The Effect of Dynamic Geometry on Developing

LEBANESE UNIVERSITY Faculty of Education Deanery

The Effect of Dynamic Geometry on Developing Students’ Spatial Abilities to Draw and Analyze Two-Dimensional Figures in Grade Five in a Private School in Lebanon

Professional Master in Education and Training Specialization: Educational Technology

Submitted by Reine F. SAKR

Supervised by Dr. Rita ZGHEIB

Beirut, 2017

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON ii Abstract Geometry is often underestimated as being a simple branch of mathematics, when it is actually an essential aspect of the everyday life of students who more often find a hard time comprehending it. This research was conducted with the purpose of finding a way to render geometry apprehensible to students, more specifically fifth graders. It aims to examine the possibility of using dynamic geometry and check its effect on the development of the students’ spatial abilities to draw and analyze two-dimensional figures. The research explores the effect of integrating technology into geometry teaching at four different levels: (1) students’ spatial abilities; (2) their results in geometry; (3) their ability to construct a two-dimensional figure; and, last but not least, (4) their ability to analyze a two-dimensional figure. The implementation phase stretched to one month, and was applied to two grade five classes of a Lebanese private school located in Beirut. One served as a control and the other as an experimental group. The former consisted of thirteen students while the latter consisted of ten. The experimental group students were taught using dynamic software, mainly GeoGebra. On the other hand, the control group students were subjected to the regular previously adopted method of instruction. Students sat for four math tests, two of which were regular tests, while the other two were administered at the beginning of the implementation phase as pretests and then repeated at its end as posttests. Data was thus collected, entered into SPSS, and analyzed both quantitatively and qualitatively. First, the quantitative results of the spatial ability tests showed that the experimental group scores increased from the posttest to the pretest while the control group scores decreased. In addition, the qualitative results back up those results too, the experimental group maintained the same percentage of correct answers that remained correct during the posttest in addition to a valuable percentage of wrong answers that became correct (13.75% and 14%) being compared to the control group (6.73% and

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON iii 9.45%). Second, the experimental group scored higher than the control group on both regular math tests. Third, the quantitative and qualitative results of the ability to construct showed that the experimental group scored higher and attained more objectives than the control group. The same results were repeated for the purpose of analysis. Therefore, all the stated hypotheses were confirmed and dynamic geometry proved to be beneficial to students’ spatial abilities and performances in geometry. Consequently, educators are advised to use and integrate technology into their classrooms. However, future researchers willing to repeat similar researches are advised to extend the implementation period for an entire school year and to apply it at higher levels. Keywords: integrating technology, dynamic geometry, spatial ability.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON iv Acknowledgments

I would like to thank my thesis advisor Dr. Rita Zgheib at the Lebanese University, as well as all the students who participated in this study. Secondly, I must express my very profound gratitude to my parents, my family, and to my fiancé for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. Thank you.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON v Table of Contents Abstract ........................................................................................................................................... ii Acknowledgments.......................................................................................................................... iv Table of Contents ............................................................................................................................ v List of Tables ................................................................................................................................. ix List of Figures ................................................................................................................................ xi Chapter 1: Introduction ................................................................................................................... 1 1.1 Overview ............................................................................................................................... 1 1.2 Problem statement ................................................................................................................. 2 1.3 Purpose of the Study ............................................................................................................. 4 1.4 Possible Causes ..................................................................................................................... 4 1.5 Possible Solutions ................................................................................................................. 5 1.6 Research Questions ............................................................................................................... 7 1.7 Operational Objectives .......................................................................................................... 7 1.8 Hypotheses ............................................................................................................................ 7 1.9 Definition of Terms ............................................................................................................... 8 Chapter 2: Literature Review .......................................................................................................... 9 2.1 Students’ Performance in Geometry in Lebanon .................................................................. 9 2.2 Importance of Geometry ....................................................................................................... 9 2.3 Importance of Technology in Education ............................................................................. 10 2.4 Technology and the Levels of Geometry ............................................................................ 12 2.4.1 The cognitive geometric thinking model. ......................................................................12 2.4.2 Dynamic Geometry .......................................................................................................13 2.5 Spatial Ability ..................................................................................................................... 14 2.5.1 Definition of spatial ability. ...........................................................................................14

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON vi 2.5.2 Importance of spatial ability. .........................................................................................15 2.5.3 The relation between spatial ability and mathematics. ..................................................16 Chapter 3: Methodology ............................................................................................................... 18 3.1 Type of Research ................................................................................................................. 18 3.2 Participants .......................................................................................................................... 19 3.2.1 Students. ........................................................................................................................19 3.2.2 Teachers. ........................................................................................................................20 3.3 Procedure ............................................................................................................................. 21 3.3.1 Administrative Protocol.................................................................................................21 3.3.2 Teaching periods and methodology...............................................................................23 3.3.3 Resources. ......................................................................................................................27 3.4 Instruments .......................................................................................................................... 30 3.4.1 Choice of instruments. ...................................................................................................30 3.4.2 Quality of instruments. ..................................................................................................31 3.4.3 Test design and directions. ............................................................................................33 3.4.4 Testing conditions. ........................................................................................................35 3.4.5 Validity of instruments. .................................................................................................36 3.4.6 Testing dates. .................................................................................................................37 3.5 Timeline .............................................................................................................................. 38 3.6 Data Collection .................................................................................................................... 38 3.7 Data Analysis ...................................................................................................................... 39 Chapter 4: Results ......................................................................................................................... 41 4.1 Spatial Ability Test .............................................................................................................. 41 4.1.1 Quantitative analysis of spatial ability test results. .......................................................41 4.1.2 Qualitative analysis of spatial ability test results. .........................................................43

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON vii 4.2 Regular Tests’ Quantitative Results .................................................................................... 48 4.3 Ability to Construct ............................................................................................................. 50 4.3.1 Quantitative results ........................................................................................................50 4.3.2 Qualitative Results.........................................................................................................51 4.4 Ability to Analyze ............................................................................................................... 55 4.4.1 Quantitative results ........................................................................................................55 4.4.2 Qualitative Results.........................................................................................................56 4.5 The use of Software............................................................................................................. 58 Chapter 5: Conclusion................................................................................................................... 59 5.1 Introduction ......................................................................................................................... 59 5.2 Conclusions ......................................................................................................................... 60 5.2.1 Research question 1. ......................................................................................................61 5.2.2 Research question 2. ......................................................................................................61 5.2.3 Research question 3 .......................................................................................................62 5.2.4 Research question 4. ......................................................................................................63 5.3 Limitations of the Study ...................................................................................................... 64 5.4 Recommendations ............................................................................................................... 65 5.4.1 Recommendations for future research. ..........................................................................65 5.4.2 Recommendations for curriculum designers and book authors. ...................................67 References .................................................................................................................................... 68 Appendices .................................................................................................................................... 74 Appendix A: Letter to the principal .......................................................................................... 74 Appendix B: Minutes of Meeting.............................................................................................. 75 Appendix C: Parents’ Consent Form ........................................................................................ 77 Appendix D: Student’s Consent Form ...................................................................................... 79

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON viii Appendix E: Administrational Permission Form ...................................................................... 80 Appendix F: Spatial Aptitude Test ............................................................................................ 82 Appendix G: Spatial Ability Practice ........................................................................................ 88 Appendix H: GeoGebra Online Resources ............................................................................... 95 Appendix I: PowerPoint Presentation ....................................................................................... 96 Appendix J: Class Worksheet ................................................................................................... 98 Appendix K: First Regular Test ................................................................................................ 99 Appendix L: Second Regular Test .......................................................................................... 102

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON ix List of Tables Table 1 Distribution of Students in the Two Groups with Details Regarding the Gender ............ 20 Table 2 The Difference between the Number of Periods Assigned by the Lebanese Curriculum and the Number of Periods Allocated for this research .................................................................. 24 Table 3 Number of Questions and Pages in Each Test .................................................................. 34 Table 4 Difference between the Researcher’s Duration and the Students’ Duration of Each Test35 Table 5 Descriptive Statistics of all Pre- and Post-tests for Students in the Experimental Group 42 Table 6 Descriptive Statistics of Pre- and Post-tests for Students in the Control Group .............. 42 Table 7 Percentage of Variation of Answers of Spatial Aptitude Test for Experimental and Control Groups................................................................................................................................ 44 Table 8 Percentage of Variation of Answers of Spatial Ability Test for Experimental and Control Groups ............................................................................................................................................. 45 Table 9 Number of Illegible and Unanswered Questions of Spatial Aptitude Test for Experimental and Control Groups .................................................................................................. 46 Table 10 Number of Illegible and Unanswered Questions of Spatial Ability Test for Experimental and Control Groups ......................................................................................................................... 46 Table 11 Descriptive Statistics of Regular Tests for Students in the Experimental Group ........... 48 Table 12 Descriptive Statistics of Regular Tests for Students in the Control Group .................... 48 Table 13 Descriptive Statistics of Construction Questions in the First Regular Test for Students in the Experimental Group .................................................................................................................. 50 Table 14 Descriptive Statistics of Construction Questions in the First Regular Test for Students in the Control Group ........................................................................................................................... 50 Table 15 Cross tabulation of Objective One for Construction ...................................................... 52

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON x Table 16 Cross tabulation of Objective Two for Construction ...................................................... 53 Table 17 Cross tabulation of Objective Three for Construction .................................................... 54 Table 18 Descriptive Statistics of Analysis Question in the First Regular Test for Students in the Experimental Group ........................................................................................................................ 56 Table 19 Descriptive Statistics of Analysis Question in the First Regular Test for Students in the Control Group ................................................................................................................................. 56 Table 20 Cross tabulation of the Analysis Objective..................................................................... 57

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON xi List of Figures Figure 1. Screenshots taken from the formulas' booklet. ................................................................ 27 Figure 2. Calculating areas of right triangles. ................................................................................. 28 Figure 3. Area and perimeter of a circle. ........................................................................................ 29 Figure 4. Demonstration of an area of a right triangle. ................................................................... 29 Figure 5. Area of rectangles and squares. ....................................................................................... 30 Figure 6. Mean difference of spatial aptitude pretest and posttest for experimental and control groups. ............................................................................................................................ 43 Figure 7. Mean difference of regular tests for experimental and control groups. .......................... 49

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 1 Chapter 1: Introduction 1.1 Overview Mathematics is the science of numbers, quantities and shapes and the relations between them (Merriam-Webster, 2016). Although defining mathematics seems easy and can be done in one sentence, mastering all lessons in this area can never be accomplished. However, development of these abilities is a must; and it is more efficient if it is done in a chronological way according to the age of the student. Mathematics encircles many sectors such as patterns, measurement, fractions, time, multiplication, equations, systems, functions, integrals and it gets more sophisticated in advanced mathematical courses. Geometry is definitely one item on this list, but unfortunately, insufficient time is assigned to it by the Lebanese curriculum. After spending a few years in the teaching field, and working as a private tutor specifically, the researcher experienced different complications than those that would arise in a normal classroom. Assuming that difficulties do not occur in the classroom is definitely not the case, but the researcher claims that when a teacher spends more time faced to one or two children “privately”, s/he can see the problem from a different point of view. In such cases, students are more likely to express themselves and highlight on their non-acquired concepts. This led the researcher to a point where she had to relate between the new complications and those similar that were taking place in the classroom: geometry is not a safe zone for most students. On the other hand, geometry is highly important to prepare students for their future life and career. For example, you could use the geometrical concept of volume unconsciously every day to decide the appropriate size of a container that should be used for the leftovers. Allowing students to fail

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 2 in this section is identical to dropping teachers’ conscience. Teachers are here to accompany students in their journey and provide them with the opportunity to thrive, now and in their future life. After investigating through her extensive teaching experience, the researcher has come to the conclusion that an exemplary class is where students are engaged, passionate about the subject, thrilled to take part in the analysis phase and exhilarated to come up with a solution. Two and three-dimensional concepts are covered since grade one in the Lebanese curriculum and remain there throughout the secondary level and college. On the other hand, the National Council of Teachers of Mathematics (NCTM) declares that two-dimensional and threedimensional spatial visualization and reasoning are core skills that all students should develop (Basham, 2007). To that avail, this research was introduced and applied in grade five; because at this level, students can improve their understanding of the difference between diverse geometrical concepts and have the ability to visualize and interpret them. Geometry demands understanding, visualizing, applying and reasoning; and it is definitely not going to be taught through rote learning. Therefore, this research is about the absolute opposite. 1.2 Problem statement According to Martin, Mullis & Foy, students were only able to answer 46.2% of the geometry test items correctly (Keuroghlian, 2013). This is an enormous indicator that students face problems when encountering geometry. In addition, the researcher’s personal teaching experience in regular classrooms and as a private tutor made this situation more detectible. Perceiving this problem is mainly due to the researcher’s teaching experience as a private tutor,

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 3 and then as a classroom teacher. It is worth noting that students tutored privately by the researcher are enrolled at the school subject of this thesis. After substantial observations, the researcher concluded that students do not know how to use basic mathematical tools such as protractors and compasses. They do know that this tool is called a setsquare and it is used to draw right angles, but when it comes to construction, learners face problems on where to put this setsquare, on which side, how to rotate it, and where to draw. Furthermore, learners lack the ability to comprehend what they have been asked to draw because they are not able to differentiate between different geometric terminologies. For example, asking students to draw a rhombus will lead them to draw a square, while they are totally convinced that this is a rhombus even though they know that a rhombus has no right angles. Alternatively, sometimes when the question demands the length of the radius of a circle, students calculate the diameter’s length instead, although they know that there are two different concepts here, but they lack the ability to associate which definition is to which terminology. Moreover, it gets harder when more constructional instructions are given on the same figure. This leads to a poor ability to analyze, interpret, and construct a geometrical figure because of a poor ability to visualize. Nowadays, it is confirmed that learners have eight multiple intelligences, and most have the spatial intelligence and they can learn through visuals (Moore, 2012, p. 55). Knowing that not all students are able to construct mental structures, they do need to visualize those geometrical shapes in front of them. And sometimes real life objects won’t serve, especially in two-dimensional geometry. For example, there is nothing in real life that could be similar to a line; since nothing can extend from both extremities to infinity. Students need other methods that

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 4 can approach them to the real construction of the figure; in order to develop their own spatial abilities. 1.3 Purpose of the Study The research has more than one purpose that will be discussed in the following paragraph. 1- Help students understand geometric terminology by associating geometrical definitions to real shapes. 2- Teach students about different geometrical tools and the way to use them in constructing figures. 3- Provide learners with the ability to visualize, and analyze two-dimensional geometrical figures using dynamic geometry. 4- Investigate whether dynamic geometry will increase or decrease students’ spatial abilities. 5- Improve the spatial abilities, by modifying the teaching style and adopting dynamic software. 6- Compare the results of the traditional teaching method to those of the dynamic geometry method. 1.4 Possible Causes The following section is the researcher’s analysis of the probable sources of the problem noted earlier.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 5 

According to the Educational Center for Research and Development [ECRD], the Lebanese Curricula consecrate less time to geometry as compared to any other mathematical component (Keuroghlian, 2013).

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Students are unable to relate geometrical definitions to real shapes.

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Geometry is not taught to students taking into account their multiple intelligences.

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Students are incapable of visualizing figures.

1.5 Possible Solutions Due to the fact that there are different causes that lead to this problem, no single solution can be taken into consideration. After reviewing several past researches stated above namely Keuroghlian, (2013), we believe that a mix of solutions proves to be more efficient. The following is the list of possible solutions: 

Use a dynamic software to explain all geometry lessons.

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Allow the students to use and experiment that dynamic software.

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Use educational games.

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Spend more time on geometry lessons.

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Highlight the importance of defining geometric terminology.

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List and clarify the specificity of all geometrical tools that a grade five student would use.

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Conduct a systematic analysis of students’ abilities to construct a geometrical figure.

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Assess students’ spatial abilities as a pre- and post-test.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 6 Applying all those solutions is beyond the scope of the researcher’s abilities, that is why solutions were limited to fit the researcher’s ability more efficiently and the need of the students. The main idea is to use a dynamic software, because “multiple intelligences” are an established fact (Moore, 2012, p. 55), and students need to visualize the rotation of a shape to be able to relate as much as possible geometrical shapes to real life. In addition to that, the researcher will try to spend more time on geometry. However, because the Lebanese curriculum is very dense, increasing the teaching periods would prove a little bit inconvenient. On the other hand, the researcher chose not to use educational geometrical games because her main goal is to assess whether the dynamic geometry method might yield better results than the normal teaching method. Students need to define specific geometric terminology; however, this step will not be accomplished through rote learning. However, students should not memorize definitions but demonstrate an ability to relate between geometric terminologies and real figures. The same thing would happen with the geometrical tools such as setsquares, protractors etc. Students are not expected to define a certain concept using exactly the same words that were provided by the teacher or the book. Students must demonstrate their understanding and analysis of the definition in order to be able to formulate their own personal accurate definition.

Furthermore, the last two points listed earlier will definitely be investigated in depth and will be analyzed in detail in the methodology section of this thesis.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 7 1.6 Research Questions Throughout this study, the main question that will be examined is: “Does dynamic geometry develop students’ spatial abilities to draw and analyze two-dimensional figures?”. However, answering this question is not easy; so a number of questions have been derived from that main question and will be examined also.

▪ Do students’ difficulties regarding geometry change over time? ▪ Do technological methods help students distinguish between various geometric terminology?

▪ Does dynamic geometry ameliorate students’ spatial abilities? 1.7 Operational Objectives Two operational objectives are set for this action research. 1- Assess students’ initial understanding of two-dimensional figures. 2- Evaluate students’ understanding of two-dimensional figures after applying dynamic geometry. 1.8 Hypotheses 1- Changing the teaching methodology and integrating technology will help improve the grade five students’ spatial abilities. 2- The dynamic geometry method will improve students' results more than the regular adopted teaching method.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 8 3- The dynamic geometry method will improve students’ ability to construct level appropriate two-dimension geometric figures more than the regular adopted teaching method. 4- The dynamic geometry method will improve students’ ability to analyze twodimension geometric figures more than the regular adopted teaching method. 1.9 Definition of Terms Geometric Terminology: special words or phrases that are used in the field of geometry. (Merriam-Webster, 2016). Geometry Education: the study of size, shape and location of objects in reference to one another (Brudigam & Crawford, 2012). Dynamic Geometry: a type of computer software used to analyze the geometry topics. It transforms geometry in a dynamic model by moving the shapes and is used to rebuild the shapes (Koparan & Yılmaz, 2015, p. 196). Spatial Ability: cognitive function that makes it possible for people to deal effectively with spatial relations, visual spatial tasks, and orientation of objects in space (Basham, 2007).

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 9 Chapter 2: Literature Review 2.1 Students’ Performance in Geometry in Lebanon Keuroghlian’s (2013) study found the following: Together, the teaching experience and test results conducted on an international level, show that Lebanese students are having difficulties with the learning and using of geometry (Keuroghlian, 2013, p. 11). He also noted that one of the reasons behind this issue is that the curriculum itself does not emphasize geometry. According to Martin, Mullis & Foy (2008), Lebanon ranks twenty-forth in geometry out of the participating forty-eight countries, seventh after the US, and first among the Arab countries (Keuroghlian, 2013). Although Keuroghlian (2013) discusses that Lebanese scores in some domains are relatively more acceptable than in other domains, he advocates that further investigation is undeniably essential. 2.2 Importance of Geometry Studying geometry reveals – in some way – the deepest true essence of the physical world. And teaching geometry trains the mind in clear and rigorous thinking (Bursill-Hall, 2002, p. 1). This is an advanced explanation of the role of geometry in our daily life. In simpler terms, Uba (2013) argues that geometry is found everywhere indeed, in all types of jobs. Geometry takes place when you schedule your day, prepare your meals, or even divide your money (Uba, 2013). It is true that geometric terminologies; such as line, point, segment, etc. aren’t used in our daily life but human’s reasoning and logical thinking are shaped by these concepts. Geometry is related to different fields such as, architecture, engineering, and astronomy. However, it is important to know that a human’s first encounter with geometry occurs early in life; like for instance when babies play educational games, including shapes, areas, and volume.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 10 Unfortunately, people tend to forget it. In short, geometry should not be underestimated because humans do need it in their daily lives (Uba, 2013). According to Naraini Idris (2006), students are often discouraged when they lack the ability to understand geometry. She adds that lacking this ability can be caused by different factors such as geometry language, visualization abilities and ineffective instruction (Abdul Saha, Ayub, & Tarmizi, 2010). 2.3 Importance of Technology in Education Twenty first century students spend most of their time immersed in popular technologies such as Facebook, Snapchat, Twitter, Instagram, and more (Moore, 2012, p. 91). They are considered digital natives since they were born and are growing up in a technological society (Moore, 2012, p. 19). In other words, they are quite knowledgeable and proficient in using it. The use of technology improves the effectiveness and the quality of instruction by capturing the student’s attention (Moore, 2012, p. 91). According to Healy (1998) and Kirpatrick & Cuban (1998), review of different studies proved that technology improves students’ results in all subjects at all levels and even more so for students with special needs (Moore, 2012). The Lebanese ministry has acknowledged the fact that technology is essential as they created an educational technological strategic plan back in 2012. The plan was supposed to be applied among Lebanese schools by the end of 2017. Unfortunately, this plan was not implemented, but this does not diminish the importance of technology. This strategic plan defined the mission and purpose of using Information and Communication Technology (ICT) in

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 11 schools since ICT can improve teaching and learning (Ministry of Education and Higher Education, MEHE, 2012). On the other hand, discussing the importance of technology does not diminish the importance of teachers and teaching because focusing only on technology in teaching and learning showed failure results across the globe (Ministry of Education and Higher Education, MEHE, 2012). In addition, a study applied in Lebanon shows that students were more motivated and engaged in the classroom, as it gave teachers the opportunity to modify the curriculum according to the students’ needs (Al Chibani & Hajal, 2017). To foster the process of integrated technology in classrooms the role of teachers and their attitudes are major. Many teachers prejudge the use of technology in education as if it would replace them. In other words, they refuse to integrate technology into their classroom and lose the interest of digital native students. Alternatively, it is important to mention that some teachers do employ technology in their classrooms. Nevertheless, some of them fight powerlessness since they are utilizing it inappropriately: they use it themselves, without providing with the students the option to interact and experience it. The referenced ideas discussed earlier highlight the importance of educating using technology, as it grasps students’ attention and could help teachers reach a wider audience. However, research shows, relying on technology only, decrease students’ scores (Ministry of Education and Higher Education, MEHE, 2012). Therefore, teachers must give themselves credit and acknowledge the fact that technology is not a competitor. Teachers should take the role of a facilitator instead of a lecturer, and students must be allowed to use and interact with that

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 12 technology. Teachers should plan their lessons effectively, to engage students while benefiting from technology for a precise allocated duration (not the entire period). 2.4 Technology and the Levels of Geometry 2.4.1 The cognitive geometric thinking model. Pierre van Hiele and Dina van Hiele-Geldof explored the development of geometric ideas in children and adults. They divided the path of geometric thinking into five continuous steps. The levels are: visualization, analysis, informal deduction, deduction, and rigor. Prior to the intermediate classes, students need to accomplish the first three levels only (Ministry of Education, 2008, p. 18). In the first stage of the model, students observe the object and treat it without its traits, definitions and description; they just learn the geometric vocabulary while decisions are mostly based on perception rather than reasoning. In the second level, students name, identify and analyze the traits of the shape, without observing the mutual relationships between their traits (Khalil, Farooq, Çakıroğlu, Khalil, & Khan, 2018). For example, students understand that all squares have four sides, with opposite sides parallel and congruent. At the third level, students use informal, logical reasoning to deduce properties. They can explore and test arguments about the traits of a shape. For example, if a quadrilateral has three right angles than the fourth one is definitely a right angle (Ministry of Education, 2008, p. 18). Khalil et al’s, (2018) deliberate on the van Hiele’s levels and state that “the geometrical understanding of students depends on their active participation in a well-designed activity, the proper objectives of the lesson, context of study, and involvement in discussion rather than memorization; all lead to raising the levels”. Furthermore, the variety of representations is the

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 13 only factor that make mathematical ideas and concepts comprehendible. And the power of understanding depend on the functional relationships between these representations (Khalil, Farooq, Çakıroğlu, Khalil, & Khan, 2018). Therefore, accomplishing the first three levels of the van Hiele’s cognitive geometric thinking model can definitely be easier with the use of technology (Ministry of Education, 2008, p. 20). 2.4.2 Dynamic Geometry Bayazit and Aksov (2010), categorized representation as visible or invisible. Representing a concept in a concrete way using symbols, graphs, models, drawings or algebraic expressions define the visible representation. On the contrary, mental manipulation on the bases of external representation is the invisible one (Khalil, Farooq, Çakıroğlu, Khalil, & Khan, 2018). Drawing programs, dynamic geometry computer applications, and applets are effective representational tools that help students virtually manipulate geometric concepts and the invisible representation. It supports relational thinking instead of instrumental thinking. This type of dynamic representations, grants students flexibility in reasoning, reflecting, and problem solving (Ministry of Education, 2008, p. 18). Khalil et al’s (2018) study showed that Geogebra aided instructions have significantly affected the mathematical achievement of students in comparison with the traditional instruction method. Another research showed a positive impact of the employment of Geogebra software on the academic achievement among eighth-grade students. The study resulted in a significant statistical difference between the mean scores of the achievement posttest in favor of the experimental group as compared to those of their peers in the control group (Farrajallah, 2016)

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 14 2.5 Spatial Ability 2.5.1 Definition of spatial ability. Spatial ability was defined in different ways over the years (Abdul Saha, Ayub, & Tarmizi, 2010). While working on primary mental abilities, Thurstone was the first to define spatial ability as he classified it as one of the primary mental abilities (Kayhan, 2005). According to Allyn & Bacon (1989), Thurstone defined spatial ability as “the ability to mentally manipulate shapes, sizes and distances in the absence of verbal or numerical symbols” (Kayhan, 2005). In 1979, Mc Gee introduced the spatial visualization, and it was defined by “the ability to create a mental image of an object and then to manipulate it mentally” (Kayhan, 2005). Additionally, Tartre (1990) defined spatial ability “as the mental skills concerned with understanding, manipulating, reorganizing or interpreting relationships visually”. While Gardener (1985) defined the Visual Spatial Intelligence as “the ability to perceive the visual world accurately, to perform transformations and modifications upon one’s initial perceptions, and to be able to re-create aspects of one’s visual experience, even in the absence of relevant physical stimuli” (Kayhan, 2005). After reviewing several studies, the researcher examined several definitions for the spatial ability that will not be mentioned; however, it is worth mentioning the three stages of spatial ability. In the present study the term spatial ability was considered as the cognitive function that makes it possible for people to deal effectively with spatial relations, visual spatial tasks, and orientation of objects in space (Basham, 2007). According to Piaget, people develop their spatial ability through three stages: (a) topological spatial visualization; (b) projective representation; and, (c) projective representation

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 15 and measurement. The first stage is mainly regarding the relation between several objects; the ability to determine the distance between objects as well as an object’s location in reference to other objects. The second stage represents the ability of a person to visualize the appearance of an object from various perspectives. The third stage is when a person can combine between the projective representation and the idea of measurement (Brudigam & Crawford, 2012). According to Piaget and Inhelder (1967) children’s spatial ability reaches an adult level at the age of twelve (Yilmaz, 2009). 2.5.2 Importance of spatial ability. Spatial ability contains several components that were defined by a number of researchers. According to Maeda et al’s (2011), spatial visualization and mental rotation abilities are the two fundamental components of spatial abilities. Spatial abilities are fundamental for interpretation, representation, manipulation of information, understanding, imagining, constructing, and problem solving (Basham, 2007). In his thesis, Basham stated, “spatial competence is basic to daily activity, higher level activities”. Those abilities help people visualize the movement of objects and manipulate them mentally via maps, graphs, diagrams, and more (Kösa, 2016). A high degree of spatial abilities aid people excel in various fields related to math and science, for instance such as architecture, chemistry, medical surgery, astronomy, music, and geology (Basham, 2007). In addition, Abdul Saha, Ayub & Tarmizi (2010) claim that “without spatial ability, students cannot fully appreciate the natural world”. Using their spatial ability, students can see three-dimensional figures since they will be able to picture, in their minds, the shape of an object in space, a figure on a paper, an area, a sketch, a street or a building (Yurt & Tünkler, 2016).

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 16 According to Brudigam & Crawford (2012) state that spatial ability is not a skill that you either have or you do not; instead, it is a skill that can be developed, even at a later age. Basham (2007) also agrees on that fact as he specified that improving those skills can be accomplished with children and adults. Yet, waiting to reach adulthood to improve spatial ability is not an acceptable solution and since spatial ability helps people remarkably in their career, developing it during school years is crucial. Therefore, it is valuable to consider diverse techniques that can help students develop their spatial ability. 2.5.3 The relation between spatial ability and mathematics. Basham (2007) found that “a potential benefit in improving spatial abilities is the improvement of academic achievement in areas of mathematics, science, and problem solving”. As he declared, the lack of development of this ability due to various beliefs is innate. However, according to Keller, new standards for school mathematics have been established by the National Council of Teacher of Mathematics (Basham, 2007). Those standards maintain that twodimensional visualization, three-dimensional spatial visualization, and reasoning are core skills that all students should develop (Basham, 2007). To excel in different subjects such as science, technology, engineering, and especially math, spatial visualization skills are critical. And those skills are also related to success in mathematics and geometry (Kösa, 2016). Moreover, according to Brudigam & Crawford, (2012) the development of spatial ability is related significantly to geometry education. In addition, according to Nik Azis (2008), Information and Communication Technology (ICT) should not be considered as an extra component to teaching; instead, it should be integrated into the Mathematics curriculum in both formal and informal ways (Abdul Saha,

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 17 Ayub, & Tarmizi, 2010). Hohenwarter & Hohenwarter (2009) also states that integrating ICT into everyday teaching practice offers creative chances for students to learn various mathematical knowledge and skills (Abdul Saha, Ayub, & Tarmizi, 2010). Voorst (1999) discusses the fact that ICT provides students with the opportunity to visualize certain mathematical concepts. Moreover, Voorst (1999) and Hohenwarter (2009) agree, “the ability to visualize and explore mathematical objects and concepts in multimedia environments can foster understanding in new ways” (Abdul Saha, Ayub, & Tarmizi, 2010). Furthermore, Abdul Saha, Ayub, & Tarmizi (2010), presents the outcomes of two different researches, where both used a computer software while teaching, the results show an enhancement in students’ spatial abilities, and geometry. Those findings were consistent with the results of the research presented by Abdul Saha, Ayub, & Tarmizi (2010); a computer assisted instruction is more effective than traditional instruction alone, in addition to enhancing students’ mathematical learning and understanding, as it also increases students’ spatial visulization. There are several activities that could develop spatial abilities. Velichova defines three levels to develop it through geometry education: “(a) views of elementary solids and drawing simple plane figures; (b) calculation of geometric properties such as surface area and volume, with extension of these properties to the more complex ones involving calculus, coordinate geometry, and constructional problems with solids; and, (c) use of upper level calculus, geometric modeling, and computers to aid in the constructing and visualizing of models” (Brudigam & Crawford, 2012).

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 18 Chapter 3: Methodology 3.1 Type of Research This thesis attempts to improve the methods and approach of teaching geometry through a mixed action research. Kemmis and McTaggart (1992: 10) define an action research as follows: ‘to do action research is to plan, act, observe and reflect more carefully, more systematically, and more rigorously than one usually does in everyday life’ (Cohen, 2011). Action research is a small-scale intervention in the functioning of the real world to address practitioners’ own issues, and a close examination of the effects of such an intervention (De Marrais & Lapan, 2004). More specifically, in education, the goal of action research is to create an inquiry stance toward teaching where questioning one’s own practice becomes part of the work and of the teaching culture (Matthews & Ross, 2010, p. 539). According to Western Oregon University (n.d.): “action research encourages change in schools, empowers individuals through collaboration with one another, encourages teacher reflection, and examines new methods and ideas”. It also states that an action research is an issue that is inspected in a particular school, and results are usually limited to that given school, department, or classroom. On the other hand, a mixed research combines both quantitative and qualitative analysis. According to Creswell & Garrett (2008), quantitative research gather data from many individuals and trends assessed across large geographic regions. While, qualitative research “yields detailed information reported in the voices of participants and contextualized in the settings in which they provide experiences and the meanings of their experiences” (Creswell & Garrett, 2008).

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 19 Teachers’ action research stems from the gap that teachers find between what is intended and what actually occurs in their teaching. In addition, such type of research improves student learning and teacher effectiveness (Western Oregon University, n.d.). The researcher, a reflective practitioner, reflected and questioned her work with her class students and private tutorees. The problem discussed in this thesis is a problem that the researcher’s students faced, and in order to help them this action research was designed adopting a mixed method in which both quantitative and qualitative data were analyzed. 3.2 Participants 3.2.1 Students. The study took place over the second half of the 2016 – 2017 school year. The population of the study consisted of all grade five students (eleven years old) divided between the two branches of a Lebanese/Armenian school. Both schools are located in Beirut and the researcher is employed in one of them. Due to the fact that in both branches each class has only one section; the population constituted the sample. The latter was divided into two groups: the experimental group formed of students in the school where the researcher teaches and the control group of students at the other branch. The experimental group is formed of ten students: five boys and five girls. It is to be noted that 50% of these students have various difficulties; three of them were presented to psychologists and diagnosed with learning difficulties. However, their parents were not following up with the psychologist, although it was recommended. The other two students came from two different Arab countries where math is taught in Arabic and thus were challenged by the English language. Yet all these students took part in this study.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 20 The control group consisted of fourteen students: seven boys and seven girls. However, one of these students was withdrawn from the study; for although he solved both pretests (Appendix F and G) and got 37.5% and 71.43% respectively and he got 50% on the first posttest, he seemed not to take the second posttest seriously as he answered all the questions in alphabetical order and ended up scoring 8.57% only. Subsequently, this student’s data was discarded as it could not contribute to the study for it was considered irrelevant. The control group finally consisted of thirteen students: six boys and seven girls. Table 1 represents the distribution of students into the two groups with details regarding the gender. Table 1 Distribution of Students in the Two Groups with Details Regarding the Gender Group

Boys

Girls

Total

Experimental

5

5

10

Control

6

7

13

3.2.2 Teachers. Two teachers took part in the field work of this research, including the researcher. The latter, with three years of teaching experience, was teaching the experimental group, while the control group had a teacher with several years of experience who is also the school principal of that branch.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 21 3.3 Procedure To ensure rigor, below are a number of procedures that were set including all the tests and resources necessary for the study as well as the required administrative protocol. 3.3.1 Administrative Protocol. The researcher meticulously followed the administrative protocol upheld in action research. The action research plan was presented to the school principals by the researcher for discussion and approval in a meeting during the month of January. Appendix A is the letter of request for a meeting and Appendix B is the recorded minutes of the meeting. The researcher discussed the variety of dynamic software available for such work, including free and payable options, such as GeoGebra, Calques 3D, Cinderella, Cabri 3D, Archimedes Geo 3D or others with the school administration. The school did not provide any dynamic software, as they did not interfere in this decision. As a result, the researcher chose GeoGebra not only because it is free, but because students can use it on their computers, iPads, or even cellphones. In fact, it is to be noted that GeoGebra has become the leading provider of dynamic mathematics software. It is user-friendly with an easy-to-use interface and can be used to create interactive learning materials that can be exported as an html webpage or embedded in a personal web page (International GeoGebra Institute, 2017). It is available in many languages and it provides several free, online, and downloadable resources (GeoGebra Community, n.d.). Although dynamic geometry software has been related with providing effective results in the education of mathematics as it promotes student-centered learning and active learning; research

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 22 on the effectiveness of integrating GeoGebra in teaching and learning mathematics is still limited (Abdul Saha, Ayub, & Tarmizi, 2010). Then, the researcher communicated with the teacher of the control group through different means: direct meetings, phone calls, and messages to transmit a clear idea regarding this thesis. The control group teacher was fully informed of the confidentiality needed for this thesis along with all the duties requested of her including tests, duration, and teaching method. All this information will be discussed in the following paragraphs of the methodology section. Moreover, to ensure compliance with the action research administrative protocol, two general meetings at each school were scheduled to be held. The participants of the meeting were to be the students of grade five themselves along with their parents. The meeting was to include a PowerPoint presentation in order to transmit enough information about this thesis. The aim of this meeting was to provide parents with sufficient information that would enable them to approve of their child’s participation in this study by signing the parental consent form (Appendix C). In addition, a student consent form (Appendix D) was also designed to be signed by the student as s/he also has the right to accept or reject being part of the study. The presentation was to include the following information:

▪ Participation is voluntary, and that their child will become part of the study only if both the parents and student agree to the student’s participation.

▪ Students may decide to withdraw from the study at any time. ▪ Parents can request that their child withdraw from the study at any time.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 23

▪ Students will neither receive any special benefit for participating nor will they be penalized for deciding not to participate.

▪ Students’ privacy is guaranteed. ▪ The students of the experimental group will be taught using a dynamic software, while the other students will continue to learn as usual.

▪ All the students will solve two tests at the beginning of the study and then repeat them again at the end, as they will be having their normal tests on regular school days. Unfortunately, due to a number of unforeseen complications (listed in the limitations of the research); there was no possibility to organize the above stated meeting. Instead, an administrative consent form was drafted and sent to the schools’ principals to sign (Appendix E). The form included all the information regarding the study. Once signed, the form confirmed the participation of all grade five students in this thesis. Both principals signed the form during the first week of May, 2017. 3.3.2 Teaching periods and methodology. The fieldwork extended from May 2 until June 9, 2017. Although the stated period consists of five weeks, the fieldwork effectively was covered in one month as one chapter of arithmetic was inserted in between the geometry lessons. In other words, the work was about nineteen periods of mathematics allocated specially for geometry. This period of time was dedicated to applying the dynamic software in class and collecting results. The nineteen periods were distributed accordingly: two periods for regular tests, one period for the pretests and one period for the posttests, which left fifteen periods of effective teaching.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 24 Note that, the Lebanese curriculum provides around fourteen periods of teaching for those three lessons including the testing periods (Center for Educational Research and Development, 1999). Table 2 represents the difference between the number of periods assigned by the Lebanese curriculum and the number of periods allocated for this research. Table 2 The Difference between the Number of Periods Assigned by the Lebanese Curriculum and the Number of Periods Allocated for this research Radius & Diameter of a circle

Area & Perimeter of Polygons

Area & Perimeter of Disks

Testing

Total

Lebanese Curriculum

1

13

–

14

Present research

4

11

4

19

The math book assigned by the school is “Mathematics, Elementary level, Fifth Year, Collection: Puissance, Al Ahlia, 2010” (Nassar, et al., 2010). Both teachers were using the same book and solving the same exercises that were agreed on at the beginning of the school year. During the implementation period, the teachers taught the last three geometry lessons stated in the yearly plan: radius & diameter of a circle, area & perimeter of polygons, and area & perimeter of disks. Those three lessons were divided, according to the yearly plan, into two different units separated by a unit on the division of decimals, a unit that is not included in this thesis. The researcher set a certain methodology according to what was reviewed in past research. Being inspired with Velichova’s definitions of geometry education levels (discussed in section 2.5.3), the researcher tried to apply those levels as much as possible. The third level was

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 25 not attainable as it was out of the students’ cognitive proximity zone, since advanced calculus level and geometric modeling are not part of grade five mathematics in the Lebanese curriculum. Therefore, the researcher aimed to apply the first two levels defined by Velichova “(a) views of elementary solids and drawing simple plane figures; and, (b) calculation of geometric properties such as surface area and volume, with extension of these properties to the more complex ones involving calculus, coordinate geometry, and constructional problems with solids (Brudigam & Crawford, 2012). However, since grade five did not cover three-dimension lessons, the researcher disregarded solids and volume and applied the first two levels according to what best fits students of grade five. Subsequently, GeoGebra and other software were used to provide students’ with the opportunity to visualize the simple plane figures in addition to including the lessons of area and perimeter into this study. The researcher taught her students using a dynamic software called GeoGebra. They also had the opportunity to use it in class. Meanwhile, the control group students were taught using a regular teaching strategy. It is to be noted that the control group teacher did not use any rote learning method; rather, she let her students experience the concepts but without any use of technology. Even though both classes included an active board, the control group were using it as a white board only for writing purposes. In addition to the GeoGebra software, the researcher used PowerPoint, ActivInspire with its built-in geometrical shapes, connectors, highlighters, tools, and a video about area & perimeter to teach her students with the perspective of diversifying strategies so as to reach all types of learners.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 26 Four sessions were spent on teaching the first geometry lesson: radius & diameter of a circle. The researcher used three tools: GeoGebra, PowerPoint, and ActivInspire. The first two sessions consisted of an interactive PowerPoint presentation made by the researcher herself (Appendix I). She would pass around the students as she was explaining the lesson, and ask them several questions, in addition to using a built-in option to point on (with a red pointer) the needed things. Then, GeoGebra was used to show the students the relation between the radius and the diameter of a circle. The researcher evaluated her students’ understanding with a small quiz on PowerPoint; in addition, students had to read the questions, choose the correct answer, and if they answered wrong they had to click again and select another answer. The other two sessions, were assigned to solving some exercises of the book using ActivInspire. The remaining two lessons were somehow related; therefore, the researcher allocated eleven periods to the two of them. For those lessons, GeoGebra, ActivInspire, and the video were used. The video was made by the researcher herself using the After Effect software and it is available on the following link: https://www.youtube.com/watch?v=VpSNLpb_5Is. The video was not shown to the students all at once. The relevant part was shown depending on the objectives and the shape that were to be covered during that session. Moreover, the experimental group had two periods of group work. The students were divided into three groups according to their abilities. In other words, the groups were comparable, with each including one leader, high achievers and low achievers. Two of the groups were given an iPad while the third group used the school computer. The three groups had the same GeoGebra lesson and a worksheet to be filled, students were supposed to draw several rectangles and squares using the software, calculate their area and perimeter by counting the squares and then coming up with the formula of each. During those group work sessions, the teacher behaved as a facilitator; she did not

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 27 interfere in their work. However, she was moving around the classroom to ensure the participation of all the group members. At the end of each session, one student had to present their group’s findings in front of the class. The formulated formulae were tested and the correct one was adapted. Note that the same worksheet was used during the two periods; it is a simple worksheet made by the researcher just to serve its purpose (Appendix J). After introducing all the formulae and applying them in a few exercises, each member of the experimental group received a booklet including all the formulae. The booklet was made by the researcher herself using the Illustrator Software (Sakr, 2017). The following are few screenshots.

Figure 1. Screenshots taken from the formulas' booklet. 3.3.3 Resources. Before the implementation phase, the researcher tried to come up with new lessons on GeoGebra. The problem was that in order to create interactive resources that serve the purpose of

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 28 the lessons to be taught, a high level of JavaScript (programming language) is required. The researcher did not have that programming background, so instead of creating new resources, she used the online materials available on the GeoGebra website. All chosen resources were content and level appropriate interactive exercises of the lessons discussed above. Although materials can be accessed online, the researcher downloaded the files, and modified a few of them for the reasons mentioned above. The modification consisted of removing extra information that overpassed the comprehension ability of the students; the lesson’s objective were not altered. The figures below show a few different used resources, and the links of all the resources are available in Appendix H.

Figure 2. Calculating areas of right triangles.


Figure 3. Area and perimeter of a circle.

Figure 4. Demonstration of an area of a right triangle.


Figure 5. Area of rectangles and squares. 3.4 Instruments 3.4.1 Choice of instruments. This thesis aims to study the effect of dynamic geometry on developing students’ spatial ability to draw and analyze two-dimensional figures. To that avail, it was judicious to revert to the work of Keuroghlian (2013), and Aydogan (2007) who both developed their own test instruments with the supervision of several mathematicians to ensure accuracy. The researcher of this thesis was the only person (working on this study) having a mathematical background, so creating tests to evaluate students’ abilities to draw and analyze two-dimensional figures was not the most relevant or reliable solution, since they could not be validated accurately. Instead, the researcher found international tests that could serve the objectives of this thesis. According to Basham (2007), a visual rotation test is used to evaluate students’ spatial abilities; he chose to use the Purdue Visualization of Rotations Test (PVRT). Basham stated that “according to the test developers, Bodner & Guay (1997), this test is among the spatial tests least

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 31 likely to be confounded by analytic processing strategies” (Basham, 2007). The PVRT named also PSVT:R was developed in 1976. Although it might seem a little bit old, the test has been primarily used for three decades on educational researches in science, engineering, technology, and mathematics (Maeda, Yoon, Kim-Kang, & Imbrie, 2011). After further discussion, with the thesis advisor, the researcher did not use this test because it evaluates three-dimensions: something out of the “zone of proximity” of the students; in other words, something they cannot solve and that is not even included in the Lebanese curriculum for grade five. All tests including three-dimensional analysis such as paper folding and mental cutting were also eliminated. 3.4.2 Quality of instruments. As she was still concerned about mental rotations, the researcher considered twodimensional rotation tests. Various instruments were inspected; some of which are online and the rest are a set of paper-pencil tests. The online tests were not that advanced, and people would do them for fun as they can share their results on social media and that is why they were not adopted. Instead, the researcher chose two tests made by different authors that had published several books. Mark Alan Stewart wrote the first chosen test, named “Spatial Aptitude”. He wrote several books published by Peterson and McGraw-Hills publications. An attorney and top-selling author, Mark Stewart wrote several books of graduate-level entrance exams. Mr. Stewart served as consultant to schools in graduate-level entrance exam programs in the University of California and California State University systems (GRE Answers to the Real Essay Questions, 2009). The chosen test is available in “Master the Mechanical Aptitude and Spatial Relations Test” book, in the sixth edition published by Publishers Group West (PGW) publications (Stewart, 2004).

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 32 Paul Newton and Helen Bristoll wrote the chosen second test, named “Spatial Ability – Practice Test 1”. Psychometric success: an internet company that sells psychomotor eBooks published the test. Both tests were chosen because they serve the objectives by testing the students’ ability to mentally manipulate shapes and rotate them. Those tests were used as pretests and posttests for both control and experimental groups. However, before choosing those tests, the researcher had to make sure that they were both level and content appropriate. Even though the spatial aptitude test is created for graduate-level entrance exam, they were relevant to the elementary students subject of this research. Basham (2007), Brudigam & Crawford (2012), and Kayhan (2005) used the same type of tests but harder ones as they demand three-dimension analysis. For the sake of this thesis, the researcher chose those two tests while taking into consideration twodimensional analysis only according to the Lebanese curriculum. Therefore, the original test was modified as discussed above. The duration (discussed in section 3.4.4) of the test was also modified in order for the test to be level appropriate. The Lebanese curriculum is a condensed one, and students are already stressed simply by sitting for regular tests at school. Due to that, we will not put students into more testing stress, and regarding the ability to draw and analyze geometrical figures, data was collected from regular tests. Two regular tests were taken into consideration in this thesis. Grade five teachers usually take turns to prepare those tests; however, the lessons enclosed in this research were parts of two units where the tests were prepared by the experimental group teacher (the researcher). However, those tests were only finalized after both teachers had solved them and agreed on their content and duration.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 33 3.4.3 Test design and directions. Both the spatial ability and the spatial aptitude tests were published online and were downloadable for free from https://www.geogebra.org/materials (Appendix H). Reproducing the questions in both files was prohibited since the files are secured and only printing was allowed. The researcher designed a cover page including the name of the test, its author(s), copyright, the researcher’s name, a black line for the students to write their name and a note for the students that they should not turn the page until they are instructed to do so (Appendix F and G). The second page of all the tests included two sections: directions and material needed. All the pages, excluding the cover page included a page number under the following format “Page … of …” to make sure that students acknowledge how many pages are left for them to cover. The spatial aptitude test was used as designed by its author: eight consecutive questions consisting of a numbered picture showing a solid shape and a set of four-lettered cut up pattern pieces. Students are supposed to circle the combination (the picture or the letter) of cut-up pieces that, when put together, will make up the shape shown. Pieces may need to be turned around or turned over to make them fit. On the other hand, the researcher took only a part the ability test. The test includes fortyfive questions and the last ten were about three-dimensional rotation. As discussed earlier, this part is out of the students’ cognitive proximity zone so the last ten questions were eliminated. The final test version consisted of thirty-five questions divided into three sections with specific directions for each section. Section one included the first twenty-five questions where students were supposed to match the shape from the first box (having a certain number) with the appropriate shape from the second box (having a certain letter). A list of numbers from one to

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 34 twenty-five was written on the right side of the two boxes, students had to write the corresponding letter for each number. The second section involved questions twenty-six to thirty including a figure on the left, and four-lettered possible answers on its right (A to D). Students were supposed again to circle the correct answer that would be the result of the rotation of the main figure. The third section included the remaining five questions (questions thirty-one to thirty-five), those questions were similar to the spatial aptitude test but harder. A main figure is given and below it four-lettered possible solutions (A to D) are found, students were supposed to circle the correct solution of the group of shapes that can be assembled to make the shape shown. The difference between this section and the spatial aptitude test is that the first test included shapes that were divided into two shapes only, while this test included shapes that were divided into three or four shapes. Table 3 draws a comparison between the number of questions and pages of the tests. The layout discussed earlier was not used in regular math tests; instead, the format of math test was used with the header designed by the school. Table 3 Number of Questions and Pages in Each Test Test

Number of questions

Number of pages

Spatial Aptitude

8

6

Spatial Ability

35

6

First Regular Test

9

3

Second Regular Test

6

2

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 35 3.4.4 Testing conditions. The spatial aptitude test was designed without setting a specific duration for it, while the spatial ability test was initially designed with forty-five questions to be solved during twenty minutes (Newton & Bristoll, n.d.). However, as discussed earlier, this timing is not for grade five students. To solve this issue, the researcher solved all tests and multiplied her time approximately by four for the students to solve. The researcher solved the spatial aptitude test in one minute twenty-four seconds, and the spatial ability test in nine minutes thirty-five seconds. The students were given five minutes, and forty minutes for each test respectively. As for the regular tests; they were both designed to be solved in forty minutes (duration of one period) and it took the researcher six minutes fifty-seven seconds to solve the first test and five minutes thirty-two seconds to solve the second one. Table 4 explicates the different duration of each test solved by the researcher and assigned to the students. Table 4 Difference between the Researcher’s Duration and the Students’ Duration of Each Test Test

Researcher Duration

Students’ Duration

Spatial Aptitude

1 minute 24 seconds

5 minutes

Spatial Ability

9 minutes 35 seconds

40 minutes

First Regular Test


40 minutes

Second Regular Test


40 minutes

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 36 All those tests were paper-pencil tests, which students solved during classroom time and in their regular seating plan. None of the students of the experimental or the control group faced a time issue; they were all able to finish ahead of time. For the pretests and posttests, both teachers, prior to administering the tests, explained that they would not be graded, and that, consequently, students would not be granted extra points for solving them. The first two pages of all the tests were printed on one side of the paper only to make sure that when the students turn the page to read the directions they cannot see the questions, while the other pages were printed two sided. The students had to wait until everyone had received their test papers, written their names, before the teacher and they could turn to the directions page, read it together, explain it, and make sure that everyone had the required materials. After finishing all those mentioned steps, the timer starts. The students were not shown the time in order not to make them anxious, but the teacher kept on reminding them every now and then how many minutes were left as per regular school procedures. As for the regular tests, no directives were given because students were already used to such tests. The papers were double sided printed and distributed to the students. Both tests were done in their grade five classroom, with students seated in their regular places. It is to be noted that the control group solved the pretests and both regular tests with their teacher present in the classroom. As for the posttests, the control group teacher had traveled and left the tests with the supervisor. 3.4.5 Validity of instruments. As discussed earlier in section 3.4.2, the rigor of instruments is maintained. However, the main challenge was to ensure the validity of the tests when used for grade five students following

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 37 the Lebanese curriculum. To solve this matter, the advisor of this thesis checked all the tests for any intrinsic ambiguity. Then the researcher solved the tests and gave them to a fellow mathematics teacher to verify the adaptability of their content. It was agreed that the test items were adequate for the age range and the specified allotted timing is reasonable. In addition, a university coach and a student enrolled in a master’s degree in didactic math checked the tests. Thus content and face validity were secured. As said earlier, the regular tests were prepared by the researcher: a grade five teacher. In other words, a teacher who teaches those classes, knows the curriculum, and knows her students’ abilities. In addition, the teachers of both groups had to solve every test together and approve of its content. 3.4.6 Testing dates. The implementation of the field work started on Tuesday, May 2, 2017 when both students of the experimental and control group solved the two pretests: spatial ability and spatial aptitude test. On Wednesday, May 9, 2017, grade five students solved their first regular test that is included in this thesis. On Monday, June 5, 2017, grade five students solved their second regular test that is included in this thesis. Friday, June 9, 2017 both students of the experimental and control group redid the spatial ability and spatial aptitude test as a posttest.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 38 3.5 Timeline The following is the factual time spent carrying out the field work of this thesis.

January 2017

April 2017

May 2017

June 2017

• Meeting with the school principal

• Getting all required signatures on official papers

• Pretests • Applying lessons (started) • Applying lessons (continued) • Regular tests • Posttests

• Analyzing data July 2017

August 2017

• Analyzing data

3.6 Data Collection The researcher and the teacher of the control group gathered data through four different tests. The data collection consisted of the pretests, the posttests, and two regular tests for both groups.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 39 The questionnaires were sent by email, the pretests and posttests were not discussed, and the teacher of the control group printed and used them as they are. However, further discussion occurred when setting the regular mathematics tests. Since the researcher had no data regarding the students of the control group, their teacher numbered them in a certain order, and she used the same number for the same student for all the tests. The pretests and posttests of the control group were sent to the researcher in a sealed envelope. The regular mathematics tests could not be gathered, because the students needed to take them back. The teacher of the control group scanned all the tests, numbered them according to her list, and sent them by email. As for the researcher, she also scanned the tests of her students in order to keep a copy for the analysis phase. 3.7 Data Analysis This study is a mixed-method research involving both qualitative and quantitative analyses, and is thus, based on several techniques of data collection and processing. The following main variables were considered:

▪ Independent variables: dynamic geometry. ▪ Dependent variables: spatial ability, student average, ability to draw, ability to analyze. The researcher had no previous data regarding the averages for students of the experimental and the control group. Instead, the pretests were used to determine each student’s level. Furthermore, the pretests and the posttests were used to measure the spatial ability. Time and gender were not considered variables in this study due to the following reasons. The researcher aimed to check if students had acquired a certain objective; she was not

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 40 concerned on how fast they were able to cover it. The tests that were made were not speed tests, meaning that time is irrelevant and it is definitely not essential to consider it as a variable. Two researches made by Aydogan (2007) and Basham (2007) shows that females and males were statistically equivalent in terms of their geometrical performance and mental rotations. Therefore, gender was not considered as a variable in this thesis. Data was entered into the Microsoft Office Excel software since the number of the participants is small. However, extended analysis was to be done by the researcher herself in a written method. To minimize the error, needed data was entered into the SPSS statistical software, version 24. For the quantitative analysis, the researcher created nine different variables. The group type was the only nominal variable used to determine if the participant was in the experimental group or the control group; it was coded experimental = 1 and control = 2. Six scale variables were also added. Those variables were students’ scores on the pretests, posttests, and the regular tests that were done in this thesis. In addition to the students’ scores of construction and analysis in the first regular test that will be discussed later, all scores are over 100 and they are represented with two decimals. As for the qualitative analysis, the researcher added four ordinal variables regarding the objectives of the first regular test, following this code: not acquired = 1, partially acquired = 2 and acquired = 3; no decimal were used. Descriptive statistics were used for the quantitative variables. As for the qualitative data, a cross tabulation was made to represent the difference between the percentage of students, experimental and control, who acquired, partially acquired or did not acquire a certain objective.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 41 Chapter 4: Results The purpose of this study was to inspect if dynamic geometry had an influence on students’ spatial ability, two-dimensional figure construction, and analysis abilities. In addition, it would determine whether dynamic geometry would yield better results than the regular teaching method. In this chapter, test results will be analyzed in order to come up with answers regarding those questions; scores of pretests, posttests, and regular geometry tests with attitude scale towards computer instruction are given in details. Moreover, the qualitative analysis regarding the tests’ objectives and the analysis of students’ answers in both pretests and posttests are presented. 4.1 Spatial Ability Test As discussed earlier, a quantitative and a qualitative analysis were conducted to analyze spatial ability tests. The quantitative analysis focuses on the mean difference between the pretests and the posttests of both groups. While the qualitative part was not done by SPSS, instead, the researcher analyzed the difference among the answers of each student between the pretest and the posttest of both spatial ability tests. Further explanation is available in the following parts. 4.1.1 Quantitative analysis of spatial ability test results. Spatial ability and spatial aptitude tests were administered as pretests and posttests to both experimental and control group. Descriptive statistics for the pretests, posttests, and the regular tests for experimental and control groups are given in Table 5 and Table 6 respectively.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 42 Table 5 Descriptive Statistics of all Pre- and Post-tests for Students in the Experimental Group Descriptive Statistics Spatial Aptitude Pretest Spatial Aptitude Posttest Spatial Ability Pretest Spatial Ability Posttest Valid N (listwise)

N 10 10 10 10 7

Mean 61.25 65.00 62.57 65.43

Table 6 Descriptive Statistics of Pre- and Post-tests for Students in the Control Group Descriptive Statistics Spatial Aptitude Pretest Spatial Aptitude Posttest Spatial Ability Pretest Spatial Ability Posttest Valid N (listwise)

N 13 13 13 13 12

Mean 62.50 53.85 71.21 67.91

On the spatial aptitude pretest, the experimental group scored 61.25% while the control group scored 62.50%. In addition, on the spatial ability pretest, the experimental group scored 62.57% while the control group scored 71.21%. Both pretests show that the students of the control group are at higher level than the students of the experimental group. In other words, comparing scores between the two groups is insignificant; instead, students’ score variation will be analyzed. The experimental group scored more on both posttests than their pretests. Their scores increased from 61.25% to 65.00% on the spatial aptitude test and from 62.57% to 65.43% on the spatial ability test, while the control group scored less on both posttests than their pretests. Their

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 43 scores decreased from 62.50% to 53.85% on the spatial aptitude test and from 71.21% to 67.91%. Figure 6 illustrates the expressed differences graphically.

62.5 61.25

Spatial Aptitude Pretest

53.85

Spatial Aptitude Posttest

65

71.21

Spatial Ability Pretest

62.57 67.91 65.43

Spatial Ability Posttest 0

10

20

Control

30

40

50

60

70

80

Experimental

Figure 6. Mean difference of spatial aptitude pretest and posttest for experimental and control groups. 4.1.2 Qualitative analysis of spatial ability test results. As stated above, qualitative analysis represents the difference in the answers of each student between the pretest and the posttest of both spatial ability tests. The researcher analyzed the answers of each student for the pretest of the spatial aptitude test, and checked if it was a correct or a wrong answer. She replicated the same work for the spatial aptitude posttest, the spatial ability pretest and posttest, and again for the control group. The results were added into a list that shows the change of the students’ answers between the pretest and the posttest. At this time, the researcher counted the following:

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 44 

The number of students that answered correctly in the pretest and their answer remained correct in the posttest.



The number of students that answered correctly in the pretest and their answer changed to a wrong one in the posttest.



The number of students that answered wrongly in the pretest and their answer remained wrong in the posttest.



The number of students that answered wrongly in the pretest and their answer changed to a correct one in the posttest. The same results were made for the spatial ability test and then replicated it for the

control group, as the results of each group were analyzed separately. Table 7 shows the percentage of variation of answers of spatial aptitude test for experimental and control group while Table 8 represents the same results but for the spatial ability test. Table 7 Percentage of Variation of Answers of Spatial Aptitude Test for Experimental and Control Groups Spatial Aptitude Test Answers

Experimental

Control

Correct  Correct

51.25%

47.12%

Correct  Wrong

10%

15.38%

Wrong  Wrong

25%

30.77%

Wrong  Correct

13.75%

6.73%

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 45 Table 8 Percentage of Variation of Answers of Spatial Ability Test for Experimental and Control Groups Spatial Ability Test Answers

Experimental

Control

Correct  Correct

51.43%

58.46%

Correct  Wrong

11.14%

12.75%

Wrong  Wrong

23.43%

19.34%

Wrong  Correct

14%

9.45%

The experimental group maintained a percentage higher than 50 for answers that were correct and remained correct in both tests (51.25% and 51.43%). While a percentage of 10 and 11.14 for answers that were correct and changed to wrong. Although the control group had a high percentage of answers that were correct and remained correct (47.12% and 58.46%) for both tests, they had a high percentage of answers that were correct and changed to wrong (15.38% and 12.75%). On the other hand, the experimental group scored 25% and 23.43% for the answers that were wrong and remained wrong along with 13.75 % and 14% for the answers that were wrong and changed to correct. While the control group scored 30.77% and 19.34% for the answers that were wrong and remained wrong along with a low percentage for the answers that were wrong and changed to correct (6.73% and 9.45%). In addition, the researcher analyzed the questions that had no answer or had an illegible one. For the spatial aptitude test, the experimental group had no missed answers or illegible ones in both pretest and posttest. As for the control group, in the pretest, one student missed two

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 46 answers and three students missed one answer. In the posttest, one student missed one answer only. On the other hand, in the spatial ability pretest, two students of the experimental group missed one question and two others wrote one illegible answer. In the posttest, two students of the experimental group missed one answer, one student missed three answers, and two wrote one illegible answer. As for the control group, in the pretest, one student missed one answer only. In the posttest, one student missed one answer, one student missed two answers, and one student missed thirteen answers. Table 9 Number of Illegible and Unanswered Questions of Spatial Aptitude Test for Experimental and Control Groups Spatial Aptitude Test Test

Experimental

Control

Pretest

0

5

Posttest

0

1

Table 10 Number of Illegible and Unanswered Questions of Spatial Ability Test for Experimental and Control Groups Spatial Ability Test Test

Experimental

Control

Pretest

4

1

Posttest

7

17

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 47 While taking into consideration the different number of questions in each test and the difference between the two groups’ population, the researcher calculated the percentages of illegible and unanswered questions for both the experimental and the control group. The results state that the experimental group answered 1.53% of all the questions either illegibly or not at all, while the control group had 2.15% of these. The quantitative results presented in this section aim to show that the fact that the experimental group scored higher on their posttests than their pretests while the control group scored lower did not occur randomly, or arbitrarily. These percentages show that the experimental group did not increase their scores haphazardly, but instead they preserved an acceptable level of answers that were correct and maintained correct while they improved several other questions, by changing the answer from wrong to correct. In addition to the fact that a higher percentage of the tests’ questions were not answered or written legibly by the students of the control group. As stated earlier spatial ability is very important, it is claimed that it helps students fully appreciate the natural world (Abdul Saha, Ayub, & Tarmizi, 2010) . Although Piaget and Inhelder (1967) stated that children develop their spatial ability by the age of twelve (Yilmaz, 2009), the participants of this study, aged eleven, were able to develop their spatial ability. Even though the results are different from Piaget and Inhelder’s study in the term of age, students were able to develop their spatial ability due to the use of technology. One can therefore, state that dynamic geometry helped students increase their spatial ability levels more than the regular teaching method.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 48 4.2 Regular Tests’ Quantitative Results As for the regular tests, the experimental group showed higher results than the control group. On the first regular test, the experimental group scored 81.25% while the control group scored 76.37%. As for the second regular test, the experimental group scored 75.55%, while the control group scored 60.08%. Tables 11 and 12 represent the SPSS output for the regular tests’ means and Figure 7 represents graphically the stated values. Table 11 Descriptive Statistics of Regular Tests for Students in the Experimental Group Descriptive Statistics First Regular Test Second Regular Test Valid N (listwise)

N 10 10 10

Mean 81.25 75.55

Table 12 Descriptive Statistics of Regular Tests for Students in the Control Group Descriptive Statistics First Regular Test Second Regular Test Valid N (listwise)

N 13 13 13

Mean 76.37 60.08


76.37

First Regular Test

81.25

60.08

Second Regular Test

75.55

0

10

20

30

Control

40

50

60

70

80

90

Experimental

Figure 7. Mean difference of regular tests for experimental and control groups. The first regular test was about the radius and perimeter of circles and several other algebraic lessons. The scores discussed here contain the students’ grades for the geometric part only. Fortunately, students of the experimental group showed better results as they scored 4.88% higher than the control group on the first test. The second regular test was about the area and perimeter of polygons and circles. In other words, the test assessed concepts that were entirely taught using the dynamic software. Again, students of the experimental group scored higher than the control group with an elevated difference of 15.47%. The above results show that dynamic geometry increases students’ results better than the regular teaching method does.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 50 4.3 Ability to Construct As discussed earlier, the first regular test assesses geometrical and algebraic concepts. For the sake of this thesis, the researcher analyzed the geometrical questions only (Appendix K). The first regular test contained three geometry exercises but only one of them included three questions of construction and one question of analysis. Only those questions will be analyzed quantitatively and qualitatively in the two following sections. 4.3.1 Quantitative results The researcher counted the students’ scores of the construction section only and converted it to a percentage. The experimental group scored 86.25% while the control group scored 81.73%. The difference of the two tests is 4.52% in favor of the experimental group. Table 13 and Table 14 portray the descriptive statistics for the test means of both groups. Table 13 Descriptive Statistics of Construction Questions in the First Regular Test for Students in the Experimental Group Descriptive Statistics First Regular Test - Construction Valid N (listwise)

N 10 10

Mean 86.25

Table 14 Descriptive Statistics of Construction Questions in the First Regular Test for Students in the Control Group Descriptive Statistics First Regular Test - Construction Valid N (listwise)

N 13 13

Mean 81.73

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 51 These results show that the integration of technology does in fact yield better results than the regular teaching method when it comes to the students’ abilities to construct an appropriate two-dimension geometrical figure. 4.3.2 Qualitative Results The scores discussed in the upper section regarding construction are divided upon three different objectives: 1- Construct a circle with given radius dimensions. 2- Construct a line. 3- Construct a perpendicular line to a given line and passing through a certain point. The researcher inspected each student’s drawing and analyzed her/his construction. Each student has either acquired, partially acquired, or not acquired a specific objective. After converting those results into a numerical value and entering them into SPSS as discussed in section 3.7, the researcher evaluated the difference between the experimental and the control groups using a cross tabulation as can be seen in Tables 15, 16 and 17 .

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 52 Table 15 Cross tabulation of Objective One for Construction Construct a circle with given radius dimensions. Cross tabulation Construct a circle with given radius dimensions. acquired Count 10 % within type 100.0% experimental % within Construct a circle with 43.5% given radius dimensions. % of Total 43.5% type Count 13 % within type 100.0% control % within Construct a circle with 56.5% given radius dimensions. % of Total 56.5% Count 23 % within type 100.0% Total % within Construct a circle with 100.0% given radius dimensions. % of Total 100.0%

Total

10 100.0% 43.5% 43.5% 13 100.0% 56.5% 56.5% 23 100.0% 100.0% 100.0%

Students of both groups were able to cover this objective. 100% of the students acquired the objective of constructing a circle with a given radius dimension.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 53 Table 16 Cross tabulation of Objective Two for Construction Construct a line. Cross tabulation Construct a line. not acquired partially acquired

acquired

4 40.0%

1 10.0%

5 50.0%

10 100.0%

33.3%

100.0%

50.0%

43.5%

% of Total

17.4%

4.3%

21.7%

43.5%

Count

8

0

5

13

% within type

61.5%

0.0%

38.5%

100.0%

% within Construct a line.

66.7%

0.0%

50.0%

56.5%

% of Total Count % within type

34.8%

0.0%

21.7%

56.5%

12 52.2%

1 4.3%

10 43.5%

23 100.0%

% within Construct a line.

100.0%

100.0%

100.0%

100.0%

% of Total

52.2%

4.3%

43.5%

100.0%

Count % within type experimental % within Construct a line. type

control

Total

Total

In the experimental group, 40% of the students did not acquire the concept, 10% partially acquired it, and 50% acquired it. While in the control group, 61.5% of the students did not acquire the same concept, 0% partially acquired it, and 38.5% acquired it. In other words, 60% of the experimental group and 38.5% of the control group were able to cover this objective. Fortunately, more students of the experimental group were able to acquire or partially acquire the objective, and less students of the same group did not acquire that objective.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 54 Table 17 Cross tabulation of Objective Three for Construction Construct a perpendicular line to a given line and passing through a certain point. Cross tabulation Construct a perpendicular line to a Total given line and passing through a certain point. not partially acquired acquired acquired Count 0 3 7 10 % within type 0.0% 30.0% 70.0% 100.0% % within Construct a 0.0% 75.0% 50.0% 43.5% perpendicular line to experimental a given line and passing through a certain point. % of Total 0.0% 13.0% 30.4% 43.5% type Count 5 1 7 13 % within type 38.5% 7.7% 53.8% 100.0% % within Construct a 100.0% 25.0% 50.0% 56.5% perpendicular line to control a given line and passing through a certain point. % of Total 21.7% 4.3% 30.4% 56.5% Count 5 4 14 23 % within type 21.7% 17.4% 60.9% 100.0% % within Construct a 100.0% 100.0% 100.0% 100.0% perpendicular line to Total a given line and passing through a certain point. % of Total 21.7% 17.4% 60.9% 100.0%

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 55 In the experimental group, 0% of the students did not acquire the concept, 30% partially acquired it, and 70% acquired it. While in the control group, 38.5% of the students did not acquire the same concept, 7.7% partially acquired it, and 53.8% acquired it. In other words, 100% of the experimental group and 61.5% of the control group were able to cover this objective. Fortunately, all students of the experimental group were able to acquire or partially acquire the objective, while 38.5% of the control group were not able to acquire it. The data available in Tables 15, 16 & 17 shows that while all students of the control group were able to acquire the first objective similar to the experimental group results, they could not achieve the same results for the following two objectives. The experimental group obtained better results, in both the quantitative and qualitative results, thus the dynamic geometry method does improve students’ ability to construct a twodimension geometrical figure. 4.4 Ability to Analyze 4.4.1 Quantitative results As discussed at the beginning of section 4.3, the first regular test had one exercise that included constructing and analyzing a two-dimension geometrical figure. Again, the researcher counted the students’ scores of the analysis question only and converted it to a percentage. The experimental group scored 60% while the control group scored 38.46%. The difference of the two tests is 21.54% in favor of the experimental group. Table 18 and Table 19 check the descriptive statistics of both groups.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 56 Table 18 Descriptive Statistics of Analysis Question in the First Regular Test for Students in the Experimental Group Descriptive Statistics First Regular Test - Analysis Valid N (listwise)

N 10 10

Mean 60.00

Table 19 Descriptive Statistics of Analysis Question in the First Regular Test for Students in the Control Group Descriptive Statistics First Regular Test - Analysis Valid N (listwise)

N 13 13

Mean 38.46

The above results show that the integration of technology yield better results than the regular teaching method when it comes to the students’ abilities to analyze a two-dimension geometrical figure. 4.4.2 Qualitative Results The one question of analysis included in the first regular test was summarized by the following objective: analyze a two-dimension geometrical figure. Once again, the researcher viewed each student’s answer and assessed according to the following levels: acquired, partially acquired, or not acquired. And since the objective was assessed in one simple question only, the students were either able to acquire the objective or not, the partially acquired option was not used in this case.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 57 After converting those results into numerical values and entering them into SPSS as discussed in section 3.7, the researcher evaluated the difference between the experimental and the control groups using a cross tabulation as shown in Table 20 below. Table 20 Cross tabulation of the Analysis Objective Analyze a two-dimension geometrical figure. Cross tabulation Analyze a two-dimension geometrical figure. not acquired acquired Count 4 6 40.0% 60.0% % within type 54.5% % within Analyze a two- 33.3% experimental dimension geometrical figure. 17.4% 26.1% % of Total type 8 5 Count 61.5% 38.5% % within type control

Total

% within Analyze a twodimension geometrical figure. % of Total Count % within type % within Analyze a twodimension geometrical figure. % of Total

Total

10 100.0% 43.5%

43.5% 13 100.0%

66.7%

45.5%

56.5%

34.8% 12 52.2% 100.0%

21.7% 11 47.8% 100.0%

56.5% 23 100.0% 100.0%

52.2%

47.8%

100.0%

In the experimental group, 40% of the students did not acquire the concept and 60% acquired it. While in the control group, 61.5% of the students did not acquire the same concept and 38.5% acquired it.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 58 Fortunately, a higher percentage of students of the experimental group were able to acquire the objective than the students of the control group. Both quantitative and qualitative results show that dynamic geometry effectively affects students’ abilities to analyze a two-dimensional geometrical figure. 4.5 The use of Software This thesis aimed to examine the effect of dynamic geometry on developing students’ spatial abilities to draw and analyze two-dimensional figures. The four results discussed earlier shows that technology affects students’ ability, ability to draw, and ability to analyze twodimensional figures effectively. Since Hohenwarter & Hohenwarter (2009) state that integrating Information and Communication Technology (ICT) into everyday teaching practices offers creative chances for students to learn various mathematical knowledge and skills (Abdul Saha, Ayub, & Tarmizi, 2010), the researcher chose to use more than one software as she eventually used GeoGebra, PowerPoint and ActivInspire. However, since no evaluation was done according to each tool separately, the researcher cannot conclude their particular effect. However, the researcher concluded that those software work in synergy all together and their combination is the key of this research’s success. Students’ progress cannot be directed to a single tool; instead, the joint use of the three software delivered a clearer message to the students.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 59 Chapter 5: Conclusion 5.1 Introduction Learning geometry is very important and not just for the sake of mathematics but for real life, since geometry is found everywhere, in all types of jobs and fields: architecture, engineering, and astronomy (Uba, 2013). According to Martin, Mullis & Foy (2008), Lebanon ranks in the twenty-fourth place in geometry out of the participating forty-eight countries, seventh after the US, and first among the Arab countries (Keuroghlian, 2013). The researcher’s teaching experience and the stated results highlight several difficulties faced by the Lebanese students when dealing with geometry. Thus, a number of previous research studies were consulted to find a suitable solution to this matter. The use of technology improves the effectiveness and the quality of instruction by capturing the students’ attention (Moore, 2012, p. 91). According to Healy (1998) and Kirpatrick & Cuban (1998), review of different studies proved that technology improves students’ results in all subjects at all levels (Moore, 2012). Drawing programs, dynamic geometry computer applications, and applets are effective representational tools that help students virtually manipulate geometric concepts. This type of dynamic representations, grants students flexibility in reasoning, reflecting, and problem solving (Ministry of Education, 2008, p. 19). Hence, this research was designed to help students overcome their difficulties when it comes to constructing and analyzing two-dimension geometrical figures while integrating technology into the teaching method. As she was concerned about assessing students’ abilities to

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 60 construct and analyze two-dimensional figures, the researcher found that spatial ability has a huge influence on geometry. Basham (2007) defines spatial ability as the cognitive function that makes it possible for people to deal effectively with spatial relations, visual spatial tasks, and orientation of objects in space. Moreover, the development of spatial ability is significantly related to geometry education (Brudigam & Crawford, 2012). While Hohenwarter & Hohenwarter (2009) state that integrating Information and Communication Technology (ICT) into everyday teaching practice offers creative chances for students to learn various mathematical knowledge and skills (Abdul Saha, Ayub, & Tarmizi, 2010). Therefore, the main scope of this research is to examine the effect of dynamic geometry on developing students’ spatial abilities to draw and analyze two-dimensional figures. The study was applied in grade five classes of the two branches of a public school in Lebanon, with ten students in the experimental group and thirteen students in the control group. Spatial ability was assessed by two tests that were done for both experimental and control groups as pretests and posttests. The ability to construct and analyze two-dimensional figures was measured through the administration of two regular math tests. 5.2 Conclusions The main purpose of this study is to help students construct and analyze two-dimension geometrical figures. However, achieving such a result cannot be conquered in a single step, thus this thesis included four different hypotheses listed in section 1.8. Therefore, in conclusion, the results of this study will be divided into four steps each related to its hypothesis.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 61 5.2.1 Research question 1. The first hypothesis of this thesis discussed whether changing the teaching methodology and integrating technology will help improve the grade five students’ spatial abilities. Basham (2007) states that spatial ability can be improved and is related to students’ performance in mathematics and science. Questioning whether changing the teaching methodology could improve students’ spatial abilities and having two different level groups made it impossible to compare their results. Instead, the variations of each group’s scores were analyzed independently. The quantitative results of the spatial ability test and the spatial aptitude test discussed in section 4.1.1 show that the experimental group scores’ increased from 61.25% (the pretest) to 65.00% (the posttest), while the control group results decreased from 62.57% to 65.43%. In addition, the quantitative analysis discussed in section 4.1.2 shows that the experimental group students did not achieve better results haphazardly. Instead, they demonstrated more application than the control group while maintaining a high percentage of questions that were correct and remained correct and low percentage of illegible or unanswered questions. Therefore, the first hypothesis may be accepted. 5.2.2 Research question 2. Al Chibani & Hajal (2017) states that a research applied in Lebanon shows that using technology in the classroom motivated the students as they were more engaged. The second hypothesis debated whether the dynamic geometry method would improve students' results more than the regular adopted teaching method.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 62 Students of both the experimental and the control groups solved two regular tests on the same assigned day. These tests were analyzed by comparing the results of the experimental group with the results of the control group for each test respectively. On both tests, the experimental group scored higher that the control group with a difference of 4.9% on the first test and 15.47% on the second test. Analyzing the students’ geometry results in two separate tests shows that the experimental group scored higher than the control group in both tests. Therefore, dynamic geometry method does improve students' results more than the regular adopted teaching method, so the second hypothesis may also be considered to have been validated. 5.2.3 Research question 3 According to the Ministry of Education (2008, p. 20) visualization is the first step of the five levels needed to develop geometrical ideas, and it is also accomplished more easily with the use of technology. Therefore, the third hypothesis was formulated to examine whether the dynamic geometry method would improve students’ ability to construct level appropriate twodimension geometric figures more than the regular adopted teaching method. To be able to answer this question, the researcher reverted to a combination of quantitative and qualitative analyses. The quantitative analysis showed that the experimental group scored 4.52% higher than the control group on the three construction questions that were asked in the first regular test. The qualitative analysis discussed the three different construction objectives covered in those same questions. Each student had either acquired, partially acquired or not acquired a

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 63 certain objective. All students of the experimental group and the control group were able to acquire the first objective, while 60% of the experimental group and 38.5% of the control group were able to cover the second objective and 100% of the experimental group and 61.5% of the control group were able to cover the third objective. According to both results of the quantitative and qualitative analyses, it is obvious that the dynamic geometry method does improve students’ ability to construct level appropriate twodimension geometric figures more than the regular adopted teaching method does. Therefore, the third hypothesis is validated. 5.2.4 Research question 4. The third hypothesis argues that the dynamic geometry method will improve students’ ability to analyze two-dimension geometric figures more than the regular adopted teaching method does. The same analyzing method that was used for research question 3 was also used for research question 4, but only one question from the first regular test was analyzed since it is the only analysis question. The quantitative analysis shows that the experimental group scored 21.54% higher than the control group on the analysis questions that was asked in the first regular test. Since the analysis part was about a small question in the first regular test, the student was classified according to whether s/he acquired the concept or no. The partially acquired option was not used. The qualitative results show that 60% of the experimental group and 38.5% of the control group were able to cover the analysis objective.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 64 The quantitative and qualitative results show that the experimental group scored better than the control group therefore, dynamic geometry method does improve students’ ability to analyze two-dimension geometric figures more than the regular adopted teaching method does and, consequently, the fourth hypothesis may also be accepted. 5.3 Limitations of the Study Although this research was carefully designed, referenced and actualized and it seems to have attained its objective, there were some unavoidable limitations. However, listing those limitations does not diminish the value of this study; instead, it helps with paving the way for future researches. This thesis aimed to study the effect of dynamic geometry on developing students’ spatial abilities to draw and analyze two-dimensional figures. In other words, this thesis targeted geometry lessons only. However, the yearly plan of mathematics for grade five was set at the beginning of the academic year. To apply the thesis, the researcher had to wait, to reach a geometry lesson to be able to apply the study, since altering or modifying the chapters was not accepted by the teacher of the control group. In addition to that, the research was supposed to start during the month of March, set a general meeting with the parents and the students, and then start implementing. Unfortunately, circumstances beyond the researcher’s control impeded her work as the researcher was diagnosed with a severe contagious virus that also strongly reduces the body’s energy. Thus, the researcher had to stay in bed for about three weeks, which were also followed by the Easter vacation. This time constraint led to postponing the research until the beginning of May and, consequently, the researcher had no choice but to apply the research on the three remaining geometry lessons that were discussed previously and substitute

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 65 the parents’ consent form (Appendix C) and student’s consent form (Appendix D) with an administrational permission form (Appendix E). The same mishap created a problem regarding the measurements part to collect data. The researcher was not able to apply more assessments to evaluate students’ ability to construct and analyze two-dimensional figures. In addition, the study was limited by the population sample and the inability to generalize. Due to the narrow number of students attending this school, the population sample was very small, thus the findings of this thesis cannot be generalized. The availability of the software was not a problem as it was accessible and free, but the researcher’s inability to write the JavaScript code limited the resources, as there was no other choice but to revert to the resources available online. 5.4 Recommendations 5.4.1 Recommendations for future research. Based on the findings of this research and the limitations that were listed above a variety of steps may be recommended to be followed in future researches. First, it is very important to enlarge the field work duration, in order to cover as many geometry concepts as possible; the implementation phase should take an entire academic year, excluding the time allocated for planning and analyzing data. On the other hand, expanding the implementation period could give teachers more time to explain the concept, while giving students an adequate opportunity to apply those concepts extensively in class and even at home using GeoGebra on their computers, tablets or cellphones.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 66 Second, for the sake of validity for potentially considering generalization, the sample size should be substantially through implementing the research on more schools and more classes. Third, include intermediate and secondary classes in the study to be able to assess the construction and analysis phase more effectively. In addition, including secondary students in the study could help evaluate the effect of dynamic geometry when it comes to three-dimension, while using three-dimension spatial ability tests. Fourth, choose schools from different environments, so students included in the study would have different social backgrounds. The schools taking part in the study should be a mix between private, official, and free schools. Fifth, normally I would have thought of recommending a study that would examine the effect of dynamic geometry on developing students’ spatial ability according to their gender. However, two past researches made by Aydogan (2007) and Basham (2007) showed that females and males were statistically equivalent in terms of their geometrical performance and mental rotations. Therefore, a similar research based on gender difference is not recommended. Finally, researchers need to have adequate background information regarding the students participating in the study. Students with impeaching difficulties should not be included in such a study because they could alter the results. However, if students with various learning difficulties must take part in the study, they should not form more than 10% of the entire sample since such students need special approaches and specific teaching methods that are set by professional teachers in the special education field.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 67 5.4.2 Recommendations for curriculum designers and book authors. Since the major difficulties are due to students’ inability to visualize and interpret figures mentally and since students’ of this era are digital natives, the following steps are advised to be followed by the curriculum designers. The curriculum should allocate more time for geometry. Unfortunately, most of the Lebanese schools assign forty-five or fifty minutes for each period instead of sixty minutes. Yet, increasing time, does not mean more time to teach new concepts, but more time to give the students the ability to apply that same objective and visualize it in various ways. Curriculum designers should acknowledge the importance of geometry and come up with a suitable organized time plan as they can solve the issue by reducing the number of objectives allocated for each class level. On the other hand, the curriculum designers need to acknowledge the fact that this generation is more motivated to learn through technology. And, according to Moore (2012, p. 55) multiple intelligences are an established fact, then it is advised to integrate technology, interactive lessons, and software into the curriculum, so students can visualize the geometrical concepts and relate them to real life. Subsequently, book authors would do well to follow the steps advised to the curriculum designers, in order to write books that combine between the curriculum and nowadays students’ needs. For example, geometry software could be introduced in the mathematics book while providing the reader with enough information on how to use it.

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THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 72 Merriam-Webster. (2016). Dictionary by Merriam-Webster. Retrieved July 2016, from MerriamWebster: http://www.merriam-webster.com/ Ministry of Education. (2008). Geometry and special sense, grades 4 to 6. Ontario, California, United States of America: Queen’s Printer for Ontario. Retrieved July 2016, from http://eworkshop.on.ca/edu/resources/guides/Guide_Geometry_Spatial_Sense_456.pdf Ministry of Education and Higher Education, MEHE. (2012, July 31). Teaching and Learning in the Digital Age: Lebanon’s National Educational Technology Strategic Plan. CERD Publications. Retrieved September 13, 2017, from Ministry of Education and Higher Education,: http://www.mehe.gov.lb/Uploads/file/TLSP.pdf Moore, K. D. (2012). Effective Instructional Strategies: From Theory to Practice (3rd ed.). California, CA, United States of Amercica: SAGE Publications. Retrieved July 2016 Nassar, H., Badr, N., Karroum, G., Moarbes, A., El Asmar, M., Atieh, K., & Merheb, C. (2010). Mathematics, Elementary level, fifth year, collection: puissance. (Al-Ahlia, Ed.) Zouk Mikael, Lebanon: Al Ahlia. Retrieved July 2016 Newton, P., & Bristoll, H. (n.d.). Spatial Ability Practice Test 1. Retrieved July 2016, from http://www.psychometric-success.com/practicepapers/psychometric%20success%20spatial%20ability%20%20practice%20test%201.pdf Sakr, R. F. (2017, May). Grade 5 Area & Perimeter Booklet. Beirut, Lebanon. Retrieved July 2017, from http://en.calameo.com/read/0048487825ce157121c1f?authid=mnxIYyRTAE1Q

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 73 Stewart, M. A. (2004). Spatial Aptitude Practice Problems. In M. A. Stewart, Master the Mechanical Aptitude and Spatial Relations Test (6th ed.). Arco, Idaho, United States of America: PGW. Retrieved July 2016, from http://www.pgworks.com/files/pdfs/SpatialAptitude.pdf Uba, C. (2013, October 27). How geometry affects our lives. Retrieved July 2016, from Newswatch times: http://www.mynewswatchtimesng.com/geometry-affects-lives/ Western Oregon University. (n.d.). Retrieved August 28, 2017, from http://www.wou.edu/~girodm/library/ch9.pdf Yilmaz, B. H. (2009, March). On the development and measurement of spatial ability. International Electronic Journal of Elementary Education, 1(2), 83–96. Retrieved September 26, 2017, from https://eric.ed.gov/?id=EJ1052049 Yurt, E., & Tünkler, V. (2016, June). A Study on the Spatial Abilities of Prospective Social Studies Teachers: A Mixed Method Research. ESTP - EDUCATIONAL SCIENCES: THEORY & PRACTICE, 16(3), 965-986. doi:10.12738/estp.2016.3.0324

‫‪THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL‬‬ ‫‪ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE‬‬ ‫‪IN A PRIVATE SCHOOL IN LEBANON‬‬ ‫‪74‬‬ ‫‪Appendices‬‬ ‫‪Appendix A: Letter to the principal‬‬

‫جانب مدير مدرسة ‪--------------------------------------------------------------------------‬‬ ‫‪.------------------------------‬‬‫الموضوع‪ :‬مناقشة إمكانيّة تنفيذ بحث اجرائي في مؤسستكم الكريمة‪.‬‬ ‫المستدعية‪ :‬رين صقر‪ ،‬طالبة ماستر مهني‪ /‬سنة ثانية ‪ -‬اختصاص تكنولوجيا التّربية ‪ -‬الجامعة اللبنانيّة‪.‬‬ ‫تحيّة طيبة وبعد‪،‬‬ ‫انطالقا ً من مبدأ التّعاون بين المؤسسات التّعليميّة‪ ،‬ورغبةً منا باالستفادة من خبرتكم الواسعة في مجال‬ ‫التّربية والتّعليم‪ ،‬ألتمس من حضرتكم تسهيل مهمتي التّعليميّة بما يتعلق بمتابعتي لشهادة الدراسات العليا‬ ‫في تكنولوجيا التربية‪.‬‬ ‫لذا‪،‬‬ ‫أطلب من حضرتكم تحديد موعد للتّشاور بإمكانية القيام ببحث إجرائي يتمحور حول موضوع قد‬ ‫تعتبرونه حاجة أولويّة في سبيل تطوير العمليّة التّعليميّة‪-‬التعلمية في مؤسستكم‪.‬‬ ‫دمتم ودامت مؤسستكم منارة للعلم والتّربية ‪.‬‬ ‫تفضلوا بقبول فائق اإلحترام‬ ‫رين صقر‬ ‫بيروت في ‪2017/1/16‬‬

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 75 Appendix B: Minutes of Meeting --------------------------

Tuesday, January 17, 2017

Call to order: A meeting to discuss the possibility of implementing a research at the school was held at ------------------------------------ in the principal’s office on Monday, January 16, 2017 at 3:00 pm. Attendees: -------------------------Miss Reine Sakr (minutes keeper) Minute Taker: Miss Reine Sakr Items for Discussion 1.

Decisions

Students problems in

We agreed that students have such difficulties and

constructing and analyzing

such problems should be resolved from its roots.

abilities.

We decided on applying the research on grade five students.

2.

GeoGebra

We discussed the software that will be used and it is a good choice since it is for free and usable on laptops, cellphones and tablets.

3.

Duration of research

------------- emphasized on the possibility of trying the software for a week then applying it. Miss Reine explained that to have an accurate research no trying period should occur ahead of time.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 76 4.

Tests

Miss Reine explained that the students will have a pre and posttest regarding paper folding and dimensional rotations. Further analysis will be gathered from students’ regular tests.

5.

Conditions of applying the

------------- highlighted on the fact that using this

research

software must not keep us late, and that the curriculum must be finished while making sure that each student has enough time to practice the construction section.

6.

Control group

Miss Reine clarified that the control group should be from the other school branch. ------------- didn’t give a clear answer about this issue since it needs an approval from --------- the principal of the other school.

Unfinished business: Meet with -------------------- to discuss the opportunity of sharing grades of students of the second school in order to consider them as a control group.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 77 Appendix C: Parents’ Consent Form Parental Permission Form – Spatial Abilities Study Project Title: The effect of dynamic geometry on developing students’ spatial abilities to draw and analyze two-dimensional figures in grade five in a public school in Lebanon. Investigators: The investigator, Miss Reine Sakr, is available to answer questions on Tuesdays from 1 p.m. until 2 p.m. or upon prior appointment. Purpose of Study: The effect of dynamic geometry on students’ abilities to draw and analyze two-dimensional figures. Inclusion Criteria: Grade five students. Description of Study: Students of grade five of both schools will have different tests to be done as pretests and posttests. Teachers will keep on with their regular curriculum and regular tests will be done throughout the research during fixed math periods. The only difference is that students of ------------ school will be taught using the chosen program to be able to visualize geometrical figures. Risks: There are no known risks. Right to refuse: Participation is voluntary, and your child will become part of the study only if you as her/his parent agree to her/his participation. At any time, either the student may withdraw from the study or the student’s parent may withdraw the student from the study. The student will not receive any special benefit for participating and the student will not be penalized if s/he does not participate.

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 78 Privacy: Student privacy is guaranteed. Results of the study may be published, but no names or any personally identifiable information will ever be included in any publication. Financial Information: There is no cost for participation in the study, nor is there any compensation to the students, parents, or teachers for participation in the study. Signatures: I have read and understand the information in the parental information form. I understand that I may direct additional questions regarding study specifics or about my child’s rights or other concerns to the investigator Miss Reine Sakr or any of the schools’ principals. I __________________________ allow my child __________________________ to participate Parent Name Student Name in the study described above and acknowledge the investigator’s obligation to provide me with a signed copy of this consent form. _____________________ on _____________________ , 2017 Parent Signature

Date

The parent/guardian has indicated to me that s/he is unable to read. I certify that I have read this consent form to the parent/guardian and explained that by completing the signature line above s/he has given permission for her/his child to participate in the study. _____________________, _____________________ on _____________________ , 2017 Reader Name

Reader Signature

Date

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 79 Appendix D: Student’s Consent Form Student Permission Form – Spatial Abilities Study I, __________________________ (print your first and last name in the blank), agree to be in a study to determine if there is an effect of dynamic geometry on students’ abilities to analyze and construct figures. I will take two pretests, then at the end of May I will take two posttests. I can decide to stop being in the study at any time without getting into trouble. __________________________ on __________________________, 2017 Student Signature Date

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 80 Appendix E: Administrational Permission Form Administrational Permission Form – Spatial Abilities Study Project Title: The effect of dynamic geometry on developing students’ spatial abilities to draw and analyze two-dimensional figures in grade five in a public school in Lebanon. Researcher: Miss Reine Sakr. Purpose of Study: The effect of dynamic geometry on students’ abilities to draw and analyze geometrical figures. Inclusion Criteria: Grade five students. Description of Study: Students of grade five at ---------------------------------------------------------------------------- will have three different pretests. Teachers will keep on with their regular curriculum but students at ------------ school will be taught using the chosen program to be able to visualize geometrical figures. Students will be given the opportunity to use the software in class, and at home if applicable. Regular tests will be done throughout the research during regular math periods. Students of both schools will take the posttest at the end of May. Risks: There are no known risks. Comments: Students will not receive any special benefit for participating. Privacy: Student privacy is guaranteed. Results of the study may be published, but no names or any personally identifiable information will ever be included in any publication. In

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 81 addition, the researcher will not be aware of students’ names from the control group (------------- school); anonymous test papers will be sent for analysis. Financial Information: There is no cost for participation in the study, nor is there any compensation to the students, teachers, or administration for participation in the study. Signatures: I have read and understand the information in the administrational information form described above and acknowledge the researcher’s obligation to provide me with a signed copy of this consent form.

I ______________________________ director of ______________________________ accept Director Name School Name that grade five students participate in this study.

______________________________ on ______________________________, 2017 Director Signature Date

______________________________ on ______________________________, 2017 Researcher Signature Date

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 82 Appendix F: Spatial Aptitude Test






THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 88 Appendix G: Spatial Ability Practice







THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 95 Appendix H: GeoGebra Online Resources The list below contains direct links to all the GeoGebra lessons used in this thesis. Note that some of these links were downloaded and modified to be used in class. The following links directs to the original version made by the creator her/himself, retrieved on May 2017. 

https://www.geogebra.org/m/H2t62qqR



https://www.geogebra.org/b/U2wgQKUA#material/ccaGeS3u



https://www.geogebra.org/b/U2wgQKUA#material/Ppq2GzEa



https://www.geogebra.org/m/MyVgfMAg



https://www.geogebra.org/m/wcE8Ajbn



https://www.geogebra.org/m/kGQ9f2pf



https://www.geogebra.org/m/fyqAUV22



https://www.geogebra.org/m/gjmRkFpr



https://www.geogebra.org/m/zkqGC5Tz



https://www.geogebra.org/m/ZStRVQjg



https://www.geogebra.org/m/aDeT8X5D



https://www.geogebra.org/m/QAvaqhWU



https://www.geogebra.org/m/umDM8ndh

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 96 Appendix I: PowerPoint Presentation


THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 98 Appendix J: Class Worksheet

Mathematics Lesson 42 Surfaces and their Area Worksheet #15 Teacher: Miss Reine Sakr Class: Grade 5 Group #: ______ Names: ________________________________________________________________________ Use the iPad or the computer to draw different rectangles and then complete the following tables. Length

1

2

3

6

7

Width

2

4

5

2

1

Perimeter

Write a formula that helps you find the perimeter of a rectangle. Prectangle = ______________

Length

1

2

3

6

7

Width

2

4

5

2

1

Perimeter

Write a formula that helps you find the area of a rectangle. Arectangle = ______________

THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 99 Appendix K: First Regular Test



THE EFFECT OF DYNAMIC GEOMETRY ON DEVELOPING STUDENTS’ SPATIAL ABILITIES TO DRAW AND ANALYZE TWO-DIMENSIONAL FIGURES IN GRADE FIVE IN A PRIVATE SCHOOL IN LEBANON 102 Appendix L: Second Regular Test
