of relationships between variables (Driscoll 1999). According to Kieran (2004), algebraic reasoning focuses on letters, operations, representing and solving problems rather than merely numbers or solution. Algebra based topics are considered as significant for middle school mathematics classrooms since recognizing relationship, generalizing beyond specific examples and investigating and analyzing pattern form the basis of algebra for the middle school students (Magiera, Van den Kieboom & Moyer 2013). Walkoe (2013) said that “algebra concepts provide the foundation for much advanced mathematical content and it serves as a gatekeeper to opportunities such as admission to college” (p.3). The integration of algebraic reasoning into middle school mathematics curriculum provides advantage in terms of building conceptual and deeper understanding of mathematics topics (Blanton & Kaput, 2005).
When the students start to learn algebra, it is expected them to have transformation from rule based fact and algorithm to solve lower thinking problems to the higher level of algebraic reasoning (Patton & De Los Satos, 2012). Because of that, in national and international contexts algebraic reasoning is taken into consideration while revising the curriculum and designing the lessons (Ministry of National Education [MoNE], 2013; National Council of Teachers of Mathematics [NCTM], 2000). Although algebra concept has an extensive coverage in Turkish mathematics curriculum and is considered to be a gatekeeper in mathematics teaching and learning (Knuth, Alibali, McNeil, Weingberg & Stephens, 2005), Turkish elementary students’ success in this topic is low (Dede & Argün, 2003; Kriegler, 2008). One of the reasons for the low success of students might be that they have inadequate algebraic reasoning ability. Hence, it would be significant to examine students’ algebraic reasoning in Turkish context.
Literature review showed that the number of studies investigating Turkish students’ algebraic thinking is limited. For this reason, in an attempt to examine Turkish students’ algebraic reasoning is believed to make contribution to the literature. Parallel to this idea, teachers can take a feedback regarding teaching algebra by means of this study and they can rearrange their methods of teaching. Moreover, this study gives opportunity to make a comparison between Turkey and other countries in terms of students’ algebraic reasoning. From this point of view, the aim of this study was to investigate 7th grade students’ algebraic reasoning ability. More specifically, their competency level in algebraic reasoning was investigated by looking their solution strategies on algebra questions regarding pattern generalization.
Method: In this study, we aimed to examine 7th grade students’ solutions on pattern generalization and identify their competency level according to their solution strategies. Thus, in order to get in-depth exploration of students’ algebraic skills qualitative case study was conducted. Data were collected from one hundred and thirty three 7th grade students who were enrolled in a public school in Kırıkkale, Turkey at 2017-2018 academic year. In order to collect data, a questionnaire containing two open-ended questions was used. The questions were taken from the previous research studies in accordance with the objectives given in the Turkish middle school mathematics curriculum (MoNE, 2013). The first and second question in the questionnaire were taken from the study of Radford (2000, p.243,244) and Kriegler (2008, p.5), respectively. Both questions were translated into Turkish. In the first question, students were expected to explore a pattern and find 25th term using the general term. In the second question, it was asked to find near term of pattern, to write general term, to investigate the position of a figure when the number of elements in pattern was given and to explore the position of a figure in pattern when its corresponding value, which is 152, is given. The validity of the questionnaire was ensured by expert opinions. Data Analysis: Data were analyzed by using Maria Luz Callejo and Albert Zapatera’s (2016) framework which is based on competencies of numerical and spatial structures, functional relationship and inverse process. The first competency is numerical and spatial structures, that is, the ability of finding corresponding value to each element in pattern according to physical location of them (Callejo & Zapatera, 2016). Students who have this ability are able to identify one of initial terms by using first three or four terms with numerical and spatial structures. The second competency is functional relationship, which is, establishing a relationship between position of term in pattern and corresponding value to this term (Rivera, 2010, Warren, 2005). Students who have this ability can find nth term of the pattern. The more complex competency is inverse process (Rivera, 2010, Warren, 2005). Students with the competency of inverse process are expected to explore the position of any term in the pattern when its corresponding value is given. As can be understood from the definitions, the numerical and spatial structure is the most basic competency while functional relationships and inverse process are more complex.
Findings: Analysis of data showed that in the first question, although 20 students (15 %) had ability of constructing numerical and spatial structures, only 4 students (3%) can construct functional relationship Moreover, the analysis of the second question showed that only one student (0,75 %) could solve the 1st sub-item of the 2nd question. According to his solution, it could be stated that he had both numerical and spatial structures competency. Furthermore, no one could answer both the 2nd sub-item and 3rd sub-item of the 2nd question. Thus, data revealed that participants have neither competency of functional relationship nor competency of inverse process. In other words, it can be said that students could solve questions that involve arithmetic only. Discussion: Based on the analysis of the data it could be deduced that 7th graders could not solve the questions regarding the generalization of algebraic terms although algebraic concepts is taught in 6th grade. The reason of students’ failure might be algebra concepts were introduced to them through memorization of algorithmic procedures (Dede & Argün, 2003).In addition, most of the students did not use multiple representations although they were required to draw table, use numerical and algebraic expression while solving questions. From this result, it can be stated that teachers might not focus on using multiple representations although there are objectives about using various models in Turkish middle school mathematics curriculum.. However, teachers may revise their lessons to give opportunities for students to develop their algebraic reasoning ability. Also, curriculum developers and textbook writers might design different activities containing representations and generalization of patterns. As a further research, a similar study could be conducted with 6th, 7th and 8th grade students to compare students’ algebraic reasoning ability at different grade levels.
Blanton, M. L., & Kaput, J. J. (2005).Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446. Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J. S. (2015). The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39-87. Callejo, M. L., & Zapatera, A. (2017).Prospective primary teachers’ noticing of students’ understanding of pattern generalization. Journal of Mathematics Teacher Education, 20(4), 309-333. Dede, Y., & Argün, Z. (2003). Cebir, öğrencilere niçin zor gelmektedir?. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24(24), 180-185. Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10. New York: Heinemann. Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8(1), 139-151. Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students’ understanding of core algebraic concepts: Equality & variable. ZDM, 37(1), 68–76. doi:10.1007/ bf02655899 Kriegler, S. (2008). Just what is algebraic thinking. Retrieved from January 20, 2018 from www.math.ucla.edu/~kriegler/pub/algebrat. html. Magiera, M. T., Van den Kieboom, L. A., & Moyer, J. C. (2013).An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking. Educational Studies in Mathematics, 84(1), 93-113. Milli Eğitim Bakanlığı [Ministry of National Education](MoNE) (2013).. Ortaokul matematik dersi (5, 6, 7, ve 8. Sınıflar) ögretim programı [Middle school mathematics curriculum grades 5 to 8]. Ankara, Turkey: Author. National Council of Teachers of Mathematics.(2000). Principles and Standards for School Mathematics. Reston, VA: Author Patton, B., & De Los Santos, E. (2012).Analyzing algebraic thinking using. Online Submission, 5(1), 5-22. Radford, L. (2000). Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237-268. Rivera, F. D. (2010). Second grade students’ pre-instructional competence in patterning activity.In M.F. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81–88). Belo Horizonte, Brazil: PME Walkoe, J. D. K. (2013). Investigating teacher noticing of student algebraic thinking. (Unpublished doctoral dissertation), Northwestern University, USA ). Warren, E. (2005). Young children’s ability to generalise the pattern rule for growing patterns. In H.L. Chick & J. L. Vincent (Eds.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 305–312). Melbourne: PME.
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