ISSN 1303-0485 • eISSN 2148-7561 DOI 10.12738/estp.2015.4.2743 Received | December 1, 2014 Copyright © 2015 EDAM • http://www.estp.com.tr Accepted | April 17, 2015 Educational Sciences: Theory & Practice • 2015 August • 15(4) • 1137-1150 OnlineFirst | August 24, 2015 Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown* Makbule Gozde Didisa Gaziosmanpasa University Ayhan Kursat Erbasb Middle East Technical University Abstract This study attempts to investigate the performance of tenth-grade students in solving quadratic equations with one unknown, using symbolic equation and word-problem representations. The participants were 217 tenth-grade students, from three different public high schools. Data was collected through an open-ended questionnaire comprising eight symbolic equations and four word problems; furthermore, semi-structured interviews were conducted with sixteen of the students. In the data analysis, the percentage of the students’ correct, incorrect, blank, and incomplete responses was determined to obtain an overview of student performance in solving symbolic equations and word problems. In addition, the students’ written responses and interview data were qualitatively analyzed to determine the nature of the students’ difficulties in formulating and solving quadratic equations. The findings revealed that although students have difficulties in solving both symbolic quadratic equations and quadratic word problems, they performed better in the context of symbolic equations compared with word problems. Student difficulties in solving symbolic problems were mainly associated with arithmetic and algebraic manipulation errors. In the word problems, however, students had difficulties comprehending the context and were therefore unable to formulate the equation to be solved. Thus, it is concluded that the differences in the structural properties of the symbolic equations and word problem representations affected student performance in formulating and solving quadratic equations with one unknown. Keywords: Mathematics education • Algebra • Quadratic equations • Quadratic word problems • Students’ difficulties * An earlier version of this manuscript was presented at the 10th Congress of National Science and Mathematics Education held in Nigde University, Turkey on June 27-30, 2012. Ayhan Kursat Erbas is supported by the Turkish Academy of Sciences through the Young Scientist Award Program (A.K.E./TÜBA-GEBİP/2012-11). a Corresponding author Makbule Gozde Didis, Elementary Education Department, Faculty of Education, Gaziosmanpasa University, Tokat Turkey Research areas: Mathematics teacher education; Mathematical modeling; Teaching and learning algebra Email: [email protected] b Ayhan Kursat Erbas, Secondary Science and Mathematics Education Department, Faculty of Education, Middle East Technical University, Ankara Turkey Email: [email protected] Educational Sciences: Theory & Practice Quadratic equations have been a fundamental topic, requires students to find factors rapidly, factoring not only in secondary mathematics curricula around simple quadratics becomes quite a challenge, while the world but also in the historical development of non-simple ones (i.e., ax2 + bx + c where a ≠ 1) algebra. Various approaches for solving quadratic become harder still. Factoring quadratics can be equations were used at different stages in this histor- considerably complicated when the leading coeffi- ical development, through representations including cient or the constant term has many pairs of factors arithmetic or numerical, algebraic or symbolic, and (Bossè & Nandakumar, 2005). Lima (2008) and visual or geometric (Katz & Barton, 2007). From a Tall et al. (2014) suggest that students’ lack of un- contemporary perspective, quadratic equations are derstanding on the procedures of linear equations, considered important in school mathematics cur- and their understanding based on “procedural ricula because they serve as a bridge between math- embodiments,” affects students’ work on quadrat- ematical topics such as linear equations, functions, ic equations. Students tend to allocate meaning to and polynomials (Sağlam & Alacacı, 2012). Further- equations and solving methods, however, the given more, like linear equations, quadratic equations are meaning is related to the movement of the symbols powerful representations used in other disciplines, rather than the mathematical concept. They also such as physics, engineering, and design, due to their documented that students perceive quadratic equa- usefulness in solving many kinds of word problems tions as mere calculations, without paying attention and for modeling realistic or real-life situations. to the unknown as a fundamental characteristic of an equation. Students mostly focus on the symbol- ic world to perform operations with symbols. For Student Performance in Solving Quadratic example, students used procedural embodiment Equations associated with the exponent of the unknown, and Various researchers (e.g., Vaiyavutjamai & Clem- solved the equation by transforming it into m = 9 2 ents, 2006) have illustrated that very little attention to solve m = 9. In this case, students’ use of the pro- has been paid to quadratic equations in mathemat- cedural embodiments “switching power to roots” ics education literature, and there is scarce research (p. 15) resulted in failing to recognize the other root regarding the teaching and learning of quadratic (i.e., m = -3). Moreover, they reported that students equations. A limited number of research studies attempted to transform quadratic equations into focusing on quadratic equations have document- linear equations. For instance, while attempting 2 ed the techniques students engage in while solving to solve m = 9, some students applied the expo- quadratic equations (Bossé & Nandakumar, 2005), nent associated with the unknown as if it were the 2 geometric approaches used by students for solving coefficient; that is, m equals to 2m, and students quadratic equations (Allaire & Bradley, 2001), stu- showed a tendency to use the quadratic formula dents’ understanding of and difficulties with solv- as the only valid method in solving every quadrat- ing quadratic equations (Kotsopoulos, 2007; Lima, ic equation. Although it is expected that students 2008; Tall, Lima, & Healy, 2014; Vaiyavutjamai, El- would use factorization for solving equations such 2 2 lerton, & Clements, 2005; Zakaria & Maat, 2010), as t - 2t = 0 and 3k - k = 0, studies have shown the teaching and learning of quadratic equations in that few students would do so. Vaiyavutjamai and classrooms (Olteanu & Holmqvist, 2012; Vaiyavut- Clements (2006) proposed that students’ difficul- jamai & Clements, 2006), comparing how quadratic ties with quadratic equations stem from their lack equations are handled in mathematics textbooks in of both instrumental understanding and relational different countries (Sağlam & Alacacı, 2012), and understanding of the specific mathematics associ- the application of the history of quadratic equations ated with solving quadratic equations. They suggest in teacher preparation programs to highlight pro- that while teacher-centered instruction with strong spective teachers’ knowledge (Clark, 2012). emphasis placed on the manipulation of symbols, rather than on the meaning of symbols, increases In general, for most students, quadratic equations student performance regarding solving quadratic create challenges in various ways such as diffi- equations, their (relational) understanding would culties in algebraic procedures, (particularly in still be quite low, and they could develop miscon- factoring quadratic equations), and an inability ceptions. For example, they found that many stu- to apply meaning to the quadratics. Kotsopoulos dents had an inadequate understanding of the “null (2007) suggests that recalling main multiplication factor law”. In solving (x - 3)(x-5) = 0, although facts directly influences a student’s ability while en- most students gave the correct answer; x = 3 and gaged in factoring quadratics. Furthermore, since x = 5, they considered two xs in the equation as solving the quadratic equations by factorization 1138 Didis, Erbas / Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown representatives of different variables, and thus, they According to Cummins et al. (1988), inadequate must take different values. That is, when they were mapping of phrases causes student to fail at solving asked to check their solutions, they simultaneously word problems. Similarly, Nathan et al. (1992) sug- sub stituted x = 3 into (x - 3) and x = 5 into (x - 5) gested that students make errors because they fail and found that 0.0 = 0 and in doing so, decided that to see how the situational aspect of the problem is their solutions were correct. This misconception related to the formal expressions, in their attempts also appeared in students’ solution of x2 - x = 12. to produce the intended internal representations. Although the linguistic form of the problem’s text conveys the significant factors that affect the com- Student Performance in Solving Algebra Word prehension process, Stacey and MacGregor (2000) Problems claimed that a major reason for students’ difficulties Application problems traditionally appear in the with word problems arises from not understanding form of word problems (Verschaffel, Greer, & de the algebraic logic of a problem. They argue that