Teaching and learning how to write proofs in Concepts of Geometry

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1 Dec. 2009, Volume 6, No.12 (Serial No.61) US-China Education Review, ISSN , USA Teaching and learning how to write proofs in Concepts of Geometry WANG Hao-hao (Department of Mathematics, Southeast Missouri State University, MO 63701, USA) Abstract: This paper concerns my experience teaching Concepts of Geometry, an inquiry-based course that emphasizes discovery learning, analytical thinking, and individual creativity. The author discusses how to guide the students to recognize the connections among different mathematical ideas; to select the types of reasoning and methods of proof that apply to the problem; to organize their mathematical thinking through communication; to make their own mathematical conjectures; and to analyze and evaluate each other s mathematical arguments and proofs. In particular, the comments and the feedback from students are included. Key words: inquiry based teaching; discovery learning; constructivist based approach The Mathematics Association of American s Committee on the Undergraduate Program in Mathematics (CUPM) made recommendations to guide mathematics departments in designing curricula for their undergraduate students. The following statement is quoted from the CUPM Curriculum Guide 2004: Recommendation 2: Develop mathematical thinking and communication skills. Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind. More specifically, these activities should be designed to advance and measure students progress in learning to: (1) State problems carefully, modify problems when necessary to make them tractable, articulate assumptions, appreciate the value of precise definition, reason logically to conclusions, and interpret results intelligently; (2) Approach problem solving with a willingness to try multiple approaches, persist in the face of difficulties, assess the correctness of solutions, explore examples, pose questions, and devise and test conjectures; (3) Read mathematics with understanding and communicate mathematical ideas with clarity and coherence through writing and speaking. In responding to the CUPM Curriculum Guide, the author started a project to use an inquiry-based teaching style in Concepts of Geometry, a junior- and senior-level college geometry class. This class emphasizes discovery learning, analytical thinking and individual creativity. The goal was to guide the students to recognize the connections among different mathematical ideas; to select the types of reasoning and methods of proof that apply to the problem; to organize their mathematical thinking through communication; to make their own mathematical conjectures; and to analyze and evaluate each other s mathematical arguments and proofs. In this report the author present the results of a study investigating the effectiveness of teaching proof writing via an inquiry-based teaching style in his Concepts of Geometry class during a two-year period, Fall 2006-Fall The author would like to thank the Inquiry-Based Learning Project Grant from EAF and the SoTL Grant from Southeast that allowed him to establish and continue this project in the Concepts of Geometry class. The author would like to thank Dr. Shing So, Dr. Craig Roberts, and all the teaching associates and SoFL fellows for their helpful conversations and suggestions. In particular, the author would like to give special thanks to Joe and Linda Dunham and Dr. Jake Gaskins for their kind assistance in improving this paper. WANG Hao-hao, Ph.D., associate professor, Department of Mathematics, Southeast Missouri State University; research field: algebraic geometry and commutative algebra. 74

3 assistance and small grants to faculty to study how their pedagogy is linked to learning. In the year 2006, the author was awarded a grant to establish a pilot project for an inquiry-based learning mentoring program in the Department of Mathematics at Southeast Missouri State University. We developed inquiry-based learning material (a problem sequence) suitable for the three-credit-hour Concepts of Geometry course in the summer of 2006 and used the material in the Fall 2006 Concepts of Geometry class. The author taught the Concepts of Geometry class for the first time in Fall In Spring 2007, the author was awarded the Center for Scholarship in Teaching and Learning (SoTL) grant and became a SoTL fellow, which allowed me to continue the pilot project in the Concepts of Geometry class through the Fall This project was designed to investigate the effectiveness of using an inquiry-based teaching approach to enhance students learning proof writing. 2. Method First, before the fall semester started, we developed inquiry-based teaching-learning material and a problem sequence, including material in both Euclidean and non-euclidean geometry. This material was designed to help all students develop a deep understanding of mathematical concepts and how to apply them. It also prepares them for using their newfound skills to further their education or the jobs. Each section of the material starts with basic definitions and examples, then moves to lemmas, theorems, and corollaries, and ends with exercises and reflection questions. It challenges students to explore open-ended situations actively. The aim of using the material is to help students understand basic background knowledge, routinely investigate specific cases, look for and articulate patterns, and make, test, and prove conjectures. The problem sequence was provided to the students free of charge at the beginning of the semester. Second, the author discussed his teaching progress with his mentor in Fall 2006, attended SoTL fellow monthly meetings, and communicated with mentors and other SoTL fellows to exchange teaching experience and ideas to improve his teaching in the classroom in Fall Third, in my daily classroom teaching, the author arranged opportunities for student presentations, observed students performance and recorded their progress. The purpose was to enable the students to learn more by discovering more for themselves. Typically, a lesson would begin with the giving definitions; then the students would explain the meaning of the definitions, draw the geometric figures, and provide examples. Then, the students built the rest of the body of knowledge as problems and proofs were assigned. Collaboration with classmates was encouraged. Students with mixed abilities were teamed together to work on group activities; each group made its own group work report. As the teacher, the author was available to assist students, providing hints to the problems or the proofs. Fourth, the author provided classroom activities and projects that use hands-on learning. In particular, the author assigned a project that required the students to construct geometric objects, explain the geometry involved in their constructions, make appropriate conjectures, and possibly prove their conjectures. Fifth, the author gave periodic survey questions to obtain students feedback and to monitor their learning. Moreover, the author used answers to the survey questionnaires to analyze the effectiveness of his teaching and their learning. the author compared the final examination score, the numerical course average, and the course letter grade for each student in the experimental section with the average final examination score, course average, and course letter grade of students in the section taught with the traditional lecturing approach. However, the author did not use the test scores for statistical analysis for significant differences between the different teaching approaches since he had only eight students. 76

6 material to prove the theorems later. The book is sometimes hard to understand. (b) The sequence does help when we are learning the proofs. (c) The course material is very helpful. The lists of theorems and definitions helps when proving new theorems because I can look back on what has already been proven or defined to assist in the new proof. (d) I am starting to like the handouts more because they are in a logical sequence. (6) Question 6: As a student, do you feel you learned in this class? If you were a teacher, what would you have done differently? Answer 6: (a) I did learn a lot from this class. I never thought about geometry in an abstract frame of mind. It is difficult, but I am learning. (b) It is definitely the class in which I learned the most new things. It was the only class that I took this semester which I have never taken before in Germany. And therefore I learned some new theorems and I even learned a complete new geometry that I have never heard of before. (c) As a student, I have learned a lot from this class. Once you break down all the theorems to prove them, it makes you look at other proofs in another way. I think this class will help me in the long run when I become a teacher, because we proved every theorem. (d) I am satisfied with the knowledge that I am building through thinking through the proofs assigned for homework. I become familiar with the theorems of geometry and can apply them to real life situations. (7) Question 7: If you were the teacher of this class, what instructional style would you use? Answer 7: (a) I enjoy the instructional style used in this course. It keeps the student engaged in learning by requiring the student to complete proofs on his own with definitions and examples given in class. It allows the student to be inspired to complete proofs on his own and gain experience with the material. (b) I would use repetition. I like when I do things over and over to truly understand them. (c) Using manipulative would aid my learning. It is beneficial to my study habits. (d) Although it takes time to go through the proofs, it forces me to think through the theorem myself and gain a better understanding of each theorem. 5. Conclusion In the author s traditional lecture style geometry class, the students simply memorize the proofs without clear understanding. In this inquiry-based class, the student either solved problems by himself/herself, or was guided by the instructor without being provided with the explicit method. This strategy is very demanding on the instructor, as the instructor must know the material extremely well in order to guide the students to a solution and effectively correct missteps along the way. It is very demanding for the students as well. It is significantly different from instructional strategies that students have typically encountered in their mathematics classes as it requires them to be much more active learners. This certainly moves students away from their comfort zone in the short term, but it has long-term benefits in that the students develop their own proofs and solutions and have truly experienced the manner in which mathematicians work. This helps the students to become mathematically independent. Based on the material and data collected, the following conclusions can be made for students in this Concepts of Geometry: 79

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