Investigation into First-Year College Students' Misconceptions about Limit Concept: A Case Study Based on Cognitive Style

The cognitive style has an important role in determining the characteristics of students on the conceptualizat ion. One of the two types of cognitive styles are field independent (FI) and field dependent (FD). The purpose of this study was to identify the fo rm of conceptual misconceptions about limit and their causes. This type of research is a case study. The data was obtained through interviews based on the conceptual problem-solving tasks. Correspondents used in this study were from first-year college students (95 people) who were studied at Universitas Jambi. The results showed that (1) FI forms of students' misconceptions, namely: misconceptions about interpreting the concept of function limit and function conditions which have limits, such as: left limit and right limit . This problem occurred due to incomplete reasoning and wrong intuition. The other result of this research was to obtain (2) FD forms of students’ misconception, namely, (a) misconceptions about prerequisite material: drawing graphs of functions, misconceptions about determining the domain of a function, misconceptions about definit ions and forms of function notations, misconceptions about determining function values on function graphs (b) misconceptions about the concept of limits: misconceptions about interpreting the concept of limits in a function, the function requirements (left limit, right limit) and misconceptions about understanding the meaning which is close to the limit function (limit-x) approaching infinity.


Introduction
Mathematics is the science of logic regarding form, structure, quantity, and concepts that interconnected between one to another. Thus, one of the competencies needed by students in mastering mathematics is a deep understanding of mathematical concepts. The theory of constructivism stated that students have to find and transform co mp lex informat ion, check new information with o ld rules, and rev ise them if the ru les are no longer appropriate by themselves [1]. The theory was the basis of learning act ivities. It also emphasized the fact that if students do a cognitive process, it will fulfill students' cognition in the form of structuring concepts fro m the learning environ ment delivered by the teacher. When students constructed a concept understanding in the cognitive structure which was obtained from interactions with the learning environment, it would highly produce errors in interpreting concepts. Mistakes in interpreting mathematical concepts are often called misconceptions [2] Misconception was referred to a concept that was not followed by a scientific understanding or expert agreement in the field [3]. Furthermore, [4] and [5] defined misconception as a cognitive structure in the form of a strong understanding that was different fro m what should be according to scientific princip les in general. Besides, it would disturbed the acceptance of new understandings. Misconceptions in understanding mathematical concepts have been found in many studies [5], [6], [15], [16], [7]- [14].
First-year college student need to adapt to entering university level [17]- [23]. Usually, they still carry the previous concept of learning, namely, h igh school level [18]. Early detection related to misconceptions could facilitate teachers to provide appropriate learning instructions. Limit concept underlay all concepts in calculus, thus it made limit become extremely fundamental in this learn ing [5]. If not properly being prepared, then limit concept could be difficult to be understood in calcu lus [24].
Specifically, a misconception about the limit concept has been carried out by [5] for secondary school level. They also exp lained how cognitive style affects the misconceptions. Misconceptions would occurred when students received informat ion about a concept. Each students has a unique way or fondness in obtaining, organizing information (cognition) and processing informat ion (conceptualization), which we usually called cognitive style [25]- [30]. Thus, it could be presumed that the cognitive style would play an important role in making students' misconceptions. The research had the same results with Lawson & Thompson (1988) who mentioned that one of the factors which caused students' misconceptions was the field-independent (FI) and field-dependent (FD) cognitive styles.

Research Design
This study identified forms of misconceptions about the limit concept in first-year college students based on their cognitive style. Therefore, th is research used a qualitative paradigm with a case study approaching.

Subject Research
This research was conducted in The Mathematics Education Study Program in Universitas Jambi. The research subjects were first-year college students (95 people) who took calculus course by considering the following reasons for research subjects selection: (1) registered first-year college students who had to take calculus courses, (2) first-year college students who had a cognitive style of FI and FD, (3) first-year college students who had misconceptions about the concept of limit based on the Certainty of Response Index (CRI) category with CRI value > 2.5 and wrong answer category, but high CRI value, wh ich could be interpreted as experiencing misconception.

Data Collection
The data collection is through a pretest before learning is carried out. This means that students' knowledge of the concept of limits is inherited fro m h igh school. Research data collection was carried out by the follo wing stages: (1) A diagnostic was conducted through a mu ltiple Certainty of Response Index (CRI) test. The aim of the test was to obtain data on students experiencing misconceptions [4]. (2) Cognitive style tests that using GEFT was carried out [30], [32]to determine the type of congressional style of research subjects based on CRI tests that were distinguished based on cognitive style. (3) After finding the research subject that experienced misconceptions based on the clasification of the positive force, then the limit concept understanding test would be performed in the form of an essay test (as example in Figure 1). Based on the results of the third stage of the test, further identification of the form of misconception was carried out. (4) The forms of misconception from the concept understanding test were also exp lored to find out the forms and causes of misconceptions that occurred, through semi-structural interviews and video records. This activity was a fo rm of data triangulation aimed at testing the validity and reliability of the data that had been obtained.

Data Analysis
The data consisted of: (1) misconception classification tests using CRI TEST. The misconception criteria adopted from [33] are explained in Table 1 as follows. Correct answer The answer is correct, but the CRI value is low, meaning it doesn't know the concept (Lucky guess) Correct answer and high CRI value mean mastering the concept well.
Incorrect answer Wrong answers and low CRI values mean it doesn' t know the concept. Incorrect answer but high CRI value mean misconception.
The CRI scores in Table 1 were taken fro m the average CRI scores for every student. There were groups of students who knew the concept (TK), did not know the concept (TTK) and those who had misconceptions (MK); (2) data fro m the interview activ ity was recorded and then mapped based on the misconception indicator code to the concept of limit, as shown in Table 2 below.

Domain Concept Indicator
The concept of prerequisite limit of functions Understanding the definition of a function Understanding the meaning of notation in a function

Drawing a graph of a function
Determining the value of a function Determining the area of origin of a function.
Determining the result area of a function.

T he concept of limit of functions
Understanding the` definition a limit of function Understanding the meaning of notation in the limit of functiona Understanding the meaning "close" to the limit of function Determining the right limit of a function Determining the left limit of a function Understanding the properties continuity of a function Determining the conditions for the existence a limit of function Determining the value limit of function

Misconception about Understanding the Concept Definition of the Limit on a Function
We know that the formal definit ion of a → ( ) = , is function approaches the limit as approaches c. thinking about limits is related to the behavior of a function near c not at c. And the function f does not require to be defined in c [34]. FI students understood the concept of lim → ( ) = , as approached , the value of did not approach to . Then, misconceptions occured when understanding the value of , where subjects understood that the value of L is right at ( ) and required the function to be defined in c. FI students understood that the value of left limit was the same as the right limit, which was 1, but due to an empty circle at the graph function, lim →1 ( ) did not have a limit value. So, it could be interpreted that FI students' understanding of lim → ( ) = , required a value in ( ), or in other words, the value of L was exact on ( ). FI students understood that the requirements of a limit function that has a limit were correct if the left limit and right limit were the same. M isconceptions occured about understanding left or right function limits that we undefined as = (blank round). FI students understood that functions did not have limit. To find out the misconceptions that occured in FI students, researchers provide questions as seen in Table 4.

Determining the Conditions for the Existence a Limit of Function (Left and Right-Hand Limit)
Student FI stated that the right limit has no limit value, because lim → ( ) = ( ) , wh ile ( ) = (−4) does not exist or is an empty circle in the graph function.

Understanding the Meaning "Close" to the Limit of Function
The misconception that occurred in the next FI student was interpreting the close meaning of a function. Students understood that in an empty circle, ( ) is undefined when = , and then it would define that the function has no limits. In question number 2 part " ", limit ( ) when approaches −2, students answered "its limit does not exist" for the reason that the left limit and right limit are the same (found). But because (−2) is an empty round, then the limit does not exist. So based on the previous description, it can be interpreted that FI students understand that a limit must be exact when = , ( ) exists. The concept of a limit that uses language approaches, does not have to be right at = not owened by FI student. FI students stated that there was no limit approaching infinity because they did not know where it ends.

Misconception about Sketching Function Graphs
Misconceptions that occurred in FD students was found in sketching graphs of functions. Students understood that in drawing a graph of a conditional function, the closed and opened intervals at the intervals/limits of a given function did not affect (continuous and non-continuous functions at = ) the graph functions. This could be seen in the sketch that drawn by FD students as follows,

Misconception about Understanding the Concept Definition of the Limit on a Function
The researcher asked the FD students about both of their knowledge or their understanding about the concept definit ion of the limit on a function. The FD students replied that the function is like a sequence. The value of L is the limit or the limit value. Like a sequence, if there is no limit the sequence has no limit. The students' statement could be seen in Table 6. The value of L is the limit or the limit value. Like a sequence, if there is no limit the sequence has no limit.

The Misconception about Determining the Function Value of a Function Graph
In determining the function value of a function graph, FD students experienced misconceptions about determining the function value of a function that had an asymptote line. Students understood that the asymptote line was the value of a function. M isconceptions could be seen fro m the answer sheet FD in question number 2 part "e". FD students answered that the value of (3) was 3. Based on the interview in Table 7, it appeared that FD students understood that the asymptote line was x = c, so f (x) = c. Determine the value of ( ) T eacher: Why is your answer (3) = 3 ? Student : Actually I'm confused, but my feeling is the dotted line at 3, so I answered Table 8. Determination of the conditions for the existence a limit function (left and right -hand limits)

Q uestions Student's Answers
Determine the limit value of Then the researchers assumed if at f (x), x would approached -4 fro m the right limit with a full round whether the limit had a value, and then the FD students answered no value for it. The researcher asked the value of limit f (x), when x approached -4, and FD1 replied 2 and 4.

Discussion, Conclusions and Suggestion
This study discusses the forms of misconceptions experienced by FD and FI students about understanding the concept of limit functions. [35] stated that it was not enough to know only about the misconceptions, but it must be understood in detail and strategies were needed to solve it. Based on the results, FD students experienced misconceptions starting from the learning materials delivery fro m the the teachers in learning act ivities. FD students experienced not only misconceptions of concept of limits, but also misconceptions about the prerequisite concept, such as: sketching graphs of functions. [36] stated that individual knowledge, related to understanding the problem situation, would cause variat ions in mental models that were formed and were constructed. The misconceptions stemmed fro m a wrong understanding of a concept. Sche ma Cognitive structure that has been formed into an understanding would continue to be used in the formation of further cognitive structure schemes.
The term cognitive was used to refer the processes that occured in the individual b rain, wh ich assisted in the process, manipulat ion, storing, and retriev ing information about the outside world [37]. Piaget in [38] also exp lained that schemata (schemas) were a cognitive structure. An individual could bind, understood, and responded to a stimulus due to the operation of this scheme. In other wo rds, cognitive skills we re the processes or skills that could help us think, solve problems, collaborate, and create schemes.
Incorrect understanding of the concept of prerequisites for FD students made an understanding scheme. When an external stimulus (new matrices schemes) was given, FD students would tend to assimilate it. Assimilat ion is the process of directly integrating new stimu li into established schemes [38], which means that FD students do not modify the schemes they have. For examp le, FD students understand the value of a function in c or f (c), and there is a new concept that is given that is the limit of infin ite functions. The understanding of the concept of infin ite functions formed into misconceptions in which FD students tended to assimilate the prior knowledge schemes that they had before, namely the concept of function values. FD students understood that lim →∞ ( ) then the function had no limit, and students assumed that the limit existed if the function was defined = . Misconceptions of FD students occurred because they were confused about the term "close" to the function limit. [39] stated that the delivery of ideas (ways of conveying concepts) fro m the teacher in learning that was beyond the ability of students' thinking could cause misconceptions about understanding the concept of limits. Th is is an important factor for teachers to know what students' abilities and characteristics are in learning The characteristics of FD students who tended to be influenced by complex contexts and tended to view a concept as a whole made acceptance of student concepts became unobstructed, so that understanding concepts could form misinterpretation because of the limitations of the appropriate concept structure. For examp le, one of th e causes of FD students' misconceptions was the students' cognitive style, which occurred in the concept of the definit ion of the limit function. FD students understood that lim → ( ) = , required using limits, and if there were no restrictions, then it was infinite and had no limit on that function. This ind icated that in the understanding of a complex nature, cognitive style influenced students on understanding the concept of "close" to the limit , thus causing misconceptions.
Misconceptions about the function limit concept also occurred in FI students. However, misconceptions about prerequisite material did not occur in FI students, and it appeared due to incomp lete reasoning and wrong intuition [31]. It is possible to happen because the FI students tend to be good at understanding complex concepts. As exp lained by [30] , FI students were not easily fooled by elements that were not relevant to the context. They also could determine the part -simple parts separate from the original context.
With the understanding of teachers related to detecting and preventing misconceptions, it was expected to minimize the occurrence of misconceptions in students. [40] stated that teachers would be able to arrange their instructions if they knew possible misunderstandings and misconceptions that students might have. M isconceptions could be minimized by providing visual displays and animations when learning limit functions [41]. However, [5] stated that the use of single animat ion might not be effective, because an explanation should be given wh ile using it. Visual learning method alone was not sufficient in developing the concept of limits.