Employing Contextual Examples and Graphing Activities to Enhance Students’
Understanding of the Relationship Between Differentiation and Integration in Calculus
Kinley11,*,#, Wararat Wongkia22, Parames Laosinchai32,#
1
Master Program in Science and Technology Education. Institute for Innovative Learning,
Mahidol University. Thailand
2
Institute for Innovative Learning, Mahidol University. Thailand
*#kinleay@hotmail.com, #parames.lao@mahidol.ac.th
Abstract
Calculus is an important subject for science, engineering and other fields of studies but
phenomenally it is abstract and difficult to learn. Introductory calculus courses were basically
taught using traditional teaching methods focusing on algebraic computation. Students mostly
focused on rote learning and the relationship between differentiation and integration was
difficult to visualize. Due to the abstract nature of calculus, students found it difficult to
understand the concepts and teachers rarely taught calculus using activities to help visualize
the concepts. Therefore, contextual examples and graphing activities were developed based on
the learning cycle approach and the lesson was taught for an hour in the classroom. This study
investigates the effectiveness of the lesson in helping the students understand the relationship
between differentiation and integration in Calculus and to measure the students’ attitude
towards the learning unit. Fourteen undergraduate students from two colleges were selected
for the study. A pre-test was conducted prior to the implementation of the learning unit and
after the intervention of lesson, a post-test and an attitude test were conducted. The test scores
showed that there was significance improvement in the post-test scores compared to the pretest scores. It was also found that the lesson was effective and enriching.
Keywords: calculus, differentiation, integration, graphing activity
Introduction
A calculus course provides the basis for higher learning in several fields of studies, e.g.
physics, chemistry, biology, engineering and business in order to understand deeply the
concepts of rate of change of motion and its applications (1). However, a lot of students do not
achieve basic understanding of calculus and find that calculus is abstract and difficult. Several
studies revealed that students developed routine techniques and manipulative skills rather than
understanding of the theoretical concepts and its applications (2, 3, 4, 5, 6, 7, 8). Orton (2)
explained how students encountered the problems with reasoning behind integration especially
when the connection was made with definite integrals. He also described the difficulty of
understanding differentiation (3). Tall (9) also found that students formed a weak concept
image of differentiation and integration when the teaching was focused on algebraic
computation of symbols and notations.
Teaching that emphasized procedural methods led students away from the conceptual
understanding of differentiation and integration (10). Moreover, a vast number of calculus
textbooks are available, seemingly covering every conceivable approach, but many problems
remain and that make the understanding and teaching of calculus difficult (11). Textbooks used
in higher secondary levels contain a lot of abstract symbols and notations which are more of a
hindrance than a help to the learning. The symbols and notations are not explained in contextual
applications (12). Therefore, students followed the procedures rather than trying to understand
of the concepts and a wide range of problems in the textbooks were solved using memorized
formulas and procedural steps.
Several studies examined the learning of calculus and focused on students’ errors,
misconceptions, inflexibility and inability to make the connections between what they learn
and apply (2, 3, 9). The teaching methods need to be modified by focusing on activities based
on context, so that the students can visualize and make connections between the contextual
activities and the algebraic notations. Due to the applicability nature of differentiation and
integration, the visualization of the phenomenon problem is a key concepts for learning and
teaching process (13).
Milovanovic et al. (14) suggested to the using multimedia to enhance the conceptual
learning of calculus and to remove the difficulties encountered in learning calculus, because
the visualization helped to better explain mathematical ideas, abstract terms, theorems,
problems, etc. However, the relationship between differentiation and integration was not
focused after the students were taught the differentiation and integration. The students
recognized that “integration is the inverse process of differentiation” but they still encountered
the difficulty in explaining how differentiation and integration related algebraically and
graphically. This was because the teaching stressed the procedural steps to solve problems (15,
16, 17). Most students have learned calculus ‘how’ rather than ‘why’ due to extensive use of
notations and symbols in teaching and learning of calculus. The real meanings of symbols and
notations students learned in the classrooms are not interpreted explicitly in the context of real
world situations and students have vague concepts of algebraic notations in relation to
geometric interpretations (18). Mundy and Graham (19) also reported that the discrepancy
between the performances on the procedural items and conceptual ones was due to the separate
understanding of geometrical and algebraic context in calculus. Therefore, teaching calculus
using contextual examples and graphing activities and connecting activities to algebraic
notations would help students in visualization and easily apply in the real world situations.
This study focused on developing contextual examples and graphing activities based
on the learning cycle approach help establish the relationship between differentiation and
integration in calculus. The main objectivities of this study were to find out the effectiveness
of the developed learning unit on the students’ understanding of the relationship between
differentiation and integration in Calculus and to measure the students’ attitude towards the
learning unit. This study aimed to address the following research questions;
i). To what extent can the learning unit enhance the students’ understanding the relationship
between differentiation and integration in Calculus?
ii). What is the students’ attitude towards the learning unit?
The Learning Cycle Approach
The Learning cycle approach was seen as an effective hands-on, minds-on, inquirybased scientific pedagogy, especially for enhancing students’ understanding by the ways in
which they learned the nature of the world (20). The learning cycle approach can result in
greater achievement in learning, better retention of concepts, improved attitudes toward
learning subjects, improved reasoning ability, and superior process skills (21, 22). Atkin and
Karplus (23) believed that textbook teaching alone did not give students at any age the
integration of conceptual understanding and process skills. Therefore, the guided-discovery
system would make the complexity of concepts easier to learn by involving themselves in the
activities (24). Later, Lawson (25) recognized that teachers can give students new knowledge
but students must actively invent or generate the concepts. The learning cycle approach appears
to be a potential means of promoting students’ understanding of difficult mathematical
concepts besides difficult those scientific ones.
Lawson and Abraham’s model of learning cycle consists of three distinct phrases of
instruction. In Exploration Phase, students learn through their own actions and reactions with
minimum guidance in activities designed to expose students to the concepts. The students are
expected to raise questions they can’t answer with their present ideas or reasoning patterns. In
Concept Introduction Phase, the concepts are introduced and explained with help from the
teacher. The concepts are usually derived from the data or classroom discussions. In Concept
Application Phase, the students explore the usefulness of the concepts they have learnt and then
applied to new situations. Its range of applicability is extended to different contexts and
situations related to the concepts.
Methodology
1. Participants
Single group pre-test post-test research design was used in this study. A pre-survey test
for finding out the students’ basic knowledge in differentiation and integration in calculus was
conducted on preparatory college students. Out of forty five of these students, eight students
were selected after the pre-survey test. In addition seven voluntary first year undergraduate
mathematics major students also participated.
2. Procedures
A pre-test was conducted for 30 minutes prior to the intervention. The learning unit on
the establishment of relationship between differentiation and integration in Calculus was taught
for one hour. At the end of the instruction, a post-test and an attitude test were conducted.
A same set of four open-ended questions was used for both the pre-test and the posttest. The first question examined whether students could relate the graph of an anti-derivative
to that of its derivative as well as the units in those graphs in the straight-line motion context.
The second question examined whether students could relate the graph of a derivative to that
of its anti-derivative in the same context. The third and fourth questions examined the students’
conceptual understanding of integration and differentiation respectively.
The attitude questionnaire consisted of thirteen Likert-type items and three open-ended
questions. The purpose of the questionnaire was to find the students’ attitude towards the
relationship between differentiation and integration and towards the learning unit. The Likert
scale in the questionnaire included “1 = Strongly disagree”, “2 = Disagree”, “3 = Neutral”, “4
= Agree” and “5 = Strongly agree”.
A paired sample t-test was used for data analysis to determine whether significant
difference between the pre-test and post-test scores exists. The Cronbach’s Alpha reliability
coefficient of the post-test was 0.58. The frequencies of the responses to each questionnaire
item were separately tabulated and interpreted. The Cronbach’s Alpha reliability coefficient of
the questionnaire was 0.79.
3. Contextual examples and graphing activities learning unit
The Lawson-Abraham model of learning cycle was used to frame the development of
a learning unit on the relationship between differentiation and integration in calculus, which
employed contextual examples and graphing activities. The students were divided into groups
of 3–4 students. All the instructions for the group activities were provided on the worksheet
given to the students. The details of the activity in each phase of the learning cycle are described
below:
i) Exploration phase
In this phase, the context was a car moving for five hours at a constant speed. The
students were asked to sketch the graph of the constant speed and to divide the area under the
line into five equal parts (see figure 1(a)). They were then asked to find the unit and the meaning
of those areas, which should help them realize the graphical relationship between speed and
distance. The students then plotted the distances on another graph paper (figure 1(b)) and
compared the two graphs, which were actually the graphs of a derivative and its anti-derivative.
They were further asked to find the equations of the two graphs. Being more familiar with
algebraic notation, the equations should help them in confirming the relationship of the two
graphs. To help them see that the graphical relationship worked for non-integers as well, they
were asked to find the areas and the distances after 0.5, 1.5, 2.5, 3.5 and 4.5 hours. Finally, they
were asked to directly calculate the area under the line in figure 5(a) from t = 0 to t = 0.5 by
integration and to confirm that the result agreed with the area and the distance in the two graphs.
They were also asked to find the slope of the line in figure 2(b) at t = 0.5 hours.
Figure 1(a). Area under the line for t =1, 2, 3, 4,
and 5 hour.
Figure 2(a). Speed-time graph
Figure 1(b). Distance at t =1, 2, 3, 4, and 5 hours
Figure 2(b). Distance-time graph
ii) Concept introduction phase
From the exploration phase, students should begin to have an idea about the
relationship between differentiation and integration. The concept introduction phase should
help them formulate the idea more completely. To that end, the students were asked the
following questions. What do you get if you find the area under the graph in figure 2(a) by
integrating the equation of the line from t = 0 to t = 5 algebraically? What is the unit of the
area? And what does the unit of the area tell you? The students should be able to see that finding
the area under the line and integrating the line of the equation would give 100 km, which would
indicate the distance travelled by the car in 5 hours as shown in figure 2(b). Then, the students
were also asked the following questions. What do you get if you find the slope of the line
graphically and differentiate the equation of the line algebraically of the graph in figure 2(b)?
What is the unit of the slope? The students should be able to see that the finding the slope of
the line graphically and differentiating the equation of line algebraically would give 20 km/hr
which would indicate the speed of the car on the graph as shown in figure 2(a). Then, the
students were asked; do you see any relationship between the graphs in figures 2(a) and 2(b)
in terms of differentiation and integration in calculus? Explain? Now, the students should be
able to see the relationship between differentiation and integration graphically, algebraically
and contextually from the activity and to conceptually understand that integration is the inverse
process of differentiation.
iii) Concept application phase
In this phase, the context was still a moving car but accelerating at 2 m/s2 for 10 seconds
instead of travelling at a constant speed. The students were asked to sketch its graph, to find
the area under the line as shown in figure 3(a) and to plot the area, which was in fact the speed,
on another graph paper as shown in figure 3(b). Then, finding the area under the line in figure
3(c) would give the distance, whose graph is shown in figure 3(c). Finding the derivatives at
instances of time on the distance-time graph in figure 3(c) would give back the speed-time
graph in figure 3(b) and finding the derivative of the speed-time graph would give the
acceleration-time graph as shown in figure 3(a). The students could use both algebraic and
graphical methods of integration to find the speed and distance from the given acceleration and
the same was true for the reverse process of differentiation. It should be noted that the
nonlinearity of the distance-time graph could also be used to emphasize the instantaneous
nature of a derivative, both graphically and algebraically.
Figure 3(a). Acceleration-time graph
Figure 3(b). Speed-time graph
Figure 3(c). Distance-time graph
Results
1. Students’ performance
A paired sample t-test showed that the average post-test score was significantly greater
than the average pre-test score (see table 1). After the intervention, there was significant
improvement in students’ performance. The mean score in the pre-test was extremely low as
many students found the questions difficult which indicated that students had no conceptual
understanding in regard to differentiation and integration in general and to the relationship
between differentiation and integration in particular. Some of the students scored much higher
in the post-test despite the fact that the intervention lasted for only one hour.
Table 1. Paired sample t-test of pre-test and post-test result
Test
Mean
Standard deviation
t
Pre-test
1.75
0.83
5.46
Post-test
4.54
1.89
Number of students = 14 and total score of the test = 17
Sig. (2-tailed)
0.00
2. Students’ attitude towards the learning unit
Nine students indicated that they had learnt differentiation and integration in an
introductory calculus course before but five students were not sure about it. Thirteen students
found that the activities were interesting and enriching and helped them understand the
relationship between differentiation and integration better. The majority of the students found
that the lesson was well organized and the instructor encouraged the students in the learning
process.
Table 2: Students’ attitude questionnaire responses
Items
I have learnt about differentiation and integration in calculus before
My Mathematics teacher never taught me the relationship between
differentiation and integration in calculus
I understood the relationship between differentiation and integration in
calculus from the activities in the lesson
I found difficult to see the relationship from area under line in integration to
slope of line in differentiation
I found difficult to see the relationship using mathematical notation used in
differentiation and integration in calculus
Did the lesson improve your understanding of the topic?
The lesson was well organized in a way that helps me understand the
relationship between differentiation and integration in calculus
The lesson was useful to understand the differentiation and integration better
The instructor has been well-prepared for the class
The instructor has encouraged the students to participate actively in class
The instructor has made an effort to enhance learning
The instructor has been open to students’ opinion
I found the class very interesting and enriching
0 0 5 8 1
Average
Response
3.71
3 3 7 1 0
2.43
0 0 6 8 0
3.57
0 8 2 4 0
3.14
0 7 2 5 0
3.21
0 1 4 6 3
3.79
0 1 4 6 3
3.64
0
0
0
0
0
0
3.79
4.29
3.93
4.07
4.07
4.29
1 2 3 4 5
2
0
0
0
0
0
4
1
4
2
2
1
3
8
7
9
9
8
5
5
3
3
3
5
Discussion and Conclusion
The pre-test result showed that the students often lacked certain conceptual
understanding in differentiation and integration from traditional mathematics teaching. The
students learnt calculus without actually understanding differentiation and integration as well
as their relationship. The concepts of differentiation and integration were traditionally taught
by focusing only on algebraic methods in an introductory calculus course at the higher
secondary level. The students saw calculus as a series of process associated with algorithms
and could not apply the concepts in the contextual situations. This agrees with Tall’s (26)
findings that students instead of having conceptual view of the symbols and notations in
differentiation and integration, they focus only on a process-oriented view. The students
encountered difficulty in relating the functional notations of differentiation and integration to
the context of motion.
The average score of the post-test was significantly higher than that of the pre-test,
indicating that the developed learning unit could enhance the learning achievement of the
students. The hands-on graphing activities helped the students think logically, develop their
own reasoning skills, and ultimately invent their own concepts of the relationship between
differentiation and integration. Sokolowski et al. (17) and Orhun (18) also employed graphing
activities in contextual settings to enhance students’ understanding of calculus. The reliability
of the posttest was rather low due to the difficulty in learning calculus and to the openendedness of the questions.
The students found the activities interesting and enriching probably because they took
active roles in the lesson and felt motivated to learn calculus, hence leading to better
performance. To really understand the relationship between differentiation and integration,
students obviously need to understand both differentiation and integration which are
themselves based on more fundamental concepts like limit and continuity and discontinuity of
a function. To understand these concepts, the lessons must focus on context-based activities
rather than on algebraic methods (26). Students usually learn and retain better when they are
actively involved in the lessons and the concepts can be visualized. The coverage of the
fundamentals of calculus is a must before the learning of the relationship between
differentiation and integration can take place. Such a coverage will require a much longer
intervention duration. It is hoped that a coherent lesson on calculus based on contextual
activities will be developed.
Acknowledgement
The first author would like to thank Thailand International Development Cooperation Agency
(TICA) for providing scholarship to pursue Master Program in Science and Technology
Education in Mahidol University.
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