Volume 7, Issue 1, January – 2022 International Journal of Innovative Science and Research Technology

ISSN No:-2456-2165

Learning Analytic Geometry with the aid of

Wolfram Alpha
Maria Gabriela Campuzano, Tania Crisanto,
Universidad Estatal Península de Santa Universidad de las Fuerzas
Elena – UPSE, La Libertad, Ecuador Armadas – ESPE, Sangolquí, Ecuador

Abstract:- The active use of ICT for math education An important area of mathematics is analytic geometry
offers known advantages like cre-ating interactive which is also called coordinate geometry. Usually, it is
environments to facilitate learning, improved monitoring studied in pre-calculus because it involves key concepts
and detailed feedback. The objective of this study is to which will aid single variable and multivariate calculus
determine the effective-ness of using Wolfram Alpha tool understanding. Mathematical representation is a key concept
on academic achievement and students’ attitude towards for learning analytic geometry and complex problems can be
it when learning analytic geometry. Wolfram Alpha web solved easily with the right representation [7]. The use of
re-source was chosen because it has a good visual ICTs can help to represent elements of analytic geometry.
representation, shows step-by-step solutions, returns Math software used for math classes may play a key role for
detailed outcome and is easy to use. A quasi-experimental a meaningful permanent learning. Previous research for math
methodology was used dividing students into learning includes the following software or web re-sources:
experimental and control groups. Academic achievement GeoGebra, Mathematica, Matlab, Cabri Geometry and
was assessed with two sample t-test to pre and post tests’ Wolfram Alpha [8-15].
scores. Questionnaires were applied to Wolfram Alpha
group to gather students’ perception of using this tool This study was performed during a distance learning
and tabulated under descriptive statistics. Research was course of pre-calculus due to pandemic. Wolfram Alpha web
applied to an online course of pre-calculus during resource was selected for teaching analytic geometry because
disruption of face-to-face classes due to COVID spread. this tool requires little programming knowledge and offers
Re-sults suggest that Wolfram Alpha improves students’ visualization and step-by-step resolution for math problems.
academic achievement generating positive and interactive Despite that Wolf-ram Alpha has been evaluated for
conditions where students can benefit from the use of ICT mathematics before by other studies, it has not been reported
in different ways. Students have optimistic attitudes its use for analytic geometry before. How students’ academic
about this tool and support it to be included in regular achievement is influenced was determined using a quasi-
pre-calculus courses. Teachers should consider the use of experimental methodology. Students’ perception towards
Wolfram Alpha for their math courses. Wolfram Alpha was assessed using questionnaires.

Keywords:- analytic geometry, Wolfram Alpha, conics, II. WOLFRAM ALPHA

parabola, quasi-experiment.
Wolfram Alpha is a search engine which computes
I. INTRODUCTION answers and provides knowledge. Using algorithms and
expert knowledge, it automatically answers questions,
As Information and Communication Technologies
performs analysis and creates reports. [16,17]. This unique
(ICT) keeps developing, its use for education keeps growing.
online computation tool was built on Wolfram language and
Worldwide, schools, colleges and universities have invested
launched in 2009 by Wolfram Research company. It can be
on computers, tablets, software, internet and training[1]. It
accessed by typing https://www.wolframalpha.com on a web
has been demonstrated that when computers and mobile
browser, through its mobile application or by Mathematica
devices are properly used, academic achievement can be
software. Wolfram Alpha’s interface is presented in Fig. 1.
increased by the use of ICT at all education levels [2]. ICT
Wolfram Alpha is used for mathematics, science and
can provide more flexibility, improve feedback and
technology, society, culture, everyday life among others
monitoring, create interactive learning and motivate students
areas [18]. This powerful tool can solve math exercises of
[3,4]. The advantages and disadvantages of using ICT on
various topics as geometry, limits, differentiation, integrals
education depends on the subject and how it is implemented.
and even Laplace transformations for differential equations
The impact technology will have on students achievement is
[19-21]. Wolfram Alpha is also used for physics, for
not only influenced by infrastructure and software but also
instance to convert units, calculate solid and liquid
influenced by pedagogical practices [5]. Face-to-face classes
properties, calculate forces and search physical constants
disruption due to COVID outbreak forced ICT into education
[22]. Step-by-step solution is offered for math, chemistry
in unexpected ways. Students were forced to attend classes
and physics problems (Fig. 1).
online using computers, tablets or mobile phones and
teachers to provide the best tools available for a high-quality
education under adverse circumstances. Math teach-ers
worldwide had to adapt to the new reality and develop
rapidly and with little training their distance courses [6].

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Volume 7, Issue 1, January – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
how pre-service teachers increased significantly their
understanding of the subject [27].

Cabri 3D was analyzed for a spatial analytic geometry

for prospective math teachers. All participants had previously
worked with Cabri 2D. Using the work-sheets created for this
study and the software, participants successfully solved all
the problems. This study concluded that Cabri 3D aids
students to understand spatial analytic geometry as it offers a
good visualization and generates positive attitudes towards
learning this subject [28].

B. Wolfram Alpha Studies

Despite that Wolfram Alpha has not been studied for
analytic geometry it has been studied for other math areas.
The following literature review relates to Wolfram Alpha
studies applied to math courses.

Researchers reported having taught mathematics with

the aid of Wolfram Al-pha for two years, in the Czech
Fig. 1: Wolfram Alpha step-by-step solution Republic. The university created a system called “TRIAL”
where mathematics problems can be retrieved. Examinations
III. RELATED WORK took place using Wolfram Alpha and the most common
mistakes were related to syn-tax errors and incorrect analysis
A. Analytic Geometry Teaching with Math Software of results [29]. Wolfram Alpha was evaluated for single
The following section presents a literature review of variable calculus course with the aid of questionnaires.
previous studies for teaching analytic geometry with Students responses aided the study to conclude that this
GeoGebra and Cabri software. Most of previous re-search software adds interaction to class, motivates students to learn
has been performed using Geo Gebra software. and it should be used permanently for math courses [30].
Wolfram Alpha was utilized for a calculus course. Students
Geo Gebra was studied with a quasi-experimental answered questionaries to assess the effectiveness of this
methodology in analytic geometry courses for first-year complementary tool. Results indicate that most of the
university students when studying the circle topic. Pre and
students believed Wolfram Alpha to be a useful resource to
post tests’ scores analysis demonstrated that when students
learn and prepare for exams [15]. Researchers reported the
learnt using GeoGebra they increased their achievement and feasibility of replacing traditional computer algebraic system
understanding. Surveys performed to students showed that (CAS) with Wolfram Alpha for differential and integral
students have positive attitude towards the software and that calculus; however, they considered this tool not powerful
their spatial visualization improved due to the software use
enough to fully replace CAS in multivariate calculus.
[23]. Researchers performed a quasi-experimental study
Surveys demonstrated that students prefer to use Wolfram
where students from the experimental group were taught with Alpha as it has natural language programming and is al-ways
the aid of GeoGebra for learning circles. T-tests on scores updated. When this tool showed deficiencies, laboratory
revealed that GeoGebra aided students to perform better than practices were complemented with content from Wolfram
control group. Surveys to students evidenced positive attitude Demonstration Project and Math-World [31]. Wolfram Alpha
towards this software [24]. Using a similar methodology,
was compared to MathCad, analyzing their functionality and
GeoGebra was used for a geometry course with positive
examples for function integration. MathCad needs students to
outcomes for post-tests for experimental group; however, in a have prior knowledge to present results in an adequate form
delayed post-test, students who were taught under traditional and this software offers experience to migrate to more
methodology scored better than the ones taught with software
sophisticated software as Matlab and Mathematica. Wolfram
[25]. GeoGebra has been applied to higher secondary
Alpha shows step by step resolution for problems and results
analytic geometry too. Twelve grade students were divided are presented in different ways [32].
into control and experimental groups where pre and post
treatment evaluations were applied. T test revealed that The previous research for Wolfram Alpha analyzed in
students instructed with GeoGebra have betters scores this section used methodologies different to the one used for
specially for low achievers [26]. GeoGebra was studied for the study presented in this paper. Only the following two
learning cylindrical and spherical coordination in analytic studies used a quasi-experimental methodology; however, to
geometry courses for pre-service teachers. Most of the different math areas. Wolfram Alpha and HD calculator apps
studies reviewed were performed using a quasi-experimental were evaluated in a differential equations course where pre
methodology analyzing pre and post test scores; however, and post tests were conducted for control and experimental
this study applied a different methodology. Semi-structured groups. Data analysis revealed academic achievement
interviews were analyzed to assess the contribution of the improvement, positive perception towards the course and
software to improve students’ understanding. Results showed anxiety reduction [33]. For pre-service teachers, Wolfram
Alpha was used to help to improve performance in pre-

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Volume 7, Issue 1, January – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
calculus problems. Pre and post tests analysis showed an The main tool used for this course was Wolfram Alpha;
improvement as errors were reduced [34]. While there are however, other Wolfram content was also used for explaining
many studies under quasi-experimental methodology for theoretical concepts to the experimental group like Wolfram
GeoGebra, there are few works with this methodology for Demostration Project which has a collection of interactive
Wolfram Alpha and there is no prior study with this material.
methodology for analytic geometry using Wolfram Alpha.
V. METHOD
IV. MATERIALS USED FOR THIS STUDY
A. Research Method and Data Collection
Wolfram Alpha can be used for analytic geometry or This study used a quasi-experimental methodology (Fig.
coordinate geometry to determine algebraic equations, 4). It is called quasi because participants are not selected
compute geometric properties and plot many geo-metric randomly. This method had been used for other educational
figures. The examples presented in Wolfram Alpha web site studies described in the section of related work [23-26, 33-
correspond to the following groups: lines, planes, conic 34]. Students were divided into two groups, control and
sections, intercepts and quadrants (Fig. 2). experimental groups. Groups were created randomly from
students of a pre-calculus course. Control and experimental
groups were taught through Zoom as part of distance
education due to Covid pandemic. Experimental group learnt
with the aid of Wolfram Alpha while control group without
this tool. The same pre and post tests were implemented for
both groups to assess their academic achievement. Students
from experimental group answered Likert-scale
questionnaires to assess their perception of using Wolfram
Alpha. Questionnaires were available through Google Forms.

Fig. 4: Methodology used for this study

Fig. 2: Analytic or coordinate geometry examples presented This study was conducted for a pre-calculus course
by Wolfram Alpha during online education due to Covid outbreak. This course
took place at Universidad Estatal Península de Santa Elena
This study used the content for lines and the following from Ecuador. Each group had a similar number of
conic sections: circle, parabola, ellipse and hyperbola. participants. Control group had 35 students and experimental
Examples of the content used is presented only for parabola group had 36 students. Female participants accounted for
geometry. Fig. 3 shows input and output for the query almost 17% in both groups. Most of previous studies
parabola followed by an equation. Despite the query is short performed for analytic geometry used GeoGebra software.
the output is extended as it presents the graph and the main This research chose Wolfram Alpha tool rather than
properties of parabola. GeoGebra software because Wolfram Alpha does not require
to know programming language, offers step by step solutions
for queries, presents detailed outcome for short instructions
which can be written in natural language or structured math
format syntaxis and displays a good visual representation.
Wolfram Alpha can be used at universities and high schools.
Partial step-by-step solution is available for free version and
full step-by-step solution for paid version. Students had
access to full paid version. Syllabus was updated and
teaching materials prepared. Even tough writing instructions
in Wolfram Alpha is not difficult, as students’ native
language is Spanish, it was necessary to introduce some
commands in English. Control and experimental groups were
taught by the same lecturer. The analytic geometry’s topics
included in this study were line, circle, parabola, ellipse and
hyperbola. Students accessed the course through their
computers or mobile devices at home. Wolfram Alpha was
Fig. 3: Parabola equation problem solved by Wolfram Alpha. accessed through a web browser. Pre and post tests had the

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Volume 7, Issue 1, January – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
same difficulty level and included multiple choice questions Null hypothesis (H0) = no significant difference
and problems. Students taught with Wolfram Alpha were not between the means of post-test scores for experimental and
allowed to use this tool during examinations. control groups (p>0.05).

B. Objectives and Research Questions Alternative hypothesis (Ha) = significant difference

One of the objectives of this study is to assess the between the means of post-test scores for experimental and
effectiveness of using the web resource Wolfram Alpha for control groups (p<0.05).
teaching analytic geometry as part of a pre-calculus course
taught online during the pandemic. Another objective is to Group No of Mean S. D. t- P
evaluate students’ perceptions of using this tool and whether students value
they recommend its further use or not. Control 35 5.09 2.96
-3.66 0.000
Experimental 36 7.58 2.78
This work addressed these research questions: Table 2: Post-evaluation’s t-test for control and experimental
 Do students who learnt analytic geometry with the aid groups
Wolfram Alpha tool have better academic achievement
than the ones who learnt without this tool? The obtained p value is 0.000 (Table 2) which is less
 What are student’s perceptions of using Wolfram Alpha than 0.05; therefore, the null hypothesis is rejected, and the
web resource during an online course due to pandemic? alternative hypothesis accepted; consequently, there is a
significant difference between means of control and
C. Data Analysis experimental groups. This result means that students who
Pre and post tests’ scores were analyzed using descriptive learnt analytic geometry with the aid of Wolfram Alpha
statistics. Normality test was performed so t-test can be achieved significantly better than the ones who were taught
applied. Two-sample t-test was applied to evaluate the with traditional methodology, supporting the fact that this
difference between means and the effectiveness of the tool contributes to improve understanding and learning of
treatment. For surveys’ answers, Likert scale was translated mathematics. This finding is coherent with related previous
to the following numerical scale: 0 = strongly disagree, 1 = research using GeoGebra and Wolfram Alpha tools [23-26,
disagree, 2 = neutral, 3 = agree, 4 = strongly agree. 33-34].
Questionnaires’ results were analyzed with descriptive
statistics. Minitab v18 Statistical Software was used. Study Surveys applied to students will aid to analyze in deep
results were compared to previous research cited in the the effect of Wolfram Alpha and what students think about
related work section. this web resource. In general, questionnaires results (Table 4)
show that students have positive perception towards Wolfram
VI. RESULTS AND DISCUSSION Alpha which is aligned with previous Wolfram Alpha
research on other math topics [15, 30, 32-33]. The question
A two-sample t-test was performed to pre test scores for
with the least mean is the one which relates to cellphone or
both groups to determine if there is a significance difference tablet users. This may relate to the fact students working on
between means with an alpha value of 0.05 (Table 1).
these devices had to access the tool through a web browser as
Null hypothesis (H0) = no significant difference the mobile application had an extra cost. The question with
between the means of pre-test scores for experimental and the highest mean relates that student strongly support that
control groups (p>0.05). Wolfram Alpha should continue to be used for analytic
geometry teaching in precalculus. Students felt confident that
Alternative hypothesis (Ha) = significant difference this tool helped them during distance learning during
between the means of pre-test scores for experimental and pandemic. In face-to-face education, teachers were physically
control groups (p<0.05) in the classroom to clarify their doubts during the whole class
aiding students to do exercises; however, in distance learning
Group No of Mean S. D. t- P zoom sessions cannot last so long to avoid students to get
students value tired, distracted or bored. Wolfram Alpha contributed to
Control 35 1.43 1.24 understanding and problem solving during online learning.
1.15 0.252 This web resource helps to improve visual representation of
Experimental 36 1.11 1.06
equations and to distinguish between conic types. For a
Table 1: Pre-Evaluation’s T-Test For Control And conic, even though equation is well written, sometimes it can
Experimental Groups be difficult to determine correctly all its properties and its
graph. Students found this tool useful to verify their results
The means for both groups have similar values. As p and avoid making mistakes. Step-by-step solution was
value is 0.252, which is greater than 0.05, the null hypothesis particularly useful for line problems. The use of Wolfram
is accepted, which suggests that there is not a significant Alpha with other Wolfram resources as Wolfram
difference between means of control and experimental Demonstration Project and Wolfram Math World allowed the
groups. This implies that control and Wolfram Alpha taught teacher to introduce ICT in precalculus classes offering
groups are homogenous and have similar academic interactive and interesting activities to develop their learning
achievement before treatment. For post-test scores a two- process. Students who used this tool answered positively
sample t-test was performed to evaluate the effect of using towards permanent learning; however, further research
Wolfram Alpha.

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Volume 7, Issue 1, January – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
should be done around this topic like delayed post-test like of Wolfram Alpha had significant better academic
the ones performed by Bakar [25]. achievement than the ones taught without it when learning
analytic geometry in a pre-calculus course. Students’
perceptions about this web resource are positive and suggest
Item Min. Max. Mean S. D.
that this tool should be implemented in regular courses
I think Wolfram
because aids student to improve visual representation,
Alpha helped me to
2 4 3,677 0,541 understanding of theoretical concepts and step-by-step
learn analytic
problem solution. Wolfram Alpha is available through a web
geometry
browser and allowed students to performed interactive
I believe Wolfram
activities. This tool provided support to distance learning as
Alpha allowed me
students could obtain immediate feedback for many
to achieve 2 4 3,335 0,608
problems. Wolfram Alpha allowed TICs actively use during
permanent
math learning.
learning.
I think that Further research should be performed to determine if
Wolfram Alpha Wolfram Alpha allows permanent learning using delayed
2 4 3,516 0,677
adds interaction to post-tests. Other research should relate how Wolfram Alpha
geometry classes helps students to use more complicated software like
I liked using Wolfram Mathematica.
Wolfram Alpha for 2 4 3,613 0,558
this course ACKNOWLEDGMENT
I support that
Wolfram Alpha The authors acknowledge “Universidad Estatal
should continue to Peninsula de Santa Elena” for allowing this study to be
3 4 3,807 0,402 conducted and for providing access to Wolfram Alpha
be utilized for
geometry online licenses through “Corporación Ecuatoriana para el Desarrollo
classes de la Investigación y la Academia” (CEDIA).
I support that
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