(Final report of the University of Stellenbosch Experiment in Mathematics Education (USEME)-project: 1977-78;

published in Afrikaans in 1987)

In co-operation with

A FreeTranslation from Afrikaans of the Theoretical Study

By

Michael De Villiers

4 Orientation

56. REFERENCES
57. TABLES

OrientationThe research reported here focused on the conceptualisation, experimental implementation and evaluation of an alternative teaching approach for formal Euclidean geometry at the secondary school level. Broadly the experimental teaching approach consisted of:

The strategy was conceptualised in relation to certain identified shortcomings in the traditional practice of geometry education, as well as certain perspectives regarding the nature of axiomatic-deductive systems.

In the past two decades dramatic changes in geometry education have occurred in almost all developed countries.

In the first place there is a school of thought that wants to completely do away with the teaching of formal geometry (i.e. a deductive system of definitions, axioms and theorems) at school level. This school already existed at the end of the previous century in the so-called "Reform Movement", who under the leadership of Perry pleaded for the abolishment of the Euclidean version of geometry in favour of an applications orientated approach (compare Wolff, 1915). This school of thought had a limited impact in that formal geometry was excluded from certain mathematics courses within the context of differentiated education at secondary school level (e.g. the British "Mathematics for the Majority" and the South African "Practical Mathematics"), as well as the reduction of the amount of formal geometry in South African mathematics syllabi. In the "Mathematics for the Majority" curriculum, proof in either a formal or informal form is completely avoided (Wilson, 1977:42) while even in the British SMP curriculum very little attention is given to proof (op cit., 120).

Another tendency has been the abolishment of the Euclidean version of formal geometry in which congruency and similarity is a central theme. Wilson (1977:19-20) distinguishes four other content approaches to formal geometry, some of which have been implemented on a large scale in Western countries as an alternative to "congruency geometry". In transformation geometry, an approach already proposed in the previous century by Klein and followed by the influential British SMP-curriculum, the study of isometric transformations of the plane play the same role as congruency in the traditional approach. The motivation for transformation geometry (amongst others) lies in that it emphasises the relationships between Geometry and Algebra and that an introduction to transformations provides a powerful conceptual frame of reference. A related approach of whom Jeger is a leading exponent, is described as follows by Wilson (loc.cit.):

A third content alternative to congruency geometry is vector geometry, in which the axioms of a vector space are taken as the departure point for a strict axiomatic-deductive approach to plane geometry. This approach has strongly been propagated by the influential French mathematician Dieudonné and was implemented in the late 70's on a limited scale in South Africa as a supplement to congruency geometry. Freudenthal provides an expansive criticism of this approach (1973: 442-445) and describes it as an effort to circumvent the problems of geometry education by presenting geometry as part of algebra. His criticism, amongst others, is based on the argument that such an approach does not provide for a pupil's need for geometric orientation and that vector geometry can only be meaningfully interpreted by pupils if they have already become acquainted to geometry in another way:

(It is noticeable that vector geometry in South Africa was precisely introduced after pupils had already become acquainted with geometry in the traditional way. The question is whether the poor success with this attempt was not due to a limited interpretation of vector geometry as merely an alternative proof technique, while in reality it has as primary function an alternative deductive organisation. The latter meaning would probably only surface if the process of deductive systematisation is emphasised during the teaching of geometry).

The fourth content alternative distinguished by Wilson is analytical geometry which was taught prior to vector geometry, also on a limited scale in South Africa and alongside congruency geometry (and is again done so presently in the 90's). Analytical geometry is nowadays normally integrated with vector geometry and can also via matrix algebra be integrated with transformation geometry (although this is not presently done in the South African context).

In the midst of these changes South Africa has remained one of the few bastions of traditional congruency geometry. The mentioned changes in geometry education were aimed at eliminating certain didactical shortcomings in the traditional approach, e.g. the low success rate regarding the teaching of proof, as well as the deficiencies of congruency geometry as an example of an axiomatic-deductive system (compare Freudenthal, 1973:401-402; 420). Not all mathematics educators are convinced that the alternatives are really better than the traditional approach (compare Freudenthal, as well as Van Dormolen, 1974:184), and it may be that in due course it is shown that the maintenance of the traditional approach in South Africa has had advantages.

In this research, various possibilities were investigated to eliminate certain shortcomings in the traditional congruency based approach without radical content changes. The alternative strategies which form the basis of this research is however radically different in the way in which the learning material is treated, as will become clearer later on.

Attempts at improving congruency geometry as a theme in school mathematics is not new. In fact, the present approach is in many respects different from Euclid's original version.

Since the compilation of Euclid's Elements as a systematisation of the geometric knowledge in 300 BC, it has had a compelling influence on geometry-education, first at university level, and later at school level. This influence became particularly strong when parts of the Elements (usually the first six books) started being used in the 14th century as prescribed books in European universities and from the 18th century in European schools (Shibli, 1932:17-24). The influence of the Elements at school level was the greatest in Britain and in the nineteenth century the practice of geometry education in British schools was characterised by the study of axioms, definitions, construction techniques and theorems in a fixed order, without any prior informal or practical introduction, and with no practical applications and little or no exercises (Shibli, 1932:22-23; Bell, 1836; Todhunter, 1882).

In France and Germany, the Elements were less slavishly followed at school level, although it also played an important role. In both these countries practical, applications-oriented geometry education formed part of the secondary school curriculum since the 16th century. In Germany the practical textbooks were replaced at the beginning of the 17th century by textbooks based on Euclid, although they still contained practical applications. From the middle of the nineteenth century Germany started deviating from Euclid's order of theorems, and an informal, practical course as preparation for formal geometry was introduced (Shibli, 1932:20). In France during the 18th century there was a strong debate regarding a choice between applications-orientated geometry and formal Euclid-based geometry, and both forms appeared in practice. With the publication of Legendre's "Elements de geometrie" in 1794 a new direction came to the fore in the form of a practice of geometry-education which was still formal, but deviated from Euclid's in important aspects. This work had a strong influence on geometry education throughout Europe and also in the United States.

Euclid's Elements is for example characterised by the fact that theorems regarding figures are not proved, unless it has already been proved that the figures can be constructed, that no arithmetic or algebra is used in the proofs, and that the ordering (presentation) of theorems is strictly determined by the logical relationships between theorems. For example the conditions for congruency of triangles are developed early in Euclid, i.e. theorem 4 (side, angle, side), theorem 8 (three sides) and theorem 26 (two angles and a corresponding side) (compare Heath, 1908:247, 260, 301).

Legendre deviates in all these aspects from Euclid's by treating figures without proof of constructibility, by using arithmetical and algebraic arguments in geometric proofs and arranging theorems around specific themes (Shibli, 1932:21-22).

In Britain geometry was first introduced to secondary schools in the early part of the nineteenth century (Shibli, 1932, 1932:23) and the Euclidean rigour and order in some textbooks was maintained until the first part of the twentieth century. In MacKay's preface to his rendition of Euclid, he writes (1887):

Nevertheless some changes did occur in the British geometry textbooks of the nineteenth century. One of the first changes was the inclusion of exercises which required pupils to prove some results themselves.

Playfair's 1843 version of Euclid (like the original) contained no exercises (Shibli, 1932:165). The versions of Bell (1836), Todhunter (1882) and MacKay (1887) all contained a variety of exercises. This change was an important broadening of the aims of geometry education. MacKay describes this as follows in his preface:

Shibli (1932) shows that the inclusion of exercises also occurred in the latter part of the nineteenth century in the USA. This change is particularly interesting since it implies that there was a stage when Geometry only consisted of formal "bookwork". This is one of the earliest signs of a shift in emphasis from purely knowledge aims with geometry education to process or skills aims.

A second change occurred in regard to the way in which proofs were set out. In Euclid's original Elements, as well as in the earlier versions such as Playfair and Bell, proofs were given in "essay form". To illustrate this, Euclid's Theorem 4 with its proof from Bell (1836:8-9) and Heath (1908:247-248) is given below:

This particular excerpt from Euclid is also given for other reasons which will be discussed further on. The (*) was inserted to indicate that the argument was based on one of Euclid's axioms, namely the axiom that states that objects which coincide are equal (Bell, 1836:5).

In Todhunter (1862, consulted resource is 1882 edition) the presentation of proofs are different in two important aspects from the more original versions. In the first place references are given to preceding theorems, definitions or axioms which are used for specific steps in the proof. Todhunter motivates this as follows in his "Introductory remarks":

The second change was that every step in the arguments started on a separate line. Todhunter indicates in his preface that in this regard he was basing his text on a French text of 1762 by Koenig & Blassiere (Todhunter, 1862:viii-ix), and refers to a viewpoint of De Morgan as motivation:

It is interesting to note that De Morgan was actually recommending something different from the provision of ready-made analyses of proofs in constituent parts to pupils; he recommended that pupils should do such analyses themselves.

In so far as we could ascertain, this recommendation has never seriously been implemented anywhere. This is an example of a frequent occurrence in different contexts in the theory and practice of geometry education, namely the difference between the execution of a particular process (e.g. proving, analysing proofs) by pupils, and the provision of completed products of such processes (e.g. proofs, proofs in analysed form) to pupils. When in a text like that of Todhunter, proofs are provided for pupils in which:

the pupils obviously are not given the opportunity for identifying the different steps in the proofs and the reasons for these steps. The gain in terms of access to knowledge in this case therefore represents a loss of exercise in the ability to read and make sense of an axiomatic-deductive presentation.

Shibli refers to further developments in the same direction in the U.S.A. In Wentworth's text of 1899 each step in the proof is started on a new indented line, and the reason for that rule is given directly below, also indented and in brackets. In Schultze's Geometry of 1913 the steps of the proof are given on separate lines in a column on the left hand side of the page, while reasons for each step are given in the column on the right hand side of the page (Shibli, 1932:145). Shibli writes as follows in this regard:

Shibli (1932:141-146) also discusses several other developments in American geometry textbooks which will not be discussed here.

It has already been pointed out earlier that Legendre deviated from Euclid's order and grouping of theorems. Hereby the way was paved for other authors to use their own ordering and grouping of theorems. Since Legendre's direct influence was more felt in France and the U.S.A. than in Britain, and since geometry education in South Africa in the nineteenth and early in the twentieth century was strongly under British influence, we shall not here discuss Legendre's approach further. Of more direct interest, is to note changes in the ordering of theorems in British and South African textbooks.

The first six books of Euclid's Elements contain 173 theorems which include constructions. The themes of these six books are as follows:

(Books VIII to X deal with numbers while books XI to XIII deal with three dimensional geometry).

As orientation for the reader, an excerpt from Book I from Heath (1908) and Todhunter (1882) is now given below. Only 2 of the 23 definitions are given, but all the postulates and "common notions", as well as the first 32 theorems are presented.

When a straight line is drawn to another straight line and the two adjacent angles formed in this manner are equal, then these angles are called right angles.

Parallel lines are straight lines which lie in the same plane, and can be indefinitely extended in both directions without meeting (intersecting) each other.

As already mentioned, Euclid's Elements with additional exercises and a more systematic exposition of proofs was used for the largest part of the nineteenth century in Britain. Two more radical changes similar to that initiated by Legendre in other Western countries, followed in due course, namely, changing the order of the theorems and a relaxation of the logical rigour by de-emphasising the role of constructions. One of the first manifestations of these changes was the Geometry course proposed in 1878 by the "Association for the Improvement of Geometrical Teaching " (A.I.G.T.). In this proposal which was formulated in the form of a syllabus, the constructions are separated from the theorems and a more thematic approach is followed with the theorems. These proposals were among others implemented in the textbook of Baker & Bourne (1905), which was also used in South Africa, and locally in Du Toit (1921). The differences in organization between Euclid's and the AIGT-proposals are presented in Table 1 (given after the references at the end).

The mentioned changes were taken further in other textbooks, and eventually crystallised out in a form of which Durell (1939) is a classical example. This work was still used in the 70's in some British schools. In Durell the theorems are ordered completely thematically and constructions are treated separately from the theorems.

Reeve's (1930) description of these changes which also occurred in the U.S.A. are significant:

And further on (op cit., 25):

The above changes did however not make a substantial change in relation to the axiomatic foundation and type of proofs, and a book like Durell's, apart from a little informal/practical introduction and exercises, is nothing but a re-arrangement of Euclid's Elements.

The 1978 South-African geometry syllabus represents a far more drastic change, as can be seen in the schematic comparison with Durell in Table 2 (given after the references at the end). A deeper lying difference between Durell's geometry and "South African Geometry" is situated in the nature of proofs for the elementary theorems. Theorems 1 and 2 of Durell requires subtle proofs while theorems 5 and 6 require reductio ad absurdum (see Durell, 1976:540-541 & 544-545). To illustrate this the proof of Theorem 1 is given below.

If a straight line stands on another straight line, the sum of the adjacent angles formed is equal to two right angles.

In contrast, the proofs of Theorems 1 and 2 in the 1978 South-African geometry syllabus are direct and logical. From this it would appear that the primary reason for the different choice of axioms in the 1978 South African syllabus is to avoid the more difficult proofs in the initial stages of formal geometry. Durell apparently also saw it as problematic to confront pupils in the initial stages of formal geometry with these proofs, as he did not give these proofs in his main text, but provided them in an appendix.

In a similar way Durell proves the different tests for triangle congruency (as in Euclid), while the side, angle, side and side, side, side congruency tests in the South African syllabus are given as axioms (and the other tests for congruency as "theorems without proofs").

The proofs of these congruency tests as theorems are not only difficult for beginners, but the principle of "superposition" (e.g. compare example on p.7 of this report) is logically problematic (compare Heath, 1905:226, as well as Shibli, 1930:103-105). Shibli (loc. cit.) refers to the following reasons for postulating one or more of the congruency tests as axioms:

The 1978 South African syllabus represents only one of the possible ways of avoiding the complicated proofs in the initial stages of formal geometry. A comparable alternative is that of Seidlin in which it is taken as an axiom that when straight lines intersect, then the directly opposite angles are equal (NCTM, 1930:54-63).

This alternative choice of axioms clearly deviates from Euclid. This deviation has to be seen against the changed conception of the nature of axioms which developed during the last century. Shibli (op cit., 99-100) writes as follows:

A distinction can be made between a classical view of axioms in which axioms are seen as self-evident truths and a modern view of axioms in which axioms are seen as useful assumptions (compare Human, 1978:159-168). Atiyah (in Athen & Kunle, 1977:66) describes the classical view as follows:

In terms of this distinction it is clear that the 1978 South African geometry syllabus utilises a modern view of axioms. It is therefore also not strange or unacceptable that some of the theorems (e.g. that directly opposite angles are equal or that the base angles of an isosceles triangle are equal) are as "self-evident" as the axioms.

Not all South African textbook authors recognized this transition from a classical to a modern view on axiomatisation, and therefore did not adapt their formulation. So for example we find in Wait & Bester (1973), as well as Gonin, Archer, Slabber & Nel (1981), the description of axioms as "self-evident truths". However, in Schnell & Shapiro (1974) and Dreyer (1973) axioms are described in accordance with the modern view.

It is an open question whether pupils in South African schools understand the nature of the axiom system they are studying. In the empirical part of this investigation it was indeed found that a large percentage of Std.8-pupils (Grade 10) do not understand the nature of axioms. The following remarks by Birkhoff & Beatley is probably applicable to the present practice of geometry education in South Africa (NCTM, 1950:86-87):

Birkhoff & Beatley's reservations and suggestions are an accurate description of the underlying philosophy of our current research. With this very condensed historical perspective on the teaching of geometry we conclude. We will now give attention to some viewpoints regarding geometry education which have not been reflected in practice. Critical views and suggested alternatives regarding the teaching of formal congruency geometry consists of substantial literature, and we will therefore here only provide cursory reference to some viewpoints that relate directly to the current study.

In the construction of axiomatic-deductive systems mathematicians normally try to use as few axioms as possible, or more precisely, to ensure that none of the axioms can be deduced from the others. In the nineteenth century in Britain and the USA the viewpoint was held that this economy of assumptions should be presented to pupils from the beginning (compare Shibli, 1932:100-104). Towards the end of the nineteenth century several reservations regarding this viewpoint and corresponding practice were being raised, mainly on the grounds that this implied that pupils were confronted with difficult proofs of propositions that were self-evident to them. Shibli (1932:101) formulates this argument as follows:

The alternative is to ignore the principle of economy of axioms in the first contact with formal geometry. Shibli (1932:102) quotes Carson as follows:

This viewpoint is also reflected in the 1923 report on geometry education by a committee of the Mathematical Association in Britain:

This viewpoint therefore implies that there should be a stage in the geometry curriculum in which a classical perspective on axioms as self-evident truths is maintained. This means that proof in this phase should only be applied to propositions which are not seen as self-evident by pupils. It is therefore about proof in the classical sense of the word as described by Hull (1968:30):

It should be pointed out that this approach has not been reflected in South African syllabi. The changes regarding the choice of axioms that have occurred in syllabi during the last few decades, have only involved the avoidance of difficult proofs in the early stages - propositions which can easily and directly be proved (e.g. the equality of directly opposite angles) are still proved irrespective of how self-evident it may appear to pupils. It would appear that the South African involvement (by way of syllabus revision) with the formal system has only been in relation to the logical complexity of proofs, and not in relation to the meaningfulness, or not, of proof to pupils.

In the 1923-report of the Mathematical Association the following five phases for geometry education were suggested (Mathematical Association, 1923:14-25):

The first two phases are described as follows in the report (op cit.: 8-9):

Phase B is described in more detail further on (op cit. : 15):

Phase C is described as follows (op cit.: 9):

As far as it could be ascertained phase B has not really been implemented anywhere in the practice of geometry education. The 1978 (and present) South African syllabus is characterised by phases A and C, although some textbooks virtually skip over phase A and already start in Grade 8 (Std 6) with phase C. (Compare De Jager & Fitton, 1981:68-80).

The compilers of the 1923-report motivate their phase plan with the cryptic comment (p.14) that the monolithic plan of Euclid "made little allowance for the change in a boy's mind between 12 and 18." More recently Van Hiele provides a psychological foundation for phases in geometry education. In the following paragraph we will briefly give attention to this.

Van Hiele (1973:91-97) distinguishes in his "niveau-theory" different levels of mathematical thought. He describes these levels with reference to the concept "rhombus". On the so-called Ground Level (Level 0) of thought a person would visually recognise an object as a rhombus on the grounds of its shape, without being able to describe any of its properties. It is typical of persons at this level to exclude the squares from the rhombi (a phenomenon which is typical of Grade 8 (Std 6) South African pupils and is easily empirically verified).

On the so-called First Level of thought the term rhombus now acquires a different meaning for a person, namely that it is now being associated with a large set of properties:

At this level it can now be recognised that a square is a rhombus, since it is clear that a square has all the properties of a rhombus.

When a person has reached the First Level of thought in regard to a given concept (e.g. the rhombi), it is possible that he could gradually understand the logical inter-relationship between the different properties of that concept. Understanding the logical relationships between these properties form the substance of the Second Level of thought. At this level it is no longer a question what properties a specific figure has, but about which properties can be deduced from one another. Proof and deductive description of properties in the form of formal, economical definitions now become the focus.

Van Hiele (1973:92) also uses the concept of "network of relations" to highlight the differences between the different levels as follows:

Van Hiele further describes the network of relations which appear on the First Level: it boils down to a system of descriptive concepts by which the properties of figures and the relationships between figures can be described. In a certain sense the transition from the Ground Level to the First Level involves a transition from an inactive-iconic handling of concepts to a more symbolic one, to use Bruner's familiar concepts. More simply put, the attainment of the First Level involves the acquisition of the technical language by which the properties of the concept can be described.

However, the transition from the Ground Level to the First Level involves more than just the acquisition of language. It involves recognising certain new relationships between concepts and the refinement and renewal of existing concepts. In the case of the rhombi it becomes logical (in terms of a list of properties of the rhombi) to include the squares under the rhombi which implies a hierarchical classification which is in contrast to the partition classification of the Ground Level. Such a generalisation of a concept can occur since the verbal description of concepts often have unforeseen implications. For example, with a verbal description of the kites the question will typically for the first time arise whether a concave quadrilateral with an axis of symmetry along one diagonal, should be considered as a kite. Together with this question a need for clarifying the meaning of the concept diagonal arises.

It is clear that when a pupil progresses from the Ground Level to the First Level regarding a particular topic (e.g. the quadrilaterals), a significant re-arrangement of relationships and a refinement of concepts occurs. There is therefore far more in this transition than merely a verbalisation of intuitive knowledge; the verbalisation goes together with a restructuring of knowledge. According to Van Hiele this restructuring must first occur before one starts exploring the logical relationships between these properties. Van Hiele (1973:94) puts it as follows:

The Second Level of thought also represents a completely different network of relations than the First Level. Where the network of relations at the First Level involves the association of properties with types of figures and relationships between figures according to these properties, the network of relations at the Second Level involve the logical relationships between the properties of figures. The network of relations at the Second Level no longer refer to concrete, specific figures, nor do they form a frame of reference in which it is asked whether a given figure has certain properties. The typical questions that are asked at the Second Level are whether a certain property follows from another, or can be deduced from a particular subset of properties (in other words whether it could be taken as a definition) or whether two definitions are equivalent.

The network of relations for the First and Second thought levels are therefore quite different (Van Hiele, 1973:90):

Van Hiele views deductive proof as an activity which belongs to the Second Level. Together with this, concepts like axioms and theorems belong to the Second Level, and has no real meaning at the lower levels.

Pupils at Levels 0 or 1 with regard to a particular topic (e.g. the quadrilaterals as plane figures) will therefore according to the Van Hiele theory not understand teaching which is focused on the activities and meanings of Level 2. When a teacher provides a deductive proof that the diagonals of a rectangle are equal, or that the directly opposite angles formed by two intersecting lines are equal, then the real meaning of the activity and of the proof as content is found in the explication of the logical relationships, and the empirical validity of the propositions being logically organised, is not relevant. For a pupil at Level 1, however, the network of logical relations does not yet exist, and such a pupil can only interpret a deductive proof as a search for conviction/certainty regarding already observed facts. He however does not doubt the validity of the observed facts and therefore experiences the teaching as meaningless. In this regard Van Hiele provides a radical critique of any form of geometry education where proof is introduced before pupils have seen its meaning in terms of systematisation.

Since proof in mathematics can also have other meanings (functions) than systematisation, Van Hiele's critique should be seen as a bit of an oversimplification. This issue is discussed in more detail in the next paragraph, in conjunction with the views of other mathematics educators regarding proof.

In the previous two paragraphs a distinction was made between different phases in geometry education, and that "proof" as an activity does not play the same role in the different phases. We will now look in more detail at the problem of proof in geometry education.

The question of when proof should be introduced in school mathematics, is according to Wilson (1977:121-122), on the one hand, dependent on the maturity (cognitive development) of pupils, and on the other hand, dependent on how much the teacher tries to emphasise the meaning and usefulness of proof.

Wilson also recommends a particular strategy for making proof more meaningful to pupils:

Afanasjewa warns against a one-sided view in which only cognitive development is seen as the source of the problem (in Freudenthal (Ed), 1958:29):

Vredenduin (in Freudenthal (Ed), 1958:31) also singles out the problem of the construction of meaning for proof as follows:

Boermeester (in Freudenthal (Ed), 1958:23) reports as follows in regard to a research project in geometry education in 1955 - 1958 in the Netherlands:

The question is what meanings does proof have in geometry and what meanings does it have for children at different levels. One danger in this regard is that proof is one-sidedly seen only as a means towards conviction or certainty. In this regard Van Zijl (1942:104) writes:

Bell (1976:24) writes in similar vein:

Bell (1976:24-25) then continues by distinguishing between three meanings of proof in the mathematical sense of the word:

We shall here refer to these three functions of proof respectively as verification, explanation and systematisation. This distinction can be utilised to refine the critical remarks of Afanasjewa, Vredenduin and Wilson regarding proof in traditional geometry education.

The term "proof" in every day language carries a meaning which is associated with "convincing", "making sure", "removing doubt", etc. These meanings therefore reflect a function of verification for proof, but does not refer to the functions of explanation and systematisation. When children are then introduced to proof within the context of geometric propositions for which they have no doubt, while it is not made clear to them that in these cases one is more concerned with explaining or systematising them, it can be expected that pupils will view proof as a meaningless activity. The typical situation is where pupils' first introduction to proof occurs within the context of an axiomatic-deductive system of geometric concepts and propositions. Here proof primarily fulfils a function of systematisation and the first propositions that are proved are usually ones about which pupils hardly have any doubt.

One way of avoiding this problem, is by the introduction of proof not within a formal axiomatic-deductive system, but in relation to propositions which appear doubtful to pupils or for which they would like an explanation. This would then mean assuming a far larger number of propositions without proof than is normally the case in a formal-deductive system. Indeed, this is precisely the intention of the distinction of Phase B in the 1923-report of the Mathematical Association (see earlier paragraph on phases in the geometry curriculum).

It appears further that Van Hiele assigns a rather limited function to proof by indicating it only as a systematising (ordering) activity on Level 2, and by not assigning some function to it at the lower levels. For example, it would appear to us that proof as a means of verification and/or explanation can feature on Van Hiele's Level 1. Van Hiele's niveau distinctions also agree strongly with the phase distinctions which were made in official documents in 1909 in Britain, and is referred to as follows in the 1923-report (Mathematical Association, 1923:14):

The traditional practice of geometry education is characterised by a strong emphasis on proof as a mathematical activity. There are however a variety of other important mathematical activities that could be emphasised in elementary geometry. In this regard we must distinguish between informal activities in which proof and systematisation does not play any substantial role, and formal activities which centre around proof and systematisation.

Regarding informal activities, there have been great changes in the last half century in most countries (including South Africa), in the sense that a considerable amount of informal geometry has been included in the geometry curricula at the primary and junior secondary levels. This issue will therefore not be discussed any further in this document.

The attention will now be focused on the spectrum of mathematical activities which can be emphasised within the context of formal geometry at school level. In the previous paragraph, different functions of proof were distinguished, and the point was made that the functions of explanation and systematisation are under-utilised in the traditional practice of geometry education.

Formal geometry however involves at least two other extremely important mathematical activities, namely, defining and axiomatizing.

Freudenthal (1973:418) writes as follows in this regard:

Blanford (1908:216-217) already pleaded in 1908 that pupils should be involved in defining as a mathematical activity:

Freudenthal (1973:416) writes in a similar vein as follows:

Among other things, it is here about a distinction between two different teaching strategies, namely, the direct presentation and the reconstructive presentation of mathematical content (see Human, 1978:77). With a direct presentation is meant that definitions, theorems, proofs, axiom systems and other types of mathematical content are presented as finished products of mathematical activity without involving pupils in those mathematical activities by which the content is typically developed.

The criticism against a direct presentation is of course not only aimed at the handling of definitions, but is also aimed at the direct presentation of an axiomatic-deductive system. Freudenthal (1973:451) writes as follows (also compare Bunt in Freudenthal (Ed), 1958:97):

The alternative strategy which is suggested in criticising a direct presentation can be described as a reconstructive presentation; in other words, a presentation in which the content (definitions, axiom systems, propositions, proofs, etc.) are not from the beginning presented in a finished (prefabricated) form, but are developed or reconstructed during teaching in a typical mathematical way (compare Wittmann, 1973:106). A reconstructive presentation should not simply be equated with discovery learning. A reconstructive approach does not necessarily imply learning by discovery; it may just be a reconstructive explanation by the teacher or the textbook. The essential characteristic of the reconstructive approach does not lie in self-discovery by pupils, but in the manner in which the ways and reasons why mathematical content and certain formulations or organisations of mathematical content is developed or changed, is brought to the surface. The motivation for the reconstructive approach also does not claim that it is more effective than the direct approach regarding the mastery of knowledge and skills related to specific topics. (In fact, it is probable that the reconstructive approach may often in this regard be less efficient as it takes more time). The didactical functionality of the reconstructive approach rather lies in that it is (by definition) that form of presentation which demonstrate certain mathematical activities, as well as providing some opportunity for pupils to participate in those activities. De Vries (1980:i) quotes Parr as follows regarding the importance of some pupil participation:

The preceding remarks do not imply that the direct approach does not or should not have any place in the teaching of mathematics. It is for example possible within a reconstructive approach (e.g. the classification of quadrilaterals) that certain content (e.g. some properties of the parallelograms) may be presented directly to ensure the smooth unfolding of the global reconstructive approach. Furthermore, the reconstructive approach is aimed at empowering pupils with the mathematical skills or processes which are used in the reconstruction. For persons (pupils, students) who already have some competency regarding certain processes, for example, defining and axiomatizing, an approach which is globally direct may be quite appropriate, especially if the primary aim of that section is knowledge acquisition.

The development of refined axiomatic-deductive presentations often have precisely the direct approach in mind with the view of maximum knowledge acquisition in a short space of time (compare Human, 1978:98). The present research however started from the premise that an understanding of the function and nature of axiomatizing and defining as mathematical processes can only be developed via a reconstructive approach, and that this understanding was a further prerequisite for most pupils (and students) to later effectively study other axiomatic-deductive systems which are presented directly. Geometry at school level can therefore from this perspective be seen as an opportunity to develop pupils' understanding of the nature and role of axioms and definitions to enable them to make sense of the later direct presentation of other axiomatic-deductive systems.

We are now in more detail going to give attention to the nature of defining as a mathematical activity.

In order to place defining into perspective, it is first necessary to distinguish between certain types of mathematical content.

Apart from construction processes (algorithms), one can among others, also distinguish between propositions, class concepts and relations as important types of mathematical content.

With propositions (statements) is not just meant conjectures as possibly implied by its everyday interpretation. With propositions is meant mathematical sentences which are implicitly or explicitly of the form "if p, then q". A specific proposition depending on the context in which it is used can have the status of a conjecture, theorem or axiom.

Class concepts are sets of mathematical objects, for example, parallelograms, cyclic quadrilaterals, closed plane figures, vector spaces, points, lines and so on. In contrast to propositions, class concepts are not described in full sentences (linguistically speaking), but by way of single nouns.

With relations is meant concepts in terms of which objects are compared by, for example, congruency, equality, similarity, parallelness, perpendicularity corresponding, etc.

Apart from the above types of content, the content of geometry (amongst others) also consist of technical terms which denote parts of a whole, for example, diagonal, base angles, diameters, as well as a variety of notations such as AR, angle ABC and PQ¤ // RS.

Propositions are general declarations which frequently describe a great variety of specific situations. Propositions typically involve class-concepts and relations. Propositions can be subjected to the process of proving. The result of proving is a logical arrangement of propositions which is called a proof. Proofs in themselves can be seen as a separate type of content.

For classconcepts we can typically construct definitions. Since classconcepts can however exist independently from conscious or explicit definitions, definitions are here also viewed as a separate type of content.

A person can for example visually distinguish between different types of quadrilaterals without being at all aware of explicit definitions for these quadrilaterals or using (or constructing) such definitions in distinguishing between them. (This can easily be demonstrated by giving a set of quadrilateral shaped pieces of cardboard to a child between 5 and 11 years with the task of sorting them). Defining, by which is meant the construction or creation of a definition and not the reproduction or repetition of a definition, is therefore not necessarily the same activity as that of constructing a class concept.

Different forms of defining can be distinguished in terms of logical nature, as well as the functions it fulfils. In the first place defining can involve the formulation of characteristics by which certain objects can be distinguished from one another. The function of such a description (definition) is not necessarily that of distinction (which may done purely visually), but to reveal the basic criteria of distinction to other persons. Such a clarification of meaning was often observed during Grade 9 (Std 7) pupil discussions regarding the classification of quadrilaterals in a reconstructive approach to quadrilaterals (compare De Vries, 1980).

Defining in this sense of the word, is an activity which forms part of the transition from the Van Hiele Ground Level to the First Level. The logical relationships between the different properties of a class concept does not play a role in this type of defining. In fact, it is quite typical that a definition which is constructed in this way, is a complete description of all the distinguishing characteristics which are recognised in examples of the concept. For this reason we shall refer to this kind of defining as intuitive defining.

Defining can however occur in a totally different context, namely, as the conscious selection of some of the properties of a concept, from which all the other properties of the concept have to be deductively deduced. Such formal defining is an activity that can only occur on Van Hiele Level 2, since it involves the logical relationships between the properties of a concept. The question is why are mathematical concepts formally defined.

One of the reasons is that the hierarchical classification of class concepts which are typical of Van Hiele Level 2, usually require different definitions than the intuitive definitions which are typically formulated to facilitate a partition distinction (classification). On Van Hiele Level 2, knowledge regarding the logical relationships between properties are available and also insight that not all properties have to be included in a definition (with regard to geometrical class concepts this insight can also exist at the Ground Level via suitable construction and measurement activities).

It is typical of mathematics that there is in more than one respect a desire for logical economy in formal defining. The term economy in the first place refers to the properties of the concept being defined. A definition is economical when none of the properties that have been included in the definition can be deduced from one another. The term "redundant" definition will be used further on to indicate definitions that are not economical.

The term economy can also refer to the influence of a particular choice of definition on the proofs of the other properties of the concept not included in its definition. One definition of a concept may in this respect be more economical than another definition if the aforementioned leads to simpler proofs of the other properties. For example, the following definition of the parallelograms is more economical in this regard than the one normally used:

The experimental course in the empirical part of this study was (among others), aimed at the following objectives:

(The results of the empirical research indicate that the attainment of these objectives are indeed possible by using certain appropriate teaching strategies).

Apart from the distinction between intuitive and formal defining, one must also distinguish between descriptive and constructive defining. Freudenthal (1973:461) puts it as follows:

Krygowska (in Servais & Varga, 1971:129-130) describes the difference between these two types of defining as follows:

Descriptive defining involves the defining of a concept which already exists in the mind of the person defining; in other words, it is a descriptive report of an already existing concept. On the other hand, constructive defining involves the new combination of properties (by addition, deletion, replacement, etc.), and the question which objects (if any) possess these properties and what other properties they have. Dienes (1963:60) describes defining in this form as follows and considers it as one of the ways in which mathematical concepts are formed.

Constructive defining can for example occur when the reader tries to distinguish between hexagons which respectively correspond to parallelograms, rhombi and squares.

The experimental course strongly focused on the ability to descriptively define as an objective by using a variety of defining exercises related to various quadrilaterals (e.g. compare Chapter 2 in Appendix A).

We shall now give some brief attention to the theoretical underpinning of another experiment in geometry education which was undertaken with similar objectives n mind as the present study.

In 1938 Fawcett reported on an experimental geometry curriculum which was presented to a group of 25 pupils in the USA. The experiment was done against the background of a movement in American education to develop critical and reflective thought among pupils. Fawcett (1938:1) writes:

Fawcett's investigation was aimed at developing and evaluating a teaching approach for proof in geometry, with the development of critical and reflective thought as spelt out above as primary objective. In order to reach this goal with the teaching of proof, it is necessary that pupils understand the nature of deductive proof. In this regard he writes (1938:10):

Fawcett also designed his experimental curriculum with the following important guideline:

The experimental curriculum allowed for the development of these logical processes of the pupil as follows (Fawcett, 1971:416-417):

This approach eventually culminates in each pupil developing/organising his/her own geometric knowledge in a logical system.

Traditionally the Russian geometry curriculum consisted of two stages: an intuitive stage for Grades 1 to 5 and a systematisation (deductive) stage from Grade 6 (11 or 12 years old).

For a period of about 40 years Soviet researchers conducted a comprehensive and intensive analysis on both the intuitive and systematisation stages, in order to find an answer to the question why so many pupils, who make good progress with other subjects, show little progress in the understanding and mastery of geometry.

In their analysis of the traditional curriculum the niveau structure of Van Hiele played an important role. It was for example determined that only 10 - 15% of the pupils at Grade 5 were at Level 1 (Compare Wirzup, 1976). Consequently the Soviet researchers described the period between Grade 1 and 5 as a prolonged period of geometric inactivity.

The traditional curriculum was made up as follows: During the first five years pupils were expected to become acquainted with 12-15 geometrical objects (e.g. names of figures, terms which indicate relationships, drawing instruments, etc.). In stark contrast, in the first topic in Grade 6 geometry it was expected of pupils to assimilate about 100 new terms and concepts.

The research further brought to light that there were often direct jumps in the teaching material from the Ground Level to Level 2. It was clear that the handling of learning materiel during the first five years was only at Ground Level. In Grade 6, however, it was expected of teachers to suddenly present work, from the very first lessons, at all three simultaneously; e.g.:

The research indicated that an important factor for the improvement of curricula and teaching methods could be found in the continuous development of topics for the formation of geometrical concepts, already commencing in Grade 1. Investigations indicated that pupils in experimental groups between Grades 1 and 4 were capable of developing a good understanding of geometry, without using deductive formalization. The Soviets however warn that the period in which facts are inductively accumulated should not be stretched out too long. In the new curriculum elementary deductions are systematically made by pupils from Grade 1.

Another important finding for which the new curriculum had to provide was that pupils in their investigation of geometric objects first approach the work qualitatively (the study of form, corresponding position, relationships, etc.) and that quantitative operations (e.g. measurement) only gradually develop at a later stage.

Since teaching a deductive system requires a lot of time and patience, it was felt that it should be started earlier and spread out more evenly over the grades. Interestingly, much greater success was achieved with an experimental course, based on Van Hiele and presented to Grade 2 pupils than what was achieved in the traditional course with Grade 7 pupils.

The above accelerated approach made it possible to already start in Grade 4 with the formal study of deductive geometry. In all stages of this formalisation, geometric transformations are emphasised, and in the higher classes vector representations are used. Sufficient information was gathered to indicate that the average Grade 8 pupil with the new course, exhibited the same level of geometric understanding as the average Grade 12 pupil with the traditional course (see Wirzup, 1976:75-96).

Two American mathematics educators, Schuster & Stone, made some proposals in 1971 for the renewal of geometry education which is directly applicable to the South African curriculum, and which served as a leitmotif for the curriculum experiment which is described in Part 2 of this report (not translated here).

Schuster describes the traditional practice of geometry education in the U.S.A. (where mathematics in the tenth school year is almost exclusively congruency geometry) as a failure (Steiner (Ed), 1971:301-302):

As alternative he suggests (op cit., 305-306):

Stone gives a concise characterisation of what a reconstructive approach to geometry may entail (Steiner (Ed), 1971:318):

The idea of local axiomatization is an important addition to the earlier frame of reference which appears in the 1923-report of the Mathematical Association and the works of Van Hiele. The idea is particularly important since it allows at least in theory, the treatment of axiomatization in geometry education without having to deal with the logical subtleties of proving elementary propositions at the start of Euclidean geometry.

In the experimental course which was designed for the purposes of the project reported here and implemented in a number of schools, local axiomatizing formed a critical phase within the following structure:

This structuring involves a refinement of phases A, B and C in the 1923-report in the sense that phase C is subdivided into two phases, namely, a phase of local axiomatizing (3 and 4 above) and a phase of global axiomatizing. Where the last mentioned phase within the context of the South African syllabus involves the uncritical acceptance of the constructibility of a variety of configurations, as well as the acceptance of the congruency conditions without proof, one could look in a further (6th) phase at a more critical level at the underlying assumptions and proofs of more "difficult" theorems (which were simply assumed earlier). This aspect has however fallen outside the field of the current project.