Students’ Learning Difficulty in Infinite Sequence and Series
Lisa and Khairani Idris
Department of Mathematics Education, State Institute for Islamic Studies of Lhokseumawe, Indonesia
Keywords: Calculus, Concept and principles, Difficulty analysis, Infinite sequence and series.
Abstract: This research aims to describe a difficulty encountered by students of Mathematics education department of
State Institute for Islamic Studies of Lhokseumawe in solving problems of infinite sequence and series,
particularly problems related to concepts and principles. Students’ performance is low in a test about infinite
sequence and series. The students’ answer sheets of the test are subsequently used as data sources for a
difficulties analysis by identifying the errors they made related to concepts and principles. Qualitative
descriptive analysis is applied as the analysis methods. The result shows that there are three types of
difficulties related to understanding concept and two types of difficulties related to understanding principles
which could be categorized as high-level difficulty. The difficulties related to understanding concepts
included: (1) identifying example and non-example of concepts; (2) using models, figures, and symbols to
represent concepts; and (3) translating from one presentation model into another presentation models. The
difficulties related to understanding principles included: (1) recognizing when a principle is required; and (2)
appreciating the roles of principles in mathematics. Based on the results, possible future research and
suggestions on effective teaching and learning for infinite sequence and series will be discussed.
1 INTRODUCTION
Calculus is a branch of mathematics which plays an
important role in the development of technology and
other knowledge fields. It is a means in the
knowledge world to elucidate changes (Purwanto,
Indriani, &Dayanti, 2005), which has been used in
various life aspects. Almost all applied science such
as engineering, agriculture, medical, pharmacy etc.
make use of calculus concepts in their development.
Besides, the social science such as economy,
psychology, etc. also require calculus concepts.
Despite the strategic role of the calculus in the
development of technology, many studies showed
students’ low performance and learning difficulties in
the course. More particularly, students’ learning
difficulties in calculus have been among the research
focuses in the area of mathematics education research
since around 1990s (e.g., Anderson &Loftsgaarden,
1987; Tall, 1993). Some topics which have been
found as common difficulties faced by students in
calculus include the concepts of limit (Maharaj, 2010)
and infinity (Kim, Sfard, &Ferrini-Mundy, 2005).
Several factors, included affective and cognitive
ones, have also been identified with relate to the
difficulties of learning calculus among university
students in Indonesia. The affective factors included
low of interest (Mutakin, 2015), metacognition,
motivation, and attitude (Hidayat, 2013) in learning
the course. On the other hand, the cognitive factors
included low of prior knowledge in algebra and limit
(Wahyuni, 2017), and in trigonometric concepts and
principles (Abidin, 2012).
The findings from the latter two studies showed
that the cognitive factors affecting the difficulties in
learning calculus were the low of prior knowledge in
basic concepts of mathematics. This might be
explained by the fact that the topics in mathematics
are interrelated to each other. The lack of
comprehension in one concept might affect the
learning in the more advanced concepts. Therefore,
students need to acquire complete knowledge in basic
concepts of mathematics, which are known as
competency standards and basic competency of
mathematics in the curriculum, in order to be able to
learn more advanced mathematics concepts. In fact,
many calculus students encountered difficulties in
comprehending learning materials in calculus(e.g.
Wahyuni, 2017)and in achieving the specified
competency standards and basic competency of
mathematics. In the calculus classes in Mathematics
Department of State Institute for Islamic Studies
284
Lisa, . and Idris, K.
Students’ Learning Difficulty in Infinite Sequence and Series.
DOI: 10.5220/0008520802840289
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 284-289
ISBN: 978-989-758-407-7
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
(IAIN)Lhokseumawe, for instance, there have been
always more than a half of students who could not
achieve the minimum mastery learning score of 65. If
such phenomena are ignored, students would
encounter more obstacles in other advanced courses
which require calculus as a prerequisite knowledge,
such as differential equation, real analysis, complex
analysis, etc.
Based on the above arguments, the goal of this
study was to analyze students’ learning difficulties in
the topic of infinite sequence and series, particularly
in solving problems related to understanding concepts
and principles. The topic of infinite sequence and
series was the beginning topic in the Advanced
Calculus Course for students of Mathematics
Education Department of IAIN Lhokseumawe. By
knowing the difficulties students encountered in the
topic, it may shed light on the appropriate steps to be
taken in improving the teaching and learning of the
topics.
2 METHODS
The qualitative descriptive approach was used to
describe students’ difficulties in learning the topic of
infinite sequence and series in this study. The
participants involved, and the instruments used in this
study are described in the followings.
2.1 Participants
The 28 third year students of Mathematics Education
Department of IAIN Lhokseumawe were involved as
the participants of this study. These students were
selected from the total of 38 students who were taking
the Advanced Calculus Course based on their low
performance (i.e., whose grades were C+ or below)
on a test in the topic of infinite sequence and series.
The Advanced Calculus course is a compulsory
course for these students, which consists of several
topics including Infinite Sequence and Series,
Positive Series of Integral Test and Other Tests,
Alternating Series, Absolute Convergence, Power
Series, Operations for Power Series, Taylor and
McLaurin Series, and Multiple Integral. The students
were separated in two classes, no difference in their
ability. One of the researchers was the lecturer of the
course. The research was conducted in the first
semester of academic year of 2017/2018.
2.2 Instruments
A test on the topic of infinite sequence and series was
used as the instrument in this study, which was
administered to the participants after they learned the
topic. The test consisted of six items designed to
examine students’ difficulties in understanding
concepts and principles related to the topic.
Six indicators of concept understanding and three
indicators of principle understanding proposed by
Cooney, Davis, and Henderson (1975) were referred
to analyze students’ answers to the test items. The six
indicators of concept understanding include:
▪ assigning, describing by using words and
defining a concept;
▪ identifying example and non-example of a
concept;
▪ using models, figures, and symbols to represent
a concept;
▪ translating from one representation model into
another representation model;
▪ identifying the characteristics of a given
concept;
▪ comparing and emphasizing concepts.
The three indicators of principle understanding
include:
▪ recognizing when a principle is required;
▪ using a principle properly;
▪ appreciating the roles of principles in
mathematics.
3 RESULTS AND DISCUSSIONS
The results of the analyses on students’ answers on
the test showed that students encountered different
level of difficulties in solving the given problems.
The percentages of students who performed errors in
solving the problems were used to categorize the
levels of difficulties for each indicator of
understanding concept and principle. We elaborated
these difficulties for the six indicators of
understanding concepts and those for the three
indicators of understanding principles in the
followings. The test item related to each indicator and
students’ errors in solving the item would be used to
discuss the difficulties.
Students’ Learning Difficulty in Infinite Sequence and Series
285
3.1 Difficulties in understanding
concepts
There were different levels and types of difficulty
encountered by students related to the six indicators
of understanding concepts.
3.1.1 The indicator of assigning, describing
by using words and defining concept
The test item used related to this indicator was: “What
are the definitions of infinite sequence and infinite
series?” The answer should be that infinite sequence
is a function whose domain is the set of natural
numbers. Infinite series is a sequence of numbers
from which a new sequence can be produced by
adding its elements successively.
Translation: Infinite sequence is all sets of natural numbers.
Infinite series is the numbers which are added together.
Figure 1: One of student’s error in assigning, describing by
using words and defining a concept.
One of the students answered to this question as
illustrated in Figure 1. From her answer, it could be
inferred that the student understood the idea of
infinite sequence and series. However, she had
difficulty to describe the definitions by using proper
mathematical words.
There were about half of students (50.98%) who
encountered such difficulty, which means that less
than a half of students who could describe the
definitions correctly. Thus, the indicator of assigning,
describing by using words and defining concept was
categorized into the medium level of difficulty.
3.1.2 The indicator of identifying examples
and non-examples of a concept
The test item for this indicator was: “Identify the
convergence of the sequence
”
A sample of student’s answer is given in Figure 2.
The error in the student’s answer in Figure 2 was that
directly replacing n with ∞, which might indicate that
the student did not have full understanding on how to
solve limit problems.
Figure 2: One of student’s error in identifying examples and
non-examples of a concept
There were many students (83.33%) in this study
encountered difficulty related to solving limit
problems. Therefore, the indicator of identifying
examples and non-examples of a concept was
categorized as very high level of difficulty.
3.1.3 The indicator of using models, figures,
and symbols to represent a concept
The test item related to this indicator was: “Determine
the explicit formula for
from
”.
Figure 3: One of student’s error in using models, figures,
and symbols to represent a concept
A sample of student’s answer is shown in Figure
3. The error in the above answer was that the formula
would result in the negative values. The student might
refer to the first number in the sequence in assigning
(-1) for the formula and ignore the negative or
positive value in the subsequent numbers in the
sequence.
Such kind of errors was performed by many
students in this study (80.39%). Hence, the indicator
of using models, figures, and symbols to represent a
concept was categorized as very high level.
3.1.4 The indicator of translating from one
representation model into another
representation model
The test item for this indicator was: “Change into the
mathematics words:
is convergent
and explain how if
is divergent.”
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286
Translation: If
is convergent, the value of
and
if
is divergent, the limit is not available.
Figure 4: One of student’s error in translating a model.
Figure 4 shows a sample of student’s answer. The
answer was improper, since there is no further
explanation what it means that the limit is not
available. There were many students (86.27%)
encountered difficulty in solving this problem, which
indicated that the indicator of translating from one
representation model into another representation
model was in the very high level of difficulty.
3.1.5 The indicator of identifying the
characteristics of a given concept and
recognizing the conditions specified by
a concept
The test item used related to this indicator was:
“Specify whether or not the
sequence
is monotonic.”
Figure 5 presents a sample of student’s answer,
which was actually true that the sequence is
monotonically increasing. However, to specify that
the sequence is monotonic, the definition of
monotonic sequence should be used. Based on the
answer, the student seems misunderstood about the
characteristics of monotonic sequence. It is not
always true that when a sequence is increasing then it
is monotonically increasing.
Translation: Since 2/3 is larger than 1/2 then the value is
monotonically increasing.
Figure 5: One of student’s error in identifying the
characteristics of a given concept and recognizing the
condition specifies by a concept.
There were more than a half of students (56.86%)
incorrectly answer this problem. Thus, the indicator
of identifying the characteristics of a given concept
and recognizing the conditions specified by a concept
was categorized as the medium level of difficulty.
3.1.6 The indicator of comparing and
emphasizing concepts
The test item related to this indicator was: “Prove that
is convergent to½.”
Figure 6: One of student’s error in comparing and
emphasizing concepts.
Based on the procedure used in the above answer,
it could be inferred that the student had lack of basic
knowledge related to solving limit problems. We
found that there were some students (54.36%)
encountered such difficulty with this item. Hence, the
indicator of comparing and emphasizing concepts
was categorized in the medium level of difficulty.
3.2 Difficulties in understanding
principles
There were different errors and different difficulty
levels encountered by students related to
understanding principles in each indicator. These
different errors are elaborated for the three indicators
below.
3.2.1 The indicator of recognizing when a
principle is required
The test item related to this indicator was: “State the
definition of the convergence of infinite sequence.”
One of the student’s answer is shown in Figure 7.
Translation: The term
are
Since the
terms have negative and positive values, then the sequence is
divergent.
Figure 7: One of student’s error in recognizing when a
principle is required.
We could see that the above answer did not
correspond to the definition of convergent and
divergent sequence. The student showed an example
of infinite sequence to explain the meaning of
divergent sequence, which might indicate that she did
not understand how to state the definition
There were almost seventy percent of students
(68.62%) showed errors in solving this problem.
Students’ Learning Difficulty in Infinite Sequence and Series
287
Hence, the indicator of recognizing when a principle
is required was categorized in high level of difficulty.
3.2.2 The indicator of using principles
properly
The test item related to this indicator was: “Specify
whether the sequence
is convergent or
divergent.”
Figure 8: One of student’s errors in using principle
properly.
Figure 8 presents one of the student’s answer to
the item. The error shown in the answer is related to
the incorrect defining of
. It should
be
, not –1.
There were some students (59.80%) found
difficult in solving infinite sequence and series with
regard to this item. Thus, the indicator of using
principles properly was in the medium level of
difficulty.
3.2.3 The indicator of appreciating the roles
of principles in mathematics
The item related to this indicator was the same
item used for the previous indicator. Figure 9 showed
one of student’s answer. From the above answer, the
student solved the limit correctly. However, when
deciding whether the sequence is convergent or
divergent, the student did not apply the definition of
convergent sequence, which resulted in the incorrect
of drawing conclusion about the convergence of the
sequence.
Figure 9: One of student’s error in appreciating the roles of
principles in mathematics.
There were many students (70.58%) performed
error related to this indicator. In this case, they found
difficult in solving infinite sequence and series by
using definitions and theorems which have been
learned. Therefore, the indicator of appreciating the
roles of principles in mathematics was categorized in
the high level of difficulty.
3.3 Discussions
The analysis results presented in the previous
subsections showed different types and levels of
difficulty encountered by students in solving
problems in the topic of infinite sequence and series.
The error performed by most of students related to
understanding concepts was when they need to
determine an explicit formula of
. When the
formula is incorrectly determined, specifying its
convergence or divergence will be also incorrect.
In specifying convergence and divergence of
in test item 2, many students encountered difficulty
with finding limit value. It seems that they did not
have adequate knowledge related to solving limit
problems, which should has been learned in Calculus
I. Student’s difficulty related to limit problems has
been acknowledged in several studies (e.g.,Wahyuni,
2017). Therefore, in order to overcome this difficulty,
more attention in the learning of topic of limit in
Calculus I is required. For instance, conceptual
understanding of limit can be developed by using
algorithmic contexts (Pettersson & Scheja, 2008).
The use of principles in each step of solving a
mathematics problem is mostly interrelated. Thus, the
incorrect use of the principle in the previous step
would continue to the following steps. Despite the
concepts and principles of infinite sequence and
series given in the test items in this study have been
taught to students, many students failed to recall and
use them in solving the test items. This might be the
result of the students not having appropriate mental
structures (Maharaj, 2010) or that they had the
instrumental understanding (Skemp, 1976) of the
concepts and principles related to infinite sequence
and series. Since the desired goal of calculus should
be relational understanding (Skemp, 1976), providing
students with a range of experiences in the Advanced
Calculus class that develop the ideas of the topic so
that they both knows and understands might be an
alternative way of teaching and learning the course.
Based on the findings, the effective ways to
facilitate and improve students’ conceptual
understanding of limit might be one of the topics of
interest for future research in the area of mathematics
education. Although such study has been conducted
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288
by Pettersson & Scheja (2008), more recent studies
involving diverse participants from different cultures
could enrich and contribute on a more fruitful
discussions. Other directions for future research
might be about how to improve students’ memory
related to definitions and theorems and their skills in
using definitions and theorems to solve problems in
calculus.
4 CONCLUSIONS
The results of this study have shown the different
types and levels of difficulties encountered by the
second-year students of Mathematics Education
Department of IAIN Lhokseumawe in solving
problems in the topic of infinite sequence and series.
The difficulties included three types of difficulties in
understanding a concept and two types of difficulties
in understanding a principle which can be categorized
into high level of difficulty. The difficulties in
understanding a concept included the difficulty in
identifying example and non-example of concepts
(83.33%), the difficulty in using models, figures, and
symbols to represent concepts (80.39%), and the
difficulty in translating from one presentation model
into another presentation model (86.27%). On the
other hand, the difficulties in understanding a
principle included difficulty in recognizing when a
principle is required (68.62%) and the difficulty in
appreciating the roles of principles in mathematics
(70.58%).
The errors performed by most of the students related
to understanding a concept was due to their low of
prior knowledge related to solving limit problems and
in determining the convergence or divergence of an
infinite sequence or series. Besides, they could not
recall the definitions and theorems related to infinite
sequence and series. With relate to understanding
principles, many students encountered difficulties in
using definitions and theorems which have been
learned in solving problems.
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