Misconception In Mathematics

Analysis of Mathematical Errors in Primary Schools See Kin Hai , Hjh Jamilah Yusuf Universiti Brunei Darussalam Negara Brunei Darussalam Abstract The skills necessary to identify and analyze errors made by pupils are needed by teachers at all levels especially at the primary schools level in Brunei Darussalam. If pupils are to be successful in tackling mathematical problems later in their schooling, the one prerequisite is the mastery of the basic concepts in their primary mathematics. Despite the best efforts of the teachers, pupils still develop mathematical misconceptions. Is it possible to reduce or eliminate these misconceptions? In this study a survey on 15 classes with a total sample of 450 pupils from five Primary schools in Brunei in the area of Muara-Bandar Seri Begawan was conducted for a period of six weeks. The aim of this study is to look for patterns of misconceptions made by pupils in the addition (15 different types of skills) and subtraction (8 different type of skills) of fractions in primary five and also to look for possible causes for these errors and to come out with some suggestive teaching strategies. Systematic errors (six types identified) of pupils have been examined in an effort to determine the causes of pupils’ failures and to provide some possible remedial steps and suggestions to rectify them. It is hoped that the findings in identification and analysis of mathematical errors would draw the attention of classroom teachers to the need to consider more than just the number of incorrect responses made by the pupils when teaching primary mathematics. Introduction In this study, systematic computational errors are classified into six different categories namely, grouping error, basic fact error, defective algorithm, incorrect operation, identity error and zero error. It will be interesting to determine if the different kinds of errors are relevant to Bruneian children. Thus, the objectives of this study are (i) to look for patterns of computational errors in addition and subtraction of fractions exhibited by a sample of 402 Primary five pupils in five Primary schools in Muara-Bandar Seri Begawan district; (ii) to analyze and classify the different types of errors made in the computation of fractions in addition and subtraction; (iii) to look for possible causes of these errors, and (iv) to suggest strategies for teaching fractions. It is expected that the analysis would help to identify the kinds of wrong strategies that lead to these incorrect responses. Hopefully, such identification could provide teachers with some definite guidance for remedial education of the children when such errors occur. Hart (1984) and Steffe (1988) focus their studies on errors made by children and then construct models to describe the conceptual operations or the thought patterns employed by the child in arriving at the answers. Other studies examine the role of teaching instructions such as games (Onslow, 1990) and alternating teaching strategies (Hart, 1984; Booth, 1984) that could help children overcome the observed errors. In this study, the error patterns were observed in the answer scripts of the Primary 5 children from five Primary schools in Muara-Bandar Seri Begawan areas who answered the questions set by the authors. Possible causes for these errors and suggested teaching strategies will be discussed after analysis of the results.

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Methodology The Sample Approximately 450 answer scripts for each test from the 15 classes of the 5 schools were examined. The total number of pupils is 450. Primary 5 pupils were chosen because it is thought desirable to investigate the effect of errors on pupils who are not involved in the PCE or Government Primary Six Examination. Instrument The instrument used a modified version of Brueckner Diagnostic Test in Fractions. The instrument has been validated by Yap (1985) and a local pilot study. It consists of problems similar to the exercises given in their textbooks. Fractions used are restricted to denominator not larger than 100. No mixed numbers or decimals are used in the question although mixed numbers are required in their responses. In each algorithm, the fractions are divided into the same denominator and unlike denominator. The category is further subdivided into reducible and nonreducible fractions. In this example,

2 3 5 + = 7 7 7

1 3 4 1 + = = =1 4 4 4 1

fraction type (N) while

is the same denominator and non-reducible

is an example of the same denominator but of

reducible fraction type (R). Fifteen distinct types of skills are distinguished in addition and eight in subtraction. As for the unlike denominator, Type A is where the LCM uses the denominator of one of the fractions. Type B is where the LCM is derived from the product of the 2 denominators. Type C is where the LCM is obtained by factoring the denominators. The classification for 15 types in addition is shown as follow: Same denominator (S) Type A Reducible (R) Non-reducible (N)

E1, E2, E3, E4 E12

Unlike denominator (U) Type B Type C

E5, E6, E7 E13

E8 E14

Skill E1:

Type SR

Skill involved Sum improper fraction to 1

Example

E2:

SR

Sum improper fraction to mixed number

3 8 11 2 + = = 1 9 9 9 9

E3: E4:

SR SR

Sum proper fraction to simplest form Sum improper fraction to mixed number in simplest form

2 1 3 1 + = = 6 6 6 2

E5:

UR

Sum improper fraction to mixed number

E9, E10, E11 E15

2 5 7 1 + = = =1 7 7 7 1

3 5 8 2 1 + = =1 = 1 6 6 6 6 3

1 5 7 1 + = =1 3 6 6 6

3

E6: E7: E8: E9: E10: E11:

UR UR UR UR UR UR

Sum reducible proper fraction to simplest form Sum reducible improper fraction to mixed number in simplest form Sum non-reducible improper fraction to mixed number in simplest form Sum improper fraction to mixed number in simplest form

1 1 3 + = = 3 6 6 3 7 16 + = = 4 12 12

NS

Sum proper fraction in simplest form

E13:

NU

Sum proper fraction in simplest form

E14:

NU

Sum proper fraction in simplest form

E15:

NU

Sum proper fraction in simplest form

4 1 = 1 3 3

7 1 37 1 + = = 1 9 4 36 36 1 5 13 11 + = = 1 4 6 12 12 1 2 9 3 + = = 6 15 30 10

Sum proper fraction to simplest form Sum improper fraction to mixed number in simplest form

E12:

1 2

1 6 2 7 1 2 2 5 1 6

+ +

+ +

+

9 23 16 1 = = = 1 10 30 15 15 1 3 = 7 7 3 7 = 8 8 3 31 = 8 40 2 7 = 9 18

As for the subtraction of fractions, eight different types of skills are identified as follows: Type

Same denominator (S)

Reducible (R) Non-reducible (N) Skill E16 E17

Type SR UR

E1 E5

Skill involved Subtract proper fraction to Lowest form Subtract proper fraction to Lowest form

E18

UR UR

E21 E22

NS

NU NU

E3 E7

Example 6 2 4 − = = 8 8 8

1 2

5 1 2 1 − = = 6 3 6 3

7 2 10 5 − = = 9 4 36 18

Subtract proper fraction to Lowest form

E20

E2 E6

Subtract proper fraction to Lowest form

E19

Unlike denominator (U) Type A Type B

1 3 5 − = = 15 10 30

Subtract proper fraction in Simplest form

6 3 3 − = 7 7 7

Subtract proper fraction in Simplest form Subtract proper fraction in

1 5 1 − = 3 6 6

1 6

Type C E4 E8

4

Simplest form E23

NU

Subtract proper fraction in

7 1 19 − = 9 4 36 5 1 7 − = 6 4 12

The instrument used in this study consists of 10 different test papers. Fifteen different types of computational skills in addition for fractions and eight distinct types for subtraction of fractions were tested. There are twenty three distinct types of questions. Each test paper consists of two to three different types of skills. There are 5 questions set for each skill (depending on the difficulty level) giving a total of 10 to 15 questions per set of test. In a pilot study carried out to correct the ambiguities and mistakes in the tests papers, a 30 minute or a period was found to be sufficient for the pupils to complete each test paper. The number of each set of test papers against the different skills are shown in the table below: Skills Type E1, E2 E3, E4, E5 E6, E7, E8 E9, E10, E11 E12, E13

Test Paper No. Test 1 Test 2 Test 3 Test 4 Test 5

Skills Type E14, E15, E16, E17 E18, E19 E20, E21 E22, E23

Test Paper No. Test 6 Test 7 Test 8 Test 9 Test 10

Procedure for data collection Tests were given to the class of about 20 to 30 pupils. A record of the test paper taken by the class will be kept to ensure that all the pupils from the class sat for all the 10 test papers. Two test papers were given per week lasting about 5 weeks. The testing schedule is as follow: Week 1

First half Test 1, Test 2

Second half Test 3, Test 4

2

Test 5, Test 6

Test 7, Test 8

3

Test 9

Test 10

Student teachers supervised the tests. No assistance either verbal or non-verbal was given to any pupil. Pupils’ responses and some interviews of about 5% of each class were selected to allow the authors to have a better understanding of the pupils’ approaches in making errors. This will help the authors in their analysis of the pupils’ errors. The pupils responses were analysed according to the following categories: 1. No error or all correct: Meaning the pupils got all correct in the 5 questions for each skill.

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2. Systematic error: The pupil missed at least three out of five questions for each skill, repeatedly recording the same mistakes for all the other skills. 3. Careless error: The pupil only missed 1-2 out of 5 questions for each skill. Basically the pupil knew how to solve the problem. 4. Random error: The pupil missed at least 3 out of 5 questions for each test. However, as for the other skills no similar pattern was found. 5. Incomplete responses: The pupil did not complete all the 5 questions for each skill. Only systematic errors made are examined. Errors made in the addition and subtraction algorithm will be identified and analysed. Percentage and number of cases will be calculated. Systematic errors are further classified into another six different sub-categories. Burton (1981) and Reys (1996) had also found that most of the errors involving 2500 American children in subtraction of fractions were systematic errors . The authors classified the errors as in the followings: 1. 2. 3. 4. 5. 6.

grouping error basic fact errors defective algorithm incorrect operation identity errors, and zero errors. Results and Analysis of Data

Error Type

Low 1

Quartile 2

3

High 4

Total No (%)

Grouping

28

27

17

16

88 (21.9)

Basic Fact

46

37

33

17

133 (33.1)

Defective Algorithm

26

28

17

1

72 (17.9)

Incorrect Operation

40

16

16

8

80 (19.9)

Identity

2

1

2

0

5 (1.2)

Zero

11

5

7

1

24 (6.0)

Total

153

114

92

43

402

From the above table, examples of systematic errors found are as follows: 1. Grouping error: For the subtraction of fractions, errors occurred at all skill levels requiring regrouping (borrowing) and this involved a large numbers of about 21.9% of pupils from the sample of 402. This error decreases as the ability level of pupils increases. The ability levels of pupils have been classified into High, Medium (second and the third quartile) and Low

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from their school academic records. Cox (1975) also discovered that most frequent error occurred in the subtraction of smaller digit from the larger digit. 23 17 14 = 24 24 24 Ward (1979) reported that most of the mistakes made in this grouping error were due to a lack of place value concepts which he found by using items directly relating to testing of place value ideas. 2. Basic fact errors: As for the addition of fraction, many errors were found involving carrying or renaming and some basic fact. This involved about 33.1% of the whole sample under investigation. Engelhardt (1977) also found that most of this kind of error occur in higher number of digit operation not because the children misremembered the number fact. This problem of number sense was reported by McIntosh, Reys, Reys, Bana and Farrell (1997). 24 8 212 + = , 17 17 17

26 18 34 + = and 29 29 29

2 1 1 = 3 9 6

3. Defective algorithm: About 17.9% of the pupils make mistakes involving incorrect application of an algorithm. There were virtually no error of this type for the highest quartile group. In this kind of error, the pupils normally started with the correct operation and then seemingly lost tract of what they were doing and resorted to a different operation. Roberts (1968) found about 20% of errors made by children were due to defective algorithm. However, he found that there was little difference across the ability range in making this error. He stated that: Although much emphasis is apparently placed on teaching the fundamentals of decimal system of numeration and positional numeration systems in other bases… it seems clear that we have not been uniformly successful in making this knowledge relevant to the processes children use in computation.

123 38 138 = 120 120 120 The authors suggest that this is in part due to the children’s perception of a computation task. The children felt that they may be expected to perform in a rote manner given a stimulus which was essentially non-meaningful to them. Therefore, the authors suggest that beside rote learning, the children might be taught the meaning behind each of these concepts in order to overcome the occurrence of this type of error. 4. Incorrect operation: This error is quite common not because of misremembered basic facts. This involved about 19.9% of the sample under investigation.

1 5 5 = 3 6 18 Errors are seldom random. The pupils’ error in this may have some sensible origins, perhaps a misinterpretation or some misunderstanding of standard procedure that the teacher has taught them.

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5. Identity error: Error that the children perform in a way suggesting some confusion in the computation of identity number. This error occurs at a low percentage (1.2%) and mostly from those low ability group. As the pupils may perceive that a number subtrating or adding the identity will remain the same number.

2 1 2 = 7 7 7 6. Zero error: Children having some difficulty with the concept of zero.This made up about 6.0% of the sample. Again there were virtually no error of this type for the highest quartile group

35 10 20 = 6 6 6 Again, the child may be wrong because of a lack of understanding of zero as the identity in the operation of subtraction of fraction or maybe because of some inadequacy in his concept of zero. So, to overcome this error, in order not to misjudge the children’s erroneous approach, an interview with the child concerned may reveal the main cause of this error. Oesterle (1959) and Reys (1996) in their reports stated that it is dangerous to regard zero as being synonymous with ‘nothing’ as most of the Bruneian Lower Primary school children may have been taught in this way. The implication is that if zero is nothing then it need not be considered. Rather zero should be looked upon as being ‘ not any of something’. Oesterle (1959) stated that: The significance of zero as a place holder and as a vital part of our number system appears, as many writers have stated, when we begin addition and multiplication of two-digit numbers. Until this time there seems to be no real reason for introducing the zero facts in any of the basic operations. Oesterle offered some ways to overcome this problem by stating that: …….. the student should be given specific practice with these processes both in isolation and as integral parts of real problems. Generalisations from numerous contacts with zero should be derived from the student’s personal experience with these facts as they occur in real problems. Wilson (1951) also offered a way of solving this zero error problem by stating that: …..specific attention to zero in the teaching programme rapidly eliminates the zero errors… In Bruneian context, the reason for this error may be that teaching does not dwell sufficiently on the zero combinations during the four operations of two-digit numbers. The pupil is being confronted with zero within a context of the symbolic, abstract notation of arithmetic demanding an understanding of place value to make it meaningful. It seems that place value is a complex idea which the pupils find difficult to grasp. The pupils might have little concrete foundation in terms of their earlier counting and measuring experiences. With the added problem of coping with zero, this might be the cause of the error. Suggestions and conclusions

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From the results obtained, it is clear that a good understanding of the basis of algorithms may possibly help pupils’ retention in the long term (relational understanding) as opposed to rote learning (instrumental learning) which is more efficient in the short term. Suydam and Dessart (1976) had offered the same suggestion when they stated that: A cardinal rule has been evolved through experience and affirmed through research: Develop mathematical ideas and skills from the concrete physical basis….. Generally, researchers have concluded that understanding is best facilitated by the use of concrete materials, followed by semi-concrete materials (such as pictures) and finally by the abstract presentation with symbols. It is important to use concrete materials in introducing algorithms as it is in introducing basic facts. The difficulty of operating with fractions is because they have a multiplicity of meaning. The authors suggested that it is therefore more meaningful to teach pupils to understand the various interpretations in symbolic or concrete forms. Ginsburg (1977) suggested that fractions can be taught in many ways. For example, the fraction of

1 as the authors suggest can be interpreted 4

and taught as: (a) A sub-area of a defined ‘whole region’: Here the whole region is divided into 4 equal parts and take 1 of them (as in Fig 1). Pupils’ earliest encounter with fractions are likely to be of a spatial kind and 3-dimensional in nature. Hart (1980) taught fractional problems by giving the pupils a paper and asking them to divide the paper by folding, cutting or drawing. He noted that the pupils improved significantly in their problems involving fractions. He also found that pupils find this spatial ‘parts of a whole’ notion the easiest aspect of fractions to grasp. Hopefully, this strategy will also work for the Bruneian pupils. Reys (1996) suggested that the part-whole meaning and the region model provided a good start in teaching fraction.

Fig 1

Fig 2

This method can be used in addition and subtraction of fractions. For example

3 3 3 + = 8 8 4

can be traditionally developed using an area representation. However, if the pupil will to represent the fractions on two separate diagrams, this method may cause some problem giving an answer of 6/16 and not as 6/8 or ¾ as in Fig 3 and Fig 4 below.

Fig 43

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(b) A comparison between a subset of a set of discrete objects and the whole set: As 1 out of 4 dots are black as in Fig 2. This is quite similar to (a) when the 4 sectors in (a) are separated. Novillis (1976) found that methods (a) and (b) are significantly no different from each other in the enhancement of pupils’ performance in solving fractional problems. However, Payne (1976) found that method (b) using the ‘sets’ approach was significantly more difficult than other methods in teaching fractions. (c) A point on a number line which lies between 0 and 1 as in Fig 3: 0

0.25

1 4

0.50

0.75

1.00

Fig 3

This strategy possesses some advantages. It makes improper fractions more important as an extension of a set of natural numbers helping to fill in the holes in between the number line. However, Novillis (1976) found that the difficulty of operating with number line was aggravated if the number line was extended beyond one. For example, to mark the fraction of 3/5 on the number line from 1 to 5. Many of the primary children cannot mark a point on this line. Here, the fraction is represented as a dot on a line as 0 and 1. (d) The result of a division operation such as one object divided between 4 people. Here the meaning of fraction is associated with the operation of dividing one whole number by another. This strategy was used by Hart (1984) with some success; example such as “ A bar of chocolate is to be shared equally between four children. How much should each child get?”

(d) A way of comparing the sizes of two sets of objects such as A has 1 and Trolley A is

1 as many dots as B in Fig 4

1 of the length of trolley B in Fig 2. Here, the basis of application of 4

fractions which occur in real-life, particularly fractions that involved the ideas of ratio or

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scale could easily be demonstrated to the children. However, Hart (1984) and Karplus et al. (1977) revealed that children tend to revert to using additive comparisons such as 5 is more than 4 and not as a ratio.

A

B Fig 1

0

1

2

3

4

5

0

1

2

3

4

5

Fig 2

As the concept of fraction is complex and cannot be grasped all at once by the pupils, it has to be acquired through a long process of sequential development through a carefully planned sequences of teaching. Hopefully, through these, the pupils can link fractions in everyday context with the abstract number they are working on in schools. Since, a pupil who is asked to cut a piece of 2 m ticker timer tape into 5 equal pieces would arrive at 40 cm per piece without any real understanding of the result that the fraction of

2 = 0.4 5

.

There is still much scope for further research in this area. The authors could only speculate on the possible reasons for pupils’ wrong responses. Further work is needed through interview to determine the precise nature of these misconceptions and the testing of these strategies in schools as suggested by the authors. Teachers should delay using fractions until fundamental concepts in mathematics are well established in pupils. This was also recommended by Swedosh (1996).

References Booth, L. (1984). Algebra: Children’s strategies and errors. A report of strategies and errors in secondary school mathematics project. NFER-Nelson. Burton, R. B. (1981). DEBUGGY: Diagnosis of errors in basic mathematical skills. In Intelligent Tutoring Systems (Ed) Sleeman, D. H. and Brown, J. S. New York:

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Academic Press. Cox, L. S. (1975). Systematic Errors in the Four Vertical Algorithms in Normal and Handicapped Populations. Journal for Research in Mathematics Education, 6, (4), 202-220. Engelhardt, J. M. (1977). Analysis of Children’s Computational Errors: A Qualitative Approach. British Journal of Educational Psychology, 47, 149-154. Ginsburg, H. P. (1982). Children’s Arithmetic: how they learn it and how you teach it. Austin, TX: PRO-ED, 1982. Hart, K. M., (1980). Secondary school children’s understanding of mathematics- Research Monograph (A report of the Mathematics Component of the Concepts in -Secondary Mathematics and Science Programme). Chelsea College, University of London. Hart, K. M., (1984). Ratio: Children’s strategies and errors. A report of the strategies and errors in secondary school mathematics project. London. NFER-Nelson. Karplus, R; Karplus, E; Formisano, M. and Paulsen, A. (1977). A survey of proportional reasoning and control of variables in seven countries. Journal of Research in Science Teaching, 13, 411-417.

McIntosh, A., Reys, B., Reys, R., Bana, J, & Farrell, B. (1997). Number sense in school mathematics: Student performance in four countries. Perth: MASTEC, Edith Cowan University. Novillis, C. F. (1976). An analysis of the fraction concept into a hierarchy of selected subconcepts and the testing of the hierarchial dependencies. Journal of Research in Mathematics Education, 7, 131-144. Oesterle, R. A. (1959). What about the ‘Zero Facts’? The Arithmetic Teacher, 6, (2), 109-111. Onslow, B. (1990). Overcoming conceptual obstacles: The qualified use of a game. School Science and Mathematics, 90 (7), 581-592. Payne, J. N. (1976). Review of research on fractions. In Number and Measurement: Papers from a Research Workshop (Ed) Lesh, R. A. Ohio, USA: ERIC/SMEAC. Reys, R. E. (1996). Helping children learn mathematics. Allyn and Bacon, Boston. Roberts, G. H. (1968). The Failure Strategies of Third Grade Arithmetic Pupils. The Arithmetic Teacher, 15, 442-446. Steffe, L. P. (1988). Children’s construction of number sequence and multiplying schemes. In J. Hiebert and M. Behr (Eds.), Number concepts and operations in the middle grade. Hillsdale, NJ/Reston, VA:LEA/NCTM.

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Suydam, M. N. and Dessart, D. J. (1976). Classroom ideas from research on computational skills. Reston, Virginia: National Council of Teachers of Mathematics. Swedosh, P. (1996). Mathematical misconceptions commonly exhibited by entering tertiary mathematics students. In P. Clarkson (Ed.), Technology in Mathematics Education, 534-541. Melbourne: Mathematics Education Research Group of Australasia. Ward, M. (1979). Mathematics and the 10-year old, Working Paper 61, School Council. Evans/Methuen. Wilson, G. M. (1951). Teaching the New Arithmetic. New York:McGraw-Hill. Yap Yee Khiong (1985). Analysis of errors in fractions of a National Primary School. An unpublished Master dissertation, Universiti Malaya, Kuala Lumpur.