2013. Vol.4, No.4, 254-258
Published Online April 2013 in SciRes (http://www.scirp.org/journal/ce) DOI:10.4236/ce.2013.44037
Calculus for Coloring
Sergio A. David1, Carlos A. Valentim Jr.1, Juan L. Linares2
1Department of Biosystems Engineering, University of São Paulo, Pirassununga, Brazil
2Department of Basic Sciences, University of São Paulo, Pirassununga, Brazil
Received December 21st, 2012; revised January 20th, 2013; accepted February 7th, 2013
Copyright © 2013 Sergio A. David et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
Professors who administer the differential and integral calculus classes in the hard sciences courses and in
the most diverse kinds of engineering, not rarely, are faced by the lack of motivation to learning and dif-
ficulties in absorbing the concepts of the discipline. With the purpose of contributing to the teach-
ing-learning process related to calculus classes, this paper looks for synthesizing a process of elaboration
and trial of an artwork destined for coloring, which content is specifically related to the discipline “Cal-
culus with Applications IV”, from the Biosystems Engineering course, of the University of Sao Paulo
(USP)-Brazil. The material, always prepared in the form of pairs of pages, one displaying the picture to be
colored (the artwork) and the other one like its twin (the text), was individually rated by the apprentices
students. The obtained results point to the possibility of facing the paradigm and showing that projects
like “calculus for coloring” are not only possible, but also a quality complement in the learning-teaching
Keywords: Calculus; Coloring; Education; Learning Process; Mathematics
The mathematics education is highlighted by its complexities
and challenges. It is possible to think about different views/
aspects of mathematics education implying challenges that can
to call attention as a common theme. This is addressed by
(Chapman, 2012). Chapman makes an analysis based on arti-
cles that deal on different challenges involving mathematics
education such as: implementing a new curriculum (Ponte,
2012), to establish connections between real-world contexts and
mathematics (Taylor, 2012), sustainable change in learning
process (Zoest et al., 2012) and implementing of technology
(Doğan, 2012). Ponte’s article (Ponte, 2012), provides insights
about a national professional development program that was
used in the preparatory stages of the introduction of a new
mathematics curriculum in Portugal indicating two main pur-
poses for mathematics education: to promote positive attitudes
and appreciation of mathematics that emphasizes three mathe-
matical processes: problem solving, mathematical reasoning
and mathematical communication. Edd Taylor (Taylor, 2012),
addresses connections to students’ out-of-school experiences
and provides insights of challenges with teachers’ learning and
a possible way of addressing them. On the other hand, (Zoest et
al., 2012), provides insights regarding sustainability of learning
trough the use of sociomathematical and professionals norms.
Describes a study that investigated the extent to which three
sociomathematical and four professional norms intentionally
fostered in an early mathematics pedagogy course re-emerged
in a similar context. In his article, (Doğan, 2012), Doğan deals
with prospective teachers views of technology (computer in
mathematics education) in their learning and teaching. He con-
sidered mathematics teachers from two different universities in
Even in accordance, in greater part, with Chapman’s analysis,
our experience in Brazil has permitted to note a low motivation
generalized of the mathematics apprentices since primary school
and even in university degree. For this reason, we have re-
searched about the stimulation and motivation in mathematics
We believe that our experience contributes with the chal-
lenge of provide a complementary alternative in order to facili-
tating learning in mathematics education. In this paper, we
applied the idea, particularly, involving the differential and
integral calculus. However, we believe that the experiment
could be applied with a similar success to other levels of
mathematics education. This is the idea or experiment, “Calcu-
lus for Coloring”, that can be extended in the near future, for a
“Mathematics for Coloring”, also considering other level of
The practice of coloring has been proved itself a very effi-
cient method on facilitating learning and memorization, such as
a visualization method. Visualization seems indeed to result in
a positive effect on the learning of students (Brandt et al., 2001).
In fact, there are a great number of the works involving re-
search on visualization in learning and teaching Mathematics.
In the Handbook edited by Gutierrez and Boero (Gutiérrez &
Boero, 2006), for instance, N. Presmeg wrote a chapter dedi-
cated exclusively to this theme. Mayer (1997) has elaborated
the dual coding theory in order to explain why pictures support
under specific conditions the understanding of technical or
physical phenomena. Mayer assumes that verbal and pictorial
explanations are processed in different cognitive subsystems
and that they result in the construction of different mental mod-
els. In the Mayer’s work, students who received coordinated
Copyright © 2013 SciRes.
S. A. DAVID ET AL.
presentation of explanations in verbal and visual format (multi-
ple representation group) generated a median of over 75% more
creative solutions on problem-solving transfer tests than did
students who received verbal explanations alone (single repre-
We believe that the using of a color code raises the con-
sciousness of connections, makes the visualization of com-
plexes designs easier and can contribute for, the almost always
required, process of abstraction of concepts and ideas through
When the act of coloring is requested to the apprentice, as a
complementary or additional job in the process of learning
about a determined concept, such act implies in a really active
way of consolidating these concepts. With hands in activity,
there is a large chance of focusing attention in a much more
intense way while concentrating on simple topics, on at a time.
At the same time, the mind associates shapes with concepts,
transferring them from the short time memory to a more lasting
It is well known that we are visual apprentices, in other
words, a lot of what we learn originates from our sense of vi-
sion. Even tasks like, for instance, those executed by a sports-
man, must be “visualized” before being fulfilled.
That is the reason why it is so important to “visualize con-
cepts” to understand and consolidate contents of a pre-estab-
lished discipline, for example.
It is possible to find in the Brazilian publishing market three
excellent books of the series “the coloring book”: Anatomy
(Elson & Kapit, 2001), Physiology (Kapit, Macey, & Meisami,
1999) and Microbiology (Alcamo & Elson, 2004). On this
works, authors manage to teach successfully to the lay public
the essence of math concepts with a series of sheets and posters.
Traditionally, courses as Anatomy or Microbiology can be
thought as more “visual” than courses like Physics and, above
all, Mathematics. Notwithstanding, an example that other courses
or disciplines are capable of “visualization” is the case of the
volume “The Physiology Coloring Book” already mentioned
above. Physiology may be understood, in certain way, as the
application of Physics, Chemistry and Math to Medicine. In this
case, is not about just describing static figures as in Anatomy. It
is substantial to explain process as forces, chemical reactions,
fluxes, states, signals and feedbacks.
In this work, we challenge the paradigm and show that a
“Calculus for Coloring” isn’t only possible as it can also be a
quality complement on the teaching-learning process.
If we compare old math textbooks to recent ones, it will be
immediate to note that, a major increase in the number and
quality of pictures to motivate the readers. In this project, we
believe that the pictures should be almost 50% of the content of
Having this in mind, the objective of this work is developing
teaching and researching skills and then testing the graphical
for coloring material created and the explanatory text about the
topics of the discipline “Calculus with Applications IV” of the
Biosystems Engineering course, at the Pirassununga campus,
from University of São Paulo (USP-Brazil).
Furthermore, this work has as its main goal to contribute to
the developing of knowledge in the teaching-learning field in
Calculus. On the edge, this work may be faced as an attempt of
planting an embryo that would contribute to the Brazil’s im-
perative challenge to improve the quality of teaching, especially
in mathematics education.
Selected Topics and Layouts
In the Biossystems Engineering course of the University of
São Paulo, the discipline “Calculus with Applications IV” has
on its syllabus, the study of ordinary differential equations (Boyce
& Di Prima, 2008)—solved by series of potentials method—as
well as the study of partial differential equation (Kreyszig, 2006)
and the Laplace transform. In the preparation of this project,
from the contents of the referred discipline, the main topics
were selected and subdivided in eleven parts, which were named
“sheets”. A sheet was composed of two pages.
The left page displays the illustration to be colored about the
topic of the referred sheet (the artwork page). The right page
(the text page), on the other hand, has a “summary text” con-
taining the main information about the topic covered. Each
sheet refers to one of the selected topics of the discipline,
1) Laplace Transform;
2) Special functions;
3) Power series;
4) Fourier series;
5) Euler Equations;
6) Legendre Equations;
7) Bessel Equation;
8) Partial Differential Equations;
9) Heat Equation;
10) Wave Equation;
11) Laplace Equation.
The transfiguration of mathematical concepts relatively ab-
stract in illustrations which can be colored and quickly under-
stood didn’t come as a trivial task. Aiming the maximization of
the receptivity by its potential users (the apprentices), the mate-
rial was built in order to attract the more attention and interest
In a way of attending those demands, a bibliographical re-
search (with emphasis on the classical books of calculus) was
Some sheets created on this project were submitted to stu-
dents that attended the referred discipline on the last semester,
for the purpose of evaluating the efficiency of the work. Armed
with crayons, the students painted each illustration and read all
the informative texts, establishing relations between the con-
tents and, at the very end, filled a form writing their individual
opinions about the project. The results of this form may be
observed on details in Section 3.2.
In the confection of each sheet, the start point was always the
artwork (illustration). From the main concept about the subject
matter, a sketch was drawn with the intention of expressing the
central idea of the topic or some important particularities. At
the same time that the art confection was made, annotations
were also written with the objective of connecting it to the art-
After the sketch of the artwork, its twin part was elaborated.
That means the written part that, besides bringing the most
important information as objectively as possible relatively to
Copyright © 2013 SciRes. 255
S. A. DAVID ET AL.
Copyright © 2013 SciRes.
the sketched artwork, also brings orientations related to the
painting procedure suggested to the subject matter.
e) Did you know? (or Stay alert!)
The last section is practically a footnote, with different colors
and font. The section “Did you know?” always contains some-
thing relevant to the apprentice, many times connecting the
subject matter in the sheet to different topics located in other
sheets. This section can also have the name “Stay alert!”. On
this case, the section contains tips about other contents which
may improve performance of the apprentice on the subjects
referred to that sheet.
The written part, as the informative text present in each sheet,
was organized in five sections (questions), which are: a) What
is it? b) How does it work? c) What is it for? d) Curiosities, e)
Did you know? (or Stay alert!).
Each one of these questions is described as follow:
a) What is it?
The answers to this question aimed to provide the student, in
a few lines, the basic concept and the nature of the topic cov-
ered by the sheet. This way, the apprentice gets a quick notion
about the subject immediately in the first lines of reading.
It’s worth remembering that distributed on these five sections
there are always information connecting the illustration of the
left page (the artwork) to the corresponding theory on the right
page (the text). In Section 3.1, both parts can be viewed in detail.
b) How does it work?
The second section of the informative text is, frequently, the
longest and most elaborated. In this section, the most important
equations and mathematical definitions of the subject matter are
In this section, two sheets are presented, as examples, as well
as the results of the experimental evaluation submitted to the
students of the referred discipline. More specifically, in the
Section 3.1, two examples of the set—artwork and text—in its
final state (already colored) may be viewed. In Section 3.2, in
the means of Figures, a brief of the impression and evaluation
that the students of Biosystems Engineering, of the University
of São Paulo, provided about the material.
c) What is it for?
In this section, an answer to one of the most frequent ques-
tionings of the students of the courses of Engineering is pro-
vided. Allowing the students to get to know the usefulness of a
determined subject may be and additional motivation for
In this section, like its own name suggests, complementary
information about thetopic may be found. Usually, it’s in this
part of the text that the pioneer author(s) of the subject of the
sheet is/are reminded, with a short description of her major
works. Other characteristics about the subject matter may be
found in this section.
The Sheets-Two Examples
We would like show two sheets examples provided to the
apprentices, such as: Fourier Series (Figure 1) and Heat Equa-
tion (Figure 2).
Example of the sheet “Fourier Series”.
S. A. DAVID ET AL.
Example of the sheet “Heat Equation”.
In order to test the efficiency of the completed job, a copy of
the sheets referring to the “Fourier Series” and “Heat Equation”
was given to the students of the course of Biosystems Engi-
neering, of the University of São Paulo, that were attending to
the discipline of “Calculus with Applications IV”. Together
with the sheets, a questioning with three questions regarding
the quality of the work was also given to the students. It is im-
portant to highlight that, to the volunteer students nothing was
offered in exchange of the notes of their impressions and
evaluations. It’s worth saying also that the professor who coor-
dinated this work and the professor of the referred class that
evaluated the material were different.
The obtained results on this evaluation are listed below:
Question 1: Did the illustrations presented in this mate-
rial contribute to the absorption of the concepts?
Question 2: Are the pictures chosen appropriately to il-
lustrate the proposed concepts in each topic?
Question 3: Would the idea of implementing this com-
plementary material with all topics ministered in calculus
The answers obtained and represented by the Figures 3-5
may be synthesized in a pretty objective way. Almost 85% of
the students believe that the sheets contribute significantly to
the absorption of contents, what leads us to conclude that the
method of coloring has great chances of obtaining efficiency in
its objective of improving and making easier the absorption of
basic concepts. Half the students (50%) classify the choice of
the images presented in the sheets for the representation of the
concepts as “Excellent”, while 36% acknowledge the choice as
“Very good”. This positive reception is and indicative that the
correct choices regarding the illustrations of each sheet are
indispensable. Finally, the expressive parcel of 72.8% classifies
as “Excellent” the idea of extrapolating this complementary
material to all the other topics in the area of Calculus.
There are several methodologies of teaching and entertaining
at the same time. We believed that “Calculus for Coloring” is
among these methodologies. The act of coloring, in addition to
providing an approach relatively easy compared to the tradi-
tional methods of study, intends to provide a greater interest
and amplify the capacity of absorption of concepts.
Considering, fundamentally, the results obtained by the an-
swers that the apprentice students provided, it can be realized
that that the initial impression that it is possible to face the
paradigm and show that “Calculus for Coloring”, is not only
possible, it is also a good complement in the process of teach-
Unfortunately, to our knowledge, until now only two professors
Result of question 1.
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S. A. DAVID ET AL.
Result of question 2.
Result of question 3.
(the authors) tested this method, which limit the conclusions
stated. It would be wishful a major enrolment of the math
teacher community to further verify the validity of the thesis
Finally, it’s worth highlighting that the results reached so far
brave and stimulate to give sequence and amplification to this
The authors wish to acknowledge the financial support pro-
vided by the Dean’s Office for Cultural and Extramural Activi
ties (PRCEU), University of São Paulo (USP-Brazil).
Alcamo, I. E., & Elson, L. M. (1997). The microbiology coloring book.
New York: Harper Collins.
Boyce, W. E., & Di Prima, R. C. (2008). Elementary differential equa-
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Brandt, L., et al. (2001). The impact of concept mapping and visualiza-
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Copyright © 2013 SciRes.