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Teachers’ Knowledge about Students’ Errors in Word Problems at Elementary Mathematics in Bangladesh

andSome Reflections on How to Solve These Issues in

Bangladesh

Khan Md Mazaharul Islam( M110593)

Supervisors: Prof. Baba Takuya Prof. Shimizu Kinya

Prof. Kusakabe Tatsuya

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1. Definition of the terms 2. Background of this research 3. Objectives of this research 4. Guiding questions 5. Literature review 6. Research framework 7. Research method 8. Results of preliminary survey 9. Findings of preliminary survey 10. Results of final survey 11. Discussion and conclusion１ 2 ． Recommendations

Outlines of the Presentation

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1 Definition of terms

1.1 Mathematical knowledge for teaching: Knowing mathematics from the perspective of helping others learn it, and being mathematically ready to teach an idea, method, or other aspect of maths (Ball, 2011).

1.2 Word Problem: By ‘’word problem” is meant a text(typically containing quantitative information)that describes a situation assumed familiar to the reader and poses a quantitative question, an answer to which can be derived by mathematical operations performed on the data provided in the text, or otherwise inferred (G.C. Leder, E. Pehkonen & G.Torner,2002).

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2 Background

Word problem solving is one of the mathematical school tasks most practiced around the world (Hiebert et al., 2003: Stigler & Hiebert, 1997). Teachers’ have to methodically analyse students’ errors, explain what they consist in, and find what causes them. In order to achieve that, teachers need certain knowledge about errors and the method of response to errors (Maria, 2008).

In Bangladesh, according to the few survey results, pupils’ are weak in solving problems in the real life, in the four fundamental rules and in the measurement (Uddin-2005).

To my knowledge no research have been done on Teachers’ Knowledge about Students’ Errors in Mathematical Word Problems at Elementary Level in Bangladesh.

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3 Objectives of this Research

1.To find out what kinds of errors Bangladeshi students make in word problems at elementary mathematics.

2.To identify Bangladeshi elementary teachers' knowledge regarding 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 by Bangladeshi elementary teachers to help students who make errors in word problems in mathematics.

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4 Guiding Questions

1.What errors do Bangladeshi students’ make in word problems in mathematics?

2. What are the Bangladeshi elementary teachers' knowledge about students' errors in word problems in mathematics, as well as on the reasons why students make such errors?

3. What teaching strategies are recognized as useful by Bangladeshi elementary teachers to help students who make errors in word problems in mathematics?

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5 Literature Review

• Errors can be used as a motivational device and as a starting point for creative mathematical explorations, involving valuable problem solving and problem posing activities (Borasi, 1997).

• An error might teach a lot both students and teachers if it evokes a reflection (Maria, L, 2008).

• The importance of teaching students how to understand diagrams and consider the use of diagrams as a communication tool has been stressed in different national curricula (NCTM, 2000).

• A significantly higher percentage of students in New Zealand obtained the correct answer in word problems by using diagrams, versus the Japanese students (Uesaka et al., 2007).

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5 Literature Review (cont..)

Common Content Knowledge(CCK)

Horizon Content Knowledge (HCK)

Specialized content Knowledge (SCK)

Knowledge of the Content and Student (KCS)

Knowledge of Content and Teaching(KCT

)

Knowledge of Content and Curriculum (KCC)

Subject Matter Knowledge

Pedagogical Content Knowledge

Framework of Mathematical Knowledge for Teaching(MKT)

(Ball,2011)

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5 Literature Review (cont..)

• KCS is ‘knowing that combines knowledge about students and knowing about mathematics’ (Ball et al.,2008, p.401). Familiarity with common errors and deciding which of several errors students are most likely to make are examples of Knowledge of Content and Student (KCS).

• KCT is ‘knowing that combines knowledge about teaching and knowing about mathematics’ (Ball et al.,2008, p.401). Many of the mathematical task of teaching require a mathematical knowledge of the design of instruction (Ball et al.,2008, p.401).

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6 Conceptual framework

Knowledge of Content

and Students Students’

Errors in Word

Problems in Mathematics

Teachers 'Pedagogical Content Knowledge (PCK)

Knowledge of Content

and Teaching

Summary information about survey sample

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Location:Preliminary survey: 2 government primary schools from two up-zilas’ of Mymensingh district in Bangladesh.Final survey: Four primary schools(3 government & 1 non-government reg.) from four up-zilas’ of Mymensingh district 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

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7 Research Method

Preliminary survey

Data analysis and revise tools

Final survey

One rural & one urban

schools

Data analysis

Summarize the findings

Discussion and conclusion

• Students’ test• Teachers

questionnaire

• Students’ test• Students’ interview• Teachers

questionnaire• Teachers’ interview

Three government &

one non-government

schools.

Recommendations

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7 Research Method (cont…)Tools Preliminary

/FinalQuestions/Items Analysis procedures

1 Students’ test Preliminary

10 questions (4 simple calculation, 2 one-stage and 4 two-stage word problems)

Rubric (based on Charles, Lester and O’Daffer, 1987))

2 Teachers’ questionnaire

Preliminary

6 questions based on 2 word problems of students’ test.

Cross thematic analysis (based on Miles and Huberman, 1994)

3 Students’ test Final 4 questions (same 4 two-stage word problems of preliminary survey)

Rubric (based on Charles, Lester and O’Daffer, 1987)

4 Students’ interview

Final 4 main questions Based on Newman’ error analysis

5 Teachers’ questionnaire

Final 6 questions based on 2 word problems of students’ test

Cross thematic analysis (based on Miles and Huberman, 1994)

6 Teachers’ interview

Final 3 main questions Cross thematic analysis (based on Miles and Huberman, 1994)

8.1 Results of Preliminary SurveyStudents’ correct answers/errors among the questions

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  Simple Calculation

 

Word problems

Question no.

  Q1 (91+82)

Q2 (415-213)

   √ ×  √ ×  √ ×  √ ×

 √         63      

 ×         1 5    

 √             57  

 ×             7 5

Q7(13+11)+(37+11)

 √ 30              

× 36 3            

Q8575+(575-170)

 √ 31   31          

× 35 3 36 2        

 √     36   36      

 ×     31 2 28 5    

 √ 36           36  

× 30 3         28 5

8.2 Results of preliminary survey

Q C1 C2 C3 C4

Q7 0 21 18 30

Q8 2 26 10 31

Q9 2 29 2 36

Q10 0 21 12 36

Q- 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)

Analysis of students test items by rubric

Q 7 Q8 Q9 Q100

10

20

30

40

50

60

70

80

Interpretation of students’ answerC1 C2 C3 C4

Num

ber

of s

tude

nts

Question

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8.3 Results of preliminary surveyTeachers’ Questionnaire

Types of errors Teacher (T) Students’ errors

T1 T2 T3 T4Spontaneous x x   11 students

Carelessness x x x  8 students

Linguistic   x   x x 14 students

Calculation x x 3 students

Causes        Lack of language comprehension skills x x x xLack of carefulness     x  Lack of calculation skills x x   x

Solution        Drill practice x x x  Follow the teacher x   x  

Group discussion x x   x

Using pictures or diagrams x x x x

Identify causes        x

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9 Findings from preliminary survey

Students are skilled in simple calculation and one-stage word problems but most of them made errors in word problems

Different categories of errors are found in students’ answers, which are known by surveyed teachers.

Teachers’ mentioned causes of students errors are lack of language comprehension skills, lack of carelessness and lack of calculation skills

As solutions of the students’ errors, teachers’ knowledge are using pictures and diagrams, drilling practice, group discussion, explain the problems with real life examples 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 and conducted final survey

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Q C1 C2

C3 C4

Schools Schools Schools Schools

A B C D A B C D A B C D A B C D

1 10 8 11 13 5 13 7 5 16 15 12 9

2 2 13 11 12 12 5 12 4 3 11 13 14 12

3 13 6 17 11 4 10 3 4 14 20 10 12

4 1 1 9 12 6 7 12 7 1 7 10 16 14 12

10.1 .1 Analysis of students test item by Rubric

10 Results of final survey

Q- 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 View

Interpretation of students’ answers

Y- No. of students regarding questionsX- Schools (A, B, C, & D)

X

Y

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C2Wrong answer (misinterpreted major part of the problem)

Daughter’s age 13 years Father’s age 37 years

Total age 50 years Their number of age 11

years

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C3Examples of partial

answer (misinterpreted minor part of the problem)

Daughter’s age=13 yearsAfter 11 years daughter age=(13+11)years

24 yearsFather’s age=37 years

After 11 years father’s age=(37+11)years =48 years

10.1.2 Examples of students answer

C4Examples of

correct solution (No error made)

Daughter’s age 13 yearsAfter 11 years daughter’s age will be

(13+11)years=24 yearsFather’s age 37 years

After 11 years father’s age will be (37+11)years=48 years

After 11 years their total age will be (48+24)years=72 years

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10.2.1 Students’ Interview

No. of Stude

nts

Q. No.

Frequency of failure on problem solving level No. of students with errors

    Reading

Understanding 

Process (Can say

how to get the

answer)

Explain the solution with

No. of students

with errors

Words diagrams

36 1 3 20 20 24 32 20

4 3 22 22 26 32 22

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Performance Frequency of failure on solving word problems

  School-A (Govt.)

School-B(Govt.)

School-c(Govt.) School-D (Reg.)

Difficult key words

      

Reading

Understanding

Process

Reading

Understanding

Process

Reading

Understanding

Process

Reading

Understanding

Process

High - 1 1 - - - - 1 1 - - - After, together

Medium 1 1 1 - 2 2 - 2 2 - 2 2 After, together,their total number

Low - 2 2 1 3 3 1 3 3 - 3 3 After, together,their total number

10.2.2 Students’ interview

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10.2.3 Example of students answer with pictures in Q1

Daughter III I

Father IIIIIII

I

24 years

48

72

24 +48

72 Answer: 72

years

The daughter is 13 years old and the father is 37 years old. What will be their total number of ages after 11 years?

Types of errors Teachers’ Knowledge about Students’ errors Teacher(T) Errors found in students’ test

T1

T2

T3

T4

T5

T6

T7

T8

Linguistic Some students will answer only 13+11=24 and 37+11=48.

x x x x x x x x 21 students

Some students can do (13+ 37)-11=39. x 9 students

Spontaneous Some students will make common error as 13+37+11=61

x x x x x x x x 8 students

Calculation Some students will make mistake in doing addition with carrying.

x x x 6 students

Carelessness Some students will make different mistake x x x x x x x x 28 students

Causes Students’ interview

Lack of language comprehension skills

Some student can’t understand the language x x x x x x x x 20 out of 36

Lack of carefulness Some students will not carefully and attentively.

x x x x

Curriculum and textbook

Due to evaluation system Students can’t use different solution process beside the textbook.

x x x x .

Lack of calculation skills

Some students have problem in doing addition.

x x x

SolutionDrill Practice Ask the students to read the problem

attentivelyx x x x x x x x

picture/diagram Help them to understand the problem with picture/diagram

x x x x x x x x 4 students use picture in the interview

Explain with example Help them to understand the problem with examples

x x x x

Key word explanation Explain the key words with examples x x x x

10.3.1Teachers’ questionnaire

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10.4.1 Summary of the teachers’ responses to the interview

 Types of students’ difficulties in teaching word problems

Teacher Problems found in

Students’ interview

T1 T2 T3 T4 T5 T6 T7 T8

Linguistic x x x x x x x x 24 students

Carelessness x  x    x        

Spontaneous     x    x x  x  

Calculation             x  x 6 students

Causes                

Lack of language comprehension skill x x x x x x x x 24 students

Lack of carefulness x   x x x   x  

Lack of calculation skills   x     x     x

Curriculum and textbook    x x  x x  

Solution                

Drill practice   x x x x   x x

pictures/diagrams x x x x   x x x

Group discussion   x   x   x    

Key words explanation     x          

Explain the problems with examples  x       x x x x

Identify common problems and discuss the solution

        x      ‘X’= in the Colum indicate the category of the row

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9.5.2Teachers’ knowledge about students’ errors

T1 T7

Questionnaire Interview Questionnaire Interview

Errors -Linguistic-Carelessness-Spontaneous

-Linguistic-Carelessness

-Linguistic-Calculation-Spontaneous

-Linguistic-Calculation-Spontaneous

Reasons

-Lack of language comprehension skill-Lack of carefulness

-Lack of language comprehension skill-Lack of carefulness

-Lack of language comprehension skill-Lack of carefulness-Lack of calculation skill

-Lack of language comprehension -Lack of carefulness

Solution

-Drill practice-Picture/diagrams-Explain with examples

-Picture/diagram-Explain with examples

-Drill practice-Picture/diagrams-Key word explanation

-Drawing picture/diagram-Group discussion-Explain with examples

10.5.1 Logical relation of teachers’ knowledge about students’ errors

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Teachers’ questionnaireErrors- Linguistic-Carelessness-Spontaneous-CalculationCauses- Lack of language comprehension skill-Lack of carelessness-Lack of calculation skill-Due to curriculum & textbooks

10.5.2 Relation among the findings

Teachers’ interviewDifficulties-Linguistic-Carelessness-Spontaneous-Calculation-Curriculum & textbooksCauses-Lack of language comprehension skill-Lack of carelessness-Lack of calculation skill-Due to curriculum & textbooks

Students’ test-Linguistic-Carelessness-Spontaneous-Calculation

Students’ interview-Reading-Understanding-Processing

Most prominent error-Linguistic

Most prominent cause-Lack of language comprehension skill

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11 Discussion and conclusion

The majority of the students made different type of errors in solving word problems.Students’ misinterpretation major part of the word problem is found prominent.Teachers’ knowledge about students errors are linguistic, spontaneous carelessness and calculation, but most common one is linguistic errors.All of the teachers have the knowledge that the language would be the main reason to commit errors in word problems. Students’ test and interview findings supported 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 word problems understandable to the students (4 Ts). Most of the teachers have the knowledge that they will adopt different solution strategies. But most prominent one is drawing pictures/diagrams. Neither of the teachers were specified nor categorized students errors and apply proper attempts according to the errors.

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12　Recommendations

• Students’ different kinds of mathematical errors and how to deal with these errors should be incorporated in the teacher education curriculum.

• Students’ mathematical errors and how to deal with errors should be incorporated in the primary teachers’ guide of Bangladesh in order to help and encourage teachers to use students’ errors as spring board for inquiry.

• Different solution techniques should be included as examples in the mathematics textbooks including solution with diagrams.

• Mathematics curriculum of teacher education should be revised in order stresses the importance of teaching mathematical problems with diagrams.

• Pictures, diagram, number line should be assimilated in the solution process of the mathematical problems in the textbook in order to make the word problems more interesting and easy understandable for the students.

ReferencesBall, D. L., & Forzani, F. M. (2011, summer). Building a common core for 　　 learning 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 Makes

it Special? Journal of Teacher Education. 59(5), 389-407. Sage Publications

Borasi, R. (1987). Exploring mathematics through the analysis of errors. For the Learning of

Mathematics, 7(3), 2-8

Charles, Randall, Lester, Frank and O‘Daffer, Phares, (1987). How to Evaluate Progress in 　　　 Problem Solving. Reston, VA: National Council of Teachers of Mathematics, 1987.

G.C. Leder, E. Pehkonen, & Torner, (2002), Beliefs: A Hidden Variable in

Mathematics Education? 271-292 2002 Kluwer Academic Publishers. Printed in the

Netherlands.

JICA, (2009) Study on Bangladesh Primary Mathematics & Science Curriculum and

Textbooks

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the Teaching-Research Process, Pedagogical University of Krakow (Poland)

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International Educational and Professional Publisher

Thousand Oaks London New Delhi

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ReferencesNCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Newman, M.A. (1977). An analysis of sixth-grade pupils’ errors on written mathematical tasks. In M. A. Clements & j. Foyster (Eds.), Research in mathematics education in Australia, 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 learning in science education. In D. F. Treagust, R. Duit, & Fraser (Ed.), Improving teaching and learning in science and mathematics. (pp. 1-14). College press, New York.Uddin Md, Mohshin,(2005).The Impact of Competency-Based Curriculum on Student Achievement at Primary School BangladeshUesaka, Y.,Manalo, E., & Ichikawa, s. (2007). What kinds of perceptions and daily learning behaviors promote students’ use of diagrams in mathematics problem solving? Learning and Instruction, v17 n3 p322-335 Jun 2007.