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The conversion of teaching to distance learning was a major challenge for all those involved in education. For elementary school-aged students, in particular, the transition was enormous, especially in mathematics. A research project in Austria investigated which features of activity-based, tools-supported mathematics instruction were helpful for mastering mathematical tasks during distance learning. The results of the research project show that action-oriented, tools-supported teaching, in which Maria Montessori’s didactic mathematics tools were used, was helpful for mastering mathematical tasks during the time of distance learning. By working with tools, the students were able to build up inner images of mathematical content and contexts. These inner images favored the process of abstraction, and they could fall back on these ideas during the time of distance learning.
University Teacher College Lower Austria, Baden, Austria
*Address all correspondence to: s.apfler@ph-noe.ac.at
1. Introduction
As in many other countries, the COVID-19 pandemic led to school closures throughout Austria. To ensure that all students were taught during this time, classes were switched to distance learning without teachers and students learning how to deal with this situation. For three school years, teachers, students and parents had to react flexibly due to the constantly changing regulations. This caused a number of problems for all of them. One of the main problems arose from the organization of school, teaching and education, which was still based on the nineteenth-century factory model [1].
Another problem arose from the technical equipment of Austrian elementary schools, although media education is included in the Austrian curriculum for elementary schools [2]. Therefore, neither the teachers nor the students had the necessary skills to cope with distance learning at the beginning.
Especially in the subject of mathematics, the implementation of distance learning was a challenge for many teachers. In mathematics, the curriculum calls for activity-based, tool-supported instruction through which children can develop an understanding of mathematical principles and relationships [3]. However, during distance learning, in many cases, there was a lack of opportunities to provide didactic materials for action-oriented instruction. This could have negative effects on the development of mathematical skills in elementary school children.
Therefore, a research study addressed whether students who could work with didactic materials in an action-oriented way in mathematics classes before the school closures could benefit from the acquired skills during distance learning.
After an examination of the theoretical background, the results of this research are presented in this article.
In order to consider the theoretical foundations of the research field, it is necessary to deal with mathematics education in Austrian elementary schools. This is constantly confronted with changes and new requirements. For example, the implementation of educational standards and competence orientation took place in 2012. Starting in the school year 2023/24, there will be new curricula that are based on the current state of research and are designed to focus on competence orientation and action orientation [3]. The current state of research and the events resulting from the necessary school closures during the COVID-19 pandemic will be discussed in detail subsequently.
2.1 Mathematics instruction in elementary school in Austria
International comparative tests [4] have shown for years that the mathematical competencies of Austrian students do not match the level of students in other countries. The question, therefore, arises as to what mathematics instruction students need in order to improve their competencies in mathematics. Research like that of Krajewski [5] emphasizes the importance of mathematical precursor skills already at preschool age and of basic mathematical competencies in the first years of schooling. For example, Krajewski [5] was able to show a connection between children’s understanding of number concepts in preschool and performance in mathematics as a subject several years later. In order to be able to build up sustainable basic ideas in all mathematical areas, but especially an idea of numbers and quantities, children need many experiences with didactic materials. For Kuntze [6], it is of great importance that the children actively deal with these materials and can act with them. These materials, in turn, can be mapped in the next step, translated into a pictorial representation, and later into a symbolic representation. The connection of action, illustrations, linguistic expressions and symbolic representations enables the development of sustainable ideas and insight into mathematical laws [7, 8]. It is essential that the children develop inner conceptions of the actions through the active, acting engagement with didactic materials, which are essential for further abstractions. For Lorenz [9], mathematical learning processes are based on these actions and mathematical thought structures are precisely these internalized actions. In order to successfully build up these mental images, it is necessary to select suitable materials. As an example of didactic materials for the development of a sustainable understanding of numbers and place values, decimally structured materials can be mentioned, such as those that Montessori made available to children as early as preschool age [10].
The development and training of the child’s senses was of great importance to Montessori because, with them, the child grasps its environment and learns to find its way in it. In the course of his development, the child is confronted with an infinite number of sensory impressions and objects. However, the properties of these objects are limited, which makes it necessary for the child to learn to distinguish and order these properties. For Montessori, there is a connection between this ordering of properties and the development of a mathematical understanding. Montessori refers to this need for structure and order as the “mathematical mind” [10]. She developed didactic materials to best support and encourage children’s mathematical development and education. Central to these materials is that the children are given the opportunity to actively engage with mathematical content, which can lead to a deeper understanding of mathematical content.
The materials developed by Montessori cover all mathematical areas of arithmetic and geometry. They are designed in such a way that the level of abstraction increases, allowing children to eventually succeed in detaching from the material [4, 10, 11].
Current research shows that students who have the opportunity to explore mathematics with materials designed by Montessori in the mathematics classroom show positive skill development [4, 12, 13, 14]. In a longitudinal study, Apfler [4] investigated whether there is a difference in the level of competence in mathematics when students in public elementary schools are taught by teachers with or without Montessori training. At the end of the third school year, those students taught by teachers with Montessori training showed significantly higher levels of competence in the competence areas of working with numbers and communicating. Whether there is a relationship between the higher level of competence and Montessori education cannot be clearly demonstrated with this study. Nevertheless, it was found that the teachers with Montessori training used a variety of Montessori materials, especially for building a viable understanding of numbers in small number spaces and for an understanding of the decadal system.
As an example of a material for building a sustainable understanding of numbers and operations, the Golden Bead material can be mentioned. In this material, which represents the decadic system, the ones is represented as a bead. The ten is represented as a bar with ten individual beads joined together. The number 100 is represented by ten tens bars joined together. The thousand is represented by a cube consisting of ten plates of hundreds. The beads can also be replaced by small cubes, as shown in the following Figure 1.
Figure 1.
Material for building the decadic number system.
In the active confrontation, the students place quantities and numbers with this bead material and thereby recognize the decadic structure of the number system under the guidance of the teacher (see Figure 2).
Figure 2.
Number sets with the decadic material.
However, not only numbers and quantities but also arithmetic operations can be placed and understood with this material. Figure 3 shows how students use the bead material to place a division. Here, the total quantity of 642 is divided into two figures. The simultaneous notation provides insights into the algorithm of written division.
Figure 3.
Written division with the decadic material.
Due to the freedom of methods that the legal framework in Austria allows, teachers who have completed Montessori training can incorporate elements of Montessori education into their teaching. Studies show that in public elementary schools, the concept of Montessori education is not implemented in its entirety, but individual elements are used. The materials in mathematics are often used by Montessori teachers to enable the children to actively engage with the materials. Comparing the findings of current research in the field of mathematics didactics, it can be seen that the methodological approach of Montessori education is consistently confirmed [4, 15].
Of course, digital media do not yet play a role in Montessori’s writings. Nevertheless, it must be questioned whether the use of digital media can also be justified in Montessori classes in the 21st century. Thus, Burow [16] also pleads to supplement the prepared learning environment, which in Montessori classes consists of analog materials, with digital offerings and virtual learning spaces, but not to replace them. The importance of action orientation, active doing, and acting engagement with the environment is thereby maintained, but expanded.
2.2 Distance learning in Austria
As in many other European countries, schools in Austria were closed in March 2020 due to the COVID-19 pandemic. Teachers only had the weekend to change lessons to distance learning. The Ministry of Education gave instructions not to develop any new subject matter during the first weeks of the school closures but only to practice, repeat and consolidate already known content [17]. This situation, unique in Austrian history [18], was an attempt to fulfill the educational mission of schools even during the pandemic [19]. In the course of the next almost 3 years, schools were permanently confronted with new demands and framework conditions and were challenged to always adapt to the circumstances in a new and flexible way.
Schools were repeatedly closed for several weeks in 2020 and 2021, and students were cared for at home by parents. Teachers in the elementary schools primarily prepared workbooks in print format, which parents picked up from school, worked on with the children at home, and brought back to school at the end of the week. Teachers corrected students’ work over the weekend and prepared the papers for the next week. Especially in elementary school, few digital offerings were used [17, 20].
Teachers played a special role in the design of distance learning. As has been confirmed in studies on face-to-face learning [21], teachers also had the most effective influence on student learning in distance learning [22]. Teachers were particularly challenged by the ever-changing environment. They had to completely change how they had been teaching, which meant significantly more time spent on school-related work [17, 23]. Despite the high workload, which resulted mainly from the lack of experience with distance learning, elementary school teachers showed a high level of commitment [24]. In addition to the high workload, teachers perceived the feeling of permanent accessibility and the dissociation of professional and private life as stressful. How teachers coped with the transition to distance learning was a function of their digital competencies and previous teaching styles. A large proportion of teachers tried to take their face-to-face teaching styles and map them digitally during distance learning. This was especially difficult for those teachers who typically work in a reform pedagogical, holistic, and student-centered manner [23].
The impact of school closures on children and young people has been enormous, but the long-term effects are not yet even foreseeable [25]. In addition to social and psychological symptoms among the children and adolescents, negative effects on learning development were evident early on. Particularly, educationally disadvantaged children achieved hardly any learning gains during distance learning. In the Netherlands, for example, which has a high standard of schools compared to Austria, “The findings imply that students made little or no progress while learning from home, and suggest losses even greater in countries with weaker infrastructure or longer school closures” [26]. There are currently few studies for Austria, but they show that there are scissor effects in the development of basic cognitive skills and that many children have large gaps [27]. Unfortunately, the results of the IKM Plus study, which will be conducted nationwide in spring 2022, are not yet available.
The research findings presented in the theoretical part of the article show that there is already a lot of research on the conversion of teaching to distance learning and its effects on children and adolescents. It shows that some children have come through the pandemic well, while others have major deficits [26, 27, 28]. In mathematics, it is shown that students benefit from activity-based, material-supported instruction and especially from activity-based engagement with Montessori materials to achieve high levels of competence in mathematics [4, 12, 13, 29, 30]. When teachers have completed Montessori training, they implement elements of the concept in their teaching and use Montessori didactic materials, especially in mathematics [4, 31, 32].
During distance learning, the teaching methods had to be changed and adapted; the lessons took place at home and the students could not work with the materials in a proven way. Research in the field of mathematics is scarce for the German-speaking countries. The question arises whether students who were already able to build up mathematical competencies during face-to-face teaching through active engagement with Montessori materials could benefit from this during distance learning [11].
The research desideratum gives rise to the following research question for the study presented here: What features of action-based, materials-supported mathematics instruction do Montessori teachers judge to be conducive to developing mathematical competencies during distance learning in times of the COVID-19 pandemic? [11].
The purpose of this study is to find out how adaptable Montessori’s educational concept is to the needs of children in the 21st century, especially in the area of mathematics, and whether recommendations for school and classroom development can be derived from it.
Within the framework of the study, a second question concerning the self-organization of learning processes was also addressed. However, this cannot be addressed in this article.
3.1 Method
A qualitative research design is best suited to answer the research question in order to ensure the subject-relatedness and subjective assessments of teachers on this issue. An expert survey was chosen as the research method, which was conducted in the form of guided interviews. A guideline enables a partial standardization and comparability of the individual interviews [33, 34]. The quality criteria of qualitative research, according to Mayring [34], were observed in the presented study.
The study was conducted in spring 2022, i.e., after the end of the distance learning phases. [11].
3.1.1 Study participants
To answer the research question, an attempt was made to select a representative sample. According to Hug and Poscheschnik [33], a sample is representative if it reflects the population in terms of essential characteristics. This resulted in several characteristics related to the sample.
The selected teachers had completed Montessori training.
The selected teachers implement individual principles of Montessori’s concept in their teaching.
The selected teachers were leading an elementary school class or elementary school from September 2019 to the time of the study.
The selected teachers taught at different Austrian elementary schools in different regions of the country.
The selected teachers were willing to participate in the study [11].
Due to the limitations of the research field based on the aforementioned criteria, nine female teachers from four Austrian provinces finally participated in the study.
What is interesting about the sample is that one teacher manages a public school and leads a class at the same time. A second teacher does not teach herself but manages a school with a total of 12 classes. In this school, half of the classes are taught by teachers with Montessori training, while the other half of the teachers have not had Montessori training [11].
Table 1 shows an overview of the participants and whether they were teaching an elementary class or managing a school and in which Austrian region they were working:
Region
Teacher
Head of school
1.
Vorarlberg
X
2.
Lower Austria
X
3.
Styria
X
4.
Lower Austria
X
5.
Lower Austria
X
6.
Vienna
X
7.
Lower Austria
X
X
8.
Vienna
X
9.
Vorarlberg
X
Table 1.
Overview of the participants.
Due to the selection criteria, especially based on their training (elementary school teaching degree, Montessori training) of the teachers participating in the study, they can all be called experts. Thus, they can be recognized as having expert status in their field of activity [33, 35].
3.1.2 Survey tool
The interviews with the experts were conducted as guided interviews. This method restricted maximum openness, allowing for all possibilities and expressions. The guide was designed to still allow for openness, but still be as structured as possible [35]. The guide consists of exploratory questions, guiding questions relevant to answering the research question, and the opportunity to ask ad hoc questions if the course of the interview necessitates it [34].
The questions about the subject area of mathematics arose from the research area presented [4, 12, 14, 26, 27, 30]. Teachers were asked about what materials they used prior to the conversion of instruction to distance learning and what role action orientation played in their mathematics instruction. Another question was related to the implementation of mathematics instruction during the distance learning phase. Finally, teachers were asked how they experienced the development of their students’ mathematical performance and what conclusions they could draw from it [11].
3.1.3 Evaluation of the data
The interviews were conducted as video conferences via Zoom, recorded, and subsequently transcribed. The transcripts were subjected to a comprehensive data analysis and evaluated in a rule-based and theory-based manner. The qualitative content analysis according to Mayring [34] was applied. In order to be able to compare the statements of the interviews with each other, categories were deductively formed, which resulted from the questions [11].
3.2 Results of research
In order to trace the development of teaching from face-to-face teaching before the school closures through the different phases of distance learning, the teachers described their teaching, specifically including their mathematics teaching [11].
Prior to the school closures, all teachers interviewed facilitated open instruction, although to varying degrees. In no class, frontal teaching was mentioned as the primary method of instruction. All students of the interviewed teachers had many opportunities during open lessons to work on, practice, repeat, and consolidate mathematical lesson contents in an activity-oriented way with Montessori materials, but also with other didactic materials. Although the teaching of the interviewed teachers was very heterogeneous, child-centered, action-oriented, open teaching was important to all teachers [11].
3.2.1 Lesson organization during distance learning
During distance learning, all teachers provided workpacks of notebooks, books, and worksheets for the children, which parents picked up from school on Monday and returned on Friday for correction. The tasks contained in them served to repeat and practice already-known lesson content, especially in mathematics [11].
Over time, some teachers changed distance learning and included videoconferencing, explainer videos, or other digital offerings, and increased contact with children. The personal accompaniment of the children by the teachers was important to them, as the answer of one participant 8 shows: “We did not want to leave the children alone, we wanted to accompany them” [11]. The teachers showed great commitment in designing distance learning in general and tried out different methods and possibilities over time. Here, too, a very heterogeneous picture emerges, since due to the lack of guidelines from the Ministry, the teachers tried to design distance learning according to their own possibilities and the needs of the children [11].
3.2.2 Teaching mathematics during distance learning
A particular challenge for the teachers interviewed was teaching mathematics in distance learning. The greatest difficulty for them was the lack of material, which they used in face-to-face lessons, especially for the activity-oriented development of new content, and which the children did not have access to in distance learning. The teachers tried to deal with this problem in different ways [11].
One teacher provided Montessori materials for the students to use at home. The children could pick these up at school and share them. However, she found that the students could not handle them at home without an introduction by the teacher and thus did not benefit from the materials. In order to still be able to teach new learning content with the support of materials, she provided illustrations of materials and worked more with finger pictures. According to this teacher, the children not only lacked haptic experiences with the didactic materials, they were also unable to build up internal images. When the students were allowed to return to school after the distance learning phase, the teacher repeated the learning content again with the materials [11].
Another teacher also gave Montessori materials to take home but had a good experience with them. She combined the work with the plans in print format with video conferences in which a lot was discussed about mathematics, mathematical problems and possible solutions [11].
Another teacher made her own materials for the children to work on fractions in 4th grade and sent them home with the work packets. In this way, the students were able to use these materials to work on fractions and fractional arithmetic. She postponed topics that could not be worked on in an activity-based manner at home until after the distance learning phase and thus to the face-to-face classroom [11].
One teacher offered video conferences on mathematical topics. In these, she worked with a document camera and showed the work with the materials. The students were not able to work with the materials themselves, but they got an idea of how the material could be used. This teacher explicitly referred to the fact that the active work with the materials before the school closures was very valuable for the students. Students had developed internal images of the materials and what they could do with them that they could draw on during distance learning [11].
Other teachers emphasized the importance of inner images about actions with materials. During the videoconferences, the children referred to these actions that they performed in face-to-face lessons and were thus able to build a deeper understanding of mathematical actions. This was especially evident for those students who were not affected by school closures until the third or fourth grade and who were able to do a great deal of work with mathematical materials in their first years of school [11].
Several teachers reported that they worked on as little new content as possible during the distance learning period. They planned very carefully for what content could be worked on later during face-to-face periods when students had access to the materials again [11].
The study presented in this article explored the question of which features of activity-based, materials-supported mathematics instruction Montessori teachers judge as conducive to developing mathematical competencies during distance learning in times of the COVID-19 pandemic [11].
In summary, it can be seen that all Montessori teachers interviewed placed great emphasis on the use of didactic materials, especially Montessori materials, during face-to-face instruction. This led the teachers to find the transition to distance learning in mathematics particularly challenging. However, it was found that students who were able to learn and experience mathematics in an activity-based and material-supported way before the school closures were able to develop inner images of materials and the actions with them. They were able to use these internal images during distance learning to describe mathematical processes and communicate about mathematics. These results connect to previous research findings by Apfler [11] showing that an action-based engagement with didactic materials, especially Montessori materials in mathematics enable high levels of competence in the areas of working with numbers and communicating. Likewise, the connection with the current subject didactic discussion in mathematics, which thematize the importance of action orientation with suitable didactic materials for the construction of inner images as well as the detachment from the material [6, 8, 9, 36, 37].
There are currently no research findings on long-term consequences in the mathematical development of students in Austria.
The teachers emphasize the importance of Maria Montessori’s concept also for children in the 21st century and underline the necessity to raise the status of Montessori education in Austria [11].
The limitations of the research presented arise from the small sample that resulted from the criteria. Due to the small number of study participants, no generally valid statements can be made, but only the results of the sample can be described. Furthermore, it should be critically noted that no well-founded statements can be made about the actual mathematical competencies of the students. The results are merely subjective assessments by the teachers. Standardized test procedures with pre- and posttest would have been necessary here, but this was not possible due to the unpredictability of the events [11].
Nevertheless, an interesting picture of the teachers’ assessments emerges and the research question can thus be answered as follows: “The present study provides evidence that action-oriented, material-supported mathematics instruction favors the development of internal conceptual images and that these could be drawn upon during distance learning. This was helpful for the processing of mathematical tasks in distance learning as well as for the further development of the mathematical competencies of the students.” [11].
These results provide relevant information for the training and further education of teachers, because the new curriculum for elementary school, which will come into force in 2023, also focuses on the active examination of didactic materials in mathematics [3]. It will be necessary to address this issue even more in initial and in-service training.
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Written By
Sabine Apfler
Submitted: 01 August 2023Reviewed: 27 September 2023Published: 16 October 2023
Open access peer-reviewed chapter - ONLINE FIRST
