Advances in Social Science, Education and Humanities Research, volume 231 5th International Conference on Community Development (AMCA 2018) Understanding Logarithm: What are the Difficulties That Students Have? Mayang Dintarini University of Muhammadiyah Malang [email protected] Abstract. This study aims at determining students’ mathematics, and it is followed by the decreasing interest difficulties in understanding logarithm material. The to continue learning math to higher levels, such as subject of this study are 10th grade students. The type research conducted [4], and[5]. Other researcher states of research is qualitative descriptive, and it applied that "The most commonly offered reason for declining several steps in the form of providing test to the participation in mathematics (given by almost 60% of students, determining the difficulties and interviewing respondents) is that the respondents considered students who had difficulty on the material. The mathematics difficult"[6]. Similar studies focus on result of the study showed that there were several several factors such as pedagogical factors and classroom difficulties such as applying definition, determining practice have been made to identify that mathematics is logarithmic properties to solve problems, applying "boring" and "difficult" by many students[7], [8]. logarithmic properties and applying the prerequisite According to Gagne, difficulties can lie in the four material. These difficulties are due to the lack of mathematical objects namely facts, concepts, principles understanding of logarithmic definitions, the lack of and skills. Facts are agreements in mathematics, such as ability to see the facts relating to problems, over-focus symbols and notations. For example the notation of the 푥 on facts of rote and technical procedures, relying on inverse exponent function 푦 = 푎 is 푦 = log푎 푥. The improper intuition, and inconsistencies in symbolic concept is an abstract idea that allows one to classify an writing and inaccuracy. object and explain whether the object is an example or not an example of the abstract idea. The example of 푐 Keywords: difficulty, logarithm, causes concept in logarithm is log푎 푏 = 푐 ⇔ 푎 = 푏. The principle is a composite of two or more concepts, which INTRODUCTION we usually call by theorems or postulate or properties. The example of principle in logarithms is log푎 푏푐 = The logarithm is important in learning since it can be log푎 푏 + log푎 푐. While skills are procedures or rules to applied in various fields such as science, computer, obtain or obtain a certain result. finance, industry and others. For example in finance, If these difficulties occur at the elementary level logarithm is used in compound interest problems. If M0 students, and ongoing to the next level, students will feel is the initial deposit, Mn is the deposit in the nth month, i unconfidence, frustration, boredom, panic, fear and so is the interest rate, and n is the mth month, where m is on. Students will also feel difficult to understand the next the original number, then Mn = M0 (1 + i) n. To find n, material. This is expressed by [9] that is many reasons to we need a logarithm to facilitate the calculation, ie = get a bad score in any subject. However, math such as log 푀푛 house of cards, cards at the very bottom should be built = log(1+푖) 푀푛 . log(1+푖) solidly and accurately so it will support the next level. Logarithms is one material that is difficult for Each level depends on the previous level. students [1]. Another study on the difficulties in learning Other causes of learning difficulties are categorized logarithms said that students are more focused on the in epistemological causes (referring to the natural procedural approaches and depended too much on rules difficulties and the abstracts of mathematical concepts), rather than the concept of logarithm itself[2].Other study psychologically causes(referring to the students' self- revealed that students often see log notations as an reasons), and pedagogical causes (the reasons for object, not an operation[3]. Therefore, students often do teaching)[10].Based on the above explanation the cancelation on a logarithmic form. For example, ln (7x - researcher is interested in examining the difficulties of 12) = 2 ln x, becomes(7x - 12) = 2x. Students seemed to students in understanding the logarithm with the purpose be able to perform the routine calculation but they were of describing the causes of difficulties experienced by unable to answer the questions that required a higher students in understanding logarithm. level of thinking[1]. Many errors were not due to lack of knowledge, but it seemed to be based on the over- METHOD generalization of algebraic rules. Difficulties and frustrations to mathematics have been The type of this research is descriptive qualitative. At felt by many people long ago. Many students feel not the beginning of the study, researcher conducted confident even afraid of dealing with math. Many studies observations to determine the subject of research. The have written about the students’ difficulties in learning Copyright © 2018, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). 239 Advances in Social Science, Education and Humanities Research, volume 231 subject of this research was 10th grade students of subject 2 wrotelog6 15 = log2 3 . log2 5 + log3 3 . log3 5. SMAN 5 Malang who faced difficulties in logarithm, had Subject 1 and Subject 2 are connected only by what is the ability to write the process of the problem clearly and known and intuitively asked. It is similar to Posamentier communicate orally well. A test was given to the who stated that students have small theories about students to determine the difficulties in understanding mathematics related to their learning[11]. According to logarithm. Then an interview was given to students to Fischbein, an intuitive interpretation based on a make sure of the difficulties and it causes. primitive, limited, and entrenched experience is the one that destroys the formal control or the terms of the RESULT algorithmic settlement, and then precludes the correct mathematical reactions[11]. Based on the results of tests and interviews, there are The ability of students to see these facts is one of the several causes of students' difficulties in understanding stages of mathematical reasoning. The ability to see the logarithms. The cause of the difficulty is as follows. interconnected facts in solving this problem was the first stage of reasoning. This indicates a lack of reasoning Less understanding of the definition of logarithms levels of Subject 1 and Subject 2. Hiebert and Carpenter In this study, researcher found some students did not say that the degree of understanding is determined by the fully understand the definition of logarithms. Eventually, number and strength of connections[12]. the students found it difficult to apply the logarithmic definition to the problem. At first, the students seemed to Over focus on the memorizing facts and procedural able to calculate the logarithm value of a number, but While working on the problem of logarithms, when they failed when they confronted more complex students are often constrained by the many logarithmic logarithms. properties and difficult for them to memorize. The For example Subject 4, Subject 4 was able to explain constraint makes it difficult for students to understand log4 16 = 2, but he was unable to explain how he got logarithms, due to over-focusing on the theorems or 1 properties they need to memorize. Students are ultimately log4 = −1. Subject 4 was able to explain log4 16 = 4 not trying to develop their reasoning. 2with the definition "4 to the power of what equal to 16, All subjects have difficulty in applying logarithms on then his answer was 2". But when he tried to explain tests. These was because the subject was still thinking to log 1 = −1, Subject 4 found no matching rules to 4 4 memorize formulas and procedures, and not trying to support his answer. Even Subject 4 tried to make its own reasoning. For example Subject 3, he has difficulty in definition, which was "4 times (-1) ...". "4 times (-1) ..." applying the properties of logarithms. Subject 3 changed 1 log 6 was not the answer he wanted to explain why log4 = log 15 to . The same error was done by Subject 2, 4 6 log 15 −1. Because Subject 4 knew that 4 x (-1) = -4. So he log2 2 he changedlog2 5 to . Similarly, Subject 4 and admitted that he did not understand the definition of log2 5 logarithms yet. Subject 5, researchers found the same error. Subject 1, This finding is similar to previous study which Subject 2, Subject3, Subject4, and Subject 5 tried to use described that the students often failed to see logarithms his memory of the properties of logarithms that he wants as a number. It means that students often do not to use. But, they were failed in remembering that understand the meaning of logarithm[1]. When there is a properties. The subject did not try to seek or rediscover "logarithmic expression, the small number is the base, the properties. They were over focused and overconfident the number after the log is the result of the logarithm and in their memories. the number on the other side of the equal sign is the The finding matched the assumption that the exponent". The expression gives the students a wrong inaccuracy may be the cause due to the lack of idea about logarithm; they perceive a logarithmic form understanding and skill of the algorithm[13]. The in some separate parts. Another study found that the lack condition refers to an error due to lack of understanding, of understanding on basic knowledge had become one of where students used incorrect rules[14].