IJLRES - International Journal on Language, Research and Education Studies ISSN: 2580-6777 (p); 2580-6785 (e) DOI: 10.30575/2017/IJLRES-2018010401 Vol. 2, No. 1, 2018 Page: 1 - 18 1 USE OF DIAGRAMS AS INSTRUCTIONAL AIDS IN TEACHING OF GEOMETRICAL CONCEPTS AT SECONDARY SCHOOL LEVEL Muhammad Rizwan Department of Education, University of Education Township Lahore, Pakistan rizwan52400@gmail.co Sobia Naseem Department of Education, University of Education Township Lahore, Pakistan srizwan.afaq@gmail.com Syed Wasif Raza Department of Education, University of the Punjab Lahore, Pakistan swraza33@gmail.com Abstract: Angle is a complex topic defined in a variety of contexts; some define angle to be as a pair of rays coming from a single point, as a rotation about a single point, or in a curve. Due to the multiple definitions of angle students get confused as to what an angle truly consists of. This study paid close attention to the misconceptions high school geometry students’ hold about the concept of angle and how to help them gain a more conceptual understanding of this essential concept with the use of different Diagrams. The main idea behind the conduction of this study is to investigate the effect of use of diagrams as instructional aids on the conceptual understanding of angles. The study hypothesizes that angle can be learned using interactive Diagrams. The results of this study showed that diagrams which based on different A.V aids was shown to be more beneficial for the students who used to clear the concept of angles. Keywords: Diagram, A.V Aids, Angle, Interactive Geometry Diagrams (IGD), Geometrical Understanding INTRODUCTION Geometry is also called the science of logical reasoning. In it, we approach everything with a question mark in our mind. As locked has said “Geometry is a way to settle in the mind a habit of reasoning 1. Here the results are developed of reasoning. There are only a few premises on which we base our reasoning. The conclusions follow naturally from the given facts when logical reasoning is applied to the same. The reasoning in geometry is of peculiar kind and possesses a number of characteristics such as simplicity, accuracy, certainty of results, originality, similarity to the reasoning of life, and verification. 1 Clements, D. H., & Burns, B. A. (2000). Students’ development of strategies for turns and angle measure. Educational Studies in Mathematics, hal. 41 31-45. mailto:rizwan52400@gmail.co mailto:srizwan.afaq@gmail.com mailto:swraza33@gmail.com Muhammad Rizwan 2 An important characteristic of geometry which distinguishes it from other subjects is its language and symbolism2. Geometry language and symbols cut short the lengthy statements and help the expression of ideas or things in the exact form. Geometry language is free from verbosity and helps in to the point, clear and exact expression of facts 3. Geometry results in their symbolic form helps in solving numerous complicated problems. Most of the later progress in geometry depends heavily on the learner’s ability to employ geometrical language and symbolism. Geometry is a way to settle in a mind a habit of reasoning. It train or disciplines the mind. Due to its very nature it possesses, a real disciplinary values. It is exact, true and to the point knowledge and therefore creates a discipline in the mind. Its truths are definite and exact. It develops reasoning and thinking powers more and demand less from memory4. Reasoning in geometrical angles possesses certain characteristics which are suitable for the training of the learner’s mind. If properly emphasized and streamlined, these characteristics are likely to develop the corresponding habits in the learner. There is a vast scope for simple reasoning in this subject. It teaches that definite facts are always expressed in a simple language and are always easily understandable5. The main thing is not the acquisition of knowledge but the acquirement of the power of acquiring knowledge. A child before coming to school has already developed a system of concepts and peculiar way of perceiving and organizing the stimuli from his surroundings. During his stay in the school the new learning experiences extend, diversify and reorganize his conceptual system, his personality also change6. The different concepts can be learnt formally as well as informally. In the educational institutions, the learning is more formal 2 Close, G. S. (1982). Children’s understanding of angle at primary/secondary transfer age. Master of science, Polytechnic of the South Bank, London, UK 3 Euclid. (2014). Elements is by far the most famous mathematical work of classical... The Elements consists of thirteen books. Book 1 outlines the fundamental propositions of plane geometry 4 Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35 (4), 258-291. 5 Hollebrands, K., Laborde, C., & StraBer, R. (2008). Technology and the learning of geometry at the secondary level. Research on Technology and the Teaching and Learning of Mathematics, 1, 155-205. 6 Bass, L. E., Charles, R. I., & Johnson, A. (2004). Prentice Hall mathematics: Geometry. Upper Saddle River, NJ: Pearson Prentice Hall. Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 3 than informal. The concepts are used to interpret the subject matter of different areas of knowledge. As we can safely say that all learning and knowledge is based upon the concepts, therefore, their importance is quite obvious7. Good result in teaching can be achieved if the students are correctly evaluated. The traditional style of teaching has badly failed to yield better result of teaching of angles. While researcher feels that if use of interactive diagrams technique, students gain conceptual understanding of angles8. The present study paid close attention to the misconceptions high school geometry students’ hold about the concept of angle and how to help them gain a more conceptual understanding of this essential concept with the use of different Diagrams9. REVIEW LITERATURE Teaching with Technology In the mathematics classrooms envisioned in NCTM’s Principles and Standards, every student has access to technology to facilitate his or her mathematics learning under guidance of a skillful teacher”10. This would be ideal yet is often unsupported by individual schools due to constraints and other outside influences. Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions. Electronic technologies, calculators and computers, are essential tools for teaching, learning, and doing mathematics11. They furnish visual images of mathematical ideas, they facilitate organizing and analyzing data, and they compute efficiently and accurately. They can support investigation by students in every area of mathematics, including geometry, statistics, algebra, measurement, and number12. Technology is expensive and takes more work for the teacher to learn and use 7 Mitchelmore, M. (1997). Children’s informal knowledge of physical angle situations. Learning and Instruction, 7 (1), 1-19. 8 Mitchelmore, M., & White, P. (1998). Development of angle concepts: A framework for research. Mathematics Education Research Journal, 10 (3), 4-27 9 Dr Mike Mitchelmore (2016). Development of angle concepts by progressive abstraction and generalization. Site Publisher: Macquarie University, Sydney Australia. 10 Daniel J. Brahier. (2016). Teaching Secondary and Middle School Mathematics 11 Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience, and school. Washington DC: National Academy Press. 12 Daniel J. Brahier. (2016). Teaching Secondary and Middle School Mathematics Muhammad Rizwan 4 the technology. Students can not only create diagrams and gain a deeper understanding through investigation but they often show different ways of thinking about mathematics with the use of technology not detected in normal classroom instruction13. The use of technology may be used to help students understand the concepts of angle by being able to manipulate the angle and see what does and does not change. Teaching with Interactive Geometry Diagrams (IGD) Students should be involved in their educational experience; they can gain a much deeper understanding of the concepts they are covering by doing the mathematics instead of simply watching teachers deliver the information to them. One good way to get students involved in their education and “doing” mathematics is with the use of diagrams. “Students’ engagement with, and ownership of, abstract mathematical ideas can be fostered through diagram A.V aids 14. MacGregor and Thomas conducted a study in 2002 involving the value of IGD in the classroom. The study consisted of a control group with a lecture based learning environment and the other group of students explored the material through an investigation with IGD. Although the study concluded that “the instructional model where the teacher provided structure and directed the problem solving activities of the students resulted in learner outcomes characterized by greater understanding of the concepts and less frustration. Although the students were frustrated with the investigation, which the study reported might be due to the fact that the teacher and the students were uncomfortable with the pictures, they “expressed a sense of self-confidence and pleasure with their accomplishments” and learned a great deal of material with the IGD through discussion and exploration15. METHODOLOGY For collecting relevant information and data in order to arrive at reliable conclusions 13 Daniel J. Brahier. (2016). Teaching Secondary and Middle School Mathematics 14 O Zaslavsky (2012). The Need for Proof and Proving: Mathematical and... - NYU Steinhardt. 15MacGregor, K. S. & Thomas, R. W. (2002). Learning geometry dynamically: Teacher structure or facilitation NECC 2002: National Educational Computing Conference Proceedings (23rd, San Antonio, Texas, June 17-19, 2002), 1-14. Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 5 certain method and procedure are required for all educational investigations. This chapter deals with the methodology and procedure adopted to conduct the study. The present study was under taken to find out the Secondary school students level of understanding in the field of geometrical angles. This study was experimental in nature. So the researcher adopted Experimental Research for his study. This section includes a description of selection of the subject, measuring instrument, construction of test, data collection procedure and data analysis. Selection of the Subject ‘Sixty’ 10th class students of Mathematics science group of Govt. H/s Baghbanpura Lahore, were selected as subjects of this study. A test was given to the subjects as Pre-Test. On the basis of test scores, pairs of subjects were made keeping in view their matching and or similar ability. Each member of each pair was included in a group A or group B randomly by using toss technique. Then group A and group B were selected as experimental group and control group through random assignment. Measuring Instrument Keeping in view the nature of the study, test was considered the best instrument for the collection of data. So a test was developed for students. There were 25 items for students in the test to identify the level of understanding or concept of the secondary school students of ‘Government High School Baghbanpura’ Lahore, in the field of Geometrical-Angles. Analysis of the content (subject matter) was made and then the test items were constructed accordingly. The test was classified into two different categories according to the nature of study. Test contains the following items a) Multiple choice items b) Short answer. The percentage of Multiple Choice items was 64% (16 items) and Short answer percentage was 36% (9 items). These test items based upon the geometrical angels who described in rays, segments, triangles, polygons, and circle etc. Muhammad Rizwan 6 FINDINGS AND DISCUSSION After the collection of data in the pre-test, instructional activity and post-test, the written responses from the pretest, instructional activity and post-test were then color coded to represent the subjects understanding of the concept being asked. The color coding was as follows: green was given to a response that was conceptually strong, yellow was given to a response that showed some conceptual understanding yet contained some misconceptions about the topic, and pink represented a lack of understanding of the topic being discussed. Once the responses were coded they were scored based on their color code. Table 1. Pre-test (Control Group). Short Questions Subject Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 MCQ s Total Marks % age M. Muzammal 2 2 1 2 1 1 2 0 1 8 20 59 M. Junaid 2 2 2 1 1 1 1 zero Zero 6 16 47 M. Zohaib 1 2 1 1 1 zero N.A zero 2 6 14 41 Mudassar 2 2 1 1 1 2 1 1 2 7 20 59 M. Zikrya 1 1 N.A 1 2 zero 2 zero 1 4 12 35 M. Haroon 2 zero 1 2 1 2 2 1 Zero 3 14 41 M. Atique 1 zero zero Zero N.A 2 2 1 Zero 4 10 29 M. Waheed 2 1 2 2 1 1 1 1 1 5 17 50 Ali Rehman 2 2 1 Zero 1 zero 2 2 1 6 17 50 M. Shakeel 2 1 2 2 2 1 zero zero N.A 3 13 38 M. Umair 1 1 1 1 2 1 1 zero Zero 5 13 38 Shahzaib 2 2 1 2 1 1 1 1 1 6 18 53 M. Zubair 1 1 2 2 2 zero 1 N.A 2 3 14 41 M. Adil 2 2 1 1 1 N.A 2 zero 1 4 14 41 M. Hamza 2 2 1 2 1 1 2 1 2 6 20 59 M. Haseeb 1 1 2 Zero Zero zero 2 zero Zero 4 10 29 Abrar 2 1 1 Zero 1 zero 1 1 1 5 13 38 Umer Ali 1 2 2 2 1 1 2 zero N.A 6 17 50 M. Zaraq 2 2 2 1 Zero zero zero 1 N.A 1 9 26 M. Asad zero 1 1 1 1 zero zero 1 1 3 9 26 M. Tayyab 1 zero Zero 1 2 1 1 zero 1 5 12 35 Shoaib Raza 2 1 2 2 1 zero N.A zero 1 3 12 35 M. Saif 1 2 2 1 1 2 zero 1 zero 6 16 47 Adnan Aslam 2 2 1 1 1 1 zero 1 1 2 12 35 Mohsin Iqbal 1 2 1 1 Zero zero 1 N.A zero 3 9 26 M. Fahid 2 1 2 1 1 1 zero 1 zero 4 13 38 M. Awais 2 2 1 2 2 2 1 1 1 5 19 56 Khawer Butt 1 2 Zero Zero 2 1 2 1 1 4 14 41 M. Azhar 2 1 2 2 2 1 zero 1 zero 4 15 44 Khalid 2 2 2 2 1 1 2 N.A 2 4 18 53 Table 2. Post-test (Control Group). Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 7 Short Questions Subject Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 MCQs Total Marks %age M. Muzammal 2 1 1 1 2 1 2 zero 1 8 19 56 M. Junaid 2 2 Zero 1 Zero 1 1 zero zero 6 13 38 M. Zohaib 1 2 1 1 2 zero zero zero 2 6 15 44 Mudassar 2 2 2 1 2 2 1 1 2 9 24 71 M. Zikrya 1 1 2 2 2 zero 2 zero 2 6 17 50 M. Haroon 2 zero Zero 2 1 2 2 1 1 4 16 47 M. Atique 1 zero 1 Zero 1 2 2 1 zero 6 14 41 M. Waheed 1 1 1 2 1 2 1 1 zero 4 14 41 Ali Rehman zero 2 2 1 1 zero 2 2 zero 3 13 38 M. Shakeel 2 1 Zero 1 2 1 zero 1 zero 4 12 35 M. Umair zero 1 1 1 2 1 1 1 zero 6 14 41 Shahzaib zero 2 1 2 1 1 zero 1 1 7 16 47 M. Zubair 2 1 1 2 2 1 1 zero 2 6 18 53 M. Adil zero 2 1 1 2 zero 2 zero 2 5 15 44 M. Hamza 1 2 2 2 Zero 1 zero 1 2 5 16 47 M. Haseeb 1 1 1 Zero 1 zero 2 1 1 6 14 41 Abrar 2 1 2 Zero 1 zero 1 1 1 4 13 38 Umer Ali 1 2 2 2 1 1 2 zero zero 6 17 50 M. Zaraq 2 2 1 1 Zero zero zero 1 zero 2 09 26 M. Asad zero 1 1 1 1 zero zero 1 1 3 09 26 M. Tayyab zero zero Zero 1 1 2 1 zero 2 5 12 35 Shoaib Raza 1 1 2 2 Zero zero zero 1 2 3 12 35 M. Saif zero 2 1 Zero 1 2 1 1 zero 5 13 38 Adnan Aslam 1 2 1 1 1 1 zero 1 zero 2 10 29 Mohsin Iqbal 1 2 1 Zero Zero 1 1 1 zero 5 12 35 M. Fahid 1 1 1 1 1 1 zero 1 zero 5 12 35 M. Awais zero 2 Zero 1 2 1 1 1 2 6 16 47 Khawer Butt 1 2 1 Zero 1 1 2 1 2 6 17 50 M. Azhar 2 1 1 2 2 1 0 1 0 4 15 44 Khalid zero 2 1 2 Zero 1 2 zero 2 6 16 47 Table 3. Pre-test (Experimental Group). Short Questions Subject Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 MCQs Total Marks %age M. Sawal zero 2 1 2 zero 1 1 zero 1 10 18 53 M. Umar Zero 2 1 1 2 1 zero 1 zero 08 16 47 M.Usman 1 1 2 1 2 1 zero 2 1 11 22 65 M.Usama 1 1 2 Zero 1 zero 1 zero 2 7 15 44 Zubair zero 2 2 Zero 1 2 zero 1 1 9 18 53 Waseem 1 2 1 Zero 1 2 zero 2 1 8 18 53 Hamza zero zero Zero Zero 1 1 2 zero 1 6 11 32 Bilal 1 1 2 Zero 1 1 1 zero zero 6 13 38 Ali Sher 1 zero 1 1 zero 2 zero zero 1 5 11 32 M. Azhar zero 2 Zero 1 1 1 1 zero 1 7 14 41 Junaid 1 1 1 Zero 1 1 zero 1 2 4 11 32 Rehman zero zero 2 Zero 1 2 zero 1 zero 6 12 35 Shehbaz zero 1 Zero 1 1 1 1 1 2 9 17 50 Ahsan Ali zero 1 Zero 1 1 1 1 1 1 5 12 35 M. Afzal zero zero 1 1 1 1 2 zero 2 6 14 41 M. Khan 1 2 Zero 1 1 2 1 1 1 3 13 38 M. Saeed zero 1 2 Zero 2 1 1 1 2 5 15 44 Ahmad zero zero 1 Zero 2 1 1 2 1 5 13 38 Tahir zero 1 1 2 1 1 zero zero zero 2 08 24 Younas 1 zero 1 Zero 1 1 1 zero zero 6 11 32 Muhammad Rizwan 8 Ali Raza zero zero 1 2 1 zero zero zero 2 4 10 29 Naeem zero 1 Zero 2 1 zero 2 zero zero 7 13 38 Daniyal 1 1 2 1 1 zero 1 zero 2 3 12 35 Zeshan zero 1 2 Zero zero zero zero 1 1 4 09 26 M. Tariq zero 1 Zero 1 1 1 zero zero 1 5 10 29 Fahad 1 2 1 Zero zero 2 zero 1 zero 7 14 41 M. Ali 1 zero Zero 2 1 1 zero zero 1 4 10 29 Dilawar zero 2 1 2 1 1 2 1 zero 8 18 53 Arsalan zero 1 2 Zero 1 zero 1 zero 1 4 10 29 Asad Hussain zero 1 1 1 1 1 1 2 zero 3 11 32 Table 4. Post-test (Experimental Group). Short Questions Subject Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 MCQs Total Marks %age M. Sawal 2 2 1 2 2 1 1 0 1 12 24 71 M. Umar 0 1 1 1 2 2 1 1 1 07 17 50 M. Usman 2 2 2 2 2 1 1 2 2 12 28 82 M. Usama 2 2 2 1 1 2 1 0 2 07 20 59 Zubair Ali 2 2 2 1 2 2 1 1 2 14 29 85 Waseem 2 2 2 2 1 2 2 2 2 09 26 76 Hamza N.A N.A 0 0 1 1 2 0 2 08 14 41 Bilal 2 2 2 2 1 2 2 0 2 10 25 74 Ali Sher 2 2 2 2 1 2 2 N.A 2 10 25 74 M. Azhar 1 2 1 2 2 2 1 N.A 2 12 25 74 Junaid 2 2 2 1 1 2 2 1 2 06 21 62 Rehman 2 1 2 1 2 2 2 1 2 08 23 68 Shehbaz 2 2 2 2 1 2 2 1 1 09 24 71 Ahsan Ali 2 1 2 1 2 2 2 1 2 08 23 68 M. Afzal 2 2 2 1 2 2 2 2 2 11 28 82 M. Khan 2 2 2 1 2 2 2 2 2 07 24 71 M. Saeed 2 2 2 2 1 2 1 1 2 08 23 68 Ahmad 2 2 1 2 2 2 2 2 2 10 27 79 Tahir 2 2 2 1 2 2 2 2 2 10 27 79 Younas 2 0 1 2 1 2 2 0 2 07 19 56 Ali Raza 1 2 2 2 2 2 1 1 2 10 25 74 Naeem 2 2 2 1 1 2 2 1 2 08 23 68 Daniyal 2 2 2 2 2 2 1 1 2 06 22 65 Zeshan 2 2 1 1 2 2 2 2 2 08 24 71 M. Tariq 2 2 2 2 2 2 1 2 1 07 23 68 Fahad 2 2 1 2 2 2 2 1 1 09 24 71 M. Ali 1 2 1 1 1 1 0 N.A 2 07 16 47 Dilawar 2 2 2 2 2 1 1 2 1 09 24 71 Arsalan 1 2 2 0 0 2 2 1 2 08 20 59 Hussain 2 2 1 2 2 1 2 1 1 10 24 71 The green code received two points, the yellow code received one point and the pink code received no points; this action was taken so the researcher could analyze the students’ gain scores. The coding was also used to help discover patterns of misconceptions held by the students throughout the study. After the researcher identified the common misconceptions the students possessed before the study by analyzing the coding of the pre-test, the researcher started considering categories or Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 9 levels of some sort to demonstrate students’ conceptual understanding of angle. While studying the data, the researcher needed to decide whether categorization or levels of understanding would best fit this study; the researcher considered these possible categorizations and levels described below in order to make her decision: a) Piaget’s stage theory of development: “They defined three stages of development between the ages of two and 11 years. At state 1, children can only recognize familiar objects, not shapes. At Stage 2 (pre-operatory level), they grasp topological (inside-outside, open-close) as well as rectilinear and curvilinear relations, but not metric or Euclidean relations. At the final stage (operatory level), children start to apply metric and Euclidean ones” 16. b) van Hiele’s levels: Similar to the Piaget’s stages yet “progressing from one stage to the next depends more on the teaching method adopted than on age, and this makes the geometry-related experiences of the child a determining factor”17. c) Mitchelmore & White (1998) categories, which can be found in figure 1. After consideration, the author chose to use a modification of the Mitchelmore & White 16 Munier, V., & Merle, H. (2009). Interdisciplinary mathematics - physics approaches to teaching the concept of angle in elementary school. International Journal of Science Education, 31 (14), 1857-1895 17 Munier, V., & Merle, H. (2009). Interdisciplinary mathematics - physics approaches to teaching the concept of angle in elementary school. International Journal of Science Education, 31 (14), 1857-1895 Muhammad Rizwan 10 categories due to the fact that one cannot say if students’ understands angle better using any of the multiple definitions. Since there are multiple definitions, multiple categories are needed to assess students’ conceptual understanding of angle18. Interpretations of Data The study was conducted to identify the use of diagrams as instructional aids in teaching of geometrical angles at secondary school level. For this purpose an objective type test was taken from the students. The data was analyzed by calculating the percentage and difference of pre and posttest of the students. From the table 5, the mean of control group pre-test and post-test is 14.33 & 14.47 and its standard deviation is 3.346 & 3.060 similarly the mean of experimental group pre-test and post-test is 13.27 & 23.33 and its standard deviation is 3.373 & 3.491, so the average score of control group test is less than experimental group test. In the comparison of control group and experimental group in pretest, the results shown that mean value and standard deviation are slightly higher than experimental group. This indicates the equivalent mental attitude towards geometrical angles without using diagrams as instructional aids among students. In contrast to pre-test, post –test results of experimental group shows tremendous results/remarked difference/huge variation than control group. Experimental group has greater value of mean (23.23) as compare to control group (14.47) while standard deviation value of experimental group exceeds from 0.431 units than control group. Similarly standard error mean values of experimental group are greater than control group in both pre and post –test. Table 5. Group statistics. Group N Mean Std. Deviation Std. Error Mean Pre-test Marks Controlled Group 30 14.33 3.346 .611 Experimental Group 30 13.27 3.373 .616 Post-test Marks Controlled Group 30 14.47 3.060 .559 18 Mitchelmore, M., & White, P. (1998). Development of angle concepts: A framework for research. Mathematics Education Research Journal, 10 (3), 4-27 Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 11 Experimental Group 30 23.23 3.491 .637 Table 6. Independent Samples Test t-test for Equality of Means 95% Confidence Interval of the Difference t df Sig. (p- value) Mean Diff. Std. Error Diff. Lower Upper Pre-test Marks Equal variance assumed 1.230 58 .224 1.067 .867 -.670 2.803 Post-test Marks Equal variance assumed -10.345 58 .000 -8.767 .847 -10.463 -7.070 Figure 2. T-test of pre and post-test Comparison between Pre and Post-Test Results of Control Group Pre and post-test of control group shows random results because some pupils have higher and some have lower percentages in posttest than in pre-test. Lower percentages may be due to lack of attention and non-serious behavior towards demonstration so that they don’t have much difference in %ages. Table 7. Comparison between percentages of pre and post –test of Control group. Subject Pre-test Post-test %age %age M. Muzammal 56 59 M. Junaid 38 47 M. Zohaib 44 41 Mudassar 71 59 Muhammad Rizwan 12 Subject Pre-test Post-test %age %age M. Zikrya 50 35 M. Haroon 47 41 M. Atique 41 29 M. Waheed 41 50 Ali Rehman 38 50 M. Shakeel 35 38 M. Umair 41 38 Shahzaib 47 53 M. Zubair 53 41 M. Adil 44 41 M. Hamza 47 59 M. Haseeb 41 29 Abrar 38 38 Umer Ali 50 50 M. Zaraq 26 26 M. Asad 26 26 M. Tayyab 35 35 Shoaib Raza 35 35 M. Saif 38 47 Adnan Aslam 29 35 Mohsin Iqbal 35 26 M. Fahid 35 38 M. Awais 47 56 Khawer Butt 50 41 M. Azhar 44 44 Khalid 47 53 Figure 3. Comparison between percentages of pre and post –test of Control group. Comparison between Pre and Post-Test Results of Experimental Group It is clear from table results that experimental group students have shown remark Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 13 difference towards conceptual understanding of geometrical angles with and without using diagrams as instructional aids in pre and posttest. They proved that learning with using diagrams is better for them and it played a greater role in developing concepts/enhancing their cognition level for geometrical angles. Students express their views that diagram made a clear concept and help for easy to understand than without diagrams. Table 8. Comparison between Percentages of pre and post –test of Experimental group. subject Pre-test Post-test %age %age M. Sawal 53 71 M. Umar 47 50 M. Usman 65 82 M. Usama 44 59 Zubair Ali 53 85 M. Waseem 53 76 Ali Hamza 32 41 Bilal Tariq 38 74 Ali Sher 32 74 M. Azhar 41 74 Junaid Iqbal 32 62 M. Rehman 35 68 M. Shehbaz 50 71 Ahsan Ali 35 68 M. Af NGzal 41 82 M. Khan 38 71 M. Saeed 44 68 Ali Ahmad 38 79 Tahir 24 79 Umer Younas 32 56 Ali Raza 29 74 Naeem Amir 38 68 Daniyal 35 65 Zeshan 26 71 M. Tariq 29 68 Fahad 41 71 M. Ali 29 47 Dilawar 53 71 Arsalan 29 59 Asad Hussain 32 71 Muhammad Rizwan 14 Figure 4. Comparison between Percentages of pre and post–test of Experimental group. Comparison between Pre-test of Experimental and Control Group In the case of comparison made between pre-test of control and experimental groups, there is no much difference in percentages of pupils. It means both groups have almost same level of understanding the lecture without using diagrams as instructional aids. Minor up and down values can be ignored due to individual intellectual level. Table 9. Comparison between Pre-test of Experimental and Control Group. Control group Pre-test Experimental group Pre-test Subject %age Subject %age M. Muzammal 56 M. Sawal 53 M. Junaid 38 M. Umar 47 M. Zohaib 44 M. Usman 65 Mudassar 71 M. Usama 44 M. Zikrya 50 Zubair Ali 53 M. Haroon 47 M. Waseem 53 M. Atique 41 Ali Hamza 32 M. Waheed 41 Bilal Tariq 38 Ali Rehman 38 Ali Sher 32 M. Shakeel 35 M. Azhar 41 M. Umair 41 Junaid Iqbal 32 Shahzaib 47 M. Rehman 35 M. Zubair 53 M. Shehbaz 50 M. Adil 44 Ahsan Ali 35 M. Hamza 47 M. Af NGzal 41 M. Haseeb 41 M. Khan 38 Abrar 38 M. Saeed 44 Umer Ali 50 Ali Ahmad 38 M. Zaraq 26 Tahir 24 M. Asad 26 Umer Younas 32 M. Tayyab 35 Ali Raza 29 Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 15 Control group Pre-test Experimental group Pre-test Subject %age Subject %age Shoaib Raza 35 Naeem Amir 38 M. Saif 38 Daniyal 35 Adnan Aslam 29 Zeshan 26 Mohsin Iqbal 35 M. Tariq 29 M. Fahid 35 Fahad 41 M. Awais 47 M. Ali 29 Khawer Butt 50 Dilawar 53 M. Azhar 44 Arsalan 29 Khalid 47 Asad Hussain 32 Figure 5. Comparison between Pre-test of Experimental and Control Group. Comparison between Post-test of Control and Experimental Group The use of instructional aids promotes learning interest and cognition level. It was clearly proved by the graph of comparison of post-test of control and experimental group that explain the higher values of percentages of experimental group due to use of diagrams as instructional aids. Table 6 also satisfies this statement effectively. Table 10. Comparison between Post-test of Control Group and Experimental Group. Control Group Post-test Experimental group Post-test Subject %age Subject %age M. Muzammal 59 M. Sawal 71 M. Junaid 47 M. Umar 50 M. Zohaib 41 M. Usman 82 Mudassar 59 M. Usama 59 M. Zikrya 35 Zubair Ali 85 Muhammad Rizwan 16 Control Group Post-test Experimental group Post-test Subject %age Subject %age M. Haroon 41 M. Waseem 76 M. Atique 29 Ali Hamza 41 M. Waheed 50 Bilal Tariq 74 Ali Rehman 50 Ali Sher 74 M. Shakeel 38 M. Azhar 74 M. Umair 38 Junaid Iqbal 62 Shahzaib 53 M. Rehman 68 M. Zubair 41 M. Shehbaz 71 M. Adil 41 Ahsan Ali 68 M. Hamza 59 M. Af NGzal 82 M. Haseeb 29 M. Khan 71 Abrar 38 M. Saeed 68 Umer Ali 50 Ali Ahmad 79 M. Zaraq 26 Tahir 79 M. Asad 26 Umer Younas 56 M. Tayyab 35 Ali Raza 74 Shoaib Raza 35 Naeem Amir 68 M. Saif 47 Daniyal 65 Adnan Aslam 35 Zeshan 71 Mohsin Iqbal 26 M. Tariq 68 M. Fahid 38 Fahad 71 M. Awais 56 M. Ali 47 Khawer Butt 41 Dilawar 71 M. Azhar Khalid 44 53 Arsalan Asad Hussain 59 71 Figure 6. Comparison between Post-test of Control Group and Experimental Group. Use of Diagrams as Instructional Aids in Teaching of Geometrical Concepts at Secondary School Level DOI: 10.30575/2017/IJLRES-2018010401 17 CONCLUSION The importance of IGD A.V aids in the mathematics classroom has been investigated through a variety of venues about a variety of topics throughout mathematics education research. This study is designed to use this research and expand upon it while investigating the importance of angle comprehension in the geometry classroom. Concerns about angle comprehension were discussed through different studies and the mathematics curriculum; angle needs to be researched further to comprehend the level of misunderstanding which students hold when it comes to understanding the use of angle. The introduction of Interactive Geometry Diagrams (IGD) into the geometry classroom has evolved and can now involve students in a more active learning environment. It is found experimentally that 92% students attempt correct response during test. In multiple choice items the percentage of students who gave correct response was 96% after the post test. In short answer type question the percentage of students who gave correct answer were 90%. 4% students got zero marks in multiple choice questions. 10% students got zero marks in short answers type questions. In pretest the Mean value of the correct response was “14.33” and standard deviation was “3.346”. In posttest the Mean value of the correct response was “23.33” and standard deviation was “3.491”. Finally the researcher was collected the significance (P) value 0.0001. This study is not looking to see if the diagram is important, this has already been established. The question to answer is not if the diagram is important but how the use of the diagram is important and beneficial for students’ conceptual understanding of angle. BIBILIOGRAPHY Bass, L. E., Charles, R. I., & Johnson, A. (2004). Prentice Hall mathematics: Geometry. Upper Saddle River, NJ: Pearson Prentice Hall. Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience, and school. Washington DC: National Academy Press. Clements, D. H., & Burns, B. A. (2000). Students’ development of strategies for turns and angle measure. Educational Studies in Mathematics, 41, 31-45. Close, G. S. (1982). Children’s understanding of angle at primary/secondary transfer age. Master of science, Polytechnic of the South Bank, London, UK. Daniel J. B. (2016). Teaching Secondary and Middle School Mathematics Muhammad Rizwan 18 Euclid. (2014). Elements is by far the most famous mathematical work of classical... The Elements consists of thirteen books. Book 1 outlines the fundamental propositions of plane geometry. Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35 (4), 258-291. Hollebrands, K., Laborde, C., & StraBer, R. (2008). Technology and the learning of geometry at the secondary level. 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