laila khalid khudhair the effect of visual thinking ...acta-academica.com/51-2/14_231-261.pdf ·...

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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232 The effect of visual thinking strategy on students' ….. Acta Academica / 2019:51 (2)

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?


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.


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.


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

http://macsaigteacher.weebly.com/


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).

https://shenandoahmiddle.com/


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).


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:

https://www.usf.edu/


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;


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.


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


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).


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


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


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


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.


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


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).


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).


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.


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.


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).


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)


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.


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).


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


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.


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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