Thesis: The Zone Of Proximal Development Theory In The Learning Of Operations In Mathematics At Primary Level

Title: The Zone of Proximal Development Theory in the
Learning Of Operations In Mathematics At Primary Level
Submitted By:
Marhama Younus
Coordinated To:
Ms. Nadia Thalho
And
Supervised
Ms. Zubaida Jamali
Assistant Professor
Gece (W) Hyderabad
Academic Year 2019
GOVERNMENT ELEMENTARY COLLEGE OF EDUCATION (W) HYDERABAD

i
Table of Contents
CERTIFICATE ........................................................................................................iii
Acknowledgement.................................................................................................... iv
Dedication................................................................................................................. v
Abstract.................................................................................................................... vi
Chapter One ..........................................................................................................- 1 -
Introduction...........................................................................................................- 1 -
1.1 Overview of the Chapter ..............................................................................- 1 -
1.2 Introduction..................................................................................................- 1 -
1.3 Background of the Study..............................................................................- 2 -
1.4 Statement of the Problem .............................................................................- 2 -
1.5 Objectives ....................................................................................................- 3 -
1.6 Research Question........................................................................................- 3 -
1.7 Scope of the Study .......................................................................................- 4 -
1.8 The Purpose of the Study .............................................................................- 4 -
1.9 Ethical Consideration Research....................................................................- 5 -
1.10 The limitations Of The Study .....................................................................- 5 -
1. 11 Definitions ................................................................................................- 5 -
1.11.1 Zone ....................................................................................................- 5 -
1. 11.2 Proximal .............................................................................................- 6 -
1.11.3 Development........................................................................................- 6 -
1.11.4 Operations ...........................................................................................- 6 -
1.11.5 Achievement........................................................................................- 6 -
1.12 Conclusion..............................................................................................- 7 -
Chapter Two..........................................................................................................- 8 -
Review of Literature..............................................................................................- 8 -
2.1 Overview of Chapter....................................................................................- 8 -
2.2 Theoretical Framework Of Research ............................................................- 8 -
2.3 Historical Background Of ZPD In Mathematics (Operations).......................- 9 -
2.4 Methods and Strategies Applied In Operations Of Mathematics .................- 11 -
2.4.1 Use of Pattering Method ......................................................................- 11 -
2.4.2 Use of Feedbacks Method....................................................................- 11 -
2.4.3 Organize Students' Responses Method.................................................- 12 -
2.4.4 Use Of Instrumental Instructors Method ...........................................- 12 -
2.4.5. The Use Of Student As Instructor Method...........................................- 12 -
2.5 Themes Of Memorization In Mathematics (Operations) .............................- 13 -
2.6 Main Features of ZPD ................................................................................- 14 -
2.7 Challenges during Uses Of Operations (MATHEMATICS) .......................- 15 -
2.8 Concussion.................................................................................................- 16 -
Chapter Three......................................................................................................- 17 -
Research Methodology ........................................................................................- 17 -
3.1 Overview of Chapter..................................................................................- 17 -
3.2 Research Methodology...............................................................................- 17 -
3.3Research Design..........................................................................................- 17 -
3.4 Time Span..................................................................................................- 19 -
3.4.1 Tools.......................................................................................................- 19 -
3.4.2 Checklist .................................................................................................- 19 -
3.4.3 Action Plan .............................................................................................- 20 -
3.5 Research Approach ....................................................................................- 20 -

ii
3.6 Population for Research .............................................................................- 21 -
3.6.1 Sampling size.......................................................................................- 21 -
3.7.2 Sampling Method.................................................................................- 22 -
3.7.4 Quota Sampling...................................................................................- 22 -
3.8 Data Collection ..........................................................................................- 22 -
3.9 Data Analysis and Interpretation.................................................................- 23 -
3.10 Conclusion...............................................................................................- 23 -
Chapter Four .......................................................................................................- 24 -
Data Analysis and Result.....................................................................................- 24 -
4.1 Overview of Chapter..................................................................................- 24 -
4.3 Reports on Action Research .......................................................................- 24 -
4.3.1 Analysis of Reconnaissance.....................................................................- 24 -
4.3.2 Analysis of Cycle one: Checklist.............................................................- 25 -
4.3.3 Analysis of Cycle Two ............................................................................- 28 -
4.3.3 Analysis of Cycle Three: Evaluate...........................................................- 29 -
Result...............................................................................................................- 30 -
4.4 Conclusion.................................................................................................- 31 -
Chapter Five........................................................................................................- 32 -
Summary, Discussion Recommendation and Conclusion.....................................- 32 -
5.1 Overview of The Chapter.........................................................................- 32 -
5.2 Summary ..................................................................................................- 32 -
5.3 Discussion..................................................................................................- 33 -
5.4 Recommendations......................................................................................- 34 -
5.5 Conclusion.................................................................................................- 34 -
6. References ...................................................................................................- 35 -
Appendix A .....................................................................................................- 41 -
Consent Letter..................................................................................................- 41 -
Appendix B......................................................................................................- 42 -
Checklist..........................................................................................................- 42 -
ppendix C ........................................................................................................- 43 -
Picking Sign Activity.......................................................................................- 43 -
........................................................................................................................- 44 -

iii
CERTIFICATE
This is to certify that Ms. Marhama Younus has conducted this
Action Research Project entitled The Zone of Proximal
Development Theory In The Learning Of Operations In
Mathematics At Primary Level" during the academic year 2019,
under the supervision of Ms. Zubaida Jamali in partial
fulfillment for the degree of B.Ed. (Hons).
Research Supervisor
Ms. Zubaida Jamali
Assistant Professor
GECE (W) Hyderabad
Principal
Ms. Nayyar Rafique
GECE (W) Hyderabad

iv
Acknowledgement
First of all, I am grateful to Allah Almighty and His
Prophet Muhammad (Peace Be upon Him) who provided me strength and health to
complete my Action Research. I also must acknowledge those people who encouraged
and supported me during my research study
This is also dedicated to my family and the many friends who
Supported me on this journey
Thank you
I would like to express my deepest gratitude to my co-coordinator Ms. Nadia Thalho,
my supervisor Miss Zubeda Jamali, for their support, collegiality, and mentorship
Assistance and supervision show me the way which is highly appreciated.
And, sharing precious opinions, experience during data collection in my research
project.
Finally, I wish to thanks to my family for their patience and support.

v
Dedication
This modest effort is dedicated to
My parents
Who brought me up to?
The point of fineness
Where we are footing today, looking for
Most prominent, and gleaming
Future at the forefront, for which they
Sacrificed most of their past,
And to
My respected teachers who guided
Me throughout
Academic life

vi
Abstract
The zone of proximal development theory in the learning of operations in
mathematics had so many importance s in teaching and learning. The main theme of
the study was to analyze the ZPD theory and differentiate during use of operations
technique or strategies in mathematics. Also, determined the learning of mathematics
operation through problems in operations and their solutions by using ZPD theory.
The area of study focused on Grade 3 in Government P.C Urdu section attached with
GECE (W) Hyderabad, Where the populations were 11 students but sampling/worked
on 4 students. The whole research was based on qualitative method and use quota
sampling methods . The main findings were analysis that misconception in operations
of mathematics because teachers not use the appropriate strategies according to the
needs of students. Its concluded that learn with the help of different strategies to
overcome the problem for futuristic needs/ recommendations for researches should be
use 21 century tools and methods to students developed logical thinking / critical
thinking .
Keywords: Learning, Operations, Zone Of Proximal Development, Theory

The zone of proximal development theory in the learning of operations in mathematics at primary level/i
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Chapter One
Introduction
1.1 Overview of the Chapter
In this chapter, there is totally concerned with the integration of table of content
which is totally forced in the area of background of the study, objective of the study,
statement of the problem, scope of the study, research question, ethical consideration
and limitation of the study, definition of key term and last is conclusion.
1.2 Introduction
The zone of proximal development is the gap between what a student can do
independently and what they can potentially do with the help of a "more
knowledgeable other.(Vinney, 2019). Furthermore, for ZPD Hudsondefned that Math
educator knows that unless students are challenged by and actively engaged in the
course material, students will quickly lose attention and taking the different learning
paces, habits of mind, and prior knowledge of individual students into account in
mathematics can be challenging. Because the concept of ZPD involves connecting
unlearned material to what is familiar, teachers play a pivotal role in the application of
ZPD in the classroom. (Hudson, 2013). Where, the design has been persistently poor
for students in the zone of proximal developmental performance in Mathematics. In
this theory, ZPD mathematical achievement could be improved as central to
instructional improvement. This is the distance between actual developmental level by
independent problem and level of potential development. Determine by problem
solving misconceptions in activities during mathematics learning under adult
guidance. The proximal development zone is the difference between what a student
can do without help and what they can do with help. They provide acceptable

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strategic social relationships, learning experiences, and instruction based on the
previous results, intuition, and present thinking of a student which guide efficient
learning and growth and investigate the capacity of student teaching to operate and
then explore the capacity to implement the suitable techniques and strategies to
achieve the objectives.
1.3 Background of the Study
Le-Vygotsky (1896) had given the concept, zone of proximal development which
developed for Soviet psychologist and social constructivist (ZPD, n.d). Engel Brecht,
Harding and Phiri (2010) assert that several lecturers who taught first-year
mathematics in 2009 reported under-preparedness of students. Maji and Makgati
(2006) assert that outdated teaching practices and lack of basic content knowledge
have resulted in poor teaching standards. Reddy (2004) stated that: There is no
single cause of South Africa's poor and diverse performance. Preliminary
explanations could be linked to multiple, complex and connected sets of issues,
including the following: issues of poverty, resources and infrastructure of schools,
low teacher qualification, and poor learning cultures in schools.
1.4 Statement of the Problem
The Statement of the Problem describes the declaration issue. The Researcher
observed the issue during the practicum because students have problems with
mathematical activities like addition, subtraction, multiplication, and division. And
the identification of operations also faces problems. The Results of the evaluation
would characterize the area of tasks between what the learner could accomplish alone
and what could be accomplished and finally mastered with help. With that knowledge,

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teachers would not have a clear understanding of where instruction can be balanced to
extend learning. Demonstrate solving the issue with the checklist and action plan as
well as many other strategies and methods to reduce the mathematics operations issue.
These strategies are helpful for solving grade 3 math issues.
1.5 Objectives
These objectives are helped to resolves the problems in operations (addition,
subtraction, multiplication, and division) of mathematics, to investigate the student
skills through ZPD theory in grade 3 and to explore the ability of the student in
mathematics operations. and then to apply the appropriate methods and strategies of
operations on students learning. Objectives in third-grade math classes are
tackled toward developing an understanding of the base-ten numeration
system, which is essential to the understanding of all future mathematical
concepts. There are also objectives that focus on fractions and fractional
relationships, as well as manipulating whole numbers via basic arithmetical
processes. Objectives also include a focus on helping students to develop
critical thinking and reasoning skills.
1.6 Research Question
Research question is useful in solving the issues in mathematics activities based on
the grade 3 issue in mathematics activities and ZPD theory (the zone of proximal
development).
1) What are the appropriate strategies for applying in student teaching
mathematics (operation) activities?

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2) How to analyze the abilities/skills of students in mathematics operations?
3) What are the ways used to explore the theory of ZPD in the operations of
mathematics?
1.7 Scope of the Study
The scope of the study is that ZPD (Zone of Proximal Development Theory) has
developed the skills and ability of students to solve problems without any help and
instruction from teachers, this study is important because the distance between the
actual level of development by the independent problem and the level of potential
development of students I have argued that the effective use of ZPD can improve
mathematics. It needs to be just enough of a challenge to help the student deepen
conceptual understanding and develop new skills that build upon the students existing
knowledge and understanding.
1.8 The Purpose of the Study
The purpose of this research is to learn, about Vygotsky's theory to develop
conceptual knowledge and create fresh abilities in mathematics operation and
basically in this research to express the fresh thoughts that teach mathematics and
confronted so many issues in mathematics. Implement the ZPD (the zone of proximal
development theory) in mathematics teaching. The real purpose of learning math is
the researcher have a variety of answers, but usually, it s straightforward: much of
the math you learn is applied to real-life situations, and the ability to do it yourself
with no need for a calculator makes sure you’re independent of technological devices
to a point. Also, even if it’s not necessarily math you’re using day-to-day, it activates
a part of your brain that other subjects aren’t as adept at doing. Problem solving, using

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the tools in your hypothetical tool belt, and applying the skills you’ve learned and
strategies you’ve acquired to solve the problem leaves you in a better place to create,
not simply ingest.
1.9 Ethical Consideration Research
Ethics in mathematics is a field of applied ethics, the inquiry into ethical aspects of
the applications of mathematics. It deals with the professional responsibilities
of mathematicians whose work influences decisions with major consequences, such as
in law, finance, and the military and environmental science. Many research
mathematicians see no ethical implications in their pure research but assumptions
made in mathematical approaches have real consequence. Actions in their pure
research but assumptions made in mathematical approaches have real consequences.
Of (club Rome, 1968). The researcher also has developed Ethical Considerations
where collected formal data way with consents of Institution and priority of Teacher
and students and also cared for Time management as well.
1.10 The limitations Of The Study
The limitations of this research /study is that to collect the data in the limitations of
time period and conduct in pc Urdu section with attached in G.E.C.E(w) Hyderabad.
in which solved the problem which one i observed and faced in during practice .I
resolve the issues of mathematics operations addition, subtraction , and multiplication
in Grade 3.
1. 11 Definitions
1.11.1 Zone

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The word zone is used to mean any area that is separated from another or is
distinguished from another based on distinctive circumstances. In computers,
networking, and the Internet, the meaning of the word zone changes depending on
the specific environment By Vengai Beal.
1. 11.2 Proximal
Proximal: Toward the beginning, the nearer of two (or more) items. For example, the
proximal end of the femur is part of the hip joint, and the shoulder is proximal to the
elbow. The opposite of proximal is distal by William C. shell. 2018.
1.11.3 Development
Development is a process that creates growth, progress, positive change or the
addition of physical, economic, environmental, social and demographic components.
1.11.4 Operations
A mathematical process, The most common are added, subtract, multiply and divide
(+, −, ×, ÷). But there are many more, such as squaring, square root, logarithms, etc.
If it isn't a number it is probably an operation.
1.11.5 Achievement
Identifying impediments to learning as well as facilitators of successful acquisition,
retention, and transfer of mathematical competencies are critical goals. A better
understanding is needed in areas such as instructional sequencing and timing so as

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to minimize prior learning interference, as well as a means to enable conceptual
shifts (David C. Geary, 2017).
1.12 Conclusion
The conclusion of this chapter one is that to resolve the issue of mathematics
operation in this the researcher designed the objects and research questions with the
help of this problem statement which one I observed in a students learning and during
teaching in which I focused on grade 3 level of students I described the benefits of our
study it increased the knowledge of students and skills also and define the purpose of
my study toward the chapter two.

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Chapter Two
Review of Literature
2.1 Overview of Chapter
In this chapter it is fully concerned with the integration of table of contents which is
totally forced in the area of background, theoretical framework of research,
background of ZPD in mathematics (operations), method and strategies applied ZPD
in mathematics (operations), memorization themes in mathematics (operations),
salient features of ZPD, challenges during operations (mathematics), and finally add
conclusion.
2.2 Theoretical Framework Of Research
The theoretical framework is the structure that can hold or support a theory of a
research study. The theoretical framework introduces and describes the theory that
explains why the research problem under study exists (Borgatti, 1999). Lev Vygotsky
and soviet in 1934 discussed in their theories that the zone of proximal development
theory used in the learning of operation in mathematics. I have used this theory (zone
of proximal development) in the learning of operations in mathematics given by
LeVygotsky, By the integration of this theory is totally recommended with the level
of that underachievement and lower performance characterize the learning of
mathematics. teaching mathematics in the Childs zone of proximal development
viewed as one way of improving performance in mathematics .literature argued that
teaching is the zone of proximal development where the Childs learning is mediated
and scaffold by the teacher or expert adult or knowledgeable peer makes learning
more meaningful, easier, manageable, effective and efficient .because from this every

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learner can learn by self and teacher instruction and through this theory the level of
student performance to be diagnosed easily in the reference of ZPD(zone of proximal
development ) theory in the operations of mathematics.
2.3 Historical Background Of ZPD In Mathematics (Operations)
Globally, the poor performance of students in mathematics is a major concerned. Van
der Walt, Maree and Ellis (2008) note that researchers agree that most learners in
South Africa are familiar with the subject matter. They further explained that South
African learners have issues with their limited technical mathematics vocabulary. In
1995, 1999 and 2003, the South African Human Sciences Research Council (HSRC)
conducted studies to assess students in mathematics and science at Grade 8 level
South Africa had the lowest score in science and mathematics among the six African
countries (Egypt, Tunisia, Morocco, Botswana, Ghana, and South Africa) that
participated in the International Mathematics and Science Study Trends [TIMSS]
(2003). The results of the South African National Study in mathematics and science,
Reddy (2004) reveals that there is a difference in performance among provinces, with
the Western Cape, Northern Cape, and Gauteng being the three highest performers.
KwaZulu Natal, Eastern Cape and Limpopo were the three lowest performers. The top
provinces had almost double the lowest-performing province ratings. Analysis of the
performance of the learners reflects the highest scores of learners in the former white
schools, whereas the lowest scores of learners in African schools. In the former white
schools, the learners have a score just below the global average. Reddy (2004) stated
that: There is no single cause of the weak and complex quality of South Africa.
Preliminary explanations could be related to multiple complex and interconnected sets

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of problems, including issues of poverty, school resources and infrastructure, low
teacher qualifications and poor school learning cultures.Language skills are a
contributing factor but it occurs that the issues of conceptual and cognitive demands
placed on students in classrooms are significant. This is confirmed by researchers
such as Ndlovu (2011) in their argument that the reasons for South African learners'
poor performance in mathematics include poor learners' socioeconomic background
(small incentive to study at home), lack of appropriate learner support materials,
general school poverty. The generally poor performance of teachers and educators
(including poor knowledge of subjects and lack of motivation), in language
instructions (often not the same as the mother tongue of learners), and insufficient
orientation of learning.
The South African Minister of Basic Education announced at a media conference held
in Pretoria on 4 January 2012 that in 2011, the pass rate for mathematics in
matriculation examination was 46.3 percent, down from 47.4 percent in 2010. This
indicates that mathematics in South Africa is still a concern.
Despite the poor matriculation pass rate, South African researchers like Engelbrecht,
Harding, and Phiri (2010) say that "several lecturers who taught mathematics for the
first year in 2009 reported under-preparedness of students." This is due to various
reasons, such as at-risk students entering universities.
Students at' risk' are students with poor performance in their school system. They are
usually low-confidence academic attainders. This is supported by Padayachee,
Boshoff, Olivier, and Harding (2011) when they explain that "from the first-year
student teaching of mathematics, they observed that several first-year students are

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being prepared for mathematics. Identifying students at risk and developing programs
to avoid their failure are necessary components of university education reform
(Donnelly, 1987).
2.4 Methods and Strategies Applied In Operations Of Mathematics
Pictures, tape diagrams, number lines, ten frames, and base-ten blocks. In this section,
we introduce suggestions for using the scaffolding method in mathematics classes.
These suggestions can promote mathematical learning and motivation in classes.
2.4.1 Use of Pattering Method
Teachers or instructors should use of patterning. While teaching a mathematical
problem, the first one need not denote expressly but make to creative potential and
encourage solving the problem. These interactions between teachers and self-students
can be of effective scaffolding.
2.4.2 Use of Feedbacks Method
When you have designed the problem and have begun the scaffolding then you should
use the feedback. If you did not use feedback then students will not notice whether
their suggestions were effective on problem-solving. If students introduced the
corrective solution then you indicate positive feedback and if it was not this form then
you should use negative feedback carefully. Of course, positive feedback will be more
effective.

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2.4.3 Organize Students' Responses Method
When you started the scaffolding, you must be noticed that students' responses do not
stray because it makes that students confused in scaffolding phases and they will not
comprehend ultimately. Then try to organize and predict their responses until you
have effective scaffolding phases.
2.4.4 Use Of Instrumental Instructors Method
In the scaffolding method, Teachers or instructors began this process often and
students will learn through the process. But we can use other instruments as teachers
or instructors! For example, in the present decade, we use computers, software or
electronic instruments. Mathematical software can be applied in mathematical class.
In these classes, the scaffolding process will be different. Human communication will
become one party. Many students cannot communicate with their teachers but they
can better communicate with electronic instruments such as software. Notice that all
students are not the same and we must not neglect them. Of course, teachers must
supervise their learning process after using these electronic instruments whether these
students have learned.
2.4.5. The Use Of Student As Instructor Method
When learning occurs in the scaffolding process, students who learned can teach
others. Instructors or teachers can make learned students teach other students. This
process motivates students to learn further. When the student teaches his/her own
classmates, self-mistakes are corrected easily.

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providing multiple opportunities for ELLs to notice and then use the new
mathematical language (Ellis, 1991)
rephrasing and helping students to notice the differences in phrasing
using guided questioning
using wait time after asking a question so that students have an opportunity to
translate the question, if necessary, and to form an answer
building on what students say, that is, amplifying not simplifying (rather than using
only basic language for ELLs, it is essential to model more sophisticated or
complex language use)
Beginning with oral tasks and then moving to written responses. The oral task is a
rehearsal for the written response. Through talk, students are able to formulate
ideas, evaluate hypotheses, and reach decisions in a context that is not constrained
by the more formal demands of written language (Gibbons, 1991).
2.5 Themes Of Memorization In Mathematics (Operations)
Nothing may be more feared in the minds of young children and their parents than
learning basic math facts. Just hearing the times, a table takes many of us back to our
own childhoods, staring at a blank page and trying to remember the dreaded 9 x 8 =
72. The good news is that our own children should not have to suffer the same fear. A
substantial amount of mathematics education research shows that children do not
master their math facts through memorization alone. Instead, true mastery comes from
being equipped with quick and effective strategies for finding the solution. By using

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these strategies, children will always have the mental tools needed to find the correct
answer and the confidence to use them. With a strategy-based approach to the basic
math facts, children use what they already know to figure out what they don't know.
Rather than racking their brains to remember the answer to a basic math fact, they can
simply find a helping fact and use it as a jumping-off point. Some of the most
common strategies for basic fact mastery include: Skip Counting, Make 10, Doubles
and Near Doubles, Nine, Commutative Property, Use Fact Families, Make it Real.
2.6 Main Features of ZPD
The zone of proximal development (ZPD) is the range of abilities that an individual
can perform with assistance but cannot yet perform independently. These skills are
called "proximal" because the individual is close to mastering them but needs more
guidance and practice in order to perform these actions independently.
The presence of someone with the knowledge and skills to guide the learner
Social interactions that allow the learner to observe and practice their skills
Scaffolding, or supportive activities provided by the mentor or teacher that help
guide the learner through the ZPD
It is the more knowledgeable other who provides critical guidance and instruction
during the sensitive learning period. While a child might not yet be capable of
doing something on her own, she is able to perform the task with the assistance of a
skilled instructor.

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Vygotsky believed that peer interaction was an essential part of the learning
process. In order for children to learn new skills, he suggested pairing more
competent students with less skilled ones.
It is important to realize that the zone of proximal development is a moving target.
As learner gains new skills and abilities, this zone moves progressively forward.
Teachers and parents can take advantage of this by continually providing
educational opportunities that are a slight stretch of a child's existing knowledge
and skills.
By giving children tasks that they cannot quite do easily on their own and
providing the guidance they need to accomplish it, educators can progressively
advance the learning process.
2.7 Challenges during Uses Of Operations (MATHEMATICS)
Mathematics education researchers and practitioners, policymakers, and parents seek
answers to Grand Challenges that will ultimately result in the enhancement of
mathematics teaching, learning, curriculum, and assessment ((NCTM, 2014,)
1) Represent complex and extremely difficult questions that are solvable (potentially
within 10 20 years);
2) Improve the quality of life through positive educational, social, and economic
outcomes potentially affecting millions of people;
3) Involve multiple research projects across many sub-disciplines to be
satisfactorily addressed;

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4) Require measurable outcomes so that progress and completion can be identified;
5) Compel popular support by encouraging the public to relate to, understand, and
appreciate the outcomes of the effort.
2.8 Conclusion
The conclusion of this chapter is that this material is totally concerned with my
subject. And perpetual researchers worked on a zone of proximal development their
researches are helping to solve problems in mathematics (operations), while studying
in the classroom and the theoretical background of ZPD is linked to mathematical
operations. The researcher used during the study applied all methods and strategies to
help my research to solve the student problem in mathematics and faced many
challenges during the application of methods and strategies and also well-defined the
memorization of mathematical operations. This all literature is only substituted step
toward the third chapter.

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Chapter Three
Research Methodology
It is completely concerned in this chapter with the integration of a table of contents
that is totally put in the research design, research approaches, time span/horizon, tool/
instrumental used for data analysis, sampling design, and final conclusion.
3.2 Research Methodology
A research methodology or involves specific techniques that are adopted in the
research process to collect, assemble and evaluate data. It defines those tools that are
used to gather relevant information in a specific research study. Surveys, checklist,
and interviews are common tools of research. The qualitative research method is used
in this research problem the main goal of the study analyzes the effect of ZPD theory
on the mathematical operations in student learning. This study is descriptive.
Qualitative research is a process of naturalistic inquiry that seeks an in-depth
understanding of social phenomena within their natural setting. It focuses on the
"why" rather than the "what" of social phenomena and relies on the direct experiences
of human beings as meaning-making agents in their everyday lives. Rather than by
logical and statistical procedures, qualitative researchers use multiple systems of
inquiry for the study of human phenomena including biography, case study, historical
analysis, discourse analysis, ethnography, grounded theory, and phenomenology.
3.3Research Design
Research design is defined as a system of methods and techniques chosen by a
researcher to integrate various research components in a reasonably logical way to

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effectively manage the research problem. The research design sets the parameters of
your project: it determines exactly what will and will not be included. It also defines
the criteria by which you will evaluate your results and draw your conclusions.
(Combes, 2019). In this research design to integrated design a cycle for collecting the
data and then through following conditions such as plan them act to design a tool then
observe the tools that are used in research and write the reflection and also analyze the
data in research design.

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3.4 Time Span
The period of time between two events or during which an event continues.
S/no Date Cycle Action Plan Tool Planning success
1 11-09-
2019
Reconnaissance Meeting
with
HM and
Teachers
Pc.
Urdu
section
Consent
Letter
meeting Achieve
ment
2 12 to
19-Nov-
2019
1 Observe Pc.
Urdu
section
Checklist Observation Achieve
ment
3 26.10.19 2 Deliver
ZPD
method
Pc.
Urdu
section
Action
plan
activity Success
4 28.10.19 2 Assess Pc.
Urdu
section
Picking
sign
Activity Achieve
ment
3.4.1 Tools
Anything that becomes a means of collecting information for your study is called a
research tool or a research instrument. For example, observation forms (checklist),
Interview, schedules, are all classified as research tools.
3.4.2 Checklist
The researcher has adopted a checklist and Picking Sign activities from the different
research papers and used in the context of research, and the reliability of tool is
checked by Supervisor as well as Expert s. the Checklist is a one-page, non-technical
checklist for the key points of good practice in research. Use of the Checklist can
assist researchers and organizations in fulfilling the requirements of regulatory,
funding and other bodies and ensure that important issues have not been overlooked

- 20 -
Checklists, rating scales, and rubrics are tools that state-specific criteria and allow
teachers and students to gather information and to make judgments about what
students know and can do in relation to the outcomes. They offer systematic ways of
collecting data about specific behaviors, knowledge, and skills.
3.4.3 Action Plan
An action plan is a systematic way of defining a goal, figuring out strategies for
meeting the goal, and deciding how you will assess whether you have met the goal. In
education, teachers can use action plans in relation to our own professional goals, and
we can also use them to work with students in the area of; Plan, Act, Observe and
Reflect.
3.5 Research Approach
The research approach is a plan and procedure that consists of the steps of broad
assumptions to the detailed methods of data collection, analysis, and interpretation.
(Priya, 2016). In this study, the deductive approach used to describe the research
generally descriptive way. When conducting deductive research, you always start with
a theory, reasoning, deductively means-testing these theories. If there is no theory yet,
you cannot conduct deductive research. (Raimo, 2019). For deductive approaches,
the emphasis is generally on causality, the aim is usually focused on exploring new
phenomena or looking at previously researched phenomena from a different
perspective. Also followed the cross-sectional where focused under the time
boundaries. A cross-sectional study is defined as an observational research type that
analyzes data of variables collected at one given point of time across a sample

- 21 -
population. This study type is also known as cross-sectional analysis, diagonal study
or occurrence study where the time is short bounded.
3.6 Population for Research
The population for research, a complete set of elements (persons or objects), that
possess some common characteristic defined by the sampling criteria established by
the researcher. The researcher has selected the number of population from the
estimated confident Level 95% Table adopted on the margin size 10 (0.5%). For
the action research and Focused area the total population is 10/10 students in Grade 3
at the primary level. PC Urdu section attached to government elementary college of
education (women) Hyderabad.
3.6.1 Sampling size
Sampling is a process used in statistical analysis in which a predetermined number of
observations are taken from a larger population. The methodology used to sample
from a larger population depends on the type of analysis being performed, but it may
include simple random sampling or systematic sampling. I have forced approximately
four students in my study. (Alicia, 2019). The selected sample size of a survey most
typically refers to the number of units that were chosen from which data were
gathered. The selected Sample size is 4/10 because these are those students who are
facing Problem in Mathematics (operation) Grade 3. However, the sample size can
be defined in various ways. There is the designated sample size, which is the number
of sample units selected for contact or data collection. There is also the final sample
size, which is the number of completed interviews or units for which data are actually
collected. (LA Vrakas, 2008)

- 22 -
3.7.2 Sampling Method
Sampling is a method that allows researchers to infer information about a population
based on results select from a part of the whole population, without having to
investigate every individual.
3.7.4 Quota Sampling
Quota Sampling is a sampling method in which researchers can form a sample
involving individuals that represent a population and are chosen according to traits or
qualities. Researchers can decide the trait as per which the sample subset selection
will be conducted so that the sample can be effective in collecting data that can be
generalized to the entire population. The final subset will be decided only according
to the interviewer's or researcher's knowledge of the population.
3.8 Data Collection
Data collection is a process of collecting information from all the relevant sources to
find answers to the research problem, test the hypothesis and evaluate the outcomes.
Data collection methods can be divided into two categories: Primary data methods
of data collection from P.C school and Secondary methods of data collection
from different webs, research papers etc. I also have collected data in this study
through the above instruments such as checklist, activities, reflection and student
record.

- 23 -
3.9 Data Analysis and Interpretation
Data analysis is a process used by researchers for data collection data analyzing and
interpreting help in reducing a large chunk of data into smaller fragments which
makes sense. Where the researcher used Ms. Office and Excel for interpretation used
graphs and Tables and as well descriptive in the way.
3.10 Conclusion
This chapter relies on the research design, planning of research study which is related
to the zone of proximal development theory in the learning of operations in
mathematics at the primary level. And concern to the Methodology which is
qualitative is used in my study. And using the time span time table of the study and
also show the population and sampling and sampling method which is used in my
study along with the data collection tools and data analysis these all are concerned in
this chapter for moving toward chapter four.

- 24 -
Chapter Four
Data Analysis and Result
It is totally concerned with the data analyses and the analyzing of the Action research
cycles which are interpreted for the purpose of qualitative research.
4.2 Data Analysis
Data analysis is a process used by researchers for reducing large data in a short,
interpreting for comprehensions. The data analysis process helps in reducing a large
chunk of data into smaller fragments, which makes related sense to achieve aim and
objective. (Schensul, 2019). Data analysis as an ambiguous, and time-consuming, but
a creative and fascinating process through which a mass of collected data is being
brought to order and focused toward the outputs, (Rossman,2019).
4.3 Reports on Action Research
4.3.1 Analysis of Reconnaissance
Report
The report of Reconnaissance in action research is to follow such parts of analysis as;
planning, acting, observing and reflecting these parts implemented on the 11-09-2019
date.
Planning
In the step of planning of action research, the researcher will do meeting with
Headmasters (HM) in the preparation of action research to the zone of proximal
development theory in the learning of operation in mathematics at primary level.

- 25 -
Act
This step/phase is involved met with HM, PC Urdu attach to GECE (w) Hyderabad
was discussed on the research on the Zone Of Proximal Development theory in main
mathematics operations teaching. I shared my study topic's objectives, I informed her,
that the significance of mathematics operation in student teaching. Also, meet with the
teachers and students
Observation
At this meeting, the HM listened closely, motivated the study subject and gave some
guidance on improving and improving action research power. During recognition it
was noted that the student did not solve the operation on which they depended. In this
part, the HM said yes, we need to work to improve student s maths, and she also said
motivating learners and involving teachers and saying that you should do this study
on mathematics operation.
Reflect
From the Reconnaissance, personally reflected during meeting with HM in this action
research step and I was provided consent letter and objective of my further data
collection on my study subject. They received a favorable reaction and listened
closely, as well as provided some guidance on the ZPD implementation. From above
all discussion I will go for the next step in my action research for cycle one is to
conduct a study and create a student checklist tool and to conduct data.
4.3.2 Analysis of Cycle one: Checklist
Report

- 26 -
For cycle one follow these steps of action research as; as plan, act, observe and
reflection these parts implemented on the date 12-09-2019 and 19-09-19.
Plan
In this cycle of action research, I planned to make an observational tool (checklist) to
diagnose the problem of students in the learning of operations in mathematics and
also planned a strategy that I used in Learning operations in mathematics.
Act
I made a checklist and filled it. I diagnosed the problems of students in the learning of
operations in mathematics in 3 rd. grade and also applied strategy and ZPD theory.
No Somehow Yes
1 3
2 2
3 2
4 1
5 2
6 3
7 2
8 2
9 3
10 1

- 27 -
Result
This above pie chart is shown the observational tool (checklist) that I observed in
grade 3 to analyze the problems that students faced in Learning of operations in
mathematics. I analyzed the checklist that I observed students are facing problems
with developing logical thinking in mathematics. Where the, 1.2% shows that yes,
Students are facing problems in mathematics operation, and 2.4% somehow, Students
are interested in the relevant subject.3.5% somehow, Students are solving the all
problems of mathematics operation his/herself.4.7% is No, interactive method is used
for teaching.5.9% the somehow, traditional method is avoided for the conducting
observation. Yes, 6.11%Teacher facilitates the poor learner. Somehow, 7.13%
Environment of the classroom is according to the required subject of teaching

- 28 -
mathematics.8.15% somehow, the Teacher developed logical thinking during learning.
Yes, 9.16% using models for teaching mathematics according to the need of
students.10.18% is No to provide clear instruction. From all observed that student is
facing problem during mathematics (operation).
Observe
I observed through a checklist that the students had a problem in the operations of
mathematics in the classroom due to the reasons of teacher did not instruct them in the
learning of mathematics. Mostly student was solving questions by their self. And
some students need instructions. And I was applying a strategy student was to solve
questions by themselves.
Reflect
I have observed students through a checklist and diagnosed their problems in
mathematics and I solve the problem modern strategy with the using of ZPD theory in
the learning of operations in mathematics.
4.3.3 Analysis of Cycle Two
Report
These steps of action research to make reflections follow such parts as plan, act,
observe and reflect those parts implemented on the 26-09-2019 date.
Plan
I planned different strategies to overcome the issues in Learning of operations in
mathematics and also apply the ZPD theory according to my research topic.
Act
In Grade 3, I applied the strategies that were base the theoretical framework
introduces and describes the theory that explains why the research problem under

- 29 -
study exists (Borgatti, 1999) The operations (Addition, Subtraction, Multiplication,
and Division). I delivered different examples according to the daily life of students.
Observe
I was observed that students take an interest in the learning of mathematics and
students were also giving good responses .they were well understood and at the same
time, we're solving questions of operations in mathematics.
Reflect
I was to work and oversee Grade 3, I used the technique that was useful in my
research and students enhanced their learning in mathematics operations (Addition,
Subtraction, Multiplication, and Division) and students also explaining their ideas
about mathematics operations. It was great successes that students have given
improvement but for the next I observed to take an assessment by using Pick Sign
activity.
4.3.3 Analysis of Cycle Three: Evaluate
Report
These steps of action research to make reflections follow such parts as plan, act,
observe and reflect those parts implemented on the 28-09-2019 date.
Plan
I planned a picking sign activity that is useful in this research .and also integrated the
theory of ZPD and I got a result of 4 students.
Act
In grade 3 I was assessing, Students through a worksheet of picking signs and check
their worksheets and also scored them.

- 30 -
Picking Signs Activity
This activity I applied 4 students those who are effective and have a problem in
mathematics in grade 3 and got a result as;
Result
In this activity Students takes good marks
Hira takes 19 out of 20 marks (95%)
Noman takes 18 out of 29 marks (90%)
Raza takes 20 out of 20 marks (100%)
Subhan takes 19 out of 20 marks (95%)
In the analysis of operations in mathematics Subhan got a 19 score out of 20 and got a
95% improvement and Hira take a 19 out of 20 and got a 95% also they improve in
the same percent and Noman takes a 90% improvement and they achieve score 18 out
of 20 and Raza got a 100% he gets a 20 out of 20 and improved well and give a well-
done performance in achievement of operations for an assessment activity these
readings I take through the percentage with the looking at the observation tool
findings I use this strategy and firstly student learn about mathematics operations
Pick and Sign Activity
Noman
18
Hira
19
Subhan
19
Raza
20

- 31 -
then to assess the students and they give a good responses in an improvement of
students in mathematics operations.
Observe
I observed during the checking of worksheet students improve in operations of
mathematics and solve themselves and got an excellent Mark's.
Reflect
I planned activities and Observe grade 3 I used the strategy that was useful in my
research and students has improved their learning in operations (Addition, Subtraction,
Multiplication, and Division) of mathematics. And students also clear their concepts
about the operations in mathematics.
4.4 Conclusion
In this chapter analyzing Research action cycles 1,2, and 3 with the using of pie chart
find out the percentage of how much percent students can develop interest, where,
students facing problems in mathematics, and through discussions. evaluate the
percentages of student's progress and improvement about the topic of research the
zone of proximal developmental theory in the learning of operations in mathematics
and after diagnosed problems of students in operations of mathematics in this applied
different strategies to overcome the issues that they Had faced problems after
applying strategies assessed students through picking the sign activity and evaluate. In
this students had improved their learning in operations of mathematics.

- 32 -
Chapter Five
Summary, Discussion Recommendation and Conclusion
5.1 Overview of The Chapter
This chapter totally concerned with a discussion that based on results, summary,
conclusion the collected data and recommendation for the future research on this topic
5.2 Summary
The summary of this whole research is based on Zone of Proximal Developmental
performance in Mathematics by students has persistently been poor. In this theory
could improve mathematical achievement ZPD as central to instructional
enhancement. This is the distance between actual developmental level by independent
problem and level of potential development. objectives are helped to resolves the
problems in operations (addition, subtraction, multiplication, and division) of
mathematics, to investigate the student skills through ZPD theory in grade 3 and to
explore the ability of the student in mathematics operations. and then to apply the
appropriate methods and strategies of operations on students learning. Objectives in
third-grade math classes are tackled toward developing an understanding of
the base-ten numeration system, which is essential to the understanding of
all future mathematical concepts. There are also objectives that focus on
fractions and fractional relationships, as well as manipulating whole
numbers via basic arithmetical processes. Objectives also include a focus on
helping students to develop critical thinking and reasoning skills. this study is
limited on Pc Urdu section with attached in G.E.C.E(w) Hyderabad. in which I
solved the problem which one I observed and faced during practice. According to the
Borgata theoretical framework introduces and describes the theory that explains why
the research problem under study exists. About the ZPD theory in the learning of
operations in mathematics and the ways of memorization of operations in
mathematics also included methods and strategies. I used a qualitative method in
research methodology and quota sampling method in sampling design. I integrated

- 33 -
the detective approach in this study and in research design action research cycles
where I started from the situation analysis (reconnaissance) and then step by step
(Plan, act, observe and reflect) the situation and solved the problem of students in the
selected area. I adopted tools as a checklist and activities related to ZPD. For all also
used a Pie chart for interpretation. I also have observed that it all depends on students
learning where children unable to develop interest because of Teacher is not using
proper strategies during the study as well as has no proper training related subject.
There should need to integrate activity-based things as well as need to provide
training for professional development. The importance of ZPD theory is more in
learning and teaching students for the development skills according to the surrounding.
5.3 Discussion
This study delivers an opportunity for introducing mathematical learning and teaching
in a scaffolding method. ZPD is defined as the teaching-and-learning process, wherein
the students sets conceptual scaffolds through conversation, present an indirect form
of perception in order for the child to be able to transfer into his/her Zone of Proximal
Development (ZPD) and development from the skills to the self-governing ability.
This process continues with the students slowly with sketch the mental scaffolds
as the child presents increasing self-governing skills and finishes when the child is
capable to do on their own whatever, up to that point, was likely only with the help of
the students.

- 34 -
5.4 Recommendations
1. Additional research should be undertaken on the nature, development, and
assessment of mathematical operations.
2. The teacher should be applying interesting and effective strategies.
3. The teacher should provide appropriate instructions for the increasing of
students' critical thinking.
4. The teacher should be giving a daily life example.
5. The teacher should be Make lessons activity-centered (hands-on). This helps
to get students actively involved.
6. The teacher should enhance poor learners' skills
7. Future Researcher should work on the same topic to solve students daily life
problems and the improving student mental level
5.5 Conclusion
I have argued that effective use of the ZPD can improve Mathematics success in the
schools. Critical to this theoretical notion is the help which is given to operationalize
the idea and this is just defined as scaffolding. The teacher takes major responsibility
for arranging the communication important the other through the steps of a math
assignment and providing the necessary support until the learner is able to do the task
freely. Research has shown that more often than not teachers do not offer the support
compulsory in Mathematics and as a result, children fail in spite of having the
potential to perform such tasks. In light of all this, I conclude that the ZPD theory if
well applied, can lead to advanced mathematical achievement.

- 35 -
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- 40 -
Appendices

- 41 -
Appendix A
Consent Letter

- 42 -
Appendix B
Checklist
PARTA
Information :
Iam prospective student for the purpose of study and presently observation for
research on The zone of proximal development theory (ZPD) in the learning of
operation in Mathematics at primary level I request you to spare time for
providing the details indicating statement given below.
PARTB
Class: 3 School name: P.c school
Date: 13 - 09 - 2019 Time: 12:00 pm
PART B
A: (1) No B: (2) Somehow C: (3) Yes
Statement No Somehow Yes
1 Students are facing problems in mathematics operation. 3
2 Students are interested in the relevant subject. 2
3 Students are solving the all problems of mathematics
operation his/ herself.
2
4 An interactive method is used for teaching. 1
5 A traditional method is avoided for the conducting
observation.
2
6 Teacher facilitates the poor learner in the classroom 3
7 The Environment of the classroom is according to the
required subject of teaching mathematics.
2
8 Teacher developed logical thinking during the learning. 2
9 Using models for teaching mathematics according to the
need of the class
3
10 Provide clear instruction that is understandable. 1

- 43 -
Appendix C
Picking Sign Activity

- 44 -

- 45 -

- 46 -

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