Visual Thinking of Definite Integral
Visual Thinking of Definite Integral
Visual Thinking of Definite Integral
4, 476-482
Available online at http://pubs.sciepub.com/education/3/4/14
© Science and Education Publishing
DOI:10.12691/education-3-4-14
Department of Electrical Engineering, Ming Chi University of Technology, New Taipei City, Taiwan, ROC
*Corresponding author: huangch@email.mcut.edu.tw
Received March 18, 2015; Revised April 02, 2015; Accepted April 08, 2015
Abstract Visualization as both the product and the process of creation, interpretation and reflection upon pictures
and images, is gaining increased visibility in mathematics and mathematics education. The use of diagrams to
visualize definite integral concept, however, is problematic for many students and may actually hinder their
problem-solving efforts. The purpose of this study, not only extends our understanding of students’ difficulties and
strengths associated with visualization but also identifies types of visual image they utilized while solve integral
problems. Through the detailed analyses of students' work and verbal protocols, the students with high visualization
ability use of imagination images in high percentages along with algebraic representations and linking these two
representations lead to the success of problem solving. The students with low visualization ability use of memory
images. It is discovered that students can produce imagination images that play a significant role in a problem
solving process. As such, a process of visualization allows an articulation between representations to produce
another representation that could help students solve given problems.
Keywords: calculus students, definite integral, representation, visual thinking, visualization
Cite This Article: Chih Hsien Huang, “Calculus Students’ Visual Thinking of Definite Integral.” American
Journal of Educational Research, vol. 3, no. 4 (2015): 476-482. doi: 10.12691/education-3-4-14.
Representation is an indispensable tool for presenting of symbols. A growing body of research supports the
mathematical concepts, communicating and considering or assertion that understanding of mathematics is strongly
thinking. Hiebert and Carpenter [22] assessed students’ related to the ability to use visual and analytic thinking.
understanding of concepts based on the relationships Researchers contend that in order for students to construct
between the representations they created. They contended a rich understanding of mathematical concepts, both visual
that “the mathematics is understood if its mental and analytic reasoning must be present and integrated
representation is part of a network of representations. The [2,24].
degree of understanding is determined by the number and According to Duval [6], visualization can be produced
the strength of the connections (p. 67)”. This perspective in any register of representation as it refers to processes
supports that proposed by Duval [6]. Duval maintained linked to the visual perception and then to vision.
that the process of mathematical thinking required not Zimmerman and Cunningham [26] contended that the use
only the use of representation systems (which Duval of the term “visualization” concerned a concept or
called registers) but also cognitive integration of problem involving visualizing. Nemirovsky and Noble [16]
representation systems. Based on Duval’s analysis, defined visualization as a tool that penetrated or travelled
learning and comprehending mathematics require back and forth between external representations and
relatively similar semiotic representations. From this learners’ mental perceptions. Goldin [8] and Hitt [12] both
perspective, the understanding of a mathematical concept emphasized the relationships among representation,
is built through tasks that imply the use of different mathematical visualization, and conceptual understanding.
systems of representation and promote the flexible Dreyfus[5] contended that what students “see” in a
coordination between representations. Therefore, learning representation would be linked to their conceptual
mathematics implies “the construction of a cognitive structure, and further proposed that visualization should be
structure by which the students can recognize the same regarded as a learning tool. Noss, Healy, and Hoyles [17]
object through different representations” [6]. described mathematical thinking as being characterized by
The learning of calculus movement emphasizes the use the ability to move freely between the visual and the
of multiple representations in the presentation of symbolic, the formal and the informal, the analytical and
concepts—that concepts should be represented the perceptual, and the rigorous and the intuitive.
numerically, algebraically, graphically, and verbally Visualization involves both external and internal
wherever possible—so that students understand representations (or images), and thus following Presmeg
connections between different representations and develop [19], we define visualization as processes involved in
deeper and more robust understanding of the concepts constructing and transforming both visual images and all
[10,13]. of the representations of a spatial nature that may be used
The essence of the concept of integrals (including other in drawing figures or constructing or manipulating them
mathematical concepts as well) is that the process concept with pencil and paper. This definition emphasizes that in
and object concept can be presented by connected but mathematical thinking and problem solving, an
different representations. A number of studies have appropriate graph can be drawn to represent the
indicated that the representations used by students to solve mathematical concept or question, and that the graph can
an integral problem are related to the meanings they be used to understand a concept or as a problem-solving
attribute to the concept of integrals [4]. The graphical tool. In this study, we investigated the visual images that
representation of definite integrals is typically used in students used to resolve specific problems and how they
calculations that involve areas under a curve, whereas managed given visualizations.
numerical representations are used for Riemann’s
cumulative addition problems [20]. Solving integrals
using common integration techniques demonstrates the 3. Method
need for symbolic representations.
In this study, we situated our investigation of 3.1. Participants and Instruments
representation theory within the context of integrals.
The 15 first-year engineering students who participated
Specifically, we examined students’ ability to use the
in this study were enrolled at a university of technology
relationship between representations to solve integral
and had learned the basic rules of integration using
problems. In our specific case - the understanding of the
primitives, as well as their relationship to the calculation
concept of integral- research conducted with this
of a number of areas under curves. These students’
representational approach highlights as a cause of these
calculus grades in the top 10% among 352 students. The
difficulties the lack of coordination between both the
instruments used for data collection were a questionnaire
graphic and algebraic representations and the
containing problems and interviews. The questionnaire
predominance of the latter in the students’ answers. This
comprised five problems in definite integral (Figure 1),
leads us to pay special attention to the use of the graphical
some of which were referenced from other studies [7,15].
representation and to visualization.
These problems enabled the students’ performance
regarding the visual thinking to be analyzed. The results of
2.2. Visualization in Mathematics Learning the questionnaire necessitated further investigation into
Visualization is a critical aspect of mathematical the visual thinking of students. The clinical interviews
thinking, understanding, and reasoning. Researchers argue were carried out after the answers to the problems had
that visual thinking is an alternative and powerful resource been analyzed [9]. Each interview lasted about 40-50
for students to do mathematics, it is different from minutes and was video- and audio-taped. In order to
linguistic, logico-propositional thinking and manipulation prepare the script for the interview, we analyzed the
478 American Journal of Educational Research
Tasks 4 and 5, however, he could not solve the problems J: (He think about 15 seconds) I can not solve this
using graphical representations. problem by the graph.
R: Can you solve this problem by drawing a graph? R: OK, can you evaluate the sum of the areas of the two
P: No, I do not know what is f(t). triangles?
R: If I assume that the graph of f(t) is a straight line, can J: Of course, 25/2 plus 1/2 equal to 13.
you solve this problem? R: Good, can you tell me what the relationship between
P: (draw a straight line on the coordinate plane) But I do the area of triangles and the integral?
not know how to draw the graph of f(t-1), and I think that J: I know the integral is the area, and the results are the
the graph has nothing to do with this problem. same for this problem, but this problem is to evaluate the
R: Do you know the relationship between the integral and integral…. Do you mean that I can evaluate the area
area? instead of the integration?
P: I know that I can use the integral to evaluate the area. R: Why not? Just you have said that the integral is the area.
R: But you say that the graph has nothing to do with this J: But…
problem. R: Can you tell me what do you think about the integral is
P: Because this problem is to evaluate the integral not the the area?
area. J: I do not know exactly, I just remembered what the
In Task 3, he stated that the proposition was true and teacher said.
provided specific examples of functions as evidence The students at the non-visual level generally used a
without giving graphic representations, failing to provide single representation, and symbolic representation was
suitable justifications. He provided a specific example that used to solve all types of problems. This indicates that
defined two functions f ( x=
) x 2 + 1 and g ( x ) = x 2 , and students consider symbolic representation as a support
tool. Additionally, students in this group were inclined to
calculated two integrals between x=1 and x=2 to obtain rely on analytical thinking instead of visual thinking. This
10/3 for f and 7/3 for g. Subsequently, the interview leads to they tend to be cognitively fixed on standard
progressed as shown below. figures and procedures instead of recognizing the
R: Can you provide a geometric or numerical example? advantages of visualizing the tasks. Presmeg [19] showed
P: (Draws the two curves of f ( x=
) x 2 + 1 and that, visualization could be a hindrance for solving a
mathematical problem, especially when a mental image of
f ( x=
) x 2 + 1 ) Like this? a specific subject controls the student’s thinking. In this
R: Can you do this in graph form without an equation? group students’ cases the mental image of standard figure
P: How do I draw graphs without equations? has dominated their thinking when trying to draw a figure
R: OK, can you draw graphs represent the definite integral? to solve problems.
P: No, I do not know how to do it.
Drawing graphs based on the two functions provided, 4.2. Visual Thinking of Definite Integral at
Porter was unable to think using graphical representations the LV Level
without algebraic formulae. Therefore, Porter’s thinking
style tended to analytic but not visual, he do not The next level of visual thinking of the concept of
understand algebra and geometry as alternative languages, definite integral regarding the existence of cognitive links
represent and interpret problem graphically, and and awareness of these links is the local-vision level. We
understand mathematical transformations visually. Porter categorized nine students into this group. These students
used visual representations are memory image, and the understood the relationships between representation
graphical representations in his memory as standard systems and could change or transfer the representations
graphics. An image of a standard figure may cause in some of the representation systems. However, these
inflexible thinking which prevents the construction of a students had difficulty coordinating these relationships.
non-standard diagram.
Another student, John, was also categorized into the
non-visual level. In Task 5, John defined the step function
and then established two integrals ([–3, 0] and [0, 3]).
When he was asked:
R: Why do you separate the integral into these two
integrals?
J: Because of absolute value, one integral represents the
value to the left of 0, and the other represents the one to Figure 2. Helen’s problem-solving process for Task 1.
the right of 0.
Helen was one of the students in this group. She could
R: Can you draw the graph of f ( x )= x + 2 ? use correct symbolic representations to perform
J: (Drawing the graph correctly) Is it right? mathematical thinking and could manipulate the area
R: Good, according to the graph, can you examine the using graphical representations according to the changes
integrals that are right or not? in integral symbols in Tasks 1, 4, and 5. Consider Task 1
J: (He think about 25 seconds) Oh, I know, the interval of for example, Helen assumed that F ′ ( x ) = f ( t ) , then
integrals should between [–3, -2] and between [-2, 3].
2
R: Good, complete this problem.
J: (After 87 seconds) It is 13.
∫1 f ( t )dt = F ( 3) − F (1) = 8.6 . Consequently,
4
R: Good, can you solve this problem by the graph that you
have drawn? ∫2 f ( t − 1)dt = F ( 3) − F (1) = 8.6 . Figure 2 shows Helen’s
480 American Journal of Educational Research
graphical representation. She knows that because the two this group used visual representations are induced image,
integrals represent the same area, so the two integrals have they induced the visual images mainly from the analytic
the same values. thinking. Although their visual thinking inclines toward
However, for Task 3, she says that the proposition is local not global thinking, this restricted visualization
false and gives graphic representations (Figure 3) but fails actually hinders their solving of the tasks. Additionally,
to make suitable justifications. their chosen visualization only reflects one aspect of the
R: Can you explain what you think of this task? integral concept, which has a number of significant
H: The area enclosed by f, x = a, x = b, and the x axis is consequences on their ability to solve the other tasks.
greater than the area enclosed by g, but the function value According to Zimmerman [25], an important component
of f is smaller than g. of visual thinking is the ability to recognize that an answer
R: But the question involves the integral of f being greater obtained algebraically is false based on geometric grounds;
than that of g. the interviews show that this component is lacking in the
H: The integral value is the area; therefore, a greater minds of many students.
integral means a greater area.
R: Does this has any relevance to the area being above or 4.3. Visual Thinking of Definite Integral at
below the x axis? the GV Level
H: It is irrelevant to the area being above or below the x
axis. Two students were categorized into this group. These
students could recognize the relationships among
representation systems and convert representations
between representation systems. In Tasks 1, 4, and 5,
Keven used correct symbolic representations to perform
mathematical thinking. He also manipulated the area using
graphical representations according to the changes in
integral symbols. Consider Task 4 for example, Keven
actually employed three methods to solve the problem.
The first method was the standard algorithm
∫1 ( f ( x ) + 2=
)dx ∫1 f ( x )dx + ∫1 =
5 5 5
2dx 18 ; the second
method was the mean value theorem for integral
Figure 3. Helen’s problem-solving process for Task 3