## Epistemological Obstacles in Coming to Understand the Limit Concept at Undergraduate Level: A Case of the National University of Lesotho

##### Abstract

The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective. Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for./{x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate ./{O) as the limit value. This method is Within the context of a sequence everyday language acted as an epistemological obstacle in interiorising actions into processes. For example, in finding lim (_1)n ,the majority of x~oo n the subjects obtained the correct answer O. It was however revealed that such an answer was obtained by using an inappropriate method. The subjects substituted one big value for n in the formula. The result obtained was the number close to O. Then 0 was taken as the limit value because the subjects interpreted the word 'approaches' as meaning 'nearer to'. Other subjects rounded off the result. In everyday life when one object approaches
another, we might say that they are nearer to each other. It seems that in this case the appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually. Within the context of a sequence everyday language acted as an epistemological obstacle in interiorising actions into processes. For example, in finding lim (_1)n ,the majority of x~oo n the subjects obtained the correct answer O. It was however revealed that such an answer was obtained by using an inappropriate method. The subjects substituted one big value for n in the formula. The result obtained was the number close to O. Then 0 was taken as the limit value because the subjects interpreted the word 'approaches' as meaning 'nearer to'. Other subjects rounded off the result. In everyday life when one object approaches another, we might say that they are nearer to each other. It seems that in this case the subjects used this meaning to get 0 as the limit value. We also round off numbers to the nearest unit, tenth, etc. The limit value is however a unique value that is found by using the limiting process of 'tending to' or 'approaching' which requires infinite values to be
considered. Some are computed and others are contemplated. In constructing the coordinated process schema, f(x) ~ L as x ~ a, over-generalisation and everyday language were still epistemological obstacles. Subjects still perceived the limit value to exist where the function is defined. The limit was also taken as a bound, lower or upper bound. In a case where the function was represented in a tabular form, the
first and the seemingly last functional value that appeared in the table of values were chosen as the limit values. Limit values were also approximated. In constructing the coordinated process an ~ L as n ~ 00, representation, generalisation and everyday language also acted as epistemological obstacles. An alternating sequence was perceived as not one but two sequences. Since the subjects will have met situations where convergence means meeting at a point, as in the case of rays of light, a sequence was said to converge to a number that did not change in the given decimal digits. For example, the limit of the sequence {3.1, 3.14, 3.141, 3.1415, ... } was taken to be 3 or 3.1 as these are
the digits that are the same in all the terms. In encapsulating processes into objects, everyday language also acted as an epistemological obstacle. When subjects were asked what they understood the limit to be, they said that the limit is a boundary, an endpoint, an interval, or a restriction. Though these interpretations are correct they are however, inappropriate if used in the technical context such as the mathematical context. While some subjects referred to the limit as a noun to show that they refer to it as an object, other subjects described the limit in terms of the processes that give rise to it. That is, it was described in terms of either the domain process or the range process. This is an indication that full encapsulation of processes into objects was not achieved by the subjects. The role of language and symbolism has been identified in making different connections in building the concept of limit as: representation of mathematical objects, translation between modes of representation, communication of mathematical ideas, manipulation of surface or syntactic structures and the overcoming of epistemological obstacles. In representation some subjects were aware of what idea some symbolism signified while other subjects were not. For example, in the context of limit of a sequence, most subjects took the symbolism that represented an alternating sequence, an = (-lr, to represent two sequences. The first sequence was seen as {I, I, 1, 1,... } and the second as {-I, -1, - 1, -1, ... }.This occurred in all modes of representation. In translating from one mode of representation to another, the obscurity of the symbol lim/ex) = L was problematic to the students. This symbol could not be related to its X~a equivalent form lex) ~ L as x ~ a. The equal sign, '=', joining the part lex) and L does not reflect the process ofj{x) tending to L, rather it appears as if it is the functional value that is equal to L. Hence, instead of looking for the value that is approached the subjects chose one of the given functional values. The part of the symbol lim was a x~a
source of difficulty in translating the algebraic form to the verbal or descriptive. The subjects saw this part to mean "the limit of x tends to a" rather than seeing the whole symbolism as the limit of j{x) as x tends to a. Some subjects actually wrote some formulae in the place of L because of this structure, e.g., lim/ex) = 2x. These subjects x~a seemed to have concentrated on the part lex) = .... This is probably because they are used to situations where this symbolism is used in representing functions algebraically. In communicating mathematical ideas the same word carried different meanings for the researcher and for the subjects in some cases. For example, when the subjects were asked
what it means to say a sequence diverges, one of the interpretations given was that divergence means tending to infinity. So, over-generalisation here acted as an epistemological obstacle. Though a sequence that tends to infinity diverges, this is not the only case of divergence that exists and therefore cannot be generalised in that way. The manipulation of the surface structures was done instrumentally by some subjects. For 1 . ti di 1·.J x 2 examp e, m in mg im +29 - 3 ,urdmugri the mam.pu 1ati.on some subjiects 0 bttaaime d x.... o x 2 part of the expressions such as ~ by rationalising or .:;- by using L'Hospital's rule 2x x which needed to be simplified. Instead of simplifying the expressions further at this stage, the substitution of 0 was done. So, .o2. = 0 was obtained as the answer. This shows that neither the reasons for performing the manipulations, nor the process of rationalising for example was understood. The result was still an indeterminate form of limit. The numerator was also not yet in a rational form. In using language to overcome epistemological obstacles, subjects were exposed to a
piece of knowledge that falsified the knowledge they had so that they could rethink replacing the old with the new. In some cases, this was successful but in others, the subjects did not surrender these old pieces of knowledge. For example, when asked what they understood the 'rate of change' to mean, the majority of the subjects associated the rate of change with time only. However, when referred to a situation that required them to find the rate of change of an area with respect to radius, some subjects changed their minds but others did not. Those who did not change their minds probably did not make any connections between ideas under discussion. The implications for practice of the findings include: In teaching one should discuss explicitly how answers to tasks concerning limits are obtained. The idea of the limit value
as a unique value can only be recognized if the process by which it is obtained is discussed. It should not be taken for granted that students who respond correctly understand the answers. It is evident from the study that even when correct answers are given, improper methods may have been used. Hence, in investigating epistemological obstacles attention should also be paid to correct answers. Also beyond this, students should be exposed to different kinds of representation of the limit concept using simple functions and using a variety of examples of sequences. Words with dual or multiple meaning should also be discussed in mathematics classrooms so that students may be aware of the meanings they carry in the mathematical context. Different forms of indeterminate states of limit should be given attention. Relations should also be made between the surface structures and the deeper structures.