A STUDY ON TEACHING GROUP AND SUBGROUP CONCEPTS Serpil Konyalıoğlu Atatürk University, K.K. Education Faculty, Department of Secondary Science and Mathematics Education, Erzurum / TURKEY [email protected] ABSTRACT The aim of this study is to investigate the effectiveness of computer‐based/assisted teaching method using ISETL programming language on students’ understanding of subgroup and group concepts. The data was collected from a group of students by means of written assesments and clinical interviews. Analysis of student’s written responses and clinical interviews seems complexities in their understanding of group and subgroup concept. Key words: ISETL (Interactive SET Language), Learning Group, Learning Subgroup GRUP VE ALTGRUP ÖĞRETİM KAVRAMLARININ ÖĞRETİMİ HAKKINDA BİR ÇALIŞMA ÖZET Bu çalışmanın amacı, ISETL programlama dilinin kullanıldığı bilgisayar destekli öğretim yönteminin öğrencilerin altgrup ve grup kavramları ile ilgili başarılarına etkisini incelemektir. Çalışmanın verileri yazılı ve klinik görüşmelerle toplanmıştır. Öğrencilerin yazılı ve klinik görüşmelerinin analizi sonucu grup kavramını anlamada karmaşa yaşadıkları görülmüştür. Anahtar kelimeler: ISETL, Grup Öğrenimi, Altgrup Öğrenimi epistemological: “the problems from which INTRODUCTION these concepts arose in an essential manner Abstract algebra has traditionally thought to are not accessible to students who are beginning be the most difficult course because of abstract to study (and expected to understand) the nature of the course, and particularly many concepts today (Nardi, 2000). students have a shock in group theory. There has been a large number of research on The concept of group is an example of a new the learning and teaching of group theory. mental object the construction of which causes Some would like to make modern algebra and fundamental difficulties in the transition from group theory more visual (Bardzell and school to university mathematics. As Robert Shannon 2002, Cox, Little and O’shea 1997). and Schwarzenberger (1991) point out, one They use some structures to visualize certain root of this difficulty is historical and concepts from abstract algebra including Number 18, 2006 155 Serpil Konyalıoğlu groups, subgroups, quotient groups, and mathematics program in Atatürk University automorphism. of Turkey. There are a number of studies to integrating I used the text “Learning Abstract Algebra technology into the abstract algebra and with ISETL” by Ed Dubinsky and Uri Leron group theory class. For example Rainbolt (1994). The course ran for 6 weeks with 4 (2001) used GAP software as a tool in abstract hours of lectures per week. The data was algebra class. Keppelmann and Webb (2000) collected from two main sources: Written used Finite Group Behavior (FGB) program assessment and clinical interviews. and they discuss the features and philosophy Near the end of the course, a written of FGB. Cannon and Playoust (1995) used the assessment was given to determine the Magma computer algebra system in their class. students’ understanding of group and Scientists such as Selden and Selden (1978), subgroup concepts. The questionnaire was Hazzan and Leron (1999) described miscon‐ given in Appendix. ceptions in abstract algebra. Based on the results of the written assessment, However, the current trend in pedagogy in 9 students were selected to discuss their abstract algebra has focused on the learning results. While selecting students to interview, perspective. And an important attack on this special attention was given to interviewing approach has been made by Ed Dubinsky and some students who gave correct, partially his colleagues. According to this approach, correct and incorrect answers on the learning is central and the research focuses on assessment. The interviews are about 15‐20 how students learn mathematics. minutes in duration. In fact, replacing the lecture method with FINDINGS and DISCUSSION constructive, interactive methods involving computer activities and cooperative learning My analysis focused on the following issues: can change radically the amount of meaningful Group and Subgroup. learning achieved by average students (Leron Group Concept : and Dubinsky, 1995). Table 1. Scores on Group An individual’s knowledge of the concept of Correct Partially Incorrect No Part 1 group should include an understanding of Answer Correct Answer Response various mathematical properties and 1 14 1 0 0 constructions independent of particular 2 9 4 1 1 examples, indeed including group consisting 3 10 1 4 0 of undefined elements and a binary operation 4‐ A 15 0 0 0 satisfying the axioms (Dubinsky et al. 1994). 4‐B 15 0 0 0 I used ISETL programming language in this 5 8 4 0 3 study. ISETL has several important properties. It is designed as a tool for teaching and Many students had no difficulty describing learning mathematics, particularly abstract the group concept, that is, a set with a binary algebra. ISETL’s syntax resembles to standard operation and four axioms. But only one mathematical notation. student wrote commutative property as a group property. METHOD As stated above, abstract group is a conceptual The students used in this study were 15 object like other concepts of abstract algebra. students enrolled in abstract algebra course When students try to understand this designed for the professional teaching of unfamiliar concept, they can make some 156 Journal of Qafqaz University A Study on Teaching Group and Subgroup Concepts mistakes. For example they see group as a set; All students in this study described subgroup because, the set concept is familiar for students. concept correctly. Some students considered And they do not take a consideration operation. that every subset was subgroup. For example In some cases group operation changes from Cemal said that {0,1} was a subgroup of Z8. an unfamiliar one to a familiar one. In the A subgroup of Z8 is {0,1}. Because this set is a 7 following excerpt, Ali considered that Z – {0} subset of Z8 and it is a group. is not a group. Cemal did not consider that a subgroup was If Z7 – {0} is a group then it has to have all of an independent group. He firstly considered a axioms of groups, which means it has to be subgroup as a subset. closed, it has to be associative, it has to have identity and inverse. Z7 – {0} consists of Perhaps the most common confusion was in {1,2,3,4,5,6} and it is not a group. Because it is question “Is Z3 a subgroup of Z6?” (see Hazan, not closed. For example 3 * 4 = 12. 12 is not in Leron, 1996). In the following excerpt, Gül this set. Hence Z7 – {0} is not a group. tried to determine whether Z3 is a subgroup of Z6. She said: Ali considered the operation of Z7‐{0} as an ordinary multiplication. Ali was not aware of “Well. One minute. A subset of a group is a the operation. subgroup if it has all of the group axioms. Some students confused global properties of an This subset has to be closed, it has to be operation on a set. Sometimes associative, associative, it has to have an identity, an commutative are inherited from a larger group. inverse. These properties do not change in a smaller set. “Z3 is a subset of Z6. The elements of Z3 are Some students confused this shortcut and they 0,1,2. Z3 has to be closed. It is already closed.” considered that the operation was inherited from a larger group. Sevgi considered that Z7‐ (Writes as follows: {0} is a group. Because she thought that inte‐ 1+1= 2 , 2 ≡ 1 (mod 3) , 1 Є Z3 gers are a group under ordinary multiplication. 2+2 = 4, 4 ≡ 1 (mod 3) , 1 Є Z3 Z7 – {0} is a group. Because integers are a 1 +2 = 3, 3 ≡ 0 (mod 3) , 0 Є Z3 group with ordinary multiplication, operation in Z7 –{0} is inherited from integers. Hence it is closed.) Sevgi considered the operation as ordinary Gül made several errors in her attempt to multiplication. She didn’t know that it required verify group axioms. First she stated: particular attention on the operation. However 1+1= 2, 2 ≡ 1 (mod 3) , 1 Є Z3 she didn’t know that integers were not a group under multiplication. In addition to confusion about modular Subgroup Concept: arithmetic, she automatically considered the subgroup operation as modulo 3. Table 2. Scores on Subgroup Our students could not see the operation in Z3. Correct Partially Incorrect No Part 1 Answer Correct Answer Response Many knew that Z3 with addition mod 3 and 1 15 0 0 0 Z6 with addition mod 6 was a group. But 2 8 0 0 7 when I asked this question (Is Z3 a subgroup 3 7 1 1 6 of Z6?), the students exhibited disequilibra‐ 4 7 3 5 0 tion. 5 6 3 6 0 Number 18, 2006 157 Serpil Konyalıoğlu CONCLUSION Rainbolt, J., 2001. Using GAP in an abstract algebra class, In A.C.Hibbard and E.J.Maycock (Eds), My primary concern in this study was how Innovations in Teaching Abstract Algebra, MAA the students learned and understood group Notes, 60, The Mathematical Association of America. and subgroup concepts. Robert, A.and Schwarzenberger, R.,1991. ‘Research in teaching and learning mathematics at an advanced I strongly believe that students’ active level’, in D. Tall (Ed.), Advanced mathematically involvement is essential to understanding Thinking, Kluwer Academic Publishers, Dordrecht / Boston / London, 127‐139. these concepts. I used ISETL for this purpose. ISETL can be a useful tool in teaching and learning abstract algebra. But the students need more time. I feel that students acquire a deeper understanding of these two concepts. REFERENCES APPENDIX: Bardzell, M.J. and Shannon, K. M., 2002. The Written Assessment Questions PasGalois Triangle: Visualizing abstract algebra, Focus, 22 (3), 4‐6. Part 1: Cannon, J. and Playoust, C., 1997. Using the Magma 1.