reasoning in quadratic equations with one unknown - Cerme 7

STUDENTS’ REASONING IN QUADRATIC EQUATIONS WITHONE UNKNOWNM. Gözde Didiş, Sinem Baş, A. Kürşat ErbaşMiddle East Technical UniversityThis study examined 10 th grade students’ procedures for solving quadratic equationswith one unknown. An open-ended test was designed and administered to 113students in a high school in Antalya, Turkey. The data were analyzed in terms of thestudents’ foci while they were answering the questions. The results revealed thatfactoring the quadratic equations was challenging to them, particularly whenstudents experienced them in a different structure from what they are used to.Furthermore, although students knew some rules related to solving quadratics, theyapplied these rules thinking about neither why they did so, nor whether what theywere doing was mathematically correct. It was concluded that the students’understanding in solving quadratic equations is instrumental (or procedural), ratherthan relational (or conceptual).Key words: Quadratic equations, instrumental understanding, relationalunderstandingINTRODUCTIONFor many secondary school students, solving quadratic equations is one of the mostconceptually challenging subjects in the curriculum (Vaiyavutjamai, Ellerton, &Clements, 2005). In Turkey, where a national mathematics curriculum for elementaryand secondary levels is implemented, the teaching and learning of quadraticequations are introduced through factorization, the quadratic formula, and completingthe square by using symbolic algorithms. Of these techniques, students typicallyprefer factorization when the quadratic is obviously factorable. With this technique,students can solve the quadratic equations quickly without paying attention to theirstructure and conceptual meaning (Sönnerhed, 2009). However, as Taylor and Mittag(2001) suggest, the factorization technique is only symbolic in its nature. Sincestudents simply memorize the procedures and formulas to solve quadratic equations,they have little understanding of the meaning of quadratic equations, and do notunderstand what to do and why. This can be described using Skemp’s (1976)categorization of mathematical understanding as either instrumental or relational. Hesimply described instrumental understanding as “rules without reasons” and relationalunderstanding as “knowing both what to do and why” (p. 20). Using the language ofSkemp, it can be said that students can perform instrumentally to solve the quadraticequations by applying the factorization technique; however, they become deprived ofrelational understanding.Although quadratic equations take an important role in secondary school algebracurricula around the world, it appears that studies concerning teaching and learning1

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

incorrect and omitted questions were calculated. The aim of this process wasdescriptive analysis. Afterwards, qualitative data analysis was conducted. Thesubjects’ responses were studied in order to provide substantial information abouttheir type of understanding. In this analysis, it was attempted to identify the commonmistakes that students made while solving the quadratic equations. Therefore, theincorrect answers for all questions have been analyzed item by item with respect tothe students’ focus when they solved the questions in the test situation. In thisprocess, students’ types of mistakes were coded by two researchers of this study whoworked initially separately. Next, the mistakes were both combined and renamedbased on their common features, and then they were classified by two researcherstogether. Lastly, these mistakes were interpreted in terms of students’ instrumentalunderstanding and relational understanding.RESULTThe first item in the instrument was related to finding the roots of a quadraticequation given in standard form (e.g., ax 2 +bx+c=0 where a, b, c R). Almost allstudents 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 thesetypes of questions, just 64% of them solved the equation ax 2 -bx=0, correctly. Whenthe solution processes of students who made mistakes (36%) were analyzed, it wasrecognized 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 mistake4

“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 fromleft 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 secondtype 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 theconstant term of the quadratic equations. Even, when just the form of the equationwas changed instead of the structure (e.g., ax 2 +bx=c where a, b, c≠0), 12% of thestudents incorrectly solved the quadratic. Because the constant term was in the rightside, they didn’t perceive that the equation was in standard form (see Figure 3). Inthis type of solution, they were able to find only one of the roots, 4.StatementsQuestion 5To solve the equation“(x-3)∙(x-2) = 0”for real numbers, Alianswered in a singleline that:“x=3 or x=2”Is this answercorrect? If it is correct,how can you show itcorrectness?Students’ types of responses with their reasoningI. II. III. IV.“The answer isRight”Since I wrote(x-3)∙(x-2) = 0as x 2 -5x+6=0and then I factorizeto find roots of it.from (x-3)=0 andfrom (x-2)=0“x=3 and x=2”Table 1: Common examples of students’ types of responses with their reasoning forquestion 5.Although all of the students stated that Ali’s answer was correct by choosing eitherone of the statements I, II, III, and IV, the ways they justified for the correctness ofAli’s solution were different. For instance, in statement I, students first transformedthe factorized expression into the standard form, and then factorized the expressionagain in the same way and found the roots by rote. In statement II, studentsunconsciously applied the null factor law. In statement III, the way of justification forsolution was based on substitution method. In all of these three statements, they couldnot clearly justify the correctness of the solution. In statement IV, students substitutedx=3 into (x-3) and x=2 into (x-2) simultaneously, and concluded that their solution5“The answeris Right”Because(x-3)=0(x-2)=0x=3x=2“Theanswer isRight”.Since wesubstitute“3 and 2”into x, theequation isprovided.(explanationmade onlywith words)“The answer isRight”.If the x=2and x=3 aresubstituted intothe equation(3-3)⋅ (2-2)=00.0 = 0

were correct since 0∙0=0. Namely, they thought that the two x’s stood for differentnumbers.StatementsQuestion 6A student hands in thefollowing work for thefollowing problem.Solve ;x 2 -14x+24=3(x-12)∙(x-2)=3(x-12)∙(x-2)=3∙1x-12=3 x-2=1x=15 x=3Ç.K= {3, 5}Is the student correct?Explain your answer withits reasons?“The answeris Wrong”Because,firstly, 3 mustcarry the leftside of theequation andequalize the0. Then, theotheroperationsmust be done.In this way,the equationx 2 -14x+21=0Students’ types of responses with their reasoningI. II. III. IV.“Theanswer isWrong”.Becausewhen wesubstitute 3and 15 forx, theequation isnotprovided.“The answer isRight”Since the result isequal to 3, weequate 3 rather than0 while factoring it.Therefore, theresult is true.Students againsolve as:“x 2 -14x+24=3”(x-12)∙(x-2)=3(x-12)∙ (x-2)=3∙1x-12=3 x-2=1x=15 x=3(3,1)“The answer isWrong”Since theequations areseparated as(3,1) there is noerror when(x-12)=3however, thereis error when(x-2)=1.It must be(x-2)=3 then,x=5. Therefore,the solution willbe {5, 15}rather than{3, 15}.Table 2: Common examples of students’ types of responses with their reasoning forquestion 6.In statements I and II (see Table 2), students were aware of the error in the solution ofthe given question. However, to explain the reasons for the mistake, they presentedprocedural explanations like the responses in statements I, II, III for question 5 (seeTable 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 0while factoring it”, it can be said that they wrongly tried to transfer the null factor lawto this context. That is, they equated the factors of equation x 2 -14x+24 with theinteger factors of 3. In statement IV, students correctly claimed “the answer of thequestion wrong”; however, their explanations were fully erroneous. Similar tostatement 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 factorsof 3. In both statements III and IV, students did not check whether the roots theyfound were appropriate or not.6

StatementsQuestion 7The solution of thequadratic equation“2 x 2 =3x”is given in the following;According to you, is thissolution correct or not?Explain your answer withits reasons?Solution:I. step 2 x 2 =3xII. step 2∙x∙x=3∙xIII. step 2∙x =3IV. step x= 3/2Ç.K = {3/2}Students’ types of responses with their reasoningI. II. III. IV.“The answer isWrong”Because 3x mustbe carried the leftside of theequation andequalized the 0.Then,2 x 2 =3x2 x 2 - 3x=0x∙(2x-3)=0x=0, x=3/2.“Theanswer isRight”Thesolution isright;however, itmust beadded 0 tothe solutionset.“The answerRight”Because whenwe substitutethe value forx, the equationis satisfied.“The answer isRight”2 x 2 =3x and x 2 isopened.2∙x∙x=3∙xYes the x issimplified.2x=3 so x=3/2.Table 3: Common examples of students’ types of responses with their reasoning forquestion 7.In statement I (see Table 3), students stated that the answer was correct. Theyexplained an appropriate procedure required for solving the equation. Since theymemorized the rule without its reasons, they could only exhibit how the proceduremust be worked. In statement II, on the other hand, students were aware that the rootsof the equation were 0 and 3/2. However, they did not recognize that when wassimultaneously canceled from both sides, the root 0 disappeared. Furthermore, instatement III, the explanation for solution was just based on the substitution method.In statement IV, students incorrectly stated that the answer was right. Like instatement II, students were not aware of the missing root 0 when canceling an in theequationDISCUSSIONThe results indicate that most of the students used the factorization technique to solvequadratic equations. This result supports Bosse and Nandakumar (2005), whoclaimed that a large percentage of the students preferred to apply the factorizationtechniques to find the solutions of quadratic equations. Also, in parallel with theresults of Bossé and Nandakumar (2005) and Kotsopoulos (2007), the result of thisstudy revealed that factoring the quadratic equations was challenging when they werepresented to students in non-standard forms and structures. After looking at theexamples of students’ solutions (see Figures 1, 2, and 3), it can be said that thestudents 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 ifwhat they were doing was mathematically correct. These results give some cluesabout students’ instrumental understanding of solving quadratic equations with oneunknown. However, to make an exact judgment about students’ relational or7

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

Having instrumental understanding does not generally cause trouble for students. It ismuch easier to obtain and use than relational understanding, just because it requiresless knowledge, and with instrumental understanding, students can generally obtainthe right answers more quickly. However, it necessities memorizing, and withoutrelational understanding the learning cannot be adapted to new tasks, and studentscannot give real reasons for their answers (Skemp, 2002). For that reason, greaterattention should be given to how the concept is introduced to reduce the possibility ofstudents learning the subjects/rules/procedures by rote. Any mechanism of solutionmust allow students to understand the meaning of the process that they apply in orderto arrive at the correct answer; otherwise, the mechanism they learn will be a sourceof error (Blanco & Garrote, 2007).RecommendationAs a result of this study, several suggestions can be made to contribute to improvingstudents’ understanding of quadratic equations. Since factoring the quadraticequations was challenging when they are presented in non-standard forms andstructures, it would be better if teachers introduce various kinds of quadraticequations in different structures rather than just in the standard form. On the otherhand, it would be also helpful for students to understand the factorization techniquesas relational when teachers clearly emphasize meaning of the null factor rather thanpresenting it just as rule. In addition, because the students can attribute differentmeanings to the symbols (Küchemann, 1981), their understanding of the meanings ofthe algebraic symbols needs to be taken into account. Therefore, if teachersemphasize the meaning of the algebraic symbols, it would also useful for students tounderstand what the symbols represent in quadratic equations. Moreover, whenteachers encourage students to use different techniques while solving quadraticequations, students’ learning may improve, and they may also gain a conceptuallyunderstanding. 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 learnrelationally. This should be the most important part of teachers’ pedagogical contentknowledge. However, research studies demonstrate a lack of secondary schoolmathematics teachers’ pedagogical content knowledge in this respect (Vaiyavutjamai,Ellerton, & Clements, 2005). Indeed, there is a need to research teachers’ knowledgeabout students’ difficulties concerning quadratic equations.REFERENCESBlanco, L. J., & Garrote, M. (2007). Difficulties in learning inequalities in students ofthe first year of pre-university education in Spain. Eurasia Journal ofMathematics, Science & Technology Education, 3(3), 221-229.Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics:Motivation for more techniques (section A). Teaching Mathematics and itsApplications, 24(4), 143-153.9

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 throughcollege levels. In F. Lester (ed.), Second Handbook of Research on MathematicsTeaching and Learning: A project of the National Council of Teachers ofMathematics. Vol II (pp. 669-705). Charlotte, NC: Information Age Publishing.Kotsopoulos, D. (2007). Unraveling student challenges with quadratics: A cognitiveapproach. Australian Mathematics Teacher, 63(2), 19-24.Küchemann, D. E. (1981). Algebra. In Hart, K., Brown, M. L., Küchemann, D. E.,Kerslake, D., Ruddock, G., & McCartney, M. (Eds.), Children's understanding ofMathematics: 11-16 (pp. 102-119). London: John Murray.Lima, R. N. (2008). Procedural embodiment and quadratic equations. RetrievedApril 1, 2010, from http://tsg.icme11.org/document/get/701.Reason, M. (2003). Relational, instrumental and creative understanding. MathematicsTeaching, 184, 5-7.Skemp, R. R. (1976). Relational understanding and instrumental understanding.Mathematics Teaching, 77, 20-26.Skemp, R. R. (2002). Mathematics in the primary school. London: Routledge Falmer.Sönnerhed, W. W. (2009). Alternative approaches of solving quadratic equations inmathematics teaching: An empirical study of mathematics textbooks and teachingmaterial or Swedish Upper-secondary school. Retrieved April 5, 2010, fromhttp://www.ipd.gu.se/digitalAssets/1272/1272539_plansem_wei.pdf.Taylor, S. E. & Mittag, K. C. (2001). Seven wonders of the ancient and modernquadratic world. Mathematics Teacher, 94, 349-351.Vaiyavutjamai, P., Ellerton, N. F., & Clements, M. A. (2005). Students’ attempts tosolve two elementary quadratic equations: A study in three nations. RetrievedApril 1, 2010, from www.merga.net.au/documents/RP852005.pdf.Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction onstudents’ understanding of quadratic equations. Mathematics Education ResearchJournal, 18(1), 47-77.10