STUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN


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1 STUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN M. Gözde Didiş, Sinem Baş, A. Kürşat Erbaş Middle East Technical University This study examined 10 th grade students procedures for solving quadratic equations with one unknown. An openended test was designed and administered to 113 students in a high school in Antalya, Turkey. The data were analyzed in terms of the students foci while they were answering the questions. The results revealed that factoring the quadratic equations was challenging to them, particularly when students experienced them in a different structure from what they are used to. Furthermore, although students knew some rules related to solving quadratics, they applied these rules thinking about neither why they did so, nor whether what they were doing was mathematically correct. It was concluded that the students understanding in solving quadratic equations is instrumental (or procedural), rather than relational (or conceptual). Key words: Quadratic equations, instrumental understanding, relational understanding INTRODUCTION For many secondary school students, solving quadratic equations is one of the most conceptually challenging subjects in the curriculum (Vaiyavutjamai, Ellerton, & Clements, 2005). In Turkey, where a national mathematics curriculum for elementary and secondary levels is implemented, the teaching and learning of quadratic equations are introduced through factorization, the quadratic formula, and completing the square by using symbolic algorithms. Of these techniques, students typically prefer factorization when the quadratic is obviously factorable. With this technique, students can solve the quadratic equations quickly without paying attention to their structure and conceptual meaning (Sönnerhed, 2009). However, as Taylor and Mittag (2001) suggest, the factorization technique is only symbolic in its nature. Since students simply memorize the procedures and formulas to solve quadratic equations, they have little understanding of the meaning of quadratic equations, and do not understand what to do and why. This can be described using Skemp s (1976) categorization of mathematical understanding as either instrumental or relational. He simply described instrumental understanding as rules without reasons and relational understanding as knowing both what to do and why (p. 20). Using the language of Skemp, it can be said that students can perform instrumentally to solve the quadratic equations by applying the factorization technique; however, they become deprived of relational understanding. Although quadratic equations take an important role in secondary school algebra curricula around the world, it appears that studies concerning teaching and learning 1
2 quadratic equations are quite scarce in algebra education research (Kieran, 2007; Vaiyavutjamai & Clements, 2006). Therefore, this study was designed to widen the research considering students reasoning when engaging in different types of quadratic equations in one unknown. In particular, this study investigated students processes for solving quadratic equations with one unknown by using the factorization technique. The findings of this study may provide teachers with insight into the reasoning that leads to the common mistakes that students make while solving quadratic equations, and hence guide them in creating a more efficient pedagogical design for teaching how to solve quadratics. Challenges faced by Students in Solving Quadratic Equations According to Kotsopoulos (2007), for many secondary school students, solving quadratic equations is one of the most conceptually challenging aspects in the high school curriculum. She indicated that many students encounter difficulties recalling main multiplication facts, which directly influences their ability to engage in quadratics. And, since the factorization technique of solving quadratic equations requires 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 nonsimple 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 are presented in nonstandard forms. For example, factoring x 2 +3x+1=x+4 is challenging for students, since the equation is not presented in standard form (Kotsopoulos, 2007). Similarly, Bossè and Nandakumar (2005) stated that the factoring techniques for solving quadratic equations are problematic for students. They indicated that students can find factoring the quadratics considerably more complicated when the leading coefficient or constant in the quadratic has many pairs of possible factors. Skemp s (1976) description of instrumental and relational understanding can be used as a framework to discuss the difficulties students have with factoring quadratic equations. While an instrumental understanding of factorizing quadratic equations with one unknown requires memorizing rules for equations presented in particular structures, relational understanding enables students to apply these rules to different structures easily (Reason, 2003). That is, when students have relational understanding, they can transfer knowledge of both what rules (and formulas) worked and why they worked from one situation to another (Skemp, 2002). Lima (2008) found that students may perceive quadratic equations just like they do calculations. Since they focus mostly on the symbols used to perform operations, they may not be aware of the concepts that are involved. Vaiyavutjamai and Clements (2006) explain that students difficulties with quadratic equations arise from the lack of both instrumental and relational understanding of the associated mathematics. They found several misconceptions regarding variables which were obstacles to 2
3 understanding quadratic equations. For example, students thought that the two x s in the equation (x3) (x5) =0 stood for different variables, even though most of them obtained the correct solutions x=3 and x=5. Hence, they concluded that students performance in that context reflect rote learning and a lack of relational understanding. METHODOLOGY Participants and the Instrument The sample of this study consisted of 113 students in four 10 th grade classes, and this study was performed in a high school in Antalya, Turkey during the spring term For the purpose of the study, a questionnaire was formed by the authors since no test to specifically explore students errors and understanding was available. The test questions were carefully selected from secondary mathematics textbooks and from research regarding quadratic equations (e.g., Crouse & Sloyer, 1977). All questions used in this questionnaire were selected to measure the study objective of determining how students determine the roots and solution set of [a] quadratic equation in one unknown. During the selection process, two mathematics educators and a mathematics teacher were consulted about whether the content of the selected questions 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 the questions was openended, they varied in type so as to be consistent with the objective of the study. Questions 1 to 4 were in the standard format in which students were expected to find the solution set of the given quadratic equation. These questions were based on procedural skills, and they were mostly used to detect students procedural abilities in solving quadratic equations in different structures. On the other hand, questions 5 to 7 introduced a mathematical scenario that included both a quadratic equation and a solution belonging to it. In these type of questions, students were expected to determine whether the solutions [belonging to] the equations were correct or not, and to make judgment about their decision. Therefore, in addition to procedural skills, these questions were used to detect students understanding of and reasoning level when dealing with quadratic equations. The mathematics teacher administered the questionnaire during the regular class period and the students were given 30 minutes to complete it. Analysis of Data Initially, the responses given to each question were givens scores of either 1 or 0. A score of 1 was given for answers that were mathematically correct in terms of both solution process and final answer. A score of 0 was given for answers that were either omitted or incorrect in terms of either solution process or final answer. Then, in order to obtain a general view of the students performance, the percentage of correct, 3
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 (x3) (x2) = 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 (x3) (x2) = 0 as x 25x+6=0 and then I factorize to find roots of it. from (x3)=0 and from (x2)=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 (x3) and x=2 into (x2) simultaneously, and concluded that their solution 5 The answer is Right Because (x3)=0 (x2)=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 (33) (22)=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 214x+24=3 (x12) (x2)=3 (x12) (x2)=3 1 x12=3 x2=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 214x+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 214x+24=3 (x12) (x2)=3 (x12) (x2)=3 1 x12=3 x2=1 x=15 x=3 (3,1) Wrong Since the equations are separated as (3,1) there is no error when (x12)=3 however, there is error when (x2)=1. It must be (x2)=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 214x+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
7 Statements Question 7 The solution of the quadratic equation 2 x 2 =3x is given in the following; According to you, is this solution correct or not? Explain your answer with its reasons? Solution: I. step 2 x 2 =3x II. step 2 x x=3 x III. step 2 x =3 IV. step x= 3/2 Ç.K = {3/2} Students types of responses with their reasoning I. II. III. IV. Wrong Because 3x must be carried the left side of the equation and equalized the 0. Then, 2 x 2 =3x 2 x 23x=0 x (2x3)=0 x=0, x=3/2. The answer is Right The solution is right; however, it must be added 0 to the solution set. The answer Right Because when we substitute the value for x, the equation is satisfied. Right 2 x 2 =3x and x 2 is opened. 2 x x=3 x Yes the x is simplified. 2x=3 so x=3/2. Table 3: Common examples of students types of responses with their reasoning for question 7. In statement I (see Table 3), students stated that the answer was correct. They explained an appropriate procedure required for solving the equation. Since they memorized the rule without its reasons, they could only exhibit how the procedure must be worked. In statement II, on the other hand, students were aware that the roots of the equation were 0 and 3/2. However, they did not recognize that when was simultaneously canceled from both sides, the root 0 disappeared. Furthermore, in statement III, the explanation for solution was just based on the substitution method. In statement IV, students incorrectly stated that the answer was right. Like in statement II, students were not aware of the missing root 0 when canceling an in the equation DISCUSSION The results indicate that most of the students used the factorization technique to solve quadratic equations. This result supports Bosse and Nandakumar (2005), who claimed that a large percentage of the students preferred to apply the factorization techniques to find the solutions of quadratic equations. Also, in parallel with the results of Bossé and Nandakumar (2005) and Kotsopoulos (2007), the result of this study revealed that factoring the quadratic equations was challenging when they were presented to students in nonstandard forms and structures. After looking at the examples of students solutions (see Figures 1, 2, and 3), it can be said that the students knew some rules (or procedures) related to solving quadratics. However, they tried to apply these rules thinking about neither why they did so, nor whether if what they were doing was mathematically correct. These results give some clues about students instrumental understanding of solving quadratic equations with one unknown. However, to make an exact judgment about students relational or 7
8 instrumental understanding as Skemp (2002) defined, indepth interviews with individual students are required. Furthermore, results also indicate that students incorrectly tried to transfer some rules from one form of equation to another (e.g., in Figure 2). This can be considered another clue to students instrumental understanding (Reason, 2003). When students were asked to examine a solution process of a quadratic equation and judge whether it was correct (i.e., in questions 5, 6, and 7), the results give additional clues about their reasoning in solving quadratics. In question 5, for example, although most of the students were aware of the correctness of the result, they did not explain the underlying null factor law used to solve the quadratics by factorization. The responses also reveal their misunderstanding of the unknown concept in a quadratic equation (see the statement IV, in question 5), which is consistent with the results of Vaiyavutjamai and Clements (2006). Students were not aware that the two s in the equation represent a specific unknown when dealing with equations in the form (xa) (xb) =0. All of these can be regarded as clues to students instrumental understanding. As stated by Lima (2008), and Vaiyavutjmai and Clements (2006), students knew how to get correct answers but were not aware of what their answers represented. Similar interpretations can be made for the responses of students to question 6. There are 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 solution process was wrong, they could not detect the conceptual errors in the solution. They just presented some rules or procedures to solve the quadratic. Second, as was clear from statements III and IV (see Table 2), due to their lack of conceptual understanding of the null factor law in solving quadratics given in standard form, they wrongly transferred this principle to a quadratic in a nonsuitable form. This can also be a clue for students instrumental understanding. Because when students relationally 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 they did not offer any explanation for why canceling s was wrong. In other words, they did not recognize that when x was simultaneously canceled from both sides, the root 0 disappeared. Also, consistent with the results reported by Bossé and Nandakumar s (2005) and Kotsopoulos (2007), although students knew the null factor law, they could not apply it appropriately when the structure of equation was changed. Collectively, all these results reveal that students attempted to solve the quadratic equations as quickly as possible without paying much attention to their structures and conceptual meaning (Sönnerhed, 2009). Although we cannot be sure if their reasoning was based on instrumental or relational understanding without indepth interviews with students, their written answers provide clues to their reasoning, and it can be said that their reasoning underlying solving quadratic equations was based on instrumental understanding. 8
9 Having instrumental understanding does not generally cause trouble for students. It is much easier to obtain and use than relational understanding, just because it requires less knowledge, and with instrumental understanding, students can generally obtain the right answers more quickly. However, it necessities memorizing, and without relational understanding the learning cannot be adapted to new tasks, and students cannot give real reasons for their answers (Skemp, 2002). For that reason, greater attention should be given to how the concept is introduced to reduce the possibility of students learning the subjects/rules/procedures by rote. Any mechanism of solution must allow students to understand the meaning of the process that they apply in order to arrive at the correct answer; otherwise, the mechanism they learn will be a source of error (Blanco & Garrote, 2007). Recommendation As a result of this study, several suggestions can be made to contribute to improving students understanding of quadratic equations. Since factoring the quadratic equations was challenging when they are presented in nonstandard forms and structures, it would be better if teachers introduce various kinds of quadratic equations in different structures rather than just in the standard form. On the other hand, it would be also helpful for students to understand the factorization techniques as relational when teachers clearly emphasize meaning of the null factor rather than presenting it just as rule. In addition, because the students can attribute different meanings to the symbols (Küchemann, 1981), their understanding of the meanings of the algebraic symbols needs to be taken into account. Therefore, if teachers emphasize the meaning of the algebraic symbols, it would also useful for students to understand what the symbols represent in quadratic equations. Moreover, when teachers encourage students to use different techniques while solving quadratic equations, students learning may improve, and they may also gain a conceptually understanding. Similar recommendations can also be found in the related literature (e.g., Bossè & Nandakumar, 2005; Sönnerhed, 2009). Undoubtedly, teachers play an important role in encouraging students to learn relationally. This should be the most important part of teachers pedagogical content knowledge. However, research studies demonstrate a lack of secondary school mathematics teachers pedagogical content knowledge in this respect (Vaiyavutjamai, Ellerton, & Clements, 2005). Indeed, there is a need to research teachers knowledge about students difficulties concerning quadratic equations. REFERENCES Blanco, L. J., & Garrote, M. (2007). Difficulties in learning inequalities in students of the first year of preuniversity education in Spain. Eurasia Journal of Mathematics, Science & Technology Education, 3(3), Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics: Motivation for more techniques (section A). Teaching Mathematics and its Applications, 24(4),
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, 57. 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 Uppersecondary 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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