Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky’s defectology Alunos com deficiência intelectual aprendendo

This research aimed to understand the learning process of fractions for two students with intellectual disabilities, collectively with students without any disability. The type of procedure used was interventionist research. There were eighteen subjects in this research, which comprised of Elementary School 6th-grade students, two of whom had intellectual disability/focus subjects. The following data collection instruments were used: 10 observations, totaling 8h20m, and 10 documentary analysis documents. The results showed that the students with intellectual disabilities learned scientific concepts, which involved the concept of fractions, mainly in relation to the idea of part-whole. The study concluded that the understanding of mathematical content, especially scientific concepts about fractions in the educational process of students with intellectual disabilities, can be effective when applied in processes involving intentional pedagogical actions – based on concepts of Vygotsky’s defectology.

INTRODUCTION
The informal dialogues that are established with some teachers of varying types of schools in Brazil about the educational process of students with intellectual disability (ID) (AMERICAN ASSOCIATION OF INTELLECTUAL AND DEVELOPMENTAL DISABILITIES, 2011;HATTON, 2012) often result in the affirmation that these students cannot keep up with the others [without disabilities] in solving the proposed activities in class, which means, they do not have the same learning pace (BEZERRA; ARAUJO, 2011;MUSIS;CARVALHO, 2010;CROCHÍK et al., 2009;RODRIGUES, 2017). As a consequence of this opinion, these teachers hardly ever invest in pedagogical actions for these students. Thus, in a classroom with the other students, ID students, who among the disabled are the majority in Brazil Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology (MELETTI;BUENO, 2013), seem already to have succeeded or benefited greatly if they can socialize (which represents a limited comprehension of the possibilities of learning for ID students).
Statements like these can be heard on a daily basis in Brazilian educational institutions and are consistent with a hegemonic view of intellectual disability, "based on the medical model, whose focus is clinical and focused on the difficulties and limitations imposed by organic conditions" (CARNEIRO, 2015, p. 3). These pessimistic conceptions were already raised as issues by Vygotsky (1983b), in the early twentieth century. For this author, [negative] manifestations that qualify the student with ID as someone who only has to live with the other students at school is not only incorrect, but also hinders the potential learning possibilities of the disabled student as they can influence the teaching practices: Под влиянием пессимистического взгляда на глубоко отсталых детей происходит обычно снижение требований, заведомое сужение, сокращение границ и пределов, которые ставит себе воспитание этих детей. Под влиянием такого взгляда, естественно, возникают минималистские тенденции, стремление свести воспитательные задачи по отношению к этим детям до возможного минимума, ограничиться самым необходимым. [...] [when a person with intellectual disability is deprived of school education, properly organized] теряет в своих возможностях и страдает из-за этого не меньше, но больше, чем нормальный ребенок. [Under the influence of a pessimistic view about severely retarded children, a lowering of demands usually takes place, a knowing narrowing of the boundaries and limits which the educators of these children set for themselves. Under the influence of such a view, minimalist tendencies arise, an attempt to reduce the educational tasks placed before these children to a minimum, to limit them to what is most necessary. [...] [when a person with intellectual disability is deprived of school education, properly organized] loses his/her potential to advance and suffers not less, but more, than a normal child. (VYGOTSKY, 1983b, p. 224).
Vygotsky studied the psychological constitution of people with disabilities and their relation to the world and to the school through their defectology, "moving from practice to theory and vice versa" (DAFERMOS, 2018, p. 27). Stetsenko and Selau (2018) cite that "defectology" is an expression that today seems to be outdated. It was not created by Vygotsky; however, he used it in a theoretical-practical original reframing, which was included in all his work (GIEST, 2018;TOOMELA, 2018;COSTAS;PASSERINO;COMA;BALDASSARRI, 2018;BARROCO, 2018;VEGGETTI, 2018;DAFERMOS, 2015;EVANS, 2003;STETSENKO, 2016STETSENKO, , 2017. Taking into account the term defectology, Gindis (2003, p. 200) points out that [d]efectology is the term that reflects the area of Vygotsky's research and practice that is relevant to contemporary special education and school psychology. [...] Ironically, the negative undertone of the term itself is in no way present in the inspiring and positive attitude of Vygotsky writings. The word defectologia (or defectology in the English transliteration) literally means the study of [the] defect. (emphasis in original).
Although, regrettably, Vygotsky's defectology is not part of the official documents originating from the Brazilian Ministry of Education (COSTAS; BRASIL, 2008), Sannino (2018) and Stetsenko and Selau (2018) emphasize that the in-depth understanding of Vygotsky's defectology is not only useful for researchers who attempt to study the inclusion of the disabled in school; defectology also provides a fundamental basis "to confront and debunk all forms of biological reductionism" (STETSENKO; SELAU, 2018, p. 318).
The present research was carried out considering a selection of some of the vygotskyan premises. Looking at the teacher's responsibility in elaborating teaching strategies that achieve the objective of ID students learning the curricular contents, or more specifically, fractions, this research has been defined, and it was developed with 6 th -grade Elementary School students, among whom were two female students with ID. The basis of the study was to find an answer to the question: do students with ID learn fractions collectively with students without any disability? [Or, objectively: to understand the learning process of fractions by two students with intellectual disabilities, collectively to students without any disability].
In order to relay the results of this research, there will be presented in the following sections: the research methodological procedures; the findings of the category "Fractions in Scratch: learning obtained via the intervention research" and the data discussion; the final considerations; and references.
Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology

METHODOLOGICAL PROCEDURES
The type of procedure used was interventionist research (LAITINEN; SANNINO;ENGESTRÖM, 2016;DANIELS, 2016;STETSENKO, 2016;GONZÁLEZ REY;GOULART;BEZERRA, 2016), pedagogical intervention research-type (DAMIANI et al., 2013). Interventional research has been used in a variety of fields: Business Administration (BONNEFOND et al., 2016;QUEROL;JACKSON FILHO;CASSANDRE, 2011), Psychology (SANNINO;SUTTER, 2011), Education (CASTRO, 2014PINHEIRO, 2014;ROCHEFORT, 2012;CENCI, 2016), among others. Taking a stand on this methodological procedure, Stetsenko (2016, p. s37) states: [i]n my interpretation, this position implies that methodological tools, strategies and techniques have to be tailored to, and to result, not in the uncovering of facts "as they are" at the present moment and within the given status quo. Rather, it implies intervening with and co-constructing phenomena and processes which we investigated in non-neutral, historically determinate ways in line with the ontological, epistemological and ideological commitments and goals that the researchers deem worthy and take on as guides for action.
The participants of this interventionist study were eighteen students at an Elementary Municipal School in the town of Pelotas, Brazil, with eight boys and ten girls. Their ages ranged from eleven to thirteen years old. The focus group participants were two female students with ID from the same class and were selected according to the following criteria: to be a student with ID, to be enrolled in the 6 th -grade of Elementary School and to be sent by the teacher for tutoring at the Mathematics and Computer Science Laboratory. All due care was taken regarding research ethics, in all the phases of research and with all the subjects. To protect the ID students' identities, the definitions "S + number" were used.
The axis of the teaching process to carry out this intervention research was the use of a digital educational game, mediated by the teacher-researcher, with content related to the concept of fractions, called Fractions in Scratch. Scratch is a free software program developed in the Massachusetts Institute of Technology 5 5 Scratch is a project of the Lifelong Kindergarten Group of the MIT Media Lab. It is available free of charge. The Scratch project received financial support from the following organizations: National Science Foundation, Scratch Foundation, Siegel Family Endowment, Google, LEGO Foundation, Intel, Cartoon Network, Lemann Foundation, MacArthur Foundation. For more information, visit: (LIFELONG KINDERGARTEN DO MIT MEDIA LAB. SCRATCH SOFTWARE, s.d.) which is a visual programming language and allows the users to engage with their own stories, animations, games, simulators, visual learning environments, music, and arts. The game Fractions in Scratch was created by the researchers and it was carried out for a period of four months. As the software is free, no patent was required.
The intervention research consisted of 10 meetings, each one with a limited time of fifty minutes, which took place once a week from September to December 2016, in the Computer Laboratory of the school, locus of this research, and involved all eighteen students in the class. Each of the 10 meetings adhered in sequence to the following script: a) Resumption of activities developed during the previous meeting: The objective of this part was to provide thinking on the conceptual approaches developed; it was when students would orally report their experiences in the previous meeting. It was the stage of the process in which the students realized their learning with the help of their teacher (VYGOTSKY, 1982a;BREITENBACH, 2018); b) Game activity: With the mediation of the researchers (WERTSCH, 2007), the students played games composed of fraction activities. The game consisted of 20 (twenty) screens. On the home screen, there was a greeting and invitation for students to participate in the activity. Subsequently, a menu for the participant choice was presented. Each screen had an activity/exercise to be solved. The feedback occurred through discussions during task execution and synthesis. This feedback consisted of a collective discussion about each exercise, which was coordinated by the teacher. Next, there are some of the screens that make up the game Fractions in Scratch. Picture 1 corresponds to the home screen of the game: https://scratch.mit.edu/about. Recently, this platform was used in the research of Teixeira, Furini and Caimi (2018).  After accessing the home screen and interacting with the character, the students were directed to a menu, in which they should choose the game, according to the proposal of the meeting. Picture 2 presents this menu: Teaching mediation involved helping students to play, to talk to their classmates and to understand fraction characteristics. With this proposal, an endeavor was made to take into account Vygotsky's theory (VYGOTSKY, 1982a) that the educator should take part in the developmental zone closest to the student, attempting to make advances that do not happen spontaneously. The mediation followed the idea that "development and learning are not in the games themselves, but in what is triggered by the interventions" (DRUZIAN, 2007, p. 69). c) Report on the activities developed: At this moment, the students gave an oral report about the activities developed on that day to their classmates (VYGOTSKY, 1982a(VYGOTSKY, , 1983cLURIA, 1966LURIA, , 1971. In this part, mediation between the students and the conceptual approach/fractions was also promoted. In order to infer the results obtained with this research, two instruments were used for data collection: observation and documentary analysis. The observation (LÜDKE; ANDRÉ, 1986) was performed while playing the game. Video recording was included along with the observation (KOHATSU, 2007) because the meetings were recorded for later transcription. Ten observations were made, which totaled 8h 20m of recordings. The documentary analysis (LÜDKE; ANDRÉ, 1986) was performed with diagnostic assessment (two drawings, two qualitative diagnostic evaluations, two school assessments) and post-intervention assessments (two final exams and two descriptive reports). Table 1 summarizes the data collected: The procedure of discursive textual analysis was used for the research data analysis, proposed by Moraes (2003), resulting in the construction of the emerging category "Fractions in Scratch: learning obtained via the intervention research". The methodological process (pedagogical intervention research-type) is outlined below (figure 3):

FINDINGS AND DATA DISCUSSION
3.1 What did the students know before?
In order to plan this intervention research and to implement it and later to understand the effects of the intervention process, it became necessary to have prior knowledge about the everyday concepts of fractions that all the students had, especially the focus participants.
Anache and Resende (2016) understand that diagnostic assessment is an important element in working with all students, including ID students, in order to elaborate pedagogical action to deal with different situations that arise in the educational context. They state that "the results The documentary analysis was performed with diagnostic assessment (2 drawings, 2 qualitative diagnostic evaluations, 2 school assessments) and post-intervention assessments (2 final exams and 2 descriptive reports).
The procedure of discursive textual analysis was used for the research data analysis, proposed by Moraes (2003).

What did the students know before?
In order to plan this intervention research and to implement it and later to understand the effects of the intervention process, it became necessary to have prior knowledge about the everyday concepts of fractions that all the students had, especially the focus participants. Anache and Resende (2016) understand that diagnostic assessment is an important element in working with all students, including ID students, in order to elaborate pedagogical action to deal with different situations that arise in the educational context. They state that "the results of the assessment are only valid if they guide the teachers in educational planning" (ANACHE; RESENDE, 2016, p. 573).
Following these formulations, a diagnostic assessment was implemented which aimed to verify the performance of the focus participants in relation to the content of fractions. Fragments from these diagnostic assessments follow (Documentary Analysis): Researcher: Girls, what does the bar mean? S1: What did you say? The bar is a bar. S2: Bar is the line. One number has to stay on top and one on the bottom. Researcher: -But why does one number have to stay on top and another on the bottom? S1: I do not know. The teacher did it so. S2: That's the way it is on the board. The teacher did it so. (Qualitative Diagnostic Assessment) Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology Realizing that her answer was incomplete, S2, for example, added: "A fraction is a different form of writing/representing calculation" (Qualitative Diagnostic Assessment). According to the fragments, in this same class, S1 and S2 did not even understand the meaning of the word "fraction".
The difficulties of understanding the concept of fractions were also found in the number of correct answers in the students' school assessment (Documental Analysis): out of 15 questions proposed, S1 scored four and S2 6 six. Based on the students' answers, it was observed that the understanding of the concept of fractions was not consolidated (according to Nunes et al. [2005], it is necessary that the students understand that the parts taken from a rectangle, for example, are equal, establishing a relationship between the concept of fractions and the division).

Learning fractions based on teacher actions
After three meetings, there were signs that demonstrated a small understanding about the concept of fractions as a result of the teaching process collectivity implemented, through the collateral method with the digital Fractions game in Scratch. The following transcription is a dialogue that illustrates a situation proposed in the game, in which an attempt was made to work on the ideas of whole and half, through an event commonly experienced in daily life: According to Ciscar and García (1988, p. 33), using everyday situations in teaching allows students to recognize mathematics in the world around them, and the teacher's task is to assist them in building the concept involved. In this report, we also observe the attempts to challenge S2 to realize the difference between one whole and one half (NUNES; BRYANT, 1997). Thus, "one" was the answer of S2 to associate it with the whole and "piece" to the other answer, referring to the half.
The understanding of part of a quantity was also evident during the intervention research in the fact that it was understood as fundamentally important in this context. To do so, the following activity was proposed: from an unlimited number of balls of three different colors (orange, blue and green), the students were to place them in baskets, according to the teacher's command. During the activity, the students were asked if they could initially identify the total elements of the set. Once this was achieved, they were asked to identify how many were orange and how many were blue among the components of the set. This moment is described in the fragment below: Researcher: S2, read the question, please. S2: "Place the balls in the basket. One quarter of the balls should be blue, three quarters should be orange". Researcher: Thank you. S1, how many balls do we want in total in the basket? S1: 4. Researcher: Okay! S2, how many will be blue? S2: 1. Researcher: Why? S2: You asked S1, how many balls she should have in the basket, and she answered 4. Researcher: Right! And? S2: There are 4 balls: 3 are orange. There will be 1 left. S1: What's left is a fraction. (Observation 4) It was understood that the situation involving discrete 6 quantities facilitated the students understanding, leading them to perceive the total amount of elements of the set of balls that should be placed in the basket.

Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology
During the intervention research, the students were encouraged to talk about their findings. In the following example, the students verbally reported what was done during the meeting: Researcher: Our time is almost over. Now, I need you to tell me everything we did today. You talk and I write on the board, right? You can start! S2: Today, we made a toy to guess the future. S1: The origami oracle and then we played. S2: We played a game in Scratch of answering questions. S1: The questions were about a bar of chocolate. S2: We talked about fractions ... a fraction is a part of a whole.
Researcher: And what else? S2: There's more, but I can't tell.
Researcher: -What about you S1, what else can you say? S1: That fractions divide. Researcher: -Do you want to add anything else? S2: No. S1: No. (Observation 5) It is possible to see that the students struggled in doing what was asked and that they did not present in their description elements that indicated awareness about the process experienced or that learning had taken place. During the intervening process, this situation changed: this moment contemplated the expectation, which consisted in observing, through the narrative carried out by the students, if a restructuring of the relationships was taking place in the intrapsychological framework as those which occur in the interpsychological framework. The following fragment illustrates this situation: Researcher: Let's report our work today? S1: All right. Researcher: Who starts? S2: Me. Researcher: You can start. S2: We played the race of fractions game. Researcher: Right. And what else? S1: We saw that in the fraction there are two numbers separated by a line. S2: The numerator and the ... S1: Denominator. S2: The denominator is the number below the line. The number above the line is called the numerator. S1: The numerator indicates how many parts have been taken. Researcher: From where? S1: From the whole. S2: The denominator indicates in how many equal parts it was divided Researcher: Then the terms of the fraction are: the numerator and the denominator. S2: Yeah. S1: Teacher, let's name them correctly now. Researcher: Correctly? Now there won't be a top number and bottom number anymore. S1: No. Now it's going to be numerator and denominator. (Observation 7) We can observe the students' evolution because the synthesis they made lead to the understanding of the terms that compose a fraction as well as their meanings.
The content of the seventh meeting was addition with fractions, involving, besides the explanation, routine activities. During the explanation, it was pointed out that there are different ways of proposing and performing an addition. It was requested that a way be worked out of accomplishing the task in groups in which everyone understood the process and no longer added up all the terms. The fragment below shows the synthesis formulated: S2: When I have to add fractions, I look at the terms. If the denominator is equal, I only count the numerator. S1: If the terms are different, I have to remember. Researcher: -I do not understand! Say it again. S2: If the denominators are the same, I only count, sum up, the numerators. S1: Yeah! It only applies if the denominators are the same. (Observation 7) The ID students said that there is a rule in the process of adding fractions. This rule is an exception to spontaneous knowledge, therefore, it can only be learned in school. This intervention procedure, which was the only subject that was working with the concepts of fractions, is a scientific concept.
The aspects that could have contributed to the learning of the ID students within the collectivity were also observed. Vygotsky (1983d) has drawn attention to the fact that the higher psychological functions, which both disabled children and children considered normal, have a social origin, both in phylogenesis and also in ontogenesis. Vygotsky points out that
The understanding expressed by Vygotsky has an influence on the way the teacher plans and implements the practice of the teaching process in the classroom (SANNINO; SUTTER, 2011). The collective form of teaching cannot be represented only by the grouping of students without planning: collective forms of teaching take place to the extent that there is collaborative work among students and between them and the teacher. In the situation transcribed below, it is possible to observe that the activity performed in pairs consisted of associating the graphic representation of a fraction with its numerical representation; S1 partner and a classmate [who has a level of development superior to S1] realized her difficulty in accomplishing the task and offered to help her: S1: I don't know. Classmate: I'll help you. S1: It's okay. Classmate: First, you're going to count how many pieces are in the whole figure. S1: Right. One, two, three, four, five, six. Classmate: Did you count? S1: Yes. Classmate: Now you have to see how many pieces are painted, colored. S1: One, two, three. Classmate: Then you have to put them together. S1: Together? Classmate: The drawing with the number. Look here! S1: I'll try ... I did it! (Note 6) The following excerpt is an example of a situation that portrays the value of interactions in the learning process, now starting from the ID student: S1: I don't get it. I can't fit them in. S2: Come on, S1. Look how I do it. You have to drag them to put them together. S1: Yes. S2: Drag. Count how many pieces there are. Then you count how many are painted. Researcher: And then? S1: Do I have to look for the number to put them together? Researcher: -Number 7 or fraction 8 ? S1: Find the fraction number. (Observation 4) In this brief report, it was observed that there are two people assisting student S1 (teacher and S2, who is the other student with ID). The participation of these two people in the relationship with S1 was fundamental in motivating her to accomplish the task. It should be noted that the zone of proximal development was created in this triadic relationship, triggered by the teacher.
The evaluation of the learning process is completed with the presentation of the data referring to the new "final assessment" and "descriptive report" (Documental Analysis). In the end, in the final assessment [Documentary Analysis] S1 scored 10 and S2 scored 13 correct answers out of 15 [in the assessment given before the intervention research, S1 student scored 4 and S2 6 out of 15 questions]. The descriptive report (Documental Analysis) of the third quarter also changed: "Regarding the content of fractions, and rational numbers, [S1] seems to have understood the basic concepts, which is fundamental for understanding this numerical set" (Descriptive Report S1); "S2 was able to understand the basic notions regarding this numerical set, which will help her to learn contents of this set" (Descriptive Report S2). The results showed an evolution in the students' learning process, regarding the concept of fractions. 7 Number is a mathematical concept to represent measure, order or quantity. Order numbers are called ordinals and quantity ones are called cardinal numbers. In mathematics, numbers are classified as natural, integer, rational and real (BRASIL, 1998). 8 Fraction belongs to the set of rational numbers. This set can be presented as a fractional or decimal representation (BRASIL, 1998).

S2
School After the intervention '2' 1. "Regarding the content of fractions and rational numbers, [S1] seems to have understood the basic concepts, which is fundamental for understanding this numerical set" 2. "S2 was able to understand the basic notions referring to this numerical set, which will help her to learn other contents of this set" Source: Authors.

Relating planning decisions and some results
The aspects that helped in the theoretical decisions to organize the teaching process in interventionist research became clear from the data presented: first, it was understood that people [students] need to think mathematically about everyday situations. This decision involved planning the classes, as described. It is important to highlight that it is not sufficient to only include everyday situations for teaching fractions, or in other words, just learning procedures is not enough for students with ID: "it is necessary to turn these procedures into thinking tools" (NUNES; BRYANT, 1997, p. 31). The subject needs to access new ways of thinking, in other words "learn new conventional systems of representation" (NUNES; BRYANT, 1997, p. 32), enabling the connection with the knowledge already acquired with new situations. Thus, still in the light of the cited authors: [...] the child's thinking will be expanded through his/her attempts to struggle with new types of invariants. In this case, he/she may use their own symbols and features for some time before being asked to use conventional forms of representation. The conceptual development in mathematics, therefore, is not equivalent to the mastery of a list of procedures, as some approaches to curriculum development have assumed in the past. Progress can come from understanding new invariants, from the ability to learn new forms of mathematical representation, and from connecting old ways to new situations that will enrich the child with meaning. (NUNES; BRYANT, 1997, p. 32).
The teaching of rational numbers (fractional representation) is formally initiated between the 4th and 5th grades of elementary school through basic dayto-day ideas such as division of pizzas, cakes, and chocolate (whole representation) and activities involving the joining of the parts (insert idea/concept of half or ½). The purpose of the study of fractional numbers is the expansion of the students' mathematical language because the numerical set they know up to this moment (natural numbers) can no longer solve present situations within the school context and in everyday life. Pereira (2011, p. 12) explains that fractions "were created to represent a part or parts of a whole that were divided into equal parts". The fraction thus refers to an "all-reference", which can be taken as a collection of anything. Pereira (2011) warns that there are different meanings for the concept of fractions: "for the same meaning, there are different resolution strategies used by students and the way of teaching the concept of fractions used in schools gives privilege to some meanings, as part-whole and multiplicative operator, to the detriment of others" (PEREIRA, 2011, p. 12). For this reason, in teaching of fractions, it is important to investigate before the teaching process the students' spontaneous concepts (what is the significance of the students' previous experiences in their lives -in which places they use them and in what forms they are represented -see ESTADO DO RIO GRANDE DO SUL, 2009). According to Pereira (2011, p. 14), [...] the teacher must create learning situations that provide the development of mathematical thinking, logical reasoning, the appropriation of mathematical language and the understanding of the world around them, through the application of new teaching methods and the use of new pedagogical practices.
The second aspect considered in this research was the following: we supported the idea that the ID students, who were participants of this research, presented all the requirements for the development of abstract thinking, and of Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology the higher psychological functions through the learning of a mathematical concept, particularly fractions. This is a fundamental comprehension, which must lead the teaching work since knowing the students diagnosis might not be a reason for labeling them. Vygotsky (1983f, p. 231) says "[н]и одна из интеллектуальных способностей не может считаться вполне отсутствующей у этих детей" ["not a single intellectual aptitude can be considered as altogether absent in these children"]. Vygotsky observed some students who had profound ID who turned out to be capable of rational action (VYGOTSKY, 1983c); in some cases, it has also been shown that "процессах психического насыщения у умственно отсталых детей в возрасте 8-11 лет не наблюдается никаких существенных отличий от нормальных детей в скорости насыщения" ["no essential differences in the speed of saturation can be observed during the process of psychological saturation between mentally retarded children at the age of eight to twelve years and for normal children"] (VYGOTSKY, 1983f, p. 235). Therefore, pessimistic and stigmatizing understandings about the abilities of the students with ID interfere in the class planning and the way they are challenged and as a result, in the goals set for their schooling processes.
The third fundamental aspect is that the teaching proposal [so the game denominated Fractions in Scratch was used] followed the notion of collateral pathways, proposed by Vygotsky (1983a). It is important for the teacher to consider that some contents will be learned by students with disabilities through other ways different from those used for the so-called typically developing children. Vygotsky (1983a) draws attention to the fact that collateral pathways, understood as paths or cultural resources that allow the accomplishment of a task and have a significant pedagogical and creative character for the child, should be considered. Vygotsky (1983a) states that the cultural development in a child's conduct is not necessarily related to an organic function. In such a comprehension, Vygotsky states that human-cultural development occurs by collateral pathways, as explicitly stated: [...] в процессе культурного развития у ребенка происходит замещение одних функций другими, прокладывание обходных путей, и это открывает перед нами совершенно новые возможности в развитии ненормального ребенка. Если такой ребенок не может достигнуть чегонибудь прямым путем, то развитие обходных путей становится основой его компенсации. Ребенок начинает на окольных путях добиваться того, чего он не мог достигнуть прямо. Замещение функций -действительно основа всего культурного развития ненормального ребенка [...]. [[...] in the process of cultural development, there is a replacement of some functions in the child by others, a construction of detours, and this opens before us completely new possibilities in the development of the abnormal child. If such a child cannot attain something directly, then the development of detours becomes the basis for compensation. The child begins to attain in roundabout ways what he could not attain directly. Replacement of functions is actually the basis of all cultural development of the abnormal child […]]. (VYGOTSKY, 1983a, p. 147-8.) The importance is clear, therefore, for teachers to develop distinguished pedagogical strategies that enable the development of scientific concepts among students with ID.
The fourth aspect can be found in Vygotsky (1982a) and Luria (1966Luria ( , 1971, who draws attention to the fact that the spoken word is the mediator of consciousness. All the students, therefore, were made to speak as much as possible in the course of the lessons. Castro, Damiani and Selau (2016, s68-s9) explain this notion, from the following: From a historical-cultural point of view, it is through words, through the enunciations of others, that we act voluntarily and consciously. We become responsible for our own conduct. But how does it happen? The word, as a sign and epistemologically based on historical and dialectical materialism, duplicates the world in our minds. It encodes experiences and meanings and allows us to become aware of them. Through the word, in human sociability, we develop our consciousness, the matrix of our thinking.
For Vygotsky (1983c), the 'possible' inability of the ID student in terms of abstraction is much more due to the lack of cultural interference as a consequence of primitivism in his/her life than to the primary derivation of the disability itself: "Его неспособность к высшим формам, абстрактного мышления не есть прямое следствие его интеллектуального […] Он лишь не овладел словом как орудием абстрактного мышления: эта неспособность есть следствие и симптом его примитивности, но не его слабоумия" ["His inadequacy for higher forms of abstract thought is not a direct result of an intellectual defect […] He simply has not mastered the use of words as tools for abstract thinking: this incapacity is a result and a symptom of his primitivism, but not of his mental Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology retardation"] (VYGOTSKY, 1983c, p. 28). Based on the Vygotskyan studies, Breitenbach (2018) points out that [f]or the process of cultural development, it is necessary for the child [with ID] to be able to use the language, the psychological-cultural tools, and the natural forms of the psychological functions to develop the higher psychological functions. In this way, it is not only the development of functions that will differentiate a more developed child from a less developed one but also his/her cultural equipment and the way he/she uses it.
It is worth noting that all the students, including those with ID, were always encouraged to speak. Although the student's speech about the content studied is a hint that the processes of mental development are going well, the dialogue between teacher and students is also a fundamental element. Wertsch (2007) draws attention to the fact that mediation of Vygotsky can be understood, fundamentally, as explicit and implicit mediation. Explicit mediation can be exemplified as one that shows the intervention as a whole (didactic decisions, social organization of teaching, definition by virtual play, classroom location), and one which is of the fundamental elements as a process of double stimulation; but implicit mediation, more difficult to comprehend, involves semiotic relations in that context (which defines the importance of the teacher as being the person best prepared for the job).
The fifth consideration can be found in the text К Вопросу О Компенсаторных Процессах В Развитии Умственно Отсталого Ребенка [Compensatory Processes in the Development of the Retarded Child], Vygotsky (1983e) presents some principles of the compensatory development of ID that were fundamental for the planning of this research: the principle of the substitution of functions, common to all children, had priority importance: in other words, none of the psychological functions is performed in only one way, but each of them is done in different ways (see TOOMELA, 2016); the principle of collectivity as a factor of development of the higher psychological functions of the normal child and ID, to which Vygotsky draws attention, indicate that mental functions first appear as a form of collaboration or interaction and, secondly, as an internal process of conduct; the relationship between the higher psychological functions are formed differently from those which manifest in the normal child: Эта сфера психологического развития, изменение межфункциональных связей и отношений, изменение внутренней структуры психологической системы есть главнейшая сфера приложения высших компенсаторных процессов складывающейся личности. Моторные связи и межфункциональные отношения характеризуют не столько сами функции, сколько то, каким образом эти функции приведены к единству.
[This sphere of psychological development, the change in interfunctional connections and relationships, the change in the internal structure of the psychological system, is the most important area for the operation of compensatory processes in forming personality. The functions themselves do not determine the motor connections and interfunctional relationships as much as does the way in which these functions are brought into unity]. (VYGOTSKY, 1983e, p. 126-7).
Finally, the author highlights affection, which encourages the child to overcome obstacles. The affection that the teacher gives to the student "они приводят к обходному пути развития" ["that will lead to a roundabout path of development"] (VYGOTSKY, 1983e, p. 127).

FINAL CONSIDERATIONS
Finally, what can we say about the question do students with ID learn fractions collectively with students without any disability? The analysis of the data collected led to the conclusion that one of the fundamental elements observed for the learning of fractions by ID students refers to the work of the teacher. The teacher's mediation work, as an intentional, conscious pedagogical action based on the theory underlying his/her actions -Vygotsky's defectology -, is the aspect that creates the best conditions for the students' learning in a group. Based on the presented analyzes, the grouping that involved students with and without difficulty in the regular school classroom, without the correct intervention of the teacher, was insufficient for children to learn fractions.
This result is very important because the proposal of inclusive education in Brazil suffers from propositions, such as those advocated by Mantoan (2017, p. 45), who apparently believe that the teacher only has to bear the presence of the disabled ones in class, as she cites: In this educational perspective, there is no place for teacher mediation as this is an affirmation of the intelligence hierarchy that sustains the control, Students with intellectual disabilities learning fractions: intentional pedagogical actions based on concepts of Vygotsky's defectology the conduction, and the superiority of the mediator, having nothing to do with the process of creative construction of knowledge and of disability as an attribute fixed in the person, which wholly defines him/her. (MANTOAN, 2017, p. 45, emphasis added).
Brazilian teachers or psychologists who take ideas such as this to support inclusive education can only reduce this proposal (due to its empty theoretical foundation) and make it widely questioned or denied by parents (who believe in a special school exclusion as the only way of schooling for the child with ID) and to weaken the action of Pedagogy in collaboration with Special Education. This ends up being inpractical (a fundamental basis for the path of a Marxist Psychology and Pedagogy, defended by Vygotsky (1982b) in "Исторический Смысл Психологического Кризиса ["The Historical Meaning of the Crisis in Psychology"]. By questioning Mantoan (2017), Breitenbach (2018, p. 143) argues that "[t]o deny the need for teacher mediation, we limit the possibility of student learning, especially those with some intellectual disability, forgetting one of the Vygotskyan principles [...], which foresee the need for special means of education for people with disabilities".
Nonsensical statements such as Mantoan (2017) not only demonstrate incredible ingenuity but also do not understand the fundamental task of the teacher, as a link to reach the zone of proximal development of the student: Учитель, работая с учеником над темой, объяснял, сообщал знания, спрашивал, исправлял, заставлял самого ученика объяснять. Вся эта работа над понятиями, весь процесс их образования прорабатывался ребенком в сотрудничестве со взрослым, в процессе обучения. И когда ребенок сейчас решает задачу, чего она требует от него? Умения по подражанию, с помощью учителя решить эту задачу, несмотря на то что в момент решения мы не имеем актуальной ситуации и сотрудничества. лежит в прошлом. Ребенок должен на этот раз самостоятельно воспользоваться результатами прежнего сотрудничества. [The teacher, working with the school child on a given question, explained, informed, inquired, corrected, and forced the child himself to explain. All this work on concepts, the entire process of their formation, was worked out by the child in collaboration with the adult in instruction. Now, when the child solves a problem, what does it requires of him/her? It requires the ability to imitate and solve the problem with the help of teacher even though we do not have an actual situation of collaboration at this moment. The situation lies in the past. Here, the child must make independent use of the results of that earlier collaboration]. (VYGOTSKY, 1982a, p. 257).
The research has highlighted the relevance of the teachers' work, wich is understood as essential for strengthening the working conditions so that the teachers can plan their educational activities, basing them on scientific knowledge. This is therefore included in the political agenda, which, though not the focus of this work, can be inserted in discussions in the field of Educational Psychology.
According to the observations and the interventions made, it seems that the sources of development of the psychological processes are always social. Thus, it can be inferred that the experiences provided through the activities enabled student-student/teacher-researcher-student interactions, which demonstrated significant elements for the learning process. However, it is necessary to further develop this type of study to be able to verify the acquired extent of knowledge as well as its effective generalizations in different places and times.

Funding
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brazil (CAPES) -Finance Code 001.
National Council for Scientific and Technological Development -CNPq.


School Performance and Leadership Styles Nexus: Exploring the Mediation Effect of Teacher Job Satisfaction in the Public Senior High Schools in Ghana

The increasing complexities of school settings and global educational reforms, school leadership has been recognized to play a key role in school performance worldwide. Nonetheless, the mechanisms through school leaders contribute to school performance has not been fully established till date especially due to difference in context of study, regional and cultural convolutions. To address this research lacuna, the study investigated the extent to which leadership styles directly and indirectly contribute teacher job satisfaction and school performance among selected Public Senior High Schools in the Kumasi Metropolis in Ghana. A sample size of 500 respondents were randomly selected and the Structural Equation Modelling (SEM) approach with Smart PLS software version 3.0 were employed for the analysis the data from the study. The study revealed that instructional, transactional, and the transformational leadership styles are the predominant leadership styles employed by the heads of the Public Senior High Schools in the study area. The instructional, transactional, and the transformational leadership styles have statistically significant and positive direct impacts on teachers’ job satisfaction and school performance. Furthermore, teachers’ job satisfaction only played statistically significant partial mediation effects in the relationship between transactional and transformation leadership styles and school performance. The study recommend that leadership in the senior high schools should put premium emphasis on the kind or leadership styles employed and teacher job satisfaction in their effort to improve school performance.

Introduction
Improving school performance fundamentally take into account the school teachers, leaders, societies and resources in general. However, all these factors are largely influenced by leadership style and practice  and teachers' job satisfaction (Hosseinkhanzadeh, Hosseinkhanzadeh & Yeganeh, 2013;Michaelowa, 2002;Somech, Drach-Zahavy & Anat, 2000;Treputtharat & Tayiam, 2014). The extent to which schools improve depend so much on the leadership styles employed by the head or leader of the school. Several studies have been conducted on the impact of educational leadership on academic performance at different levels of academic institutions. Among these studies include Lezotte and McKee (2006), Kouzes & Posner (2012) and . Leadership styles and practice have been found to directly influence performance either positively or negatively. According to (Afful-Broni, 2005), school success also depends on the commitment level of all stakeholders of the school, which invariably affects the atmosphere for teachers and other staff to commit to their respective roles and performances.
Leadership is a concept that has been widely studied and researched across a variety of domains including both the business and educational worlds. Cezmi Savas and Toprak (2014) stated, "Leadership is known as an effort that directs organizational activities to achieve a common goal". With the ever-changing educational landscape, principals must incorporate a wide range of leadership skills and styles in order to direct their school organization towards common goals and a well-directed vision. The principal is the most influential person in a school. "The difference between more effective principals and their less effective colleagues is not what they know, it is what they do" (Whitaker, 2003).There have been numerous education reforms intended to raise the achievement standards of students for the past 30 years.
As a result of the complexity and constantly evolving school setting these reforms have created, school leadership becomes of great interest in international education as it is increasingly recognized as having a key role in improving student outcomes (Day, Gu & Sammons, 2016;Mulford, 2008). Transformational leadership, transactional leadership and instructional leadership are the most regularly cited theories in education related literature (Robinson, 2008).
Transformational leadership focuses on establishing school culture and vision to enhance the quality of school teaching and learning, develop people, and improve the organization (Shatzer. 2014). While transformational leadership can strongly influence teachers, numerous studies (Leithwood & Jantzi, 2006;Ross & Gray, 2006) have concluded that these positive impacts have a much weaker effect on student achievement.
This model infers that a principals' efforts should be concentrated on the promotion of better outcomes for students, and the importance of improving the quality of classroom teaching and learning.Research on instructional leadership (Robinson, 2008, Shatze, 2014 has concluded that instructional leadership can influence student achievement, primarily through improvements to teacher's work conditions and school culture. These studies have concluded that it can have a more noticeable effect on student achievement than transformational and transactional leaderships, primarily because it places more of an emphasis on the quality of teachers and their teaching. Transactional leaders identify primarily tasks of the followers, establish the structure, emphasis on planned and scheduled work. Followers are rewarded or punished to achieve organizational goals (Hoy & Miskel, 2010). Ezeh (2008) observed that transactional leadership is more effective when organization desire to achieve their aims and objectives. Aydin, Sarier, and Uysal, (2013) show that transactional leadership affects job satisfaction at a broad level and in a positive way.
Extant literature has reported that leadership style and teachers' job satisfaction are closely related to school effectiveness including school performance (Ali, Sharma & Zaman, 2017;Somech, Drach Zahavy & Anat, 2000). It has also been shown that leadership style and school culture influences teachers' job satisfaction. Thus, leadership style, teachers' job satisfaction, and school performance can be assumed to be interrelated, though there is little known about the precise relationship among them especially in the context of Ghanaian education settings.
The extent to which schools improve depend so much on the leadership styles employed by the head or leader of the school. Several studies have been conducted on the impact of educational leadership on academic performance at different levels of academic institutions. Among these studies include Lezotte and McKee (2006), Kouzes & Posner (2012) and . Leadership styles and practice have been found to directly influence performance either positively or negatively. According to (Afful-Broni, 2005), school success also depends on the commitment level of all stakeholders of the school, which invariably affects the atmosphere for teachers and other staff to commit to their respective roles and performances. Educational performance is the ultimate goal of educational policy makers and governments which serve as a key indicator to move economies. Some previous studies in Ghana such as Mensah, Okyere and Kuranchie (2013) and Boampong, Obeng-Denteh, Issaka and Anamuah-Mensah, (2016) have shown a falling standards of performance among the senior high schools especially in the case of mathematics yet among these studies, leadership styles of the heads of these institutions were found to be good. Therefore, several stakeholders have questioned the cause(s) of the falling standards of performance of the educational institutions especially the senior high schools in Ghana.
Based on this, this study ventured an argument in the following premises: Firstly, the study argued that even though leadership styles and practices of educational leaders and administrators impact on performance, the impact is mediated by job satisfaction of their subordinates. Secondly, the effect of leadership style employed by the educational leaders on school performance is also mediated by the teacher job satisfaction.
However, studies focused in this direction of these arguments is less explored particularly in Ghana among senior high school educational setting. Investigating the mediation role of school culture and job satisfaction in this study employed a novel statistical computation approach such as the structural equation Modelling (SEM) to conduct path analyses which is among the pioneering studies in Ghana to utilize this novel computational approach to study this kind of interactions. This study therefore aimed to narrow the existing gap and shortage of literature pertaining to heads leadership styles in relation to school culture, teachers' job satisfaction and school academic performance in Ghanaian Senior High School settings. Secondly, the findings can be used to help practicing administrators determine possible areas of focus for improving teacher effectiveness and student achievement.
These findings may be particularly relevant for practicing educational leaders in schools identified as low performing schools in need of improvement. Additionally, the findings of this study may be used by state educational leaders and policy making officials to assess current systems of evaluation for principals and other school level executives.

Literature Review and Hypothesis Development
Several studies demonstrated the relationship styles between school performance from different contexts, organizations, and methodology. In the educational settings, leadership styles such as the transactional, instructional and transformation are predominantly uses and have proved to be effective and support performance. In his view explains transactional leadership style as a form of leadership in which relationships with teachers are based upon an exchange for some valued resource. Through the transforming process, the motives of the leader and follower merge. Transformational leadership contributes positively to school performance. Educational leaders need not lead their institutions based on give-and-take relationships, but rather on the leader's personality or character traits, ability to cause change by being a leader's role model, as well as always committing to a shared vision and goals. It is incumbent upon the heads to emulate the kind of leadership style that will help them remain efficient in fulfilling their leadership obligations.
Nonetheless, Wangihti (2014 ) also believes that with regards to the leadership styles used by school principals the autocratic leadership style of school head was found to have a negatively influence on school teacher's levels of job performance. Again, the author articulate that Headmasters' laissez faire leadership style Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol.11, No.15, 2020 moderately influences school teacher's levels of job performance whilst transformational leadership style positively influences school teacher's levels of job performance as compared to transactional leadership style. It suggest that, headmasters need to mix more than one method of leadership style so that a positive atmosphere can be achieved. Heaven and Bourne (2016) investigated on instructional leadership and its effect on students' academic performance in secondary educational institutions in Kingston and St. Andrew, Jamaica. The findings indicated that a positively weak statistical correlation existed between the performance of students and instructional leadership. Boampong, Obeng-Denteh, Issaka and Anamuah-Mensah (2016) studied the effect of leadership styles of head teachers on the academic performance of students in Ghana. The findings from these authors indicated that the leadership styles of head teachers have some influence on academic performance of students. Based on this background, the following hypothesis are developed: H1a: Instructional leadership has significant positive impact on school performance H1b: transactional leadership has significant positive impact on school performance H1c: transformational leadership has significant positive impact on school performance Studies such as Krishnan, (2005) and Huang & Chou, (2005) have demonstrated the significant relationship between leadership styles and employees' satisfaction in business organization. In this sense it is established that, the kind of leadership style exhibited in an organization determines the level of employee job satisfaction and their commitment towards the duty. As some leadership styles improve employee job satisfactions, others are seen to reduce job satisfaction of employees (Fernandes & Awamleh, 2004;Rad & Yarmohammadian, 2008;Lock & Crawford, 2004;Griffith, 2004).
According to Rad and Yarmohammadian's (2008) leadership styles such as democratic, transformational and transaction contribute positively to employees' job satisfaction and suggested significant correlation between the employees' job satisfaction and the leadership styles of managers. Nonetheless, the study by Lok and Crawford (2004) found no significant differences with the impact of leadership styles on job satisfaction and commitment. According to Hungra, Chuni, Aslam, Azam and Rehman, (2005), there is a positive relationship between autonomy, leadership behavior, team work environment and job satisfaction.
Transformational leadership provides some indications on effective leadership practices that may contribute to teacher job satisfaction and work commitment. For example, one study by Griffith, (2004) suggests that transformational leadership, with its insights on instilling values and morals in an organization, had an indirect effect on job satisfaction and teachers' commitment to stay in the profession. The principal's leadership practices may influence job satisfaction and work commitment as a result of personal interactions to build the teacher's selfesteem, which is closely related to social identity, providing a sense of belonging to a group.
Specifically, it appears principals' effect on teachers' job satisfaction and work commitment may be mediated through collaborative relationships by providing teachers with professional recognition and appreciation of their work. This is in agreement with Mees (2008) study of leadership behavior in public school leaders in relation to teacher job satisfaction in Cyprus. Evidently, perceived individual considerate behaviors of school principals satisfies an employee's needs and thus leading to the satisfaction of teachers on their job. Hence in this study, it is hypothesized that: H2a: Instructional leadership styles has significant and positive impact on teacher job satisfaction. H2b: Transactional leadership styles has significant and positive impact on teacher job satisfaction. H2c: Transformational leadership styles has significant and positive impact on teacher job satisfaction.
Job satisfaction plays a key role in the success of an organization's business, whereby the relationship between job satisfaction and performance has been shown to be significant (Usop, Kadtong & Usop, 2013). And high employee job satisfaction is important in order for an organization to function and achieve its goals. In a similar manner, in an educational context the job satisfaction experienced by teachers is likewise crucial for schools' successful operation. Teacher Job satisfaction has been found to have a positive and significant effect on teaching efficacy (Collie, Shapka & Perry, 2012), student learning (Michaelowa, 2002), student academic performance and school effectiveness (Somech et al., 2000).
For instance, Horn-Turpin (2009) investigated the job satisfaction levels of 121 teachers from special education and showed that this was positively correlated with their perceived teaching efficacy. Here, how satisfied the teachers were with their performance in the school, the more likely their practice was successful, and, therefore, the more likely the academic progress and achievement of their students met the educational targets (Treputtharat &Tayiam, 2014). Liang, Chan, Lin and Huang (2011) take job satisfaction as mediator between transactional and transformational leadership style and articulate that these styles create strong correlation between employee's performances. In all, it has been shown that teacher job satisfaction appears to have a mediating role between school leadership and school effectiveness. Based on this, it is hypothesized that: H3a: Teacher job satisfaction has positive and significant impact on school performance H3b: Teacher job satisfaction mediate the relationship between Instructional leadership and school performance Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol.11, No.15, 2020 H3c: Teacher job satisfaction mediate the relationship between transactional leadership and school performance H3d: Teacher job satisfaction mediate the relationship between transformational leadership and school performance Based on the above literature review, the conceptual framework below was developed.

Methodology
The paper investigated the extent to which leadership practices directly and indirectly influence job satisfaction and school performance in Public Senior High Schools in the Kumasi Metropolis in the Ashanti region of Ghana. Based on the objectives of the study, teacher job satisfaction variable served as the mechanism and the mediating variable through which the leadership styles influence school performance. In this study, the quantitative descriptive survey research design was employed to elicit adequate information to answer the research questions of the study. The study was conducted in Fifteen (15) different selected public Senior High schools in the Kumasi Metropolis in Ashanti Region in Ghana.
The Kumasi Metropolis capital city of the Ashanti Region in Ghana. The region is centrally located in the middle belt of Ghana and lies between longitudes 0.15W and 2.25W, and latitudes 5.50N and 7.46N. The region shares boundaries with four of the ten political regions including Brong-Ahafo in the north, Eastern region in the east, Central region in the south and Western region in the South west (GSS, 2012).
The study used the total sample size for the study was 500 respondent's Head masters, assistant Headmasters and teachers. The multi stage sampling method used were the stratified sampling, simple random sampling and the purposive sampling techniques. First, the researcher used proportionate stratified sampling technique. With this, the senior high schools in the Kumasi Metropolis were divided into four strata based on the type of school namely Category A, B, C and D. These categories are determined based on the development of infrastructures, resources and others factors which have the potential to affect school performance. Based on this, the researcher randomly selected five schools from Category A, four schools from Category B, three (3) schools from Category C while three schools were selected from Category D. Schools in the same category have similar features and characteristics. Moreover, since the respondents were Teachers in these senior High schools, the simple random sampling method was again used to select 25% of teachers from each of the schools selected.
In the final stage, the purposive sampling technique was then used to select the headmasters and their immediate assistant headmasters to be part of the respondents for the study. This constituted 15 headmasters and 15 assistant head masters. In all the total sample size of 500 respondents were selected and administered with questionnaires for the data collection.
The study used the structured questionnaires consisting of Likert scale type questions for the data collection from respondents. The Five-Point Likert Scale type responses were used for the study. The questionnaires were developed based on the conceptual framework of the studies, literature review, and the theoretical framework. This Likert scale type Reponses ranged from 1-Strongly Agree to 5-strongly Agree. The measuring instrument items were from the following sources. School Performance, Transactional leadership practices, Instructional leadership practices and Transformational leadership practices and teacher satisfaction (Hoy & Miskel, 1996;De Nobile, 2007). The study employed the path analysis where latent variables were used for the study. Since the variables used for the study cannot be measured directly, a latent variables were used for the path analysis particularly to test the relationship between the variables using the observed variables. The study will utilize the partial least squares Structural Equation Modeling (PLS-SEM) for the analysis of the data. This method was used to examine various pathways and mechanisms through which the leadership practices influence school performance and teacher job satisfaction. . The use of structural equation modeling, the Smart PLS has been proven to be effective software for such analysis involving latent variables and mediation effect (Hair, Ringle & Sarstedt, 2013).

Results and Discussion
The study examined the extent to which Instructional leadership, transformational leadership, transactional leadership, influence school performance in the case of selected senior high schools in the Ashanti region in Ghana. In this study, Teacher job satisfaction served as mechanisms through which the leadership styles among the selected senior high schools influence the school performance. In this sense the structural equation model pathway analyses using Smart-PLS software was used in the study. The structural equation model approach help to simultaneously run multiple regression by exploring the direct and indirect effect of leadership styles through a mediation analysis. This section comprises the discussion of demographic characteristics of respondents, the evaluation of measurement model and evaluation of the structural model.

Demographic Characteristics
The respondents of the study constituted teachers, head of departments, headmasters, assistant headmasters in selected Public senior high school in the Kumasi Metropolis in Ghana. The results from the study indicated that, out of the five hundred (500) respondents who participated in the study, 386(77.2%) were males while 114 (22.8) were females. Respondents who took part in the study were predominantly males and this confirms the fact that, in the Kumasi Metropolis teachers in the Public senior high schools are predominantly males where their counterpart female teachers constitutes the minority.
In terms of age of respondents, the findings reveal that, participants below 30 years constitute 14.8%, 30-40 years were 36.2%, while 39.85 comprised those within 41-50 years. The results show that. Only few of the respondents representing 9.2 % were within 51-60 years. This implies that, majority of the respondents were within 30-50 years. In the case of their highest educational attainment, the results show that, 412 of the respondents representing 82.4% had their first degree while 88 representing 17.6% had their post graduate degrees in different disciplines. This results confirm that the minimum qualification of the teachers in the public senior high schools in Ghana is the first degree.
Leadership styles practiced by headmasters of the selected senior high schools. In terms of working experience of the respondents, the results demonstrate that, 20.8% had working experience between 2-5years, whereas 32.4% had 6-10 years of working experience. Similarly, the results show that, 24.85 had 11-15 years working experience while 22% had more than 15years work experience. This shows that, most of the respondents had more than 5 years' work experience.

Measurement Model
The study examined the latent variables including Instructional leadership, transformational leadership, transactional leadership, school performance, teacher job satisfaction, and school culture. The purpose of the study was to investigate the extent to which leadership styles include school performance. Teacher job satisfaction served as mechanisms through which the leadership styles among the selected senior high schools influence the school performance. For a successful utilization of the PLS-SEM, the Confirmatory factor analyses were conducted. These latent variables were measured with observed construct items as shown in Table 1.
The confirmatory factor analyses were computed for the factor loadings of each observed construct indicator which combined to measure each of the latent variables. Results from Table 1 below presents the factors loading of the various observed measurement construct items, standard deviations, the variance inflation factors and the significant values measured as p-values. The results indicate that, all construct items had factors loadings greater than 0.7 which showed a good factor loadings form the confirmatory analyses. However, prior to these results, all the construct items with factor loadings less than 0.5 were dropped from the analyses since their loadings were not within the acceptable thresholds. The results present a minimal standard deviations which implies that, the responses were not widely dispersed and were closed to the average response to the selected construct items.
The P-values represent the significant level of the factor loadings for the observed construct indicators. The findings from Table 1   The measurement model below demonstrate the individual observed constructs indicators measuring their latent variables. The estimated measurement model presented in figure 1 below presents the path ways through which the latent variables such as the leadership styles influence the school performance. The latent variables including school culture and teacher satisfaction serves as the mediating variables and the mechanisms through which the leadership styles play a role in school performance. The measurement model in figure 3 below also shows the factor loading, the R-square of the pathways model and the estimated coefficient of the direct pathways regression of the structural relationships among the variables.   Table 2. These measurements demonstrated the latent variable factors and indicators that were significantly consistent and reliable in measuring the structural model. The explanatory power of the model measured by R-square was also computed. According to (Hayduk and Littvay 2012) if the AVE and CR have loading less than 0.5 and 0.7 correspondingly, the results will imply weak convergence validity. As a results, are not desired to be maintained for the analysis.
The findings from the study show that the Cronbach's Alpha for all the latent variables used are above 0.7 which implies that, the latent variables are more reliable and consistent and appropriate for the structural equation modeling. Similarly, the Rho_A, CR, and AVE for all the indicators are also above 0.7 which shows that, the items indicators were more reliable and valid for the analysis. The R-squared value which represent the explanatory power of the model revealed that, when school performance was used as the dependent variable, the R-square was 0.428. This implies that, the model explains about 42.8% of the changes in the school performance are explained by the model. Simply when teacher satisfaction was used as dependent variable, the R-square was 0.670 which implies that about 67% of the changes in the teacher job satisfaction was explained by the model. The results are presented in the table 2 below.

Discriminant validity test
The results from the discriminant validity among these measured variables have been presented in Table 3. The discriminant validity test was conducted using the Fornell-Larcker criterion to examine the extent to which the constructs differ from one another as shown in Table 3 below. The results from the Table 3 below presents the latent variables, numerical values in a major and minor diagonals. The values in the major diagonal represents the square root of AVE of the latent variable constructs while the values in off-diagonals are the cross-correlations between the constructs. The discriminant validity was then measured by comparing the differences between the overlapping constructs and the square root of the AVE. The findings below from demonstrate that, the difference between the cross correlations and the square root of the AVE are minimal and acceptable to justify for further analyses of the variables and their structural relationships. This consequently show that, the variables are valid Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol.11, No.15, 2020 99 and consistent to produce efficient outcome for the study. The results are presented in Table 3 below.

Pathway Analyses of Direct impacts of Leadership Styles and Teacher Job Satisfaction on School Performance.
To address the research objectives, the study examined the direct effect of leadership styles namely the instructional, transformational and transactional on school performance and teacher job satisfaction. Again, the direct effect of teacher job satisfaction on school performance was empirically examined. Results from Table 4 presents the presents the coefficient of direct pathways of the structural relationship between the leadership styles, teacher job satisfaction and school performance. The results reveal that, instructional leadership (β=0.329, P<0.01), transactional leadership (β=0.164, P<0.01), and transformational leadership (β=0.223, P<0.01) have statistically significant and positive impact on school performance. The findings imply that, headmasters of the senior high schools who employ predominately any of these leadership practice have the likelihood of improving school performance. Nonetheless, among these leadership styles, the results demonstrate that, instructional leadership styles tend to have greatest direct impact on school performance compared to other leadership styles. Furthermore, the results show that, (β=0.112, P<0.05), transactional leadership (β=0.197, P<0.01), and transformational leadership (β=0.661, P<0.01) have statistically significant and positive impact on teacher job satisfaction. The implication is that, the more school heads adopt these leadership styles the more likelihood that they would improve teacher job satisfaction in the senior high schools. More importantly, the results suggest that, transformational leadership (β=0.661, P<0.01) tend to contribute greater improvement to teacher job satisfaction than other leadership styles considered in the study. These results are presented in the structural model in figure 3 below. The structural relationships presents the weights or the statistical significance of the relationships.

Figure 3: Structural Equation Model
Similarly, the results show that teacher job satisfaction (β=0.163, P<0.10) also have statistically significant positive and directly impact on school performance. Increasing teacher job satisfaction in the school contributes to increasing school performance. School type resented by category of school being category A, B, C and D was used in the model as a control variable. The results show that, type of school (β=0.06, P<0.05) has significant and positive impact of school performance. The results from the analyses are presented in table 4 below.

Indirect Effect of Leadership Styles on School Performance
The study further investigated the indirect effect of leadership styles practiced by the headmasters of the various senior high schools. In this direction, teacher job satisfaction serves as the mechanism through which leadership styles practiced in the schools affect school performance. The results from the mediation analysis presented in the table 4 below show that teachers job satisfaction significantly served as a partial mediation between transactional leadership (β=0.032, P<0.10) and school performance. This implies that, teacher job satisfaction serves as a partial medium through which transactional leadership practices affect school performance. This simply means that, transactional leadership can have both direct and indirect effect on school performance through teacher job satisfaction. In this case teacher job satisfaction becomes the partial pathways or channel through which transaction leadership style may affect school performance.
Likewise the results reveal that, teacher job satisfaction also have significant partial mediation effect in the relationship between transformational leadership style (β=0.107, P<0.10) and school performance. The results suggest that, even though transformational leadership can directly influence school performance, there is a more likelihood that, the channel of the effect can pass through teacher job satisfaction. However, the results reveal that, teacher job satisfaction does not have a significant mediation effect in the relationship between instructional leadership style and school performance. The results from the mediation analysis are presented in the Table 5 below. Note: → denotes the direction of effect from the pathways analysis Note: ***, **.* denote 1%, 5% and 10% levels of significant significantly

Discussion
The study tested the various hypotheses of the structural relationships between the leadership styles, teacher job satisfaction and school performance. In all four key hypotheses with sub-constructs were tested. The results are presented in table 4 which demonstrated the hypotheses that were supported from the structural equation model analyses conducted. The results show that, the hypotheses H1a, H1b and H1c were supported by the findings from the study. The hypothesis (H1a) which states that, instructional leadership style has positive and significant impact on school performance was supported. The findings from this study are agree with findings from (Robinson, 2008, Shatze, 2014 who conclude that instructional leadership can influence student achievement, primarily through improvements to teacher's work conditions and school culture. This as a results contributes significantly to improving school performance. Likewise Hypothesis (H1b), which states that, transactional leadership style has positive and significant influence on school performance was supported.
In the same way, the findings show that, the hypothesis (H3a) which states that, transformational leadership style has positive and significant impact on school performance was supported.
Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol.11, No.15, 2020 In the views of Day (2016) the concentration of leadership should be on promotion of better outcome, quality teaching and learning environment. Transformational and transactional leaderships, primarily focus on student achievement and quality teaching. The findings from this study justifies that, these leadership styles have positive effect on school performance including improvement in teaching and learning, improving infrastructure, maintaining disciplines among students and staff. All these translate to improve the overall school performance Furthermore, the hypotheses H2a, H2b and H2c were also supported by the results from the study. Similarly, the hypothesis (H2a) which posit that, Instructional leadership style has significant positive impact on teacher job satisfaction was also supported by the results. Hypothesis (H2b), which posits that, transactional leadership style has significant positive impact on teacher job satisfaction was also supported by the results.
Hypothesis (H2c), which posits that, transformational leadership style has significant positive impact on teacher job satisfaction was also supported by the results. The results from the study reveal that, good school leadership style has strong relationship with teacher job satisfaction. The findings show that, improving leadership styles such as instructional, transactional and transformational leadership styles contribute significantly to improving teachers' commitment, hardworking abilities and their entire job satisfaction in the school. The results of the hypothesis testing are presented in table 6 below. Table 6: Results from hypotheses Tests Note: → denotes the direction of effect from the pathways analysis Note: ***,** and * denote 1%, 5% and 10% levels of significant respectively. Moreover, the hypotheses H3a, H3b, H3c and H3d were also supported by the results from the study. Consequently, the hypothesis (H3a) which posits that teacher job satisfaction has significant and positive impact on school performance was supported. Again, the hypothesis (H3b) which opine that teacher job satisfaction has a significant and positive mediation effect in the relationship between instructional leadership style and school performance was not supported.
In the same way, hypothesis (H3c) which says that teacher job satisfaction has a significant and positive mediation effect in the relationship between transactional leadership style and school performance was supported. In the same way, hypothesis (H3b) which says that teacher job satisfaction has a significant and positive effect in the relationship between transformational leadership style and school performance.
Results from the mediation also demonstrate that, as teacher's satisfaction is increased, the resulting effect is improvement in school performance. The findings from the study and hypothesis testing conclude that teachers job satisfaction serves as a partial mediation through which leadership styles such as instructional, transactional and transformation contribute to school performance. This indicate that, the leadership styles practiced by the headmasters in the various senior high schools have both direct and indirect effect on school performance. To achieve the significant indirect effect of leadership on school performance, focus should be directed to improving teachers' job satisfaction in the school. The findings are also in line with Ali, Sharma and Zaman (2017) who found that leadership style and teachers' job satisfaction are closely related to school effectiveness including school performance. This implies that, for headmasters to achieve a meaningful; school performance teachers' job satisfaction should not be ignored. It must be part of the organization goals to be achieved especially by the school administrative board.
The findings also support the study by Afful-Broni, (2005), who argued that school success also depends on the commitment level of all stakeholders such as teachers of the school who contribute directly to the achievement and outcomes of the school activities particularly through teaching and learning activities. As concluded by Collie, Results from the study suggest that the kind of leadership styles employed in the senior high schools can significantly influence the quality of teaching, learning, student achievement and improvement in the overall school performance. The findings consequently demonstrate that student leadership styles utilized by the heads of the senior high schools can improve the working conditions of their teachers, teachers' job satisfaction. These findings are in line Shatzer, Caldarella, Hallam & Brown, 2014) who argued that, good leadership has a noticeable impact on environment of their school which may result in their positive job satisfaction.
For example, Transformation school principals identify and share school vision, led and inspire others by example, create a culture of learning, and encourage staff members to undertake professional development. In the case of Instructional leadership, the emphasis is usually on the academic progress and achievement in the school. This indicates that, heads who employ this leadership styles to some extent achieve desired value of creating, clear educational goals, planning, curriculum objectives, and quality of teaching and learning performance.

Conclusions and Recommendations
The increasing complexities of school settings and global educational reforms, school leadership has been recognized to play a key role in school performance worldwide. Nonetheless, the mechanisms through school leaders contribute to school performance has not been fully established till date especially due to difference in context of study, regional and cultural convolutions. To address this research lacuna, the study investigated the extent to which leadership styles directly and indirectly contribute teacher job satisfaction and school performance among selected Public Senior High Schools in the Kumasi Metropolis in Ghana. Based conceptual framework of the study, teacher's job satisfaction served as the mechanism and a mediating variable through which leadership styles indirectly contribute to school performance. The study used a sample size of 500 respondents who constituted teachers, headmasters, and head of departments from 15 public senior high schools randomly selected from the Kumasi metropolis in Ghana. The structural equation model (SEM) approach was used to simultaneously examine the structural pathways through which school leadership styles and teachers job satisfaction influence school performance. The smart PLS software version 3.0 was employed for the analysis the data from the study.
Based on the analysis, the findings from the study revealed that instructional, transactional, and the transformational leadership styles are the predominant leadership styles employed by the heads of the public Senior High Schools in the study area. Results from the study show that, instructional, transactional, and the transformational leadership styles are have statistically significant and positive direct impact on teachers' job satisfaction. Nonetheless, the findings reveal that transformational leadership styles has the greatest influence on teacher job satisfaction among the senior high schools as compared to the other leadership styles.
Furthermore, the findings suggest that, instructional, transactional, and the transformational leadership styles had direct significant and positive impacts on school performance. Improvement in the use of these leadership styles contribute directly to significant increase in school performance. In terms of the extent of the impacts, the results demonstrate that, instructional leadership styles had the greatest direct significant positive impact on school performance while transactional had the least significant positive influence on school performance.
Moreover, the results show that, teachers' job satisfaction did not serve as full mediation in the relationship between the leadership styles and school performance. However, the findings subsequently reveal that, teacher job satisfaction played a partial mediation role in the structural relationships between leadership styles and school performance. This implies that, for heads of these public senior high schools to improve school performance, the job satisfaction of the teachers crucial and must not be ignored since the impact of the leadership practice could also pass through teacher job satisfaction. As the teachers become more satisfied, they are more committed, give out their best and contribute positively to school performance. In the other hand, if these teachers are not satisfied, their commitment levels may be low hence their contribution to school performance might be low.
In line with the above findings, the study recommend that, we recommend that, school administrators and heads especially in the senior high schools should employ more of transformational leadership styles. In exercising this kind of leadership, schools emphases on quality teaching and learning, developing human capital, and encourage staff to undertake professional development programs. Transformational leaders pay much attention to individual employee development, idealized influence, and intellectual stimulation. These are necessary to improve school performance which constitute academic achievement, improvement in school enrolment, discipline, reduction in staff absenteeism, school drop among others.
The findings from this study has a relevant contribution to practicing educational leaders in schools identified as low performing schools which need improvement in their performance. As the study identified the direct and indirect pathways through which leadership contributes to school performance, these findings are useful for school leaders to employ transformational leaders' styles, transaction in tandem with much emphasis on teacher job satisfaction in order to increase school performance. Additionally, the findings of this study may be used by state educational leaders and policy making officials to assess current systems of evaluation for principals and other school level executives. Findings from this study may provide additional validation for the inclusion of certain leadership practices, especially those related to teacher satisfaction, in the evaluation instrument of school level administrators.
Further, the study recommend that, leadership in the senior high schools should place much emphasis on improving teachers' job satisfaction. The findings revealed that, as teachers' job satisfaction is improved, they tend to show high level of commitment, dedication and unbridled zeal towards achieving the set goals in the school and high level of school performance. In conclusion, this study extend the empirical and conceptual evidence on the extent to which school leadership styles directly and indirectly contributes to school performance.
The study further suggest that, future studies may expand this study in terms of sample size and may employ other empirical analysis techniques such as hierarchical regression. In this direction, future research may consider other variable which may moderate the impact of leadership styles on school performance which constituted the limitation of this current study it considered only mediation variable. Examining the factors that moderate the relationship between leadership styles and school performance may strengthen the understanding of the current knowledge on the mechanism through which leadership styles influence school performance. Moderation variables such as resource allocation, government policies and school climate may be worthy of studying and included in the future model to study the nexus between school leadership and performance.