THE ROLE OF ANGLE IN UNDERSTANDING THE VECTORS

THE ROLE OF ANGLE IN UNDERSTANDING THE VECTORS Pavlakos Georgios* - Spyrou Panayotis** - Gagatsis Athanasios*** (*) University of Athens, Department of Mathematics, pavlakosgeor@hotmail.com (**) University of Athens, Department of Mathematics, pspirou@cc.uoa.gr (***) University of Cyprus, Department of Education, gagatsis@ucy.ac.cy ABSTRACT The main objective of this article is the study of how the concept of the angle comes in, in the understanding of the concept of vector by the Lyceum students. We perceive vector as procept, as defined by Gray & Tall (1994). In this context, it has been regarded that the concept of vector is constructed, in the mind of children, with the compression of two fundamental procepts, the procept of length and the procept of angle. To define our hypothesis, a questioner has been passed out to students of the second class of Lyceum, in schools both in Athens and the countryside. INTRODUCTION THEORETICAL BACKGROUND The concept of vector is a multidimensional one and concerns many levels of knowledge construction. This combined with the many contexts the term appears in algebra, geometry and physics, create many problems in teaching as well as in its comprehension. In contemporary research about vectors, in secondary education there are very few references about the development of the conceptual thought from a mathematical point of view. The concept of vector is mainly encountered in papers concerning physical sciences, e.g. Mechanics (Aguirre & Erickson, 1984, Graham & Berry, 1990, 1991, 1992, 1993, 1996, 1997, Rowlands, Graham & Berry, 1998), without examining exclusively the vectors. They looked at every considered vector (force, velocity etc) independently and did not attempt to find their common elements, their common problems occurred and finally their common mathematical structure. In the last few years, some interesting attempts have been made for a systematical research of the concept of vector in the mathematical education (Forster, 2000, Forster και Taylor 2000, Demetriadou & Gagatsis, 1995, Gagatsis & Demetriadou, 2001, Demetriadou & Tzanakis, 2003, Watson, 2002, Watson, Spyrou & Tall., 2002). Specifically, during the research, it was found interesting the difference of the length of the vector from its corresponding direction, an observation made by Foster, even though its conclusions did not lead to any significant results. An improved and more systematic research has been presented in recent papers by A. Gagatsis, E. Dimitriadou and K. Tzanakis, which had many conclusions concerning teaching and learning of vector analysis. For example, certain conclusions have been presented that support the view that a non-clear, incomplete and partial perception about the mathematical idea of the vector (appeared in elementary physics), may not be available in any abstraction form, because it does not directly relate with the particular properties of the pure mathematical concept of the vector. Finally, in their research work A. Watson, P. Spyrou and D. Tall examine the relationship between the perceptual world of embodiment and conceptual world of

mathematical symbolism with particular reference to the concept of vector. It appears that the embodiment of vector as a journey leads more naturally to the use of triangle law, while the embodiment of vector as force leads more naturally to the parallelogram law. This may lead many students at the first stages of learning to significant misinterpretations especially in Mechanics. In the last research work, this connection of difficulties that students face with experiences and actions from the everyday life, caused the motivation of our research effort. Our research work is based on the idea of procept of Gray & Tall (1994), which reduces the understanding of a concept to the set of equivalent procedures that constitute the process of construction in the mind of that concept and the symbol by which it is evoked. This cognitive category gives us the possibility to confront analytically the construction of the concepts by their decomposition to elementary actions on the objects of real world. Specifically: An elementary procept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object. A procept consists of a collection of elementary procepts which have the same object. (Gray & Tall, 1994, page 120) Our basic research hypothesis is that the procept vector is the result of compressing two different actions: the action on the lengths and the action on the angles (Appendix 1). Specifically in this study, we have tried to examine the possible connections between the concept of angle and the concept of vector and finally to show up its essential role in the understanding of vector by the students. THE STUDY The study took place on April 2004. A total of 100 students of the second grade of 7 Lyceums of the Athens region and countryside participated in the study. It is noted that the students of this grade encounter for the first time the concept of vector in Mathematics courses. The questionnaire (Appendix 2) was handed out to the students and they had to give their answers in a half an hour period. The presentation of the Questionnaire The questionnaire contains three different kinds of questions. More specifically: The objective of the first question is to check if the students are familiar with the various types of angles, if they can recognize them and identify them in the pictures that are shown to them. Furthermore, with this procedure we intend to find out and spot any connections between angles and other tasks. The third question is the only one that does not closely relate to the angle. This will be useful in determining if the conclusions drown from the other questions are due to the introduction of the angle or it is a result of a general ignorance of vectors and the corresponding calculus. In the remaining questions, the angle of two vectors plays a central role and is mainly related to their sum. With these tasks we wish to investigate the possible misinterpretations that occur from such interrelations. The variables of the research The following table contains the variables that were defined for analyzing the data of our research. Ria Identifying an acute angle in the i-picture Rir Identifying a right angle in the i-picture

Rio Identifying an obtuse angle in the i- picture Ris Identifying a straight angle in the i- picture V2 2 nd question V3a 3 rd question, sub question a V3b 3 rd question, sub question b V3c 3 rd question, sub question c V4 4 th question V5 5 th question V6 6 th question V6p Attempt to confront the 6 th question with the parallelogram law V6t Attempt to confront the 6 th question with the triangle law V7 7 th question Grading Criteria The variables that concern the identification of angle take the value of 1, if the angle was identified in the corresponding picture and 0 if that it was not true. The variables concerning the remaining questions, excluding V5, are equal to 1, if the corresponding question was answered correctly or 0 if a wrong answer or no answer was given. For V5 the following values were given: the value 1 was given, if the existence of the obtuse angle was mentioned in the justification, the value 0,66 if another correct justification was given, 0,33 if the answer was correct but without any justification, and the value 0 if a wrong answer or no answer was given. Finally, the variables V6p and V6t were graded with 1, if an answer with the use of the corresponding rule was attempted and 0 in any other case. Statistical techniques Except the descriptive and qualitative processes of the given data, the Implicative Statistical Model of Gras was used. With the use of the software program CHIC, the implicative analysis of data, resulted in the following two graphs: 1. Similarity Tree, in which the variables are interconnected accordingly to the similarity (or no similarity) that they posses. 2. Implicative Diagram, in which the implicative connections that exist among variables are shown. RESULTS Descriptive and qualitative analysis In the framework of the descriptive analysis of the data of this research work, the results that arose from the group of students that participated in the research are presented. In the following table the identification percentages of acute, right, obtuse and straight angles respectively are presented, for each of the pictures given to the students. R1 R2 R3 R4 R5 R6 R7 R8 R9 a 0.42 0.72 0.34 0.49 0.88 0.82 0.12 0.74 0.55 r 0.09 0.2 0.58 0.16 0.01 0.62 0.75 0.18 0.8 o 0.29 0.49 0.44 0.28 0.55 0.37 0.21 0.65 0.33 s 0.31 0.03 0.14 0.38 0 0.22 0.02 0.14 0.03

We observe that the students can recognize, as seen in the piechart (this concerns the recognition of the angles in all pictures), in a more easy way first the acute angles, then the right angles and finally the straight angles. It is of special interest the fact that there was no child who recognized directly the zero or full angle, despite their age (17 years old) and their expected mathematical maturity. This could be explained by the fact that the acute angle in particular is consider to be a prototype of angle. Besides that, it is not difficult for someone to find out that the mathematical school textbooks are dominated by acute angles, and then follow the right and obtuse types of angles. This may have certain side effects in the cognitive development of the concept of angle in student s minds. obtuse straight Piechart right acute Average 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.28 0.59 0.58 Barchart 0.78 0.8 0.34 0.38 0.26 0.19 0.24 V2 V3a V3b V3c V4 V5 V6 V6p V6t V7 Questions Let us consider now the barchart, in which is shown the average of every variable concerning the other questions. As it can be observed the easiest question is the 4 th, something that can be due to that the correct answer preassumes only knowledge of the basic theory and does not require any complex reasoning. The subquestion c of the third exercise follows, with a percentage 78% and then sub questions a and b with percentages 59% and 58% respectively. At this point it is necessary to explain why the first two subquestions had considerably lower success percentages compared to the third one. Especially in a, i.e. the obtained percentage appears to be (exactly) the same with c, i.e. OA = OB Ο = ΟΓ. Their specific difference is that the existence of the symbol - in the text of question 3c leads the student to convert Ο to Ο and then to correspond it to ΟΓ. On the contrary, in subquestions 3a and 3b, in order to be answered correctly, the student must investigate all the possible combinations, something that requires extra cognitive effort. Conclusively, the two sub questions appear to be mathematically equivalent, but they are not cognitively equivalent. The 6 th question (recommended by Anna Watson) follows with a percentage 38% of success. Some students assumed that the vectors are opposite, but their largest difficulty was in the computation of the angle of the two given vectors. So even though they answered that the sum of the vectors is not 0, however they made many mistakes in attempting to draw the sum of vectors. It is of special interest also the answer of two students that seems to have used their intuition to answer the question:

Maybe these two students perceived the two vectors as forces and drew the vector of the force that produces the same result with the two given vectors. However, because they did not indicate that they are familiar with the mathematical context, their answer has been marked as incorrect. The percentage 34% that corresponds to question 5 does not represent the correct answers that amount to 49%. The majority of the students that answered correctly did not justify their answer or gave the expected reasoning such as: Correct. Draw parallel lines to sum up. It is not wrong. When we add vectors we do not add the corresponding magnitudes. It is correct according to the parallelogram law. It is of special interest the attempt of a student to justify what he was looking at It is wrong, the vectors α and β have been lengthened and the vector α + β has been shortened, which should be longer than the other two. This student, who was not able to justify the result with mathematical means, tried to find a way to avoid the forthcoming cognitive conflict. In the 2 nd question the students faced many difficulties. This fact combined with the previous results shows that students cannot make assumptions about the sum of two vectors when their angle changes. It is important at this point, to stress that students have learned a year ago in the Physics course that only when the angle is 0 o the equation F ολ = F 1 + F 2 is valid. However, as it already has been stated, in order the concept of the vector be available for abstraction, its mathematical structure must be presented, something that obviously does not happen in the Physics class. Finally, the 7 th question required that the students had understood very well the previous concepts as well as the concept of the free vector and its parallel translation. So, it was not surprising to us that it gave the lowest success percentage. IMPLICATIVE STATISTICAL ANALYSIS OF R. GRAS Similarity relations between questions The Similarity Tree presents the grouping of questions according to the subject behavior during the solution process. In the following graph there are only the variables that concern questions that are related with vectors (questions 2-7). As we can observe, two main groups of similarity are formed which are interrelated in a 99% level of significance. Similarity Tree

The first group includes the variables V2, V4, V5, V6, V6p and V7. These are variables that concern the questions in which the angle of two vectors plays the basic role. It is of special interest the subgroups that include the variables (V4, V7) and (V6, V6p). The first subgroup relates questions that concern the straight angle. The second subgroup is even more interesting since it relates the correct solution process of the 6 th question with the attempt to use the parallelogram law. This phenomenon may be attributed to the fact that those who tried to apply the parallelogram law had to detect the angle of two vectors, something that forced them to get over the problem of the originality of the question and understand its mathematical structure. So, they were more likely to succeed in relation with those who tried to apply the triangle law, in which the definition of the angle of vectors is not required, but only their conversion to be successive, in other words the application of the process. The second similarity group includes the variables V3a, V3b, V3c and V6t, which are not immediately related with the determination of an angle. All the connections between these variables are statistically significant at a 99% level. The fact that the sub questions concerning exercise 3 presents certain similarity in their solution process, especially the grouping of 3a and 3c, which simply confirms their mathematical similarity, may not be a surprise, but presents special interest in explaining their connection with V6t. As previously mentioned, the triangle law does not require the calculation of the angle of two vectors and this must be the key for this question similarity. Many students expressed their intent to convert vectors to successive ones, but not being able to understand the mathematical context of the question, they failed to do so. Implicative relations between questions The Implicative Diagram shows only the implications that resulted between the variables that relate angle recognition and questions relating to the vectors. The bold black arrows relate to implications that are 99% significant, while the rest have a significance of 95%. Implicative Diagram The implicative relationships (R8s V4), (R3s V4), (R6s V4) are of special interest. It is noted here that in the analysis which has been conducted the variables R2s, R5s, R7s and R9s were not included, due to their very low average rate. Consequently, it is really amazing the fact that three out of five variables, that participate in the research work, and concern the identification of a straight angle, relate to the question which concerns the definition of the angle of opposite vectors and the implication (R6s V4) is statistically significant at 99%. Here, perhaps another connection, if somewhat risky, could be made. The 6 th picture of the questionnaire:

presents a similarity with the representation of opposite vectors, which the 4 th question is referred to: Furthermore, the entailment (V7 V4) connects two questions that relate to opposite vectors and therefore to the straight angle. V4 is the second term of the entailment, because the 4 th question has fewer cognitive requirements in relation to the 7 th question, as it has already been mentioned. Finally, the absence of V3a, V3b and V3c from Implicative Diagram is not surprising, since this is due that these question do not directly relate to angles, something that verifies our previous conclusions. CONCLUSIONS Many explanations about the difficulties that students encounter with vector calculus can be found in the bibliography, e.g. the misinterpretations about the vector addition are due to the fact that the students used to the manipulation of real numbers, fail to realize that vectors are different mathematical objects with their addition and multiplication. From the results of this study, important indications have aroused to support that in this kind of methodology the role of the angle in the cognitive procedures concerning vectors has been underestimated, a fact that we have tried to show up in our research study. More specifically, it has been observed that students face the questions that are directly related with an angle differently than those in which the role of the angle is only indirect. Likewise, implicative connections between variables arose, that implicate the recognition of straight angles and tasks that involve the straight angle, even at a 99% significance level. On the other hand, it is of special interest the implications between variables that relate to questions that involve the same kind of angle. Furthermore, it would be useful to point out the low recognition percentages of angles, such as the straight and obtuse, and the non-recognition of zero or full angle. The fact that students face the greatest difficulties and make the most of mistakes in the questions that involve such kinds of angle, must not be a random one. It would be interesting to investigate similar questions with younger students, in a mathematical environment as well as in a Physics framework. Such a research effort, should include experimental teaching of the vector subject, with the planning and use of suitable exercises, which will help the children in the understanding of the mathematical context of the concept of length, the concept of angle and their compression in a new mathematical subject entitled the vector.

REFERENCES 1. Aguirre, J. & Erickson G. (1984). Students conceptions about the vector characteristics of three physics, Journal of Research in Science Teaching, 21, 5, 439-457. 2. Berry, J. & Graham, T. (1991). Using concept questions in teaching mechanics. Int. J. Math. Educ. Sci. Technol., 22, 5, 749-757. 3. Crowe, M. J. (1985). A history of vector analysis : the evolution of the idea of a vectorial system. United States of America: Dover Publications. 4. Demetriadou, H. & Tzanakis, C. (2003). Understanding basic vector concepts: Some results of a teaching approach for students aged 15. Proceedings of the 3 rd Mediterranean Conference on Mathematical Education, ed. A. Gagatsis & S. Papastavridis, Hellenic Mathematical Society & Cyprus Mathematical Society, Athens, pp. 665-673. 5. Demetriadou, H. & Gagatsis, A. (1995). Teaching and Learning Problems of the Concept of Vector in Greece. In Gagatsis (eds), Didactics and History of Mathematics, Erasmys ICP-93-G-2011/11, Θεσσαλονίκη. 6. Forster, P. (2000). Process and Object Interpretations of Vector Magnitude Mediated by Use of Graphics Calculator. Mathematics Education Research Journal, 12, 3, 269-285. 7. Forster, P. & Taylor, P. (2000). A multiple-perspective analysis of learning in the presence of technology. Educational Studies in Mathematics, 42, 35-59. 8. Gagatsis, A. & Demetriadou, H. (2001). Classical versus vector geometry in problem solving. An empirical research among Greek secondary pupils. International Journal of Mathematical Education in Science and Technology, 32, 1, 105-125. 9. Graham, T. & Berry, J. (1990). Sixth Form Students Intuitive Understanding of Mechanics Concepts: Part 1. Teaching Mathematics and its Applications, 9, 2, 82-90. 10. Graham, T. & Berry, J. (1992). Sixth-Form Students Intuitive Understanding of Mechanics Concepts: Part 2. Teaching Mathematics and its Applications, 11, 3, 106-111. 11. Graham, T. & Berry, J. (1993). Students intuitive understanding of gravity. Int. J. Math. Educ. Sci. Technol., 24, 3, 473-478. 12. Graham, T. & Berry, J. (1996). A hierarchical model of the development of student understanding of momentum. Int. J. Math. Educ. Sci. Technol., 18, 1, 75-89. 13. Graham, T. & Berry, J. (1997). A hierarchical model of the development of student understanding of force. Int. J. Math. Educ. Sci. Technol., 28, 6, 839-853. 14.Gray, E. M. & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 25, 2, 115-141. 15. Ioannou, I. & Gagatsis, A. (2003). Study on the role of representations in the understanding of vectors. Proceedings of the 3 rd Mediterranean Conference on Mathematical Education, in A. Gagatsis & S. Papastavridis (ed), Hellenic Mathematical Society & Cyprus Mathematical Society, Athens, pp. 187-195. 16. Rowlands, S., Graham, T. & Berry, J. (1998). Identifying stumbling blocks in the development of student understanding of moments of forces. Int. J. Math. Educ. Sci. Technol., 29, 4, 511-531.

17. Tall, D. O., Gray, E M., Ali, M. B., Crowley, L. R. F., DeMarois, P., McGowen, M. C., Pitta, D., Pinto, M. M. F., Thomas, M., & Yusof, Y. b. (2001). Symbols and the Bifurcation between Procedural and Conceptual Thinking, Canadian Journal of Science, Mathematics and Technology Education, 1, 81 104. 18. Watson, A., Spyrou, P. & Tall, D. (2002). The relationship between physical embodiment and mathematical symbolism: The concept of vector. Mediterranean Journal for Research in Mathematics Education, 1, 2, 73-97. 19. Watson, A., (2002). Embodied action, effect, and symbol in mathematical growth. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4, 369 376. Norwich: UK. APPENDIX 1 Modification of the initial fig. 9 of Tall et al., 2001 : Spectrum of outcomes procedural proceptual To DO routine mathematics accurately To perform mathematics flexibly & efficiently To THINK about mathematics symbolically Progress Procept of vector Procedures acting on lengths Procedur Process Procedure(s) Procept Process(es) Procedure(s) Procept Process(es) Procedure(s) Procedures acting on angles Procedur Process Procedure(s)

APPENDIX 2 1. Find and underline as many kinds of angles as you can find in the following pictures. If you wish, you can write their name next to them. Example: acute angle

2. Compute the size of the angle of two vectors if the magnitude of their sum is equal to the sum of their magnitudes, i.e. if α + β = α + β ; α. 0 ο β. 60 ο γ. 90 ο δ. 120 ο ε. 180 ο στ. 3. If the point O is the middle of the straight segments ΑΒ and ΑΓ, then fill in the following equalities, by selecting every time one of the table s elements. α. OA = β. AB = γ. O = 2OA ΒΟ ΟΒ 2ΟΓ i) ii) iii) iv) v) vi) 2ΟΒ vii) viii) ix) ΓΟ Γ 1 ΟΓ 2 ΑΒ 4. The angle that is formed by two opposite vectors is: α. 0 ο β. 60 ο γ. 90 ο δ. 120 ο ε. 180 ο στ. 5. Sofia has designed the sum of two given vectors by using the parallelogram law. However, the length of the sum α + β it was found to be smaller than the length of each of the vectors α and β. She believes that has made a mistake. What is your point of view? 6. Nikos thinks that the sum of the following vectors is 0. What do you think? If you agree with Nikos, then justify your answer. If you disagree, then draw the sum of the two given vectors. 7. Is it possible that two non-collinear vectors have a sum that equals to the zero vector 0?