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Evaluating the Use of Wolfram Alpha
for Linear Algebra Learning in
Teleducation
Evaluación del uso de Wolfram Alpha en el
Aprendizaje de Algebra Lineal en Teleducación
Gabriela Campuzano
*
Esther Gonzabay
*
Abstract
The use of ICTs in mathematics learning has the potential to improve
academic performance in an interactive environment. Teleducation
in COVID times demands the active use of technological tools, thus
generating the need to determine which tools are suitable for the
different branches of mathematics. This work evaluates the use of
the Wolfram Alpha tool in the teaching of Linear Algebra in
teleducation in COVID times. This tool was selected because of its
ease of use, detailed answers and because it does not require software
installation. The research was applied to second semester
engineering students, dividing the students into a control group and
an experimental group. Pre- and post-treatment test results were
statistically analyzed to detect whether there was an increase in
academic performance. Students using Wolfram Alpha completed a
survey and interview to assess their perception of the tool. The
experimental group performed better than the control group. The use
of Wolfram Alpha was well accepted by the students suggesting its
* Magister, Universidad Estatal Península de Santa
Elena, Ecuador, mcampuzano@upse.edu.ec,
https://orcid.org/0000-0003-1522-0228
* Magister, Universidad Estatal Península de Santa
Elena, Ecuador, egonzabay@upse.edu.ec,
https://orcid.org/0000-0002-3897-3700
Article
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future use in Linear Algebra courses for tele-education and even
face-to-face education in post-pandemic times.
Keywords: Wolfram Alpha, Algebra Lineal, teleeducación,
vectores, matrices
Resumen
El uso de las TICs en el aprendizaje de matemáticas tiene el potencial
de mejorar el rendimiento académico en un ambiente interactivo. La
teleducación en tiempos de COVID demanda el uso activo de
herramientas tecnológicas, por lo que se genera la necesidad de
determinar que herramientas son adecuadas para las distintas ramas
de las matemáticas. Este trabajo evalúa el uso de la herramienta
Wolfram Alpha en la enseñanza de Algebra Lineal en teleducación
en tiempos de COVID. Se seleccionó esta herramienta por su
facilidad de uso, respuestas detalladas y porque no requiere
instalación de software. La investigación se aplicó a estudiantes de
segundo semestre de ingeniería, dividiendo al estudiantado en un
grupo de control y uno experimental. Se analizaron estadísticamente
los resultados de exámenes pre y post tratamiento para detectar si
hubo un incremento en el rendimiento académico. Los estudiantes
que usaron Wolfram Alpha respondieron una encuesta y entrevista
para evaluar su percepción sobre la herramienta. El grupo
experimental obtuvo un mejor rendimiento que el grupo de control.
El uso de Wolfram Alpha tuvo buena aceptación entre los
estudiantes sugiriendo su uso futuro en cursos de Algebra Lineal
para teleducación e incluso educación presencial en tiempos post
pandemia
Palabras clave: Wolfram Alpha, Algebra Lineal, teleeducación,
vectores, matrices
Introduction
In the last decades, the number of people attending university has
increased and digitalization has penetrated the educational field
resulting in the introduction of distance or remote education methods
through digital technologies. (Kalmar et al., 2022). Teleducation is
understood as distance training through the use of information and
communication technologies (ICTs) offering an interactive and
flexible environment without requiring the physical presence of
students facilitating distance learning. (Benítez et al., 2021; Suescun
Valero, 2021). Traditionally, teleducation has offered advantages
such as ease of mass education, diversity of content and ease of
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access to information so that the student can read, listen and watch
the content several times until it is fully understood. In addition, the
student does not need to travel to the study center, saving time and
economic resources. (Lares & Peña, 2021). Among the negatives,
Benítez et al. (2021) The absence of physical contact, the fear of
teachers and students before a new learning model and access to
computers, tablets, smartphones and limited internet in some sectors
of society.
Due to the development of the COVID-19 pandemic, the face-to-
face classroom had to be replaced by virtuality in whole or in part,
with teleducation being implemented around the world at all levels
of education (El-Soussi, 2022; Fretheim et al., 2021; Rodríguez-
Rivas, 2021; Yau et al., 2022). The lack of physical contact, which
was seen as a disadvantage of teleducation, became the main
advantage of this form of learning to avoid the escalation of SARS-
CoV-2 cases. In some countries, education became entirely virtual,
while other countries opted for a hybrid model where some students
attend classrooms more than others. (Fretheim et al., 2021). In some
countries, where access to technological resources and the Internet
is limited, virtuality imposed new challenges to be overcome.
(Fretheim et al., 2021). Education has been forced to innovate and
opt for flexible learning modalities that allow it to be more resilient
for both COVID and post-COVID times. (Lennox et al., 2021).
Education in times of COVID requires the development of
interactive content that motivates the learner and mitigates the lack
of physical contact to which students are accustomed and the adverse
effects that the pandemic has had such as depression and anxiety
(El-Soussi, 2022; Hoofman & Secord, 2021; Suberviola, 2021).
Therefore, it is necessary to train teachers and study technological
tools to determine which tools are appropriate for each subject.
Many professionals are required to take mathematics during their
university education, which requires the study of mathematics.
(Costa & Rossignoli, 2017)This calls for a study of mathematics
teaching in the current context. Hoofman y Secord (2021) collected
evidence suggesting that students have the perception that
mathematics is assimilated in a better way when there is interaction
in a face-to-face environment with the teacher, which makes
teaching mathematics online in COVID times a challenge.
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Del Pino y Lozano (2012) state that sometimes students present
difficulties in learning mathematics attributable to the lack of
understanding of the conceptual part and lack of development of
skills to solve problems where an algorithm cannot be applied. The
correct use of ICTs for teaching mathematics can increase
usefulness, motivation, enjoyment and confidence and decrease
anxiety about these subjects. Technology strengthens the
mathematics learning process by presenting content numerically,
graphically, and symbolically without the extra burden of spending
time solving complex computational problems by hand, allowing
students to deepen conceptualization and self-assessment. (Kilicman
et al., 2010). However, when ICT is not used correctly, it can become
a major obstacle to learning mathematics. (Del Pino & Lozano,
2012; Hu et al., 2018; Real Pérez, 2013). Hardman (2019) points
out that the literature on the use of ICTs in mathematics is
conflicting, with some studies concluding that ICTs improve student
performance, while other studies show the opposite.
Despite the existence of various technological tools for learning the
different branches of mathematics, in some sectors, there have
traditionally been barriers to their implementation, such as the lack
of computer laboratories, sometimes the software is paid and lack of
teacher training. (Fernández & Muñoz, 2007; Marshall et al., 2012).
During teleducation, the student has been forced to use a computer
or a mobile device to attend classes, do homework and assignments.
Thus, in teleducation, the barrier of not having access to a computer
lab has been overcome by having personal computers, tablets and
cell phones available to use the different technological tools selected
by the teacher. The subject of linear algebra is present in the
curriculum of many careers such as those related to economics and
above all in the various branches of engineering (Costa &
Rossignoli, 2017). Studies have shown that students find the learning
of linear algebra difficult, which makes it pertinent to study its
teaching. (Harel, 2017)This makes it pertinent to study its teaching
during COVID and post COVID times. Educational research in
mathematics initially focused mainly on the subject of calculus,
while in the last thirty years the study of linear algebra has also
gained momentum. (Aydin, 2009). Most of the software and
applications for learning algebra have traditionally been studied for
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the face-to-face modality, so it is necessary to carefully analyze the
tools that are used in teleducation of this branch of mathematics. The
problem arises because there is scarce literature on the tools used in
tele-education in COVID times for teaching algebra, however, there
is available literature on tools for face-to-face education that is very
useful.
Matlab is the most widely employed paid software for teaching
linear algebra and the main program included in Linear Algebra
books as. Grossman (2008); Kolman y Hill (2006); Ortiz y Giraldo
(2018). Other paid computer algebraic systems (CAS) used in linear
algebra include Cabri, Maple (Klasa, 2010) and Mathematica, Linalg
(Aydin, 2009) while in free software we have Scilab and Octave
(Atencio, 2013). For this study it was decided to use the Wolfram
Alpha tool as an alternative to traditional CAS, because it does not
require the installation of software for its operation as the tools
mentioned above, it is easy to use as it does not require programming
knowledge and provides a detailed answer to the problems. Wolfram
Alpha (Fig. 1) is a Mathematica-based answer finder that includes
algebra processing, numerical computation, symbolic computation,
visualizations and statistics. (Abramovich, 2021; Rosly et al., 2020).
There are several ways to write the mathematical problem and it can
be written in natural language and does not require writing
programming code as in the case of computer algebraic systems.
This tool was released in 2009 by Wolfram Research and can be used
with Windows, Macintosh and Android systems through the url
www.wolframalpha.com. (González Martel et al., 2018). The
difference between the free and the paid version lies in the number
of steps shown in the step-by-step solution. During the literature
review, no studies on teaching linear algebra with Wolfram Alpha
were found, however, studies of Wolfram Alpha for other branches
of mathematics relevant to this study are analyzed.
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Figure 1 Wolfram Alpha snapshot of a linear algebra exercise
(adapted for publication)
Dimiceli et al. (2010) successfully replaced the use of traditional
CAS in teaching differential and integral calculus in their computer
labs with Wolfram Alpha. However, they suggest a combined
teaching of CAS and Wolfram Alpha for the subject of calculus of
several variables. This study involved the analysis of the results of
student surveys on the development of calculus laboratory practices.
Říhová et al. (2020) Through survey analysis, we collected positive
student responses to using Wolfram Alpha for financial
mathematics. The use of this tool was not only for solving
mathematical problems but also for financial concepts. In the Czech
Republic, Necesal y Pospıšil (2012) they taught mathematics for two
years with the help of Wolfram Alpha at the higher level. A
proprietary system called TRIAL stored the problems and
evaluations were given using Wolfram Alpha. Wolfram Alpha was
studied for one-variable calculus courses through quizzes where
student acceptance of this tool was evidenced by making classes
more interactive and motivating the student to learn and prepare for
assessments. (C Conceição & Fernandes, 2017; Palencia-González
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& Llamas, 2019). Muñoz y Porras (2018) used the same
methodology used in this study, the analysis of academic
performance with a control group and an experimental group, for the
teaching of differential calculus, integral calculus and differential
equations with Wolfram Alpha, Derive and Geogebra. The statistical
analysis of the pre and post evaluations showed a better academic
performance of the experimental group. Campuzano y Crisanto
(2021) In the experimental group, a similar methodology to this
work was used to study the learning of analytical geometry using
Wolfram Alpha, showing that the tool contributes to an improvement
in academic performance and that students have a good acceptance
of Wolfram Alpha.
This work responds to the need to study tools for learning
mathematics through tele-education as a consequence of the total or
partial replacement of classroom attendance due to the spread of the
COVID-19 virus. This study analyzes an online tool called Wolfram
Alpha as a replacement for traditional computer algebraic systems
that offers advantages such as ease of use and does not require the
installation of software as it only requires an internet connection
through a computer or mobile device. This research responds to the
following objectives: a) To evaluate the effect of Wolfram Alpha on
the academic performance of teleducation students in the subject of
engineering linear algebra b) To determine the student's perception
of Wolfram Alpha for the learning of linear algebra through
teleducation.
Materials and methods
The first component of the research corresponds to a quantitative
methodology in a quasi-experiment employing control and
experimental groups (Figure 2). Cohen et al. (2017) indicates that
this type of research is widely used in education where participants
cannot be randomly selected. This method has been employed by
other studies evaluating Wolfram Alpha for mathematics learning.
(Campuzano & Crisanto, 2021; Muñoz & Porras, 2018). To assess
academic performance, pre- and post-treatment tests were
administered to the control and experimental groups. Wolfram
Alpha was employed for the topics of orthonormal (Gram-Schmidt)
base construction and eigenvalues and eigenvectors. These topics
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were selected based on experience in previous years, taking into
account the topics in which the students had the greatest difficulty.
Figure 2 Diagram of a quasi-experiment with pre- and post-
evaluations.
Fuente. (Cohen et al., 2017)
The null hypotheses analyzed by this work are the following: a) In
the means of the pre-treatment test scores, there is no significant
difference between the control group and the experimental group b)
In the means of the post-treatment test scores, there is no significant
difference between the control group and the experimental group.
The control group received classes without the use of Wolfram
Alpha and the experimental group with the use of this tool in its free
version. Both groups were taught by the principal investigator and
took the exams without the use of this tool. The test scores were
analyzed with Minitab statistical software. Descriptive statistics and
the Student t-test for two independent samples were used once it was
determined that the data had a normal distribution through the Ryan-
Joiner test. The second component of the research corresponds to the
analysis of student perception of the use of this tool. It was carried
out through the analysis of surveys under the Likert scale conducted
only to the experimental group. The results were analyzed under
descriptive statistics using.
This study was carried out for a linear algebra course for engineering
students at the Peninsula de Santa Elena State University of Ecuador
(UPSE). This work is part of the research project "Use of ICTs in the
teaching of mathematics at UPSE in COVID and post COVID
times", which evaluates the use of these tools in subjects such as
geometry, trigonometry, calculus of one variable, calculus of several
variables, differential equations and linear algebra. A total of 80
students in the second semester of the electronics course were
randomly divided into a control group and an experimental group.
The semester where the research was carried out was the third
semester taught online. At the beginning of the semester, the students
were invited to participate in the study and the consent of the entire
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student body was obtained. The content of the face-to-face classes
was adapted for teleducation. Both groups received the same content
through the use of the Moodle platform and Zoom sessions, the
difference being that the experimental group used Wolfram Alpha
for which they received training in the use of the tool. This tool was
accessed through an internet browser on a computer, tablet or
smartphone.
To analyze whether Wolfram Alpha contributes to improved
academic performance when used for learning linear algebra, a
pretest and a post-test were administered to both groups. The
purpose of the pretest is to determine if both groups have similar
knowledge of the subject so that the same starting point exists. The
post-treatment test aims to evaluate if there is a difference in the
academic performance of the group that had sessions with Wolfram
Alpha with respect to the control group. The exams were elaborated
based on evaluations made in past semesters in face-to-face and
virtual modality, linear algebra books and solved exercises that were
made available to the students in the form of a video were also used
as a basis. The exams consisted of 10 multiple-choice questions and
were peer-reviewed by professors from other engineering careers at
the same university. The reliability of the instrument was determined
using Minitab with a Cronbach's Alpha coefficient value of 0.84.
The perception of the use of Wolfram Alpha was evaluated with
surveys conducted to the participants of the experimental group. It
was conducted through Google Forms and the link to access it was
placed in the virtual classroom. All members of the experimental
group answered it. This instrument was developed based on the
review of the literature and the objectives of the study (Dimiceli et
al., 2010; Muñoz & Porras, 2018; Říhová et al., 2020) and the
objectives of the study found in the introductory part of this
document, using the Likert scale. As in the case of the exams, this
instrument was reviewed by the university's peer professors.
Results
The first result (Table 1) that is presented corresponds to the
characterization of the study participants and the percentage of
students who used a cell phone to access the course because they did
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not have a computer at hand. In general in Ecuador, the percentage
of women studying engineering is low, so the percentage of women
is low. Given the closure of physical classrooms and the
implementation of virtuality, many students who previously did not
have permanent access to a computer were forced to acquire one or
use an existing one, sharing it mainly with family members, so the
percentage of students who access only via cell phone is low.
Table 1 Characteristics of the student body that participated in the
study
Group
Number
Percentage
of women
Average
age
Mainly cellular
use
Experimental
Control
28%
The following are the results of the evaluation of academic
performance that was performed using the pre- and post-tests in the
control and experimental groups. Employing the Student t-test for
two independent samples, the measures of the grades of both groups
before the application of the treatment were compared to evaluate
whether both groups have similar knowledge. The null hypothesis
that was analyzed is the following:
H0: There is no significant difference between the control group and
the experimental group in the means of the pre-treatment test scores
(p>0.05).
Table 2 Results of the t-test for the pretreatment examination
Number
Media
Standard
Deviation
t
P
Control
2.23
1.51
-0.52
0.602
Experimental
2.40
1.48
Table 2 shows that p has a value of 0.602, which is higher than 0.05;
therefore, the null hypothesis is accepted and it can be considered
that both the control and experimental groups have similar
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knowledge before receiving the algebra classes of the selected
topics. The scores in the pre-treatment evaluation are not so low due
to the fact that the students receive several concepts related to linear
algebra during their pre-university training.
The Student's t-test for two independent samples to analyze the
measures of the grades of both groups after the application of the
treatment to evaluate whether the Wolfram Alpha tool produces an
improvement in academic performance yielded the results shown in
Table 3. This analysis corresponds to the following null hypothesis:
H0: In the means of the post-treatment test scores, there is no
significant difference between the control group and the
experimental group (p>0.05).
Table 3 Results of the t-test for the post-treatment examination
Number
Media
Standard
Deviation
t
P
Control
5.28
2.15
-3.30
0.001
Experimental
6.75
1.84
The p-value is 0.001, less than 0.05, so the null hypothesis is rejected
and the alternative hypothesis is accepted, i.e., there is a statistically
significant difference between the means of the post-treatment
evaluations, showing that the Wolfram Alpha tool is effective in
learning linear algebra. This result is relevant because it means that
this tool could replace computer algebraic systems only by requiring
an internet connection and a minimum knowledge of writing
commands.
Table 4 presents the main results of the surveys under descriptive
statistics. The equivalence between the Likert scale and the
numerical scale is as follows 0 = strongly disagree, 1 = partially
disagree, 2 = neither agree nor disagree, 3 = partially agree, 4 =
strongly agree.
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Table 4 Wolfram Alpha Tool Usage Survey Results
Ask
Media
Standard
Deviation
After using Wolfram Alpha, I find it useful
for learning linear algebra.
3.9
0.389
Wolfram Alpha has helped me achieve long-
term algebra learning
3.6
0.527
I think Wolfram Alpha makes algebra classes
more interactive.
3.7
0.584
I liked using Wolfram Alpha for the different
parts of this linear algebra course
3.7
0.54
I think Wolfram Alpha should continue to be
used in virtual mathematics classes.
3.7
0.577
I think Wolfram Alpha should be deployed to
face-to-face math classes.
3.7
0.515
Wolfram Alpha used through the cell phone
allowed me to have the same experience as
on the computer.
1.057
In general, a positive perception of users about the use of Wolfram
Alpha can be evidenced, which is consistent with other studies of
this tool for other branches of mathematics (Dimiceli et al., 2010;
Muñoz & Porras, 2018; Říhová et al., 2020). Wolfram Alpha is
considered a tool that adds interaction to the educational
environment that should continue to be used in future linear algebra
courses through tele-education or face-to-face with the use of
computers or cell phones. Although the students responded
positively on the long-term learning of algebra, for a more detailed
evaluation of this aspect a study is needed to conduct new
evaluations after a reasonable time. The lowest scoring question was
regarding the use of Wolfram Alpha on the cell phone, possibly
attributable in some cases to the fact that during the practices some
students could not access through the default internet browser and
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had to switch to another one because the Wolfram Alpha application
for the cell phone is paid. Figure 3 shows the students' responses
about the feature they liked the most about Wolfram Alpha. The
features they liked the most are related to detailed answers to
problems and flexibility in typing commands. These responses are
related to the factor that these students have used Matlab in other
subjects previously and that when comparing Wolfram Alpha to a
CAS it is evident that no programming skills are required and that a
single command line delivers detailed answers that would
correspond to several lines of programming in a CAS
.
Figure 3. Features that Wolfram Alpha users liked the most
This study used the free version of Wolfram Alpha, however, the
paid version offers advantages such as full visualization of the step-
by-step resolution that has much potential to strengthen the learning
process. Future studies can focus on this aspect and its influence on
the student body.
Discussion
The incorporation of technology in the learning of linear algebra has
been mainly focused on the use of Matlab software, which is even
included in the textbooks of this subject. Wolfram Alpha is an
interesting alternative to this paid software since it only requires a
computer, tablet or cell phone with internet connection and has an
easy-to-use interface that does not require programming knowledge.
The statistical analysis of the grades of the students in the control
and experimental groups showed that the use of the Wolfram Alpha
technological tool through the Internet allowed the students to
improve their academic performance in a linear algebra course
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carried out through tele-education in COVID times. Survey analysis
shows a positive perception of the use of this tool and suggests that
Wolfram Alpha be used in future Wolfram courses in tele-education
or face-to-face mode. In post-COVID times, it is possible that
students will use technological tools in a more active way for
learning the different branches of mathematics in the classroom, so
the use of this tool will continue and in areas where there are not
enough computers for the entire student body, students will use
mobile devices to access the tool.
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