The promotion of spatial imagination through the active use of cube structures

Content

1 Introduction

2 The spatial imagination
2.1 Definition of spatial imagination according to THURSTONE and BESUDEN
2.2 Visual perception – the prerequisite for spatial imagination

3 The connection between spatial vision and intelligence
3.1 THURSTONE's Primary Factors of Intelligence
3.2 GARDNER's theory of multiple intelligences

4 The development of the spatial concept
4.1 The development of spatial thinking according to PIAGET
4.1.1 The distinction between perception and imagination
4.1.2 PIAGET's step theory of intelligence development
4.1.3 The stages of development of spatial operations
4.1.4 Criticism of PIAGET's step theory
4.2 Gender-specific differences in the development of spatial vision

5 Structure of the teaching unit:
5.1 Description of the learning group
5.1.1 General requirements
5.1.2 Content requirements
5.1.3 More detailed description of the children to be observed
5.2 1. Factual analysis
5.3 Didactic considerations
5.3.1 Classification of the topic in curricular requirements
5.3.2 The relevance of spatial imagination
5.3.3 Conditions under which spatial presentation can be promoted
5.3.3.1 Acting, Arguing, Mental Analysis
5.3.3.2 Head geometry
5.3.4 To the selection of teaching content
5.3.5 Competences and learning objectives of the teaching unit
5.4 Methodological considerations
5.4.1 Action experiences on the concrete material
5.4.2 The choice of forms of work and social
5.4.3 Differentiation
5.5 Tabular overview of the structure of the teaching unit

6 Presentation and reflection of selected lessons
6.1 Detailed description of the fourth sequence
6.1.1 Main intention, competences, learning objectives and learning opportunities
6.1.2 Didactic-methodological preliminary considerations
6.1.3 Planned course of lessons
6.1.4 Reflection
6.2 Detailed description of the sixth sequence (double visit)
6.2.1 Main intention, competences, learning objectives and learning opportunities
6.2.2 Didactic-methodological preliminary considerations
6.2.3 Planned course of lessons
6.2.4 Reflection
6.3 Detailed description of the seventh sequence
6.3.1 Main intention, competences, learning objectives and learning opportunities
6.3.2 Didactic-methodological preliminary considerations
6.3.3 Planned course of lessons
6.3.4 Reflection

7 Reflection and conclusion

8 Bibliography

9 Appendix

1 Introduction

"... that's when I realized that we would miss something [...] if we didn't introduce children of primary school age to geometry."¹

By imparting basic geometric knowledge and skills, geometry lessons make an important contribution to the child's ability and intellectual development, which enable him to participate in social life. As this is predominantly spatially structured, the geometric forms and arrangements surrounding us must first be understood and penetrated so that we can find our way around and orient ourselves in it. The promotion of spatial imagination through geometric content comes as "one of the ultimate goals of geometry teaching"², a particularly significant role, too.

The spatial imagination as the ability to orient oneself in space, to reproduce spatial conditions in the imagination and to operate with them mentally, is not available to the children from birth. It must therefore be developed and promoted accordingly.

If sufficient support in geometry teaching is not possible, learning difficulties can often be the result in many school areas. The effects on daily life activities would also be devastating: Catching a ball, sorting dishes into the closet or crossing a street are already tasks that require spatial imagination. We are just not aware of this in these situations, because "We have become so accustomed to space that we all too easily forget its meaning for us and its meaning for those we educate."³

With this knowledge of the need to promote spatial imagination through geometric content in the classroom, it is incomprehensible why geometry teaching to this day "in teaching practice a rather stepmotherly existence"⁴ in addition to arithmetic lessons and are often limited to a few hours before the holidays. A look into the mathematics book⁵, with which the second class works, shows that even there geometry is treated only peripherally and extends only over a few isolated sides.

In geometry lessons, geometric knowledge and skills of the children from the preschool period can be linked, in which the children "built, laid, experimented and in this way gained experience in space that must be continued."⁶ From this, the principle can also be derived that the promotion of spatial imagination should always be based on actions on the concrete material, since ideas of objects and their movements can only be established when they have been dealt with in an action. In addition, this "game character" can convey a positive attitude towards mathematics. Particularly computationally weak students can be motivated by a sense of achievement by solving geometric tasks for arithmetic content.⁷

In order to counteract the current situation of geometry teaching at primary schools and to enable the children to have the necessary acting access and the examination of geometric content, I will deal intensively with the "promotion of spatial imagination through the acting handling of cube buildings" in this term paper and the associated teaching unit. In particular, I will pursue the following questions:

- To what extent do the use of the homogeneous building material "cube" and the chosen teaching content promote the spatial imagination of the children?
- How does the process from the acting handling of cubes to purely mental reproduction take place? What do the chosen methods and didactic decisions contribute to this?
- How does the acting handling of dice affect the attitude and motivation of the children to teach mathematics?

In order to answer these questions, I will observe three children more closely during the course of the lesson and document and analyze their written and oral work results.

First, in the theoretical part (ch. 2- 4) of the present work, the scientific background of the topic is examined in more detail in order to be able to align the structure of the contents and methods with the learning group in a targeted manner. In Chapter 2, for example, a definition of spatial imagination is made, to which the representation of the connection between spatial conception and intelligence in Chapter 3 is linked. The description of the development of spatial imagination in Chapter 4 forms an important basis for the planning of the teaching unit in Chapter 5, in which didactic and methodological decisions are presented on the basis of a description of the learning group. A detailed documentation and reflection of selected lessons in Chapter 6 finally leads to a final conclusion of the entire teaching unit in Chapter 7, in which essential results are collected and consequences for further teaching practice are shown.

2 The spatial imagination

The terms spatial imagination, spatial imagination or, for short, spatial imagination, which are used synonymously in this work, refer to abilities with the help of which people mentally find their way in two- and three-dimensional space and operate with concrete, visible or imagined objects.⁸ Due to its complexity, this intelligence factor is divided into several sub-abilities in the scientific literature. Although there is no uniform definition, it is considered certain that spatial imagination is a largely independent factor of human intelligence.

2.1 Definition of spatial imagination according to THURSTONE and BESUDEN

THURSTONE, who has done important work in understanding the structure of spatial imagination, defines spatial perception as the ability to "to operate with 2- or 3-dimensional objects in the imagination"⁹ and in his 3-factor hypothesis mentions the following subfactors of this intelligence aspect:

- illustration (visualisation) includes the ability to imagine spatial movements of entire objects or parts of objects mentally.B, such as rotations, displacements and folds.
- Spatial relationship (spatial relations) denotes the recording of the spatial arrangements and relationships of objects or object parts to each other, whereby the location of one's own person is outside this arrangement.
- Spatial orientation (spatial orientation) characterizes the ability to correctly classify one's own person in a spatial situation, i.e. to find one's way around a room mentally or in real life.¹⁰

Based on the category system according to LINN & PETERSEN, MAIER proposes an expanding differentiation of THURSTONE's 3-factor theory by delimiting a subfactor spatial perception of THURSTONE's spatial orientation and additionally emphasizing the imagination of rotations as an independent factor:

- Spatial perception (Spatial perception) refers to the ability to identify the horizontal and vertical with the involvement of one's own body.
- Imagination of rotations (Mental rotation) is the ability to quickly and accurately imagine rotations of two- or three-dimensional objects. This subcomponent of the spatial presentation is in thurstone's subfactor illustration integrated according to MAIER "broad and generalized".¹¹

From a mathematical didactic point of view, BESUDEN defines spatial imagination as "an asset acquired through mental processing (internalization) of perceptions of material objects, which has become aware of the spatial references and can reproduce them."¹² Similar to THURSTONE, he also mentions three subfactors of spatial presentation, "whose independence is not clearly established"¹³ :

- Spatial orientation (spatial orientation) enables the correct spatial classification of one's own person in order to be able to move really or mentally in space.
- Spatial imagination (spatial visualisation) is the ability to reproduce spatial objects or relationships even when they are absent.
- Spatial thinking (spatial thinking) refers to the skill of dealing with spatial imagining contents in a mobile way, the prerequisite for which is the internalization of actually performed actions on objects.¹⁴

2.2 Visual perception – the prerequisite for spatial imagination

Seeing, as a purely physical process, forms the starting point of visual perception: Reflected light from three-dimensional objects located in the visual area hits the retina through the pupil, creating a two-dimensional image there. The resulting nerve impulses are transmitted to the brain and passed on there. works. The perceived is compared and interpreted with memory contents. Thus, visual perception is more than just seeing, because it also involves processing and retaining perceived objects.¹⁵ In the scientific literature, the importance of visual-spatial perceptual abilities as a necessary prerequisite of spatial imagination is always emphasized.¹⁶

FROSTIG distinguishes five areas of visual perception:

- Visuomotor coordination as the ability to adapt movements of the body or individual parts of the body to vision (e.g.: catch a ball).
- Figure-reason distinction is the ability to recognize and isolate sub-figures against a complex background or an overall figure (e.g.: to recognize a rectangle in a collection of geometric shapes).
- Perceptual constancy denotes the skill of recognizing objects despite different sizes, arrangements, spatial layers or colorings (e.g.: recognizing a cube from different angles).
- Perception of spatial relationships is the ability to recognize and describe relationships and the spatial location of objects to each other (e.g.: describe the location of a cube between other objects).
- Perception of the spatial position is defined as the ability to recognize and describe the space-position relationship of an object to the location of one's own person (e.g.: the three-mountain experiment by PIAGET¹⁷ ).¹⁸

HOFFER has formulated two further components of visual perception:

- Visual distinction he describes as the ability to recognize not only similarities but also differences between objects (e.g.: sorting and classifying geometric bodies according to their characteristics).
- Visual Memory as an ability to relate characteristic features of a no longer existing object in the imagination to other objects (e.g.: compare the properties of cube and cuboid in the imagination).¹⁹

3 The connection between spatial vision and intelligence

Psychological research has produced many theories on the structure and structure of human intelligence. Some of them contain the factor of spatial imagination as an independent and significant component, which illustrates the need for its promotion for the intellectual development of children. As an example of this, the works of THURSTONE and GARDNER are presented here.

3.1 THURSTONE's Primary Factors of Intelligence

THURSTONE assumes seven primary factors of intelligence, which "although largely different from each other, but still have low correlations with each other"²⁰ and which he mentions as basic conditions of intelligence performance. In addition to the factors V erbal (word comprehension), W ord Fluency (word fluency), N umber (computational skill), P erception (velocity of perception), M emory (memory) and R easoning (logical thinking) denotes THURSTONE a factor S pace, which includes the ability to "to operate with 2- or 3-dimensional objects in the imagination"²¹ and thus defines the spatial imagination. Later, in its 3-factor theory, THURSTONE divides this complex intelligence factor into subfactors (cf. chapter 2.1).²² THURSTONE was one of the first to recognize the ability of spatial vision as an independent factor of intelligence and thus did groundbreaking work to understand this intelligence factor.²³

3.2 GARDNER's theory of multiple intelligences

Since THURSTONE, many intelligence researchers have been able to confirm his conclusion that spatial vision is to be regarded as an independent factor of intelligence and differs from logical and linguistic abilities. So does GARDNER, who in his theory of multiple intelligences differentiates between linguistic, musical, logical-mathematical, physical-kinesthetic, intra- and interpersonal as well as spatial intelligence and defines them as the ability to to perceive the visual world correctly, to transform and modify the original perception and to reproduce images of the visual experience even when appropriate physical stimulations are missing".²⁴ This makes it clear that GARDNER also divides intelligence into several sub-abilities, which include the factor of spatial presentation as an independent factor.

4 The development of the spatial concept

Like all primary factors of intelligence, the idea of space in humans continues to evolve over the years. Figure 1 shows the estimated development of THURSTON's intelligence factors (see chapter 3.1) according to BLOOM (1973)²⁵.

Compared to the other intelligence factors, the spatial imagination (red line) shows a stronger ability to develop. At the age of about four, the idea of space is only weakly developed. However, their development increases steeply between the ages of 7 and 14. Thus, at the age of nine years, 50%, at the age of 14, about 80% of the maximum performance of the room vision are already developed, if one assumes that adults have 100% of the performance. At the age of nine to 14, there are therefore particularly good opportunities for promotion and, above all, success for geometry lessons.²⁶

4.1 The development of spatial thinking according to PIAGET

Piaget's investigations and findings still form the most important basis for understanding geometric learning. The results of his extensive experiments were confirmed in several follow-up studies.²⁷ In the following, therefore, based on a distinction between perceived and presented space, PIAGET's step theory of intelligence development is described and the development of the spatial concept according to PIAGET is presented, which takes place in three consecutive stages.

4.1.1 The distinction between perception and imagination

In connection with the development of intelligence and the development of spatial thinking in the child, PIAGET particularly emphasizes the difference between perceived space (perception of space) and imagined space (spatial presentation):

The construction of spatial relationships develops on two different levels. It begins at the level of perception and continues at the level of perception. Spatial perception arises in the direct handling of an object. A body is perceived and appears before the "inner eye". PIAGET describes this as a perceptual image, since this image cannot yet be operated on mentally. He speaks of an imaginal image only when the object can be reproduced by drawing, describing, producing or naming properties from memory. The idea is therefore not to be equated with the ability to recall spatial objects in front of the mind's eye.

The perceived space develops much faster than the imagined space. Even toddlers are already able to figure out a particular object they know, among others, by drawing on their memory. However, with this object, they can not yet perform operations in the head.

Although both spaces develop very similarly, there can be a gap of several years between the construction of perception and that of perception. For this reason, Piaget clearly separates these two rooms from each other.²⁸

Furthermore, he proves that an idea can only be built up through the real action, which illustrates the necessity of dealing with geometrical content in an acting way (cf. 5.3.3.1), because children cannot imagine the results of even simple processes until they have carried them out themselves. So the spatial notion is "an internalized action [...] and not simply the pictorial idea of some external fact, such as the result of an action".²⁹

4.1.2 PIAGET's step theory of intelligence development

PIAGET developed four stages of intelligence development based on observations on children of different ages. The train of thought when solving a task was always important to him, but not the right answer. According to PIAGET, cognitive development takes place in stages. In these subsections, the thinking and behavior of the child in various situations reflect a specific mental structure. The following step theory of intelligence development is the starting point for PIAGET's theory of the development of spatial vision, which is discussed in chapter 4.1.3.

In the sensorimotor phase (0 – 2 years) the child acquires the concept of space, starting from sensorimotor activities with objects of his immediate surroundings. At the end of the stage, there may already be some imagination,"the space [...] but can probably not yet be presented and mentally reconstructed."³⁰

At the beginning of the preoperational phase (2 – 7 years) spatial ideas are built on the basis of sensorimotor activities. The child acquires spatial schemes, "which are becoming increasingly mobile and structured."³¹ At the end of this phase, the child is already able to detect and perform some transformations, since the "internalized actions are coordinated to such an extent that their compilation and consequently each and every one of them can be handled in both directions."³²

In the concrete-operational phase (7 – 11 years) shows the ability to operate mentally with objects. The children's activities are dynamic, reversible and also transformable in concrete operations. Another characteristic of this phase is the mental change of perspective, so that children can now imagine how objects look from a different point of view or look like after a rotation in space.³³

In the formal-operational phase (from 11 years) "the view reaches its completion".³⁴ Thus, the thought processes of this phase represent on the one hand the goal of steadily growing internalization of action. On the other hand, they prepare the axiomatization of the room. The adolescent is now able to make hypotheses and formally test them.³⁵

4.1.3 The stages of development of spatial operations

Against the background of the step theory of intelligence development, PIAGET's theory of the development of the idea of space, which takes place in the succession of spatial operations and which must be constructed by the child in the course of cognitive development, is to be seen. "The general line of development in spatial thinking in the child, from topological to projective to metric space, can be considered secure."³⁶

The development of topological relations (up to 7 years). At the beginning of the development of spatial thinking, the child first uses elementary, topological relationships to orient himself in space. It recognizes and distinguishes features such as open or closed, limited, adjacent, consecutive, etc.³⁷

The development of projective relationships (from 7 years). With the onset of primary school age, the children are able to construct projective straights (shortening the straight line in an oblique image), as well as to understand shadow projections, cutting operations and surface processing.³⁸ The child is increasingly able to correctly locate objects and to correctly recognize and describe their arrangements to each other, also from a different perspective.³⁹

The development of Euclidean relations (from 7-12 years). The Euclidean idea of space develops partly simultaneously with the appropriation of projective space, "where they support each other." ⁴⁰ The child now thinks in terms of Euclidean space and can perform congruence mappings (reflection, rotation, displacement). It detects that objects and their volumes do not change by moving. Length measurements, area and volume calculations are recorded and thus also the invariance of objects. The child can correctly reproduce layers and distances, perform scale reductions and enlargements and distinguish between the concept of volume and surface.⁴¹

4.1.4 Criticism of PIAGET's step theory

Even if the general line of development of spatial thinking in the child can be considered secure, some criticisms of PIAGET's theory are expressed:⁴²

- Thus, the tests carried out by PIAGET are considered from today's point of view as not empirically proven since only a small number of subjects were tested, whose social origin and possible related differences in the level of development were not taken into account. In addition, experimental arrangements were insufficiently described and the evaluations were not sufficiently secured by controls.
- The tests are liable " Laboratory smell " because PIAGET is mostly on paper and pencil "or on measurements of spatial abilities at the green table"⁴³ Limited. The children's understanding of a larger spatial environment was thus ignored.
- PIAGETs Age of the individual stages are controversial and can only be regarded as a rough orientation.
- one high verbal proportion made the mostly verbally formulated tasks for some children inscrutable and not understandable. In some cases, even the investigators committed scientific errors through the incorrect use of mathematical terms.
- There is a close link between the results of the investigation and the Type of task, because these often did not correspond to the experiences of the children. Tasks dressed in known facts changed the results of the investigation.
- PIAGETs inferences that the cognitive development of the child is based on an improvement in mental abilities, i.e. thinking, neglect the influence of knowledge. Intellectual changes are due to thinking and knowledge from today's perspective. The development of cognitive abilities is even predominantly dependent on an increase in knowledge. "Much of what Piaget sees as fundamental changes in the understanding of the world is merely an improvement in memory."⁴⁴

Despite these criticisms, PIAGET's theory still "the most important theory of mental development in childhood and adolescence"⁴⁵ and should therefore be given special consideration in the planning of lessons in order to optimally support and challenge the children. PIAGET's findings clearly show that adults think very differently than children. Only the understanding of the child's thinking can help to influence his thinking and behavior. Completely ignoring PIAGET's findings could even lead to children being over- or under-challenged, both of which reduce motivation and learning success. In addition, PIAGET recognizes that the child's development can only progress if the practice of known actions and challenging situations (in the classroom) are equally taken into account.⁴⁶

4.2 Gender-specific differences in the development of spatial vision

In several studies on gender-specific differences in intelligence performance, it was found that female subjects are generally superior in verbal areas (word comprehension, word fluency), but the performance of male subjects is better in the areas of logical thinking, as well as in spatial conception and spatial orientation.⁴⁷ In the field of spatial presentation, the gender-specific differences are particularly pronounced compared to other cognitive abilities.⁴⁸ The reasons for these differences are socialization-related influences (the handling of typical male toys in childhood) as well as genetic, hormonal and neuropsychological causes.⁴⁹

However, these differences occur in all subcomponents of the spatial idea (cf. Chapter 2.1) only after puberty. Before that, there were hardly any differences.

Nevertheless, the effects of spatial-visual abilities on mathematical performance should not be ignored. While men are able to successfully solve mathematical problems despite little training in their spatial abilities, deficits in women's spatial vision have a negative effect on their mathematical performance. Women therefore seem less able to "to compensate for a deficit in the spatial-visual area by other qualifications."⁵⁰

Accordingly, importance should already be attached to promoting the idea of space during school hours in order to be able to correct any deficits that may occur or even to prevent them. The existing differences should be noted in a differentiated way in order to be based on this "to be able to initiate the most qualified and targeted training measures possible"⁵¹, which, in order to be effective, are characterized by playful and experimental actions and are intended to take into account the promise of spatial references.

5 Structure of the teaching unit:

In this chapter, the teaching unit based on the theoretical foundations and planned is now presented. First, the learning group in which the unit is carried out is described in more detail. This is followed by a factual analysis of the subject matter as well as the didactic and methodological considerations and finally a tabular overview of the structure of the planned teaching unit.

5.1 Description of the learning group

5.1.1 General requirements

The second class consists of 15 children, five of whom are girls and ten boys. The school is located in a more rural catchment area and the students come from predominantly socially strong families.

At the beginning of the new school year, a change of class teacher took place, which the children received very well.

The learning and working behavior of the class can be described as overall positive. Mathematics is the favorite subject for many children in the class, which is reflected in their high motivation and ambition to "crack" even difficult tasks.

In everyday lessons, I often fall back on a helper system in which students who have already completed the tasks set help other children. They take the helper task very seriously and are now able to explain tasks to their classmates in a child-friendly way without anticipating the solution. With the addition of three children, who often still need a guiding hand, all children in the class work independently. Since the beginning of the school year, they have often been working on weekly schedules in mathematics lessons, selecting their own tasks and controlling them independently.

The social behaviour of the children is also predominantly positive. They meet each other in a friendly and helpful way across genders. In mathematics lessons, I introduced some rituals, such as "the silent bunny" at the beginning of an hour, the kalimba as a reminder of the so-called "20cm language" (conversations may only be heard 20cm far) and the bell as a sign of a change of work phase. The children are familiar with these rituals and observe them.

In the class, there is a strong performance gap. This makes a differentiation necessary in order to do justice to all students and to enable individual learning progress (see Chapter 5.4.3). Five children in the class are characterized by particularly good performances in mathematics. They calculate very quickly and securely in the number space up to 20 and are also able to quickly record and implement new tasks. They already show very good skills in problem-solving thinking. On the other hand, three students in the class show very poor performance. The contents of the first class are incomplete, so they have big problems in expanding the number space. In a geometry unit already carried out, a similar distribution was found, but the performance gap was significantly smaller, since the weaknesses of the three students lie mainly in the arithmetic range.

5.1.2 Content requirements

The children know the forms of work and social work individual, partner and group work, teacher-student conversation and the work at stations or a learning counter. They are able to work cooperatively and effectively with one or more partners. They are happy to present their results to their classmates.

[...]

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¹ Radatz, H.; Rickmeyer, K. (1991), p.7

² Ibid., p. 56

³ Ibid., p. 56

⁴ Ibid., p. 56

⁵ cf. Wittmann, E.Ch.; Müller, G.N. (2004): The number book 2

⁶ Radatz, H., Rickmeyer, K. (1991), p.11 (author's note)

⁷ cf. Radatz, H., Rickmeyer, K. (1991), p.8

⁸ cf. Franke, M. (2007), p.28

⁹ Maier, P.H. (1999), p.20

¹⁰ Cf. ibid. PP.31-42

¹¹ Ibid. p.48 and p.45-49

¹² Besuden, H. (1984), p. 66

¹³ Ibid. P. 32

¹⁴ cf. Ibid. P. 32

¹⁵ cf. Franke, M. (2007), p.32

¹⁶ Cf. ibid. pp.32-52 and cf. Radatz, H.; Rickmeyer, K. (1991); pp.15-17

¹⁷ cf. Piaget, J.; Inhelder, B. (1971), p.251 f.

¹⁸ cf. Radatz, H; Rickmeyer, K. (1991), pp.15-17

¹⁹ cf. Franke, M. (2007), p.50 and Maier, P.H. (1999), p.12

²⁰ Maier, P.H. (1999), p.20

²¹ Ibid. P. 32

²² cf. Maier, P.H. (1999), pp.18-21

²³ cf. Franke, M. (2007), p. 52

²⁴ Gardner, H. (1991), p.163

²⁵ cf. Maier, P.H. (1999), p.78

²⁶ cf. Maier, P.H. (1999), p.78 and Besuden, H. (1999), p. 6

²⁷ cf. Radatz, H.; Rickmeyer, K. (1991), p.11

²⁸ cf. Piaget, J.; Inhelder, B. (1971), p.523

²⁹ Ibid. P. 32

³⁰ Maier, P.H. (1999), p.88

³¹ Franke, M. (2007), p.78

³² Piaget, J.; Inhelder, B. (1971), p.528

³³ cf. Franke, M. (2007), p.78 and Maier, P.H. (1999), p.90

³⁴ Piaget, J.; Inhelder, B. (1971), p.528

³⁵ Cf. ibid. P. 32

³⁶ Rost, D.H. (1977), p.58

³⁷ cf. Maier, P.H. (1999), p.91

³⁸ Cf. ibid. P.92 and Franke, M. (2007), pp.82-86

³⁹ cf. Rost, D.H. (1977), p.55

⁴⁰ Piaget, J.; Inhelder, B. (1971), p.31

⁴¹ cf. Hellmich, F. (2001), p.9 and Franke, M. (2007), pp.86-88

⁴² cf. Franke, M. (2007), S.91-92 and Maier, P.H. (1999), S.94-95 and Rost, D.H. (1977), S.56-58

⁴³ Maier, P.H. (1999), p.94

⁴⁴ Franke, M. (2007), p.92

⁴⁵ Ibid. P. 32

⁴⁶ Ibid. P. 32

⁴⁷ cf. Rost, D.H. (1977), p.29

⁴⁸ cf. Maier, P.H. (1999), p.204

⁴⁹ Cf. ibid. S. 24-27.

⁵⁰ Cf. ibid. P. 32

⁵¹ Cf. ibid. P. 32