Analisis Kesilapan NewmanNewman’s Error Analisys

HM 2011

pengenalan

• 1.1 Difficulties in Problem Solving• • Look at the following mathematical problems:• There are 5 rows of chairs. If every row has 8 chairs, how

many chairs are there altogether?

• What number should be added to 25 in order to get a sum of 38?

• George has 8 less apples than Ali. If George has 23 apples, how many apples does Ali have?

• These problems are normally called word problems in mathematics. Very often, your students fail to solve this type of problems. You may have wondered why?

• The procedure for diagnosing difficulties experienced by students when attempting to solve word problems described in this section will hopefully enlighten you on this.

• According to Anne Newman (1983), when solving a mathematical word problem, students will need to go through five hierarchical stages:

• • Reading the problem;(Reading)• Comprehend what is read; (Comprehension)• Carry out a mental transformation from the words of the question to

the selection of an appropriate mathematical strategy; (Transformation)

• Apply the process skills demanded by the selected strategy; (Process Skills)

• Encode the answer in an acceptable written form. (Encoding)•

CONCRETEStories in Real

Setting

ABSTRACTMathematical

Symbols

5. Encoding

4. Process Skills

3. Transformation

1. Reading

2. Comprehension

Figure 1. Stages of solving a word problem in Mathematics.

• The diagram in Figure 1 summarizes the above stages. • When a word problem is given to students in the

form of realistic settings, students must first read the problem, comprehend its meaning and then transform the problem into appropriate mathematical form which is abstract in nature. Students then need to perform the relevant mathematical skills to find the solution in mathematical symbols. Finally, the solution needs to be encoded into a suitable form that can be correctly interpreted in realistic settings.

• According to Newman (1983), students can make errors in any of these five stages when solving a word problem. Many studies had been carried out to investigate these errors. Table 1 shows the results of five such studies

Error Type Clarkson(Papua

New Guinea, 1983)

(n = 1851 errors)

%

Kaushil et al.

(India, 1985)

(n = 329 errors)

%

Marinas & Clements (Penang,

1990)(n = 382 errors)

%

Singhatat (Thailand,

1991)(n = 220 errors)

%

Gan (Miri, 1993)

(n = 685 errors)

%

Reading 12 0 0 0 5

Comprehension 21 24 45 60 41

Transformation 23 35 26 8 26

Process Skills 31 16 8 15 11

Encoding 1 6 0 0 3

Carelessness 12 18 21 16 14

• According to Newman (1983), errors can also occur because of carelessness or lack of motivation. When motivation is lacking, students normally do not bother to pay much attention in answering the questions or may just simply write down any answer. The errors based on these two factors can happen at any stage as shown in Figure 2.

Reading

Comprehension

Transformation

Process skills

Encoding

Careless mistakes

Motivation

Figure 2. Hierarchy in Newman’s Error Analysis Procedure and types of initial errors.

Newman’s Error Analysis Procedure

• Newman developed a diagnostic method using interview as an instrument to identify the categories of initial errors that happen when solving word problems. Briefly, this diagnostic method involves the following steps:

• Students take a test on solving word problems.• Teacher marks students’ answers.• Teacher carries out a diagnostic interview with each at-risk

student.• Teacher analyses the interview results.• Teacher draws up and interprets a diagnostic profile for each at-

risk student.

The interview• The main purpose of the diagnostic interview is to identify, for each

incorrectly answered item from the word-problem test, the category of initial errors made by a particular at-risk student. When carrying out the interview, you need to take note of the following points:

• The interview is conducted separately with each at-risk student.• The interview should be carried out as soon as possible after the word-

problem test.• Make sure the interview is conducted with the minimum of distracting

noises and interruptions.• During the interview, do not assist the student in any way. Remember

that the purpose of your interview is NOT to teach your students, but to identify their difficulties.

• In order to conduct the diagnostic interview, you should give a new set of the word-problem test to the at-risk student while you keep the old set for reference.

• Then, you need to go through the five categories of interview questions with the student, starting from the first category (Reading) until the last category (Encoding).

• For a particular test question, the interview will be stopped once the first error is identified and the error category is determined.

• The five categories of interview questions are described in the following paragraphs.

(a) Category 1: Reading • Instructions/Questions forwarded to the student during interview:

• “Please read the question to me. If you don’t know a word or number leave it out.”

• Student who makes mistakes at this stage cannot recognize (pronounce) or read the terms, symbols or words in the question.

• Examples of errors at the ‘Reading’ category:

• A student is asked to read the question: “There are 238 men and 47 women in the market. What is the total number of people in the market?”

• (i) The student is hesitant in reading the word total.• (ii) The student cannot read the word market in the question.

(b) Category 2: Comprehension• Instructions/Questions forwarded to the student during interview:• “What does this word/sign mean?”• “Tell me what the question is asking you to do.”

• At this stage, the student may face difficulties in understanding the specific terms, phrases or the general meaning of the given question.

• Examples of errors at the ‘Comprehension’ category:

• A student is asked to comprehend the question: “12 apples are shared between 3 girls. How many apples does each girl get?”

• During the interview, if the student is able to say the word shared confidently but he guesses its’ meaning as “take away”, then he faces difficulty in comprehending specific terminology. On the other hand, if the student can not tell a satisfactory general meaning of the question, then the error is also classified as comprehension error.

(c) Category 3: Transformation• Instructions/ Questions forwarded to the student during interview:• • “Tell or show me how you start finding an answer to this question.”• • At this stage, error occurs when the student is unable to verbalise or

show with pencil and paper how the item can be translated into a mathematical form (symbol, operation, or sentence) which will allow a correct answer to be worked out later.

• Examples of errors at the ‘Transformation’ category:• • A student is told to interpret the question: “Ali and Ahmad have a total of

9 marbles. Ali has 3 marbles. How many marbles does Ahmad have?”• The student is able to describe correctly during interview: “When we add

Ali’s and Ahmad’s marbles, we get 9 marbles” but he wrote the wrong mathematical sentence, that is: “3 + 9 = ”.

(d) Category 4: Process Skills• Instructions/ Questions forwarded to the student during the

interview: • • “Show me how you work out the answer for this question. Tell me

what you are doing as you work.”• • The student cannot implement the calculation steps correctly when

implementing the algorithm although he was successful in writing the mathematical sentence related to the question. Examples of students’ responses considered to be wrong at this stage include answering randomly (i.e. by random guess), usage of wrong operations, errors in calculation steps or not answering at all.

Examples of errors at the ‘Process skills’ category:

4 7 6 3

+ 2 5 - 1 7

6 1 2 5 4

(e) Category 5: Encoding• Instructions/ Questions forwarded to the student during interview:• • “Now write down the answer to the question.”• • At this stage, the student is unable to write the answer correctly either in

numerals, symbols or words although the student has gone through the ‘Process skills’ stage successfully and is able to verbalise the correct answer. In addition, incorrectly spelt words and reversals of numerals are also considered as encoding errors.

• • Examples of errors at the ‘Encoding’ Category:• • The student verbalises “one ringgit fifty-five sen”, but write the answer

down as “RM 1.505”.

• The student verbalises “5 kilometres” but write the answer down as “5 KM”. It is important for you to note that each error identified by the Newman’s error analysis procedure is the first error made by the student for a particular question. So, it does not mean that the student will definitely not make other errors after the first error is identified. However, Newman’s diagnostic interview stopped after identifying the first error made.

1.1.2 Carelessness / Motivation factor

• • On the second attempt in answering a question during the diagnostic

interview, if the student makes no mistake throughout the five stages and is able to answer the question correctly, then the incorrect answer given by the student in the earlier test is caused by a careless mistake. Some students can be careless due to feelings of anxiety while solving the problem.

• • If the results of the diagnostic interview show that the student’s incorrect

answer to the question on the first attempt during the earlier word-problem testing session was brought about by his or her attitude, then the error is caused by lack of motivation. The student may be under pressure and are not motivated. Some students who are under pressure or are not motivated will just write any answer in the test but can actually manage to get the right answer when they are at ease or not pressurized.

•

• Newman (1977) dalam kajiannya untuk menyiasat kesilapan yang dilakukan oleh kanak-kanak yang berumur 12 tahun dalam satu ujian 40 item memberi satu ujian bertulis kepada 917 kanak-kanak daripada 31 kelas yang berasingan.

• Seramai 124 orang daripada 917 kanak-kanak ini dikenalpasti terdiri daripada murid dalam golongan pencapaian rendah daripada setiap kelas telah mendapati membuat sejumlah 3002 kesilapan pada ujian yang awal dan sebanyak 70 % daripada mereka telah melakukan kesilapan yang sama pada sesi temubual .

• Setengah daripada kesilapan berlaku sebelum murid dapat pergi ke tahap proses. Kebanyakkannya kesilapan mereka terdiri daripada jenis bacaan dan kefahaman.

• Pada kajian Clement (1980), di mana melibatkan seramai 92 murid pencapaian rendah dan 92 murid pencapaian sederhana didapati bahawa murid pencapaian rendah telah melakukan lebih banyak kesilapan pada peringkat bacaan dan kefahaman berbanding dengan murid pencapaian sederhana.

• Kajian yang dijalankan Marina & Clement kepada murid tingkatan satu menurut Ellerton & Clement (1992) juga menunjukkan bahawa 71% kesilapan berlaku pada tahap Kefahaman dan Transformasi.

• Gan (1993) dalam kajiannya menggunakan kaedah serupa kepada pelajar tahun 5 di sebuah sekolah di Miri ,Sarawak secara keseluruhan mendapati

• 72.1% pelajar menghadapi masalah dalam• kesilapan tahap bacaan (5.3%), • kefahaman (40.7%) • transformasi(26.1%).• Yakni 66.8% adalah pada tahap kefahaman dan transformasi. • Kesilapan ini mencadangkan sama ada murid-murid tidak memahami

soalan-soalan yang melibatkan masalah berperkataan atau walaupun mereka sudah faham tetapi tidak dapat menentukan operasi-operasi yang sesuai untuk menyelesaikan masalah.

• Penyelidik ini menyatakan tidak cukup penekanan diberikan untuk membantu murid dalam kefahaman masalah matematik dan penentuan turutan operasi matematik

Haslan Manja dan Teh Ah Huat (2004)

• Berdasarkan Analisis Newmans temubual kepada 251 murid 9 buah sekolah di Tuaran, Ranau, Kota Marudu, Ranau .

• Masalah bacaan yang ketara ( 39.61%)

• Tidak memahami soalan berayat (40.04%)

• Hampir sama spt kajian Gan, Miri; Clement, P. Pinang.

• Kesukaran menjawab soalan tahap tinggi

• Masa 30 minit tidak mencukupi bagi kebanyakan pelajar- lemah( 15 soalan).

Jenis Kesilapan Bil. Keseluruhan*

Peratusan (%)

Peratusan(%)

Bacaan 490 39.61 81.65

Kefahaman 520 42.04

Transformasi 133 10.75

Operasi 54 4.37 18.35

Pengekodan 24 1.94

Kecuaian 16 1.29 1.29

Jenis Kesilapan Keseluruhan* Peratusan Peratusan

bacaan490 39.61

81.65

kefahaman520 42.04

transformasi133 10.75

18.35

operasi54 4.37

pengekodan24 1.94 1.94

kecuaian16 1.29 1.29

D. PROSEDUR KAJIAN NEWMAN (CLEMENT,1980)

• Murid diberi ujian yang terdiri daripada lima belas soalan subjektif yang merupakan masalah perkataan.

• Guru memeriksa ujian .• Guru memilih beberapa murid daripada

kumpulan pencapai rendah (30% bawahan).• Setiap murid yang dipilih disuruh mencuba kali

kedua soalan yang didapati salah semasa menjalani ujian.

PROSEDUR KAJIAN NEWMAN (CLEMENT,1980)

• Murid menyelesai penyelesaiannya, guru akan menemubual murid dengan menggunakan Prosedur Analisis Kesilapan Newman,

dengan mengemukakan lima soalan.

• Guru akan mencatatkan jawapan murid bagi soalan yang dikemukakan .• Jika dalam percubaan soalan kali kedua , didapati murid dapat jawapan

yang betul dan setelah mendengar jawapan temubual bagi soal guru berasa yakim bahawa kecuaian telah berlaku.

Maka kesilapan ini dikatakan sebagai KECUAIAN.

• Jika tidak, kesilapan mungkin terdiri daripada samada dari jenis PEMBACAAN, KEFAHAMAN, TRANSFORMASI, PROSES atau

PENGEKODAN.

BIL SOALAN TEMUBUAL JENIS KESILAPAN

1 Bacakan soalan itu kepada saya. Pembacaan

2 Beritahu saya apakah yang dikehendaki oleh soalan.

Kefahaman

3 Beritahu saya apakah kaedah atau cara yang kamu akan menggunakan untuk mengira jawapan.

Transformasi

4 Cuba anda tunjukkan kiraan langkah demi langkahdan memberitahu saya apakah yang anda fikirkan.

Proses

5 Beritahu saya apakah jawapan bagi soalan itu.Tunjukkan jawapan anda.

Pengekodan

Contoh soalan berayat

• 1. Ahmad membeli 150 biji guli. Keesokannya, Abu memberikan 270 biji guli kepada Ahmad dan hari yang berikutnya, Siti memberikan 305 biji guli kepada Ahmad. Berapa jumlah biji guli yang ada pada Ahmad sekarang?

Contoh analisis dengan Excell

Kesilapan

Q1 Q2

Q3 Q4

Q5

Q6 Q7

Q8

Q9 Q10

Q11

Q12

Q13

Q14

Q15 Total % Peratusan

1 30 32 34 32 34 32 34 32 34 33 33 33 33 32 32 490 39.61 81.65

2 23 28 30 30 43 33 29 32 36 44 39 42 37 36 38 520 42.04

3 9 11 9 12 9 10 6 12 5 7 12 9 7 6 9 133 10.75 18.35

4 1 7 9 7 2 4 4 5 4 1 0 2 3 3 2 54 4.37

5 4 4 0 0 1 3 2 1 3 2 0 0 1 2 1 24 1.94

6 6 0 0 1 0 1 4 0 0 0 1 0 1 1 1 16 1.29 1.29

Jumlah 73 82 82 82 89 83 79 82 82 87 85 86 82 80 83 1237 100.00

Tahap Kesilapan

Soalan Bacaan Kefahaman Transformasi Operasi Pengekodan Kecuaian

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Ujian Matematik Tahun Tiga Nama Kelas Sekolah

• Arahan : Jawab semua soalan dan tunjukkan semua langkah pengiraan.• Tulis jawapan di ruangan yang disediakan

• 1. Ahmad membeli 150 biji guli. Keesokannya, Abu memberikan 270 biji guli kepada • Ahmad dan hari yang berikutnya, Siti memberikan 305 biji guli kepada Ahmad. Berapa jumlah biji guli yang ada pada Ahmad sekarang?

• 2. Dalam satu Majlis Konvokesyen sebuah Universiti, terdapat seramai 1000 graduan yang • bakal menerima ijazah. Daripada bilangan itu, seramai 345 graduan perempuan.• Berapakah jumlah graduan lelaki yang bakal menerima ijazah pada hari itu ?

• 3. Halim membeli 12 bungkus gula-gula. Setiap bungkus gula-gula yang dibelinya• itu mengandungi sebanyak 120 biji setiap satu. Berapakah bilangan biji gula-gula• yang ada pada Halim?

• 4. Seorang nelayan mempunyai 1200 ekar ikan yang hendak disimpan dalam bakul

ikan. Setiap bakul ikan hanya boleh diisi dengan 40 ekor ikan sahaja. Kirakan jumlah bakul yang perlu disediakan untuk menyimpan ikan tersebut?

• 5. Sebuah bas kilang hanya boleh mengangkut seramai 45 orang penumpang.Pada satu hari 1000 pekerja kilang sedang menunggu bas kilang untuk pulang.

Kemudian, selepas lama menunggu 500 orang pekerja menggunakan kenderaan lain untuk pulang. Berapa buah bas yang boleh memuatkan bilangan pekerja kilang yang tinggal?

6 . Bilangan buah durian yang telah gugur daripada 3 batang pokok ialah 143 biji,

25 biji dan 34 biji. Berapa biji semuanya yang telah gugur?

• 7. Jumlah semua rumah di taman B ialah 485 buah. 249 buah rumah itu ialah rumah jenis teres setingkat, dan bakinya rumah jenis yang lain. Hitungkan bilangan rumah jenis yang lain itu?

• 8. Abd. Maruf membeli 5 bungkus gula-gula untuk adik-adiknya, jika setiap bungkus

mengandungi 8 biji gula-gula, berapa biji gula-gula yang dibelinya?

• 9. Dalam sebuah dulang ada 27 keping kuih. Kuih-kuih itu hendak diletakkan dalam 9

buah piring dengan sama banyak. Berapa keping kuih yang boleh diletakkan dalam

tiap-tiap piring?

• 10. Osman membeli 5 tiket bas untuk perjalanan dari Kota Kinabalu ke Tawau dengan harga RM 45.00 setiap satu. Jika dia membayar dengan RM 250, berapakah

wang baki yang diterimanya?

• 11. Dalam sebuah bekas, terdapat 245 biji guli merah dan 636 biji guli putih. Kemudian

14 biji guli merah dan 9 biji guli putih dimasukkan ke dalam bekas tersebut. Kirakan

jumlah guli putih yang ada dalam bekas itu.

• 12. Seorang penjaja membeli 2 kotak buah epal daripada pemborong. Setiap kotak masing-masing mengandungi 460 biji buah epal. Bagi kotak yang pertama, penjaja

• tersebut mendapati 431 biji epal adalah elok dan yang selebihnya telah rosak.Bagi kotak kedua pula hanya 11 biji buah epal yang telah rosak. Hitungkan jumlah buah epal yang masih elok.

• 13. Aziz menjual 271 buah buku pada hari pertama dan 162 buah buku pada hari

kedua. Kini tinggal 74 buah buku di kedainya. Kirakan jumlah buku yang ada pada mulanya.

• 14. Dalam suatu pertandingan kuiz, kumpulan A dan B masing-masing memperoleh 627 dan 592 markah. Kira beza markah antara dua kumpulan tersebut.

• 15. Sebanyak 525 keping setem dibahagikan antara Siew Ling, Marlyn dan Zarina.

Siew Ling dan Marlyn masing-masing mendapat 168 keping setem. Hitung bilangan setem yang diperolehi oleh Zarina.

Rujukan :• • Bennett, A.B. Jr. & Nelson, L. T. (2001). Mathematics for elementary teachers : A conceptual approach 5th ed. Boston : McGraw Hill.• Bitter, G.,Hatfield, M., Edwards, N. (1989). Mathematics methods for the elementary and middle school. A comprehensive approach. Boston : Allyn

and Bacon.• Burns, M. (1984). The math solution : Teaching for mastery through problem solving. Sausalito, California : Marilyn Burns Education Associates.• Cofman, J. (1991). What to solve? Problems & suggestions for young mathematicians. New York: Oxford University Press.• Ellerton, N. F. & Clements, M. A. Applying the Newman analysis procedure to Mathematics and Science education research in Asia and the Pacific.

Paper presented at the International Seminar on State-of-the-Art of Research in Science and Mathematics Education in Southeast Asia and the Pacific, held at the SEAMEO-Regional Centre for Education in Science and Mathematics, Penang, 17-19 February, 1992.

• Evan, R. & Lappin, G. (1994). Constructing meaningful understanding of mathematics content. In professional development for teachers of mathematics, pp.128-143. Reston, Virginia: NCTM.

• Gan, T. H. (1993). Mengaplikasikan prosedur analisis Newman untuk mengklasifikasikan kesilapan-kesilapan murid-murid tahun 5 semasa menyelesaikan masalah berperkataan di Sekolah Rendah Kerajaan South, Miri. Jurnal Pendidikan 1993-Maktab Perguruan Sarawak, pp.33-47.

• Henderson, K. B. & Pingry, R. E. (1953). Problem solving in mathematics. In the learning of mathematics: Its theory and practice (21st Yearbook of the National Council of Teachers of Mathematics), pp. 228-270. Washington, DC: NCTM.

• Krulik, S. & Reys, R.E. (1980). Problem solving in school mathematics 1980 Yearbook. Reston, Virginia: NCTM.• Lester, F. K. Jr. et. al. (1994). Learning how to teach via problem solving. In professional development for teachers of mathematics, pp.152-166.

Reston, Virginia: NCTM.• Marinas, B. & Clements, M. A. (1990). Understanding the problem: A prerequisite to problem solving in mathematics. Journal for Research in

Science and Mathematics Education in Southeast Asia, 13(1), 14-20.• National Council of Teachers of Mathematics (NCTM) (1980). An agenda for action: Recommendations for school mathematics. Reston, Virginia:

NCTM.• Newman, A. (1983). The Newman language of mathematics kit: Strategies for diagnosis and remediation. Sydney, Australia: Harcourt Brace

Jovanovich.• Polya, G. (1973). How to solve it. Princeton, NJ: Princeton University Press. • Singhatat, N. (1991). Analysis of mathematics errors of lower secondary pupils in solving word problems. Penang: SEAMEO-RECSAM. • Suydam, M. (1987). Indications from research on problem solving. In teaching and learning: A problem solving focus. Reston, VA: NCTM.