Understanding Students' Mathematical Errors and ... - ispacs

Mathematics Education Trends and Research 2014 (2014) 1-10

Available online at www.ispacs.com/metr Volume 2014, Year 2014 Article ID metr-00051, 10 Pages doi:10.5899/2014/metr-00051 Research Article

Understanding Students' Mathematical Errors and Misconceptions: The Case of Year 11 Repeating Students Hjh Roselizawati Hj Sarwadi1, Masitah Shahrill2* (1) Maktab Duli Pengiran Muda Al-Muhtadee Billah, Ministry of Education, Bandar Seri Begawan, Brunei Darussalam (2) Sultan Hassanal Bolkiah Institute of Education, Universiti Brunei Darussalam, Bandar Seri Begawan, Brunei Darussalam

Copyright 2014 © Hjh Roselizawati Hj Sarwadi and Masitah Shahrill. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Students' errors are causally determined, and very often systematic. Systematic errors are usually a consequence of student misconceptions. These can include failure to make connections with what they already know. There are beliefs held by students that inhibit learning from errors, such as they cannot learn from the mistakes and that mathematics consists of disconnected rules and procedures. Student errors are unique and they reflect their understanding of a concept, problem or a procedure. This study investigated how much mathematics have been understood and remembered by Year 11 repeating students, how their confidence level affects their responses to test items, and the causes for errors and misconceptions. A total of 74 Year 11 repeating students participated in this study and nine Year 11 mathematics teachers were surveyed. The sets of data were collected from the student test performance, student confidence level scale, and student and teacher questionnaires. The quantitative analyses of the students' results suggested that their test performance was not notably affected by their confidence. However, students' misconceptions seemed to have a significant impact on their progress and achievement in the test. The findings also suggested that students' errors and misconceptions were not only varied but there exist different causes as well. Keywords: Errors and misconceptions, confidence level, causes, secondary school repeaters, progress and achievement.

1 Introduction Mathematics, in its very nature, is full of abstract representations. It is a hierarchical build-up of concepts, skills and facts. The successful learning of mathematics involves a systematic building up of such a hierarchy of concepts [13], and ideas need to be understood and woven together in order for concepts to build on one another [4]. However, teaching in schools seems to be more focused on how to

* Corresponding Author. Email address: [email protected]

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get correct answers using rules or procedures rather than why procedures work. But at the same time, learning can also take place by first learning algorithms, with understanding developing later. According to Angle (2007) [3], many mathematical concepts can be understood only after the learner has acquired procedural skill in using the concept. More than often in schools, teachings of mathematics are more focused on rules, procedures and formulas used to arrive at the correct answers rather than teaching students basic concepts. Rules are reinforced through drills and practices, and in the process of doing that teachers got drifted further away from one very important component in teaching mathematics, which is teaching mathematical concepts. Consequently, learning is based heavily on the acquisition of rules and procedures. Teaching the students only procedural skills will impair learning in the classroom and will not equip students well with the necessary skills mathematically for the future. It is partly true that students will be able to do computation if they are drilled but will not be able to do problem solving and application questions properly because the latter demands both procedural and conceptual understanding. Students follow methods and rules taught to them without knowing or being concerned about why those methods work, which Skemp (1976) [18] referred to as an instrumental understanding. This approach to teaching mathematics persists even if teachers are not oblivious of the numerous errors and misconceptions made by students. Teaching simply the procedures would be like providing students the 'temporary indulgence' as opposed to teaching them the concepts which would mean to give them a 'life-long wealth', and, as the saying goes, 'Give a man a fish, you feed him for a day. But, teach him to fish, you feed him for a lifetime'. As teachers, our primary concern should be to examine each of students' written work diagnostically, to look for patterns so that we can theorise possible causes for errors and misconceptions. Teachers will develop strategies which can be used to encourage students to reflect on their understanding and make connections between new ideas and existing knowledge [1]. The research study investigated the following: Students' performance in the given test, and the correlation between their answers and confidence level, and student errors and difficulties in the mathematical learning process by analysing students' work diagnostically. The research questions investigated are given below: Is there a correlation between students' confidence level and their test answers? What type of errors and misconceptions did the students demonstrate in their test? 2 Students' Errors and Misconceptions Learning is a continuous process that involves active participation of students even though instruction clearly affects what students learn. Constructivists such as Piaget (1972) [10] and Skemp (1976) [18] viewed learning as; knowledge is not constructed solely from experience but rather, a blend of experience and present knowledge structures. Piaget (1972) [10] suggested that mental structures or schemata are constructed through interaction by processes called assimilation and accommodation. Assimilation is a process whereby new ideas are fitted to what a child already knows (existing schema). Accommodation, on the other hand, is a process whereby the existing schemata have to be restructured to fit new information. Once a schema or concept is formed, it is stable and resistant to change. A student's existing schema or concept will therefore determine what he or she learns from experience or instruction. Skemp's theory (1976) [18] suggests that a concept is activated in the mind when an example of it is encountered. He pointed out that in order to develop good concepts; good examples of the concept are required. Mathematics learning is cumulative, that is, new knowledge gained is linked to the previous knowledge. Hence, if a student is unable to ''assimilate'' and ''accommodate'' this creates a gap in the learning of the concept, and in turn, leads to mathematical errors or misconceptions. Making errors in computation is not utterly bad. It is a significant part of the learning process if these errors are dealt with diagnostically. Most student errors are not of an accidental character, but are attributable to individual problem solving strategies and rules from previous experience in the mathematics classroom [7, 8], incompatibility with teachers' instruction or techniques, or students observed patterns and inferences during instruction [4].

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According to Radatz (1980) [12], students' errors are causally determined, and very often systematic. Systematic errors are usually a consequence of student misconceptions. These can include failure to make connections with what they already know. There are instances where students connect new information with pre-conceived knowledge; only those preconceptions are wrongly understood. Students may connect pattern with a misconception and thereby learn an erroneous procedure [4]. Ashlock (2002) [4] further stated that misconceptions and erroneous procedures are results of overgeneralisation and overspecialisation of rules in an effort to make sense of new information. Unless pedagogical actions are taken or interventions done by teachers, some of these errors will persist for a very long time. There are beliefs held by students that inhibit learning from errors and one of the beliefs is they cannot learn from the mistakes [6]. Student established a robust structure that there is no connection between right and wrong ways of doing mathematics, and those such beliefs are more keen to go back to the beginning of a question and ignore errors in the solution. Another belief is that mathematics consists of disconnected rules and procedures [19, 20]. Students who hold such beliefs perceived mathematics as not meant to make sense [5]. Student errors are unique and they reflect their understanding of a concept, problem or a procedure. Analysing student errors may reveal the erroneous problem-solving process and thus provide information on the understanding of and the attitudes towards mathematical problems. Upon analysing performance tests in solving text problems, erroneous patterns demonstrated by students are due to other language difficulties, inadequate understanding of texts, or incorrect number determination [12]. Student errors are usually persistent unless the teacher intervenes pedagogically. By examining each of their written work diagnostically, teachers would be able to look for patterns and hence find possible causes for errors and misconceptions. Subsequently, teachers will develop strategies which can be used to encourage students to reflect on their understanding. Concepts and schemata are stable once they are formed and are held to be resistant to change. Thus, good examples of concepts are required in order for good concepts to be established as suggested by Skemp (1976) [18]. However, students are not always successful in acquiring or developing correct conceptual structures which resulted in misconceptions. Misconceptions and errors must not be seen as obstacles or 'dead ends', but must be regarded as an opportunity to reflect and learn. Teachers should recognise these misconceptions; prescribe appropriate instructional strategies to be more diagnostically oriented in order to avoid any subsequent major conceptual problems. Diagnosis should be continuous throughout instruction. 3 Methodology 3.1. Participants The participants in this research study included 74 students who were repeating Year 11 and nine Year 11 mathematics teachers. The participating students were chosen from three separate classes, namely X, Y and Z in the sample school. The first author had a professional connection with the sample school and thus this provided her with an easy access for convenient sampling purposes. In addition, she taught some of the students and having been their previous mathematics teacher, a rapport had been established between them. The two groups of students, from class X (N = 36) and class Y (N = 32) were the all-repeaters classes, while the other group (N = 6) was from class Z, a mixed-repeater-fresher class. There were only two all-repeaters classes in the school accommodating majority of the unsuccessful candidates from the 2009 Brunei Cambridge General Certificate of Education or the BCGCE O Level Examination1. Of the participating students, 21% were 16 years of age, 67% were 17 years of age, and the remaining 12% were in the range of 18 to 20 years of age, as of the time of study. About 35% of the sample population failed which was categorised as U, 54% only managed to get pass which was categorised as either D or E, and 1

.This group of students was not able to further their studies in the sixth form colleges because they did not fulfil the entry criteria set by the Ministry of Education which were, obtaining five BCGCE O Level credits in subjects including Bahasa Melayu (or Malay Language) and English Language; or four credits from Bahasa Melayu, English Language and two English-medium subjects such as Mathematics.



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the remaining 10% obtained credits B or C in their Mathematics BCGCE O Level Examination the year before this study took place. 3.2. Instruments Data were collected through a written test consisted of 13 questions, and surveys given to both participating students and teachers. The written test was designed to cover various topics in the mathematics syllabus, and as such provide coverage of a wide range of misconceptions. The test items covered topics on numbers (fractions and decimals), Algebra, Statistics, Geometry, measures including conversions of units of measurement and area, and also probability. The test items produced were either obtained from the past years examination question books or reinvented with reference to some mathematics textbooks or workbooks. In this paper however, only the analyses of items pertaining to fractions, decimals and algebra will be reported. It should be noted that the written test had been previously validated because it was used as an instrument for a previous mini research conducted by the first author. The Cronbach alpha value of the test was 0.9, suggesting that the test items had relatively high internal consistency. 4 Results and Discussion 4.1. General Analysis of Test Overall results of the test indicated only 4% of students passed the test, whose total marks were above 50%. Entries in Table 1 show the mean scores and the range of scores on the written test for the three classes. There was no student who got all the questions correct. The highest overall score was 37 out of 60 from Class X and the lowest was 2 out of 60 from Class Y. The class means did not vary much, being between 12.8 and 16.8. The overall mean result of the test was 14.8 (equivalent to 24.7%) with a standard deviation of 8.3. This value indicates little variation in the student test scores, suggesting this is a homogenous group.

Class X Y Z Overall

Table 1: Results on written test, for the three classes of students (N = 74) Number of Mean Standard Lowest Students Score/60 Deviation Score/60 36 16.3 8.7 4 32 12.8 7.8 2 6 16.8 7.7 6 74 14.8 8.3 2

Highest Score/60 37 29 28 37

There were also students who did not attempt (or avoided) questions in the written test, especially on items on statistics and probability. In trying to extract the reasons, some students gave responses to the first author, such as not understanding the question, forgot the formula, found the question difficult, and also to a point of being embarrassed or humiliated if the answers given were incorrect. Additional information in relation to the 74 sample students are provided as follows. Students in classes X and Y were required to register at least four subjects for the BCGCE O Level sittings during the year of this study. Students in these groups were from the 'Art Stream', which meant that they were not taking any of the three pure sciences, namely Physics, Chemistry, and Biology as their subjects. The other six students in the latter group were the only repeaters in class Z, a class that offered a non-BCGCE O Level English Language subject called 'English for International General Certificate of Secondary Education', abbreviated as the IGCSE English. All these students either voluntarily or had no choice but to re-take the mathematics subject that year.



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4.2. Analysis of Confidence Level In theory, it was believed that confidence affects performance. So ideally if a student is confident, he or she responses well to questions and hence, excel in examinations. However, in this study such correlation cannot be established. A sound conclusion cannot be drawn to validate confidence affecting responses. This is due to inconsistent patterns observed in the collected data, and there is no clear-cut division between high and low confidence level, and, correct and wrong answers. Even though statistics showed that most students who were confident, were able to answer correctly in some of the questions in the test, there were instances where students who were fairly confident but were not able to give correct answers. Similarly, there were few occurrences where students were not confident but were able to give correct responses. From the analyses, it was observed that for most items in the correct response category, the confidence level 0 was much less frequent than the higher confidence levels (1 and 2). This shows that, in general, students were quite optimistic about their correct answers. However, similar pattern was also observed in few items in the incorrect category which implies students were not always aware of their misconceptions. The confidence profiles of correct and incorrect responses to test items are shown in Figures 1 and 2.

Figure 1: Confidence examination for correct answers

Figure 2: Confidence examination for incorrect answers

The Pearson correlation coefficient for the association between correct answers and average confidence is found to be 0.62, which is considered marginal and not quite impressive. The association is still weakly positive or moderate even though there were many instances where students were quite optimistic about their correct answers. It was observed too that there were occurrences where students who were quite pessimistic gave correct answers. The association between incorrect answers and average confidence is insignificant being -0.28. The scatter plot demonstrates irregular or inconsistent dispersion of points. Hence, a concrete conclusion cannot be drawn here to verify incorrect responses correlates with low confidence. Supposedly students who scored highly in tests are very confident and those who scored poorly are less confident but the results in this research shows variability. Some students are confident even if their test scores are poor while some are pessimistic even if they score better marks than their counterparts. The scatter plot in Figure 3 reveals the weak positive association between average confidence index and total test score, and the corresponding Pearson correlation coefficient is 0.52.



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Figure 3: Confidence examination for total test score

4.3. Analysis of Individual Test Items on Fractions The responses to three questions on fractions show that they do have problems understanding concept of fraction. Question 1a in Section A (or, AQ1a) was on division of mixed number by a whole number. About 12.5% of the incorrect responses were caused by careless mistakes while others were due to errors shown in Table 2. Some of these students were observed to commit more than one type of the listed errors in their worked solutions. Table 2: Types of error identified in students' worked solutions to question 1a in section A Identified Error % Committed 'Whole number to fraction' Error 37.5% Conversion Error 33% Inverting the wrong fraction 25% Cancelling error 8.3% Incorrect Operation- cross multiplication 8.3%

Selected samples of the students' worked solutions that demonstrate the types of error stated in Table 2 are given in Figure 4. Conversion errors are detected in solution (iii); and errors on inverting fractions are observed in solution (iv). (iii)

(iv)

1 2 6 4

1  7  6 1 6

1 2 6 4



9 6 3 27   2 24 1  13

1 2

Figure 4: Samples of selected worked solutions showing error patterns in AQ1a

While most students knew how to convert mixed number to improper fraction as a basic principle when doing multiplication or division, there were still few who did. Those who knew might be able to do the conversion correctly, or might have problem remembering how to operate on the digits in order to get the new numerator. Solution (iii) shows that the students add all digits in the mixed number (2 +4+1=7) to get the improper fraction 7꞉4 and 7 (omitting the denominator). While most of the students in this study knew that they need to invert fractions when division changes to multiplication but some did not fully grasp the principle. Instead of inverting the divisor, they inverted the dividend, or some were able to recognise which fraction to invert but did not quite get it right. For instance, in solution (iv), the students thought 6: 1as the reciprocal of 6.



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4.4. Analysis of Individual Test Items on Decimals Questions 3a and 3b in Section A are on multiplication and addition of decimals respectively. These two questions are among the easiest in the test with each having more than 50% correct. However, the major problem identified in both questions is the dislocation of decimal point. In AQ3a, students got distracted by the decimal point when trying to do multiplication but ignored decimal point when trying to do addition in AQ3b. Some of the students' worked solutions are illustrated in Figure 6. AQ3a) 7.89  3.2 (i)

(ii)

AQ3b) 15.7 + 3.731 + 4.91 (i)

(ii)

7.89  3.20 15.78 234 8 250.58

15.7 4.91 20 6.1

20 6.1  3.731 20 9 .731

3 .2  7.89 25.78  23.67 49.45 15.7  3.731  4.91 157 491 648

3731 648 0.004379

Figure 5: Worked solutions showing error patterns in AQ3a and AQ3b

4.5 Analysis of Individual Test Items on Algebra The three questions on Algebra in this test were from different units in the mathematics syllabus. Question 2a in Section A was simplifying expression in index form; Question 2b in Section A was on solving quadratic equation which was already in the factorised form; and Question 4 in Section B was solving simultaneous equations. BQ4 has the most number of negligence compared to the other two algebraic questions, while AQ2a has the least negligence and the highest number of correct responses. In retrospect, solving equation, be it quadratic or simultaneous, has always been the least popular kind of question among low-achievers in mathematics. In this section, only AQ2a and AQ2b are discussed. In AQ2a, 40.5% of the incorrect responses were results of 'vanishing unknown' error, and 35.1% were due to wrong application of law of indices. Some of the students' worked solutions are shown below in Figure 6. (i)

25 x  5 x 2

4

(ii)

252 54  52 

(iii)

 25 x  5 x  5x

2

1 4

25  5  125  1252

  2  4  2

25 x 2  5 x 4 2

25 x 2  5 x 4  (25  2)  (5  4)  50  20

1 4

(iv)

25 x 2  5 x 4  25  5 ( x 2  x  4 )  1  1  5  1 x 2  4  2   5  2 x x  x 

Figure 6: Worked solutions showing error patterns in AQ2a



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The 'vanishing unknown' as illustrated in solutions (i) and (ii) was due to students overgeneralising 'cancelling common factors'. They perceived exponential form as two separate terms instead of one, that is, the base is detached from the power. Furthermore, powers or indices are only sensible to be used to numbers and not unknowns. Students with this kind of perception lack understanding of basic algebra, and of what it means to write in index form. Solution (iii) demonstrates incorrect application of the law of indices (adding indices when they are supposed to subtract) and misconceived negative index. In solution (iv) students associated negative power to denominator, or denominator therefore negative power. Students always have the tendency to stereotyped procedures with certain types of problems. In AQ2b, students thought that every bracket must be expanded or need to be eliminated; or in solving quadratic equations, they must do factorisation even though it is unnecessary to do so. About 80.4% of incorrect responses to this question were results of doing expansion and worse still, doing it incorrectly. Solutions (i), (ii) and (iii) in Figure 7 below demonstrated how students eliminated the brackets by adding terms within a bracket (observed in the first two solutions), or gathering like terms together (in solution iii). (i)

(2 x  3)( x  2)  0 5  2 7

(ii)

(2 x  3)( x  2) x  5  5x

(iii)

(2 x  3)( x  2)  0 2x  x  3  2  0 3x  1  0

Figure 7: Worked solutions showing error patterns in AQ2b

5 Conclusions and Recommendations In response to the research question on whether or not there is correlation between confidence and answers, it is suggested by the analysis of the students' results that their test performance was not notably affected by their confidence. There is no clear-cut division between high and low confidence level, and correct and wrong answers. Some students who were fairly confident were not able to give correct answers but in contrast; students who were not confident were able to give correct responses. The study has also shown that students, even at the end of their secondary schooling are still struggling with some of the fundamental concepts in mathematics. The level of misconceptions seemed to have a significant effect on students' progress and achievement in high stakes tests or examinations. Similar findings can also be found in previous studies [2, 13, 14, 15, 16]. It is also suggested that students are always not aware of having a misconception. Fundamental mathematical misconceptions may originate in the primary stage of schooling, but, students develop even more 'robust' misconceptions in the secondary level as a result of inattention. Some of the sample students' errors and misconceptions revealed in this study were maybe a result of inattentiveness on the part of classroom teachers. In the secondary class teaching, teachers assumed that students knew very well the most simple and fundamental concepts such as finding equivalent fractions before doing addition and subtraction because such were imparted when students were firstly taught at the primary level. There were various types and causes of errors as demonstrated in the students' written tests. For instance, students' failure to execute subtraction of fractions, or multiplication of decimals was due to their lack of number sense and inability to do estimation. Number sense is the most basic component of number concepts taught in the primary level, and without which students will have difficulty learning to compute. Estimation is essential not only to solve problems which do not require an exact value, but also to ensure answers are reasonable or not. Some other causes of misconceptions in this study were faulty prerequisite knowledge or faulty existing schema, misinterpreting symbols, lack of understanding on the relationship between units of measures, lack of understanding of operations, and overgeneralising and overspecialising procedures and formulas. Teachers should be made more aware of how misconceptions might arise and employ strategies that incorporate errors in their teaching. They need to examine each student's written work diagnostically rather than just scoring papers. What strategies do



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teachers in this research study employ to correct errors and misconceptions? Many of the teaching strategies stated in this research survey have been made into practice by the sample teachers. However, teachers should also provide exercises that will stimulate detecting, reasoning about and correcting errors by including, for instance false mathematical statement or erroneous worked-out solutions in tests and examinations. Students' written work allows teachers to detect errors and misconceptions and therefore will help teachers to plan diagnostic instruction. However, in tests and examinations, often there are few if not many questions that are not attempted (termed negligence) by students. The sample students provided reasons such as not being able to understand question properly; questions were too wordy or questions that needed more thinking and problem solving (refer also to Pungut & Shahrill, 2014 [11]); did not remember formula; to avoid future humiliation if their answers were wrong; or the questions were simply difficult. The omitted questions were found to be related to the topics that the sample students affirmed difficult such as mensuration, probability and statistics. The data in this research study were collected from Year 11 repeaters in a single-sex school only, and hence the results only exclusively represent that school. However, these results may have the potential of helping teachers from other schools to understand the problems faced by not just Year 11 repeaters, but low-achievers in mathematics in general (refer also to Mundia, 2010 [9]). It can be used as a tool to inform teachers of mathematical concepts that should be addressed even if they are as fundamental as units of measure, decimals or fractions, with careful thought and planning. This study provides teachers awareness of potential misconceptions so that they can be prevented in the future. The data gathered from this study comes from one of the secondary schools in Brunei Darussalam. The results of this study singularly represent the sample school and may not be a general portrayal of all Year 11 repeaters. However, these results may be useful to teachers at other schools to gain insight on the level of understanding of their respective students, particularly the more senior of the secondary level students. There is a possibility that similar misconceptions occur elsewhere as within the sample students. References [1] J. Anderson, Where did I go wrong? Teachers' and students' strategies to correct misconceptions in secondary mathematics classrooms, In H. Forgasz, T. Jones, G. Leder, J. Lynch, K. Maguire & C. Pearn (Eds.), Mathematics: Making connections, Australia: The Mathematical Association of Victoria, (1996) 1-11. [2] L. H. Ang, M. Shahrill, Identifying students’ specific misconceptions in learning probability, International Journal of Probability and Statistics, 3 (2) (2014) 23-29. http://article.sapub.org/10.5923.j.ijps.20140302.01.html [3] D. Angle, What is conceptual understanding? (2007). Retrieved from https://www.maa.org/external_archive/devlin/devlin_09_07.html [4] R. B. Ashlock, Error patterns in computation: Using error patterns to improve instruction, New Jersey: Pearson Education, Inc, (2002). [5] M. Barnes, Constructivist Perspective on Mathematics Learning, Reflections, 19 (4) (1994) 7-15. [6] R. Borasi, The invisible hand operating in mathematics instruction: Students' conceptions and expectations, In T. J. Cooney & C. R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s, Reston, Virginia: NCTM, (1990) 174-182.



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[7] S. H. Erlwanger, Benny's Concept of Rules and Answers in IPI Mathematics, Journal of Children’s Mathematical Behavior, 1 (2) (1973) 7-26. [8] H. Ginsburg, Children's arithmetic: The learning process, New York: D. Van No strand Company, (1977). [9] L. Mundia, Problems in Learning Mathematics: Comparison of Brunei Junior High School Students in Classes with and without Repeaters, Journal of Mathematics Research, 2 (3) (2010) 150-160. http://dx.doi.org/10.5539/jmr.v2n3p150 [10] J. Piaget, The epistemology of interdisciplinary relationships, Paris: Organisation for Economic Cooperation and Development, (1972). [11] Hj Mohammad Hairol Azaman Hj Pungut, M. Shahrill, Students' English Language Abilities in Solving Mathematics Word Problems, Mathematics Education Trends and Research, 2014 (2014) 111. http://dx.doi.org/10.5899/2014/metr-00048 [12] H. Radatz, Students' Errors in the Mathematical Learning Process: A Survey, For the Learning of Mathematics, 1 (1) (1980) 16-20. http://www.jstore.org/stable/40247696 [13] J. Ruberu, How Mathematical Concepts are Understood and Misunderstood, Science and Mathematics Education, 1 (2) (1992) 2-6. [14] M. Shahrill, A further investigation of decimal misconceptions held by primary and secondary students, M.Ed. Dissertation, University of Melbourne, Melbourne, Australia (2005). [15] M. Shahrill, From the general to the particular: Connecting international classroom research to four classrooms in Brunei Darussalam, D.Ed. Dissertation, University of Melbourne, Melbourne, Australia (2009). [16] M. Shahrill, Investigating decimals misconceptions: Cross-sectional and longitudinal approaches, Saarbrücken, Germany: VDM Verlag Dr. Müller, (2011). [17] M. Shahrill, Clustering of decimal misconceptions in primary and secondary classes, International Journal of Humanities and Social Science, 3 (11) (2013) 58-65. [18] R. R. Skemp, Relational Understanding and Instrumental Understanding, Mathematics Teaching, 77 (1976) 20-26. [19] D. Tirosh, Inconsistencies in Students' Mathematical Constructs, Focus on Learning Problems in Mathematics, 12 (3&4) (1990) 111-129. [20] S. Vinner, Inconsistencies: Their Causes and Function in Learning Mathematics, Focus on Learning Problem in Mathematics, 12 (3&4) (1990) 85-98.
