Algebra is described as a “mathematical language that combines operations, variables, and numbers to express mathematical structure and relationships in succinct forms” (Blanton et al., 2011, p. 67). It is one of the crucial branches of mathematics which constitutes a gateway between arithmetic reasoning in elementary school and advanced mathematics of higher grades (Blanton & Kaput, 2005). Researchers have agreed on the importance of algebraic thinking in learning mathematics (Asquith et al., 2007; Kieran, 2004). Stephens (2008) suggested that algebra in K-12 refers to “away of thinking instead of something we simply do (e.g., collect like terms, isolate the variable, change signs when we change sides)” (p. 35). Researchers emphasized that the focus should not be on understanding the rules to manipulate symbols and use algebraic procedures excellently but on developing algebraic thinking. Hence, identifying students’ conceptions, difficulties, and errors in algebra might be a good step in determining these standards.

In Third International Mathematics and Science Study (TIMMS), students were asked real-world problems to use algebraic models and explain the relationships. Despite Turkish eighth-grade students performed gradually increasing performance in algebra tests from year to year (MoNE, 2014; MoNE, 2016; MoNE, 2020), the scores of eight grade students in algebra items in TIMMS 2019 presented that Turkish eighth-grade students’ algebra scores were under the average mathematics score (MoNE, 2020). The difficulties students faced while learning algebra resulted in them becoming isolated from mathematics and stopping learning mathematics early in high school (Kaput, 2002). Thus, a nationwide movement, algebra for all, was called by U.S. educators and researchers to get all students to attain algebra (Moses & Cobb, 2001). In response to these concerns, the National Council of Teachers of Mathematics (NCTM) proposed instructional programs that enable learners “to understand patterns, relations, and functions,” “to represent and analyze mathematical situations and structures using algebraic symbols,” “to use mathematical models to represent and understand quantitative relationships,” and “to analyze the change in various contexts” (NCTM, 2000, p. 37). Thus, it might be beneficial to explore the algebraic thinking of Turkish eighth-grade students to improve their algebra performance. This study investigates the research questions:

1. What is the nature of eighth-grade students’ algebraic thinking around the issues of equivalence and equations, generalized arithmetic, variable, and functional thinking?
2. Which difficulties and errors do eighth-grade students have around the issues of equivalence and equations, generalized arithmetic, variable, and functional thinking?

This case study explores eighth-grade students’ conceptions and difficulties in algebra. Participants of the study are 267 eighth-grade students in a public middle school in Turkey. To investigate students’ conceptions and difficulties in algebra, first, the researchers prepared an Algebra Diagnostic Test (ADT) based on informal classroom observations in algebra classes, interviews with middle school mathematics teachers, and studies in the related literature. The test was prepared considering the big ideas in algebra (Blanton et al., 2015) and Turkey’s middle school mathematics curriculum (MoNE, 2018). Before conducting ADT on students, Pilot Testing I and Pilot Testing II processes were held to ensure validity and reliability issues. 140 students participated in Pilot Testing I in the spring semester of the 2017-2018 academic year, and 136 students participated in Pilot Testing II in the fall semester of the 2018-2019 academic year. Finally, ADT included 17 open-ended items in the scope of the big ideas of equivalence, expressions, equations, and inequalities, generalized arithmetic, variable, and functional thinking. ADT was administered to eighth-grade students in the spring semester of the 2018-2019 academic year. Students’ responses to the items in ADT were analyzed by coding their conceptions, solution strategies, and difficulties. Thus, students’ responses were explored in frequencies and percentages, considering their strategies and errors.

Results indicated that students successfully did simple arithmetic and symbolic manipulations and solved algebra story or word problems. However, students were unsuccessful in the tasks such as comparing two algebraic expressions, writing the general rule of an algebra story problem, and interpreting the covariation in functions. Students generally focused on using x to manipulate the symbols instead of considering the relational understanding of x. Although students could write an algebraic expression based on a word problem, most students, who could write the symbolic expression, struggled to identify what x refers to in the algebraic expressions they wrote. Asquith et al. (2007) found that more than half of the students hold a multiple-values interpretation. Conversely, 25% of the students could express a multiple-values interpretation to answer the task comparing 3n and n+6. In functional thinking items, students were asked to find the answer for a specific value and write the general rule based on the algebra story problem. Results showed that most students were unsuccessful in writing the general rule of a given problem, although they could solve the problem using arithmetic (e.g., doing a substitution, guess-and-test, modeling, & unwinding). Also, it was observed that students mainly prefer to solve the problems using arithmetic even if they could write the algebraic expression symbolically.

Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272. https://doi.org/10.1080/10986060701360910 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. https://doi.org/10.2307/30034944 Blanton, M., Levi, L., Crites, T., & Dougherty, B. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3–5. In R. M. Zbiek (Series Ed.), Essential understanding series. National Council of Teachers of Mathematics. 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. Kaput, J. J. (2002). Research on the development of algebraic reasoning in the context of elementary mathematics: A brief historical overview. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the international group for the psychology of mathematics education (pp. 120–122). ERIC. Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, M. & M. Kendal (Eds), The Future of the Teaching and Learning of Algebra The 12thICMI Study (pp. 21-33). Springer. Ministry of National Education [MoNE] (2018). Matematik dersi öğretim programı ilkokul ve ortaokul 1-8 sınıflar [Mathematics curriculum primary and middle school grades 1-8]. Retrieved on July 10, 2020 from http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=329 Ministry of National Education. [MoNE]. (2014). TIMSS 2011 Ulusal Matematik ve Fen Raporu: 8. Sınıflar [TIMMS 2011 National Mathematics and Science Report: 8th Grade]. Retrieved November 01, 2022, from https://timss.meb.gov.tr/meb_iys_dosyalar/2022_03/07135958_TIMSS-2011-8-Sinif.pdf Ministry of National Education. [MoNE]. (2016). TIMSS 2015 Ulusal Matematik ve Fen Ön Raporu [TIMMS 2015 National Mathematics and Science Preliminary Report]. Retrieved November 01, 2022, from https://timss.meb.gov.tr/meb_iys_dosyalar/2022_03/07135609_TIMSS_2015_Ulusal_Rapor.pdf Ministry of National Education. [MoNE]. (2020). TIMSS 2019 Türkiye Ön Raporu [TIMMS 2019 Turkey Preliminary Report]. Retrieved November 01, 2022, from http://www.meb.gov.tr/meb_iys_dosyalar/2020_12/10173505_No15_TIMSS_2019_Turkiye_On_Raporu_Guncel.pdf Moses, R. P., & Cobb, C. E. (2001). Radical equations. Beacon Press. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Author. Stephens, A. C. (2008). What “counts” as algebra in the eyes of preservice elementary teachers?. The Journal of Mathematical Behavior, 27(1), 33-47. https://doi.org/10.1016/j.jmathb.2007.12.002