ALTERNATIVE INSTRUCTIONAL STRATEGIES FOR GEOMETRY EDUCATION:
A THEORETICAL AND EMPIRICAL STUDY
(Final report of the University of Stellenbosch Experiment in Mathematics Education (USEME)-project: 1977-78;
published in Afrikaans in 1987)
P G Human & J H Nel
In co-operation with
M D De Villiers, T P Dreyer &
S F G Wessels
A FreeTranslation from Afrikaans of the Theoretical Study
Michael De Villiers
(Although the theoretical study is dated, and does not for example include recent research findings in geometry education nor new developments and directions in modern geometry, or technology such as dynamic geometry, it still provides an invaluable starting point for reconceptualisation the teaching of traditional, formal Euclidean geometry)
"The deductive structure of traditional geometry has not just been a didactical success. People today believe geometry failed because it was not deductive enough. In my opinion, the reason was rather that this deductivity was not taught as reinvention, as Socrates did, but that it was imposed on the learner. Anyhow some people today preach and exhibit the abolition of geometry. Among those who should introduce youth to their cultural heritage, there are quite a few who would gladly thrust geometry in the cultural incinerator. The days of traditional geometry are counted, if at all it still survives in some places. What is destined to follow it? This urgent question stands behind the title of this chapter. The case of geometry - has it been sentenced to death and did it get a fair trial? Has it rightfully been sentenced or on false evidence? Has the defendant (sic) been left without counsel? Should not the investigative procedures be reopened?"
- Hans Freudenthal (1973 : 402)
"Schemes of geometrical education ¼ are lacking in foundation, method and extent. Euclid's Scheme - itself utterly unsuitable as an introduction to the subject, has been so far tampered with that hardly any scheme remains. So long as no attempt is made to devise a connected development based on the many intuitions which are common to all civilised beings before they reach maturity - so long will the subject realise a painfully small proportion of its potential value."
- G L Carson (1913)
SOME Historical and theoretical perspectives on geometry education
7 An overview of some historical developments in geometry education
23 Phases in the geometry curriculum
27 The Van Hiele theory
31 Proof in school geometry
35 Other mathematical thought processes in geometry
40 Defining as a mathematical process
46 Fawcett's experiment
48 Russian (Soviet) research on geometry education
51 Recent proposals for renewal in geometry education
HISTORICAL AND THEORETICAL PERSPECTIVES ON GEOMETRY EDUCATION
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 presentation of proof before the presentation of a formal system of axioms and definitions in order to make proof more meaningful to pupils
·the involvement of pupils in the construction of formal definitions in order to develop their ability to define, and understanding of the nature and function of definitions in mathematics
·the involvement of pupils in the selection of axioms in order to develop their understanding of the nature and function of axiomatic-deductive systematizations of mathematical knowledge.
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.):
"... Jeger sees the transformations of the plane as the central object of study, rather than figures drawn in the plane. In the Klein approach the transformations are the means, and the properties of the figures are the end; in the Jeger approach the figures become the means and the properties of transformations are the end."
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:
"To be able to interpret geometry as linear algebra, the learner - and not only the textbook author - should first have become acquainted with geometry."
(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.
An overview of some historical developments in geometry-education
"Whenever demonstrative geometry was taught, Euclid was the master."
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):
"... no change has been made on Euclid's sequence of propositions, and comparatively little on his mode of proof."
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:
"The deductions or exercises appended to the various propositions ("riders'', as they are sometimes termed) have been intentionally made easy and, in the First Book, numerous. It is hoped that beginners, who have little confidence in their own reasoning power, will thereby be encouraged to do more than merely learn the text of Euclid."
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:
"If in two triangles two sides of the one are respectively equal to two sides of the other, and the angles included between these equal sides are also equal, then the bases will also be equal, and the remaining angles will be respectively equal.
Let ABC and DEF be two triangles with the sides AB and AC respectively equal to the sides DE and DF, namely AB to DE and AC to DF, and the angle BAC equal to the angle EDF.
I claim that the base BC is also equal to the base EF, that the triangle ABC is equal to triangle DEF, and the remaining angles are respectively equal to the remaining angles, namely those opposite the equal sides, that is, the angle ABC to angle DEF and the angle ACB to the angle DFE. Because if the triangle ABC is placed on triangle DEF and if the point A is placed on point D and the straight line AB on DE, then the point B will fall on E, since AB is equal to DE. Since AB coincides with DE, the straight line AC will also coincide with DF, since the angle ABC is equal to the angle EDF; therefore the point C will coincide with point F, since AC is equal to DF. But B also coincides with E, therefore the base BC will coincide with the base EF and is therefore equal to it.* Therefore the whole triangle ABC will coincide with the whole triangle DEF and is equal to it; and the remaining angles of the one will coincide with the remaining angles of the other, and be equal to each other; namely, the angle ABC to angle DEF, and the angle ACB to the angle DFE. And that is what was required to prove.
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":
"To assist the student in following the steps of the reasoning, references are given to the results already obtained which are required in the demonstration. Thus 1.5 indicates that we appeal to the result obtained in the fifth proposition of the First Book".
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:
"As the Elements of Euclid are usually placed in the hands of young students it is important to exhibit the work in such a form as will assist them in overcoming the difficulties which they experience on their first introduction to processes of continuous argument. No method appears to be so useful as that of breaking up the demonstrations into their constituent parts; this was strongly recommended by professor De Morgan more than thirty years ago as a suitable exercise for students, and the plan has been adopted more or less closely in some modern editions."
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:
"... each distinct assertion in the argument begins in a new line, and at the end of the lines are placed the necessary references to the preceding principles on which the assertions depend"
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:
"The parallel column arrangement seems to emphasise more strongly the necessity of giving a reason for each statement made, and it saves time when the teacher is inspecting and correcting written work."
Shibli (1932:141-146) also discusses several other developments in American geometry textbooks which will not be discussed here.
"By changing the sequence Legendre broke the Euclidean spell" - Shibli
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:
Book I: Rectilinear figures, including congruent triangles, parallel lines, construction of triangles and properties of triangles (48 theorems).
Book II: Geometrical algebra, for example, geometrical proofs for a number of algebraic identities such as the distributive property and (a + b)2 = a2 + 2ab + b2, as well as theorems related to areas.
Book III: Circles, chords and tangents.
Book IV: Polygons and circles.
Book V: Ratio.
Book VI: Similar figures.
(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.
1. It is possible to draw a straight line from any point to another point.
2. It is possible to infinitely extend a finite straight line segment into a straight line.
3. It is possible with any centre and radius to construct a circle.
4. All right angles are equal.
5. If a straight line cuts two other straight lines so that the interior angles on the same side of the transversal line is less than two right angles, then the extensions of the last two straight lines would meet on the side of the transversal where the interior angles are less than two right angles.
1. Objects which are equal to the same object, are also equal to each other.
2. When equals are added to equals, then the combined wholes are also equal.
3. When equals are subtracted from equals, then the remaining parts are also equal.
4. Objects which coincide are equal.
5. The whole is greater than any part of it.
1. To construct an equilateral triangle on a given line segment.
2. To construct a line segment from a given point equal to a given line segment.
3. To mark off on the longer line segment of two given, unequal line segments, a part equal to the shorter one.
4. If two sides and an enclosed angle of one triangle are respectively equal to the two sides and enclosed angle of another triangle, then the two triangles are congruent.
5. In an isosceles triangle the base angles are equal and if the two equal sides are extended, then the angles below the base are also equal.
6. If a triangle has two equal angles, then the sides opposite the angles, are also equal.
7. It is impossible to construct two different triangles on the same base having equal sides starting at the endpoints of the base.
8. If three sides of one triangle are respectively equal to three sides of another triangle, then these two triangles are congruent.
9. To bisect a given angle.
10. To bisect a given line segment.
11. To construct a perpendicular line to a given line at a given point on the line.
12. To construct a perpendicular line from a given point outside a given line to that given line.
13. When a line is perpendicular to another line, then the angles thus formed are two right angles, or the two angles together are equal to two right angles.
14. When two straight lines meet at the endpoint of a third line so that the adjacent angles formed are equal to two right angles, then the first two lines lie in a straight line.
15. When two straight lines intersect, then the directly opposite angles are equal.
16. An exterior angle of a triangle is greater than any of the opposite interior angles.
17. Any two interior angles of a triangle are together less than two right angles.
18. In any triangle the longest side is subtended by the largest angle.
19. In any triangle the largest angle is subtended by the largest side.
20. In any triangle two sides are together longer than the third side.
21. When from the endpoints of one side of a triangle lines are drawn that meet inside the triangle, then the line segments thus formed, will be shorter than the other two sides of the triangle, but a larger angle is formed between these two than between the other two sides of the triangle.
22. To construct a triangle with given sidelengths.
23. To construct an angle at a given point on a given straight line, equal to a given angle.
24. If in two triangles two sides of the one triangle are respectively equal to two sides of the other triangle, and one of the angles between the two equal sides is greater than the other one, then the one base will be longer than the other.
25. If in two triangles two sides are respectively equal to two other sides, but the one base is longer than the other, then the angle between one pair of equal sides would be greater than the other.
26. If in two triangles two angles are correspondingly equal to the other two angles, and one side is equal to another side, namely, either the side which is subtended by one of the equal angles or the side which is included by the equal angles, then the remaining sides would be equal to the remaining sides and the remaining angle to the remaining angle.
27. When a straight line cuts two other straight lines and alternate angles are equal, then the last-mentioned straight lines are parallel.
28. When a straight line cuts two other straight lines and corresponding angles are equal, then the last-mentioned straight lines are parallel.
29. When a straight line cuts parallel lines, then the alternate and corresponding angles are equal and the co-interior angles on the same side of the transversal line are equal to two right angles.
30. Straight lines that are parallel to the same straight line, are also parallel to each other.
31. To draw, through a given point, a line parallel to a given line.
32. An exterior angle of a triangle is equal to the sum of the opposite interior angles and the three interior angles together are two right angles.
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:
"The theorems of geometry are concerned with proving geometric statements; the problems are concerned with the construction of geometric figures in a plane, the only instruments allowed being an unmarked straight-edge and a pair of compasses ... In early days ... the writers on plane geometry did not attempt to prove any theorem until they had shown that the figure could be constructed. For this purpose they placed some of their problems of plane geometry first, and introduced others as needed. At present we generally assume that all figures in plane geometry can be constructed.
Because of this modern view ... we generally place the problems at the end of any particular book or chapter."
And further on (op cit., 25):
"For about 2000 years Euclid's sequence was the order that was universally followed. Today we have a simpler and more usable sequence, not so rigidly scientific as Euclid intended it for university students, but within the reach of high school students and better adapted to their needs.
Euclid was little concerned with the classification of propositions. He arranged his propositions in an order that seemed to him to begin with the easiest proposition. He then built the superstructure so as to construct his figures before using them. We attempt to classify our propositions, but we do not attempt to construct our figures before using them. From the standpoint of strict logic Euclid's plan is better; for teaching purposes ours is superior."
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.
Given a straight line CE meeting a straight line ACB.
To prove that angle ACE + angle BCE = 2 rt angles
If angles ACE = angle BCE, each is by definition a right angle.
\angle ACE + angle BCE = 2 rt angles
If angle ACE is not equal to angle BCE, suppose angle ACE is the greater.
Construction: Draw CN perpendicular to ACB.
Proof: angle ACE + angle BCE = angle ACN + angle NCE + angle ECB
= angle ACN + angle NCB
= 2 rt angles (Constr.)
If angle ECB is the greater, the proof is the same, except that A and B are interchanged.
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:
1. "The proofs of the congruence theorems are too long and difficult at the beginning of the course, tending to discourage the pupil.
2. The proof by superposition is a peculiar one, used only in few cases and is dropped soon after it is learned.
3. Proof by superposition gives the pupil the wrong conception of the meaning of a proof, and leads him to apply the same method of reasoning in proving exercises.
4. The postulation of these theorems would open the way immediately to the more interesting and educational work of solving easy originals that are proved by means of congruent triangles.
5. The increase in the assumptions does not make the training in reasoning less valuable."
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:
"The extensive study of assumptions during the past fifty years has had some effect on the teaching of elementary geometry. A few decades ago teachers imagined that axioms were self evident truths, that all the assumptions used were explicitly stated as such, and that Euclid's system was a rigorously logical science. The work of mathematicians in recent times has dispelled all these illusions. It is now seen that axioms are merely assumptions, that many assumptions are used in elementary geometry without being formulated, and that a strictly rigorous geometry requires the mind of a Hilbert or a Veblen."
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:
"¼ the axioms of Euclidean geometry were meant to be self-evident truths derived from physical experience."
Krygowska (in Servais & Varga, 1971:148) describes the modern view as follows:
"the choice (of axioms) is determined by simplicity and convenience in the development of the theory."
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):
"Do our students appreciate the need for certain assumptions as fundamental in our logical system? Do we not allow them to infer that in geometry we can prove everything, and that we assume certain elementary propositions not because we have to, but because we are in a hurry to get on to more important matters? Should we not rather seize the opportunity to impress on them the need of certain assumptions, and show that this need is not peculiar to geometry, but is inherent in all logical systems? Could anything but good come from our indicating the possibility of some latitude in the choice of assumptions for geometry, and showing that each such choice leads to a slightly different approach to geometry, each valid with respect to its own group of assumptions and to no others? It is neither necessary nor desirable, perhaps, to mention non-Euclidean geometries in this connection.
What is the point in telling beginners that we shall assume certain "self-evident truths" and then asking them to prove certain other propositions which they regard as equally self-evident truths? Would they not come to a quicker understanding of the nature of a proof through the effort to prove easy 'originals' which are not too plausible and which seem therefore to require justification?"
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.
Phases in the geometry curriculum
"Economy of axiom systems tends to be an aesthetic criterion of mathematics rather than a practical virtue of mathematical education"
- Griffiths & Howson
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:
"These propositions are too obvious to require a proof in the minds of high school pupils, and their very obviousness makes the proof too subtle and complicated at the beginning of the course."
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:
"Let the statements which are easily acceptable to pupils be adopted as postulates, and let deductions be made from them with full rigor. Later, when speculation becomes more natural, let the meanings of consistence and independence of a set of postulates be explained, if not fully demonstrated, and let the pupil reduce the number of postulates by actually proving some of them as propositions. By this method a better idea of the structure of a science is imparted."
This viewpoint is also reflected in the 1923 report on geometry education by a committee of the Mathematical Association in Britain:
"By the end of Stage B (the deductive stage) the boy should know the interesting theorems of plane geometry; he should be able to devise constructions and to solve easy riders (originals); he should be able to apply his knowledge to simple solid figures; he should have some grasp of logical method. On the other hand, the theorems he knows will not at present be deduced from a small number of axioms; his knowledge will rather be based on a considerable number of assumptions, assumptions which are appropriate to his present mental development."
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):
"The whole notion of proof grew out of the wish to understand the unobvious and surprising in terms of what could be accepted as axioms or commonplace."
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):
A. An experimental phase which concentrates on the solution of real life problems (e.g. land surveying) and constructions, and which provide a context in which the underlying results regarding angles at a point, parallel lines and congruent triangles are introduced. Deduction can be done if the opportunity presents itself, but formal written proofs should be left for the next phase.
B. A deductive phase in which proof is formally introduced in regard to interesting results which are not self-evident to pupils.
C. A systematization phase in which proof is utilized to deductively order familiar results, and to deduce them from a small number of independent axioms.
D. Modern geometry, that is, the study of topics such as inversive, affine and projective geometry.
E. Foundations of Euclidean and non-Euclidean geometry.
The first two phases are described as follows in the report (op cit.: 8-9):
"The study of geometry as a preliminary experimental stage in which proofs are not attempted and propositions are not formally enunciated; more knowledge is being acquired unconsciously than consciously ¼
The next stage, everything which is obvious is taken for granted, argument is used only to introduce the unexpected. Here consideration of symmetry can play a large part, and congruence still rests on the adequacy of data to give a definite figure. It is typical of this stage to accept the equality of alternate angles almost without discussion, but to prove that the angle-sum of a triangle is two right angles."
Phase B is described in more detail further on (op cit. : 15):
"The boy next learns to prove theorems and riders and to write out proofs. The subject matter of this stage is the whole of elementary plane geometry, with occasional inroads upon the easier parts of solid geometry. At first sight this programme does not seem to differ substantially from the old-fashioned plan, but a fundamental distinction in method is recommended, which we will now explain. The main interest in Euclid's work is the systematic process. Euclid (and every writer on formal geometry) is chiefly concerned to construct a chain of reasoning; the links are sub-ordinate to the chain. The links are theorems, some of them interesting and arresting, some of them of no special interest in themselves but necessary as part of the chain. At the stage we are discussing the systematising instinct is not strongly developed. On the other hand great interest may be aroused by the search for geometrical truths. It is better at this stage to steer into the unknown than to attempt the proof of properties that are judged to be obviously true; the more striking the result, the greater the interest aroused. The reasoning powers have become strong enough for the use of the deductive method. This method will be prominent, but not to the exclusion of intuition and induction; in Stage A the deductive method was weakly developed and in Stage C we shall find that it is supreme. To maintain interest a choice of theorems must be made; the fundamental theorems on angles at a point, on parallels, and on congruence will still be assumed, and to those assumptions will not be added the fundamental theorems on similarity (not to be stated dogmatically, but to be granted on the strength of common experience)."
Phase C is described as follows (op cit.: 9):
"We begin to examine the elementary propositions. The process is not proof in the natural sense, but organisation. In other words what we establish is not an individual proposition but a relation between propositions. To express the result in the form that we assume one proposition and prove another is a convention, but we are dealing now with students mature enough not to be misled by a convention that is explained to them. The mere fact that it is reasonable to organise results which it seems absurd to demonstrate forestalls a distrust of proof and a contempt for proof that the average student is apt to feel when he is told with all solemnity that he has established the symmetry of the circle: the teacher who is careful to avoid confusion between a general theorem and a particular case, and between analogy and demonstration, need not fear that the student's idea of what constitutes a proof will be less satisfactory than under the ideal of a rigid geometrical system to be mastered from the axioms forward.
The process of organisation culminates in the recognition that geometrically all the obvious propositions depend on one of two principles; an axiom of congruence and an axiom for dealing with parallels."
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.
The Van Hiele Theory
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:
"The intended figure is now a collection of properties, that he has learnt to summarise under the name 'rhombus'" (freely translated from the Dutch)
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:
"In a network of relations the words `rhombus', `side', `angle', `square', etc. have unique meanings, associated with a distinct collection of properties. Each level is associated with a different network of relations." (Freely translated from Dutch).
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 network of relations on the Second Level can only be meaningfully established, when the network of relations at the First Level are adequately established. When the first network of relations are present in such an adequate form, that its structure becomes apparent and one can talk about it with others, then the building blocks for the Second Level are ready" (Freely translated from Dutch)
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):
"The reasoning of a logical system belongs to the Second Level of thought. The network of relations which is based on a verbal description of observed facts, belong to the First Level of thought. These two levels have their own networks of relations where the one is distinct from the other: one either reasons in the one network of relations or in the other."
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.
Proof in school geometry
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.
"It has often been asserted that children are not able to reason until they reach a certain age. If this is so, attempts to introduce proof will fail until this stage is reached. When we do feel that our pupils are ripe for these ideas, it will be important to convince them of the need for proof, and indeed of its usefulness."
Wilson also recommends a particular strategy for making proof more meaningful to pupils:
"Perhaps the most effective way of doing this is to encourage pupils to make conjectures of their own, and when they have done so to challenge them to convince you of the truth of their conjectures.
To achieve this requires careful choice of subject matter; situations are needed in which the pupil is likely to be able to both make a plausible conjecture and also to argue in defence of his statement ¼ Geometry is fertile ground to search for such topics."
- Wilson (1977:121-122)
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):
"The proof of self-evident statements are not difficult for young beginners because they are not capable of logical thinking, since the same (negative) inclination towards self-evident statements are exhibited by older beginners. The problem is that the purpose (function) of proof is not clear, and in the light of pupils previous knowledge it cannot be expected to be otherwise."
Vredenduin (in Freudenthal (Ed), 1958:31) also singles out the problem of the construction of meaning for proof as follows:
"The main trouble for the pupil is to understand what proving a theorem is, and why he must give formal proofs. ¼ Why do we want to prove that the sum of the angles of a triangle is 180° when we can verify it by measuring the angles and by putting three congruent triangles around a vertex? Why do we prove the theorem of Pythagoras though it was known long before it was proved and nevertheless nobody ever mistrusted it?"
Boermeester (in Freudenthal (Ed), 1958:23) reports as follows in regard to a research project in geometry education in 1955 - 1958 in the Netherlands:
"Several teachers warn against premature proofs of evident theorems. Theorems should not be proved until the pupils are convinced of the necessity of proving them."
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:
"¼ there are cases when one is fully justified in saying: `I am convinced by intuition that it is true'. Such cases in Geometry are:
The angles at the base of an isosceles triangle are equal
The opposite sides of a parallelogram are equal.
The two tangents from a point to a circle are equal.
The proofs which school books offer for such propositions are not for the purpose of convincing any one that they are true, for everyone knows by intuition that they are true."
Bell (1976:24) writes in similar vein:
"Some teachers have said that proof, for a pupil, is what brings him conviction. Although this is a valuable remark, in that it directs attention to the need for classroom explanations to have meaning for the pupil rather than be formal rituals, it is perhaps dangerous in that it avoids consideration of the real nature of proof. Conviction is normally reached by quite other means than that of following a logical proof."
Bell (1976:24-25) then continues by distinguishing between three meanings of proof in the mathematical sense of the word:
"The mathematical meaning of proof carries three senses. The first is verification or justification, concerned with the truth of a proposition; the second is illumination, in that a good proof is expected to convey an insight into why the proposition is true; this does not affect the validity of a proof but its presence in a proof is aesthetically pleasing. The third sense of proof is the most characteristically mathematical, that of systematisation, i.e. the organisation of results into a deductive system of axioms, major concepts and theorems, and minor results derived from these"
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 Board of Education, in a Circular issued in 1909, recognised three stages in elementary geometry. The committee would amalgamate the first two of these, but would subdivide the third."
Other mathematical thought processes in geometry
"Geometry is more than proof" - Allen Hoffer
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:
"Good geometry instruction can mean much - learning to organise a subject matter and learning what is organising, learning to conceptualise and what is conceptualising, learning to define and what is a definition. It means leading pupils to understand why some organisation, some concept, some definition is better than another. Traditional instruction is different. Rather than giving the child the opportunity to organise its spatial experiences, the subject matter is offered as a pre-organised structure. All concepts, definitions, and deductions are preconceived by the teacher ¼ or rather by the textbook author."
Blanford (1908:216-217) already pleaded in 1908 that pupils should be involved in defining as a mathematical activity:
"Definitions are the working hypotheses of the child: they develop gradually with the growth of his knowledge.
To me it appears a radically vicious method, certainly in geometry if not in other subjects, to supply a child with ready made definitions, to be subsequently memorised after being more or less carefully explained. To do this is surely to throw away deliberately one of the most valuable agents of intellectual discipline. The evolving of a workable definition by the child's own activity, stimulated by appropriate questions, is both interesting and highly educational. Let us try to discover the kind of conception already existing in the child's mind - vague and crude it generally is, of course, otherwise what need for education? - let us note carefully its defects, and then help the child himself to re-fashion the conception more in harmony with the truth. This newer and more correct conception, sprung from the old, will itself subsequently be replaced by a truer one, but it has thereby played its essentially useful function as a link whereby the vague becomes slowly transformed into the more accurate and true. Only thus can we make sure that the child assimilates knowledge and is really prepared for the digestion of more complex mental food. Contrast this procedure with that of forcibly thrusting into the mind a full born definition of which the child neither perceived the need nor understands the beauty and the truth."
Freudenthal (1973:416) writes in a similar vein as follows:
"¼ the Socratic didactician would refuse to introduce the geometrical objects by definitions, but wherever the didactic inversion prevails, deductivity starts with definitions. (In traditional geometry they even define what is a definition - a still higher level in the learning process.) The Socratic didactician rejects such a procedure. How can you define a thing before you know what you have to define? This is a general principle but it is particularly important in mathematics where "definition" has its special meaning. In mathematics a definition does not just serve to explain to people what is meant by a certain word. In mathematics definitions are links in deductive chains, but how can you forge such a link unless you know in which chain it should fit?"
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):
"Should axiomatics be taught in schools? If it is taught in the form it has been in the majority of projects in the last few years, I say "no". Prefabricated axiomatics is no more a teaching matter in school instruction than prefabricated mathematics in general. But what is judged to be essential in axiomatics by the adult mathematician, I mean axiomatising, may be a teaching matter. After local organisation the pupil should also learn organising globally and finally cutting the ontological bonds. But to do so, he must be acquainted with the domain that is to be organised, and the bonds that are to be cut should exist and should be vigorous. This is an exacting demand. The pupil who has never been self reliant in organising, will overlook connections only if the number of links is small, and the bonds with reality usually are little cultivated and weak. Of course, all these demands can be dismissed if the student is confronted with a ready made axiomatic system from which he may obediently draw a few conclusions. He can be drilled in this art, but this can hardly contribute to understanding axiomatics. The only result would be to enlarge again the stock of denatured school mathematics."
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:
"Mathematics is not a spectator sports; one does not "learn" mathematics, one "learns to do" mathematics.
The essence of mathematics is doing it. Mathematics is more an experience than a body of knowledge to be assimilated."
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.
Defining as a mathematical process
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 parallelograms are the quadrilaterals with bisecting diagonals."
The experimental course in the empirical part of this study was (among others), aimed at the following objectives:
·to let pupils realise that different definitions for the same concept are possible
·to let pupils realise that definitions may be redundant or economical
·to let pupils realise that some economic definitions lead to shorter, easier proofs of properties
·to develop some level of ability among pupils to formally define concepts
(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:
"In the course of his mathematical experience, the student has got acquainted with two kinds of definition, the describing definition which outlines a known object by singling out a few characteristic properties, and the algorithmically constructive and creative definition which models new objects out of familiar ones."
Krygowska (in Servais & Varga, 1971:129-130) describes the difference between these two types of defining as follows:
"What interests us from the educational standpoint is the interplay of two fundamental aspects of thinking arising in the process of definition: analysis, which is expressed in the establishment of a report, and synthesis, which leads to the construction of a project. These aspects play different roles in different situations; their mutual relation decides the nature of the definition, which is apprehended by the definer either as a `report definition' or as a `project definition'.
For instance, Cauchy has constructed the `report definition' of the limit of a function for he has formally legalised a concept which, having been used by mathematicians in an operative and creative manner, was nevertheless vague from the standpoint of logic. The analysis of Euclidean geometry made by Hilbert, leading to his system of axioms, has the same character. A pupil who, after apprehending a concept intuitively and operatively (drawing, model, concrete activity) seeks a definition, performs similar work. The special character of this process lies in the fact that the person who looks for a definition a posteriori expresses in that definition only part of his knowledge and intuitions regarding the object thus defined. He must therefore choose and isolate from the set of properties already apprehended a subset of properties the conjunction of which is sufficient to imply all the others. This choice may be more or less free and the criteria for it may be different or even diametrically opposed (e.g. the `natural' definitions in classical mathematics or the `genetic' - constructive -definitions in teaching, or the `key theorems' as definitions in Bourbaki's work). The method of a posteriori definition is therefore not easy and calls for a creative effort and a broad, deep knowledge of the elementary structures used for defining more complicated structures.
Construction of an a priori definition involves other aspects of thinking and another type of creation. On the basis of the axioms of Euclidean geometry which are the result of establishing a report definition, new geometries have been proposed a priori, the non-Euclidean geometries. These were project definitions constructed by an operation on the given axioms: omission of one axiom, replacement of some axioms by others, etc. Classifications, generalisations, particularizations, variations in conditions and different combinations of known structures leading to intersection structures may become starting points for project definitions in general and project axioms in particular.
Rational educational organisation of the interplay of these two aspects, which transform a description into a definition, a report of intuitions and previous knowledge into a structured project, seems to us to be the essential prerequisite for this form of introduction to the axiomatic method."
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.
"¼ the class has not been formed by construction from its elements but from above, so to speak, by logical combination of attributes."
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:
"¼ demonstrative geometry has long been justified on the ground that its chief contribution to the general education of the young people in our secondary schools is to acquaint them with the nature of deductive thought and to give them an understanding of what it really means to prove something. While verbal allegiance is paid to these large general objectives related to the nature of proof, actual classroom practice indicates that the major emphasis is placed on a body of theorems to be learned rather than on the method by which these theorems are established. The pupil feels that these theorems are important in themselves and in his earnest effort to `know' them he resorts to memorisation. The tests most commonly used emphasise the importance of factual information, and there is little evidence to show that pupils who have studied demonstrative geometry are less gullible, more logical and more critical in their thinking than those who did not follow such a course."
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):
"For purposes of this study it is assumed that a pupil understands the nature of deductive proof when he understands:
1 The place and significance of undefined concepts in proving any conclusion.
2 The necessity of clearly defined terms and their effect on the conclusion.
3 The necessity of assumptions or unproved propositions.
4 That no demonstration proves anything that is not implied by the assumptions."
Fawcett also designed his experimental curriculum with the following important guideline:
"¼ the logical processes which should guide the development of the work should be those of the pupil and not those of the teacher"
The experimental curriculum allowed for the development of these logical processes of the pupil as follows (Fawcett, 1971:416-417):
"While the teacher makes it possible for each student to think about geometry in his own way, he also makes it possible for all members of the class to consider the ideas thus uncovered. The selection of undefined terms and the final statement of a definition on assumption reflect agreement of the entire group, including the teacher who serves as discussion leader and guide, responsibilities which he must not abdicate ¼
Observation, measurement, intuition, and induction lead to conjectures concerning properties of geometric figures. Such conjectures are checked for proof or disproof by logical procedures and this process replaces the highly questionable procedure of telling the student what it is he must prove. Any conjecture found to be consistent with the accepted geometric foundation is expressed by the student in his own words as an established generalisation and these statements along with their respective proofs are presented to the class for critical consideration. Through the ensuring discussion the students come face to face with the needs for rigor. Does the generalisation state what has been proved and only what has been proved? Does it go beyond the limitations of the assumed or given data? Is the accompanying proof altogether acceptable? Have any hidden assumptions been used? The combined intellectual power of the entire class is used in this process and well nourished intellectual soil promotes the growth of ideas in an atmosphere of increasing rigor, reflecting standards agreed to by both students and teacher. Revised statements are more precise. Implicit assumptions are frequently detected and the class discussion which follows may result in such assumptions becoming a part of the geometric structure, suggesting to the students, in the words of Professor Young, that mathematics is growing at the bottom as well as at the top."
This approach eventually culminates in each pupil developing/organising his/her own geometric knowledge in a logical system.
Russian (Soviet) research on geometry education
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.:
1. to acquaint pupils with new geometric figures and to recognise them according to their shape (Ground Level)
2. to study the properties of figures in a practical way, and to identify figures according to their properties (Level 1)
3. to let pupils develop definitions for the figures to relate properties to each other and to logically deduce certain properties from others (Level 2).
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.
In the new geometry curriculum the following general plan was designed:
1. an initial introduction to two- and three-dimensional figures (Grades 1 - 2)
2. the study of the properties of figures (Grades 2 - 3)
3. the study of relationships between figures (Grades 2 - 3)
4. measurement of geometrical quantities (Grades 1 - 3)
5. the use of figures and measurement in the study of numbers and operations on them (Grades 1 - 3)
6. an introduction to the concepts of set theory (Grade 3).
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).
Other proposals for renewal in geometry education
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):
"When we ask questions like `Why Euclidean Geometry' and `What purpose does it serve', the answer is twofold:
1 For historical reasons and otherwise Euclidean Geometry is regarded as a (the?) means of showing students how to build an axiomatic system, of teaching them the deductive character of mathematics, and training them to write formal proofs;
2 Euclidean Geometry is useful - indeed necessary - for science and engineering.
The first reason is stated first because it has been the most influential in the teaching of geometry. The primary focus of the teaching has been on constructing the formal systems: the postulates, definitions and formal proofs. The important thing to note is that this focus has not been changed by the reform of recent years. The reform assumed the existing philosophy, namely that tenth year geometry was to be devoted to an axiomatic development of Euclidean Geometry, and exerted its energies to preserve Euclidean Geometry as a model of logical thinking. To be sure, some of the new books have enriched the content of the course in other ways, and many of the mathematicians associated with the reform had broader goals, but the mould was cast and it was hard to break. Thus, the reformers did not change the basic aim of the high school course nor were they successful in their altering of the spirit in which the geometry was to be studied.
I wish to argue that focusing on the formal structure of geometry is a serious mistake which has had some unfortunate results. The program has failed; that is, the goal of teaching formal structures via geometry has not been achieved.
It is my experience and the experience reported by others that few students came away from high school geometry with anything beyond a superficial understanding of, and appreciation for, an axiomatic structure. Paul Kelly is said to have made an informal survey with the conclusion that the thing most people recall from their high school geometry courses is that geometry is the subject in which proofs are written in two columns separated by a line on the page.
Although I am not in possession of the data, I have been told by mathematics educators that there is substantial documentation of the fact that large numbers of high school students memorise proofs they don't really understand, come to believe that the only acceptable proofs are those given in `step reason' form (echoing Paul Kelly's finding), and in spite of the fact that they have heard of non-Euclidean Geometry, still believe that the axioms they memorised are `self evident truths' and are the axioms for Euclidean Geometry rather than a set of axioms for the subject."
As alternative he suggests (op cit., 305-306):
"Axiomatics is learned slowly, so axiomatic treatment should be spread over the entire curricula. Great concentration on formal development, as it currently exists in secondary geometry, is overwhelming to students. The compromise I suggest here is that we make geometry locally axiomatic in contrast to the globally axiomatic treatment that is now given. That is, I believe that students as they grow to mathematical maturity would benefit more from smaller axiomatic systems within the subject of Euclidean Geometry. This point of view might be regarded as a request to teach mathematics the way it existed among the Greeks prior to Euclid. The theory of congruence no doubt existed as a set of interrelated propositions; the theory of ratio, proposition and similarity was developed as a unit by Eudorus. An examination of the curriculum topics will surely turn up portions that admit to simple axiomatic development. Perhaps a unit on transversals, or one on circles and angle measure, or another on parallelograms.
And some of these could be offered at much lower levels than they are now taught precisely because of their simplicity.
Part of the motivation behind my `locally axiomatic' proposal is that I think it is most important to begin early in inculcating in students the notion that the propositions of mathematics are logically interrelated and dependent upon one another. A great deal of this can be done before axioms are ever introduced, and before we say that a logical system must rest on certain approved propositions and in terms of some undefined terms.
Another thing that motivate my proposal is that the present manner of teaching formal structure doesn't teach students where the primitive frames come from or (rephrased) how and why the mathematician creates his deductive system the way he does. If teaching at the early stages gives students experience - physical and logical - with the concepts then there is a much better basis for teaching the important matters in foundations. For example, it is important to see where the definition comes from; the study of inter-relationships of propositions will turn up necessary and sufficient conditions which, the student should learn, can often be utilised to define a concept in a convenient manner. Another thing of importance is the idea of generalising: How is this done? With a view toward what? Most important are the questions of how to abstract, how to choose axioms, how to mathematize a subject that we already have some knowledge about. This sequence of considerations would be closer to real mathematical development than exists in the current curriculum, because students would then learn that axiomatic treatment of a subject is usually the last stage in the creation of a mathematical subject. Moreover, there would be more training in mathematical creativity than can possibly come from studying the finished and highly polished product, and only filling in the gaps that textbook authors select as exercises."
Stone gives a concise characterisation of what a reconstructive approach to geometry may entail (Steiner (Ed), 1971:318):
"First of all, the student should be led on the basis of physical experience to geometrical propositions that seems to be valid in the physical world. He should then be shown that some propositions can be proved from others and he should be led to construct other proofs on his own. In the course of his experience with mathematical proofs, whether in algebra, arithmetic, or geometry, he should begin to see that several different proofs schemes are used over and over again and even that a limited number of these appears sufficient to build up all the proofs with which he has become familiar. From these observations it should be possible for the student to reach an understanding of the principles; first, that a proof consists in the repeated application of certain specified elementary proof schemes; second, that the logical structure of a set of propositions are provable from its various subsets. The problem of axiomatizing a set of propositions can then be defined as that of finding a `small' subset from which every member of the set on its negation, but not both, can be proved. The problem of axiomatizing geometry is that of finding a set of basic geometrical propositions, called the axioms, from which `all' others can be proved or disproved. The relevance of this logical structure to the `real' geometry of the physical world should also become plain: namely, that acceptance of the physical validity of the axioms entails a belief in the physical validity of the theorems proved from them."
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:
1. Informal geometry as it has been presented in South African schools till the end of Grade 9 (Std.7). (This can be viewed as being in correspondence with phase A of the 1923-report of the Mathematical Association).
2. Proof restricted to propositions which really require justification and/or explanation, and assuming without proof all propositions about which pupils have no doubts (This can be viewed as being in correspondence with phase B of the 1923-report).
3. Construction of formal, economical definitions for the different types of quadrilaterals, and logically deducing the other properties from the definitions as a first exercise in local axiomatizing.
4. Local axiomatizing regarding other groups of related propositions, for example, those related to intersecting and parallel lines.
5. Global axiomatizing (not included in the experimental course).
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.
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A I G T
1 - 3
5 - 6
7 - 8
9 - 12
13 - 14
16 - 21
24 - 25
27 - 28
29 - 31
Construction of equilateral triangles and equal line segments.
Side, angle, side congruency condition.
Base angles in isosceles triangles and converse.
Three sides congruency condition.
Bisection of line segments and construction of perpendiculars.
Angles on a straight line together are two right angles and converse.
When two straight lines cut, the directly opposite angles are equal.
Inequalities in triangles.
Construction of triangle with given sides.
Construction of equal sides.
More inequalities in triangles.
Two angles and a corresponding side congruency condition.
Equality of alternate or corresponding angles implies parallelism.
Further theorems on parallel lines.
Sum of the interior angles of a triangle and its exterior angle.
Theorems 2, 3 & 4:
Theorems 9 - 14:
Theorem 16; 18:
Theorems 21 - 24:
Angles at a point
All right angles are the same size. (This is a postulate in Euclid).
Euclid's theorems 13, 14 & 15.
Side, angle, side congruency condition.
Base angles of isosceles triangles.
Two angles and a common side congruency condition.
Converse of Theorem 6.
Inequalities in triangles.
Three sides congruency condition.
More inequalities in triangles (Euclid's theorems 25 and 17)
Two angles and a corresponding side congruency condition.
A perpendicular gives shortest distance from a point to a line. (Not in Euclid)
More congruency conditions.
Angles formed by parallel lines (Euclid's theorems 27 - 29).
Angles in triangles (Euclid's theorem 32).
1978 RSA SYLLABUS
Lines which meet & intersect.
Definition: When one line meets another line to form two equal angles, these angles are called right angles.
Axiom: All right angles are equal.
Theorem 1: When two lines intersect, than any two adjacent angles are together equal to two right angles.
Theorem 2: If two adjacent angles are together equal to two right angles, then the outer legs lie in a straight line.
Theorem 3: If two lines intersect, then any two directly opposite angles are equal.
Definition: Lines in the same plane which do not meet, are called parallel lines.
Axiom: (Playfair) Through a given point only one line parallel to a given line (not through the given point) can be drawn.
Theorem 5: Two lines are parallel if a transversal forms
a. equal alternate angles
b. equal corresponding angles
c. co-interior angles on same side of transversal which are together equal to two right angles.
Theorem 6: If a transversal cuts two parallel lines, then
a. alternate angles are equal
b. corresponding angles are equal
c. co-interior angles on same side of transversal are together equal to two right angles.
Definition: The sum of the angles around a point is 360° .
Axiom: If two lines intersect, than the sum of any pair of adjacent angles are equal to 180° , and conversely, if the sum of two adjacent angles are equal to 180° , then the outer legs lie in a straight line.
Theorem 1: If two lines intersect each other, then any two directly opposite angles are equal.
Definition: If two lines are intersected by a transversal and a pair of corresponding angles are equal, than the lines are parallel, and conversely, if a transversal intersects two parallel lines, then the pairs of corresponding angles are equal.
Theorem 2: Alternate angles.
Theorem 3: Co-interior angles on the same side of the transversal.