A ‘Tetrahedron Model of the Content of Teacher Education

 A ‘Tetrahedron Model of the Content of Teacher Education and Its Implications for Mathematics Teacher Preparation in Mongolia

Abstract

The aims of this paper are twofold. The first is to build a model of the content of teacher education through such four step procedure: (1) identifying the key attributes of teacher quality while reviewing relevant literature, (2) generating the perceived attributes of an ideal teacher through generalizing and idealizing the key attributes of teacher quality and its properties and (3) dimensionating the content of teacher education through the  perceived underlying attributes of an ideal teacher and  (4) configuring the dimensionated content of teacher education  to a tetrahedron, a three dimensional configuration which embodies  four same boundaries, each of which is identified as an equal lateral triangle. The second is to reconsider the programme of mathematics teacher preparation in the light of the tetrahedron model of the content of teacher education.  

Using a tetrahedron model of the content of teacher education, it can be proposed that there are at least 24 variants to design and restructure the content of teacher education and to organize teacher preparation. One of them presents the current practice of mathematics teacher preparation in Mongolia while the rest of them provide alternative possibilities to redesign and restructure the programmes of mathematics teacher preparation. Based on the findings from a tetrahedron model, some implications emanated from the tetrahedron model- based analysis were provided to improve mathematics teacher education and teacher preparation in Mongolia.

Key words: teacher quality, the attributes of teacher quality, an ideal teacher, an ideal teacher attributes, the content of teacher education, tetrahedron configuration, tetrahedron model of the content of teacher education

Introduction:

Mathematics teacher quality has a great deal of contributions towards improving the quality of mathematics education that in turn affects tremendously education quality. In fact, as for Mongolia, the quality of education is irrelevant.  As a matter of fact, Mongolia is placed at the top in terms of quantitative indicators of educational performance, namely enrolment rate, the ratio of boys and girls in schools and literacy rate whereas it is ranged at the end in terms of qualitative indicators such as employment rate, quality life, a human development index. Moreover, the development index of Mongolia in terms of Educational for All is 0.916 whereby listing at the 67th place out of 127 countries (EFA report 2007). On the other hand, a recent study indicates that there is no big difference between the poor peoples and the non-poor ones in terms of being satisfied with life quality. In fact, a number of people holding higher degree in education constitute more 10 percent of the whole poor (National Statistics Office 2002).  Those figure proved that the quality of education, is unsatisfactory and irrelevant to our society wherein we have practised the values of democracy and free economic market for the last two decades. Thus, we are challenged by improving education quality. By the same token, as a major component of general education, the quality of mathematics education will be judged.

In order to enhancing mathematics teaching quality, we need to examine how accurate teacher candidates are prepared. Stated differently, it is needed to ensure whether teacher candidates are prepared so that they bear such attributes that meet the characteristics of teacher quality. In respond to this question, this paper aims to build an abstract model of the content of teacher education through generalizing and idealizing the main attributes of teacher quality and dimensionating it and then to discuss the possibilities to apply it for enhancing the design and structure of the programs of teacher preparation. 

 In according with such purpose, this paper will be outlined as follows:

What attributes feature teacher quality?

             The key attributes of an ideal teacher generalized the nature of the attributes of  teacher quality

The dimensionality of the content of teacher education

A tetrahedron configuration and its compatibility with the content of teacher education

A tetrahedron model of teacher education and its contributions to increasing our awareness of teacher education and teacher preparation

The Tetrahedron Model of the Content of Teacher Education and its some Implications on Improving Mathematics Teacher Preparation in Mongolia

As a result, this paper will bring up a tetrahedron model of the content of teacher education along with its four major variants to use it in a practice whereby restructuring and redesigning the programs of teacher preparation. In addition, some interpretations and implication emanated from this model into mathematics teacher preparation in Mongolia were discussed.

What attributes feature teacher quality?

Teacher quality is an underlying concept that encompasses comprehensive and dynamic characters and attributes. The nature of comprehensiveness is manifested itself in diverse insights, attempts and understandings proposed by different scholars and agencies, each of which aims, to some extent, to recognize the constituencies of teacher quality. In fact, the multiple and dynamic constituencies of teacher quality tends to be categorized into two groups, namely, personal qualities and professional qualities. As the name suggests, the personal qualities of teacher quality tend to be personal or subjective and therefore, it is highly likely to belong to disposition of a teacher candidate. Yet, the professional ones are highly likely to cover the qualities possibly possessed by teacher candidates through teacher education services, namely teacher in-service and pre-service training. Accordingly, the latter one tends to cover teacher attributes or competencies of pedagogy, didactics and content knowledge. Taken together the various ideas and perspectives on teacher quality appeared in recent papers, it is likely to be generalized that teacher quality are characterized by teacher attributes  in pedagogy, didactics and content knowledge and his or her disposition as well.

According to Hargreaves (1988, cited in Meg 1991), teaching quality is the blend of personality attributes, sound pedagogy and a secure subject match. At the same time, Lamn (1972, 2000, cited in Arnons & Reichel 2007) proposes four aspects of work of good teacher such acculturation, socialization, individualization and disciplinary expertise. Here, personal and social aspects are emphasized in a broad sense.  What is more, a comparative study of policies in ten countries that aimed at improving teacher quality highlighted that teacher quality bears six key characteristics, namely, commitment, love of children, mastery of subject didactics and multiple models of teaching, ability to collaborate with teachers and a capacity of reflection (Hopkin & Stren 1996). The study highlighted the mastery of subject didactics along with other attributes.

 Moreover, with reference to the OECD report of Quality in Teaching (1994, cited in Fredriksson 2004), it is known that teacher quality has five dimensions such as knowledge of substantive areas and content, pedagogic skills, reflection, empathy and managerial competence.  In addition, Arnons & Reichel (2007) categorized the qualities of ideal teacher into two main categories: teacher personality and professional knowledge. Referring to the authors, teacher personality covers teacher general personal qualities such as having a sense of humor, being kind-hearted, calm, fair, optimistic, humane, stubborn and principled and empathetic and loving children whereas professional knowledge consists of subject knowledge, didactic knowledge about both educational methodology and individual pupil.  Hence, it is referentially ascertained that teacher attributes such as loving children, empathy, humor, being kind-hearted and commitment comprises particular constituencies of teacher quality which, are, apparently more personal rather than professional.

Taken together the ideas and arguments focusing on teacher quality and its constituencies, it can be seen that teacher quality embodies two major components, namely personal qualities and professional qualities. The professional qualities tend to cover teacher attributes in pedagogy, didactics and content knowledge whereas the personal qualities pertain to his or her disposition.

The key attributes of an ideal teacher generalized the nature of the attributes of teacher quality

A challenging question emanated from teaching in practice when its quality is judged is whether   teacher candidates are well prepared so that their attributes and competencies meet the features in teacher quality.  Responding to this question is, practically, controversial because the degree of such preparedness of teacher candidates depends upon candidates’ disposition albeit the sameness in programs of teacher education and its implementation. However, this question is ideally answerable as long as the following ideal conditions are provided: (1) all teacher candidates bear satisfactory dispositions that are compatible with possessing teaching profession; (2) the programs of teacher education are designed and implemented so that all teacher candidates possess the key attributes of teacher quality. In turn, such idealization of teacher education and preparation help us perceive the characteristics of a good teacher. 

Under such idealization of teacher education and preparation mentioned above, it can be seen that any teacher candidate after completing the programs of teacher education will become an ideal teacher. In other words, it is perceived that an ideal teacher is characterized by the same attributes that teacher quality is featured.  Hence, it is generalized that an ideal teacher ought to be featured by the four major attributes compatible with teaching profession such as pedagogical competence, didactical competence, content knowledge and teacher disposition.

In order to prepare teacher candidates as good as ideal, the content of teacher education ought to be designed as an amalgam of the key attributes of ideal teachers. In turn, it prompts us to dimensionate the content of teacher education so that the nature of teacher attributes is holistically represented.  This task will be considered in the next portion of this writing.

The dimensionality of the content of teacher education

In mathematics, the dimensionality of any space is dependent upon whether its properties satisfy with the criteria of dimensions. Generally speaking, the main criterion of space dimensionality can be broadly stated that a space will be dimensionated if only it contains a set of special elements in itself whereby any other element can be expressed for its particular operation. If such a particular set contains finite members, a space is labeled as a space with finite dimensions.  For instance, Euclidean plane is a two-dimensional space because there exits a set of two special vectors whereby any other vector is expressed by their linear combinations. What is more, as far as advantages to dimensionate a space are concerned, it is also known that any attributes of a dimensionated space can be perceived as a certain combination of dimensional elements. Returning to Euclidean plane, it can be said that any vector in this space is perceived as a linear combination of two dimension vectors. Keeping in mind a brief description about the dimensionality of a mathematical space and its advantages, we can examine the degree of dimensionality of the content of teacher education in an interpretative sense at least.

The constituencies of the content of teacher education are, practically, diverse, yet theoretically controversial. In fact, its diversity can be seen in different programs that function differently to prepare teachers in different contexts whereas its controversy can be also appeared in contestable views on defining the nature of teacher and teacher education. However, with regards to a challenging question posed as whether teacher candidates are well prepared so that their attributes and competencies meet the features of teacher quality, the content of teacher education might be perceived as a closed configuration that covers key attributes of an ideal teacher and other potential attributes generated by their combinations. In other words, the content of teacher education is regarded as a set of teacher attributes that are, in turn, composed of the key attributes of an ideal teacher and other potential attributes generated by their integrations and combinations. Indeed, this perception about the content of teacher education gives us to sensitize out that if the nature of the content of teacher education encompasses a set of the key attributes in the content of teacher education whereby it’s any attribute can be expressed to certain degree, it can be dimensionated. Accordingly, whether such a set exists is also an important question.  Such two questions will be dealt with below.

As perceived as a closed configuration of a set of teacher attributes, the content of teacher education contains the key attributes of an ideal teacher whereby its other potential attributes might be generated to some extent in terms of dependence or association or implication at least.  At the same, referring to the main criteria of dimensionality of an abstract space, it is proposed that the key attributes of an ideal teacher might be identified as dimensions of the perceived content of teacher education.  Thus, it is implicitly clarified that the content of teacher education has the four main dimensions such as pedagogical competence, didactical competence, content knowledge and teacher disposition.  Hence, it is importantly asked as how configure the content of teacher education through its four dimensions. The next portion will answer this question.

 A tetrahedron configuration and its compatibility with the content of teacher education

As discussed so far, the content of teacher education is perceived as a closed configuration which might be filled with a set of teacher attributes that are generated by its four dimensions such as pedagogical competence, didactical competence, content knowledge and teacher disposition.  In addition, there is no hierarchical rank among such four dimensions. That is to say, each of four dimensions must be equally considered in configuring the content of teacher education.  Taken into consideration aforementioned properties of the content of teacher education and its dimensions, it is conceived that a tetrahedron and its properties can present best the nature of content of teacher education. This conception will be justified as follows.

Now, let us see how the nature of the content of teacher education is configured to a tetrahedron.  If we suppose that one of the four boundaries of a tetrahedron represents one of the four dimensions of the content of teacher education, the tetrahedron itself in terms of its volume depicts the holistic nature of the content of teacher education. Besides, the edges of the tetrahedron represent the partial associations among four dimensions whereas its every vertex indicates a point at which any three of the four attributes intersect.  What is more, the spirit of the ideal teacher that is in turn determined by the dynamic of teacher competence acquisition might be resembled as a central point of a tetrahedron at which its four medians are intersected. 

The contributions of the tetrahedron model of the content of teacher education towards increasing our awareness of teacher education and teacher preparation will discussed in the next portion of this writing.

A tetrahedron model of teacher education and its contributions to increasing our awareness of teacher education and teacher preparation

According to a tetrahedron model of the content of teacher education, it can be seen that there are at least 24 variants or ways (mathematically, 4*3*2=24) to design and restructure the teacher education content and preparation that is in turn categorized into four major models, namely, a pedagogical competence-based tetrahedron, a didactical competence-based tetrahedron, a content knowledge based tetrahedron, and a teacher disposition based tetrahedron.

As far as the pedagogical competence-based tetrahedron model is concerned, it can be expounded that teacher candidates are initially subjected to be trained by the courses that are intended to enable teacher candidates to acquire pedagogical competence. Followed by the pedagogical courses, they will be delicately provided by other courses that are designated to provide them with the rest of teacher competences. The advantage of this kind of teacher preparation should be recognized that teacher candidates will be aware of pedagogical or managerial knowledge and skills priory to acquiring the rest of teacher competences whereas teacher candidates might be disadvantaged by not making known early how much they are challenged by acquiring the content knowledge and didactical competencies.

Apart from the pedagogical competence-based tetrahedron model, the didactical competence -based tetrahedron model will provide us with a different insight into designing the programme for teacher preparation. According to this model, teacher candidates are subjected to acquire firstly particular didactical knowledge and skills. After that, they will possess the rest of the three major competences. The advantage of this model might be recognized that teacher candidates will be acquired particular didactical approaches, theories and best practices so that they will apply them for own teaching and learning. At the same time, they will be disadvantaged by being subjected to tackle with didactical theories and practices priory to acquiring the rest of the teacher competences, in particular, appropriate content knowledge.

With regard to a content knowledge based tetrahedron model, it can be seen that teacher candidates are subjected to possess firstly the content knowledge concerning a particular teaching subject and then they will proceed with handling the rest of the teacher competencies. Seemingly, it will facilitate teacher candidates to handle teacher’s didactical competences whereas it is likely to have some drawbacks to deal with pedagogical competencies because of their attitudes embedded in one particular area.  In fact, this model seems to be a prevailing one in teacher preparation practices.

In according with a teacher disposition-based model, it can be noted that teacher candidates’ attributes will be observed and examined by the criteria of teacher disposition. Then, teacher candidates whose attributes meet teacher disposition criteria will delicately proceed with acquiring the rest of the teaching competencies. The advantages of this model is that the fitness of the attributes of teacher candidates to teaching profession is early examined whereas it might be biased against individuals who has strong interests to become teacher albeit that she or he does bear appropriate attributes of teaching profession.

To sum up, it can be noted that tetrahedron model of the content of teacher education will bring four major insights into designing and restructuring the programme for teacher preparation such as a pedagogical competence-based tetrahedron, a didactical competence-based tetrahedron, a content knowledge based tetrahedron, and a teacher disposition based tetrahedron. Each of them has disadvantages and disadvantages in teacher preparation in practice. However,   at the end of the teacher preparation processes governed by the tetrahedron model, teacher candidates will be ideally perceived to be a teacher who can bear four major teaching competencies that are in turn generalized by teacher quality attributes.

The Tetrahedron Model of the Content of Teacher Education and its some Implications on Improving Mathematics Teacher Preparation in Mongolia

As a result of analyzing the programme of teacher preparation in Mongolia in the light of the tetrahedron model, it can be exposed out that the content for teacher preparation and its designing tends be expounded by a model that is represented as a tetrahedron with the content knowledge base. Referring to the current programs for mathematics teacher preparation in Mongolia (School of Mathematics and Statistics 2000), it is known that teacher candidates are subjected to handle firstly mathematics courses in four terms that are quite separated from mathematics didactics. In fact, at an initial stage of the teacher preparation, there is no difference between student who is intended to be a mathematician and students who become mathematics teachers. In the last terms, teacher candidates are subjected to take courses dealing with didactical issues in mathematics teaching along with managerial matters related with maintaining mathematics classroom. Hence, reminding the characteristics of a content knowledge-based tetrahedron model, we can point out that the mathematics teacher preparation in Mongolia is likely to facilitate teacher candidates to handle the teacher competences of mathematics didactics whereas it is likely to neglect the equal importance of other competencies of mathematics teachers, namely, teaching competence in mathematics didactics, teaching competence in mathematics classroom management or pedagogical competence and teacher disposition in mathematics teaching.

 Keeping the tetrahedron model and its variants in mind, we will bring forth that there will be three other main variants to redesign and restructure the teacher preparation programs in Mongolia, namely, pedagogical competence-based design, didactical competence-based design, content knowledge based design, and teacher disposition based design.

Conclusion

The attributes of teacher quality will be regarded as key factors that we will take into account designing and restructuring the programs of teacher preparation. As a matter of fact, teacher quality comprises multiple constituencies that are divided into personal attributes, professional ones. Indeed, the personal qualities of teacher quality tend to be personal or subjective and therefore, it is highly likely to belong to disposition of a teacher candidate whereas the professional one tends to cover teacher attributes or competencies of pedagogy, didactics and content knowledge. Taken together, it is likely to be generalized that teacher quality are characterized by teacher attributes  in pedagogy, didactics and content knowledge and his or her disposition as well.

Someone who bears the same attributes that teacher quality is characterized, will be called here as an ideal teacher. Accordingly, it is generalized that an ideal teacher ought to be featured by the four major attributes compatible with teaching profession such as pedagogical competence, didactical competence, content knowledge and teacher disposition. In order to prepare teacher candidates as good as ideal, the content of teacher education ought to be designed as an amalgam of the key attributes of ideal teachers.

The content of teacher education is perceived as a closed configuration that might be filled with a set of teacher attributes that are in turn built upon by its four dimensions such as pedagogical competence, didactical competence, content knowledge and teacher disposition.  In addition, there is no hierarchical rank among such four dimensions. That is to say, each of four dimensions must be equally considered in configuring the content of teacher education. Taken into consideration aforementioned properties of the content of teacher education and its dimensions, it is conceived that a tetrahedron and its properties can present best the nature of content of teacher education.

 A tetrahedron model of the content of teacher education suggests at least 24 variants or ways to design and restructure the teacher education content and preparation that is in turn categorized into four major models, namely, a pedagogical competence-based tetrahedron, a didactical competence-based tetrahedron, a content knowledge based tetrahedron, and a teacher disposition based tetrahedron.  Each of such model has own advantages and disadvantages.

The significance of this is model of the content of teacher education is that it represents all possible ways and orientations to design and restructure teacher education content and preparation to meet teacher competences. In this sense, it can be implied to examine and enhance the existing programs for teacher preparation. However, the justification of building this model is needed to examine in the future study.

References

Arnons, S & Reichel, N 2007, ‘Who is the ideal teacher? Am I? Similarity and difference in perception of students of education regarding the qualities of a good teacher and of their own qualities as teachers’, Teachers and Teaching: theory and practice, vol.13, no.5 October 2007, pp.441-464.

Fredricksson,U 2004,  ‘Quality Education: The key role of Teachers’,  Education International Working Paper, no.14, September 2004. 

Hopkins, D & Stern, D 1996, ‘Quality teachers, quality schools: International perspectives and policy implications’, Teaching & Teacher Education, vol.12, pp.501-517, 1996.

Meg, M 1991, ‘Quality in teacher education’, Assessment & Evaluation in Higher Education, vol.16, no.1, April 1991.

Ministry of Education,Culture and Science 2007, EFA Report (draft), Ulaanbaatar, p.6.

National Statistics Office of Mongolia, the World Bank, UNDP 2004, Household Income and Expenditure, Living Standards Measurement Survey: Main Report, 2002-2003, Ulaanbaatar, p.41.

 School of Mathematics and Statistics, The programmes for mathematics teacher preparation, 2000, The University of Education, Ulaabaatar.