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de Villiers
M. (1996)
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Process versus product teaching in geometryThe distinction between "processes " and "product s" in mathematics education is a relatively old one. With a product is meant here the end-result of some mathematical activity which preceded it. As far back as 1978, the Syllabus Proposals of MASA regarding the South African Mathematics Project, stated: "The intrinsic value of mathematics is not only contained in the PRODUCTS of mathematical activity (i.e. polished concepts, definitions, structures and axiomatic systems, but also and especially in the PROCESSES of MATHEMATICAL ACTIVITY leading to such products, e.g. generalization, recognition of pattern, defining, axiomatising. The draft syllabi are intended to reflect an increased emphasis on genuine mathematical activity as opposed to the mere assimilation of the finished products of such activity. This emphasis is particularly reflected in the various sections on geometry. " - MASA (1978:3) Regrettably these good intentions, except for a few schools, were hardly implemented on a large scale in South African schools. Most teachers and textbook authors simply continued providing pupils with ready-made content that they merely had to assimilate and regurgitate in tests and exams. Traditional geometry education of this kind can be
compared to a cooking and bakery class where the teacher
only shows pupils cakes (or even worse, only pictures of
cakes) without showing them what goes into the cake and how
it is made. In addition, they're not even allowed to try
their own hand at baking!
Table 3 Due to limitations of space, we shall here mainly focus on the handling of definitions at Van Hiele Level 3. The direct teaching of geometry definitions with no emphasis on the underlying process of defining has often been criticised by mathematicians and mathematics educators alike. For example, already in 1908 Benchara Blandford wrote (quoted in Griffiths & Howson, 1974: 216-217): "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 memorized 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. " The well-known mathematician Hans Freudenthal (1973:417-418) also strongly criticized the traditional practice of the direct provision of geometry definitions as follows: "... most definitions are not preconceived but the finishing touch of the organizing activity. The child should not be deprived of this privilege ... Good geometry instruction can mean much - learning to organize a subject matter and learning what is organizing, learning to conceptualize and what is conceptualizing, learning to define and what is a definition. It means leading pupils to understand why some organization, some concept, some definition is better than another. Traditional instruction is different. Rather than giving the child the opportunity to organize spatial experiences, the subject matter is offered as a preorganized structure. All concepts, definitions, and deductions are preconceived by the teacher, who knows what is its use in every detail - or rather by the textbook author who has carefully built all his secrets into the structure ." From our preceding discussion of the Van Hiele theory it should be clear that understanding of formal definitions only develop at Level 3, and that the direct provision of formal definitions to pupils at lower levels would be doomed to failure. In fact, if we take the constructivist theory of learning seriously (namely that knowledge simply cannot be transferred directly from one person to another, and that meaningful knowledge needs to be actively (re)-constructed by the learner), we should even at Level 3 engage pupils in the activity of defining and allow them to choose their own definitions at each level. This implies allowing the following kinds of meaningful definitions at each level: Van Hiele 1 As can be seen from the two examples at Van Hiele Levels
1 & 2, pupils' spontaneous definitions would also tend
to be partitional , in other words, they would
not allow the inclusion of the squares among the rectangles
(by explicitly stating two long and two short sides). In
contrast, according to the Van Hiele theory, definitions at
Level 3 are typically hierarchical , which
means they allow for the inclusion of the squares among the
rectangles, and would not be understood by pupils at lower
levels. hierarchical: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Clearly in general, partitional definitions are longer since they have to include additional properties to ensure the exclusion of special cases. Another advantage of a hierarchical definition for a concept is that all theorems proved for that concept then automatically apply to its special cases. For example, if we prove that the diagonals of a parallelogram bisect each other, we can immediately conclude that it is also true for rectangles, rhombi and squares. If however, we classified and defined them partitionally, we would have to prove separately in each case, for parallelograms, rectangles, rhombi and squares, that their diagonals bisect each other. Clearly this is very uneconomical. It seems clear that unless the role and function of a hierarchical classification is meaningfully discussed in class as described in De Villiers (1994), many pupils will have difficulty in understanding why their intuitive, partitional definitions are not used. The USEME experimentIs it possible to devise teaching stategies for the teaching of the processes of defining and axiomatising at Van Hiele levels 3 and 4? This in fact was the focus of the University of Stellenbosch Experiment with Mathematics Education (USEME) conducted with a control group in 1977 and an experimental group in 1978 (see Human & Nel et al, 1989a). The experiment was aimed at the Grade 10 (Std 8) level and involved 19 schools in the Cape Province. Whereas the traditional approach focusses overridingly on developing the ability of making deductive proofs (especially for riders), the experimental approach was aimed mainly at: - developing the ability to construct formal, economical definitions for geometrical concepts The following is an example of one of the first exercises in defining used in the experimental approach (see Human & Nel et al, 1989b:21).
The spontaneous tendency of almost all the pupils in (c) was to make a list of all the properties discovered in (a); thus giving a correct, but uneconomical description (definition) of the rhombi (thus suggesting Level 2 understanding). This led to the next two exercises which was intended to lead them to shorten their descriptions (definitions), for example:
Clearly here pupils were led to shorten their
descriptions (definitions) of rhombi by leaving out some of
its properties. For example, in 3(a) pupils found that one
does not need to use all the properties to construct a
rhombus. One could for example obtain one by constructing
all sides equal. In (b) and (c) pupils typically came up
with different shorter versions, some of which were
incomplete (particularly if they're encouraged
to make it as short as possible by promising a prize!), for
example: "A rhombus is a quadrilateral with
perpendicular diagonals ". This provided opportunity
to provide a counter-example and a discussion of the need to
contain enough (sufficient) information in one's
descriptions (definitions) to ensure that somebody else
knows exactly what figure one is talking about. (1) A rhombus is a quadrilateral with all sides equal. 
Figure 12: Construction & measurement
Psychologically, constructions like these are extremely important for the transition from Level 2 to Level 3. It helps to develop an understanding of the difference between a premisse and conclusion and their causal relationship; in other words, of the logical structure of an "if-then " statement. Logically each of the above statements can be rewritten in this form. For example, the last statement could be rewritten as: "If a quadrilateral has both pairs of opposite sides parallel and one pair of adjacent sides equal, then it is a rhombus (ie. has all sides equal, perpendicular bisecting diagonals, etc)". Smith (1940) reported marked improvement in pupils' understanding of "if-then " statements by letting them make constructions to evaluate geometric statements as follows: "Pupils saw that when they did certain things in making a figure, certain other things resulted. They learned to feel the difference in category between the relationships they put into a figure - the things over which they had control - and the relationships which resultedwithout any action on their part. Finally the difference in these two categories was associated with the difference between the given conditions and conclusion, between the if-part and the then-part of a sentence. " After some experimental exploration of different
alternative definitions for the rhombi as described above,
the pupils were then led into a deductive phase where
starting from one definition they had to logically check
whether all the other properties could be derived from it
(as theorems). The same exercises were then repeated for the
parallelograms. Eventually, it was explained to pupils that
it would be confusing if everyone used different definitions
for the rhombi and parallelograms, and it was agreed to
henceforth use one definition only for each concept. (Note
that the role and function of a hierarchical classification
for the quadrilaterals was not adequately addressed at the
time of the USEME experiment, and was one of the reasons for
the subsequent study reported in De Villiers (1994)).
In attempting to answer questions 1, 2 and 3, pupils inevitably argued circularly. The following is an example:
Figure 13: A circular argument
These series of proofs can be schematically represented
as shown in Figure 13 and clearly illustrate the underlying
circular argument. The problem is that no
matter how much they try, they inevitably land up with some
kind of circularity. Although many pupils did not at first
recognize the problem, some subsequent exercises alerted
them to the underlying problem and the realization that it
is impossible to prove all mathematical statements or
properties of mathematical objects without obtaining a
circular argument. They then realized that one had to accept
one of these properties as a statement without
proof (ie. as a definition or axiom) to avoid a
circularity.
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(corresponding
angles,
)
(directly
opposite angles)
(co-interior
angles, 
)
(QP
extended forms straight line)
(APB
is straight line)
(alternate
angles, 
)
