Challenges in teaching Real Analysis classes at the University of PGRI, South Sumatra, Indonesia
aa r X i v : . [ m a t h . HO ] A ug Challenges in teaching Real Analysis classes atthe University of PGRI, South Sumatra, Indonesia
E. Septiati and N. Karjanto ∗ Department of Mathematics Education, University of PGRI, Palembang, South Sumatra, Indonesia School of Applied Mathematics, Faculty of Engineering, The University of Nottingham Malaysia CampusSemenyih, Selangor, Malaysia
Presented July 25, 2008; Submitted July 30, 2008; Updated August 4, 2020
Abstract
This paper discusses our experiences and challenges in teaching advanced undergraduate RealAnalysis classes for Mathematics Education students at the University of PGRI (Persatuan GuruRepublik Indonesia, Indonesian Teachers Association) Palembang, South Sumatra, Indonesia. Weobserve that the syllabus contains topics with a high level of difficulty for the students who arespecialized in education and intend to teach mathematics at the secondary level. The conventionallecturing method is mainly implemented during the class, with some possible variations of themethod, including the Texas method (also known as Moore’s method) and the small group guideddiscovery method. In particular, the latter method has been implemented successfully for a RealAnalysis class at Dartmouth College, New Hampshire by Dumitras¸cu in 2006. Although it is a realchallenge to apply a specific teaching method that will be able to accommodate a large number ofstudents, the existing teaching activities can still be improved and a more effective method could beimplemented in the future. Furthermore, the curriculum contents should be adapted for an audi-ence in Mathematics Education to equip them for their future career as mathematics teachers. Anyconstructive suggestions are welcome for the improvement of our mathematics education systemat the university as well as on the national scale.
There are several teaching methods and certainly teaching style varies from one lecturer to another.Several ways of teaching methods that are commonly carried out in many parts of the world in-clude the following: questioning, explaining, demonstrating, collaborating, and learning by teach-ing (Committee on Undergraduate Science Education et al., 1997; Good, 2008). In particular, thelearning by teaching (German,
Lernen durch Lehren ) is a widespread method in Germany, where thestudents take the teacher’s role and teach their peers.More specifically, we want to use certain methods in teaching mathematics, and in this context,in teaching Real Analysis classes. Methods of teaching mathematics include the following: classicaleducation, rote learning, exercises, problem-solving, new math, historical method, and reform orstandard-based mathematics (Clarke, 2003; Fan et al., 2004; Lockhart, 2009). In particular, there aresignificant research results on the implementation of the realistic mathematics education method inIndonesia (Armanto, 2002; Fauzan, 2002; Hadi, 2002; Zulkardi, 2002). Furthermore, cooperativelearning methods are now being used more and more often in teaching undergraduate mathematicsand science (Davidson & Kroll, 1991; Rogers et al., 2001; Dubinsky et al., 1997; Finkel & Monk, 1983)as well as in higher education settings (Ledlow, 1999; Milis, 2010). ∗ [email protected] ? ). The author concludes that the guideddiscovery method is an excellent modality of exposing students to mathematical research.We observe that a conventional teaching method using instruction and lecturing for the Real Anal-ysis courses presents a challenge for the students who are specialized in Mathematics Education. Thischallenge motivates us in bringing this problem into the surface.What is “real analysis” and what is the scope of the course on Real Analysis? Real analysis is abranch of mathematical analysis dealing with the set of real numbers. In particular, it deals with theanalytic properties of real functions and sequences, including convergence and limits of sequencesof real numbers, the calculus of the real numbers and continuity, smoothness, and related proper-ties of real-valued functions (Bressoud, 2007; Krantz, 2004; Stein & Shakarchi, 2009; Trench, 2013).Certainly, a course on Real Analysis should cover the aforementioned materials. This course is animportant component of mathematics curriculum for both educational and noneducational streams atthe undergraduate level.In this paper, we share our experiences in implementing different teaching methods to the RealAnalysis courses. This paper is organized as follows. Section 2 discusses the organization of thecourses, including reference textbooks being used and the method of assessment. Section 3 explainsthe challenges and difficulties that students face in following the classes. Furthermore, Section 4discusses our observations in conducting the classes and implementing several teaching methods. Thissection also provides the students’ responses toward different teaching methods. Finally, Section 5gives the conclusion and remark for future research to our discussion. The classes of Real Analysis I and II are compulsory subjects for undergraduate students in Mathemat-ics Education at the University of PGRI, Palembang, South Sumatra, Indonesia. These courses carrythree credit points and are given to third-year students or in the fifth and sixth semesters of their study.There is only one-time interaction every week and it lasts for 150 minutes, which is three times 50minutes.There are ten topics in total which are covered in the Real Analysis courses, in which six topicsbelong to Real Analysis I and the other four belong to Real Analysis II. The materials covered in RealAnalysis I are ordered set, field, Euclidean space, metric space, topological concepts in metric space,and sets in metric space. The topics covered in Real Analysis II are convergence sequence, Darbouxintegral, Riemann integral, and Rieman-Stieltjes integral.The number of students in one class ranges from 15 up to 40 students and there are 9 parallelclasses for the same course with a total number of 335 students. One senior lecturer plays a role as acoordinator for three other lecturers. Practically, the material presented in this paper is merely basedon our observation of four different classes taught by one of us (ES). One class consists of only 15students while the other three are 28, 29, and 32 students, respectively.Lecture notes are prepared and compiled from several mathematical analysis books, among oth-ers are
Introduction to Real Analysis by Bartle & Sherbert (2011),
Principles of Mathematical Analysis by Rudin (1976),
Pengantar Analisis Real by Darmawijaya (1986), and lecture notes on Real Analysisfrom Malang State University, East Java, Indonesia. For Real Analysis II, additional references havebeen used, among others are
Analisis Real by Soemantri (1993),
Real Analysis by Royden (1988) and
Fundamental Concepts of Analysis by Smith & Albrecht, Jr. (1981). There are other excellent textbookson Real Analysis at the introductory level, including (Browder, 2012; Kolmogorov & Fomin, 1975;Protter & Charles Jr., 2012; Stromberg, 2015; Wheeden, 2015).2he method of assessment is based on several components with different weights. One courseworkis assigned and is graded on an individual basis, this assignment carries 20 percent of the final grade.Two examinations–the mid-semester and final exams–carry 30 and 50 percents of the final grade,respectively.
Many students have a wrong interpretation of what mathematics subjects involve. They generallyassociate mathematics with counting, calculation, and computation which in turn restrict the disciplineinto only arithmetic. One online encyclopedia defines mathematics as the body of knowledge centeredon such concepts as quantity, structure, space, and change and also the academic discipline that studiesthem ? . That is why mathematics includes the use of abstraction and logical reasoning which involvesrigorous deduction from appropriately chosen axioms and definitions.We observe that the students are less familiar with theorems and how to prove them. Implementingmathematics symbols and terminologies is far from familiar. The concept of set theory is still weaklycomprehended. As an example, many students are not able to distinguish simple notations suchas ( a, b ) , [ a, b ) , ( a, b ] and { a, b } . In particular, the students consider proving theorem, convergence,and Riemann integral as the most difficult topics. Bear in mind that the students who specialize inMathematics Education spend only merely of 55% of the total credit points on Mathematics coursesfor the entire study period, i.e. 84 out of 154 credit points are Mathematics courses.Furthermore, the students also face difficulties in some technical issues, in particular, to find theliterature. It is rather difficult to obtain reference books since the library has a limited amount ofthese books while the number of students is quite massive. The price of these textbooks is consideredvery expensive for all of the students. Even if the students possess textbooks, yet, since English isnot the mother tongue of students, the language barrier may present another significant challenge inunderstanding the material. We have implemented several teaching methods in conducting the Real Analysis classes. These arethe conventional instruction method, Moore’s method, and the guided discovery method. The conven-tional instruction method is implemented to the majority of the class sessions, in particular, to explaindefinitions and new concepts. Moore’s method is implemented when explaining the properties of inte-gral. For instance, this method is used to prove the following theorem. If f is a bounded function andDarboux integrable on an interval [ a, b ] , show that f is also Darboux integrable on the same interval.The guided discovery method is implemented in some theorem-proving sessions.We observe that the students prefer the conventional instruction method more than the other twomethods. The students are not able to follow Moore’s method at all since none of them can answeror to give an idea in solving the theorem above. It is observed that a small number of the studentscould follow the guided discovery method, i.e. less than 20%. Regarding the preference of teachingmethod, 34.6% of the students prefer the conventional instruction method where the teacher onlylecturing, 32.7% prefer a variation in teaching method, 13.5% prefer the conventional instructionand discussion, 5.8% prefer the guided discovery method, 5.8% prefer the conventional instructionand problem-solving and 1.9% each for preference in question and answer session, task assignment,discussion, and self-study.Regarding the implementation of different teaching methods in understanding the material, 48%of the students considers it helpful, 21.2% also considers it helpful but prefers only the conventionalinstruction method, 5.8% says it can be helpful but without the discussion session, 9.6% considers itis not helpful at all and 15.4% says it depends on the material being covered. Furthermore, we wouldlike to know what kind of comprehension the students acquire after completing the courses on Real3nalysis. Almost 40% of the students (39.4%) acquires logical reasoning and improvement in theoremproving, 30.3% acquires knowledge on the topic of integral, 18.2% improves their understanding inmathematical symbols and 12.12% claims do not improve at all.Regarding the material delivery by the lecturer, almost 60% says that it is very easy to understand(59.6%), 32.7% respond that it is sufficiently easy to understand, and only 7.7% say that it is difficultto understand. Regarding the availability of the textbooks, an excellent number of 94.2% claim thatit is very helpful, 3.8% say that it is helpful but they need some other additional references and 9.6%say that it is not helpful. The following section gives a conclusion to our discussion. We have discussed that the Real Analysis courses are very important components in the curriculumof the Mathematics Education program. Nevertheless, a majority of the students consider that thesecourses are very tough and challenging. We have implemented different teaching methods to helpthe students to get a better understanding of the materials. Even though Moore’s method and theguided discovery method have been implemented successfully in some mathematics courses in severalcolleges in the US, we observe that these methods are still difficult to be implemented for the RealAnalysis courses, particularly at the University of PGRI, Palembang, Indonesia. Apart from implement-ing excellent teaching methods, we strongly believe that the curriculum for these courses should beadapted to the characteristics of students who are specialized in Mathematics Education and their ca-reers after completing their degree. Although the materials in the Real Analysis will never be given tosecondary school students, the participants in these classes are trained to think critically. In turn, theability of this critical thinking is very beneficial for a good teacher. For future research, it is importantto investigate the significant value of the students’ responses. This investigation should involve thequantitative calculation and validation test.
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