70
Revista Iberoamericana de la Educación, Vol – 7 No. 1, January - March 2023
e-ISSN: 2737-632x
Pgs 70-86
* Master's Degree, Lecturer at
Universidad Estatal Península de
Santa Elena, La Libertad,
Ecuador,
mcampuzano@upse.edu.ec,
https://orcid.org/0000-0003-1522-
0228
**Master's Degree, Lecturer at
Universidad Estatal Península de
Santa Elena, La Libertad,
Ecuador, csanchez@upse.edu.ec,
https://orcid.org/0000-0002-2965-
9189
***Doctor of Philosophy Degree,
Professor at Universidad Estatal
Península de Santa Elena, La
Libertad, Ecuador,
mbayas@upse.edu.ec,
https://orcid.org/0000-0002-0636-
495
Received: July, 2022
Approved: December 2022
DOI:
https://doi.org/10.31876/rie.v
6i4.239
http://www.revista-
iberoamericana.org/index.
php/es
How to cite:
Campuzano, G., Sánchez, C.,
Bayas, M. (2023) Use of math
software Mathematica to
learn to derivate multiple
variables functions online
Revista Iberoamericana De
educación, 7(1)
Use of math software Mathematica to
learn to derivate multiple variables
functions online
Uso de software matemático para aprender derivar funciones de varias
variables en línea
Utilização de software matemático para aprender a derivar funções de
várias variáveis online
Gabriela Campuzano*
Carlos Sánchez**
Marcia Bayas***
Abstract
Technological tools, teaching materials and students’ acceptance are
key aspects for online education at current times. The use of
computer algebra systems (CAS) may improve students’ perception
about online learning as it offers an interactive environment where
academic achievement could increase. Math software Wolfram
Mathematica under its cloud version was studied for learning second
semester multivariate calculus online. A quasi-experiment was
conducted with control and experimental groups. Pre-tests were
carried out to evaluate the similarity of groups prior treatment and
post-tests to assess scores after students learnt derivatives in
multivariable calculus. Two-sample t-test was used to determine
how similar the means of the control and experimental groups were
for pre and post-tests’ scores. Experimental group scored
significantly better than control group demonstrating the positive
effect this software had on multivariate calculus learning. Students
from experimental group answered a survey to evaluate in more
detail the experience. Results indicate that math software should be
implemented in calculus courses during online education.
Key words: math software, online learning, derivatives,
multivariable calculus.
Resumen
Herramientas tecnológicas, materiales didácticos y aceptación
estudiantil son aspectos clave para la educación en línea actualmente.
El uso de sistemas algebraico computacionales (CAS) puede mejorar
la percepción de los estudiantes del aprendizaje en línea, ya que
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Use of math software Mathematica to learn to derivate multiple variables functions online
71
ofrece un entorno interactivo donde el rendimiento académico puede
mejorar. Se estudió el software matemático Wolfram Mathematica
en su versión en la nube para el aprendizaje en línea de cálculo de
varias variables de segundo semestre. Se realizó un cuasi-
experimento con grupos control y experimental. Se realizaron
pruebas pre-tratamiento para evaluar la similitud de los grupos y
pruebas posteriores para evaluar las puntuaciones después de que los
estudiantes aprendieron derivadas en cálculo multivariable. Se
utilizó la prueba t de dos muestras para determinar qué tan similares
eran las medias de los grupos de control y experimental para los
puntajes de las pruebas previas y posteriores. El grupo experimental
obtuvo un puntaje significativamente mejor que el grupo de control,
lo que demuestra el efecto positivo que tuvo este software en el
aprendizaje del cálculo multivariable. Los estudiantes del grupo
experimental respondieron una encuesta para evaluar con más detalle
la experiencia. Los resultados indican que el software matemático
debe implementarse en los cursos de cálculo durante la educación en
línea.
Palabras clave: software matemático, aprendizaje en línea,
derivadas, cálculo de varias variables
Resumo
As ferramentas tecnológicas, os materiais didácticos e a aceitação
dos estudantes são hoje aspectos fundamentais da educação em
linha. A utilização de sistemas algébricos de computador (CAS)
pode melhorar a percepção dos estudantes sobre a aprendizagem em
linha, proporcionando um ambiente interactivo onde o desempenho
académico pode ser melhorado. O software matemático Wolfram
Mathematica na sua versão em nuvem foi estudado para a
aprendizagem on-line do cálculo multivariável do segundo semestre.
Foi realizada uma quase-experimentação com grupos de controlo e
experimentação. Foram realizados pré-testes para avaliar a
semelhança de grupos e pós-testes para avaliar as pontuações após
os estudantes terem aprendido derivados em cálculo multivariável.
Foi utilizado um teste t de duas amostras para determinar quão
semelhantes eram os meios dos grupos de controlo e experimentais
para as pontuações de pré e pós-teste. O grupo experimental teve
uma pontuação significativamente melhor do que o grupo de
controlo, demonstrando o efeito positivo que este software teve na
aprendizagem do cálculo multivariável. Os estudantes do grupo
experimental completaram um inquérito para avaliar melhor a
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Use of math software Mathematica to learn to derivate multiple variables functions online
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experiência. Os resultados indicam que o software matemático deve
ser implementado em cursos de cálculo durante o ensino em linha.
Palavras-chave: software matemático, aprendizagem em linha,
derivados, cálculo multivariável
INTRODUCTION
During the 1990s, the invention of World Wide Web increased access
to online learning for a wide range of disciplines (Harasim, 2000).
Online courses and even academic degrees increased around the
world as web sites and online community groups developed rapidly
(Rajabalee & Santally, 2021). Although online education had been
widely offered before, in 2020, COVID pandemic affected face-to-
face education forcing to convert several courses to online learning
as schools and universities were forced to close to avoid physical
contact. Some schools and universities had to overcome many
difficulties to implement this kind of education as ICTs had not been
widely implemented prior the pandemic. Education innovation
became necessary while access and engagement became key aspects.
In some cases, this sudden introduction of online education
introduced numerous barriers like insufficient connectivity, lack of
confidence, lack of experience with technological tools, financial
problems to acquire technological devices and lack of a proper
physical space free of noise and distractions (Abuhammad, 2020;
Mailizar et al., 2020; Zaharah et al., 2020). During the pandemic,
some institutions taught solely online while others used a
combination of face-to-face and online education. In some cases,
institutions which had returned to physical classrooms had to convert
online again depending on the number of people infected with
COVID 19. In some schools and universities, the so-called new
normality limited face-to-face education and some courses have
stayed online or shifted to hybrid modality. This pandemic forced
education to be delivered in flexible ways combining face-to-face
and online learning (Lockee, 2021). Traditional learning is evolving
into digital and intelligent. Education must adapt to a new normalized
environment after the pandemic and be able to adapt when new
emergencies happen again (Dong et al., 2022). Learning math online
can be complex and some students may feel that it is best learnt with
face-to-face interaction (Mukuka et al., 2021). The use of math
software may improve students’ perception about online learning as
it offers an interactive environment and where academic achievement
could increase.
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Learning mathematics is compulsory for many higher education
programs (Akugizibwe & Ahn, 2020). Learning math for university
students is important because they are required to reason, think
creatively, gather information, communicate ideas and test these
ideas critically. Math taught in engineering courses usually include
single variable calculus, multivariate calculus and differential
equations. Some students find calculus difficult because their
mathematical knowledge is not strong enough; consequently,
lecturers are required to use a wide range of interactive activities to
accomplish the expected learning outcomes of calculus (Carbonell et
al., 2012). In multivariable calculus, several variables’ functions are
derived and used to determine directional derivatives, gradients,
tangent and normal planes, among others. Operating multivariable
functions can represent great complexity for some students as they
find complicated to migrate to the two or three variable world and
represent functions in 3 dimensions (Kashefi et al., 2010).
Learning derivatives can be complex and software can aid students
to increase their understanding through interactive and dynamic
activities while handling complex graphs (Caligaris et al., 2015).
Since the 80s, when computer algebra systems (CAS) emerged,
software’s efficiency had been debated and studied. Rapid changes
in ICT, new software and updated versions of existing software
promotes to continue with this research (Sevimli, 2016). Computer-
based tools can strengthen mathematical thinking an generate an
appropriate environment to overcome difficulties in calculus courses
(Kashefi et al., 2012). There is a wide range of software for
mathematics. Computer algebra systems like Mathematica, Derive,
Matlab and Maple are proprietary packages while Maxima, Octave
and SageMath are free software. Dynamic geometry software (DGS)
includes Geogebra, Geometer’s Sketchpad and Cabri Geometry
(Ayub et al., 2012; Fluck et al., 2020). In addition, Wolfram
Research offers Wolfram Alpha which is a web service browser with
potential to replace or complement CAS (Dimiceli et al., 2010). For
statistics, computer programs like SPSS and R stand out for
education, research and commercial use (Akugizibwe & Ahn, 2020).
Technology offers tools to improve math understanding, making
necessary to assess their effectiveness on academic achievement and
students’ perception (Arbain & Shukor, 2015). Mathematica
software was chosen for this research because it has a cloud version
and provides students an experience where coding and programming
skills are developed, which is desired for undergraduate engineering
students.
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Mathematica software is widely used for symbolic calculation,
handling data, solving math problems numerically and symbolically,
modelling and simulating phenomena, among others (Ayub et al.,
2012; Zotos, 2007). This computer program has been constantly
changing to adapt to technological advances and currently offers
desktop, cloud and mobile version. New changes have produced new
research with new article and book publications. Barba-Guaman et
al. (2018) demonstrated through interviews and scores’ comparison
how Wolfram Mathematica aids students to improve calculus
interpretation. A quasi experiment with control and experimental
groups was performed, but a statistical test was not performed to
analyzed differences between mean scores. Results indicated that this
software aided theoretical abstraction and made classes more
interesting and less difficult. Baist et al. (2020)studied the effect of
Mathematica on learning outcomes for a vector algebra course for
control and experimental groups. Scores of pre and post tests were
analyzed with normality Saphiro-Wilk and with Mann Whitney test
to determine if they are significantly different between groups. They
concluded that students’ achievement increased with the use of
Mathematica and that this software aided to improve self-regulated
activities. As Jordanian universities usually lack the use of CAS for
math teaching, Hiyam et al. (2019) evaluated Mathematica and
Wolfram Alpha for teaching derivatives with satisfactory results as
students from the experimental group scored better than control
group. Means and standard deviation were analyzed with an
ANCOVA analysis of covariance. This software created a dynamic
environment, helped to understand theory through graphical
representation and motivated students to propose innovative
solutions. Hiyam et al. (2019) used the same software with a similar
methodology than the one used for this study for a face-to-face single
variable calculus course while the current study focuses on
multivariate calculus taught online. There is evidence that math
software contributes to improve learning outcomes and academic
achievement; however, there is the need to determine specific
software for specific math areas for online education.
This study implemented Mathematica software using its cloud
version for online multivariate calculus classes to improve students’
mathematical thinking trough better visualization, interactivity and
problem posing. Previous research evaluated under different
methodologies the benefits of using Mathematica software mainly in
computer laboratories during face-to-face education; however,
Mathematica’s effect on academic achievement has not been
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evaluated under a quasi-experimental methodology for multivariate
calculus for online education.
MATERIALS AND METHODS
The methodology used is a quasi-experiment with experimental and
control groups. Cohen (2017) states that this method is widely used
in educational research when participants cannot be randomly
chosen. This methodology had been used before by some studies
(Adelabu et al., 2022; Arbain & Shukor, 2015; Hiyam et al., 2019;
Takači et al., 2015; Yimer & Oromia, 2022). Pre and post tests were
applied to control and experimental groups to assess academic
achievement and evaluate the effectiveness of using Mathematica.
Literature rpeview and learning difficulties faced by students in
previous years influenced the topics selected for applying
Mathematica. The selected topics for second semester multivariable
calculus are partial derivatives, directional derivatives and gradient.
The null hypotheses related to this study are the following:ç
1. There is no significant difference (p>0.05) between average
grades of pre-test for control and experimental groups.
2. There is no significant difference (p>0.05) between average
grades of post-test for control and experimental groups.
The main researcher taught both courses and had previous training
in Mathematica. Students at their homes accessed Mathematica as
Wolfram Cloud on their computers or mobile devices. Research
process and instruments used in this work are shown in Table 1.
Experimental group had sessions with Wolfram Cloud while control
group participated of Zoom sessions without this tool. A pre-test was
applied to control and experimental groups. Experimental group had
complimentary sessions with Wolfram Cloud. First, it was necessary
to introduce the software to students and teach them how to access
and use it. After the treatment, students of both groups participated
in a post-test with similar difficulty as the pre-test. Two sample t-test
was performed to determine if there is a significant difference
between the two groups. After treatment, experimental group
answered a survey to evaluate the perception of the contribution of
the CAS to their learning process.
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Table 1. Research process and instruments
Groups
Pre-observation
tool
Process
Post-
observation
tool
Control
Pre-test
Taught with
traditional
methodology
without
software
Post-test
Experimental
Pre-test
Taught with
Mathematica
software
Post-test,
survey
Participants
This work was performed at “Universidad Estatal Peninsula de Santa
Elena”, Ecuador. Undergraduate computer engineering students were
selected for this study. A total of 59 students from second semester
participated in this research. This research took place during online
multivariate calculus. Table 2 presents number, gender and average
age of students who participated in this research. For experimental
group (30 students), 10 students (33%) accessed Mathematica
through Wolfram Cloud using a computer and 20 students (67%)
used mobile version of Wolfram Cloud.
Table 1. Students who participated in this research
Number
Female
Male
Average Age
Control Group
29
9 (31%)
20 (69%)
22
Experimental Group
30
10 (33%)
20 (67%)
23
Data Collection
Data for analyzing academic achievement was collected by pre and
post-test applied to control and experimental groups. Tests were
prepared by researchers analyzing syllabus content, problems
included in calculus books and Putnam exam materials. Post and pre-
tests had similar difficulty. Tests had 10 multiple choice questions.
Other math teachers from other careers from the same university
reviewed the tests and their suggestions were included. Test’s
reliability was determined calculating Cronbach’s Alpha coefficient
which value was 0.85.
The survey to assess students’ perception of using Mathematica was
elaborated with 10-item using Likert scale. This questionnaire was
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Use of math software Mathematica to learn to derivate multiple variables functions online
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prepared according to this research objectives and analyzing studies
for other math software. A small group of students answered a pilot
survey. The entire experimental group answered the survey published
in the university’s learning management system. Questions were
written in Spanish and translated into English for this publication.
Data analysis
Results were analyzed using MINITAB v19 statistical software.
Descriptive statistics were applied to detect difference in pre and
post-tests’ scores. A normality test was performed using Shapiro-
Wilk method. Two-sample t-test was used to determine how similar
the means of the control and experimental groups were for pre and
post-tests’ scores. Questionnaires were tabulated and analyzed under
descriptive statics to determine minimum, maximum, mean and
standard deviation values once Likert scale was converted to the
following numerical scale:
1 = strongly disagree, 2 = disagree, 3 = neutral, 4 = agree, 5 = strongly
agree
Teaching Materials
Some teaching materials used on the course were elaborated by the
teacher and others were retrieved from Mathematica software help,
Wolfram Demonstration Project and open course “Introduction to
Calculus” offered by Wolfram U. The selected materials contributed
to reach learning outcomes and involved the most important topics
of multivariate calculus. Self-elaborated materials included animated
power point slides and Mathematica notebooks. The following is an
example of the materials elaborated by the researchers, originally
written in Spanish, and later translated to English.
Multivariate calculus handles functions of more than one
independent variable called functions of several variables. The
gradient of a function of several variables
!
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is the vector
calculated by:
'!
"
#$ %
&
( !
!
"
#$ %
&
) *+ !
"
"
#$ %
&
,
(1)
Where
!
!
"
#$ %
&
represents the partial derivative of
!
"
#$ %
&
with
respecto to x and
!
"
"
#$ %
&
the partial derivative of
!
"
#$ %
&
with
respecto to y. For partial differentiation, a variable is set, and the
others are considered constant. For instance, for
!
"
#$ %
&
,
#$
#!
implies
“x” is a variable; consequently, “y” is constant. The code presented
in Fig. 1 finds the gradient of
!
"
#$ %
&
(
!
!
%
* %
&
at
"
-.$/
&
calculating
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Use of math software Mathematica to learn to derivate multiple variables functions online
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the partial derivatives. Using the command ContourPlot this graph is
obtained.
Figure 1. Gradient of a function at a given point
RESULTS
Before applying the treatment, it is necessary to evaluate students’
prior knowledge with a pre-test, so later the improvement can be
evaluated for both groups. Pre-test results will also aid to determine
if experimental and control groups have similar prior knowledge.
Table 3 shows results of the t-test performed to pre-test grades for
control and experimental groups. The null hypothesis for this test is:
H
0
: There is no significant difference (p>0.05) between average
grades of pre-test for control and experimental groups.
Since p is greater than 0.05, the null hypothesis is accepted, and it
can be stated that there is no statistically significant difference
between control and experimental group, so treatment can be applied
as control and experimental groups are similar.
Table 3. Results of independent t-test on pre-test scores
Group
N
Mean
Std.
Deviation
T
P
Control
29
1.52
1.88
-0.37
0.714
Experimental
30
1.70
1.93
A post-test was carried out after the experimental group learnt using
Mathematica on their computers or mobile devices and control group
were taught without this software. Results of the t-test applied to the
scores of the post-test for both groups are displayed in Table 4. The
null hypothesis for this test is:
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Use of math software Mathematica to learn to derivate multiple variables functions online
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H
0
: There is no significant difference (p>0.05) between average
grades of post-test for control and experimental groups.
As it can be observed, p value is less than 0.05, so null hypothesis is
rejected, and alternative hypothesis accepted. This means that there
is a significant difference between academic achievement of control
and experimental groups. Comparing mean values, we notice that
mean for experimental group is 2.23 higher, which implies that
academic achievement of experimental group is better than control
group. Therefore, it has been demonstrated that the use Mathematica
during online classes has a positive effect on students’ achievement.
This result is consistent with works performed by Baist et al. and
Hiyam et al. (2019) which concluded that Mathematica software
increased academic achievement as means for control and
experimental groups differ considerably.
Table 2. Results of independent t-test on post-test scores
Group
N
Mean
Std.
Deviation
T
P
Control
29
5.47
1.93
-3.96
0.000
Experimental
30
7.70
2.39
Experimental group answered a questionnaire to evaluate the use of
Mathematica or Wolfram Cloud during online classes and its results
are presented in Table 5. Most of the questions have an average
higher than 4, where 1 corresponds to strongly disagree and 5 to
strongly agree. The question with the least score is the one which
states that Mathematica offers the same experience in mobile devices
as computers. This may be attributed to the way menus are presented
in the mobile application.
Table 3. results of the survey for control group
Question
Minimum
Maximum
Mean
Standard
Deviation
After using
Mathematica, I
believe it is useful for
learning calculus.
2
5
4,424
0,792
I feel confident that
Mathematica has
aided me to reach
permanent learning.
3
5
4,424
0,663
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I think that
Mathematica makes
calculus classes more
interactive.
3
5
4,636
0,603
Mathematica has
aided me for better
visualization of
functions
3
5
4,727
0,517
I believe that
Mathematica should
continue to be used
for math classes
during online classes
2
5
4,667
0,692
Mathematica has
helped me in problem
posing
3
5
4,606
0,556
I enjoyed
programming math
problems in
Mathematica
3
5
4,606
0,659
Mathematica’s
mobile app allowed
me to have the same
experience as in the
computer
2
5
3,758
1.032
When schools and universities were forced to close face-to-face
education and to convert to online, there was uncertainty of how this
will affect education and which tools may be used during online
courses. Students replied positively when enquired about using this
software to aid the transition to online learning and to improve
education. Students’ responses to the questionnaire showed that they
liked and enjoyed using Mathematica as this software generates an
interactive environment and increases their programming
knowledge. Students feel confident that this software has allowed
them to reach permanent learning; however, further research is
required as delayed post-test were not performed. Students
recommend this software for online and face-to-face classes (Table 5
and Fig. 2) as mathematics becomes more interesting, more exercises
can be solved, and learning is interactive and auto-regulated.
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Figure 2. The percentage of students who think Mathematica should
continue to be used for online education
Every student considered Mathematica useful for learning calculus
online and many enjoyed using it. Calculus classes with this
computer program became more dynamic and some concepts related
to derivatives easier to understand which is aligned with
Mathematica’s previous studies (Barba-Guaman et al., 2018; Hiyam
et al., 2019). Plotting graphs helped students to understand faster
partial derivatives, directional derivatives and gradients. Visual
representation was useful in multivariable calculus as 3D graphs and
contour plots may be difficult and time consuming to do by hand.
Students using math software may increase their knowledge about
ICT applied to education and programming.
CONCLUSIONS
This study used a quasi-experiment to assess the effect that
Mathematica had during an online multivariate calculus course.
Computers and smartphones were used to access online Mathematica
as Wolfram Could. Based on the results of independent t-tests
performed to students’ scores, it can be stated that this software
through its cloud version aided students of second semester to
increase their academic achievement when compared to control
group. Students’ perceptions of the software gathered by surveys
were positive and suggest that Mathematica should continue to be
used during online learning as this software allows a meaningful
learning in an interactive environment. Visual representation was
particularly useful for multivariate calculus as some 3D plotting
3% 3%
18%
76%
0%
10%
20%
30%
40%
50%
60%
70%
80%
Disagree Undecided Agree Strongly agree
Percentage (%)
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could become very difficult to be performed by hand. Students felt
enthusiastic about programming for a math course as programming
is essential for their engineering studies. The new normality after
SARS-CoV-2 suggests that classes will not be the same anymore.
Online and hybrid education will continue to be extensively
implemented as some universities lack infrastructure to maintain
biosecurity measures and online access to education may be more
convenient for people who cannot attend in-person courses.
REFERENCES
Abuhammad, S. (2020). Barriers to distance learning during the
COVID-19 outbreak: A qualitative review from parents’
perspective. Heliyon, 6(11), e05482.
https://doi.org/10.1016/J.HELIYON.2020.E05482
Adelabu, F. M., Marange, I. Y., & Alex, J. (2022). GeoGebra software
to Teach and Learn Circle Geometry: Academic Achievement
of Grade 11 Students. Mathematics Teaching Research
Journal, 14(3), 1–16.
Agus, I., & Hadi, A. (2020). The Responses of Mathematics Pre-
Service Teachers Toward Online Lectures in the Covid-19 Era.
Edumatika : Jurnal Riset Pendidikan Matematika, 3, 113.
https://doi.org/10.32939/ejrpm.v3i2.584
Akugizibwe, E., & Ahn, J. Y. (2020). Perspectives for effective
integration of e-learning tools in university mathematics
instruction for developing countries. Education and
Information Technologies, 25(2), 889–903.
https://doi.org/10.1007/S10639-019-09995-Z/TABLES/5
Arbain, N., & Shukor, N. A. (2015). The Effects of GeoGebra on
Students Achievement. Procedia - Social and Behavioral
Sciences, 172, 208–214.
https://doi.org/10.1016/J.SBSPRO.2015.01.356
Ayub, A. F. M., Tarmizi, R. A., Bakar, K. A., & Luan Wong, S. (2012).
WxMaxima Computer Software as an Aid to the Study of
Calculus by Students with Different Learning Approaches.
Procedia - Social and Behavioral Sciences, 64, 467–473.
https://doi.org/10.1016/J.SBSPRO.2012.11.055
Baist, A., Amarullah, A., Safitri, P. T., & Enawar. (2020). The Use of
Computational Mathematics Teaching Materials aided
Mathematica Software in Vector Algebra Course. Journal of
Physics: Conference Series, 1477(2), 022013.
https://doi.org/10.1088/1742-6596/1477/2/022013
83
Revista Iberoamericana de la Educación, Vol – 7 No. 1, Enero – Marzo 2023
Use of math software Mathematica to learn to derivate multiple variables functions online
83
Baltà-Salvador, R., Olmedo-Torre, N., Peña, M., & Renta-Davids, A.
I. (2021). Academic and emotional effects of online learning
during the COVID-19 pandemic on engineering students.
Education and Information Technologies, 26(6), 7407–7434.
https://doi.org/10.1007/S10639-021-10593-1/TABLES/5
Barba-Guaman, L. R., Quezada-Sarmiento, P. A., Calderon-Cordova,
C. A., Sarmiento-Ochoa, A. M., Enciso, L., Luna-Briceno, T.
S., & Conde-Zhingre, L. E. (2018). Using wolfram software to
improve reading comprehension in mathematics for software
engineering students. Iberian Conference on Information
Systems and Technologies, CISTI, 2018-June, 1–4.
https://doi.org/10.23919/CISTI.2018.8399388
Bhute, V. J., Inguva, P., Shah, U., & Brechtelsbauer, C. (2021).
Transforming traditional teaching laboratories for effective
remote delivery—A review. Education for Chemical
Engineers, 35, 96–104.
https://doi.org/10.1016/J.ECE.2021.01.008
Caligaris, M. G., Schivo, M. E., & Romiti, M. R. (2015). Calculus &
GeoGebra, an Interesting Partnership. Procedia - Social and
Behavioral Sciences, 174, 1183–1188.
https://doi.org/10.1016/J.SBSPRO.2015.01.735
Carbonell, M. R. E., Saà-Seoane, J., & Pons, J. V. (2012). Innovative
Self-Assessment and Teaching/Learning Techniques for
Calculus within the RIMA Project (UPC-ICE). Procedia -
Social and Behavioral Sciences, 46, 686–691.
https://doi.org/10.1016/J.SBSPRO.2012.05.182
Cohen, L. , M. L. , M. K. (2017). Research Methods in Education (8th
ed.). Routledge.
Dimiceli, V. E., Lang, A. S. I. D., & Locke, L. (2010). Teaching
calculus with Wolfram|Alpha. International Journal of
Mathematical Education in Science and Technology, 41(8),
1061–1071. https://doi.org/10.1080/0020739X.2010.493241
Dong, Y., Shao, B., Lou, B., Ni, C., & Wu, X. (2022). Status And
Development Of Online Education Platforms In The Post-
epidemic Era. Procedia Computer Science, 202, 55–60.
https://doi.org/10.1016/J.PROCS.2022.04.008
Fluck, A. E., Ranmuthugala, D., Chin, C. K. H., Penesis, I., Chong, J.,
Yang, Y., & Ghous, A. (2020). Transforming learning with
computers: Calculus for kids. Education and Information
Technologies 2020 25:5, 25(5), 3779–3796.
https://doi.org/10.1007/S10639-020-10136-0
84
Revista Iberoamericana de la Educación, Vol – 7 No. 1, Enero – Marzo 2023
Use of math software Mathematica to learn to derivate multiple variables functions online
84
Harasim, L. (2000). Shift happens: online education as a new paradigm
in learning. The Internet and Higher Education, 3(1–2), 41–61.
https://doi.org/10.1016/S1096-7516(00)00032-4
Hiyam, B., Zoubi, A., & Khataybeh, A. (2019). Utilizing
MATHEMATICA Software to Improve Students’ Problem
Solving Skills of Derivative and its Applications. International
Journal of Education and Research, 7(11).
Kashefi, H., Ismail, Z., & Yusof, Y. M. (2010). Obstacles in the
Learning of Two-variable Functions through Mathematical
Thinking Approach. Procedia - Social and Behavioral
Sciences, 8, 173–180.
https://doi.org/10.1016/J.SBSPRO.2010.12.024
Kashefi, H., Ismail, Z., Yusof, Y. M., & Rahman, R. A. (2012).
Fostering Mathematical Thinking in the Learning of
Multivariable Calculus Through Computer-Based Tools.
Procedia - Social and Behavioral Sciences, 46, 5534–5540.
https://doi.org/10.1016/J.SBSPRO.2012.06.471
Kilicman, A., Hassan, M. A., & Husain, S. K. S. (2010). Teaching and
Learning using Mathematics Software “The New Challenge.”
Procedia - Social and Behavioral Sciences, 8, 613–619.
https://doi.org/10.1016/J.SBSPRO.2010.12.085
Kusbeyzi, I., Hacinliyan, A., & Aybar, O. O. (2011). Open source
software in teaching mathematics. Procedia - Social and
Behavioral Sciences, 15, 769–771.
https://doi.org/10.1016/J.SBSPRO.2011.03.181
Lockee, B. B. (2021). Online education in the post-COVID era. Nature
Electronics, 4(1), 5–6. https://doi.org/10.1038/s41928-020-
00534-0
Mailizar, A., Abdulsalam, M., & Suci, B. (2020). Secondary school
mathematics teachers’ views on e-learning implementation
barriers during the COVID-19 pandemic: The case of
Indonesia. Eurasia Journal of Mathematics, Science &
Technology Education, 1–9.
Marpa, E. P. (2021). Technology in the teaching of mathematics: An
analysis of teachers’ attitudes during the COVID-19 pandemic.
International Journal on Studies in Education, 3(2), 92–102.
Mukuka, A., Shumba, O., & Mulenga, H. M. (2021). Students’
experiences with remote learning during the COVID-19 school
closure: implications for mathematics education. Heliyon, 7(7),
e07523. https://doi.org/10.1016/J.HELIYON.2021.E07523
Purnomo, E. A., Sukestiyarno, Y. L., Junaedi, I., & Agoestanto, A.
(2022). Analysis of Problem Solving Process on HOTS Test
85
Revista Iberoamericana de la Educación, Vol – 7 No. 1, Enero – Marzo 2023
Use of math software Mathematica to learn to derivate multiple variables functions online
85
for Integral Calculus. Mathematics Teaching Research
Journal, 14(3), 199–214.
Rajabalee, Y. B., & Santally, M. I. (2021). Learner satisfaction,
engagement and performances in an online module:
Implications for institutional e-learning policy. Education and
Information Technologies, 26(3), 2623–2656.
https://doi.org/10.1007/s10639-020-10375-1
Ralston, J. (2004). ICT, learning and primary mathematics. Education
3-13, 32(2), 60–64.
https://doi.org/10.1080/03004270485200231
Saha, R. A., Ayub, A. F. M., & Tarmizi, R. A. (2010). The effects of
GeoGebra on mathematics achievement: Enlightening
Coordinate Geometry learning. Procedia - Social and
Behavioral Sciences, 8, 686–693.
https://doi.org/10.1016/j.sbspro.2010.12.095
Sevimli, E. (2016). Do calculus students demand technology
integration into learning environment? case of instructional
differences. International Journal of Educational Technology
in Higher Education, 13(1), 1–18.
https://doi.org/10.1186/S41239-016-0038-6/TABLES/2
Takači, D., Stankov, G., & Milanovic, I. (2015). Efficiency of learning
environment using GeoGebra when calculus contents are
learned in collaborative groups. Computers & Education, 82,
421–431. https://doi.org/10.1016/J.COMPEDU.2014.12.002
Tall, D. (2013). How humans learn to think mathematically: Exploring
the three worlds of mathematics. How Humans Learn to Think
Mathematically: Exploring the Three Worlds of Mathematics,
1–457. https://doi.org/10.1017/CBO9781139565202
Tarmizi, R. A. (2010). Visualizing Student’s Difficulties in Learning
Calculus. Procedia - Social and Behavioral Sciences, 8, 377–
383. https://doi.org/10.1016/J.SBSPRO.2010.12.053
Weston, C., & McAlpine, L. (1998). How six outstanding math
professors view teaching and learning: The importance of
caring. International Journal for Academic Development, 3(2),
146–155. https://doi.org/10.1080/1360144980030207
Yimer, S., & Oromia, A. (2022). Effective Instruction for Calculus
Learning Outcomes through Blending co-operative Learning
and Geogebra . Mathematics Teaching Research Journal,
14(3), 170–189.
Zaharah, Z., Kirilova, G. I., & Windarti, A. (2020). Impact of Corona
Virus Outbreak Towards Teaching and Learning Activities in
86
Revista Iberoamericana de la Educación, Vol – 7 No. 1, Enero – Marzo 2023
Use of math software Mathematica to learn to derivate multiple variables functions online
86
Indonesia. SALAM: Jurnal Sosial Dan Budaya Syar-i, 7(3),
269–282. https://doi.org/10.15408/SJSBS.V7I3.15104
Zotos, K. (2007). Performance comparison of Maple and
Mathematica. Applied Mathematics and Computation, 188(2),
1426–1429. https://doi.org/10.1016/J.AMC.2006.11.008
