Content uploaded by Seckel María José
Author content
All content in this area was uploaded by Seckel María José on Dec 10, 2021
Content may be subject to copyright.
mathematics
Article
Primary School Teachers’ Conceptions about the Use of
Robotics in Mathematics
María JoséSeckel 1, * , Adriana Breda 2, Vicenç Font 2and Claudia Vásquez 3
Citation: Seckel, M.J.; Breda, A.;
Font, V.; Vásquez, C. Primary School
Teachers’ Conceptions about the Use
of Robotics in Mathematics.
Mathematics 2021,9, 3186. https://
doi.org/10.3390/math9243186
Academic Editors: Jay Jahangiri and
Michael Voskoglou
Received: 1 November 2021
Accepted: 7 December 2021
Published: 10 December 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Departament of Didactics, Universidad Católica de la Santísima Concepción, Alonso de Ribera,
Concepción 2850, Chile
2
Departament of Linguistic and Literary Education, and Teaching and Learning of Experimental Sciences and
Mathematics, Universitat de Barcelona, Passeig de la Vall d’Hebron, 171, 08035 Barcelona, Spain;
adriana.breda@ub.edu (A.B.); vfont@ub.edu (V.F.)
3Departament of Mathematics, Campus Villarrica, Pontificia Universidad Católica de Chile,
Villarrica 4930445, Chile; cavasque@uc.cl
*Correspondence: mseckel@ucsc.cl
Abstract:
Learning about the conceptions used by primary school teachers towards the use of robotics
in class is essential as the first step towards its application in the classroom. Therefore, with the
purpose of describing the understanding applied when teaching and learning mathematics use
educational robots, research was conducted by means of mixed methods using a descriptive design
by survey. Such research consisted of closed questions (Likert-type scale from 1 to 5) and open
questions, given to 83 primary school teachers who currently teach students in the first years of
school (First to Fourth grade) in two Chilean districts. The results showed that in general, there is a
positive predisposition towards the addition of robots in the learning and teaching of mathematic
processes during the first years of school, even though teachers claim there is a struggle to incorporate
robots in their lessons due to the high number of students and the reduced space in their classrooms.
Keywords: math education; educational robotics; teachers’ conceptions
1. Introduction
The agenda of general research related to the study of conceptions determines that
such understandings are the ones which shape the meaning we give to things. This is
because conceptions are formed from an individual perspective on the one hand, because
of our own experiences, and on the other hand, from a social perspective as a result of the
confrontation of our own elaborations with others [1].
Given the fact that teaching involves both personal and social perspectives, several
studies have been carried out, focusing on investigating the teachers’ conceptions, particu-
larly the ones connected with teaching mathematics. With the intent of understanding the
teachers’ pedagogical methods and to know how they conceive teaching, many studies
have been concerned with identifying their conceptions, not only about mathematics [
2
–
5
],
but also about their teaching and learning processes [
6
–
8
]. In this view, several works
have focused on identifying the teachers’ conceptions of teaching different mathematical
objects [
9
,
10
]; affective aspects of learning and class-design [
2
,
11
,
12
]; evaluation and com-
petence [
13
]; teaching through the process of problem solving [
14
–
19
]; and the use of ICT
resources [4,20–22].
In connection with the use of ICT resources, currently we can find research that seeks
to study pedagogical practices and the teacher’s conceptions in terms of the use of robots
or to the development of computational thinking through robotics in classrooms [
23
–
30
].
Additionally, we can find examples of teachers’ training in robotics [
31
], that focused on
the STEM approach [
32
], as an active learning development [
33
]. Particularly, we can find
research on the conceptions of the use of pedagogical robots at early ages [34–36].
Mathematics 2021,9, 3186. https://doi.org/10.3390/math9243186 https://www.mdpi.com/journal/mathematics
Mathematics 2021,9, 3186 2 of 17
Those studies show ambiguous results, because even though teachers value positively
the use of robots in their classrooms, they also consider that the use of an educational
robot may affect their perceptions and practices because it includes new knowledge and a
change in the methodology of teaching and learning, different from the traditional ones.
Specifically, the perception of teachers regarding the use of pedagogical robots during the
early ages, in particular the use of Bee-bot [
34
,
35
,
37
], is positive for the technical aspects
and for the pedagogical and social aspects as well. However, they also point out that it
presents difficulties and limitations.
Moreover, different trends and approaches regarding teachers’ training, at their initial
stages as well as throughout their careers, suggest that the research on and reflection of
teachers’ practices is a key strategy to improve teaching. Among the diverse theoretical
approaches in mathematics, the Onto-Semiotic Approach (OSA) to cognition and mathe-
matical instruction [
38
], suggests the Didactic Suitability Criteria (DSC) as a tool to organize
the reflection of teachers’ training, especially when such approach is focused on improving
the teaching and learning processes of mathematics.
The DSC is a combination of six criteria, which break down in components and
indicators that guide the improvement of implemented teaching processes. Said criteria
(along with their components and indicators) can be inferred from the reflection of teachers’
training on their personal or external teaching practices, even when there has been no
development process to teach them. The DSC’s components as well as their indicators
have been proposed considering the current trends regarding the teaching of mathematics,
the National Council of Teachers Mathematics’ Principles [
39
] and the research’s results in
Didactics of Mathematics, which have an extensive consensus in the community.
The DSC is a multidimensional construct that consists of six partial criteria [
40
]:
(1) epistemic facet criteria, to value whether if the math taught is “good math”; (2) cognitive
facet criteria, to value prior to initiate the teaching process whether if what is intended to
be taught is within a reasonable distance from what is already known by the students and,
after the process, if the learnings accomplished are close to what was intended to be taught;
(3) interactional facet criteria, to value if the interactions resolve the students’ doubts and
difficulties; (4) mediational facet criteria, to value the adaptation of the materials and
temporal resources used through the training process; (5) affective facet criteria, to value
the students’ implications (interests and motivations) through the training process; and
(6) ecological facet criteria, to value the adaptation to the educational training project, to the
curricula guidelines and to the conditions of the socio-professional environment, among
others. Each criterion breaks down into components and indicators (Table 1), which makes
it possible for this construct to be useful to value the suitability of the teaching and learning
processes of mathematics [41,42].
Table 1. Didactic suitability criteria and components.
Suitability Criterion Components
Epistemic Errors, ambiguities, richness of processes, representativeness
of the complexity of the mathematical object
Cognitive Prior knowledge, curricular adaptation to individual
differences, learning, high cognitive demand
Interactional Teacher–student interaction, students’ interaction, autonomy,
formative assessment
Mediational Materials resources, number of students, class schedule and
conditions, time
Affective Interests and needs, attitudes, emotions
Ecological Curriculum adaptation, intra- and interdisciplinary
connections, social and labor usefulness, didactic innovation
Various research have observed a phenomenon that repeats frequently: the com-
ponents and indicators of the DSC proposed by the OAS function as regularities in the
teachers’ approach when they value an episode or justify that a didactics proposal signifies
Mathematics 2021,9, 3186 3 of 17
an improvement, without having been taught the usage of such tools to guide their reflec-
tion. That is to say, the teachers’ comments can be considered as proof of the implicit use
of any of the DSC components or indicators as a guide which must orientate the teachers’
practice for a correct application [
43
]. We began this research by asking teachers who do not
know about the didactic suitability criteria construct what they think of the incorporation of
robots in the teaching and learning processes of mathematics. In a certain way, we put them
in a position to reflect on a possible improvement of their teaching of mathematics, with
the sole purpose of analyzing and organizing their statements applying the components
and indicators of the DSC.
Breda, Font and Pino-Fan [
44
], explain this presence of DSC in the teachers’ approach
as having been taken from an existing consensus (from different origins) established
among the community of mathematics’ teachers regarding the aspects to be considered
to accomplish a good math class. In contrast, whenever it is possible to determine a
criterion that guides the teachers’ practice, such criterion can be interpreted as a belief,
if we understand it according to Peirce [
45
], and zero criterion as a disposition towards
action. The group of criteria, also according to Peirce, can be understood as the teachers’
conception. This way of understanding the conception as a group of criterions, is also
considered by some investigators of the Didactics of Mathematics (i.e., [46,47]).
Therefore, one of the uses of DSC in the investigational field, is the consideration of
the teachers’ conceptions, assuming the premise that when consulting about the implemen-
tation of an innovation in the mathematics’ teaching and learning processes, the teachers
are faced with a situation of analysis, in which their conceptions are revealed and such
conceptions are based implicitly in the DSC [
40
,
43
,
48
–
51
]. In that sense, in the present study
it is intended to answer the following question: Which are the conceptions that primary
school teachers have regarding the use of educational robots in the process of teaching and
learning mathematics? This question is relevant in the context of the Chilean ministerial
initiatives towards introducing Computational Thinking in the Math and Science syllabus.
To answer the matter, the study has the objective to characterize teachers’ conceptions on
the use of robotics when teaching and learning mathematics during the early ages at school.
2. Materials and Methods
This study was conducted by means of a mixed methods approach and the methodol-
ogy applied was an explorative-descriptive type [
52
]. A non-experimental research method
design was followed, descriptive by means of a survey which was characterized by being
the first approach to the phenomenon researched, providing general data of the population
of interest [53].
2.1. Participants
The sample of participants consists of a probabilistic sampling, which represents the
teachers from educational centers that have a practice agreement with a university from
Maule’s Region. Said sample was selected by means of a two-stage cluster sampling method
with probabilities proportional to size. In the first stage, 30 different size sample institutions
were obtained, and in the second stage, 2 out of 4 grades of those institutions were randomly
selected, then the teachers of each of the grades obtained were selected. Therefore, the
final sample consists of 83 teachers from First to Fourth grades of primary school in Chile
(74 women and 9 men), who work at a public and a partially state subsidy school in Talca
and Curicódistricts (Chile). Among these participants, 19 have a masters’ degree, 24 have
a post-graduate degree in mathematics and 8 of them have a training course in robotics.
Some of the participants have more of these academic formations (masters’ degree and
robotics training, masters’ degree and post-graduate degree or post-graduate degree and
robotics training). In terms of years of professional experience, 10 participants have at least
1 year of experience, 17 have between 1 and 3 years of experience and 56 have more than
3 years.
Mathematics 2021,9, 3186 4 of 17
2.2. Instruments and Variables
The instrument applied (see annex 1 in see Annex 1 in Supplementary Materials),
used during the COVID-19 sanitary confinement during the year 2020, involved a total of
42 items (see details in Seckel, Breda and Font [
54
]) that were presented in two sections; the
first one entailed closed questions (39 items) and the second one, open questions (3 items).
The validation process was conducted through the expert’s judgement technique, in
which 5 judges were involved, and had replied to an online validation survey. The items
were evaluated through four parameters: clarity (the item is easily understood, that is to
say, its semantics and syntaxis are correct), coherence (the item has a logic relation with the
dimension or the parameter analyzed), relevance (the item is essential or important, that is
to say, it must be included) and sufficiency (the items which belong to a same dimension
are sufficient to obtain the extent of the dimension) [
55
]. The closed question items were
evaluated considering the 4 parameters, whereas the open question items were evaluated
through the parameters of clarity, coherence and relevance. The process of reviewing the
questions handed out to the judges was conducted in two stages. In the first stage, the
Content Validity Coefficient (CVC) was applied, by which it was able to measure the grade
of agreement among the experts regarding the items and the instrument in general. Once
the coefficients have been obtained, their values were analyzed according to the scales:
unacceptable (CVC minor to 0.60), deficient (CVC major or equal to 0.60 and minor or
equal to 0.70), acceptable (CVC major to 0.71) and minor or equal to 0.80), good (CVC
major to 0.80 and minor or equal to 0.90) and excellent (CVC major than 0.90) [
52
]. In
the second stage, the relevance to retain those items that had accomplished a CVC major
to 0.71 and minor or equal to 0.80 (validity and acceptable concordance) was analyzed
according to the expert’s recommendations. Moreover, in this stage, the instrument was
revised and corrected following the experts’ suggestions, according to four scopes: [
53
]
(i) appropriate use of words, (ii) adequacy of the questions’ meaning so that they can
quantify only one objective, (iii) adding an item to increase the adequacy of a specific
dimension. Therefore, the instrument evaluated obtained a validity coefficient of 0.88.
The first section considered a Likert-type scale from 1 to 5 (1 completely disagree and
5 completely agree, 3 neither agree nor disagree as a middle point) with 39 affirmations
about the learning and teaching with robots (which will be referred as to dimensions from
now on) and articulated based on DSC (epistemic, cognitive, interactional, mediational,
affective, ecological), its components and indicators (Table 1).
The teaching dimension considers 21 affirmations or items referring to the impact of
the use of pedagogical robots when teaching mathematics, whereas the learning dimension
comprises the subsequent 18 items related to the learning of mathematics by using robots
as a pedagogical resource. Moreover, these affirmations are grouped according to the
didactic suitability criteria that have been taken as a foundation for its reasoning, as shown
in Table 2.
Initially, the participants’ conceptions were inferred from the annex of closed questions
considering 2 variables: the first one is the known dimension (teaching dimension and
learning dimension), and the second one is the didactic suitability criterion, which is
reflected in their score of each affirmation.
Furthermore, in terms of personal and professional characteristics of the participants,
the following information was collected: gender (1 = woman; 0 = man), academic formation
(postgraduate degree in math or initial degree, master’s degree and/or training in robotics)
and classroom experience (1 = less than a year; 2 = between one year and three; 3 = more
than three). As mentioned before, in reference to the variable of the academic formation, it
must be pointed out that the formation options are not limited—a participant may have
answered more than one option (i.e., masters’ degree and robotics training, masters’ degree
and post-graduate degree or post-graduate degree and robotics training).
The second section considered three open questions, which were supposed to be
answered after watching a video in which the Blue-Bot robot was presented along with
its programming functions. In the first open question, a math assignment was introduced
Mathematics 2021,9, 3186 5 of 17
with the following query: “What suggestions would you provide to the teacher to address
the start, development and ending of the class?”. Then, the following questions were
introduced: “What benefits would have the use of the pedagogical robot during a math
class?” “What difficulties might be experienced while using the pedagogical robot during
a math class?”.
Table 2. Items distribution according to dimension and didactic suitability criterion.
Dimension Didactic Suitability
Criterion Items
Teaching
Epistemic 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10
Cognitive 11 and 12
Interactional 13 and 14
Mediational 15, 16 and 17
Affective 18 and 19
Ecological 20 and 21
Learning
Epistemic 22, 23, 24, 25, 26 and 27
Cognitive 28 and 29
Interactional 30 and 31
Mediational 32 and 33
Affective 34, 35 and 36
Ecological 37, 38 and 39
2.3. Results Analysis
The analysis of the data collected followed a quantitative validity model; the qual-
itative data support the quantitative validity [
56
]. Therefore, the survey’s quantitative
data were analyzed first, and then the qualitative data. Finally, the quantitative data were
validated with the qualitative data.
In the quantitative study, different strategies of analysis were conducted. At first,
the interest was focused on identifying professional profiles similarly enough to define a
homogenous group internally but them being externally heterogenous. To do so, a group
analysis or hierarchical cluster was conducted through the Ward group method, with the
Euclidean distance squared as the measure of proximity among the groups or clusters. For
the identification of the groups, we used the Ward’s linkage dendrogram with a maximum
distance of around 5. Once the clusters or groups were established, the next step was to
explore the distribution (in terms of center, position, and dispersion) of the participants
item’s scores regarding the impact of the use of pedagogical robots in the mathematics
learning and teaching processes according to the different didactic suitability criterions
(Table 3), thus inferring the participants’ conceptions. Finally, a descriptive analysis was
carried out in terms of the grade of relation between the experience or previous academic
formation and the participants’ conceptions according to the different professional profiles
established. Therefore, the statistics test or Pearson’s chi-squared test is used, which allows
to contrast the independence hypothesis in a contingency table, among qualitative variables.
Moreover, said results are complemented with the force estimation association as of the
Spearman’s rank correlation coefficient. This measure sets between
−
1 and 1 and allows
to conclude in a larger grade of association if the value is near the extremes (
−
1 negative
relation and 1 positive) or lesser in cases where the value is nearly 0.
The qualitative data were analyzed through the content analysis method [
57
]; there-
fore, the didactic suitability criterion was defined as the theoretical reference from which
the participants’ discourse was analyzed. Said criterion was also considered in the design
of the closed-questions survey. Consequently, once the data were collected, the following
occurred: (1) the content corpus was set up; (2) the categorization of the units’ analysis
took place; (3) the relation between the qualitative and quantitative analysis was deter-
mined. Moreover, to secure the validity of the qualitative data the study considered the
regulatory dependence criterion [
58
], that consists of a control process of the data inter-
Mathematics 2021,9, 3186 6 of 17
pretation made by a group of examinators composed of a national collaborator and two
international collaborators.
Table 3.
Frequency (percentage) of participants according to the variables analyzed within each cluster.
Characteristic Variable Cluster 1 Cluster 2 Cluster 3
Gender Woman 47 (87.0%) 18 (94.7%) 9 (90.0%)
Academic
formation
Postgraduate 15 (27.8%) 6 (31.6%) 3 (30.0%)
Master’s 8 (14.8) 2 (10.5%) 9 (90.0%)
Course 6 (11.1%) 2 (10.5%) 0 (0.0%)
Years of
previous
experience
Less than 1 year 5 (9.3%) 4 (21.1%) 1 (10.0%)
Between 1 and 3 years
16 (29.6%) 0 (0.0%) 1 (10.0%)
More than 3 years 33 (61.1%) 15 (78.9%) 8 (80.0%)
Grades of
previous
experience
First 37 (68.5% 14 (73.7%) 8 (80.0%)
Second 39 (72.2%) 11 (57.9%) 7 (70.0%)
Third 34 (63.0%) 11 (57.9%) 6 (60.0%)
Fourth 30 (55.6%) 15 (78.9%) 4 (40.0%)
3. Results
3.1. Quantitative Study
3.1.1. Participants’ Professional Profiles
Through the group analysis, three groups or clusters of participants were identified
who had similar experience or academic formation as well as shared the concepts of
using pedagogical robots in the learning and teaching mathematics. The first cluster
was formed by 54 participants (65.1%) whereas the second one was by 19 participants
(22.9%) and the third by 10 participants (12.0%). In Table 3, some of the personal and
professional characteristics are exhibited in relation to the participants’ previous experience
and academic formation, pursuant to the three identified clusters.
As for Cluster 1, formed by 54 participants, among them 87% were women, who
mainly have a postgraduate degree in Math, even though they do not exceed the 28% of
the teachers, whereas those with an introductory course in robotics are the least frequent
variable. In terms of previous experience, most of the participants (61.1%) have more than
3 years of practice, focused on the first grade’s education (72.2% for second grade and
68.5% for first grade).
In Cluster 2, only 1 person is a man out of the 19 participants (94.7% women), and
likewise with the above-mentioned case, most of the participants have a postgraduate
degree in Math, and do not exceed a third of the grand total of the teachers in this group
(31.6%). As for the years of experience in the classroom, 78.9% of the participants have been
teaching for more than 3 years, whereas 21.1% do not surpasses the first year of practice.
Moreover, this previous experience focused mainly on the fourth and first grades (78.9%
fourth grade and 73.7% first).
Finally, in Cluster 3 again women are predominant, (90%) amongst the 10 participants
of this group, who in the majority have a master’s degree (90%) none of them have taken
an introductory training course in robotics for teaching. As for the previous experience,
80% of the participants have been teaching for more than 3 years at primary level, mainly
at first and second grades (80 and 70%, respectively).
3.1.2. Participants’ Conceptions
In this section we explore the results through a double-entry table where the measures
of centralization and dispersion of the scores related to the first variable (teaching/learning)
are shown in the columns; and in the rows scores related to the didactic suitability criterion.
The results obtained are shown in Table 4.
Mathematics 2021,9, 3186 7 of 17
Table 4. Statistics of the participants’ scores.
Didactic
Suitability
Criteria
Teaching Dimension Learning Dimension
A SD P% A SD P%
Epistemic 4.3 0.81 P10 = 3.2 4.4 0.74 P10 = 3.5
Cognitive 4.4 0.86 P10 = 3.0 4.5 0.77 P10 = 3.5
Interactional
4.4 0.88 P10 = 3.0 4.4 0.76 P10 = 3.0
Mediational
3.5 1.06 P30 = 3.0 4.4 0.78 P10 = 3.0
Affective 4.4 0.75 P10 = 3.5 4.5 0.72 P10 = 3.7
Ecological 4.2 0.85 P10 = 3.0 4.6 0.70 P10 = 4.0
Global 4.2 0.74 P10 = 3.1 4.5 0.66 P10 = 3.6
A (average); SD (standard deviation); P% percentile.
Globally, the average scores of the participants were bigger towards the learning
of mathematics with robots (4.5 points), but in both cases greater than the indifference
position. On the other hand, in reference to didactic suitability criteria, we noticed that in
the first variable, in teaching as well as in learning, the average scores surpassed the three
points of indifference. Nevertheless, the mediational didactic suitability criterion presented
a bigger variation between both dimensions, with more than one point of difference from
the average score, being the lowest towards teaching. Hence, it can be inferred that among
the participants there is a favorable conception towards the incorporation of robot in the
mathematics’ learning and teaching processes. This means that the participants have a
conception with certain balance for the value on the advantages represented by the incor-
poration of robots (mathematics that facilitates relevant processes such as generalization
and pattern guesswork, better motivation for students, better learning, among others) even
though the aspect the participants see as the less beneficial is related to the mediational
didactic suitability criterion.
Later, it was analyzed further on the results according to the three groups or clusters
previously detailed (Table 3), and said results are presented in Table 5.
Table 5. Average score of the participants according to their cluster.
Didactic
Suitability
Criteria
Cluster 1 Cluster 2 Cluster 3
Teaching Learning Teaching Learning Teaching Learning
A SD A SD A SD A SD A SD A SD
Epistemic 4.7
0.32
4.9
0.22
4.0
0.50
4.0
0.30
2.7
0.82
3.0 0.78
Cognitive 4.8
0.46
4.9
0.28
4.3
0.45
4.2
0.48
2.7
0.98
3.0 0.86
Interactional 4.8
0.38
4.8
0.44
4.1
0.64
4.1
0.50
2.7
0.88
3.2 0.94
Mediational 3.9
0.98
4.6
0.62
3.0
0.66
4.4
0.53
2.3
0.69
3.2 0.95
Affective 4.8
0.31
4.8
0.32
4.1
0.40
4.3
0.44
3.0
0.80
3.1 0.90
Ecological 4.5
0.60
4.9
0.17
3.9
0.57
4.3
0.44
2.8
0.92
3.3 1.07
A (average); SD (standard deviation).
Table 5shows that in Cluster 1, the scores were positive in every one of the criteria
according to both dimensions (teaching and learning), even though slightly minor regard-
ing teaching mathematics. Furthermore, in the latter, the mediational didactic suitability
criterion was the lowest average score (3.9 points) regardless being above the indifference
position. In Cluster 2, the scores were also positive, but less intense than in the previous
group. In this case, the greater differences among the score in regards both dimensions
were collected in the mediational criterion, showing an indifference position in terms of the
adequation of robotic material resources for teaching mathematics (3.0 points); however,
it was positive in terms of learning (4.4 points). In Cluster 3, the scores were in general
negative or indifferent regarding teaching and learning mathematics with the use of peda-
gogical robots. Additionally, as well as in the other cases, the criterion with less average
score was the mediational in the teaching mathematics with the use of robots (2.3 points).
Mathematics 2021,9, 3186 8 of 17
When grouping the participants in clusters, it could be observed that the favorable
conception of incorporation robots in the mathematics learning and teaching process varies
accordingly to the group. It can be noted that the first cluster is more favorable and the
third is not.
3.1.3. Relation between Conceptions and Experience or Previous Academic Formation
In order to analyze the relation between the conceptions of the use of robots in mathe-
matics teaching and learning process and some of the variables related to the experience
and previous academic formation of the participants, it is necessary to redefine some of the
variables. From the global results (Table 4), we noticed that participants gave, in general,
positive scores to the affirmations to both dimensions analyzed, teaching and learning
mathematics with the use of robots. This allowed us to infer on conceptions that entails
positive dispositions towards the incorporation of the robots in the learning and teaching
of mathematics, also towards the advantages of the use of robots for learning mathematics.
However, the results in Clusters 2 and 3 reflected some cases in which the disposition was
less positive or even negative. Thus, to analyze the average scores in each of the study’s
dimensions, a qualitative criterion was defined in regards the intensity of the results and it
is differentiated between the positive conceptions (average score less than 4.5) and very
positive (average score more or equal to 4.5).
Additionally, considering the also qualitative nature of the collected variables on
the academic formation (if each participant has (1) or has not (0) a postgraduate degree,
master’s, or course) and the experience (less than a year (1), between one and three (2), or
more than three (3)) we used the independent chi-squared test. The contingency tables used
in each case are presented in Tables 6and 7, together with the p-value of the independence
test and the Spearman’s rank correlation coefficient.
Table 6.
Frequency of the participants according to the crossed analysis between the conceptions and academic formation.
Dimension Variable Positive Very Positive Chi-Squared g.l. p-Value Correlation
Teaching
conceptions
Postgraduate
degree
Yes 11 13 1.601 10.206 0.139
No 36 23
Master’s
degree
Yes 6 5 0.022 10.881 0.016
No 41 31
Courses Yes 4 4 0.158 10.691 0.044
No 43 32
Learning
conceptions
Postgraduate
degree
Yes 8 16 0.002 10.961 0.005
No 20 39
Master’s
degree
Yes 3 8 0.237 10.626 0.053
No 25 47
Courses Si 1 7 1.786 1 0.181 0.147
No 27 48
Table 7.
Frequency (percentage) of the participants according to the crossed analysis between the conceptions and the
previous experience.
Dimensions Variable Years Square-Chi g.l p-Value Correlation
<1 1–3 >3
Teaching conceptions
Positive 5 7 35 2.618 2
Very positive 5 10 21 0.270 −0.153
Learning conceptions
Positive 5 2 21 5.209 2
Very positive 5 15 35 0.074 −0.068
Mathematics 2021,9, 3186 9 of 17
For the interpretation of the data present in Table 6, it is necessary to consider that a
participant could have declared academic formation in more than one of the variables, or
else could have declare dnot having any of them.
The empiric results (Tables 6and 7) show that there is not enough evidence in the
sample to reject the hypothesis of independence between the academic formation and
the previous experience, and the conceptions of the participants regarding the use of
robots in learning and teaching mathematics (see p-values in both tables, all of them
greater than the level of significance assumed at 0.05). As for the academic formation,
the conceptions regarding teaching are closely related (although slightly) to the fact of
having or not a post graduate degree in mathematics (correlation of 0.139), whereas the
conceptions of learning are related to the specialization courses of robotics (correlation
of 0.147). Furthermore, in terms of the years of experience, the association is stronger,
although negative (
−
0.153 correlation) as the conceptions regarding teaching were positive,
the participants’ professional experience was lower. In the case of learning, which was also
negative, the association was not so strong.
3.2. Qualitative Study
Below, we present the analysis of the open questions, broken down by questions and
evidence of the participants’ discourse will be presented (P1, P2, P3,
. . .
, Pi “i” stands for
the code assigned for each of the participants).
The process of categorization of the analyzed units showed evidence of suggested
actions in the participants’ discourses that derived implicitly from any of the didactic
suitability criterion (or from some of their components or indicators).
In Figure 1, the results obtained from the first open question “What suggestions would
you provide to the teacher to address the start, development and ending of the class?”
are stated.
Mathematics 2021, 9, x FOR PEER REVIEW 9 of 17
No
27
48
Table 7. Frequency (percentage) of the participants according to the crossed analysis between the
conceptions and the previous experience.
Dimensions
Variable
Years
Square-Chi
g.l
p-
Value
Correlation
<1
1–3
>3
Teaching
conceptions
Positive
5
7
35
2.618
2
Very positive
5
10
21
0.270
−0.153
Learning
conceptions
Positive
5
2
21
5.209
2
Very positive
5
15
35
0.074
−0.068
For the interpretation of the data present in Table 6, it is necessary to consider that a
participant could have declared academic formation in more than one of the variables, or
else could have declare dnot having any of them.
The empiric results (Tables 6 and 7) show that there is not enough evidence in the
sample to reject the hypothesis of independence between the academic formation and the
previous experience, and the conceptions of the participants regarding the use of robots
in learning and teaching mathematics (see p-values in both tables, all of them greater than
the level of significance assumed at 0.05). As for the academic formation, the conceptions
regarding teaching are closely related (although slightly) to the fact of having or not a post
graduate degree in mathematics (correlation of 0.139), whereas the conceptions of learn-
ing are related to the specialization courses of robotics (correlation of 0.147). Furthermore,
in terms of the years of experience, the association is stronger, although negative (−0.153
correlation) as the conceptions regarding teaching were positive, the participants’ profes-
sional experience was lower. In the case of learning, which was also negative, the associ-
ation was not so strong.
3.2. Qualitative Study
Below, we present the analysis of the open questions, broken down by questions and
evidence of the participants’ discourse will be presented (P1, P2, P3, …, Pi “i” stands for
the code assigned for each of the participants).
The process of categorization of the analyzed units showed evidence of suggested
actions in the participants’ discourses that derived implicitly from any of the didactic suit-
ability criterion (or from some of their components or indicators).
In Figure 1, the results obtained from the first open question “What suggestions
would you provide to the teacher to address the start, development and ending of the
class?” are stated.
Figure 1. Suggestions regarding the implementation of a math assignment with the use of the Blue-
Bot Robot.
Figure 1.
Suggestions regarding the implementation of a math assignment with the use of the
Blue-Bot Robot.
As it can be inferred from Figure 1, the criterion that gathers much of the percentage
analyzed units is the “interactional” one, that encloses a 70% of the action recommendations
given by the participants for the implementation of the assignment. Such recommendations
are mainly focused on the teacher-student interaction. Furthermore, some examples of the
analyzed units related to this criterion:
P13: Introduce questions connected with pattern sequences and perform an activity
with the classroom in which the students must follow patterns.
P44: Using the work of the students randomly picked out and through guided ques-
tions, make a summary of the work performed in class.
P70: Consider every idea given by the children, even if they are not correct. Let there
be communication.
P67: The teacher explains how to move Blue-Bot, with clear and concise instructions.
Mathematics 2021,9, 3186 10 of 17
P81: It will be observed if it (the robot) made all the respective stops in each flower;
if not, then some questions will be made in order to identify their errors and be able to
correct them.
Additionally, the cognitive and mediational criteria have also obtained a high per-
centage (55% in each) in the participants’ discourse. In terms of the cognitive criterion, it
can be observed that the analysis units are focused on producing actions that allow the
recognition from the students of their previous knowledge, this can be observed in the
following examples:
P15: For the start, I would find out what the students know about the objective and by
asking them questions inquire into their knowledge.
P24: Be clear that the students know the patterns and sequences.
P67: Activate previous knowledge regarding the use of the math robot.
P71: At the beginning, I conducted a brainstorm of the concepts to work on.
P79: Activated previous knowledge on the use of patterns. The students can explain
the pattern of simpler activities.
In the mediational criterion, the analyzed units are focused on the use of additional
resources beyond the use of the robot, the classroom’s conditions (physical space and
number of students) and time management, as evidenced in the following examples:
P41: Show them a video to activate the previous knowledge.
P49: A video is introduced along with a similar activity for the students to know the
use of the robot.
P7: To address the class’s objective properly, I would recommend my colleague to
consider the space defined for the activity ( . . . ).
P16: represent other paths in the classroom.
P30: For starting, I would recommend giving the students some time to observe the
image ( . . . ).
Regarding the emotional criterion, the participants made comments on actions that
can motivate the students, i.e.:
P4: Perform a pattern recreational activity, for example, create rhythmic patterns
with applauses.
P13: Additionally, they can jump and move forward from flower to flower imagining
they are the bee.
P22: Conduct an adequate motivation to challenge them to work in class.
P76: At the beginning of the class, I would recommend doing a game based on the
exercise of the students being the bee and where they must move accordingly to the
images in the classroom floor and where they’ll have to count how many jumps they have
performed in order to reach one place or the other.
Regarding the epistemic criterion, most of the comments highlighted the fact that the
use of Blue-Bot would improve the generalization and formulation of guesswork.
P3: Show them different patterns through the robot; show them how it identifies,
show them the images of the flower and the bee so that the students can position the robot
in each of the images, and then they will name the pattern that the robot followed to pass
through each image and the rest of the students can guess the formed pattern, i.e., flower,
bee, flower, bee.
P13: Use the educational robot that moves from the bee towards each of the flowers,
where the student will have to identify the pattern followed by the robot.
P22: I consider the importance of the modelling, step by step in real time (about 7 min).
P34: I would recommend the development of group thinking about the instructions
created, which could be discussed later where the students can communicate and argue
about their experience during the activity.
P37: For the ending, the students should state the different alternatives of the patterns
that they used in order to solve each of the given situations, modelling and reasoning
their choices.
Mathematics 2021,9, 3186 11 of 17
Regarding the ecological criterion, there were less comments than those reported in
the other criteria; nevertheless, they are relevant since they suggest connecting the robot’s
activity with other topics, i.e., studying bees and pollination.
P31: Comment on the bees’ characteristics and their pollination process.
P68: It cannot be overlooked the importance of coordinating with other syllabus, using
the bee activity with Natural Sciences and the importance of using them for our planet.
P84: Before presenting the activity, it would be important to associate it with the
science class and the patterns observed, giving emphasis with the bees’ life (extracts from
the movie Bee Movie could be used, where patterns of the bees’ conducts are shown).
Alternatively, in Figure 2the benefits that would derive from performing activities
that allow the introduction of robots in the mathematics’ teaching and learning process are
noted; acknowledged in the teachers’ discourse when replying to the question of “What
benefits would have the use of the pedagogical robot during a math class?”.
Mathematics 2021, 9, x FOR PEER REVIEW 11 of 17
through each image and the rest of the students can guess the formed pattern, i.e., flower,
bee, flower, bee.
P13: Use the educational robot that moves from the bee towards each of the flowers,
where the student will have to identify the pattern followed by the robot.
P22: I consider the importance of the modelling, step by step in real time (about 7
min).
P34: I would recommend the development of group thinking about the instructions
created, which could be discussed later where the students can communicate and argue
about their experience during the activity.
P37: For the ending, the students should state the different alternatives of the patterns
that they used in order to solve each of the given situations, modelling and reasoning their
choices.
Regarding the ecological criterion, there were less comments than those reported in
the other criteria; nevertheless, they are relevant since they suggest connecting the robot’s
activity with other topics, i.e., studying bees and pollination.
P31: Comment on the bees’ characteristics and their pollination process.
P68: It cannot be overlooked the importance of coordinating with other syllabus, us-
ing the bee activity with Natural Sciences and the importance of using them for our planet.
P84: Before presenting the activity, it would be important to associate it with the sci-
ence class and the patterns observed, giving emphasis with the bees’ life (extracts from
the movie Bee Movie could be used, where patterns of the bees’ conducts are shown).
Alternatively, in Figure 2 the benefits that would derive from performing activities
that allow the introduction of robots in the mathematics’ teaching and learning process
are noted; acknowledged in the teachers’ discourse when replying to the question of
“What benefits would have the use of the pedagogical robot during a math class?”.
Figure 2. Benefits of using Blue-Bot robot during a math class.
In Figure 2, it can be observed that the affective suitability didactic criterion is the
one that had more mentions in the analyzed units, followed by the cognitive criterion. As
for the analyzed units related to the affective criterion, we observed that they relate pri-
marily with the students’ needs and interests:
P3: The benefit would be the motivation towards the classes, a very much active
learning by the children, participatory, where they can learn through games and technol-
ogy, which is what nowadays motivates them the most.
P48: I think that the pedagogical robot would help a lot in my classes because stu-
dents would be more motivated.
P79: To learn through games makes learning much more amicable.
In turn, the analyzed units related to the cognitive criterion are mainly related to the
benefits of the use of the robot to enhance the mathematical learning (learning component
of this criterion):
Figure 2. Benefits of using Blue-Bot robot during a math class.
In Figure 2, it can be observed that the affective suitability didactic criterion is the one
that had more mentions in the analyzed units, followed by the cognitive criterion. As for
the analyzed units related to the affective criterion, we observed that they relate primarily
with the students’ needs and interests:
P3: The benefit would be the motivation towards the classes, a very much active
learning by the children, participatory, where they can learn through games and technology,
which is what nowadays motivates them the most.
P48: I think that the pedagogical robot would help a lot in my classes because students
would be more motivated.
P79: To learn through games makes learning much more amicable.
In turn, the analyzed units related to the cognitive criterion are mainly related to the
benefits of the use of the robot to enhance the mathematical learning (learning component
of this criterion):
P2: All those related to cognitive development in a whole range of possibilities that
comprise mathematical skills.
P47: Learning through errors and realizing mathematical concepts.
P83: It is a rich learning process, cerebrally speaking, because the logical actions provoked
in the children does not happen through the traditional process of learning mathematics.
Even though most of the benefits are related to the affective and cognitive suitability
criteria, the other criteria: mediational, ecological, interactional and epistemic, respectively,
are also valued as for their benefits.
P46: It would be a definite, visual, motivational means of interaction, worked through
trial-and-error. The game is introduced; it gathers the group within the class.
P48: It would be an innovative tool, something that will catch their attention because
it is not a common element in class.
Mathematics 2021,9, 3186 12 of 17
P70: The conversation (interaction) and the recollection of ideas will be the elements
which will solve the problem presented.
P40: It would be a powerful tool to develop mathematical skills in my students.
Finally, in Figure 3, the difficulties that would derive from conducting actions that
would allow the introduction of robots in the mathematics learning and teaching process
are presented, observed in the teacher’s discourses as a result of the question: “What
difficulties might be experienced while using the pedagogical robot during a math class?”.
Mathematics 2021, 9, x FOR PEER REVIEW 12 of 17
P2: All those related to cognitive development in a whole range of possibilities that
comprise mathematical skills.
P47: Learning through errors and realizing mathematical concepts.
P83: It is a rich learning process, cerebrally speaking, because the logical actions pro-
voked in the children does not happen through the traditional process of learning mathe-
matics.
Even though most of the benefits are related to the affective and cognitive suitability
criteria, the other criteria: mediational, ecological, interactional and epistemic, respec-
tively, are also valued as for their benefits.
P46: It would be a definite, visual, motivational means of interaction, worked
through trial-and-error. The game is introduced; it gathers the group within the class.
P48: It would be an innovative tool, something that will catch their attention because
it is not a common element in class.
P70: The conversation (interaction) and the recollection of ideas will be the elements
which will solve the problem presented.
P40: It would be a powerful tool to develop mathematical skills in my students.
Finally, in Figure 3, the difficulties that would derive from conducting actions that
would allow the introduction of robots in the mathematics learning and teaching process
are presented, observed in the teacher’s discourses as a result of the question: “What dif-
ficulties might be experienced while using the pedagogical robot during a math class?”.
Figure 3. Difficulties that might be experienced while using the Blue-Bot robot during a math class.
As for the difficulties, in general terms it is observed that the units analyzed focused
on the teaching dimension, mainly in the mediational criterion with a 57%. In said crite-
rion, it is evidenced that the analyzed units aim their discourse primarily towards the
number of students and the space of the classroom, as exemplified below:
P12: I think that the main difficulty manifests in terms of the number of students per
classroom and the little (or no) space in which we must develop the playful activities that
imply teamwork.
P19: The problem would be that due to the great number of students in the classroom,
the development of the class would be complicated. Not every student would be able to
manipulate it. Moreover, we do not have enough room to use it.
P46: The room in my classroom is small: I have 23 students, and the teacher and as-
sistant interact in the same space.
P56: The biggest difficulty is the big number of the class and the little room inside the
classroom to perform the work.
P60: The number of students is high for just one person in charge of said group of
people. Additionally, it would be ideal to have a special area designed for this type of
robotic classroom.
Figure 3. Difficulties that might be experienced while using the Blue-Bot robot during a math class.
As for the difficulties, in general terms it is observed that the units analyzed focused
on the teaching dimension, mainly in the mediational criterion with a 57%. In said criterion,
it is evidenced that the analyzed units aim their discourse primarily towards the number
of students and the space of the classroom, as exemplified below:
P12: I think that the main difficulty manifests in terms of the number of students per
classroom and the little (or no) space in which we must develop the playful activities that
imply teamwork.
P19: The problem would be that due to the great number of students in the classroom,
the development of the class would be complicated. Not every student would be able to
manipulate it. Moreover, we do not have enough room to use it.
P46: The room in my classroom is small: I have 23 students, and the teacher and
assistant interact in the same space.
P56: The biggest difficulty is the big number of the class and the little room inside the
classroom to perform the work.
P60: The number of students is high for just one person in charge of said group of
people. Additionally, it would be ideal to have a special area designed for this type of
robotic classroom.
Even though most of the difficulties are related to the mediational suitability criteria,
other difficulties associated with the rest of the criteria are valued, except for the epistemic
criterion. Next, we present some of the comments related to the affective, ecological,
cognitive, and interactional criteria, respectively.
P70: That the students are not motivated to learn.
P22: The teacher ’s preparation in regards this type of resource and maybe the costs of
its introduction.
P52: Maybe the handling of the robot is not as understandable to everyone.
P79: Not giving the instructions clearly enough may cause conflict among the students;
nevertheless, this will depend on the group management and the clarity of the explanation
from the teacher in charge.
Mathematics 2021,9, 3186 13 of 17
3.3. Confirmation of the Quantitative Data
The recommendations made by the participants (Figure 1), are basically suggestions
on actions that should be performed by the teacher. Said actions can be inferred to derive
from the principles that guide the teacher’s practice and which they considered to be
positive. Moreover, those principles can be interpreted as an implicit use of the DSC (or
of any of its components or indicators). Those principles which guide the action can also
be interpreted as dispositions towards the action that derives from a group of beliefs that
form the participants’ conceptions, as was observed in the previous analysis. In such
conception, the presence of the DSC is inferred, meaning a coherent result was obtained
when performing a quantitative analysis. Additionally, this agrees with the results existing
in the literature, which highlight good practices when using robots in the classroom,
especially in: (a) collaborative work, through active and continuous learning considering
the profile of the students; (b) the motivation, according to the profile of the students of
the training experience, where robotics helps personalized learning; (c) the use of suitable
materials; (d) the teachers’ training for a didactic innovation that begins with the planning,
the articulation of strategies of the moments in which the use of robots is integrated along
with the process of evaluating its incorporation in the classroom [31,32].
As it was observed in the above-described exploratory analysis, the average scores
that the participants gave to their valuations for the closed questions questionnaire were
bigger for learning (4.5 points) than for teaching (4.2 points). This result is coherent with the
benefits that the participants give to their answers for the second open question
(Figure 2)
about the use of robots in the classroom: essentially, more learning (cognitive suitability
criterion) and more motivation (affective suitability criterion) which both are more directly
related to the learning dimension than to the teaching dimension.
Additionally, though all the categories analyzed (criterions) from both dimensions
(learning and teaching) allowed to conclude positive conceptions about the use of robots in
math classes during the early years of school, above the three points (of indifference), it was
observed that the mediational suitability criterion in the teaching dimension obtained the
lowest score (3.5 points). In this sense, the qualitative data allowed us to confirm this result,
specifically in the data reported in Figure 3, since the negative scores to the statements
related to this category were the ones referring to the high number of students and the
reduced space in the classrooms, which complicates the teamwork.
4. Conclusions
In this study, the results which allowed to characterize the conceptions of primary
school teachers regarding the impact of the use of robotics in Mathematics were pre-
sented. The sample used consisted of 83 Chilean primary school teachers (for First to
Fourth grades).
To begin with, it was convenient to identify professional profiles according to relevant
characteristics of the participants’ professional experience and academic formation; after a
group analysis, three different clusters or groups were identified, with similar characteris-
tics among them (Table 4). The first cluster consisted of 54 (65.1%) subjects, whereas the
second was formed by 19 (22.9%) and the third by 10 (12.0%). In all the cases, there were
a predominance of women (as well as globally), whereas the academic formation was in
most of the cases a postgraduate degree in math in Clusters 1 and 2, and a master’s degree
in Cluster 3. As for professional experience, in all the groups the participants had in their
majority more than three years of experience, mainly in Second, Fourth and First grades
for Clusters 1, 2 and 3, respectively.
Secondly, the conceptions of the teachers about the use of pedagogical robots in the
learning and teaching mathematics’ process were explored. Among the global results, it
can be concluded that the participants have conceptions that entails positive dispositions
about the introduction of robots for teaching mathematics. Additionally, their conceptions
are positive towards the advantages of the use of robots for learning mathematics, although
Mathematics 2021,9, 3186 14 of 17
slightly less positive regarding teaching mathematics with robots. This being a result
consistent with other studies [34,35,37].
In turn, the results shown that these conceptions are complex, and in them the presence,
in a bigger or lesser scale, of the six DSC can be inferred, more evidently in the qualitative
analysis (Figures 1–3) than in the quantitative one. In relation to that result, it can be
objected that it was foreseeable, considering the fact of how the items in the questionnaire
were presented. Nonetheless, it can also be considered that the teaching and learning
processes are very complex because several factors intervene in them (affective aspects,
management of the classroom, use of the resources, learning evaluation, among others);
therefore, it is typical that the conceptions are a broad group of beliefs, as suggested by
several investigators, i.e., D’Amore and Fandiño [
48
], and that some of them can even
conflict according to the context in which the process of learning and teaching takes place.
This has also been observed in other studies [48,58,59].
A relevant matter is to question what the relation among the group of teachers and
their conceptions is. In Cluster 1, the conceptions were highly positive in every suitability
criterion in terms of learning and teaching mathematics with robots. In Cluster 2, the
conceptions were also positive, but less intense than in cluster 1. In this case, an indifferent
position in respect of the adequation of the robotic material resources when teaching
mathematics was observed, but this was positive in terms of its learning. In Cluster 3,
the conceptions were, in general, negative; in the other cases, the criterion with a lower
average score was the mediational one regarding the use of robots for teaching math. As
for the qualitative data, they confirm these results providing more information since the
participants’ discourse acknowledges that the difficulties to implement the use of robots
are related to the high number of students in their classrooms and the reduced space
for teamwork.
In turn, in the search of relations among the conceptions of the use of robots when
teaching and learning mathematics according to the participants’ previous experience
and the academic formation, it could be observed, after implementing the independent
chi-squared test, that the scores of the variables analyzed as for the academic formation
(postgraduate degree, master’s or courses) and the previous experience (in years) are
independent from the participants’ conceptions towards the use of robots for teaching
and learning mathematics. However, although weak and non-relevant, some empirical
correlations reflect a certain degree of negative association among the conceptions towards
teaching and the participants’ years of experience.
Finally, these results can be very useful as guidance for the development of stages of
training and professional formation of primary school teachers so that robotic resources
can be incorporated in the teaching and learning processes of mathematics, since these
results show the complexity of factors that must be considered when introducing the use
of robots in the classrooms.
Supplementary Materials:
The following are available online at https://www.mdpi.com/article/10
.3390/math9243186/s1, Annex 1: Items presented in the online questionnaire.
Author Contributions:
Conceptualization, M.J.S., A.B. and V.F.; methodology, M.J.S.; data prepro-
cessing, M.J.S.; data validation, A.B., V.F. and C.V.; writing—original draft preparation, review, and
editing, M.J.S., A.B., V.F. and C.V.; project administration, funding acquisition, M.J.S. All authors
have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the Fondecyt research project N
◦
11190547 and the Spanish
R&D project PGC2018-098603-B-I00 (MCIU/AEI/FEDER, UE).
Institutional Review Board Statement:
The study was carried out in accordance with the guidelines
of the Declaration of Helsinki, and was approved by the Ethics Committee of the Catholic University
of Maule (N◦113, 23 June 2020).
Informed Consent Statement:
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement: The data is not publicly available due to ethical requirements.
Mathematics 2021,9, 3186 15 of 17
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Guzmán, A. El Enfoque de Métodos Mixtos: Una Nueva Metodología en la Investigación Educatica; Idea Editorial: Santa Cruz de
Tenerife, Spain, 2015.
2.
Ponte, J.P. Concepções dos Professores de Matemática e Processos de Formação. In Educação Matemática: Temas de Investigação;
Ponte, J.P., Ed.; Instituto de Inovação Educacional: Lisboa, Portugal, 1992; pp. 185–239.
3. Guimarães, H. Ensinar Matemática: Concepções e Práticas. Master ’s Thesis, DEFCUL, Lisboa, Portugal, 1988.
4.
Loureiro, M.C. Calculadoras na Educação Matemática: Uma Experiência na Formação de Professores. Master’s Thesis, DEFCUL,
Lisboa, Portugal, 1991.
5.
Ponte, J.P.; Carreira, S. Spreadsheet and investigative activities: A case study of an innovative experience. In New Information
Technologies and Mathematical Problem Solving: Research in Contexts of Practice; Ponte, J., Matos, J.F., Matos, J.M., Fernandes, D., Eds.;
Springer: Berlin, Germany, 1992.
6.
Thompson, A.G. Teachers’ Conceptions of Mathematics and Mathematics Teaching: Three Case Studies. Unpublished. Doctoral
Dissertation, Universidade da Georgia, Athens, GA, USA, 1982.
7.
Moreano, G.; Asmad, U.; Cruz, G.; Cuglievan, G. Concepciones sobre la enseñanza de matemática en docentes de primaria de
escuelas estatales. Rev. Psicol. 2008,26, 299–334. [CrossRef]
8.
Feiman-Nemser, S.; Floden, R. The cultures of teaching. In Handbook of Research on Teaching, 3rd ed.; Wittrock, M.C., Ed.; Macmillan:
New York, NY, USA, 1986.
9.
Thompson, A. Teacher’s beliefs and conceptions: A synthesis of the research. In Handbook of Research on Mathematics Learning and
Teaching; Grouws, D.A., Ed.; Macmillan: New York, NY, USA, 1992; pp. 127–146.
10.
Moreno, M.M.; Azcárate, C. Concepciones y creencias de los profesores universitarios de matemáticas acerca de la enseñanza de
las ecuaciones diferenciales. Enseñanza Cienc. Rev. Investig. Exp. Didácticas 2003,21, 265–280. [CrossRef]
11.
Martínez Silva, M.; Gorgoriói Solá, N. Concepciones Sobre la Enseñanza de la Resta: Un Estudio en el ámbito de la Formación
Permanente del Profesorado. Rev. Electrónica Investig. Educ.
2004
,6. Available online: https://redie.uabc.mx/redie/article/view/
93/ (accessed on 3 October 2021).
12.
Abrantes, P. Porque se Ensina Matemática: Perspectivas e Concepções de Professores e Futuros Professores; (Provas APCC); DEFCUL:
Lisboa, Portugal, 1986.
13. Fey, J.T. Mathematics teaching today: Perspectives from three national surveys. Math. Teach. 1978,72, 490–504. [CrossRef]
14.
Dolores, C.; García, J. Concepciones de Profesores de Matemáticas sobre la Evaluación y las Competencias. Números Rev. Didáctica
Matemáticas 2016,92, 71–92.
15.
Bedoya, M.M.; Ospina, S.A. Concepciones que Poseen los Profesores de Matemática Sobre la Resolución de Problemas y Cómo
Afectan los Métodos de Enseñanza y Aprendizaje. Ph.D. Thesis, Maestría en Educación Matemática, Universidad de Medellín,
Medellín, Colombia, 2014.
16. Contreras, L.C.; Carrillo, J. Diversas concepciones sobre resolución de problemas en el aula. Educ. Matemática 1998,10, 26–37.
17.
Trejo, E.; Camarena, P. Concepciones de los profesores y su impacto en la enseñanza de un sistema de ecuaciones lineales con dos
incógnitas. In Acta Latinoamericana de Matemática Educativa; Patricia, L., Ed.; ComitéLatinoamericano de Matemática Educativa:
México, DF, Mexico, 2011; pp. 1095–1103.
18. Franco, A.M.; Canavarro, A.P. Atitudes dos Professores Face àResolução de Problemas; APM: Lisboa, Portugal, 1987.
19.
Silva, A. A Calculadora No Percurso de Formação de Professoras de Matemática. Master’s Thesis, DEFCUL, Lisboa,
Portugal, 1991.
20.
Veloso, M.G. Novas Tecnologias de Informação: Um Programa de Formação de Professores de Matemática. Master’s Thesis,
DEFCUL, Lisboa, Portugal, 1991.
21.
Ribeiro, M.J.B.; Ponte, J.P. A formação em novas tecnologias e as concepções e práticas dos professores de Matemática. Quadrante
2000,9, 3–26.
22.
Wachira, P.; Keengwe, J.; Onchwari, G. Mathematics preservice teachers’ beliefs and conceptions of appropriate technology use.
AACE J. 2008,16, 293–306.
23.
McGinnis, J.R.; Hestness, E.; Mills, K.; Ketelhut, D.; Cabrera, L.; Jeong, H. Preservice science teachers’ beliefs about computational
thinking following a curricular module within an elementary science methods course. Contemp. Issues Technol. Teach. Educ.
2020
,
20, 85–107.
24.
Sullivan, F.R.; Moriarty, M.A. Robotics and Discovery Learning: Pedagogical Beliefs, Teacher Practice, and Technology Integration.
J. Technol. Teach. Educ.
2009
,17, 109–142. Available online: https://www.learntechlib.org/primary/p/26177/ (accessed on
3 October 2021).
25.
Bouchaib, F.; Hanane, N. Pedagogical Robotics–A Way to Experiment and Innovate in Educational Teaching in Morocco. Int. J.
Educ. Learn. Syst. 2017,2, 71–75.
26.
Tang, A.L.; Tung, V.W.S.; Cheng, T.O. Dual roles of educational robotics in management education: Pedagogical means and
learning outcomes. Educ. Inf. Technol. 2020,25, 1271–1283. [CrossRef]
27.
Yang, Y.; Long, Y.; Sun, D.; Van Aalst, J.; Cheng, S. Fostering students’ creativity via educational robotics: An investigation of
teachers’ pedagogical practices based on teacher interviews. Br. J. Educ. Technol. 2020,51, 1826–1842. [CrossRef]
Mathematics 2021,9, 3186 16 of 17
28.
Casey, J.E.; Pennington, L.K.; Mireles, S.V. Technology Acceptance Model: Assessing Preservice Teachers’ Acceptance of Floor-
Robots as a Useful Pedagogical Tool. Tech. Know. Learn. 2021,26, 499–514. [CrossRef]
29.
Kennedy, J.; Lemaignan, S.; Belpaeme, T. The Cautious Attitude of Teachers towards Social Robots in Schools. In Robots 4
Learning Workshop at IEEE RO-MAN; 2016. Available online: https://core.ac.uk/download/pdf/84595376.pdf (accessed on
3 October 2021).
30.
Jormanainen, I.; Zhang, Y.; Kinshuk, K.; Sutinen, E. Pedagogical Agents for Teacher Intervention in Educational Robotics Classes:
Implementation Issues. In Proceedings of the 2007 First IEEE International Workshop on Digital Game and Intelligent Toy
Enhanced Learning (DIGITEL’07), Jhongli, Taiwan, 26–28 March 2007; pp. 49–56. Available online: https://ieeexplore.ieee.org/
abstract/document/4148831 (accessed on 3 October 2021).
31.
Ribeiro, C.; Coutinho, C.; Costa, M.F. Educational Robotics as a Pedagogical Tool for Approaching Problem Solving Skills
in Mathematics within Elementary Education. In Proceedings of the 6th Iberian Conference on Information Systems and
Technologies (CISTI 2011), Chaves, Portugal, 15–18 June 2011; pp. 1–6. Available online: https://ieeexplore.ieee.org/abstract/
document/5974210 (accessed on 3 October 2021).
32.
Schina, D.; Esteve-González, V.; Usart, M. An overview of teacher training programs in educational robotics: Characteristics, best
practices and recommendations. Educ. Inf. Technol. 2021,26, 2831–2852. [CrossRef]
33.
Scaradozzi, D.; Screpanti, L.; Cesaretti, L.; Storti, M.; Mazzieri, E. Implementation and Assessment Methodologies of Teachers’
Training Courses for STEM Activities. Tech. Know. Learn. 2019,24, 247–268. [CrossRef]
34.
Lopez-Caudana, E.; Ramirez-Montoya, M.S.; Martínez-Pérez, S.; Rodríguez-Abitia, G. Using Robotics to Enhance Active Learning
in Mathematics: A Multi-Scenario Study. Mathematics 2020,8, 2163. [CrossRef]
35.
González, Y.A.C.; Muñoz-Repiso, A.G.V. A Robotics-Based Approach to Foster Programming Skills and Computational Thinking:
Pilot Experience in the Classroom of Early Childhood Education. In Proceedings of the Sixth International Conference on
Technological Ecosystems for Enhancing Multiculturality, Salamanca, Spain, 24–26 October 2018; pp. 41–45. Available online:
https://dl.acm.org/doi/abs/10.1145/3284179.3284188 (accessed on 3 October 2021).
36.
Schina, D.; Esteve-Gonzalez, V.; Usart, M. Teachers’ Perceptions of Bee-Bot Robotic Toy and Their Ability to Integrate It in Their
Teaching. In Robotics in Education; Lepuschitz, W., Merdan, M., Koppensteiner, G., Balogh, R., Obdržálek, D., Eds.; Advances in
Intelligent Systems and Computing; Springer: Cham, Switzerland, 2021; Volume 1316. [CrossRef]
37.
Papadakis, S. Robots and Robotics Kits for Early Childhood and First School Age. International Association of Online Engineering.
2020. Available online: https://www.learntechlib.org/p/218338/ (accessed on 3 October 2021).
38.
Seckel, M.J.; Vásquez, C.; Samuel, M.; Breda, A. Errors of programming and ownership of the Robot concept made by Trainee
Kindergarten Teachers during an induction training. Educ. Inf. Technol. 2021. [CrossRef]
39.
Godino, J.D.; Batanero, C.; Font, V. The onto-semiotic approach: Implications for the prescriptive character of didactics. Learn.
Math. 2019,39, 37–42.
40.
NCTM. Principles and Standards for School Mathematics; VA and National Council of Teachers of Mathematics: Reston, VA,
USA, 2000.
41.
Breda, A.; Pino-Fan, L.R.; Font, V. Meta didactic-mathematical knowledge of teachers: Criteria for the reflection and assessment
on teaching practice. EURASIA J. Math. Sci. Technol. Educ. 2017,13, 1893–1918. [CrossRef]
42.
Sánchez, A.; Font, V.; Breda, A. Significance of creativity and its development in mathematics classes for preservice teachers who
are not trained to develop students’ creativity. Math. Educ. Res. J. 2021,13, 31–51. [CrossRef]
43.
Godino, J.D. Indicadores de la idoneidad didáctica de procesos de enseñanza y aprendizaje de las matemáticas. Cuad. Investig.
Form. Educ. Matemática 2013,8, 111–132.
44.
Breda, A. Características del análisis didáctico realizado por profesores para justificar la mejora en la enseñanza de las matemáticas.
Bolema 2020,34, 69–88. [CrossRef]
45.
Breda, A.; Font, V.; Pino-Fan, L.R. Criterios valorativos y normativos en la didáctica de las matemáticas: El caso del constructo
idoneidad didáctica. Bolema 2018,32, 255–278. [CrossRef]
46.
Peirce, C.S. The Fixation of Belief, Popular Science Monthly 12, 1–15 November 1877. Available online: http://www.bocc.ubi.pt/
pag/peirce-charles-fixation-belief.pdf (accessed on 3 October 2021).
47.
D’Amore, B.; Fandiño, M.I. Cambios de convicciones en futuros profesores de matemáticas de la Escuela Secundaria Superior.
Epsil. Rev. Soc. Andal. Educ. Matemática Thales 2004,58, 23–44.
48.
Ramos, A.B. Objetos Personales, Matemáticos y Didácticos, del Profesorado y Cambios Institucionales. El Caso de la Contex-
tualización de las Funciones en una Facultad de Ciencias Económicas y Sociales. Tesis Doctoral No Publicada, Universitat de
Barcelona, Barcelona, Spain, 2006.
49.
Breda, A.; Seckel, M.J.; Farsani, D.; Silva, J.F.; Calle, E. Teaching and learning of mathematics and criteria for its improvement
from the perspective of future teachers: A view from the Ontosemiotic Approach. Math. Teach. Res. J. 2021,13, 31–51.
50.
Garcés, W. Criterios que Orientan la Práctica del Profesor Para Explicar Matemáticas en un Curso de Ciències Básicas en Carreras
de Ingeniería en el Perú: El Caso de la Derivada. Tesis Doctoral No Publicada, Universitat de Barcelona„ Barcelona, Spain, 2021.
51.
Hummes, V.B.; Font, V.; Breda, A. Combined Use of the Lesson Study and the Criteria of Didactical Suitability for the Development
of the Reflection on the own Practice in the Training of Mathematics Teachers. Acta Sci. 2019,21, 64–82. [CrossRef]
52.
Fernández-Morales, K.; Vallejo, A.; Ojeda, M.; y McAnally-Salas, L. Evaluación psicométrica de un instrumento para medir la
apropiación tecnológica de estudiantes universitarios. Rev. Electrónica Psicol. Iztacala 2015,18, 286–306.
Mathematics 2021,9, 3186 17 of 17
53.
Hernández, R.; Fernández, C.; Baptista, P. Metodología de la Investigación, 6th ed.; McGraw-Hill: Ciudad de México, Mexico, 2014.
54.
Hernández-Nieto, R. Instrumentos de Recolección de Datos en Ciencias Sociales y Ciencias Biomédicas; Universidad de los Andes:
Mérida, Venezuela, 2011.
55.
Seckel, M.J.; Breda, A.; Sánchez, A.; Font, V. Criterios asumidos por profesores cuando argumentan sobre la creatividad
matemática. Educ. Pesqui. 2019,45, e211926. [CrossRef]
56. Bisquerra, R. Metodología de la Educación Educativa, 6th ed.; La Muralla: Madrid, Spain, 2016.
57. Latorre, A. La Investigación-Acción. Conocer y Cambiar la Práctica Educativa, 2nd ed.; Graou: Barcelona, Spain, 2004.
58.
Ramos, A.B.; Font, V. Criterios de idoneidad y valoración de cambios en el proceso de instrucción matemática. Rev. Latinoam.
Investig. Matemática Educ. 2008,11, 233–265.
59.
Hummes, V.; Sol, T.; Breda, A. Argumentación práctica sobre el Teorema de Pitágoras por profesores de matemáticas en un curso
de formación. In Investigación en Educación Matemática XXIV; Diago, P.D., Yáñez, D.F., González-Astudillo, M.T., Carrillo, D., Eds.;
SEIEM: Valencia, Spain, 2021; pp. 343–350.