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Scientia et Technica Año XXVII, Vol. 27, No. 04, diciembre de 2022. Universidad Tecnológica de Pereira. ISSN 0122-1701 y ISSN-e: 2344-7214

Learning of platonic solids by using augmented

reality and theory of didactic situations

Aprendizaje de los sólidos platónicos mediante el uso de la realidad aumentada y la

teoría de las situaciones didácticas

L.H. Carmona-Ramírez ; L.A. Montoya-Suárez

DOI: https://doi.org/10.22517/23447214.24771

Scientific and technological research article

Abstract—Augmented reality (AR) is a technology that

combines in real time environment information with the lived

information a virtual environment. This technology can be used

together with the theory of didactic situations (TDS) for the

learning of spatial geometry. The aim of this study was to evaluate

the effect of an intervention using AR and TDS on the learning of

Platonic solids (PS) in junior high school (7th degree) students. 34

students were allotted in both an experimental group (n=17) and a

control group (n=17). Students in the experimental group received

a didactic sequence related to PS using AR and those in the control

Group received a traditional study class with a 3D manipulatives.

The learning of students in both groups was assessed with a

semiquantitative scale (from 1 to 4) comparing data gathered at

the beginning of the study (pretest) with data obtained after

applying the didactic sequence or receiving the traditional study

class (posttest). Pretest data revealed that the starting knowledge

about PS was similar between both groups. Posttest data showed

that students of the experimental group learned better about PS

than students of the control Group. The use of AR and TDS could

improve the learning PS in students because they can easily

identify and interact with the common patterns of this geometric

elements as if these were real objects.

Index Terms— Augmented Reality, Geometry, Learning,

Platonic solids, Theory of Didactic situation.

Resumen— La realidad aumentada (RA) es una tecnología que

combina información del entorno en tiempo real con la

información vivida en un entorno virtual. Esta tecnología se puede

utilizar junto con la teoría de situaciones didácticas (TSD) para el

aprendizaje de la geometría espacial. El objetivo de este estudio

fue evaluar el efecto de una intervención usando RA y la TSD en

el aprendizaje de los sólidos platónicos (SP) en estudiantes de

secundaria (7º grado). Se asignaron 34 estudiantes tanto en un

grupo experimental (n=17) como en un grupo de control (n=17).

Los estudiantes del grupo experimental recibieron una secuencia

didáctica relacionada con los SP usando RA y los del grupo control

recibieron una clase de estudio tradicional con manipulativos en

3D. El aprendizaje de los estudiantes de ambos grupos se evaluó

con una escala semicuantitativa (del 1 al 4) comparando los datos

recogidos al inicio del estudio (pretest) con los obtenidos después

de aplicar la secuencia didáctica o recibir la clase de estudio

tradicional (postest). Los datos de la prueba preliminar revelaron

This manuscript was sent on June 21, 2021 and accepted on November 21,

2022.

que el conocimiento inicial sobre los SP era similar entre ambos

grupos. Los datos posteriores a la prueba mostraron que los

estudiantes del grupo experimental aprendieron mejor sobre PS

que los estudiantes del grupo de control. El uso de AR y TDS

podría mejorar el aprendizaje los SP en los estudiantes porque

pueden identificar e interactuar fácilmente con los patrones

comunes de estos elementos geométricos como si estos fueran

objetos reales.

Palabras claves—Aprendizaje, Geometría, Realidad Aumentada,

Sólidos Platónicos, Teoría de las situaciones didácticas.

I. INTRODUCTION

UGMENTED reality (AR) is an emerging technology that

combines the objects of both the real and the virtual world

[1], allowing the interaction in real time of the information

of the natural environment with the information that is lived in

the virtual environment [2]. Thus, the user can employ the

senses without leaving the real world. The implementation of

AR for teaching mathematics; particularly, in the field of

geometry in junior high school education has gained relevance

because it enables the study of three-dimensional models and

their two-dimensional representations from different points of

view [3]. This allows the students to understand the concepts of

geometric representation and their interaction in real time so

that the learner can discover the characteristics and fundamental

properties of three-dimension (3D) objects [4].

Several studies [1]; [5]; [6]; [7]; [8], have suggested that AR

favors the student motivation for learning mathematics, making

this technology a valuable resource for teachers at the time of

teaching and that can be applied in any area or level of training

due to its versatility.

On the other hand, several authors have carried out works on

the importance of AR in the learning and teaching of geometry,

because AR favors the acquisition of abilities for spatial

visualization by favoring the retention of the memory of the

objects to be studied over long term [9]; [10].

Although AR is a technological mediator, this cannot be

considered as relevant if it is not accompanied by a didactic

Scientia et Technica Año XXVII, Vol. 27, No. 04, diciembre de 2022. Universidad Tecnológica de Pereira.

263

mediator, such as the theory of didactic situations (TDS) used

in this study [11]. In didactic situations the student learns

through the adaptation to a learning environment that is a factor

of contradictions [12], difficulties, and imbalances, a little like

human society does, this knowledge is the result of the

adaptation of the student, which manifests itself through new

responses that are the mark of learning [13]. According to the

TDS, the selected learning environment (in this case AR) must

be designed in such a way that the knowledge resulting from

adaptation learning is similar to the knowledge that you want to

teach without the need to be explained by the teacher [14]

The synergy between TDS and AR can be considered as a

valuable element within the mathematics didactics that should

be taken into account by teachers at all education levels because

it facilitates the understanding of abstract concepts through the

manipulation of 3D objects with sequences of learning by

adaptation, which represent an invaluable couple for the

teaching-learning in geometry [1]; [15]

This research aimed to compare the effect produced by AR

and TDS on the learning of PS in seventh-grade high school

students. We hypothesized that the implementation of a didactic

sequence with AR mediated by TDS favors the learning of the

concepts of the PS in students.

II. METHODOLOGY

This study was approved by the local ethical committee on

education of the Master Program in Didactics of Mathematics

of the Universidad de Caldas. It was conducted for six months

and included two experimental groups of 7th-grade students in

a public urban high school from Manizales, Caldas, Colombia.

A. Study design

The approach of the present research was quantitative with a

descriptive and correlational scop, from which two hypotheses

were formulated (null and alternative) about the possible

relationships between the two variables that were included in

the problem of the study. Further, the study was considered as

quasi-experimental with two intact groups (experimental (EG)

and control groups (CG)) from 7th degree from the similar high

school education with similar characteristics and particularities

(i.e.: age, economic stratum, place, parents' education, amongst

others). To note, all the students attended the mathematics class

with the same teacher.

The EG was composed of 17 students who were taught with

the use of an AR App (AR Platonic Solids (Spanish Version

1.0)) designated by us using the academic license by Unity

Technologies, which is a Danish-American video-game

software development company based in San Francisco, CA,

USA [16], and a didactic sequence called PS Learning using

AR designed with the didactic approach of the TDS that was

composed by four didactic situations: action, formulation,

validation, and institutionalizing.

Briefly, during the first 45 min the students received the TDS

guide, were trained in the management of the AR App, and

performed the first didactic situation (action), which consisted

in solving the question (cognitive conflict): “Is it possible to

made convex polyhedrons with regular polygons?”. After a

break of 10 min, the second lapse of 45 min was used for

performing the second didactic situation (formulation), in

which the students came together to reach an agreement about

the question of the first didactic moment. A second 10 min

break was allowed to continue with a 45 min session in which

was performed the third didactic situation (validation) by using

mathematics demonstration by Euler`s theorem. Furthermore, a

45 min session was used for performing the four didactic

situations (institutionalizing), in which the students made an

oral explanation of their findings and the teacher clarify

inquietudes, mistakes, and concepts and finally made a final

conclusion for institutionalizing the concept about what a PS is.

The control group also included 17 students, who did not

receive the intervention but were taught with the same theory

through the master class and the use of manipulatives. Two

independent master classes with 90 min of duration were

performed, and additional homework for an independent task

for the students with a mean time of 90 min was recommended.

An entrance test (pretest) and an exit test (posttest) were applied

to both groups. In line with this, a rubric (Table I) was designed

for evaluating the gained learning of the students in relation to

a geometry knowledge exam that included pre and post-test

questions with both dichotomic natures (yes or no type) or of

free answering (open).

TABLE I

RUBRIC FOR ASSESSING THE ADVANCE OF LEARNING OF PLATONIC SOLIDS IN

TWO GROUPS OF 7TH HIGH SCHOOL STUDENTS

Degree of

learning

Score

Specific criteria

Low

From 0 to 5 right answers in the questionary

used for both pre and post-test.

Basic

From 6 to 10 right answers in the

questionary used for both pre and post-test.

Advanced

From 11 to 15 right answers in the

questionary used for both pre and post-test.

Superior

From 16 to 20 right answers in the

questionary used for both pre and post-test.

A summary of the design of the study is shown in Fig. 1.

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Fig. 1. Schematic representation of the study`s design.

B. Statistical analysis

Initially, data was evaluated for normality by a Shapiro-Wilk

test, after this test data exhibited a non-parametric distribution

(P < 0.05), reason why the medians of the groups were

compared by using a non-parametric ANOVA (Kruskal-Wallis

test), followed, when necessary, by either non-paired and paired

non-parametric comparisons (i.e.: Wilcoxon and U Mann-

Whitney test. A P < 0.05 was accepted as statistically

significant for all the tests.

III. RESULTS

All 34 participants could successfully finish this study. In

general, students from both groups were receptive and

motivated during the experiment, but the students from the EG

group were apparently more enthusiastic and demonstrated

interest in learning through the use of AR plus TDS.

From the statistic point of view the nature of the data

gathered in the present study was non-parametric, such as

presented in table II after performing a Shapiro-Wilk test.

TABLE II

SHAPIRO-WILK TEST FOR EVALUATED MEDIAN GROUPS AT PRE- AND POST-

TEST

Then we compared the medians from the 4 evaluated groups

by using a Kruskal-Wallis test in order to diminish the

probability of accept or reject the null hypothesis when this was

true or false (type I or III errors). This a-priori test is

fundamental before comparing the medians by non-parametric

paired tests (Table II).

TABLE III

RESULTS FOR KRUSKAL-WALLIS TEST PERFORMED TO ALL MEDIAN GROUPS AT

THE PRE- AND POST-TEST MOMENTS.

Test Statistics

a,b

Var_Rpta

Kruskal-Wallis H

38,119

Asymp. Sig.

0.000

a. Kruskal Wallis Test

b. Grouping Variable: Group_PrPo

Finally, non-paired and paired non-parametric tests were run

to compare the potential differences between the medians from

the groups. In table III appears the results from group medians

compared at the pre-test moment, which demonstrated that both

medians were statistically similar.

TABLE IV

RESULTS FOR U-MANN-WHITNEY TEST PERFORMED TO THE MEDIAN GROUPS

AT THE PRE-TEST MOMENT.

Estadísticos de prueba

a,b

Var_Rpta

Mann-Whitney U

127,500

Wilcoxon W

280,500

-0.797

Asymp. Sig. (2-tailed)

0.426

Exact Sig. [2*(1-tailed Sig.)].

unilateral)]

0.563

Grouping Variable: Group_PrPo

Not corrected for ties.

In table IV are show the output data from the paired

comparisons form both group medians at the post-test moment.

At this point, we observed and significant difference between

both groups.

TABLE V

RESULTS FOR U-MANN-WHITNEY TEST PERFORMED TO THE MEDIAN GROUPS

AT THE POS-TEST MOMENT.

Test Statistics

Var_Rpta

Mann-Whitney U

52,000

Wilcoxon W

205,000

-3,678

Asymp. Sig. (2-tailed)

0.000

Exact Sig. [2*(1-tailed Sig.)].

Grouping Variable: Group_PrPo

Not corrected for ties.

0.001

Table V shows the results of the non-parametric test

(p=0.001).

Regarding the statistical analysis from gathered data in this

study, we observed that the entrance knowledge about spatial

geometry, and particularly that related to PS (as measured by

the pretest) was similar between both student groups. Once the

treatments, either magistral class plus manipulative objects or

AR plus TDS, were applied to the student groups, we noticed

Tests of Normality

Kolmogorov-Smirnov

Shapiro-Wilk

Group

Statistic

Sig.

Statistic

Sig.

Var_Rpta

CGPr

0.497

0.000

0.470

0.000

EGPr

0.440

0.000

0.579

0.000

CGPo

0.469

0.000

0.533

0.000

EGPo

0.300

0.000

0.752

0.000

a. Lilliefors Significance Correction

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265

that those students from the EG shown significantly (P= 0.001)

better rubric posttest score than the students from the control

group (Fig. 2).

Fig. 2. Assessment of the degree of learning according to the rubric scale

in the experimental (EG) and control groups (CG) during both pre and post-

test interventions

IV. DISCUSSION

AR is an innovative technology that can complement the real

world environ-ment with a smartphone to generate sensory

inputs [15]. These virtual components seem to coexist with the

real one in the same spaces to improve the perception of reality

by students and enrich the information to be disclosed [17].

On the other hand, a didactic situation is a set of relationships

explicitly or implicitly established between a student or a group

of students, some environment (including instruments or

materials, in this case, AR), and the teacher in order to allow

students to learn some knowledge [11]. The union of both

strengthens the teaching-learning process of mathematics,

giving the student the possibility to explore and learn naturally,

while the teacher designs the activities so that the student

develops competencies and recreates learning from a novel

didactic perspective.

The present study compared the traditional didactic approach

to teach PS with a didactic method that combined TDS plus AR.

The results of our research showed that this last didactic method

was suitable and better than the traditional classroom approach

for the teaching of the PS because the degree of learning was

significantly higher in the students of the EG.

We performed an exhaustive bibliographic search focused

on the use of AR and didactic mediators such as TSD. However,

according our searching results, we only could find a paper

about these topics, which is a recent study by our research team

that evaluated the AR and the didactics engineering as a didactic

mediator [18]. In general, the results obtained in the mentioned

study showed that the students significantly learned the

concepts of quadric surfaces thanks to the use of AR, as well as

the didactic sequence mediated with the guide based on didactic

engineering. This could be a very important indicator for

researchers in this field, because although AR favors

visualization it is necessary to guide it with a mediator or

didactic approach [18].

Currently, the use of smartphones by students has become a

problem for teachers because many students lose attention

during traditional class and prefer using these devices to play

videogames, see musical videos, consult social networks, and

so forth. Thus, the teachers should change their traditional

didactic methods including the use of smartphones as didactic

tools and no as distraction elements during the teaching and

learning process.

The present study demonstrates that the inclusion of

smartphones in the teaching-learning process of PS through a

TDS plus AR method could be an important approach for the

teaching of geometric concepts and spatial learning and

consequently favor the learning of the students in a friendly

environment without the apparent deprivation of their

smartphones[16];[18]; [19].

Notably, we observed that some students from the control

group were anxious about using their smartphones during class.

In many instances was necessary to ask some of them that did

not use their smartphones during the class of PS to avoid some

distraction that could produce adverse effects during their

learning process and disturbing the learning process of those

students that were interested in this classic approach of PS

teaching. At this point, it is possible to consider that, in some

extension, the significantly lower degree of learning of the

students of the control group (as observed in the post-test) when

compared to the students of the EG was also due to the

distraction produced by the smartphones of those students that

showed less interest during the class.

Traditionally, the most complicated in teaching geometric;

particularly in the teaching solids, is that these figures need a

3D representation, thus some teachers are facing the problem of

representing this type of objects in a traditional board by using

2D geometric solid plots, and in many situations, these plots do

not represent the exact form of the 3D object (solid). The use of

manipulable (3D) solids is a common methodological approach

but this kind of didactic tool does not let to dissect and study

the entire structure, something different happens with the use of

the AR [18]. Maybe, the use of smartphones improved the

learning of PS in the students of the EG.

This study had several limitations. First, the smartphones

used in this study (owned by the students) were obsolete.

Second, our results do not allow to distinguish if the

improvements of learning were due to AR or TDS. Third,

another limitation could be the small sample of participants

enrolled in the study, which could prevent the generation of

robust conclusions. Finally, our AR App only can be played on

smartphones with Android platforms Oreo 8.1 or successive

versions. It is also important that the smartphones used for this

study had memory storage of 45 MB or more.

V. CONCLUSION

We concluded that those groups of students that were trained

with the DST guide and instructed AR App showed

significantly better comprehension of the PS when compared

with the control group. Furthermore, from a subjective point

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of view, we appreciated that the students from the EG

presented a more enthusiastic and motivated behavior in the

learning of PS in comparison to the students from the control

group.

However, the feasibility of using this approach depends on

students' access to Android devices of given minimum

specifications, which are not always guaranteed in contexts

such as Latin America.

ACKNOWLEDGMENT

The authors thank the support from the Research Group on

Education and Educators Training -EFE of the Universidad

Católica de Manizales, Manizales, Caldas. We also thank

Universidad de Caldas

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Luis Hernando Carmona Ramirez

received a Bachelor's degree in education

with emphasis in mathematics, he also holds

a specialization degree in didactics of

mathematics and physics and a master's

degree in didactics of mathematics from the

University of Caldas - Colombia. He is

currently an instructor professor of the

bachelor's degree in mathematics and

physics of the faculty of education and UAFCNyM, and of the

master's degree in science didactics at the Catholic University

of Manizales, Manizales. E-mail: lucarmona@ucm.edu.co. Her

research interests include the field of science didactics and

applied mathematics. He is a member of the Education and

Educator Training Research Group –EFE.

ORCID: https://orcid.org/0000-0002-4136-851X.

Luis Angel Montoya Suárez received a

Bachelor's degree in education with

emphasis in mathematics, he is master’s

degree in didactics of mathematics from the

University of Caldas – Colombia.

Email:luisangelmontoya@hotmail.com.

ORCID: https://orcid.org/0000-0003-0495-3709