reasoning in quadratic equations with one unknown - Cerme 7

instrumental understanding as Skemp (2002) defined, in-depth interviews withindividual students are required. Furthermore, results also indicate that studentsincorrectly tried to transfer some rules from one form of equation to another (e.g., inFigure 2). This can be considered another clue to students’ instrumentalunderstanding (Reason, 2003).When students were asked to examine a solution process of a quadratic equation andjudge whether it was correct (i.e., in questions 5, 6, and 7), the results give additionalclues about their reasoning in solving quadratics. In question 5, for example, althoughmost of the students were aware of the correctness of the result, they did not explainthe underlying null factor law used to solve the quadratics by factorization. Theresponses also reveal their misunderstanding of the unknown concept in a quadraticequation (see the statement IV, in question 5), which is consistent with the results ofVaiyavutjamai and Clements (2006). Students were not aware that the two ’s in theequation represent a specific unknown when dealing with equations in the form(x-a) ∙(x-b) =0. All of these can be regarded as clues to students’ instrumentalunderstanding. As stated by Lima (2008), and Vaiyavutjmai and Clements (2006),students knew how to get correct answers but were not aware of what their answersrepresented.Similar interpretations can be made for the responses of students to question 6. Thereare two salient points related to their reasoning in explaining the given solution. First,although students were expected to explain the reason(s) why the given solutionprocess was wrong, they could not detect the conceptual errors in the solution. Theyjust presented some rules or procedures to solve the quadratic. Second, as was clearfrom statements III and IV (see Table 2), due to their lack of conceptualunderstanding of the null factor law in solving quadratics given in standard form,they wrongly transferred this principle to a quadratic in a non-suitable form. This canalso be a clue for students’ instrumental understanding. Because when studentsrelationally understand a rule, they can use it in a different context (Reason, 2003).Similar inferences can be made for the students’ responses to questions 7 where theydid not offer any explanation for why canceling ’s was wrong. In other words, theydid not recognize that when x was simultaneously canceled from both sides, the root0 disappeared. Also, consistent with the results reported by Bossé and Nandakumar’s(2005) and Kotsopoulos’ (2007), although students knew the null factor law, theycould not apply it appropriately when the structure of equation was changed.Collectively, all these results reveal that students attempted to solve the quadraticequations as quickly as possible without paying much attention to their structures andconceptual meaning (Sönnerhed, 2009). Although we cannot be sure if theirreasoning was based on instrumental or relational understanding without in-depthinterviews with students, their written answers provide clues to their reasoning, and itcan be said that their reasoning underlying solving quadratic equations was based oninstrumental understanding.8