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231 Laila Khalid Khudhair
Laila Khalid Khudhair Lecturer, Department of Educational and Psychological Sciences, College Of Education for Humanities, University of Tikrit
The effect of visual thinking strategy on students' achievement in mathematics and their algebraic thinking
ISSN:0587-2405
e-ISSN: 2415-0479
Acta Academica • 2019, 51(2):231-261
© Creative Commons With Attribution (CC-BY)
Abstract
The current study aimed at defining the effect of visual thinking strategy (henceforth VTS) on students' achievement in mathematics and their algebraic thinking. The research sample consisted of (60) students in the third intermediate grade. They were randomly divided into two groups including (30) students who were taught using VTS and a control group including (30) students who were taught using conventional method. To achieve research objectives, an achievement test consisting of (40) items was developed. In addition, (25) items were developed for algebraic thinking test to measure these skills: recognition of patterns, representation of relationships and functions, the use of algebraic variables and symbols. The research validity and reliability were verified through using the appropriate statistical means.
Research Problem
Mathematics is one of the important sciences that enhanced development
through applications which become means for the development and
innovation of other sciences, making it an appropriate environment for
training students to think (Al-Beheiry, 2006: 165). Mathematics is the core
factor for development and an essential component of the curricula around
the world. It is an important syllabus that works to develop mathematical
thinking and prepare students who can analyze, interpret, predict, make
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decisions and solve problems. Mathematics is considered a difficult syllabus
for most students who experience low level of achievement. Hence,
mathematics teachers face many problems regarding the curriculum,
methods of teaching, as well as the learner. Many studies have agreed that
students in intermediate grades suffer from poor understanding regarding
mathematics, maybe due to the conventional methods of teaching. The
change in mathematics textbooks towards modernization and development
as found in the textbook (1st ed.) of third intermediate grade is the new
curriculum applied in Iraq in the academic year (2017-2018). This could be
a turning point in the presentation and organization of lessons according to
six items of learning, including: understanding, training and solving life
exercises, thinking, writing, as well as classroom exercises which differ from
those in the lessons because they are objective and their answers are based
on multiple-choice method. Being a new method for students in terms of
answering questions and presenting the lesson using modern ways, thus
the researcher seeks to use strategies different from the conventional
method of mathematics teaching that may lead to raising the level of
academic achievement and developing skills of mathematical algebraic
thinking for the students of the research sample.
Algebra is one of the five basic components of mathematics in accordance
with the content criteria contained in the document of principles and
standards of mathematics textbook issued by the National Council of
Teachers of Mathematics (NCTM) of the United States of America. It has
various applications both in mathematics and other sciences. Algebra is an
abstraction and generalization of arithmetic because it examines the
characteristics of numbers and the dimension of their abstraction. The 12th
conference of research on (the future of algebra) focused on algebra as
having an important effect on thinking. Hence, algebraic thinking is one of
the types of thinking. Moreover, mathematical thinking is one of the main
objectives of teaching and learning mathematics in the 21st century
worldwide and at all and different age levels (al-Mufti, 2001: 195). Most
teachers tend to use the conventional method when teaching and dealing
with algebraic relationships without attempting to develop algebraic thinking
skills. According to Al-Qahtani and Abdul Hameed (2010), algebraic thinking
is concerned with the development of algebraic skills. Based on the above,
research problem is formed as follows:
What is the effect of VTS on students' achievement in mathematics
and their algebraic thinking?
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Research Objectives
The current research aims to identify the effect of employing VTS on:
Educational achievement in mathematics among students in the
third intermediate grade.
Algebraic thinking in mathematics among students in the third
intermediate grade.
Research Hypotheses
To achieve research objectives, the following null hypotheses are
formulated:
H1. There is no statistically significant difference at significance level (0.05)
regarding the average scores of achievement in mathematics between the
experimental group taught using VTS and the control group taught using
conventional method.
H2. There is no statistically significant difference at significance level (0.05)
regarding the average scores of algebraic thinking in mathematics between
the experimental group taught using VTS and the control group taught using
conventional method.
Research Significance
The importance of research is demonstrated in the following points:
Defining the effect of VTS on academic achievement and algebraic
thinking for students under research.
The importance of VTS in providing the opportunity to perceive
forms and models visually and make visual comparisons between
the characteristics of those forms reaching directly to the learner,
which leads to confirming those characteristics in teacher's mind
and maintaining the effect of learning. In addition, algebraic
thinking skills can be acquired by providing steps for each skill, as
well as they help the learner to communicate with others through
discussions.
Emphasizing the importance of algebraic thinking as being one of
the most important types of mathematical thinking, thus attention
should be on the development of its fields and skills.
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The teaching plans prepared by the researcher according to VTS
may benefit teachers and educational supervisors in preparing the
teacher's guide for intermediate grade.
Encourage the student to employ strategies that are concerned
with algebraic thinking skills that focus on all educational grades
and call for compatibility with the content of the educational
curriculum in accordance with the requirements of the 21st century
as contained in the new common basic international standards that
include eight mathematical competencies, the essence of which is
algebraic thinking.
Research Scope
The current study is limited to students of third intermediate grade in schools
belonging to Education Directorate of Salah al-Din Province. It investigates
their achievement in mathematics during the first academic semester,
including topics of: applications, sequences, compound sequences,
absolute value inequalities, multiplying algebraic numbers, and solution of
two linear equations graphically) in mathematics textbook (1st part) for the
academic year (2019-2020).
Definition of Terms
1. Achievement: Alderman (2007: 101) defined it as the ability to
achieve the desired, acquired educational experience.Practically,
achievement is defined as the mathematical knowledge learned by
students of first intermediate grade as a result of having
educational experience related to mathematical topics. It is
measured by marks they obtain in the final test of achievement
designed for this purpose.
2. VTS is defined by Amer and al-Masry (2016: 106) as that it
includes a series of regulatory measures and focuses on
developing the vision of teaching learners based on their thinking
methods and developing their abilities in translating visual
language into written or spoken verbal language. Concerning the
practical definition of VTS, it refers to all the procedures planned
and organized by the researcher to be taught as skills for students
of the experimental group throughout the experiment period where
they are employed for the practice of learning process in the
classroom.
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3. Algebraic thinking is defined by Herbert and Brown (2000: 123) as
the use of symbols and tools to analyze different mathematical
situations by: first, extracting information from these situations;
second, representing this extracted information by words, tables,
graphs and equations; third, interpreting this information by finding
a solution for the unknowns and examining different hypotheses.
Additionally, Steele (2005: 105) defined it as the ability to analyze
and identify patterns to represent quantitative relationships
between patterns and to generalize these quantitative
relationships.
In practice, algebraic thinking is one of the types of thinking whose skills
(exploring algebraic patterns and generalizations, exploring algebraic
relationships and functions, using representation and symbols) will be taught
by the researcher during the course of the experiment for students of the
experimental group.
Research Background
1. Visual Thinking Strategy (VTS)
VTS has been developed in the United States since the mid-1970s by the
cognitive psychologist Abiyaill Housen and art educator Philip Yenawine.
They have presented a developed model of VTS in an effort to understand
the degree of difference between viewing the influence of a meaningful piece
of art (Amer and Al-Masry, 2016: 106). VTS has been developed for more
than twenty years. It is not an extension of the art class and can be used in
any group of people or students (p: 3, http://macsaigteacher.weebly.com).
Visual representation supports visual thinking more than other types of
thinking in developing the ability to understand relationships found in the
presentation and identify causal relationships within the same form. Visual
thinking is an intellectual ability associated with the sensory-visual aspects
where it occurs when there is a mutual harmony between the forms and
relationships perceived by the learner and the correlation and intellectual
output based on vision and graphic representation (Habib, 2003: 98).
VTS is a method whereby the teacher's discussions facilitate artistic images,
shapes and drawings. It has been reported to have a consecutive positive
impact on both teachers and students. Perhaps, it is the simplest method
whereby teachers can provide students with the key behaviors sought by
common core skills: thinking skills that have become common and are
changed from lesson to lesson; linguistic and verbal literacy, visual literacy,
and collaborative peer-to-peer interactions. VTS provide a way to start the
learning process for deep thinking to be applied to various topics including
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poetry, mathematics, sciences and social studies. It provides open but highly
organized discussions for forms, images and diagrams. It greatly increases
the types of thinking and language among students as well as long-term
knowledge skills (P: 2, https://shenandoahmiddle.com). This strategy
includes a series of systematic procedures and focuses on developing the
vision of educating learners. In addition, it provides a professional program
for the development of classroom teachers as being based on experimental
research. This experimental research is based on the ways of individuals'
thinking and focuses on developing their abilities in translating the visual
language into written verbal language. VTS provided for students in different
situations serves them when they identify other topics (Amer and Al-Masry,
2016: 106).
1.1 Objectives of VTS
Objectives of VTS include:
1. Developing technical skills among people through diverse
cultures and at various times and places to generate
confidence in building the meaning of a wide variety of art.
2. Developing the ability of observation among people.
3. Developing active participation among individuals by asking
unlimited questions and answers by the teacher.
4. Creating interaction among students by solving problems
through team work.
5. Developing creative thinking among students.
6. Developing communication skills, such as expression,
listening to perspectives and managing discussion.
7. Acquiring the skill of comprehensive view of the topic and its
fragmentation.
8. Developing motivation and curiosity among learners to
acquire information.
In addition, there are other objectives including: developing communication
skills and those of creative and logical thinking and mathematical thinking,
as well as providing the confidence for students to deal with complexity,
ambiguity and diversity of opinions (Amer and Al-Masry, 2016: 106).
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1.2 Steps of VTS
Visual thinking is defined as a set of visual activities that can be employed
through an intellectual strategy. The steps of this strategy are as follows:
1. Presenting the mathematical model that expresses the
mathematical problem and its contents, after identifying its
data and proof.
2. Realizing relationships in the mathematical model and
identifying their characteristics, whether logical or causal, so
that they can be limited and useful.
3. Correlating existing relationships through the model and
inferring new relationships in the light of the relationships or
data specified in the model, bearing in mind that some of
the information given may be redundant or incomplete.
4. Understanding ambiguities or gaps in the model after
examining existing and pre-inferred relationships in the
second and third steps of this strategy and defining the
positions of ambiguities or gaps under study.
5. Thinking visually about the model in light of the positions of
ambiguities or gaps identified and attempting to use
concepts, rules, theories or previous proofs to clarify
ambiguity and gaps in order to solve the problem.
6. Imagination of solution through the model presented, taking
into account the inclusion of the previous steps, as this step
is the result of the previous five steps. Imagination of
solution is done intellectually as shown in figure (1).
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Figure 1: steps of VTS
This type of thinking is associated with the ability to spatial perception as it
develops the ability to imagine, mental work and mental imagination of
situations. Hence, many scientists dealt with the problems mentally and
reached solutions during their focused mental thinking at different times and
contexts; while others revealed relationships through mental thinking first
and then investigating the validity of their findings practically or logically. This
ability requires the human being to think mentally about his desires before
implementation (Amer and Al-Masry, 2016: 106). Through the process of
solving collective problems in VTS, students develop the desire and ability
to present their own ideas, while respecting and learning from the views of
their peers. Students participate by contributing observations and ideas, and
participate in lessons based on VTS using ways that they often do not
participate in the usual lessons. This strategy is a means of discussion and
conversation and linking relationships in the model visually (p: 3,
https://shenandoahmiddle.com).
According to Alfra (2007), VTS includes a series of systematic measures
that define the role of both teachers and learners in order to develop skills
of communication and creative, mathematical and logical thinking in order
for teachers to achieve confidence when dealing with complexity, ambiguity
and diversity of opinions. Visual thinking consists of three overlapping
strategies as follows: thinking about design, thinking about vision, thinking
about perception. VTS includes educational strategies for the teacher and
the learner (Afana, 2001: 169).
Moreover, Afana (2006) believes that those who think visually employ vision,
imagination and drawing in an active way, as they examine the mathematical
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problem from different aspects and may succeed in choosing the
presumption indicating its solution. After having a visual understanding of
the mathematical problem, they imagine alternative solutions and then
attempt to express this with quick diagrams to compare and evaluate them
later. Therefore, visual thinking is completely selected when vision,
imagination and drawing merge into an active interaction. In addition, Izb
(2002) states that visual thinking includes realizing the relationships and
their association, bridging gaps and understanding ambiguity to develop a
conceptual perception to achieve the ultimate goal of the situation.
1.3 The Role of the Teacher and the Student in the Light of VTS
Many questions help to examine the visual model. The role of the teacher
lies in giving questions, facilitating discussions with students, interacting with
their ideas and suggestions, participating in the composition of meanings,
as well as listening to various perspectives. On the other hand, students
have time and opportunity to think carefully and rethink and continue thinking
where they listen to each other, guess together and discuss their ideas and
suggestions for creating new ideas and then reviewing the results. For
example, questions asked by the teacher in mathematics lesson in light of
VTS could be as follows:
What is happening with this graph, or with this problem?
What did you see / read that makes you represent symbols and
figures in this way?
What relationships can you link?
What does this pattern represent?
What can you find more?
Students contribute to the discussion by providing their own observations
and thoughts. All contributions are accepted and considered neutral by the
teacher so that students can learn from the views of others (p: 3,
https://www.usf.edu).
1.4 The Relationship between VTS and Educational Theories
VTS includes educational strategies for both teachers and students based
on the active and focused examination of learners when reviewing the
objectives VTS and found to be linked to current cognitive and pedagogical
ideas. It includes:
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Using unlimited questions, reformulation and the
association of the student's related ideas by the teacher to
facilitate the discussion of the group.
Creating a secure atmosphere for discussion whereby the
teacher actively examines each idea and observation (Amer
and Al-Masry, 2016: 112).
2. Algebraic Thinking
2.1 The Concept of Algebraic Thinking
According to Berg (2009: 62), algebraic thinking is a development based on
mathematical reasoning and is superior to it through generalization.
Algebraic thinking is based on the generalization of individual's
computational ideas and is characterized by free work in the field of
arithmetic. In algebra, addition of mathematical expressions can be done
according to the general rules of arithmetic, i.e., they are not only calculated
as in the framework of mathematical reasoning (2+3 =5); whereas in
algebraic thinking, (2+3) is a special case of (a+b) where (a and b) are two
real numbers. As for the student of intermediate school, calculations are
special cases of more general algebraic concepts. In 1997, Herbert and
Brown defined algebraic thinking as the use of symbols and tools to analyze
different mathematical situations by extracting information from these
situations first; secondly, representing this extracted information by words,
tables, graphs, and equations; and thirdly, interpreting this information by
finding a solution for the unknowns and examining the various hypotheses.
Vance (1998) defined algebraic thinking to be like generalization of
arithmetic or as a language whereby arithmetic is generalized.
However, algebra is more than just a set of rules; it is a way of thinking
(Hebei, 2008: 16). Algebraic thinking is not only concerned with the
processing of symbols, but also focuses on giving priority to what is taught
at all academic levels for the student. The students start with simple things
in the early grades and progress until being able to express and think about
the mathematical operations using algebraic symbols and process (Battista
and Van, 1998: 145). Consistent with Driscoll et al. (2003), algebraic
mathematical thinking enables the student to:
Represent different patterns using tables, diagrams and words;
Analyze and generalize these patterns;
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Compare and correlate different representations of a
relationship;
Distinguish between linear and non-linear correlations, as well
as their characteristics using table or equation;
Develop the initial perception to understand the different uses
of the variable;
Use algebraic symbols to represent different situations;
Solve problems and identify equal formulations for algebraic
formulations;
Solve life problems using different representations; and
Use diagrams to analyze differences in amounts associated
with linear relationships, and then keeping them for distance
learning.
2.2 Skills of Algebraic Thinking
Based on previous studies that dealt with algebraic thinking skills (Al-Otaibi,
2019; Al-Khateeb, 2017; Al-Ma'atham and Al-Manoufi, 2017; Al-Rifai, 2009;
Obeida, 2016), it consists of three basic skills as clarified in table (1).
Table 1: distribution of algebraic thinking skills
AlgebraicThinking
Skills
Sub-Skills
Understand patterns Repetitive patterns;
Numerical patterns;
Geometric patterns; and
Representing the pattern using tables
and coordinate level.
Representation of
relationships and
functions
Identify functions; and
Representation of functions using
graphs, tables, arrow chart, symbols
and number line.
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The use of algebraic
symbols and variables
Use variables to express life situations;
Solve life problems by representation
using table; and
Representation of quantitative
relationships.
2.3 Components of Algebraic Thinking
Shamout et al. (2018: 466-467) stated that algebraic thinking consists of two
components:
1. Mathematical Thinking Tools
They include: solution of mathematical problem, mathematical
representation and mathematical thinking. Solution of mathematical problem
consists of strategies to solve mathematical problem and multiple solutions
(Krieler, 2009). It requires algebraic knowledge through solving the
mathematical problem in the field of algebra; in addition to applying various
strategies suitable to solve the mathematical problem, especially new
mathematical problems, and validity and interpretation of results (NCTM,
2000). Concerning mathematical representation, algebraic representation of
mathematical problems helps to generalize understanding in the process of
solving algebraic mathematical problems by translating and converting
words and quantitative relationships into equations that are easy to solve.
These skills include presentation of visual, symbolic, numerical and verbal
relationships, the transition between different representations and the
interpretation of information in the representations (Amin, 2012).
2. Basic Algebraic Ideas
It expresses the content and the material which develop the mathematical
field in which the tools of mathematical thinking are created and developed.
Algebraic ideas are explored based on three different visions. First, algebra
is a generalized arithmetic that includes mathematical strategies based on
concepts, ratio and proportionality. Second, algebra is the language of
mathematics that includes the meaning of variables, expressions and
solutions; understanding and using the characteristics of the system of
numbers; reading, writing and dealing with numbers and symbols using
algebraic rules; the use of equal symbolic representations to address
algebraic expressions, formulations, equations and inequalities. Third,
algebra is a tool for the study of associations and mathematical modeling.
Hence, search for and expression of mathematical patterns and rules and
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their generalization in real contexts and representation of mathematical
ideas using equations, tables, diagrams, working with input and output
patterns and developing consistent drawing skills are mathematical
processes and procedures creating algebraic skills (Badawi, 2007: 167).
2.4 Inputs and Teaching Activities that Help Students Develop
Algebraic Thinking
a. Problem Solving
It emphasizes the analysis of algebraic problems using equations and
variables. Students can use those inputs to examine and understand the
mathematical content, formulate problems from situations inside and outside
the school, and apply various strategies to solve problems while
emphasizing new problems and changing solutions and judging their
reasonableness (NCTM,1989: 75). There are various visual strategies used
to help students solve the problem, including: diagram, model, pattern, table,
Venn diagram and graph (Badawi, 2019: 448-457).
b. Semantic Inputs
Algebraic changes are used to represent relationships and interpret
changing quantities. Semantic input is intended to understand algebraic
symbols as meanings describing functions that are represented by diagrams
and algebraic processes as meanings of performing the processes with
diagrams for these functions (Vincent, 2000; Kieraw, 1996).
c. Generalization Inputs
They emphasize the importance of algebraic changes to represent
numerical and geometric patterns and determine relationships which govern
them (Vincent, 2000). The activities of functions, patterns and
generalizations provide learners with problems that require them to
describe, expand, analyze and create patterns and relationships that
increase their ability to use graphs, tables, equations, functions, words and
build mathematical models (NCTM, 2000: 222; Perry, 2000: 196).
d. Language-Based Inputs
The language that is clearly used in teaching and learning algebra when
students discuss their ideas on a topic enables them to build models, create
diagrams, analyze and criticize verbal arguments, and write about problem
solving processes. Presenting their thinking through these various forms
provide a clear vision of how they develop concepts, trends and processes
(Dougherty and Matsumoto, 1999: 91).
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Literature Review
1. Studies on VTS
Table (2) shows a number of studies related to VTS, as follows:
Table 2: studies related to VTS
No. Researcher
and Year
Research
Objective
Research
Method
Results
1 Abdullah
(2019)
To examine
VTS and its
applications in
arts teaching
(photography
as model)
The study
employed
experimental
method on (30)
students at the
Institute of Fine
Arts,
Kadhimiyah. It
used visual
thinking test, T-
test of two
independent
samples, chi-
squared test,
Mann–Whitney
U test and
Pearson
correlation
coefficient.
There is
statistically
significant
difference
regarding the
average
scores
between the
experimental
and the control
groups in
visual thinking
test in favor of
experimental
group.
2 Hamza
(2018)
To determine
the
effectiveness
of VTS in the
topic of
"basics of
design" for
developing
students' skills
in department
of design
This study used
experimental
method on (80)
female
students from
College of Fine
Arts, University
of Babylon. It
applied visual
thinking test,
percentage of
(73.71%) is not
a low
percentage
obtained by
respondents;
however, it is
not sufficient.
Therefore,
students do
not have
sufficient
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visual thinking
test items.
visual thinking
skills.
3 Al-Diriri
(2017)
To identify the
effectiveness
of VTS in
poster design
This study
employed
descriptive
method on (4)
shocking
advertising
campaigns. It
analyzed,
interpreted and
decoded (4)
sets of shock
campaigns.
The designer
motivates the
receiver to use
thinking
strategies so
that s/he could
translate visual
images and
their
denotations.
2. Studies on Algebraic Thinking
Table (3) shows a number of studies related to algebraic thinking, as follows:
Table 3: studies related to algebraic thinking
No. Researcher
and Year
Research
Objective
Research
Method
Results
1 Al-Rifai
(2019)
To determine
the effects of
teachers'
review of
activities
about
variables and
patterns on
the
development
of algebraic
thinking and
modifying
teachers''
beliefs
towards the
nature of
algebra
teaching.
The study
employed
experimental
method on
(60)
students, the
third group,
College of
Education,
Department
of
Mathematics.
It used
research
tools
including:
achievement
test on
variables and
a. There is
statistically
significant
difference
regarding the
average scores of
before and after
test between the
experimental
group and the
control group in
achievement
about variables
and patterns in
favor of
experimental
group.
b. There is
statistically
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patterns,
algebraic
thinking test,
algebraic
belief
instrument. It
applied T-
test to two
related
samples for
the average
scores
before and
after test.
significant
difference
regarding the
average scores of
before and after
test between the
experimental
group and the
control group in
achievement
about algebraic
thinking test in
favor of
experimental
group.
c. There is
statistically
significant
difference
regarding the
average scores of
before and after
test between the
experimental
group and the
control group in
achievement
about algebraic
belief instrument
in favor of
experimental
group
2 Al-Otaibi
(2019)
To define the
effectiveness
of using
Flipped
Classroom
strategy in
the
development
of algebraic
thinking skills
This study
used
experimental
method on
(56) students
in second-
grade high
school. It
used
teacher's
There is
statistically
significant
difference
regarding the
scores between
the experimental
and the control
groups in
algebraic thinking
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for female
students in
secondary
grade.
guide to
teaching
sequences,
algebraic
thinking skills
test, storyline
tutorial,
Edmodo
educational
platform. It
applied T-
test to two
independent
groups,
multivariate
analysis of
variance,
Box's M test
of variance-
covariance
matrices,
Levene's test
of the
equality of
variances.
skills test in favor
of experimental
group.
3 Shamout et
al. (2018)
To identify
the effect of a
teaching
strategy
based on
Dubinsky
theory
(APOS) on
the
development
of algebraic
mathematical
thinking
related to
associations
This study
employed
experimental
method on
(122)
students. It
performed
algebraic
mathematical
thinking test
and T-test to
two
independent
groups.
There is
statistically
significant
difference
regarding the
scores between
the experimental
and the control
groups in
algebraic
mathematical
thinking test in
favor of
experimental
group.
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Research Methodology
1. Research Approach
The experimental method was employed for achieving research objectives.
Experiment prepares a reliable and controlled variable of the specific
conditions of an action (Shehata, 2009: 208). Also, experimental design is
one of the most controlled and rigorous designs. It is the gold standard to
which research designs are compared (Abbas et al., 2009: 192). Therefore,
the researcher adopted the experimental design characterized by post-test
partial control of the research groups, as shown in table (4):
Table 4: Experimental research design
Group Equivalence
of groups
Independent
variables
Dependent
variables
Experimental VTS Achievement
Control Conventional
method
Algebraic
thinking
2. Research Sample
The current study community consisted of students of third intermediate
grade in the morning government high schools for boys belonging to Salah
al-Din General Directorate of Education, city center, for the academic year
(2019-2020). The researcher visited the Department of Education, Division
of Planning. In order to perform the experiment, the researcher selected the
sample represented by Al-Thara intermediate school for boys for the
following considerations: the school was near to researcher's house,
cooperation of the school principal to facilitate the task for the researcher,
there were two groups for the third intermediate grade in the school and this
is one of the most important conditions to be met in terms of research. Only
the statistically failed students were excluded as shown in table (5).
Table 5: distribution of research sample to groups
Group Division Total
number of
students
Number of
excluded
students
Number of
students after
exclusion
Experimental A 33 3 30
Control B 35 5 30
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3. Equivalence between Research Groups
The two research groups were equalized regarding some variables, namely,
age as calculated in months, mathematics scores for the past year,
intelligence, and parents' qualification. The statistical equivalence of the two
research groups regarding some variables that are believed to affect the
results of the experiment are shown as follows:
a. Age as calculated in months
The researcher calculated students' ages of both groups. The information
about students' ages was obtained from the school records and the students
themselves. The ages were calculated in months. The results showed that
there is no statistically significant difference as the calculated T-value was
lower than the tabulated one. This means that the two groups are equal in
this variable, as clarified in table (6).
b. Mathematics scores for the past year
The scores were obtained from the school records and then processed
statistically using SPSS and T-test for two independent samples. The results
revealed that there is no statistically significant difference at significance
level (0.05), as shown in table (6).
c. Scores of intelligence test
Intelligence is defined as the ability to process and memorize the information
(Al-Rubaie et al., 2013: 60). Raven test was used for being a good test,
having validity and reliability, having standards suitable for the Iraqi
environment and could be applied to the age of (15) years and above. It
consisted of (60) items divided into five groups (A, B, C, D, E). T-test was
employed for two independent samples to determine the statistical
significance of differences. The results demonstrated that there is no
statistically significant difference at significance level (0.05), as stated in
table (6).
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Table 6: results of T-test of both groups with equivalent variables
Variable
s
Groups T-value
Experimental,
(30) students
Control, (30)
students
Arithm
etic
Mean
Standa
rd
Deviati
on
Arithm
etic
Mean
Standa
rd
Deviati
on
Calculat
ed
Tabulat
ed
Age 1...02 2.21 1.2.22 2..22 ..10 2
Scores
of the
past
year
33.32 6.33 30.33 0.22 ..32
Scores
of
intellige
nce
33.12 6..2 30..1 6.22 ..32
d. Parents' qualification
A form was distributed to students asking them to write the qualification of
their parents. Based on personal information available in the school records,
there were three levels of qualification including primary, intermediate/
secondary and bachelor. The calculated T-value was less than the tabulated
one after employing chi-squared test to determine the statistical significance
of differences. Thus, the two groups were equal in this variable, as shown in
table (7).
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Table 7: results of chi-squared test for parents' qualification
Group Nu
mbe
r
Chi-squared value Signifi
cance
level Pri
mar
y
Interme
diate/
Second
ary
Bach
elor
Calcu
lated
Tabul
ated
Experi
mental
(father)
03 0 2 22 1.3 5.99
at
freed
om
score
(2)
not
statistic
ally
signific
ant Control
(father)
03 2 6 23 2.2
Experi
mental
(mothe
r)
03 23 3 22 1.3
Control
(mothe
r)
03 0 2 22 1.3
4. Research Requirements
a. Identifying the content
The scientific material included topics of applications, sequences,
compound sequences, absolute value inequalities, multiplying algebraic
numbers, and solution of two linear equations graphically in mathematics
textbook (1st part, 2nd ed.) for the academic year (2019-2020).
b. Formulation of behavioral objectives
Behavioral objectives were formulated according to Bloom's taxonomy
(knowledge, comprehension, application and analysis). There were (60)
cognitive objectives for the three chapters from mathematics textbook. They
were presented to experts in the field of methods of mathematics teaching
to certify their adequacy. The researchers amended some objectives after
obtaining (85%) of experts' consent on each objective. These objectives
were used in preparing the daily teaching plans and in building the academic
achievement test and algebraic thinking.
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c. Preparation of teaching plans
The teaching plan is a real translation of the objectives of the textbook
content. The teacher must use various teaching plans to determine the
activities s/he employs and the students' responses should achieve these
objectives (Aqilan, 2002: 209). The researcher prepared the required plans
for the experimental group according to VTS and plans for the control group
according to the conventional method. A sample of teaching plans for both
methods was presented to a number of experts in the field of mathematics
teaching to express their opinions about their adequacy.
5. Research Tests
They include the preparation of tests on achievement and algebraic
thinking.
1. Achievement Test
After identifying the content and objectives of test, (40) test items were
formed through preparing the table of specifications (test map) where the
percentage of lessons and behavioral objectives was determined.
a. Test instructions: instructions are important as they guide the
students to know how to answer the test items. Students'
answers were corrected according to the typical answers
(correction key), and hence the score (2.5) was given to the
correct answer and (zero) to the wrong answer and for the
neglected items that were treated as wrong answer. Thus, the
total score became (100).
b. Test validity: it was based on two types:
Face validity: it covers the objectives it purports to measure,
i.e. the general appearance of the test in terms of the type
of items, their relevance and transparency (Kay et al., 2012:
253). A draft of achievement test consisting of (40) objective
items (as multiple choice) with behavioral purposes was
presented to a group of experts in the field of teaching
methods to express their opinions and observations on the
relevance, transparency of items and their adequacy to the
specified behavioral purposes, as well as the rationality of
alternatives and any other observations to improve the test.
They agreed on test by (81%) with amending some of its
items as all of them are valid.
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Content validity: the specification table is an indicator of
content validity for the test (Thorndike, 1989: 56). The
researcher prepared the test map, so that content validity
for the test is achieved.
c. Application of the test to the exploratory sample:
The first application: this test was applied to the first sample
consisting of (50) students of third intermediate grade in Ibn
al-Mu'tam high school for boys belonging to the Education
Directorate of Salah al-Din, Tikrit. This is to certify the clarity
of items, response instructions as well as time as the time
taken by the first student and the last student was recorded
(about 90-120 minutes). After calculating the average time
of the test, it was about (105) minutes. This test conducted
on 2/12/2019 on Monday.
The second application: this is to analyze the test items. It
was applied to the second sample consisting of (64)
students. The answers were corrected according to the
correction criterion and then the scores were arranged
downwardly. The sample was divided into two groups:
higher (32) students and lower (32) students. According to
the existing literature, it is better to divide the same score
into (50%) as higher and (50%) as lower. Therefore, items
can be analyzed statistically. Through the analysis process,
the following factors were calculated:
i. Item difficulty: items difficulty and easiness were
calculated by applying the difficulty equation. It was
found that item difficulty ranged between (0.02-0.68).
The items are acceptable if the difficulty ranges
between (20-75) (Melhem, 2012: 269). Thus, the test
items are acceptable.
ii. Discrimination power item was calculated using the
statistical equation for the objective items. It was
found to range between (0.20-0.60). This means that
the test items are acceptable in terms of their
discrimination power. Discrimination power is good
when ranging within (20%) and more (Al-Zahir, 1999:
13).
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iii. Effectiveness of distraction: this equation was used
and was found to be effective with negative value for
all items.
d. Test reliability: it means the consistency of measurement and
observation, as well as producing the same results when
repeated (al-Noor, 2007: 173). Kuder-Richardson formula was
used to measure test reliability. The literature indicated that the
tests are good if the reliability ranges between (0.60-0.85) (Hills,
1976: 152). After measuring the test reliability, it was found to
be (0.88), which is high. Thus, the test can be applied.
2. Algebraic Thinking Test
This test was prepared as follows:
a. Test objective: to measure the skills of algebraic thinking among
the students under study. Based on previous studies, the skills
included: understanding patterns, representing relationships and
functions, using algebraic variables and symbols.
b. Forming test items: the questions were formed and included items
on the above skills after reviewing previous studies (Antonio, 2003;
Columb and Pearson, 2001; Drisswell, 1999; Greens, 1999;
Kriegler, 1999; Documents of National Council of Mathematics
Teachers, 2000).
c. Test validity: a draft of test was presented to experts in the field of
methods of teaching mathematics to certify the adequacy of items.
Their opinions were considered and the items were amended
accordingly.
d. The exploratory application of test: it was applied to a sample
consisting of (52) students of third intermediate grade in Al-
Mukhief intermediate school for boys belonging to the Education
Directorate of Salah al-Din, Tikrit, on 17/11/2019, and then
reapplied to them after (14) days. Hence, the correlation coefficient
between the two applications was (0.74), expressing test reliability.
During its application to the sample, the test time was observed for
each student. The actual time taken to answer the test items was
about (45) minutes. The test instructions and items were clear to
the students, and then the answers were corrected according to
the correction criterion. Then, the scores were arranged
downwardly. The sample was divided into two groups: higher (26)
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students and lower (26) students. As stated in the literature, it is
better to divide the scores themselves into (50%) for higher and
(50%) for lower. Thus, the items can be analyzed statistically.
Through the analysis process, the following factors were
calculated:
i. Item difficulty: items difficulty and easiness were calculated
by applying the difficulty equation. It was found that item
difficulty ranged between (0.02-0.66). Thus, the test items
are acceptable.
ii. Discrimination power item was calculated using the
statistical equation for the objective items. It was found to
range between (0.20-0.63). This means that the test items
are acceptable in terms of their discrimination power.
iii. Effectiveness of distraction: this equation was used and was
found to be effective with negative value for all items.
e. The final draft of test: the test included (25) items distributed to the
three algebraic thinking skills. The objective tests used were in the
form of multiple choice which included four alternatives, one of
which is correct. Thus, the test can be applied.
Table 8: distribution of algebraic thinking skills test items
Algebraic
Thinking Skills
Sub-Skills Number of
items
Score
Understand
patterns
Repetitive patterns;
Numerical patterns;
Geometric patterns;
and
Representing the
pattern using tables
and coordinate level.
2. 2.
Representation
of relationships
and functions
Identify functions; and
Representation of
functions using graphs,
tables, arrow chart,
2. 2.
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symbols and number
line.
The use of
algebraic
symbols and
variables
Use variables to
express life situations;
Solve life problems by
representation using
table; and
Representation of
quantitative
relationships.
0 1.
Total 10 2..
6. Experiment Application
After completing the requirements of experiment and ensuring the
equalization of the two research groups, the tests of achievement and
algebraic thinking were applied to the research sample which consisted of
(60) students of third intermediate grade in al-Thara intermediate school for
boys in the first semester. It started on 9/12/2019 and ended on 11/ 12/2019.
7. Statistical Means
In this study, SPSS was applied to process the data through the following
tests, statistics and equations: T-test for two independent samples, chi-
squared test, Kuder-Richardson formula, item difficulty, discrimination
power item and Effectiveness of distraction.
Research results and recommendations
1. Results and Discussion of Achievement Test
To verify the first null hypothesis: (There is no statistically significant
difference at significance level (0.05) regarding the average scores of
achievement in mathematics between the experimental group taught using
VTS and the control group taught using conventional method) and to test
the significance of differences, T-test for two independent samples, equal in
number, was employed as shown in table (9).
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Table 9: results of T-test for the experimental and control groups in
achievement test
Group Num
ber
Aver
age
Stan
dard
devia
tion
Free
dom
degr
ee
T-value Signifi
cance
level
(0.05)
Calcul
ated
Tabul
ated
Experi
mental
30 61.3
.
3.12 58 21.22 2.30 Statistic
ally
signific
ant Control 30 32.2
.
0.21
Based on results presented in table 9, there is a statistically significant
difference in favor of the experimental group due to the following reasons:
a. VTS may have a clear role in increasing the level of students'
achievement through understanding the relationships in models
and identifying the characteristics of those relationships and
attempting to use them in solving mathematical problems.
b. VTS releases the student from the conventional method in which
s/he is merely a passive recipient and has become an active and
effective participant.
c. Perhaps the use of VTS has reduced abstraction in mathematics,
removing ambiguities or gaps through forms and attempting to use
concepts, rules or proofs to bridge the gap between the problem
and its solution.
d. The thinking strategy may have a role in increasing educational
achievement for its role in increasing emphasis and attention to
educational approaches and thus making visual comparisons
between the characteristics of the forms, facilitating them to the
learner to visually understand the mathematical problem.
2. Results and Discussion of Algebraic Thinking Test
To verify the second null hypothesis: (There is no statistically significant
difference at significance level (0.05) regarding the average scores of
algebraic thinking in mathematics between the experimental group taught
using VTS and the control group taught using conventional method) and to
258 The effect of visual thinking strategy on students' ….. Acta Academica / 2019:51 (2)
test the significance of differences, T-test for two independent samples,
equal in number, was employed as shown in table (10).
Table 10: results of T-test for the experimental and control groups in
algebraic thinking test
Group Num
ber
Aver
age
Stan
dard
devia
tion
Free
dom
degr
ee
T-value Signific
ance
level
(0.05)
Calcul
ated
Tabul
ated
Experi
mental
30 60.6
3
0.00 58 21.00 2.30 Statistic
ally
significa
nt Control 30 3..3
0
0.00
Based on results clarified in table (10), there is a statistically significant
difference in favor of the experimental group due to the following reasons:
a. VTS may help the student to use the skills of algebraic thinking and
represent the information mathematically using diagrams, tables,
graphs and equations.
b. VTS may contribute to motivate students' thinking to represent and
understand relationships by using various representations that make
it important to teach and learn algebra as a tool of thinking.
c. The existence of VTS presents the mathematical model which
expresses the mathematical problem and understanding
relationships in it and defining the characteristics of relationships.
This requires employing the representation in the learning of algebra
and the application of representations and their translation to solve
algebraic problems.
3. Recommendations
In the light of research results, the study presents the following
recommendations:
Instructing the teaching boards to pay attention to modern
teaching strategies and attempt to apply them, including VTS.
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Authors of the curriculum should provide support for mathematics
textbooks by providing activities and mathematical exercises that
enhance the skills of algebraic thinking for students.
Training courses and seminars should be held by the general
directorates of education to provide mathematics lessons in a way
that enhances the student's satisfaction and provokes motivation
to develop the skills of thinking algebraic thinking.
Describing the skills of mathematical algebraic thinking for
mathematics teachers and working to develop them for their
students.
Training mathematics teachers to use VTS in mathematics at
different stages because of its obvious effect on increasing the
level of academic achievement of students.
Inviting the teaching boards to pay attention to third intermediate
grade for qualifying the student to move to the secondary grade.
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