reasoning in quadratic equations with one unknown - Cerme 7

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quadratic equations are quite scarce in algebra education research (Kieran, 2007;Vaiyavutjamai & Clements, 2006). Therefore, this study was designed to widen theresearch considering students’ reasoning when engaging in different types ofquadratic equations in one unknown. In particular, this study investigated students’processes for solving quadratic equations with one unknown by using thefactorization technique.The findings of this study may provide teachers with insight into the reasoning thatleads to the common mistakes that students make while solving quadratic equations,and hence guide them in creating a more efficient pedagogical design for teachinghow to solve quadratics.Challenges faced by Students in Solving Quadratic EquationsAccording to Kotsopoulos (2007), for many secondary school students, solvingquadratic equations is one of the most conceptually challenging aspects in the highschool curriculum. She indicated that many students encounter difficulties recallingmain multiplication facts, which directly influences their ability to engage inquadratics. And, since the factorization technique of solving quadratic equationsrequires students to be able to rapidly find factors, factoring simple quadratics (i.e.,x 2 +bx+c=0 where b, c R) become a quite challenge, while non-simple quadratics(i.e., ax 2 +bx+c=0 a, b, c R and a≠1) become nearly impossible. Moreover,students encounter crucial difficulties in factoring quadratic equations if they arepresented in non-standard forms. For example, factoring x 2 +3x+1=x+4 ischallenging for students, since the equation is not presented in standard form(Kotsopoulos, 2007). Similarly, Bossè and Nandakumar (2005) stated that thefactoring techniques for solving quadratic equations are problematic for students.They indicated that students can find factoring the quadratics considerably morecomplicated when the leading coefficient or constant in the quadratic has many pairsof possible factors.Skemp’s (1976) description of instrumental and relational understanding can be usedas a framework to discuss the difficulties students have with factoring quadraticequations. While an instrumental understanding of factorizing quadratic equationswith one unknown requires memorizing rules for equations presented in particularstructures, relational understanding enables students to apply these rules to differentstructures easily (Reason, 2003). That is, when students have relationalunderstanding, they can transfer knowledge of both what rules (and formulas) workedand why they worked from one situation to another (Skemp, 2002).Lima (2008) found that students may perceive quadratic equations just like they docalculations. Since they focus mostly on the symbols used to perform operations, theymay not be aware of the concepts that are involved. Vaiyavutjamai and Clements(2006) explain that students’ difficulties with quadratic equations arise from the lackof both instrumental and relational understanding of the associated mathematics.They found several misconceptions regarding variables which were obstacles to2
understanding quadratic equations. For example, students thought that the two x’s inthe equation (x-3) ∙(x-5) =0 stood for different variables, even though most of themobtained the correct solutions x=3 and x=5. Hence, they concluded that students’performance in that context reflect rote learning and a lack of relationalunderstanding.METHODOLOGYParticipants and the InstrumentThe sample of this study consisted of 113 students in four 10 th grade classes, and thisstudy was performed in a high school in Antalya, Turkey during the spring term2009-2010.For the purpose of the study, a questionnaire was formed by the authors since no testto specifically explore students’ errors and understanding was available. The testquestions were carefully selected from secondary mathematics textbooks and fromresearch regarding quadratic equations (e.g., Crouse & Sloyer, 1977). All questionsused in this questionnaire were selected to measure the study objective ofdetermining how students “determine the roots and solution set of [a] quadraticequation in one unknown”. During the selection process, two mathematics educatorsand a mathematics teacher were consulted about whether the content of the selectedquestions were consistent with the objective of the test. In light of their suggestions,seven open ended question were determined. Although the format of the all of thequestions was open-ended, they varied in type so as to be consistent with theobjective of the study. Questions 1 to 4 were in the standard format in which studentswere expected to “find the solution set of the given quadratic equation”. Thesequestions were based on procedural skills, and they were mostly used to detectstudents’ procedural abilities in solving quadratic equations in different structures. Onthe other hand, questions 5 to 7 introduced a mathematical scenario that includedboth a quadratic equation and a solution belonging to it. In these type of questions,students were expected to determine “whether the solutions [belonging to] theequations were correct or not, and to make judgment about their decision”.Therefore, in addition to procedural skills, these questions were used to detectstudents’ understanding of and reasoning level when dealing with quadraticequations.The mathematics teacher administered the questionnaire during the regular classperiod and the students were given 30 minutes to complete it.Analysis of DataInitially, the responses given to each question were givens scores of either 1 or 0. Ascore of 1 was given for answers that were mathematically correct in terms of bothsolution process and final answer. A score of 0 was given for answers that were eitheromitted or incorrect in terms of either solution process or final answer. Then, in orderto obtain a general view of the students’ performance, the percentage of correct,3
Page 1: STUDENTS’ REASONING IN QUADRATIC
Page 5 and 6: “Find the solution set of the equ
Page 7 and 8: StatementsQuestion 7The solution of
Page 9 and 10: Having instrumental understanding d

reasoning in quadratic equations with one unknown - Cerme 7

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