Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown

(ProQuest: . denotes formulae omitted)Quadratic equations have been a fundamental topic, not only in secondary mathematics curricula around the world but also in the historical development of algebra. Various approaches for solving quadratic equations were used at different stages in this histor-ical development, through representations including arithmetic or numerical, algebraic or symbolic, and visual or geometric (Katz & Barton, 2007). From a contemporary perspective, quadratic equations are considered important in school mathematics curricula because they serve as a bridge between mathematical topics such as linear equations, functions, and polynomials (Saglam & Alacaci, 2012). Furthermore, like linear equations, quadratic equations are powerful representations used in other disciplines, such as physics, engineering, and design, due to their usefulness in solving many kinds of word problems and for modeling realistic or real-life situations.Student Performance in Solving Quadratic EquationsVarious researchers (e.g., Vaiyavutjamai & Clements, 2006) have illustrated that very little attention has been paid to quadratic equations in mathematics education literature, and there is scarce research regarding the teaching and learning of quadratic equations. A limited number of research studies focusing on quadratic equations have documented the techniques students engage in while solving quadratic equations (Bosse & Nandakumar, 2005), geometric approaches used by students for solving quadratic equations (Allaire & Bradley, 2001), students' understanding of and difficulties with solving quadratic equations (Kotsopoulos, 2007; Lima, 2008; Tall, Lima, & Healy, 2014; Vaiyavutjamai, Ellerton, & Clements, 2005; Zakaria & Maat, 2010), the teaching and learning of quadratic equations in classrooms (Olteanu & Holmqvist, 2012; Vaiyavutjamai & Clements, 2006), comparing how quadratic equations are handled in mathematics textbooks in different countries (Saglam & Alacaci, 2012), and the application of the history of quadratic equations in teacher preparation programs to highlight prospective teachers' knowledge (Clark, 2012). In general, for most students, quadratic equations create challenges in various ways such as difficulties in algebraic procedures, (particularly in factoring quadratic equations), and an inability to apply meaning to the quadratics. Kotsopoulos (2007) suggests that recalling main multiplication facts directly influences a student's ability while engaged in factoring quadratics. Furthermore, since solving the quadratic equations by factorization requires students to find factors rapidly, factoring simple quadratics becomes quite a challenge, while non-simple ones (i.e., ax2 + bx + c where a ^ 1) become harder still. Factoring quadratics can be considerably complicated when the leading coefficient or the constant term has many pairs of factors (Bosse & Nandakumar, 2005). Lima (2008) and Tall et al. (2014) suggest that students' lack of understanding on the procedures of linear equations, and their understanding based on "procedural embodiments," affects students' work on quadratic equations. Students tend to allocate meaning to equations and solving methods, however, the given meaning is related to the movement of the symbols rather than the mathematical concept. They also documented that students perceive quadratic equations as mere calculations, without paying attention to the unknown as a fundamental characteristic of an equation. Students mostly focus on the symbolic world to perform operations with symbols. For example, students used procedural embodiment associated with the exponent of the unknown, and solved the equation by transforming it into m = 1/9 to solve m2 = 9. In this case, students' use of the procedural embodiments "switching power to roots" (p. 15) resulted in failing to recognize the other root (i.e., m = -3). Moreover, they reported that students attempted to transform quadratic equations into linear equations. …