# STUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN

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4 incorrect and omitted questions were calculated. The aim of this process was descriptive analysis. Afterwards, qualitative data analysis was conducted. The subjects responses were studied in order to provide substantial information about their type of understanding. In this analysis, it was attempted to identify the common mistakes that students made while solving the quadratic equations. Therefore, the incorrect answers for all questions have been analyzed item by item with respect to the students focus when they solved the questions in the test situation. In this process, students types of mistakes were coded by two researchers of this study who worked initially separately. Next, the mistakes were both combined and renamed based on their common features, and then they were classified by two researchers together. Lastly, these mistakes were interpreted in terms of students instrumental understanding and relational understanding. RESULT The first item in the instrument was related to finding the roots of a quadratic equation given in standard form (e.g., ax 2 +bx+c=0 where a, b, c R). Almost all students correctly solved this equation by factorization. In the following questions, quadratic equations were given in different structures (e.g., ax 2 -bx=0, c=0). In these types of questions, just 64% of them solved the equation ax 2 -bx=0, correctly. When the solution processes of students who made mistakes (36%) were analyzed, it was recognized that their mistakes were based on two different types. Find the solution set of the equation 2 2 = 0. Figure 1: An example of students first type of mistake Find the solution set of the equation 2 2 = 0. Figure 2: An example of students second type of mistake 4

5 Find the solution set of the equation 2 = 12 Figure 3: An example of students mistake when just the form of equation changed. In the first type of wrong solution (see Figure 1), students carried the term -2x from left side to the right, and then simplified the term x in both sides of the equation. Consequently, they ignored one of the roots of the equation, which is 0. In the second type of wrong solution (see Figure 2), students tried to factorize the equation. Here, students perceived the form ax 2 -bx=0 just like ax 2 +bx+c=0 and thought -2x as the constant term of the quadratic equations. Even, when just the form of the equation was changed instead of the structure (e.g., ax 2 +bx=c where a, b, c 0), 12% of the students incorrectly solved the quadratic. Because the constant term was in the right side, they didn t perceive that the equation was in standard form (see Figure 3). In this type of solution, they were able to find only one of the roots, 4. Statements Question 5 To solve the equation (x-3) (x-2) = 0 for real numbers, Ali answered in a single line that: x=3 or x=2 Is this answer correct? If it is correct, how can you show it correctness? Students types of responses with their reasoning I. II. III. IV. Right Since I wrote (x-3) (x-2) = 0 as x 2-5x+6=0 and then I factorize to find roots of it. from (x-3)=0 and from (x-2)=0 x=3 and x=2 Table 1: Common examples of students types of responses with their reasoning for question 5. Although all of the students stated that Ali s answer was correct by choosing either one of the statements I, II, III, and IV, the ways they justified for the correctness of Ali s solution were different. For instance, in statement I, students first transformed the factorized expression into the standard form, and then factorized the expression again in the same way and found the roots by rote. In statement II, students unconsciously applied the null factor law. In statement III, the way of justification for solution was based on substitution method. In all of these three statements, they could not clearly justify the correctness of the solution. In statement IV, students substituted x=3 into (x-3) and x=2 into (x-2) simultaneously, and concluded that their solution 5 The answer is Right Because (x-3)=0 (x-2)=0 x=3 x=2 The answer is Right. Since we substitute 3 and 2 into x, the equation is provided. (explanation made only with words) Right. If the x=2 and x=3 are substituted into the equation (3-3) (2-2)=0 0.0 = 0

6 were correct since 0 0=0. Namely, they thought that the two x s stood for different numbers. Statements Question 6 A student hands in the following work for the following problem. Solve ; x 2-14x+24=3 (x-12) (x-2)=3 (x-12) (x-2)=3 1 x-12=3 x-2=1 x=15 x=3 Ç.K= {3, 5} Is the student correct? Explain your answer with its reasons? The answer is Wrong Because, firstly, 3 must carry the left side of the equation and equalize the 0. Then, the other operations must be done. In this way, the equation x 2-14x+21=0 Students types of responses with their reasoning I. II. III. IV. The answer is Wrong. Because when we substitute 3 and 15 for x, the equation is not provided. Right Since the result is equal to 3, we equate 3 rather than 0 while factoring it. Therefore, the result is true. Students again solve as: x 2-14x+24=3 (x-12) (x-2)=3 (x-12) (x-2)=3 1 x-12=3 x-2=1 x=15 x=3 (3,1) Wrong Since the equations are separated as (3,1) there is no error when (x-12)=3 however, there is error when (x-2)=1. It must be (x-2)=3 then, x=5. Therefore, the solution will be {5, 15} rather than {3, 15}. Table 2: Common examples of students types of responses with their reasoning for question 6. In statements I and II (see Table 2), students were aware of the error in the solution of the given question. However, to explain the reasons for the mistake, they presented procedural explanations like the responses in statements I, II, III for question 5 (see Table 1). In statement III, students incorrectly stated that the answer was right. Looking at the statement since the result is equal to 3, we equate to 3 rather than 0 while factoring it, it can be said that they wrongly tried to transfer the null factor law to this context. That is, they equated the factors of equation x 2-14x+24 with the integer factors of 3. In statement IV, students correctly claimed the answer of the question wrong ; however, their explanations were fully erroneous. Similar to statement III, these students tried to apply the null factor law to the equation. Nonetheless, in this case, they only equated the factors to 3 rather than to the factors of 3. In both statements III and IV, students did not check whether the roots they found were appropriate or not. 6

10 Crouse, J. R., & Sloyer, W. C. (1977). Mathematical questions from the classroom. USA: Prindle, Weber & Schmidt. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. Lester (ed.), Second Handbook of Research on Mathematics Teaching and Learning: A project of the National Council of Teachers of Mathematics. Vol II (pp ). Charlotte, NC: Information Age Publishing. Kotsopoulos, D. (2007). Unraveling student challenges with quadratics: A cognitive approach. Australian Mathematics Teacher, 63(2), Küchemann, D. E. (1981). Algebra. In Hart, K., Brown, M. L., Küchemann, D. E., Kerslake, D., Ruddock, G., & McCartney, M. (Eds.), Children's understanding of Mathematics: (pp ). London: John Murray. Lima, R. N. (2008). Procedural embodiment and quadratic equations. Retrieved April 1, 2010, from Reason, M. (2003). Relational, instrumental and creative understanding. Mathematics Teaching, 184, 5-7. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, Skemp, R. R. (2002). Mathematics in the primary school. London: Routledge Falmer. Sönnerhed, W. W. (2009). Alternative approaches of solving quadratic equations in mathematics teaching: An empirical study of mathematics textbooks and teaching material or Swedish Upper-secondary school. Retrieved April 5, 2010, from Taylor, S. E. & Mittag, K. C. (2001). Seven wonders of the ancient and modern quadratic world. Mathematics Teacher, 94, Vaiyavutjamai, P., Ellerton, N. F., & Clements, M. A. (2005). Students attempts to solve two elementary quadratic equations: A study in three nations. Retrieved April 1, 2010, from Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students understanding of quadratic equations. Mathematics Education Research Journal, 18(1),

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