Chapter – I
INTRODUCTION
Background of the study
People
have been using mathematics from the very beginning of human civilization. It
is believed that mathematics was originated along with the origin of man. Human
beings themselves created mathematics in the need for application to counting a
measuring in relation to both of quantities as well as spatial objects.
Mathematics usually develops as the changed needs of human being.
Regarding
the origin & development of mathematics, H. Preston (in Brij Lal, 2007) has
expressed his view that mathematics was developed from the need of organized
society of people. For instance the primitive tribes living by hunting &
collecting the natural harvest of forest & field, they needed rudimentary
knowledge of counting to communicate numbers to the tribes. This may be the
numbers of animals in a hand or the number of people in a hostile tribe. Also
needed were the measure of size, strength, distance & time however crudely
formulated they might be. A certain primitive awareness of similarities of the
shapes must be present in effort to duplicate arrow heads & implements,
& it is also important to have some means of describing location involving
both concepts, which later developed in to mathematics.
From
the above discussion very simple evidence about the origin of mathematics was
according to the need of human being in the process of civilization & has
been an essential component of human civilization, so, the history of
mathematics is a part of the history of human civilization. This view is
further justified from the history that mathematics was originated in the river
valley civilizations like the Tigris, the Nile, the Euphrates & Hwang Ho, and
the Yangtze etc. By supporting these versions. Eves (1983) stated about the
origin of mathematics as:
“Every
mathematics required a partial basis for it development & such a basis
arose with the evaluation of more advanced from of society. It was along some
of great rivers of Africa & Asia, that the new form society made their appearance
that Nile in Africa the Tigris & Euphrates in Western Asia, the Indus &
the Ganges in South Central Asia & the Hwang Ho & then the Yangtze in
Eastern Asia ……… Thus early mathematics can be said to have originated in
certain area of agricultural & engineering pursuits.”
The
term mathematics has been interpreted & explained in various ways. It is
the numerical & calculation part of man’s life & knowledge. It helps
the man to give exact interpretation to his/her ideas & conclusions. It
deals with quantitative facts & relationships as well as with problems
involving space & form. It enables the man to study various phenomena in
space & establish various relationships between then. It explains that this
since is a byproduct of our empirical knowledge.
Mathematics
is also known as the science of logical reasoning. As Locke has said
“mathematics is a way to settle in the mind a habit of reasoning”, Dictionary
of mathematics “Mathematics is the science of number & space”. (Sidhu 1975)
In
the broad sense mathematics is a way of thinking. Organizing, analyzing &
synthesizing a body of data.
Formal
education was started with the opening of Durbar School in 1910 B.S. the
National Education System (NESP) has realized that a well-grounded
understanding of mathematics is essential for everyday life as for higher study
in the field of science & technology. People in every part of country have
become aware of the importance of Mathematics in school was an optional
subject. The first attempt was the establishment of college of education &
the starting of training for the school mathematics teacher to bring
improvement in mathematics teaching In 2026 B.S. The ministry of education set
up an advisory committee to help revise & improve Primary School Curriculum
including mathematics & adopt effective teaching procedure based upon the
principle of psychology to teaching.
The
nineteenth century was largely devoted to effect of establishing on a firm
logical foundation. Thus creating a division between pure & applied
mathematics accompanied to growth of specialization. Upadhyay (2007) writes
applied mathematics was also developed by other repetitive from other disciplined
such a s physics & engineering. Applied mathematics is not an alternative
to pure mathematics. It is the super imposition of an extra & very
difficult problem of learning mathematics. So, applied mathematics is directly
related to physical, biological social & environment. The formulas &
principle that come from pure mathematics is applicable with physics,
Chemistry, engineering etc. in daily life application. So, for real definition
of applied mathematics is a classical application of mathematics, applied
mathematics is the mathematics with significant, applied mathematics is applied
in real life & applied mathematics is the applied means of having of
people.
“Error
analysis in mathematics teaching strives to identify the nature of error a
learner may commit in dealing with a particular type of assignment. The result
of such analysis may help to the teacher to appreciable appropriate corrective teaching
for the individual learner & make recommendations to the curriculum
developers for producing further instructional material.”
A
systematic analysis & comparison of occurrence of error made by the
students is of considerable importance. In the past, errors were considered as
a bad sign in learning & teaching process. Nowadays the scenario has been
changed completely. Errors no longer are taken as bad sign in learning. it is believed
that, it is natural to make mistake then, perhaps there is no learning. In fact
it is an important component in minimizing the errors in learning process &
developing competence. The study of learner’s error occurrence in solving word
problems has great importance on mathematics teaching & learning. In this
sense, error analysis is a step wise procedure that is used to identify the
errors.
“Matrix”
According to D.T. Finkbeiner, 1960
February, “Matrix theory, or more generally linear algebra is relatively recent
mathematical development. Its roots extend back 100 years to the work of
Hamilton, Cayley & Sylvester, but it has attracted widespread interest only
in the past two or three decades. Today matrices are effective tools in quantum
theory as well as classical mechanics;
in aeronautical mechanical & electrical engineering in statistics &
linear programming & therefore in all the social science which these
theories serve.”He further added on the importance of the concept of “matrix”.
Even a cursory glance at current
mathematical literature reveals that matrix theory is in a stage of active
growth. It is rather surprising therefore that the mathematical background
required for an understanding of matrix theory is sufficiently modest that a
substantial first course can be mastered by undergraduates courses in matrix
theory are currently presented in several ways. A computational course can be
offered in which the calculations themselves are emphasized more than their
meaning. Alternately, the study of matrices can be motivated from the familiar
problem of solving systems of linear equations, the important connection
between matrices & linear transformations being deferred until the end of
the course.” (Finkbeiner, D.T. 1960)
According
to John L. Kelly “we can consider matrix algebra as a sort of hyper complex
number system. We shall see that there are a natural addition and scalar
multiplication. Which are closely related to vector addition & scalar multiplication?
Many of the problems of vector geometry which we have studied have a natural
formulation in terms of matrix algebra & matrix. Algebra will give us
technique with which to attack a wide variety of other problems. We shall see
that each matrix corresponds naturally to a certain sort of function which we
shall call a linear transformation. We could say that the geometry of linear transformation
is identical with the algebra of matrices, although at this stage it is
difficult to label parts of our mathematical theory as algebra or as geometry
for there is no clear distinction.
One
of the new applications of mathematics is the concept of matrices at school
level. The history of the development of matrix concept started only in the
middle part of the 19th century “The matrix algebra devised by the
English mathematician, Arthuz Cayley (1821 – 1895) in 1857. Matrices arose with
Cayley in connection with linear transformation (Howard Eves, 1984).
“The
theory of matrix is a powerful tool in modern mathematics. It is especially
useful in the system of linear equations, linear transformation etc. it has
been found more & more important applications in Physics, chemistry,
engineering & social sciences etc.” (D.R.Bajracharya, R.M.Shrestha, M.B.
Singh, Y.R. Sthapit, B.C. Bajracharya, 2060 B.C.)
With
the redown democracy in Nepal in 1990 A.D. the secondary school curriculum
(CDC) has brought some improvements in school curriculum & text books,
accordingly the improvement of text books of different grades are being
implemented from 1992 A.D. But in the existing school curriculum, the concept
of matrices is introduced in the optional subject of mathematics. There is no
such study carried out on the achievement of the students with appropriateness
of this unit at the secondary level of 10th grade in context of
Nepal.
A
chapter on “Matrices” was recently incorporated in the school mathematics
curriculum as an optional at secondary level in Nepal.
As
a researcher I wanted to investigate why the students commit error while
solving problem in matrix. How could we overcome these problems, so we could
make mathematics more interesting & popular among students? Hence the study
about the errors made by secondary level students in matrix is required.
Statement of the problem
This study is mainly concerned with
the identification of errors committed by the class IX students in matrix.
Especially study6 has sought to answer the following questions:
a) What sort of error do students commit
in solving problem in matrix?
b) Does the error made by students on
solving problem in matrix are different?
c) Does the gender have any effect in
committing error while solving problem in matrix?
Objective of the study
The main objective of this study is
to identify the types of error committed by the students of grade IX in solving
problem in matrix. This study also intended to accomplish the following
objectives;
a) To identify the error committed by
students on problem solving in matrix.
b) To compare the error with respect to
gender of grade IX students in matrix.
c) To analyze the errors made by
students according to Newman Hierarchy of error analysis in solving problem of
matrix.
Significance of the study
Mathematics has been accepted as an
essential component of formal education from ancient period because
understanding & skill in mathematics are fundamental qualification for
literacy” which in inevitable for the civilized society. The students have to
apply mathematical concepts & skills in daily life, in their professional
& vocational fields as well as in their higher study. But it is generally
accepted that students are week in day to day life application of mathematics.
Here
the researcher has tried to study the error made by students of class IX in
matrix learning. This study is about the student’s error in matrix in
knowledge, skills, application & problem solving. The purpose of this
inquiry will identify, a classify by relative frequency, the most common errors
can make sample of pupils in their attempts to solve problem of matrix, so the
study have the following significance.
a) The improvement in the development in
the day to day classroom teaching.
b) To help the teacher educators,
mathematics teacher, policy maker, mathematics educationist, curriculum planner
students.
c) It attempt to analyze & improve
the teaching & learning of this topic.
Limitation of the study
The study has following Limitation.
a) This study will be concern where the
students commit error but not how & why?
b) This study will be limited to error
in skill, application & problem solving of matrix?
c) This study will be conduct to the
grade IX students of Dolakha district.
d) Only the errors made in matrix
learning are included.
Definition of related terms:
Error: - The
first mistake done by the students during the solution of the problem.
Reading Error: - Inability of student to read the words in the question such that he
can’t grasp all the information given on the questions.
Transformation Error: - If the student understood the question but couldn’t
transform it into mathematical expression, this type of error is classified as
transformation error.
Encoding Error: - If the student made the correct solution to the question but could not
express this solution into acceptable written from, this type error is encoding
error.
Comprehension Error: - If the student could not grasp the overall meaning of the
given problem this type of error is classified as comprehension error.
Process Skill Error: - If the student did not know the procedure to carry out the
operation correctly & carryout the solution this type of error is
classified as process skill errors.
Chapter – II
REVIEW OF RELATED LITERATURE
A review of related literature is the
source of the further study of research task. It provides the researcher in
making his problem more realistic precise, researchable & meaningful. It
helps to conduct the research program & gives a better idea of surveying
& research, then it guides towards conclusion. Thus the review literature
is an important & essential guideline of research planning. This chapter
deals with the study of the literature related to this study. Mainly the
literature includes previous thesis, book, journals & internet. Different
related literature would be helpful in understanding different aspect of error
analysis of matrix learning in secondary level. Some research entitles &
their purposive findings are listed below.
Ø
Newman (1977) studied on one hundred & twenty
four low achieving grade six pupils & found that reading, comprehension
& transformation errors made by low achieving pupils accounted for 13%, 22%
& 12% respectively. Thus almost half of the errors occurred in the first
three steps.
Ø
Bhattacharya (1991) prepared a diagnostic test to
diagnose the disabilities in the area of linear equation of grade VII students.
The main objective of the study was to compare the disabilities among different
areas such as knowledge, understanding & application of linear equation.
Ø
Poudel (2002) concluded in the masters degree
thesis entitled; “The occurrence of errors while solving word problems in mathematics by the lower secondary pupils”
that;
i)
There
were 4.88% reading errors, 6.37% comprehension errors, 52.44% transformation
errors, 11.89% process skill errors & 24.42% encoding errors.
ii)
Boys
committed less error then the girls.
Ø
Marahatta (2002) studied on “A case study on
computational error on fraction by grade – VI students in Chitwan district.”
The main findings were,
i)
Students
generally commit more error in addition of fractions then in introduction of
fraction. Thus the conclusion was derived from the result of null hypothesis in
which the error mean was higher in addition of fraction then in the introduction
of fraction.
ii)
Students
higher errors in divisions of fraction is higher then in multiplication of
fraction.
iii)
There
is no effect of sex to commit the errors in areas of operation of fraction
considered in this study.
Ø
Bhaat (2003) studied on “An error analysis in
quadratic equation at grade X.” This study was mainly focused with
identification & comparison of errors committed by grade – X students in
quadratic equations. The main findings were:-
i)
Students
committed more error in knowledge then understanding of quadratic equation.
ii)
Students
generally committed more errors in application of quadratic equations then
understanding of quadratic equation.
iii)
There
is no effect of location committing equal no. of errors in understanding knowledge
of solving & application of quadratic equation.
iv)
The
role of gender is less important to commit the errors in understanding
knowledge of solving & application of quadratic equation.
v)
The
type of school (private or public) is a cause to commit less or more errors in
knowledge of solving & application of quadratic equation.
Ø
Kafle (2006) studied on “error analysis of the
proof of the theorem in geometry in grade X” for this thesis in master’s
degree. The main objectives of this study were;
-
To
identify the errors.
-
To
classify the errors on the basis of a recognized theory.
-
To
indicate possible causes of errors.
Ø
Upadhyay (2007) studied on “the type of error mostly
done by the students of grade – V in Janakpur municipality.” He found that,
-
Students
were observed using their own method with confusion.
-
Students
did use their methods but could not supply enough reason while putting down in
examination paper.
-
Most
of the students could not understand the situation given in language form.
Ø
Panthi (2009) studied on “ an error analysis in
equation of grade – VIII students” aim to;
-
Identify
the error committed by students on problem solving.
-
Analyze
the error committed by the student in problem solving.
-
Compare
the error with respect to gender.
Theoretical Review
Newman Error Analysis Research
Newman (1997), an
Australian language educator who in the mid 1970’s developed a systematic
procedure for analyzing errors made by students responding to written
mathematical tasks. Since 1997 a steady streams of research paper has been
published reporting “Newman data” in the Asia-Pacific Region Australia, Brunei,
Indonesia, India, Malaysia, the Philippines and Thailand. The study was placed
on pencil and paper test. The marked price of a book was $20. However for a
sale 20% discount on the mark price was given; what is the sale price?
According to Newman (1997) a person wishing to obtain a correct solution to
word problems, this must ultimately precede the following hierarchy:
1.
Read
the problem.
2.
Comprehend
what is used.
3.
Carryout
the mental transformation from the words of the question to the selection of an
appropriate mathematical strategy.
4.
Apply
the process skills demanded by the selected strategy and
5.
Encode
the answer in an acceptable written from.
Clements
(1980) illustrated the Newman technique with the diagram shown in figure given
below. According to Clement (1980) errors due to the form of the questions are
essentially different from those in the other categories shown in figure given
below because the source of difficulty resides fundamentally in the question
itself rather than in the interaction between the problem solver and the
question. Someone who had read, comprehend and worked out an appropriate
strategy for solving a problem might decline to proceed further in the
hierarchy because of a lack of motivation.
INTERACTION
BETWEEN THE QUESTION
AND
THE PERSON ATTEMPTION IT.
CHARACTERISTICS 
OF THE QUESTION ENCODING CARELESSNESS
PROCESS
TRANSFORM
COMPREHENSION
QUESTION FORM READING MOTIVAITON
Fig. The Newman Hierarchy of Errors
Causes
Newman (1983) recommended that the
following “question” or a request be used in interviews which are carried out
in order to classify student’s errors on written mathematical tasks:
Classification
|
Typical
Questions
|
Errors
|
1.
Reading
|
Please
read the question to me
|
Do
not recognize key words or symbols.
|
2.
Comprehension
|
Tell
me what the question is asking you to do?
|
Can
read the problems well but cannot comprehend the meaning of the words,
symbols or questions.
|
3.
Transformation
|
Tell
or show me how you start to find an answer to this question.
|
Cannot
transform sentences into mathematical forms.
|
4.
Process skills
|
Show
me how you get the answer. Tell me what you are doing as you work.
|
Can
choose an appropriate operation but cannot complete the operation accurately.
|
5.
Encoding ability
|
Write
down your answer to the question.
|
Can
perform the correct operations but writes the answer incorrectly.
|
If
an incorrect response is given to the question then error is classified
according to where the first “breakdown” occurred in the attempt to get a
solution. If pupils who originally got a question wrong got it right when asked
by an interview to do it once again the interviews should still make the five
requests in order to obtain information on whether the original error would be
attributed to carelessness or motivational factors.
In
her initial study, Newman (1997) found that reading comprehension and
transformation errors made 124 low achieving grade VI pupils accounted for 19%,
22% and 12% respectively of all errors made. Thus, almost half the errors made
occurred before the application of process skills. Studies carried out with
primary and junior secondary schools children by Clement’s (1980) and Clarson
(1983) obtained similar results with about 50% of errors first occurring at the
reading, comprehension or transformation stages. Clement’s sample included 726
students in grades V – VII in Melbourne; Watson’s study was confined to a
preparatory grade in primary schools and Clarlson’s sample consisted of 95
grade VI students in two community schools Papua New Guinea.
Chapter – III
METHODOLOGY
Research
methodology is a scientific approach which deals with the systematic way of
collection data & use of appropriate research design. It describe the
desigh of procedure which is to be carried out to achieve the objectives of the
study. It explains desing of the study, sample of the study, tool of the study,
procedure of data collection & procedure of data analysis. This is a
descriptive research & aims at getting the reality through quantitative
analysis using original data.
Design of the data
This
study is an intuitionist research combined with interpretative in design.
“Newman errors on problem solving of matrix” technique for error analysis is
the theoretical base of this study. Next, the researcher use percentage to
compare the error committed by the students.
Population of the study
The
population of the study will be the students of grade IX of Dolakha district.
Sample of the study
The
data for analysis will be gathered from four government schools of Dolakha
district, selected at randomly. For the convenience of the study, only grade IX
students will accounted for the study, which is merely a purposive selection
& 24 students (3 boys & 3 girls from each school) will selected by
lottery method as a sample.
Tools of the study
The
data will be collected through the achievement test paper. The test will be
developed on the basis of mathematics textbook of grade IX prescribed by the
Government of Nepal.
Data collection Procedure.
After
making the tools ready, the research will be visited each of the selected
secondary schools. At first the researcher will be described the purpose of
respective schools. The pencil – paper test will be administered to a sample of
24 students of grade IX included in the sample. The test will be conduct in
congenial environment & there will time boundary for the students. After
completion of the test, answer sheets will be collected & only on wrongly
answered items, the incorrect steps will be considered & only error will be
counted for each wrong answered items.
Procedure of data analysis
The
researcher will analyze a interpret the data by using percentage.
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