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# Teachers’ Knowledge about Students’ Errors in Word Problems at Elementary Mathematics in Bangladesh and Some Reflections on How to Solve These Issues in Bangladesh

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### Teachers’ Knowledge about Students’ Errors in Word Problems at Elementary Mathematics in Bangladesh and Some Reflections on How to Solve These Issues in Bangladesh

1. 1. Khan Md Mazaharul Islam( M110593)Supervisors: Prof. Baba TakuyaProf. Shimizu KinyaProf. Kusakabe Tatsuya1
2. 2. 1. Definition of the terms2. Background of this research3. Objectives of this research4. Guiding questions5. Literature review6. Research framework7. Research method8. Results of preliminary survey9. Findings of preliminary survey10. Results of final survey11. Discussion and conclusion１2．RecommendationsOutlines of the Presentation2
3. 3. 1.1 Mathematical knowledge for teaching: Knowingmathematics from the perspective of helping others learn it, andbeing mathematically ready to teach an idea, method, or otheraspect of maths (Ball, 2011).1.2 Word Problem: By ‘’word problem” is meant a text(typicallycontaining quantitative information)that describes a situationassumed familiar to the reader and poses a quantitative question,an answer to which can be derived by mathematical operationsperformed on the data provided in the text, or otherwise inferred(G.C. Leder, E. Pehkonen & G.Torner,2002).3
4. 4. 2 BackgroundWord problem solving is one of the mathematical school tasks mostpracticed around the world (Hiebert et al., 2003: Stigler & Hiebert,1997).Teachers’ have to methodically analyse students’ errors, explain whatthey consist in, and find what causes them. In order to achieve that,teachers need certain knowledge about errors and the method of responseto errors (Maria, 2008).In Bangladesh, according to the few survey results, pupils’ are weak insolving problems in the real life, in the four fundamental rules and in themeasurement (Uddin-2005).To my knowledge no research have been done on Teachers’Knowledge about Students’ Errors in Mathematical Word Problems atElementary Level in Bangladesh.4
5. 5. 1.To find out what kinds of errors Bangladeshi studentsmake in word problems at elementary mathematics.2.To identify Bangladeshi elementary teachers knowledgeregarding students errors in word problems in mathematics,as well as the reasons why students make such errors.3. To analyze teaching strategies recognized as useful byBangladeshi elementary teachers to help studentswho make errors in word problems in mathematics.5
6. 6. 4 Guiding Questions1.What errors do Bangladeshi students’ make in wordproblems in mathematics?2. What are the Bangladeshi elementary teachersknowledge about students errors in word problems inmathematics, as well as on the reasons why studentsmake such errors?3. What teaching strategies are recognized as useful byBangladeshi elementary teachers to help students whomake errors in word problems in mathematics?6
7. 7. • Errors can be used as a motivational device and as a startingpoint for creative mathematical explorations, involving valuableproblem solving and problem posing activities (Borasi, 1997).• An error might teach a lot both students and teachers if itevokes a reflection (Maria, L, 2008).• The importance of teaching students how to understanddiagrams and consider the use of diagrams as a communicationtool has been stressed in different national curricula (NCTM,2000).• A significantly higher percentage of students in New Zealandobtained the correct answer in word problems by usingdiagrams, versus the Japanese students (Uesaka et al., 2007).7
8. 8. 5 Literature Review (cont..)CommonContentKnowledge(CCK)Horizon ContentKnowledge (HCK)SpecializedcontentKnowledge(SCK)Knowledge of theContent andStudent (KCS)Knowledge ofContent andTeaching(KCT)Knowledgeof Content andCurriculum (KCC)Subject Matter Knowledge Pedagogical Content KnowledgeFramework of Mathematical Knowledge for Teaching(MKT)(Ball,2011)8
9. 9. • KCS is ‘knowing that combines knowledge about students andknowing about mathematics’ (Ball et al.,2008, p.401).Familiarity with common errors and deciding which of severalerrors students are most likely to make are examples ofKnowledge of Content and Student (KCS).• KCT is ‘knowing that combines knowledge about teaching andknowing about mathematics’ (Ball et al.,2008, p.401). Many ofthe mathematical task of teaching require a mathematicalknowledge of the design of instruction (Ball et al.,2008, p.401).9
10. 10. Knowledge ofContent andStudentsStudents’ Errorsin WordProblems inMathematicsTeachers PedagogicalContent Knowledge (PCK)Knowledge ofContent andTeaching10
11. 11. 11Location:Preliminary survey: 2 governmentprimary schools from two up-zilas’ ofMymensingh district in Bangladesh.Final survey: Four primary schools(3government & 1 non-government reg.)from four up-zilas’ of Mymensinghdistrict in Bangladesh.Date : 15 to 17 February’ 2012(preliminary survey): 22 February to 22 march’ 2012(final)Grade ：３Students: 69 (36+33) (preliminary):124 （ ３ 6 ＋ ３ 1+30+27)(final)Teachers:Preliminary:4 (3 females & 1 male)Final: :8 (6 females & 2 males)Mymensingh
12. 12. Preliminary surveyData analysis andrevise toolsFinal surveyOne rural & oneurban schoolsData analysisSummarize thefindingsDiscussion andconclusion• Students’ test• Teachers questionnaire• Students’ test• Students’ interview• Teachers questionnaire• Teachers’ interviewThree government& one non-governmentschools.Recommendations12
13. 13. 7 Research Method (cont…)Tools Preliminary/FinalQuestions/Items Analysis procedures1 Students’ test Preliminary 10 questions (4 simplecalculation, 2 one-stageand 4 two-stage wordproblems)Rubric (based on Charles,Lester and O’Daffer,1987))2 Teachers’questionnairePreliminary 6 questions based on 2word problems ofstudents’ test.Cross thematic analysis(based on Miles andHuberman, 1994)3 Students’ test Final 4 questions (same 4 two-stage word problems ofpreliminary survey)Rubric (based on Charles,Lester and O’Daffer, 1987)4 Students’interviewFinal 4 main questions Based on Newman’ erroranalysis5 Teachers’questionnaireFinal 6 questions based on 2word problems ofstudents’ testCross thematic analysis(based on Miles andHuberman, 1994)6 Teachers’interviewFinal 3 main questions Cross thematic analysis(based on Miles andHuberman, 1994)13
14. 14. Students’ correct answers/errors among the questions14Simple CalculationWordproblemsQuestion no.Q1(91+82)Q2 (415-213)√ × √ × √ × √ ×√ 63× 1 5√ 57× 7 5Q7(13+11)+(37+11)√ 30× 36 3Q8575+(575-170)√ 31 31× 35 3 36 2√ 36 36× 31 2 28 5√ 36 36× 30 3 28 5
15. 15. Q C1 C2 C3 C4Q7 0 21 18 30Q8 2 26 10 31Q9 2 29 2 36Q10 0 21 12 36Q- QuestionC1-No attemptC2-Misinterprets major part of the problemC3-Misinterprets minor part of the problemC4-Complete understanding of the problem(Rubric designed based on Charles, Lester and O’Daffer, 1987)01020304050607080Q 7 Q8 Q9 Q10NumberofstudentsInterpretation of students’ answerC1 C2 C3 C4Question
16. 16. 16Types of errors Teacher (T) Students’ errorsT1 T2 T3 T4Spontaneous x x 11 studentsCarelessness x x x 8 studentsLinguistic x x x 14 studentsCalculation x x 3 studentsCausesLack of language comprehension skills x x x xLack of carefulness xLack of calculation skills x x xSolutionDrill practice x x xFollow the teacher x xGroup discussion x x xUsing pictures or diagrams x x x xIdentify causes x
17. 17. 9 Findings from preliminary survey Students are skilled in simple calculation and one-stage word problems butmost of them made errors in word problems Different categories of errors are found in students’ answers, which are knownby surveyed teachers. Teachers’ mentioned causes of students errors are lack of languagecomprehension skills, lack of carelessness and lack of calculation skills As solutions of the students’ errors, teachers’ knowledge are using pictures anddiagrams, drilling practice, group discussion, explain the problems with real lifeexamples and key word explanation. There is no remarkable differences of students’ performance and teachers’knowledge between rural and urban area schools.Based on the above findings I revised the tools andconducted final survey17
18. 18. Q C1 C2 C3 C4Schools Schools Schools SchoolsA B C D A B C D A B C D A B C D1 10 8 11 13 5 13 7 5 16 15 12 92 2 13 11 12 12 5 12 4 3 11 13 14 123 13 6 17 11 4 10 3 4 14 20 10 124 1 1 9 12 6 7 12 7 1 7 10 16 14 1210.1 .1 Analysis of students test item by Rubric10 Results of final surveyQ- QuestionC1-No attemptC2-Misinterprets major part of the problemC3-Misinterprets minor part of the problemC4-Complete understanding of the problem(Rubric designed based on Charles, Lester and O’Daffer, 1987)Graphical View18
19. 19. Interpretation of students’ answersY- No. of students regarding questionsX- Schools (A, B, C, & D)XY19
20. 20. C2Wrong answer(misinterpreted majorpart of the problem)Daughter’s age 13 yearsFather’s age 37 yearsTotal age 50 yearsTheir number of age 11 years39C3Examples of partialanswer (misinterpretedminor part of theproblem)Daughter’s age=13 yearsAfter 11 years daughter age=(13+11)years24 yearsFather’s age=37 yearsAfter 11 years father’s age=(37+11)years=48 yearsC4Examples of correctsolution (No errormade)Daughter’s age 13 yearsAfter 11 years daughter’s age will be(13+11)years=24 yearsFather’s age 37 yearsAfter 11 years father’s age will be (37+11)years=48yearsAfter 11 years their total age will be(48+24)years=72 years20
21. 21. 10.2.1 Students’ InterviewNo. ofStudentsQ.No.Frequency of failure on problem solving level No. ofstudentswitherrorsReading Understanding Process (Cansay how toget theanswer)Explain the solutionwithNo. ofstudentswitherrorsWords diagrams36 1 3 20 20 24 32 204 3 22 22 26 32 2221
22. 22. Performance Frequency of failure on solving word problemsSchool-A(Govt.)School-B(Govt.) School-c(Govt.) School-D (Reg.) Difficult keywordsReadingUnderstandingProcessReadingUnderstandingProcessReadingUnderstandingProcessReadingUnderstandingProcessHigh - 1 1 - - - - 1 1 - - - After,togetherMedium 1 1 1 - 2 2 - 2 2 - 2 2 After,together,their totalnumberLow - 2 2 1 3 3 1 3 3 - 3 3 After,together,their totalnumber10.2.2 Students’ interview22
23. 23. 10.2.3 Example of students answer with pictures in Q1Daughter IIIIFather IIIIIIII24years487224+4872Answer: 72 yearsThe daughter is 13 years old and the father is 37years old. What will be their total number of agesafter 11 years?23
24. 24. Types of errors Teachers’ Knowledge about Students’ errors Teacher(T) Errors found instudents’ testT1T2T3T4T5T6T7T8Linguistic Some students will answer only 13+11=24 and 37+11=48. x x x x x x x x 21 studentsSome students can do (13+ 37)-11=39. x 9 studentsSpontaneous Some students will make common error as13+37+11=61x x x x x x x x 8 studentsCalculation Some students will make mistake in doing additionwith carrying.x x x 6 studentsCarelessness Some students will make different mistake x x x x x x x x 28 studentsCauses Students’ interviewLack of languagecomprehension skillsSome student can’t understand the language x x x x x x x x 20 out of 36Lack of carefulness Some students will not carefully and attentively. x x x xCurriculum and textbook Due to evaluation system Students can’t use differentsolution process beside the textbook.x x x x .Lack of calculation skills Some students have problem in doing addition. x x xSolutionDrill Practice Ask the students to read the problem attentively x x x x x x x xpicture/diagram Help them to understand the problem withpicture/diagramx x x x x x x x 4 students usepicture in theinterviewExplain with example Help them to understand the problem with examples x x x xKey word explanation Explain the key words with examples x x x x10.3.1Teachers’ questionnaire24
25. 25. 10.4.1 Summary of the teachers’ responses to the interviewTypes of students’ difficulties in teaching wordproblemsTeacher Problems foundin Students’interviewT1 T2 T3 T4 T5 T6 T7 T8Linguistic x x x x x x x x 24 studentsCarelessness x x xSpontaneous x x x xCalculation x x 6 studentsCausesLack of language comprehension skill x x x x x x x x 24 studentsLack of carefulness x x x x xLack of calculation skills x x xCurriculum and textbook x x x xSolutionDrill practice x x x x x xpictures/diagrams x x x x x x xGroup discussion x x xKey words explanation xExplain the problems with examples x x x x xIdentify common problems and discuss the solution x‘X’= in the Colum indicate the category of the row25
26. 26. 9.5.2Teachers’ knowledge about students’ errorsT1 T7Questionnaire Interview Questionnaire InterviewErrors -Linguistic-Carelessness-Spontaneous-Linguistic-Carelessness-Linguistic-Calculation-Spontaneous-Linguistic-Calculation-SpontaneousReasons -Lack of languagecomprehensionskill-Lack ofcarefulness-Lack of languagecomprehensionskill-Lack ofcarefulness-Lack of languagecomprehensionskill-Lack ofcarefulness-Lack of calculationskill-Lack of languagecomprehension-Lack ofcarefulnessSolution -Drill practice-Picture/diagrams-Explain withexamples-Picture/diagram-Explain withexamples-Drill practice-Picture/diagrams-Key wordexplanation-Drawingpicture/diagram-Group discussion-Explain withexamples2610.5.1 Logical relation of teachers’ knowledge about students’ errors
27. 27. Teachers’ questionnaireErrors- Linguistic-Carelessness-Spontaneous-CalculationCauses- Lack of language comprehension skill-Lack of carelessness-Lack of calculation skill-Due to curriculum & textbooks10.5.2 Relation among the findingsTeachers’ interviewDifficulties-Linguistic-Carelessness-Spontaneous-Calculation-Curriculum & textbooksCauses-Lack of language comprehension skill-Lack of carelessness-Lack of calculation skill-Due to curriculum & textbooksStudents’ test-Linguistic-Carelessness-Spontaneous-CalculationStudents’ interview-Reading-Understanding-ProcessingMost prominent error-LinguisticMost prominent cause-Lack of languagecomprehension skill27
28. 28.  The majority of the students made different type of errors in solving wordproblems.Students’ misinterpretation major part of the word problem is found prominent.Teachers’ knowledge about students errors are linguistic, spontaneouscarelessness and calculation, but most common one is linguistic errors.All of the teachers have the knowledge that the language would be the mainreason to commit errors in word problems. Students’ test and interview findingssupported this condition.Some of the teachers presented their knowledge very logically in favor of errors-reasons-solution( T1, T3, T4, T6).Following textbook is one of the problem for the teachers to make the wordproblems understandable to the students (4 Ts). Most of the teachers have the knowledge that they will adopt different solutionstrategies. But most prominent one is drawing pictures/diagrams.Neither of the teachers were specified nor categorized students errors and applyproper attempts according to the errors.28
29. 29. • Students’ different kinds of mathematical errors and how to deal with theseerrors should be incorporated in the teacher education curriculum.• Students’ mathematical errors and how to deal with errors should beincorporated in the primary teachers’ guide of Bangladesh in order to helpand encourage teachers to use students’ errors as spring board for inquiry.• Different solution techniques should be included as examples in themathematics textbooks including solution with diagrams.• Mathematics curriculum of teacher education should be revised in orderstresses the importance of teaching mathematical problems with diagrams.• Pictures, diagram, number line should be assimilated in the solution processof the mathematical problems in the textbook in order to make the wordproblems more interesting and easy understandable for the students.29
30. 30. ReferencesBall, D. L., & Forzani, F. M. (2011, summer). Building a common core forlearning to teach, and connecting professional learning to practice.American Educators, 35 (2), 17-21, 38-39.Ball, D. L. Thames, M. H. and Phelps, G. (2008). Content Knowledge for Teaching; What Makesit Special? Journal of Teacher Education. 59(5), 389-407. Sage PublicationsBorasi, R. (1987). Exploring mathematics through the analysis of errors. For the Learning ofMathematics, 7(3), 2-8Charles, Randall, Lester, Frank and O‘Daffer, Phares, (1987). How to Evaluate Progress inProblem Solving. Reston, VA: National Council of Teachers of Mathematics, 1987.G.C. Leder, E. Pehkonen, & Torner, (2002), Beliefs: A Hidden Variable inMathematics Education? 271-292 2002 Kluwer Academic Publishers. Printed in theNetherlands.JICA, (2009) Study on Bangladesh Primary Mathematics & Science Curriculum andTextbooksMaria, L (2008.) An Analysis of Students’ Mathematical Errors inthe Teaching-Research Process, Pedagogical University of Krakow (Poland)Miles, M. B., & Huberman, A. M. (1994). Qualitative Data AnalysisSecond edition, SAGE PublicationsInternational Educational and Professional PublisherThousand Oaks London New Delhi30
31. 31. ReferencesNCTM. (2000). Principles and standards for school mathematics. Reston, VA: NationalCouncil of Teachers of Mathematics.Newman, M.A. (1977). An analysis of sixth-grade pupils’ errors on written mathematicaltasks. In M. A. Clements & j. Foyster (Eds.), Research in mathematics education inAustralia, 19977 (Volume 1, pp. 239-258). Melbourne: Swinburne Press.Treagust, D. f., Duit, R., & Fraser, B. J. (1996). Overview: Research on students’preinstructional conceptions- The driving force for improving teaching and learningin science education. In D. F. Treagust, R. Duit, & Fraser (Ed.), Improving teachingand learning in science and mathematics. (pp. 1-14). College press, New York.Uddin Md, Mohshin,(2005).The Impact of Competency-Based Curriculum on StudentAchievement at Primary School BangladeshUesaka, Y.,Manalo, E., & Ichikawa, s. (2007). What kinds of perceptions and daily learningbehaviors promote students’ use of diagrams in mathematics problem solving? Learningand Instruction, v17 n3 p322-335 Jun 2007.31