ERROR ANALYSIS IN MATHEMATICS: A SYSTEMATIC META SYNTHESIS STUDY OF 1963-2018

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International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 11, Issue 4, April 2020, pp.603-614, Article ID: IJARET_11_04_058
Available online at https://iaeme.com/Home/issue/IJARET?Volume=11&Issue=4
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
DOI: https://doi.org/10.17605/OSF.IO/TMCS4
© IAEME Publication Scopus Indexed
ERROR ANALYSIS IN MATHEMATICS: A
SYSTEMATIC META SYNTHESIS STUDY OF
1963-2018
Dr. R. D. Padmavathy
Assistant Professor, Department of Education, Tezpur University (A Central University),
Assam, India
ABSTRACT
This study was conducted with the objective to collect the classifications of
mathematics error analysis models given by Donald’s, Radatz, Hadar et al., Linchevski
and Herscovics, Hirst, Robert, Hardt, Davis, Nolting, Makonye and different
approaches in error analysis assessment. In addition to that study, a great deal of effort
has been invested by the researcher to collect the error analysis models in mathematics
as well as the collection of previous research conducted in the mathematical error
analysis were reviewed and errors were synthesized. For this researcher adopted a
qualitative approach. To map the pattern of research conducted in error analysis in
mathematics systematic Meta synthesis method was adapted to document researches
carried out to identify different types of errors in the period of 1963-2018.
Key words: Error Analysis, Mathematics, Meta- synthetic study, Classification.
Cite this Article: R. D. Padmavathy, Error Analysis in Mathematics: A Systematic
Meta Synthesis Study of 1963-2018, International Journal of Advanced Research in
Engineering and Technology (IJARET), 11(4), 2020, pp. 603-614.
https://iaeme.com/Home/issue/IJARET?Volume=11&Issue=4
1. INTRODUCTION
Mathematics is a basic science, very important subject, prerequisite for acquiring and enhancing
the other abilities widely use in all aspects of human life. But many students face difficulty in
understanding mathematical concepts and also not cognitively prepared. As a result, scores of
mathematics was less than as expected. Number of researches had been carried out to
understand the factors responsible for low achievements and attempt had been made to find the
causes for difficulties. Success of mathematics mostly relies on the effective teacher and their
teaching methods followed by students practice. As a prerequisite it is most important for the
teachers to understand their learners learning styles, difficulties and the kinds of mistakes they
do as well as the misconceptions they have because it only can help the teachers to design the
effective teaching learning process. Student’s errors often provide insight into student’s
misunderstandings about a particular mathematics concept or skill. Learning is a continuous

Error Analysis in Mathematics: A Systematic Meta Synthesis Study of 1963-2018
process, mathematics learning is cumulative and in that if students unable to assimilate and
accommodate this leads to mathematical errors and misconceptions ( Sarwadi & Shahrill, 2014)
2. ERROR ANALYSIS
Error analysis is a one of the important research strategy for clarifying the fundamental
questions of mathematics learning. It provides a rich source of knowledge about the processes
and influences involved in mathematical problem solving. According Radatz (1980) error
analysis gains two importance’s in two respects. First with regard to the requirements of
academic practice, as an opportunity to diagnose learning difficulties, as a method of
developing criteria for differentiating mathematical education and as a means to create
awareness and support for the performance and understanding of individual students. Second,
error analysis seems to be a remarkable starting point for research on the mathematical teaching
–learning process. Analyzing errors made by students provides teacher’s insight regarding their
students “procedural and conceptual misunderstandings” (Mercer & Mercer 2005). Errors that
students make can sometimes be even more informative to teachers than correct responses.
3. ERROR ANALYSIS IN MATHEMATICS EDUCATION
Error analysis has been a major theme of research for a long period in mathematics education.
A seminal study conducted by Roberts (1968) sparked off this area of investigation. To date,
the topic of error analysis has involved not only the identification of pupils' errors but also their
thinking processes which lead to mathematical difficulties (Rees and Barr, 1984). Much of the
research done on the types of errors that occur in mathematics took place in the 1980’s, but is
still very valid for present investigation.
Radatz (1979) provided a good definition of error,
• Errors in the learning of mathematics are not simply the absence of correct answers or
the result of unfortunate accidents. They are the consequence of definite processes
whose nature must be discovered.
• Errors seem to be possible to analyze the nature and the underlying causes of errors in
terms of the individual’s information-processing mechanisms.
• The analysis of errors offers a variety of points of departure for research into the
processes by which children learn mathematics (p. 170).
Student’s committing errors in mathematics are not simply the result of ignorance or
situational accidents. They are the results of student’s individual difficulties which need more
attention and concentration. It shows that the students have failed to understand the required
concepts, techniques, problems etc.
4. DIFFERENT APPROACHES IN ERROR ANALYSIS ASSESSMENT
Error analysis has been used widely by many researchers as a means of making inferences about
the nature of mental processes in mathematical thinking. Mulhern (1989) suggests four types
of approaches to study errors in mathematics.
• Counting of the number of incorrect solutions
• Analysis of the types of error
• Analysis of error patterns
• Constructing problems in such a way as to induce errors in individuals.
According LimKokSeng (2010) the process of analysis consisted of the following steps,
• Identification of errors

R. D. Padmavathy
• Description of errors
• Classification of errors
• Quantification of errors and making inference of causes of errors
This analysis of students’ works focused on procedural and conceptual errors. The
quantitative process is to identify the similarities and differences in and this process of data
provides a link to error patterns made by high, medium and low ability students. This approach
of analysis led to the causes of errors.
5. OBJECTIVE OF THE STUDY
• To collect the classifications of mathematics error analysis models and different
approaches in error analysis assessment.
• To collect, review and synthesize the mathematical errors identified in previous
research.
6. RESEARCH DESIGN
To map the pattern of research conducted in error analysis in mathematics systematic Meta
synthesis method was adapted to document researches carried out to identify different types of
errors in the period between 1963-2018. Meta synthesis is conducted by summarizing the results
of the research on analysis of student’s errors in mathematics (Naraini,et al., 2018)in the period
between 1963-2008. Documentation Techniques used for data collection. As rightly pointed
out by Naraini,et al., (2018) “The urgency of this research is to provide empirical findings for
efforts to improve the quality of elementary school mathematics learning. The data were
analyzed by focusing on the content, namely (1) conducting literature search; (2) conducting
appropriate screening and selection of research results; (3) synthesizing qualitative findings;
and (4) compiling a final report”.
7. SEARCH METHODS
For collecting the information different sources like educational research surveys published by
CASE, NCERT, Dissertations International Abstract, mathematics educational journals,
published books, published and unpublished Ph.D. thesis, dissertations, proquest dissertation,
ERIC and educational research related websites, mathematical blogs, have been surveyed.
Moreover the present study is limited to articles available on the internet based on “error
analysis in mathematics” and other studies related to mathematics and misconceptions are not
considered.
The results of the findings are presented as follows
7.1 Errors analysis and Mathematics Achievement
The review examines research that focuses on both the computation and word problem. Each
study produces its own model of error types. Though, they do have some common categories
of errors among them. One of the main methods used to analyze students’ errors is to classify
them into certain categorizations based on an analysis of students’ behaviors. Through using a
cognitive information-processing model and considering the specialties of mathematics
(Xiaobaoli, 2006). The summary of different types of errors identified from the previous studies
is given below in Table 1.

7.2 Classification of Errors
Many researchers have attempted to understand the errors and misconceptions committed by
students in learning mathematics. This led them to classify errors into categories and to list the
misconceptions in order to understand the error more deeply.
Classification 1: Donald’s Error Classification (1963)
According to Donaldson (1963) errors are frequent in mathematics teaching and learning. He
proposed three categories of generic errors in mathematics; they are structural errors, executive
errors and arbitrary errors. They were very clear even five decades later. The Donald’s Error
Classification was discussed below:
• Structural Error: This type of error arose due to mistaken perception about the nature
of mathematical concepts, some failure to appreciate the relationships involved in the
problem or to grasp some principle or essential rule to solution.
• Executive Error: This type of error involves failure to carry out manipulation or
procedures even though the required concepts have been understood. These may be
referred to as procedural errors (Orton, 1983a). Executive errors may occur with or
without understanding of underpinning mathematical concepts.
• Arbitrary Error: This type of error occurs when the learner ignores part of available
information while acting on the rest.The same type of errors identified in the works of
Orton (1983) with same name and Biggs & Collis (1982) named Pre-Structural, Uni-
Structural, and Multi Structural errors.
Classification 2: Hendrik Radatz(1979)
Hendrik Radatz (1979) classified errors based on the information processed by the learners.
This provided a cognitive model of the causes of errors and suits for all the branches of
mathematics. His categories are:
• Errors due to deficient mastery of prerequisite skills, facts, and concepts (including
ignorance of algorithms, lack of mastery of basic facts, incorrect procedures in applying
mathematical techniques, and insufficient knowledge of necessary concepts and
symbols). Chuaboon Liang and Eric Wood (2005) also identified this error.
• Errors due to incorrect associations or rigidity of thinking,
• Errors due to the application of irrelevant rules or strategies
• Errors due to difficulties in obtaining spatial information
• Errors due to language difficulties.
Classification 3: Movshovitz-Hadar, Zaslavsky and Inbar (1987)
Another broad classification of errors was done by Nitsa Movshovitz-Hadar, Orit Zaslavsky
and ShlomoInbar (1987) is informative. Their easy-to-use classification contains six categories
of errors in high school mathematics:
• Misused data (neglecting given data, adding irrelevant data or incorrectly copying data).
• Misinterpreted language (incorrect translation of facts from one language to another,
where the languages are possibly symbolic).
• Logically invalid inference (error due to false generalization of old knowledge into new
knowledge)
• Distorted theorem or definition

R. D. Padmavathy
• Unverified solution (each step is correct but the final result is not a solution to the
original question). Corder (1987) called this as ‘unsystematic errors or slips’
• Technical error (including careless errors, computational errors e.g. 7 x 8 = 54, or
incorrect data extraction, manipulation errors in algebra e.g. leaving out closing
brackets, errors due to failure to carry out calculations and process computational
algorithms, reading data from tables, algebraic errors such as reducing writing )
Classification 4: Linchevski and Herscovics (1994)
Liora Linchevski and Nicolas Herscovics (1994) identified five cognitive obstacles in learning
algebra. As their subjects had just begun algebra, the errors identified are based on arithmetic.
The list is included here to give a sense of the wide variety of possible errors, particularly in
early algebra learning.
• Problems with order of operations – giving preference to addition over subtraction and
to subtraction over multiplication.
• Not seeing the canceling effects of binary operations
• Problems with factoring out –1. (Later LioraLinchevski and DroraLivneh (1999)
• Detachment of a term from its
• ‘Jumping off with the posterior operation
Classification 5: Hirst (2003)
Hirst (2003) proposed three more error categories: procedural extrapolation, equation balancing
errors, and pseudo –linearity. This type of error classification is totally different from
Donaldson’s and Orton classification. These errors arise out of the learner's predisposition to
generalize what they have learnt to new situations by way of assimilation and accommodation
(Piaget, 1968; Siegler 1995).
• Procedural Extrapolation Error: This type of error occurs when a learner attempts to
extend a previously learnt procedure to a new situation similar to one learnt in the past.
It is an overgeneralization of a valid procedure in new situations where it causes errors.
• Equation Balancing Errors: This type of error occurs from learners emanating the earlier
learnt principle interrupt. Principle: you do the same thing to both sides of an equation
and they are still equal.
• Pseudo –Linearity Error: This type of errors occurs before students encounter linearity
in an overt, systematic manner in linear operators in differentiation and integration or
linear transformation in linear algebra. These errors emanate from simplifying
expressions like x(y + z) = xy + xz. This type of error was first identified by Fischebin
&Barash(1993) and named as linearity errors and Norman & Pritchard (1994)label it
as mis-generalized distributive.
Some common example of pseudo linearity errors are
● (a+b)2
= a 2+
b2
;
● log(x – y) = log x- log y ;
● Square root of ( 9x2
+4)= 3x+2
7.3 Others Error Classification
Other quite common mathematical errors identified by researchers were listed below: Robert
(1968) identified wrong operation, defective algorithm, random response and obvious

computation error. In addition to this error Engel Hardt (1977) proposed basic facts error,
incorrect /inappropriate operationalerror, grouping error, inappropriate inversion, identity error,
zero errors and incorrectly applying proportional reasoning by Hart (1984). Cipra(1983)
remarks one of the error learner make was failure to check one’s solution. Davis (1984) said
this error forms are related to meta-cognition or “inside critics”.Fischbein and Barash (1993);
Chuaboon Liang and Eric Wood (2005) listed errors formed due to overgeneralization.
Nolting (1997) identified six more errors that learners make while writing a test. They are
misread direction error, careless errors, conceptual error, application error, missing question
and not completing a problem. Megan Elbrink (2007) identified calculation error, procedural
errors, and symbolic error. Makonye(2011) remarks error patterns that are common across
different learners suggest that learners possess similar concepts and procedures and classify the
mathematical errors into domain general errors(DGE) which tend to be general categories in
mathematics and other subjects and domain specific errors (DSE) classification which are
subject based or topic specific. There are also some other common error categories listed in
chapter 2. It is clear that some of the error categorization discussed above has been used in this
study and some were not useful in this research and so are not being referred again.
Table 1 Different Types of Errors obtained from various studies on Error Analysis
Researcher Year Identified Error Type
Donaldson 1963 1.Structural Errors
2. Executive Errors
3. Arbitrary Errors
Robert 1968 1. Wrong Operation
2. Defective Algorithm
3. Random Responses
4. Computation Error
Engelhardt 1977 1. Basic Facts Error
2. Defective Algorithm,
3.Incomplete Algorithm
4. Incorrect Operation,
5.Inappropriate Operation,
6. Grouping Error
7. Inappropriate Inversion
8. Identity Error
9. Zero Errors.
New Man 1977 1. Reading Errors
2. Process Errors
3. Comprehension Errors
4. Question Form Errors
5. Transformation Errors
6. Motivation Errors
7. Careless/Unknown Errors
Brown and Burton's 1978 Systematic Errors in Subtraction
i.e.,"Buggy.
Hendrik Radatz
For all branches of
Maths (Cognitive
based)
1979 1. Difficulties in obtaining spatial information,
2. Deficient mastery of prerequisite facts
3. Incorrect associations or rigidity of thinking
4. Application of irrelevant rules or strategies
5.Pupils' negative 6. Language difficulties
Clement 1981 1. Syntactic Translation Errors
2. Semantic Translation Errors
3. Reversal Error

R. D. Padmavathy
Biggs & Collis–
(SOLO)
1982 1. Pre-Structural Error
2. Uni-Structural Error
3. Multi-Structural Error
Orton 1983 1. Structural Errors
2. Executive Errors
3. Arbitrary Errors .
Flart 1984 Incorrectly applying proportional reasoning
Movshovitz-Hadar,
Orit Zaslavsky ,
ShlomoInbar
1987 1. Misinterpreted Language
2. Technical Error
3. Logically Invalid Inference
4. Misused Data
5. Distorted Theorem or Definition
6. Unverified Solution
Sharma 1988 1. Arithmetic : Basic Facts, Defective Algorithm,
Wrong Operation, and Wrong Order of Operation.
2.Properties of Numbers :Associative,
Distributive Sign Error
3.Procedural: Misuse of The Property of Equality,
+ Property of =, ×Property of =, Coefficient Error
+, Coefficient Error Wrong Inverse Operation
4. Conceptual: Order of Operations, Sign, Like
Terms, Zero Annexation, and Misunderstanding
Constants as Variables.
5. Mechanical/Perceptual: Careless/Random,
Incomplete Operations.
Fong, Ho-Kheong 1993 Schematic Errors
1. Using irrelevant knowledge or procedure
2.Incomplete schema but without any errors
3.Incomplete schema but with errors
4.Complete schema but with errors
5.No solution
Categorical Errors
1.Language 2. Operational
3. Mathematical Thematic
4. Psychological
LioraLinchevski ,
Nicolas Herscovics
1994 1. Problems with Order of Operations
2. Not Seeing The Canceling Effects of Binary
Operations
3. Problems with Factoring Out –1.
4. Detachment of a Term from its Operation
5. Jumping Off with the Posterior Operation’
Tony Barnard’s 2002 1. Misunderstanding the meaning of variables
2. Errors regarding equivalent forms and the
meaning of the equals sign
3. Over-Generalizing
4. Misinterpretation of Questions
5. Confusion between Misapplied Rules
6. Confusing Similar Notation
7.Errors with minus signs
8. Operating on one Part of a Compound Term
9. Errors With Simplifying Fractions
10. Errors with the wrong application of order of
operations

Chua Boon Liang
and Eric Wood
2005 1. Errors due to over-generalization (OG) of
concepts and rules,
2. Errors due to a deficient mastery of concepts,
rules and pre-requisite skills (D)
3. Miscellaneous (M) category
Megan Elbrink 2007 1. Calculation Errors,
2. Procedural Errors,
3. Symbolic Errors
Anita Campbell 2009 1. Due to use of a distorted Theorem,
2. Defective definition or Algorithm
3. Errors from Confusion between Operations
4. Over-Generalizing, Including False-Linearity.
5. Simplifying fractions and other errors.
6. Misuse of the equals sign
7. Distributing / Factoring Minus Signs.
8. Confusing Similar Notation.
9. Misunderstanding Meaning of Variables.
10. Technical Error due to carelessness
Gunawardena
Egodawatte
2009 1. Comprehension Error
2. Encoding Error
3. Verification Error
Lim KokSeng 2010 1. Errors due to incorrect order of operation,
2. Addition of Integers Error,
3. Misinterpretation of symbolic notation
4. Co Join Errors,
5. Coefficients of 1 or -1,
6. Multiplication of Variable Error
7. Negative Pre Multiplier Error
8. Invisible Numerical Coefficient,
9. Distributive errors in bracket expansion
10. Detachment of negative sign error
11.Operations of Variables
12. Exponent Errors - (3 Types)
− Students add the exponents ,
− students misinterpreted the exponent
− Students changed while writing
exponents
Padmavathy 2016 1. Concept error, 2. defective algorithm,
3. misused data error, 4.calculation error
5. technical error
Bayu Gustama
Kurniawan
2014 1.Conceptual Error 2. Procedural/ Skill Errors
3. problem solving/ conclusion
4. Understanding the problem
5. Changing sentences into mathematical sentences
6. Computing 7. Error in planning
8. Errors in implementing the settlement plan
(as cited in Naraini,et al., 2018)
Ratih Rahayu
Agustina
2015
Dhita
Wuryaningtyas
2017
Anggun Dewi
Novita
2017
Ainul Ferianto 2017
Observations from various types of errors identified by the previous researchers were shown
in the previous paragraphs. The summarized network of categorical errors made by problem
solvers were classified in terms of four groups of knowledge: language, mathematics thematic,
operational and psychological was shown in the Figure 1:

R. D. Padmavathy
Figure 1 Summary of the Types of Errors Obtained from Various Studies on Error Analysis
Summary of the Types of Errors Analysis adopted from Ho-Kheong (1993)
Figure 1 shows the reading ability and comprehension are subsumed under language. Errors
which are due to children's inability to read word problems and/or to understand the problem
are classified under this category. Understanding the problem includes interpretation of
mathematical language. Under the operational knowledge, the external operation is concerned
with encoding of the various types of information such as verbal statements, symbols and spatial
figures. Errors which involve encoding words, symbols or spatial features in an incorrect form
are classified under this category. They are classified as external operational knowledge
because the information is retrieved from external sources i.e. the question. For the internal
operation, some activities identified are transformation, recalling, relating computation and
application of information. These activities are classified as internal operational knowledge
because they are operated within the short-term memory of the brain. The mathematical themes
include basic facts, algorithms and concepts. The psychological knowledge involves motivation
and carelessness in problem solvers' written work. Ho-Kheong (1993).
8. SUMMARY OF DIFFERENT CATEGORIES OF ERRORS IN INDIA
In India, sixty one studies done to analyze errors among them forty studies concentrate in the
area of arithmetic. Only nine studies have been done in Algebra. Few studies concentrate on
geometry and only one study was carried out in the area of trigonometry. Eight studies have
been done in the combination of a few topics. They are arithmetic algebra, geometry & modern
mathematics. One study was carried out in combination of all the arithmetic, algebra, geometry,
trigonometry, modern mathematics and statistics (Padmavathy, 2016).
In India during the period of Fourth Survey of research in mathematics education reported
during 1970-1988 only four studies were carried out to analyze errors in mathematics. The Fifth
Survey reported during 1988 to 1992 consists of 47 studies in Mathematics education but none
of them were concentrated on analyzing the mathematical errors. The Sixth Survey reported
during 1993 to 2000 seven studies carried out on errors committed and difficulties faced by
learners in learning mathematics. Few of them are given below
Knowledge
Language Operational Mathematical
Themes
Psychological
Factors
Basic facts,
Concepts,
Algorithms
Motivation,
carelessness
Internal
External
Reading Ability,
Comprehension
Encoding
Words
Symbols
Spatial
Transformation
Recalling,
Relating,
Computation,
Application

• Goel(1996) identified that the total number of errors committed by children in different
grades varied significantly.
• Pal, et al.(1997) found most of the errors were due to process of dualism, response error,
inadequacy in language used in the definition ,rules or procedure names.
• Bhatia (1998) identified around 15 types of common errors.
• Khichi (1998) in different areas of mathematics using the integration and application of
integration besides computational errors using lower level concepts
• Paria( 1999) identified conceptual and computational difficulty in the selected topics.
• Padmavathy (2016) identified five types of errors and the related cognitive factors.
Hardly few studies were conducted to show the effect of remedial measures for weakness.
Remediation was provided by 15 researchers like Das and Barua(1968), Singha(1971),
Rastogi(1983),Goel (1996b) and Subramaniam and Singh(1996) in their studies and found the
remedial measures were effective in improving mathematics achievement-Sixth Survey of
Educational Research.
9. CONCLUSION
Basis of all mathematics achievement or performance is based on the understanding of the
mathematical concept itself (Yorulmaz & Önal,2017). Every teacher gives their best to produce
the best mathematization (learner) with higher order thinking skills. Understanding these types
of errors and misconception on the right time is crucial to provide remedial on the right situation
help the learner to reach the better results in mathematics. The results based on the findings of
the study are an attempt to collect and disseminate the knowledge on Error analysis in
mathematics among the future generations who are interested in mathematics teaching learning
process and researchers. It also helps the future researchers and teachers to prepare a systematic
action plan to overcome the errors and enhance the mathematical teaching learning process.
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1895

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