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The Van Hiele
theory
The Van Hiele theory originated in the respective
doctoral dissertations of Dina van Hiele-Geldof and her
husband Pierre van Hiele at the University of Utrecht,
Netherlands in 1957. Dina unfortunately died shortly after
the completion of her dissertation, and Pierre was the one
who developed and disseminated the theory further in later
publications.
While Pierre's dissertation mainly tried
to explain why pupils experienced problems in geometry
education (in this respect it was
explanatory and
descriptive), Dina's
dissertation was about a teaching experiment and in that
sense is more prescriptive
regarding the ordering of geometry content and learning
activities of pupils. The most obvious characteristic of the
theory is the distinction of five discrete thought levels in
respect to the development of pupils' understanding of
geometry. Four important characteristics of the theory are
summarised as follows by Usiskin (1982:4):
fixed order - The
order in which pupils progress through the thought levels
is invariant. In other words, a pupil cannot be at level
n without having passed through level
n-1.
adjacency - At each level
of thought that which was intrinsic in the preceding
level becomes extrinsic in the current level.
distinction - Each level
has its own linguistic symbols and own network of
relationships connecting those symbols.
separation - Two persons
who reason at different levels cannot understand each
other.
The main reason for the failure of the traditional
geometry curriculum was attributed by the Van Hieles to the
fact that the curriculum was presented at a higher level
than those of the pupils; in other words they could not
understand the teacher nor could the teacher understand why
they could not understand! Although the Van Hiele theory
distinguishes between five different levels of thought, we
shall here only focus on the first four levels as they are
the most pertinent ones for our secondary school geometry.
The general characteristics of each level can be described
as follows:
Level 1:
Recognition
Pupils visually recognize figures by their global
appearance. They recognize triangles, squars,
parallelograms, and so forth by their shape, but they do
not explicitly identify the properties of these
figures.
Level 2: Analysis
Pupils start analysing the properties of figures and
learn the appropriate technical terminology for
describing them, but they they do not interrelate figures
or properties of figures.
Level 3: Ordering
Pupils logically order the properties of figures by short
chains of deductions and understand the interelationships
between figures (eg. class inclusions).
Level 4: Deduction
Pupils start developing longer sequences of statements
and begin to understand the significance of deduction,
the role of axioms, theorems and proof.
The differences between the first three levels can be
summarised as shown in Table 2 in terms of the objects and
structure of thought at each level.
Table 2
By using task-based interviews, Burger & Shaughnessy
(1986) characterized pupils' thought levels at the first
four levels more fully as follows:
Level 1
(1) Often use irrelevant visual properties to identify
figures, to compare, to classify and to describe.
(2) Usually refer to visual prototypes of figures, and is
easily misled by the orientation of figures.
(3) An inability to think of an infinite variation of a
particular type of figure (eg. in terms of orientation
and shape).
(4) Inconsistent classifications of figures; for example,
using non-common or irrelevant properties to sort
figures.
(5) Incomplete descriptions (definitions) of figures by
viewing necessary (often visual) conditions as sufficient
conditions.
Level 2
(1) An explicit comparison of figures in terms of their
underlying properties.
(2) Avoidance of class inclusions between different
classes of figures, eg. squares and rectangles are
considered to be disjoint.
(3) Sorting of figures only in terms of one property, for
example, properties of sides, while other properties like
symmetries, angles and diagonals are ignored.
(4) Exhibit an uneconomical use of the properties of
figures to describe (define) them, instead of just using
sufficient properties.
(5) An explicit rejection of definitions supplied by
other people, eg. a teacher or textbook, in favour of
their own personal definitions.
(6) An empirical approach to the establishment of the
truth of a statement; eg. the use of observation and
measurement on the basis of several sketches.
Level 3
(1) The formulation of economically, correct definitions
for figures.
(2) An ability to transform incomplete definitions into
complete definitions and a more spontaneous acceptance
and use of definitions for new concepts.
(3) The acceptance of different equivalent defintions for
the same concept.
(4) The hierarchical classification of figures, eg.
quadrilaterals.
(5) The explicit use of the logical form "if ...
then ' in the formulation and handling of
conjectures, as well as the implicit use of logical rules
such as modus ponens .
(6) An uncertainty and lack of clarity regarding the
respective functions of axioms, definitions and
proof.
Level 4
(1) An understanding of the respective functions of
axioms, definitions and proof.
(2) Spontaneous conjecturing and self-initiated efforts
to deductively verify them.
Russian research on
geometry education
Geometry has always formed an extremely prominent part of
the Russian mathematics curriculum in the nineteenth and
twentieth centuries. This proud tradition was no doubt
influenced by (and instrumental in) the achievements of
several famous Russian geometers in the past two centuries.
Traditionally the Russian geometry curriculum consisted of
two phases, namely, an intuitive phase for
Grades 1 to 5 and a systematisation (deductive)
phase from Grade 6 (12/13 year old).
In the late sixties Russian (Soviet)
researchers undertook a comprehensive analysis of both the
intuitive and the systematisation phases in order to try and
find an answer to the disturbing question of why pupils who
were making good progress in other school subjects, showed
little progress in geometry. In their analysis, the Van
Hiele theory played a major part. For example, it was found
that that at the end of Grade 5 (before the resumption of
the systematisation phase which requires at least Level 3
understanding) only 10-15% of the pupils were at Level 2.
The main reason for this was the insufficient attention to
geometry in the primary school. For example, in the first
five years, pupils were expected to become acquainted, via
mainly Level 1 activities, with only about 12-15 geometrical
objects (and associated terminology). In contrast, it was
expected of pupils in the very first topic treated in the
first month of Grade 6 to become acquainted not only with
about 100 new objects and terminology, but it was also being
dealt with at Level 3 understanding. (Or alternatively, the
teacher had to try and introduce new content at 3 different
levels simultaneously). No wonder they described the period
between Grades 1 and 5 as a "prolonged period of
geometric inactivity "!
The Russians subsequently designed a very
successful experimental geometry curriculum based on the Van
Hiele theory. They found that an important factor was the
continuous sequencing and development of concepts from Grade
1. As reported in Wirszup (1976: 75-96), the average pupil
in Grade 8 of the experimental curriculum showed the same or
better geometric understanding than their Grade 11 and 12
counterparts in the old curriculum.
The primary & middle
school geometry curriculum
The parallels from the Russian experience to South Africa
are obvious. We still have a geometry curriculum heavily
loaded in the senior secondary school with formal geometry,
and with relatively little content done informally in the
primary school. (Eg. how much similarity or circle geometry
is done in the primary school?) In fact, it is well known
that on average, pupils' performance in matric (Grade 12)
geometry is far worse than in algebra. Why?
The Van Hiele theory supplies an important
explanation. For example, research by De Villiers &
Njisane (1987) has shown that about 45% of black pupils in
Grade 12 (Std 10) in KwaZulu had only mastered Level 2 or
lower, whereas the examination assumed mastery at Level 3
and beyond! Similar low Van Hiele levels among secondary
school pupils have been found by Malan (1986), Smith &
De Villiers (1990) and Govender (1995). In particular, the
transition from Level 1 to Level 2 poses specific problems
to second language learners, since it involves the
acquisition of the technical terminology by which the
properties of figures need to be described and explored.
This requires sufficient time which is not available in the
presently overloaded secondary curriculum.
It seems clear that no amount of effort
and fancy teaching methods at the secondary school will be
successful, unless we embark on a major revision of the
primary school geometry curriculum along Van Hiele lines.
The future of secondary school geometry thus ultimately
depends on primary school geometry!
In Japan for example pupils already start
off in Grade 1 with extended tangram, as well as other
planar and spatial, investigations (eg. see Nohda, 1992).
This is followed up continuously in following years so that
by Grade 5 (Std 3) they are already dealing formally with
the concepts of congruence and similarity, concepts which
are only introduced in Grades 8 and 9 (Stds 6 & 7) in
South Africa. No wonder that in international comparative
studies in recent years, Japanese school children have
consistently outperformed school children from other
countries.
Although the recent introduction of
tessellations in our primary schools is to be greatly
welcomed, many teachers and textbook authors do not appear
to understand its relevance in relation to the Van Hiele
theory. Although tessellations have great aesthetic
attraction due to their intriguing and artistically pleasing
patterns, the fundamental reason for introducing it in the
primary school is that it provides an intuitive visual
foundation (Van Hiele 1) for a variety of geometric content
which can later be treated more formally in a deductive
context.
For example, in a triangular tessellation
pattern such as shown in Figure 8, one could ask pupils the
following questions:
(1) identify and colour in parallel lines
(2) what can you say about angles A, B, C , D and E and
why?
(3) what can you say about angles A, 1, 2, 3 and 4 and
why?
Figure 8: Visualisation
In such an activity pupils will realize that angles A, B,
C, D and E are all equal since a halfturn of the grey
triangle around the midpoint of the side AB maps angle A
onto angle B, etc. In this way, pupils can be introduced for
the first time to the concept of "saws " or
"zig-zags " (alternate angles). Similarly,
pupils should realize that angles A, 1, 2, 3 and 4 are all
equal since a translation of the grey triangle in the
direction of angles 1, 2, 3 and 4 consecutively maps angle A
onto each of these angles. In this way, pupils can be
introduced for the first time to the concept of
"ladders " (corresponding angles). Pupils
should further be encouraged to find different saws and
ladders in the same and other tessellation patterns to
improve their visualisation ability.
Since each tile has to be identical and
can be made to fit onto each other exactly by means of
translations, rotations or reflections pupils can easily be
introduced to the concept of congruency. Pupils can also be
asked to look for different shapes in such tessellation
patterns, eg. parallelograms, trapezia and hexagons. They
could also be encouraged to look for larger figures with the
same shape , thus intuitively introducing them
to the concept of similarity (as shown in
Figure 8 by the shaded similar triangles and
parallelograms).
Tessellations also provide a suitable
context for the analysis of the properties of geometric
figures (Van Hiele 2), as well as their logical explanation
(Van Hiele 3). For example, after pupils have constructed a
triangular tessellation pattern as shown in Figure 9, one
could ask them questions like the following:
(1) What can you say about angles A and B in
relation to D and E? Why? What can you therefore conclude
from this?
(2) What can you say about angles F and G in relation to
angles H and I? Why? What can you therefore conclude from
this?
(3) What can you say about line segment JK in relation to
line segment LM? Why? What can you therefore conclude
from this?
Figure 9: Analysing
In the first case, pupils can again see that angle A =
angle D due to a saw being formed. Also angle B = angle E
due to a ladder. It is then easy for them to observe that
since the three angle lie on a straight line, that the sum
of the angles of triangle ABC must be equal to a straight
line. They can also observe that this is true at any vertex,
as well as for any size triangle or orientation, thus
enabling generalization. In the second case, the exterior
angle theorem is introduced and in the third case, the
midpoint theorem. Such analyses are clearly just a short
step away from the standard geometric explanations (proofs);
all they now need is some formalisation. In Figure 10 the
three levels are illustrated for the discovery and
explanation that the opposite angles of a parallelogram are
equal.
  
Figure 10: Three levels
Another important aspect of the Van Hiele theory is that
it emphasizes that informal activities at Levels 1 and 2
should provide appropriate "conceptual
substructures " for the formal activities at the next
level. I've often observed teachers and student teachers who
let pupils measure and add the angles of a triangle for them
to discover that they add up to 180 . From a Van Hiele
perspective this is totally inappropriate as it does not
provide a suitable conceptual substructure in which the
formal proof is implicitly embedded. In comparison, the
earlier described tessellation activity clearly provides
such a substructure. Similarly, the activity of measuring
the base angles of an isosceles triangle is conceptually
inappropriate, but folding it around its axis of symmetry
lays the foundation for a formal proof later. The same
applies to the investigation of the properties of the
quadrilaterals. For example, it is conceptually
inappropriate to measure the opposite angles of a
parallelogram to let pupils discover that they are equal. It
is far better to let them give the parallelogram a half-turn
to find that opposite angles (and sides) map onto each
other, as this generally applies to all parallelograms and
contains the conceptual seeds for a formal proof.
Recently I had a conversation with a
teacher who quickly dismissed a fellow teacher's
introduction to tessellations who first let his pupils pack
out little card board tiles. This teacher felt that it
produced untidy patterns, was ineffective and time
consuming, and that one should just start by providing
pupils with ready-made square or triangular grids and show
them how they can then easily draw neat tessellation
patterns (see Figure 11). Although such grids are a useful
and effective way of drawing neat patterns, it is
conceptually extremely important for pupils to first have
had prior experience of physically packing out tiles, ie.
rotating, translating, reflecting the tiles by hand. The
problem is that it is possible to draw tessellation patterns
on such grids without any clear understanding of the
underlying isometries which create them, which in turn are
conceptually important for analysing the geometric
properties embedded in the pattern.
Figure 11: Using grids
  
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