Teaching concepts about autistic students

In this category are gathered the teaching conceptions about autistic students revealed in the interviews of the eight (8) teachers, seeking to understand what the same concepts about being an autistic student, its teaching and its learning. For this first category, fifty units of meaning emerged, listed by all the respondents

Five (5) of the eight (8) professors interviewed reveal that they understand the autist as a student who has very pronounced learning difficulties, with disinterest in the recording of content and specific disciplines, such as Mathematics. On the other hand, two other (2) teachers conceive such a student as a subject possessing outstanding intelligence and high skills in specified areas of their own interest.

U11PD: [...] he has a lot of difficulty in relationship [...] in learning also [...].

U23PT: The first thing that comes to mind is that the student is superdotated in some area [...].

We understand that all autists are different from each other, although some features bring them closer to each other to a greater or lesser degree. The conceptions of the interviewees, presented here, carry experiences that they had specifically with their autistic students and, considering that some of them only had this experience with one or two students, their conceptions about what they think is the autist are based on them.

Listing one of the idiosyncrasies of the autist, the disability in communication and language, relating it to learning difficulties, we see, in the studies of ^{Williams and Wright (2008}), mentioned in the research of Lamb, Resende and Thiengo (2017), that autistic children, usually, have difficulty in assimilating language. Therefore, they need more time to understand what others are saying and, consequently, to appropriate the learning.

With regard to the high performance of some autistic students, mentioned by two interviewees, we have in ^{Silva, Gaiato and Reveles (2012}) the understanding that “Certain characteristics typical of autism, such as the obsession for specific subjects, attention to detail, the hyperfocus and the ability to research a subject thoroughly, make some people of the autistic spectrum magnificent in certain issues” (^{Silva; Gaiato; Reveles, 2012}, p. 100).

According to four (4) respondents, autistic students need treatments that differ from the others and establishing differentiated educational practices is necessary for learning to happen. For them, enabling a welcoming environment and showing greater affection to these students makes them feel safe in their relationship with the teacher. The respondents understand that affectivity, combined with a structured and systematic school routine, will make it possible for the autistic student to be better positioned in space and time, which would directly reflect on the behavior and development of that student.

U4PV: It all depends. [...] depends much on the environment [...] whether it is welcoming or not [...].

U22PS: [...] when you gain their trust changes [...].

On ^{Cunha’s opinion (2011}), thinking about an inclusive school environment requires not only pedagogical resources, but also human quality involved. We communicated with the author about the important role of the teacher as a mediator in the development of affective relationships, promoting a receptive relationship to all students. In the opinion of ^{Marinho and Merkle (2009}), “[...] the fact is that there is no way to separate cognitive development from affective and its biological essence, and thus, regardless of the etiological and diagnostic view that one has about autism, it is of fundamental importance that the form of educational approach to these children is clear” (Marinho; Merkle, 2009, p. 06).

^{Cunha (2011}, p. 53) points out that “the affective relationship of the autistic student with the teacher is the beginning of the process of building their autonomy in school. Although the autist usually encounters difficulties to understand people’s feelings and subjectivity, he is not without emotions.” It is noted that there is a concern of the teachers interviewed in getting to establish a link with the autistic student to better know their specifics, aiming their teaching and their learning in a more appropriate way. Thus, we understand that the affectivity, mentioned by them, provides a less informal connection with all students and, in the case of autists, this personal connection between student and teacher can both lead to overcoming barriers and barriers that may prevent learning from developing.

With regard to the characteristics of the autistic student, difficulty in relationship, isolation and lack of socialization are autistic specificities highlighted by six respondents. According to them, autists live in a different world than other students, and, for this reason, they maintain behavioral attitudes so singular. The respondents have the autist as a ‘different’ student, an unknown being, who lives and thinks differently from others. Still, according to them, because their characteristics do not resemble the others, the autist becomes an unknown to them.

U17PV: [...] an unknown, because it needs to be unraveled [...].

U15PD: He has his own world, which is different from ours.

Corroborating Falcão (2011), we believe that, in order for inclusion to be beneficial for all, it is necessary to understand the differences as something positive, given that each has singularities that are manifested through attitudes, simple situations, like dressing, or even more pronounced differences, as is the case of disabilities.

Regarding the difficulties of relationships, even if they are not considered as rules, due to the diversity of behaviors presented by autists, we see that “not to relate with visual contact, facial expressions, relationship with peers, primarily by the routine, being that the autistic child can both isolate itself and also interact in a way strange to the usual patterns” (^{Marinho; Merkle, 2009}, p. 06-07) is a common characteristic in much of autists.

Given the existence of a heterogeneity of people within a classroom, with different feelings, characteristics and personalities, we do not always pay attention to this diversity. In dealing with the world of peculiarities of the autists mentioned by the respondents, ^{Silva, Gaiato and Reveles (2012}) point out that

When you hear the word "autism", you immediately come to mind the image of a child isolated in his own world, contained in an impenetrable bubble, who plays strangely, swings his body over and over, alien to everything and everyone. It is usually associated with someone "different" from us, who lives on the margins of society and has an extremely limited life, in which nothing makes sense. But it's not exactly that way. This look seems too narrow to us: when we talk about autism, we are referring to people with absolutely revealing abilities, who shut down deep in our soul, and make us reflect on who actually lives alienated (^{Silva; Gaiato; Reveles, 2012}, p. 03, authors' emphasis).

For ^{Silva, Gaiato e Reveles (2012}), a person with autism feels, looks and perceives the world in a very different way from the common behaviors expected of other people (although even among non-autistic people there is no rule) and it is the role of parents, teachers, professionals and society as a whole to seek to enter this particular universe and try to understand and respect the world from the point of view of autists. Furthermore, according to the authors, “[...] to understand and master the unique world of individuals with autism is to have the opportunity to participate in a daily miracle: the rediscovery of what is most human in us and in them” (^{Silva; Gaiato; Reveles, 2012}, p. 04).

According to ^{Marinho and Merkle (2009}, p. 02), “we live in a society with pre-established standards, where anyone who is outside them is in the first instance excluded”. When it comes to the inclusion of autists in common school classes, it is considered important to approach the theme of autism, to know its specifics and to bring students and the entire school group a better understanding in this regard. According to Marinho and Merkle (2009), promoting strategies capable of decreasing the delay in the social development of the autist is one of the biggest problems faced by educators in the school environment, which, consequently, entails losses in the relationship with other people and communication skills.

In a perspective opposite to the one highlighted by the respondents, the studies of ^{Marques, Barbosa and Gomes (2018}) report that there is a peaceful acceptance on the part of students in relation to fellow students with disabilities. According to the authors, students not only welcome autistic people, but also care to help them at all times. From these discussions, looking at our data, we are led to reflect that possibly what teachers think about the interaction between autistic and non-autistic students is more related to the very way these teachers conceive of students than to the way different students actually understand each other.

In general, it can be noted that the teachers interviewed characterize the autistic student as a subject that presents many difficulties. Such difficulties are listed by the lack of socialization and interaction, accentuated learning difficulty and non-acceptance of other students, becoming a challenge for teaching and learning. It draws our attention to the fact that there is no direct mention of the part that belongs to teachers to deal with these aspects, that is, they are exempt, in their conceptions listed here, from the difficulties of interaction, of learning of their students.

Teaching concepts about the inclusion of autistic students in common schools

In this category, the views of the teachers interviewed about the movement of inclusion of autistic students in common schools will be discussed. The interview questionnaire did not give prior and more explicit direction for the interviewees to talk about this topic. Meanwhile, she appeared in five units of meaning, revealed by two interviewees.

The two-teacher concept of including autists in common schools is a major challenge. In the view of the respondents, the education of autists in an inclusive perspective is challenging by the fact that the student himself notes that there are differences in involvement with school tasks.

U40PS: [...] a child with autism suffers [...] when she sees that she cannot accompany the class [...].

Again, as in the previous category, one of the teachers interviewed is excluded as a participant in the schooling process of their student, that is, if the autist suffers, it would be due to problems that do not compete with the teacher, nor the other students. He wants to know about his autistic status. Such a conception approaches the characteristics of the concept of school integration, in which the role of adapting to the environment was the sole competence of the student.

Another teacher interviewed believes that, in order for inclusion to happen, it is necessary to get the autistic student to perform the same tasks as the others, however, according to her, by knowing the specificities of the student, considers this a difficult task. This feeling of difficulty on the part of autistic students in the conception of teachers must be highlighted, as this can disadvantage a learning-seeking movement: it has difficulties, there is not much to do, “it is difficult”. According to her,

U18PT: [...] to me, to include is to make that student do the same thing that everyone is doing [...].

This idea is closer to the concept of inclusion that we are advocating, that is, participating in the same space, with the same tasks, although some of them need some support. It is noted that teachers reveal different conceptions about inclusion, which consequently results in different concerns about having an autistic student in the classroom.

According to ^{Kupfer and Petri (2000}), inclusion became a concern for Brazilian educators especially after a whole series of federal and state laws came into force. However, they are laws that have not been accompanied by measures to enable their proper implementation, as well as measures facilitating their implementation. We need to discuss, in the training sessions, the subject of the legislation, the concepts behind these documents, as well as having a continuous reflection on how we, as teachers, conceive inclusion in our role. More than that, we need to reflect on what is exposed in inclusive education policies in relation to our school contexts.

Faced with what the interviewees explained regarding the inclusion of autists in common education, and falling into the first category, in relation to what they think about the autist, ^{Camargo and Bosa (2009}, p. 69) point out that, “[...] to the extent that the subject is seen only under the angle of its limitations, the belief in its educability and possibilities of development will be associated with the impossibility of the stay of this subject in spaces such as the common school”. Still according to the same authors, “[...] these conceptions seem to influence pedagogical practices and expectations about educability of these students. The difficulties of teachers, in a general way, presented themselves in the form of anxiety and conflict in dealing with the ‘different’” (Camargo; Bosa, 2009, p. 69, grids in the original).

In another unit of meaning that follows, an interviewer draws attention to the fact that we should pay attention to which room to include the student. This goes against the idea of an inclusive school, because if it were, any classroom would present conditions for inclusion. If we have problems, they're all over the school. If we have a solution in the same way, it must be discussed with everyone. With this, the idea that we should think in which room we will include our students, any of them, needs to be rethought, consistent with the conception that each has about inclusion.

U43PT: [...] when we are going to make an inclusion, the school needs to think a lot about the room where it will put that student [...].

The conception of the respondents about having an autistic student in secondary school turns out to be a challenging fact. For them, inclusion creates a personal discomfort for the autistic student, as well as for the others. Again, the discomfort of the teachers themselves was not explicitly highlighted, leaving them in the condition of analyzing the process without being included in it. In general, the discomfort is given, according to the interviewees, by the autist, for noticing that there are differences in the treatment with him and by the students, for not accepting and not respecting the idiosyncrasies of this subject.

The teacher’s conceptions about the school’s structural support needed for the inclusion of the autistic student

In this topic, we discussed the conceptions of the teachers interviewed regarding the need for structural school support that favours the inclusion of the autistic student. 26 units of significance listed by the eight respondents were considered.

Having in the room an auxiliary teacher for the autistic student is considered by 5 interviewees an essential support that should be provided by the school in order to favouring learning, interaction and collaboration with the work of the regent teacher. In the opinion of these respondents, having an autistic student in a common room without him having an auxiliary teacher to serve him individually becomes a much more difficult task for themselves.

U15PS: [...] until the time he had no attendant [auxiliary teacher], to teach was very difficult [...].

According to ^{Souza (2019}), the presence of supporting teachers in common education is essential for the care of children supported by Special Education, as well as being a fundamental element for inclusive education. Law No. 12.764, of 27 December 2012, in its Article 3, in its single paragraph, states that “in cases of proven need, the person with autism spectrum disorder included in the common classes of regular education, pursuant to paragraph IV of article 2, shall have the right to specialized accompaniment” (Brasil, 2012, p. 01). In this way, some autistic children have the right of an auxiliary teacher and, many teachers, when they encounter autistic students in the classroom, already expect from the school management this specialized support, because in the case of failure to meet, individually, the specificities of that student, will have the auxiliary teacher to assist him. It should be noted that, even if the law guarantees the right to a specialised accompaniment, the school and family interventions with the public educational sector are often required in order to enforce the legislation, and most of the time the professional provided is a trainee teacher, often without the specific training for Special Education.

As pointed out by ^{Vigotski (2006}, p. 113), “what a child can do today with the help of adults can do tomorrow by himself”. In this sense, it is understood that, with a good mediation, you can launch challenges and qualitative results in the development and learning of new skills of the autistic student, being the teacher of support of the utmost importance for the mediation of learning and support to the teacher in the process of Learning and inclusion.

On the other hand, the role of the relationship between the regent and the supportive teacher must be discussed and, given the importance shown between the two, we understand that the teaching and learning process must take place through collaborative work, so that it is possible to a training with good quality, equity and autonomy of the autistic student. These are different tasks and so they must be understood. Above all, the student cannot be assigned as responsibility only to a few, such as the supporting teachers, but to all school staff.

The importance of having a specialist to work with the school inclusion of the autist, as well as with his idiosyncrasies of the school daily life, was pointed out by three (3) interviewees as an important structural support that should be provided by the school. The interviewees cited a psychopedagogue or a psychologist as specialists suitable for the work in the care, not only to autistic students, but to all those who interact directly with these subjects, i.e. parents, teachers, classmates and teaching staff.

U30PV: [...] the school needs to have a psychologist, [...] not only for autistic students, in general [...] to assist both teachers [...] and students.

The psychologist with specific and well-defined training, inserted in a context of knowledge of human development, will be able to detect the phased and compromised areas of the autist and, being sensitive to the reports of teachers and family, can contribute to the inclusion and learning of this student in an indirect way, since we understand the commitments of the teacher and the psychologist as different, and should be respected their guidelines of performance. Now, if there is a need for a multidisciplinary team to diagnose an autistic student, we also have to have this collective look after the diagnosis.

Healthcare or psychology professionals must not interfere with the individual commitment of other school employees, i.e. they must be a support in what they are adequately trained for. However, we cannot all give up our commitments and duties, which must be pedagogical when we are inside a school. Finally, if everyone focuses on their limits (teachers with the pedagogical issues and professionals with the issues that fit them) and together dialogue continuously and collaboratively, we can draw better paths, more inclusive. Even because, many of us do not have enough knowledge to deal with the schooling of autistic students in isolation.

The need for classrooms equipped with games, computers, multimedia projector, finally, physical and pedagogical structure that are favourable for the development, the reception and the inclusion of the autistic student in the common education is pointed out by four (4) interviewed as the necessary structure that the school should provide for the attention to inclusion of this subject.

U47PT: [...] a computer, a date-show in the classroom is the minimum that can offer [...] only the book does not provide more [...] that a student needs.

Challenges such as adequate physical infrastructure, accessibility, training of teachers, multidisciplinary staff, teacher support and didactic-pedagogical materials are necessary resources for enabling inclusive education. As an alternative to such challenges and in view of what was stated in the interviews, it is noted that the teachers claim these resources as a way to assist them in the development of learning, not only for students supported by Special Education, but for the entire school community.

In summary, the teachers conceive that, in order for there to be educational inclusion of autistic students in teaching, it is necessary structural school support in the form of specialized services with professionals such as psychologists, psychopedagogues, rooms equipped with manipulabled materials, as well as the support of the auxiliary teacher. In this sense, the concept of these services and professionals assumes a fundamental character in an inclusive space for autistic students. However, we stress that, from our point of view, these spaces and professionals should participate in a dialogue involving the entire school community, avoiding the creation of demands that are delegated to each individual. In our understanding of inclusion, for teaching design to receive more meaningful influences, we need to promote the debate of the role of all of us around a common goal.

Contribution of School Mathematics to the development of the autistic student

In this unit, teaching conceptions on the contributions of Mathematics in the development of the autistic student in common education are gathered. In other words, the interviewees seek to discuss what kind of contribution this can be, if there are differences between other students, etc. We highlight here 12 units of analysis by 5 five (5) of the eight (8) professors interviewed.

The teachers maintain that Mathematics brings positive contributions to the teaching, school development and personal of the autistic student in the common school. However, as well mentioned by Viana (2017, p. 16), “the teaching of mathematics in the perspective of inclusive education, brings a challenge to the teaching action at its core of pedagogical action: to address the differences” and, in the case of autistic students, the challenge increases even more, by the fact that TEA manifests itself in a different way in each person, requiring the teacher to adopt methodological strategies appropriate to the specificities of each student.

School Mathematics, in the development of the autistic student, is conceived by the respondents as a discipline that favours teaching and learning. However, it is necessary for the teacher to have knowledge of the specificities of the student in order to be able to serve him with better quality. The teachers understand that teaching and learning processes should include mathematics that involves everyday experiences with practical activities and logical reasoning, such as those related to Financial Mathematics, in order to make teaching more meaningful and less abstract. On this subject, ^{Cunha (2011}) reiterates that the child or adolescent with autism has great difficulties in carrying out daily activities and it is in this sense that the school, through curricular activities, will become a support to collaborate with their learning and, consequently, autonomy. It is also highlighted that the concern in some of the speeches is in the direction of a discipline that favours this autonomy, especially the topics related to the financial life of each.

U29PT: [...] it has to be applied mathematics, that makes sense in the lives of students.

U19PV: Towards independence [...]. I am very much in favour of financial mathematics [...] it can be essential for many things [...].

It is important to highlight the fundamental role of initial and continuous training to teachers who attend autistic students, because even mathematics can collaborate with the school performance of the apprentice, if the teacher does not have knowledge about their specificities, their behaviors, their learning time and the methodologies that may be adopted, in very little will be able to contribute to their learning.

The analyses of the interviews allowed us to understand that, for teaching mathematics to autists, the best are activities that aim to build meaning with functional and daily activities. In this sense, we note a certain limitation of the contribution of School Mathematics pointed out by the respondents in relation to the curriculum as a whole. It draws our attention to the fact that some aspects have not been mentioned, for example, the contribution on the use of games as a possibility of socialization, manipulabled materials as in favor of cognitive development and/or use of the Computer Laboratory and its various possibilities for learning.

To counteract this result, we can quote studies that have been carried out, such as the research of ^{Fleira and Fernandes (2019}), which uses manipulabled materials to insert the content of equations of the 2nd degree. According to the authors, the resource used not only collaborated with the autistic student in mathematical practices, but also to their effective inclusion in the classes. ^{Chequetto and Gonçalves (2015}), with the use of play materials and games, found a way to bring the student closer to the discipline of Mathematics, specifically the content of multiplication and division. The study by ^{Amaral (2018}) shows that the use of digital games, applied to an autistic student and also to other children who need Specialized Educational Care, has triggered a better understanding of concepts of the additive and multiplicative field.

The limitation of this power of scope of Mathematics, conceived by the respondents, makes us wonder whether it is related to what the autist can/will or not learn, or with the difficulties of the teachers themselves in the absence of discussions in teaching training. Such a conception does not share what we understand by Inclusive Mathematical Education, in which all students, regardless of any characteristics they possess, can learn, depending on the resources and tasks that we propose, which may or may not broaden the barriers to their participation. However, we cannot limit their possibilities in advance.

In ^{Souza and Silva’s vision (2019}), overcoming this limited vision of the school’s capacity to contribute to these students requires a change in the paradigms of inclusion, enabling a differentiated look at the autistic student, realizing his potentialities and recognizing him as a subject capable of learning. According to the authors, from the moment that there is this recognition and appreciation, the school and teachers start to develop innovative and non-exclusive educational practices, enabling the student to participate actively in the process of building knowledge.

Educational Approaches to Teaching Mathematics for Autistic Students

We present, in this category, what the conceptions of teachers about teaching approaches applied to autistic students as necessary in the teaching of Mathematics in common schools. 40 units of meaning were shown, presented by all the respondents. The use of technological resources, such as electronic games, computers, math software, educational websites and applications, was pointed out by three (3) respondents as resources associated with teaching and learning. For the respondents, the use of such resources gives students greater interest in the discipline of Mathematics and, consequently, in the process of schooling.

U29PV: [...] use the multimedia with the activities in books and PowerPoint, [...] Geogebra [...] website and games [...].

U40PT: [...] made him interested in my class.

^{Silva, Moura and Soares (2017}) highlight, in their studies, that autistic children show interest in handling and using computer resources (mobile, tablet, electronic games etc.) and for this reason are being inserted more and more often in therapies and in the teaching and learning processes of these children. Although the educators interviewed showed interest in the use of such resources, what concerns us is the precarious training and information, as mentioned earlier, that most teachers have regarding autism and these students, specifically. The act of knowing the student and his specific needs contributes to achieving the goal of learning in the applied activity. It is not enough to know only the materials, but also the students. According to ^{Orrú (2003}, p. 04), “such an individual exposed to various visual stimuli without the proper intervention of the educator, becomes stressed because of the saturation of information that remains unfunctional to him”.

The use of manipulabled materials, such as games, was mentioned by four (4) respondents as a teaching methodology that provokes the interest of the autistic student in learning. According to the teachers, the use of these recreational resources as a teaching strategy motivates them to participate better in mathematics classes, improving the student's attention to activity.

U31PS: [...] I knock on that key that when you work with concrete material [...] attracts attention and is interesting.

With proper planning, in the sense of being articulated to the teaching objectives of the teacher, the use of manipulable materials is a suggestion of methodology that can resignify the pedagogical practices in teaching Mathematics for autists. However, before presenting activities with these materials, it is necessary to know the autistic student so as to establish what skills he possesses and what he needs to develop, given that, as the teacher is getting to know his student, he can establish new strategies.

According to ^{Chequetto and Gonçalves (2015}), the use of manipulable materials can be a way to bring students closer to the discipline of Mathematics, sometimes considered to be of greater difficulty for understanding. However, the authors emphasize the importance to be given to the planning of classes, combining such resources, mainly in the work with autistic students, thinking about their specifics and taking into account their behavioral nuances and their learning time.

Some alternatives that can be used in the classroom are pointed out by ^{Brito and Sales (2017}) and ^{Gaiato (2019}), to enable the learning of autistic students. They are: use of materials of interest to the student, clear and objective explanation of the work that will be carried out, involve the student in the organization of the activity, sit him/her in front of the room, adopt positive reinforcements (such as compliments), use visual resources, among others. We realize that the use of games can enable most of these alternatives approaching in a playful way the contents to be worked on.

Even considering the commitment of educators to work with methodological strategies that involve games and manipulabled materials, we agree with ^{Nacarato (2005}), which points out that no manipulabled material constitutes the guarantee for the improvement of mathematics teaching. Its effectiveness or not will depend on the way it is used. The strategy is not only in the material, but also in the teaching leadership.

Although not mentioned in the interviews, we know that there are difficulties that may arise with the use of manipulabled materials. Some factors may interfere negatively in the application of these materials, for example overcrowded rooms, lack of aiding teacher, insufficient pedagogical resources that serve all students, etc. However, even with all these challenges, it is clear from the interviews that educators are looking for alternatives to the care of autistic students, so that they have the opportunity to learn in a playful way and aimed at interaction with the other students.

According to three (3) professors interviewed, by understanding that autists are students who present marked difficulties, who cannot accompany the class in the activities and need a more individualized follow-up, reveal the need for teaching strategies with activities that are differentiated from the others.

U24PR: [...] all activities [...] had to be different [...] did not copy, nor wrote [...] speech problem [...].

U33PD: [...] I have always sought to do in a differentiated way, the activities [...].

On the activities that autists carry out in the room, ^{Azevedo (2014}) mentions three classifications: the adapted activities in which they are part of the curricular adaptations that teachers must make when necessary with the aim of enabling the learning of all; the different activities where only autistic students carry it out; and the equal activities, that is, those that all carry out together. However, the same author stresses the importance of these activities not to be a reason for segregation, but rather to be carried out in accordance with what is being worked by the class or at times when everyone can participate. From our point of view, rather than already defining in advance whether the activities will be the same or not, we need to maintain coherence with our individual conception of inclusion.

The autistic student, like all others, already arrives at school with a baggage of interests in some objects, subjects, themes, that is, he already comes with certain interests established before the school environment. Some teachers appropriate this interest and start from this principle to try to establish a relationship between teacher and student.

As mentioned earlier in the theoretical section, the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) (^{APA, 2013}) is a document used by health professionals as a reference to diagnose mental and behavioral disorders, such as, for example, ASD, but that does not portray the pedagogical issues of the student. The document addresses the specific interest that autists demonstrate with great intensity for certain subjects, objects, themes. According to the document, we have: “[...] 2. Insistence on the same things, inflexible adherence to routines or ritualized patterns of verbal or non-verbal behavior [...] 3. Fixed and highly restricted interests that are abnormal in intensity or focus [...]” (^{APA, 2013}. p. 50).

In general, teachers understand that the use of technological resources, manipulable materials, differentiated activities and knowledge about the specifics of their autistic student are teaching approaches that contribute to the teaching of mathematics for these subjects.