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FROM GEOMETRICAL FIGURES TO DEFINITIONAL RIGOR:
TEACHERS’ ANALYSIS OF TEACHING UNITS MEDIATED THROUGH
VAN HIELE'S THEORY
LUCILLA CANNIZZARO AND MARTA MENGHINI
DIPARTIMENTO DI MATEMATICA
UNIVERSITA' DI ROMA "LA SAPIENZA" Piazzale Aldo Moro 5, 00185 Roma, Italy
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From Geometrical Figures to Definitional Rigor:
Teachers’ analysis of teaching units mediated through van Hiele's theory
Abstract: The present work deals with a curriculum design used in 8 parallel classes of
high school (with students aged 14-15) by means of worksheets dealing with the properties
of the principal plane geometric figures.
Van Hiele's theory was used to facilitate teachers' discussion of pupils' performances on
agreed worksheets, in order to enhance their didactical awareness. Our hypothesis is that
van Hiele’s theory is a proper means to provide encouragement and support to teachers for
rethinking their teaching goals and practices. This environment enables teachers to manage
their pupils' transition from the perception of a figure to its definition. In this paper we will
recount the teachers’ discussion as it unfolded in a school-university workgroup. The
analysis of pupils’ work on worksheets interweaves with an analysis of the teaching actions
that foster the process of understanding definitions.
Executive Summary: The present work issues out of a curriculum design for the first year
of high school (age 14), carried out by members of a school-university workgroup with a
long tradition of action research. The origin of the curricular project was the
acknowledgement that pupils entering secondary school, even if they know names and
shapes of many geometric figures, are not familiar with their properties and are not always
able to point out specific differences expressed in the definitions (Cannizzaro & Menghini,
2001). The teachers needed to acknowledge what the pupils already knew and to turn it into
an understanding of definitions. These needs emerge when passing from the intuitive
geometry of middle school to the so-called "rational geometry" of high school, namely to
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the logical and theoretical organization of geometry. The two levels and the problems
related with the passage from one to the other are not specifically an Italian characteristic.
They are described, for instance, by Clements & Battista (1992) with reference to the
situation in the U.S.A.. Cannizzaro & Menghini (2004) shared van Hiele’s theory of levels
with a small group of teachers, who were to change from middle school to high school
teaching, in order to explicit the difference in methods, steps and aims concerning the
teaching of geometrical figures at the two different levels.
The workgroup of the curricular project described here is composed of two researchers (the
authors of the present work) and by experienced high school teachers. From time to time
some of the teachers (in this case Carla and Linda) are in charge for specific curricular
proposals, which are then implemented by other members of the group (in this case 6 more
teachers).
Carla and Linda began to develop worksheets on geometrical figures (De Santis & Percario,
1997); in the following we refer to this work as version 0. We noticed that the stages
singled out by the two teachers seemed to be following the "levels of thought" that van
Hiele proposed as the basis of his work on geometrical understanding.
After the first implementation of version 0 of the worksheets, our contribution was to give
theoretical support to Carla and Linda. We used van Hiele's theory to encourage them to
reflect on their own practice, looking for elements underpinning their expertise. They
recognized in van Hiele's theory a framework corresponding to their purpose, i.e. to move
gradually from the sphere of visualization and description, or recognition of properties, to
the sphere of refinement of language and definitions.
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Carla and Linda considered it appropriate to introduce the theory of levels to all the other
teachers in the group. We took the first presentation by van Hiele & van Hiele-Geldof
(1958) as a basis, even if it referred to middle schools, and asked the other teachers in the
group to test their colleagues' proposal carefully and to reflect together on teaching practice
observation.
It was decided to modify the worksheets with the collaboration of the whole group, and to
reflect further on the way to use them. Nevertheless, teachers were convinced that building
up a precise sequence of exercises aimed at the acquisition of the levels (as in Usiskin,
1982) was not as relevant as the way in which exercises are presented and discussed in the
classroom. Teachers also agreed not to focus on the acquisition of different levels by the
individual pupil, but rather on class evolution.
On our part, the focus was on teachers, on how they address the issue of hierarchical versus
partition definition and on how they improve their conceptual and didactical knowledge of
this subject.
1. Conceptual Framework For Teacher Enhancement
As stressed by Jacobson & Lehrer (2000), teachers' knowledge about typical milestones and
trajectories of children's reasoning has a positive influence on learning. Our aim was the
improvement of teacher’s conceptual and didactical knowledge on definitions in order to
facilitate the cognitive growth of his/her pupils. Our interaction aimed at the development
of a “reflective practice” in a narrative perspective (Love, 1994) and of an open-
mindedness factor (Krainer, 1994) towards the thinking of others and towards changes in
one’s own knowledge, believes and didactical practices.
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1.1. Didactical Framework: from Perception to Definitions
The steps followed by a pupil in passing from the perception of geometrical figures,
through their description and definition and on to the higher levels of Euclidean deduction
represent the core of the work of van Hiele (1986) and van Hiele & van Hiele-Geldof
(1958). The interplay between the perception of an object and its written description is
discussed by Monaghan (2000). Roots of this last aspect are in Skemp's (1979) analysis of
the interplay among mental models, concepts and conceptual structures; in the terms of the
interplay between concept image and concept definition in Tall & Vinner (1981); and in the
interaction between figural and conceptual aspects of geometrical reasoning in Mariotti &
Fischbein (1997).
To stress this aspect with teachers we mainly used van Hiele’s idea (van Hiele & van Hiele-
Geldof’s, 1958; van Hiele, 1986) of the passage from the symbol (a figure with the range of
its properties) to the significant signal (which of the properties remind us of the entire
symbol) and from the signal to the definition (the property sufficient to distinguish a
figure), because Carla and Linda considered these passages very crucial.
Van Hiele's symbol represents the level of perception or visualization (level 1), into which
the pupils condense all properties of a geometrical figure with which they have had
experience. For example most internal properties of a rhombus are condensed in its symbol:
equal sides, parallel sides, equal opposite angles, diagonals that halve each other under right
angles. The figures have the character of images, i.e. a symbolic character. Van Hiele's
signal represents the level of description or analysis and exploration (level 2), when
perceptions are translated into descriptions, but without involving specific linguistic
abilities. One property emerges (the significant signal) that will be significant in the
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description of the figure (equal sides, in the case of the rhombus). From this signal the pupil
is able to anticipate other properties. The images become less important with respect to
geometric relations. The successive level (level 3) is that of definition, or descriptive
definition. One begins to observe the various relations from a logical point of view, to
order them: implication, and therefore definition, takes on significance within the realm of
geometrical relations. This, according to van Hiele & van Hiele-Geldof (1958, p. 79), is the
essence of geometry. We will not refer to van Hiele's further levels, which are concerned
with proof.
We used, in order to underline the different cognitive aspects characterizing van Hiele’s
levels, also terms suggested by the reading of authors other than van Hiele. The symbol, for
instance, can be intended as a visual image, as a “prototype”, or as a diagram with
markings. In any case a symbol is more than a vague shape; it needs to be built up through
specific didactical actions.
In our experience, it is not possible to find one single way in which teachers stimulate the
passage to the successive levels 2 and 3. These are achieved by explicit recognition of the
properties of a figure: specializing or generalizing figures (de Villiers, 1994), operating on
the figures in order to learn the possible modifications (Duval, 1988) or using van Hiele’s
five phases (Clements & Battista, 1992). Teachers observed that these different ways were
already present in version 0 of the worksheets, for instance when the pupils were asked to
complete sentences, or to operate with meccano, or to cut and fold figures in order to
distinguish the properties of two different sets of figures.
1.2. Mathematical Framework: Exclusive and Inclusive Definitions
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After the introduction of van Hiele's theory, we offered the teachers a short historical
working session about the Definitions of Euclid. The starting points were Euclid's
definitions XX and XXII in Book I of the Elements (Table 1).
Table 1
We noticed that Euclid seems to follow an aesthetic sense, first defining regular figures as
an equilateral triangle and a square, and then going on to figures that are less regular.
These definitions of triangles and quadrilaterals are not used later on in Book I. When it
becomes necessary to prove theorems, Euclid changes the definitions of the geometric
figures simply by changing some terminology: so the oblong becomes a rectangle. This
term denotes a quadrilateral with four right angles; the rhomboid becomes a parallelogram,
i.e. a quadrilateral inscribed in two pairs of parallel lines.
So, in order to deduce, Euclid passes from exclusive to inclusive definitions.
Why then does Euclid start with exclusive definitions? We would say in order to "recall"
the various shapes to the reader. Certainly, these kinds of definitions are "good definitions"
from a perceptive point of view and maybe they are "popular" among pupils for this reason.
Some middle school teachers adapt themselves to this instance using Euclid's exclusive
classification of geometric figures: the aim of middle school geometry is to observe,
recognize and collect various kinds of figures, much more than to define and to prove.
Entering high school, pupils are forced in a direction that is useful for the further work of
proving, trying to change her/his perceptions (as we will see later in the experience with the
meccano). The language of sets and the concept of set inclusion play a crucial role in this
change, but the set-theoretic standpoint is not a natural internal construction by the pupils.
For instance, from a cognitive point of view, they have to deal with the "inverse
relationship" between the number of properties of a figure and its generality.
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The authority of Euclid was very useful to push teachers a step forward into van Hiele's
theory, supporting that the point was not that of avoiding the "mistake" of exclusive
definitions, but that of leading pupils slowly towards inclusive definitions.
Nevertheless we will see that teachers will often try to anticipate the set classification using
Venn diagrams.
We also discussed the possible definitions of a trapezium. For the most two definitions can
be found in Italian textbooks: a trapezium is a quadrilateral with two parallel sides (thus
including the set of parallelograms in the set of trapezia) or, more frequently, a trapezium is
a quadrilateral with two parallel sides and two non-parallel sides (thus excluding not only
parallelograms, but even rectangles). But the same books define an isosceles trapezium as a
trapezium with two equal sides; so, in the first case, the parallelogram would be an
isosceles trapezium, even without having an axis of symmetry. The teachers in any case
declare that they prefer the inclusive definition, even though, in the subsequent discussions,
they often seemed to have in mind the exclusive definition.
We believe that the point of a correct definition, at the high school level, is not to illustrate
what an object is, but to explain "what a definition is"; and this cannot be given before the
object is known, with all of its properties. The definition expresses the properties sufficient
to distinguish a figure, i.e. the "minimal elements" with which to describe an object already
known. Here is the difference between descriptive definition (a posteriori), which should
be the norm in elementary mathematics, and prescriptive (or normative or constructive or a
priori) definition, which is more apt in advanced mathematical thinking (de Villiers, 1998,
Pinto & Tall, 1999). Any definition is felt as prescriptive, if it is "given" and does not
originate in the pupil's experience. As Freudenthal (1973, p. 417) wrote with reference to
van Hiele's work "There may be uncertainties, for example in knowing whether a square
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belongs to the rhombi, or a rhombus to the parallelograms. The teacher can impose
definitions to decide such controversies, but if he does so, he is degrading mathematics to
something like spelling that is governed by arbitrary rules".
2. Methods and Procedures
2.1. In the classroom
The activity involved 8 classes of the first year of different secondary schools (14-15 years
of age) and was introduced when the school year had already started. Some other activities
had already been done (mainly algebra, and some work in the Cartesian plane). The time
required for the whole project was about 10-12 hours, spread over more than one month.
As in version 0, the new project consists of a series of 6 worksheets, for a total of about 30
exercises, which highlight the properties of triangles and quadrilaterals. Requested
constructions or geometric manipulations are alternated with observations of figures
already drawn. The practical work is notably reduced with respect to that foreseen by van
Hiele & van Hiele-Geldof (1958) for the lower scholastic levels, but it is not totally
abandoned: one tries to work in such a way that transmission is not based only on language.
An operative aspect remains ("draw", "observe", …).
In the new format, pupils had to work on the worksheets without some preventive
intervention aside from teachers. After the work on each file, a phase of discussion in class
was to follow whose purpose was to record and compare the answers. In this phase teachers
were always to intervene following and not preceding the development suggested by the
worksheets. Some of the teachers who had the chance to use a Dynamic Geometry
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Software, such as Cabri Géomètre, in their classroom decided autonomously to use it as a
support to the discussion.
2.2 In the working group
Our report is not focused on pupils’ learning processes but on how teachers observe, act
and reflect on them. It starts from the moment in which the teachers began to re-work
version 0 and to reflect on teaching procedures and results. For each sheet or couple of
sheets we had an elaboration phase, in which the previous version was modified and some
teaching modes were decided, and a further phase about reflections on results after the
worksheet had been used in the classes. When describing this latter phase in the report we
will put into evidence teachers' Observations of pupils' performances and analysis of
worksheets in the working group (these observations will be preceded by the label O), and
teachers' Actions in the classroom (the description of these activities will be preceded by
the label A), as reported by the teachers themselves.
We devoted two working sessions to each sheet, corresponding to the elaboration phase
and to the reflections on results. In the latter teachers brought the worksheets filled in by
pupils, and a note summarizing the main quantitative data, completed by their qualitative
judgment referring to classroom discussion. We put together the quantitative data and
calculated the percentages concerning the totality of pupils involved in the project. These
were used as a support to the discussion in order to have a more concrete basis. But the
empirical data for this research consisted of our field notes and written records of the
discussions with and among teachers during the working sessions. The discussions include
teachers’ narratives of their work in the classroom.
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Of course, the observations of the teachers may be influenced by their expectations. We
must remember that in their own classes teachers have the role of teacher and of observer
contemporarily, as pointed out by Malara et al. (1996). We intend here to stress the
importance of an engagement of the teacher in designing and conducting a step of their
project, mediating theory and practice.
3. Results
The exercises that are reported in the following as a support to teachers’ discussion, are
those of the new version. Sometimes the comparison with version 0 will help in
understanding the change in the teachers’ point of view. We collected the results and
comments in five episodes. Each episode corresponds to the discussion concerning a
particular conceptual step on the way towards level 3 (Table 2). In the following we
describe the episodes 1, 3 and 4, giving only a short outline of the episodes 2 and 5.
Table 2
3.1. Episode 1: Worksheets 1 and 2 - Triangles. "Rough" Constructions and
Verbalizations.
Elaboration. The first exercise of Version 0 of this worksheet had recalled the definition of
axial symmetry and required the construction of the image of several triangles with respect
to a vertical line.
At first, in the discussion of the new elaboration of the worksheets the teachers once again
tended to "force in" the use of the definition of symmetry, but then they realized that
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symmetry is not the aim of this worksheet but merely a way to start observing triangles1. So
they decided to stimulate the pupil's intuition by drawing on the worksheets two
corresponding triangles and writing that these were symmetrical with respect to the given
line and suggesting folding the paper along the line. Now exercise 1.1 asks for two other
triangles to be drawn and the procedure used to be described.
Together with the teachers we decided to pay great attention to the way pupils are using and
verbalizing the characteristics and properties of triangles at a perceptive or technical level.
Exercise 1.2, after recalling the terms isosceles and equilateral for triangles, poses the
question of what a triangle that is not isosceles is, and asks whether an equilateral triangle
is isosceles or not.
Table 3
Vanna suggested using set representation to give the solution to the problem of defining the
isosceles and equilateral triangles. At first the other teachers agreed, but Linda warned that
this would mean introducing a hierarchy among triangles (de Villiers, 1994) and shifting to
van Hiele’s level 3. We supported her observation by underlining that our aim at this point
was to stimulate a comparison, which may or may not be linked with linguistic experiences.
In exercise 2.1 an isosceles triangle is drawn with an oblique base. Through suggesting
concrete operations of folding and cutting with scissors and through observations on the
figures, pupils are requested to recall some terminology and to observe the equality of
angles at the base of an isosceles triangle by completing sentences.
Table 4
1 When the worksheets were used geometrical transformations had not yet been dealt with in class and so reference was only made to more or less intuitive knowledge in the middle school.
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Exercises 2.3, 2.4 and 2.5 are intended to point out the distinctive characteristics of
isosceles and equilateral triangles in comparison with the other kinds of triangles, referring
to properties of symmetry, of altitudes, medians and bisectors. Such, again, stimulating the
passage to level 2.
Table 5
Teachers anticipated pupils’ difficulties in working with altitudes, and discussed how to
deal with them during the discussion phase in class after the worksheet.
Worksheet 1 and 2 - Reflections on results. The teachers were astonished when they
realized that the answer to the first exercise of worksheet 1 was different for pupils in
different classes. Nearly all Bruna’s and Linda’s pupils folded the paper and then identified
the symmetric points of the vertices of the triangle by marking the sheet with their pencils.
On the other hand, Vanna's, Ileana's and Pina's pupils used the plane construction by tracing
the perpendicular line and transferring the measures. This made Bruna realize that she often
uses mental images in her teaching and that she doubted about the ability of her pupils (first
year of a technical school) in the use of a ruler and a compass. Vanna instead admitted that
she did not encourage her pupils (first year of a high school for sciences) to fold the paper
since she thought this activity was not for pupils of this age. In the other two classes there
was a bigger variety, not influenced by the actual teacher and connected to pupils’ previous
experience in the middle school: some pupils folded, others traced an arc that would show
the rotation of the figure "in space", others made the construction, even if they could not
describe it rigorously. This variety provides evidence of the interplay between curriculum
as designed and curriculum as "wrapped" around the on-going action and interaction of
pupils and teacher.
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(O) In some pupils (less than one third) the constructive aspect prevails, and the global
consideration of the figure is missing. The construction is done point by point, but this is
sometimes a procedure of little rigor: pupils made use of some words (like perpendicular,
distance, folding), but these did not correspond to a mental or concrete action really being
undertaken.
We remark that, in these pupils, the small operational steps predominate over the global
aspects in accordance with the development of processes through repetition and
interiorisation (Dubinsky, 1991), and with a level of pre-procedure (DeMarois & Tall,
1999).
(O) Other pupils used reflection on a line at a perceptive level as a "visual structure".
This observation confirms, for the on-going work too, that visual meaning can occur
without knowing of the existence of a plane construction done using drawing instruments,
i.e. visual aspects are separate and can occur at a primitive level before process-object
construction.
(O) Again in exercise 1.2 we can see some differences in behavior. For the question
- A triangle is not isosceles if…
we had three kinds of answers. A syntactic answer, where the pupil would deny the
definition "… if it does not have at least two equal sides". A semantic answer, where the
pupil would explain the meaning of denying, "If all the sides are different". In the third
kind of answer, the pupil would explicitly include the case of the equilateral triangle: "if all
the sides are different or if all the sides are equal".
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During the discussion we assumed that in middle school the isosceles triangle could have
been defined in a different way, as Euclid defined it. The teachers at first referred to that
exclusive definition as to a wrong definition; then, re-thinking at the Euclidean definitions,
the word “wrong” became a code internal to the group with which they teased each other.
In reality the discussion made them converge on a more respectful idea of the situation, one
less dependent on didactic tradition and more tolerant towards their pupils and their
colleagues in the middle schools.
Here teachers already realized that the language of the teacher and of the curriculum is
formulated on a convention, which is based on the "inclusive" relationships. With this kind
of convention language difficulties arise for the pupils. For many pupils 'non-isosceles' is
either not clear or means equilateral.
(A) Pupils like the subsequent discussions in class, which are based on the use of the terms
'at least', 'only', etc.: they are led towards paying greater attention. Carla reported that her
pupils asked "why at least?", but after the discussion it was as if something had been
dislodged and as if they were able to 'look inside' the terminology for the first time.
(O) In exercise 2.1, teachers affirmed that the majority of the pupils (more than 80%) do
not realize that the isosceles triangle has been defined and that there are two possible
definitions (distinctive signals) for it. One teacher remarked that in any case the word
"definition", which was not used in the worksheets, had been used in the discussion phase
both by the pupils and by the teachers, but it had the same extension of meaning as
"description". There is a sort of hybrid zone in which the mathematical language has not
emerged yet, and only the language as a means for communication remains.
(O) From the analysis of pupils' work on exercises 2.3, 2.4 and 2.5 (asking them to trace the
3 altitudes of a triangle and to count the altitudes and the medians of an equilateral one and,
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respectively, those of an isosceles triangle), teachers could note the difficulties they were
expecting. More than 50% of the pupils had some form of problem.
(A) In such cases the teacher needed to prompt pupils in various ways: the teachers in our
group have used different methods; strings that hang from the vertex of a cardboard triangle
made to lean on its various sides; work on right-angled triangles; the showing of an
animation prepared by the teacher using Cabri Géomètre. These activities are designed to
allow the pupils to fold back (Pirie et al., 1994) to a conceptual and eventually a definitional
activity.
The teachers pointed out that, at this age, when working with altitudes it is better to
underline the graphic relationship between two straight lines, rather than using the images
given by a plumb line and gravity.
3.2. Episode 2: Worksheet 3 - Rectangles, Squares and Rhombi. Comparing properties.
The worksheet on quadrilaterals must push pupils to think of the characteristic properties
specific to the various figures; through it the teachers want to underline the differences
between squares, rhombi, rectangles, parallelograms and trapezia with respect to the
properties of their sides, their angles and their diagonals. The purpose is to compare a
symbol with its “verbal description”. Terms like “always”, “any”, “all” and “at least”
qualify the meaning of the word “is” (de Villiers, 1994) and allow relations to be translated
into a mathematical language which is not yet at level 3.
Exercise 3.6 was the first one intended to gradually lead to the set classification of
quadrilaterals. The essence of this passage is the movements made by quadrilaterals with
meccano (Table 6).
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Table 6
Teachers who did not have a meccano used the Cabri software to simulate the meccano
experience. One of the teachers, Mariolina, had also prepared an additional work sheet on
the set classification of quadrilaterals, and this would have helped, in her opinion, to
characterize quadrilaterals in terms of necessary and sufficient conditions. The other
teachers then diminished her enthusiasm, because this additional activity would prevent
useful critical moments.
Reflecting on the results, teachers observed that a "simple" question contained in worksheet
3 (“is a square always a rectangle?”) is not really easy for pupils: very few pupils realize at
this point that the square is a rectangle. A further exercise, starting from the observation
that the diagonals are axes of symmetry for the square, guides pupils gradually to recognize
that they are perpendicular to each other. Teachers knew that only few pupils would
understand the proof. However, in line with van Hiele & van Hiele-Geldof (1958, p. 71),
they accepted that pupils simply added this experience to the set of their perceptions. All
the concepts used in the proof are clear to the pupils, but for most of them, the consequent
logical organization of the concepts is still difficult.
3.3. Episode 3: Worksheet 4 - Rectangles, Parallelograms and Trapezia. Approaching
Definitions
Elaboration. The worksheet again proposes the meccano experience to point out the
different characteristics of the rectangle and the parallelogram (Table 7).
Table 7
The teachers discussed how much they should refer to the images in the first questions of
exercise 4.1. Then they introduced the word "generally" and the third option "it depends".
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The purpose was to elicit the classes’ consolidated ideas about the symbols (in the first
series of questions) and about the significant signals (in the second series) which describe
the various figures. The "it depends" which was inserted among the answers should let
pupils realize that the diagram on the paper is not the only representation possible and
should remind them of the experience with the meccano; otherwise the questions about the
angles and about the diagonals would be misleading.
Again in this case, some of the teachers decided to simulate the meccano and Bruna
prepared a Cabri macro that would make it possible to obtain the 'limit positions'
dynamically.
For the trapezium the symbol is recalled with a figure, and several assertions are proposed
in Worksheet 4 through exercises 3 and 4 (Table 8).
Table 8
The fourth exercise requires attention to be paid to the diagonals again. It is necessary to
compare the trapezium and the parallelogram and to decide whether or not the diagonals
cut each other in half.
These worksheets led to an essential point. The first phase of guided investigation has been
completed, the pupils have a lot of information and are beginning to organize themselves.
According to Freudenthal (1973) deduction begins at this point: the class is moving toward
level 3.
At this point the teacher can encourage this work by supplying some global considerations.
It is important that the latter do not contain anything new; they must be simply a synthesis
of what the pupils already know. Actually at this point (sometimes before the worksheet on
the trapezium) the teachers think that it is appropriate to present the set schemata for the
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inclusion of the set of squares in the set of rectangles (or rhombi), and of these in the set of
parallelograms.
Worksheet 4 - Reflections on results. (O) During the phase of class discussion the teachers
noticed that some difficulties arise when the quadrilateral gains more properties.
So for some pupils the rectangle is a parallelogram, but the square is not (presumably the
use of the meccano with unequal sides may be responsible for this).
Mariolina suggests that this difficulty could be overcome by using different concrete
artifacts, such as the "elastic bandage" suggested by Castelnuovo & Barra (2000). Cabri too
could simulate such an artifact.
Here the pupils should distance themselves from the image and think more generally.
(A) Only Carla and Linda used the meccano in class. Mariolina, Vanna and later Bruna
simulated it with Cabri. Pina and Ileana admitted they never used it, but just described it,
believing the experience unnecessary. The pupils who did not use the meccano or its
simulation did not consider the 'it depends' answer in exercise 4.1 seriously. The lack of use
of the “it depends” was so generalized that Pina and Ileana did not underline the mistake
during the discussion, but they proposed and discussed a different formulation of the
question with the pupils: "Is the parallelogram "always" a rectangle? " or, "Can the
parallelogram be a rectangle?"
In this phase Bruna started to use Venn diagrams to represent quadrilaterals. But she
realized that it was necessary to use meccano again, and decided to simulate it with Cabri.
Only after that Bruna asked for the sets of the rhombi, of the parallelograms and of the
trapezia to be inserted in a picture representing the set of quadrilaterals with the subsets of
squares and rectangles.
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(O) Even if the pupils were considering the square in the intersection of the set of
rectangles and the set of rhombi, they did not always understand that squares have the
properties of both figures. Despite having answered correctly, one female pupil said: "The
square is less complex, hence it has fewer properties!"
Here again we notice the difference between the pupil's and the teacher’s language.
(A) Bruna engaged in a discussion with her pupils about the terms "complex" and "simple".
The work on Venn diagrams required a lot of thinking and discussions in all the classes;
pupils and teachers needed to shift from the worksheet to the blackboard representation
very often.
(O) As far as the exercise on the trapezium is concerned, it is evident that many pupils have
in mind an isosceles trapezium, while others are thinking of a right-angled trapezium. In
this case the symbol (the standard perception) has more properties than the figure itself.
Pupils have the tendency to favor properties that produce figures that are as regular as
possible.
The answers to the questions in exercise 4.3 were probably given from a linguistic-logical
point of view rather than from a perceptive one: about 65% said that the parallelogram is a
trapezium. There will not be a definite answer to that question. Teachers agreed that it is
only important that the pupils compare their different symbols.
3.4. Episode 4: Worksheet 5 - Quadrilaterals. The Significant Signal: Inclusive
Definitions
Elaboration. Worksheet 5 consists of two exercises intended to be a structured synthesis on
quadrilaterals. In Exercise 5.1 pupils are requested to complete sentences in a first column,
ticking the appropriate box of a table. For instance, ticking an “X” in the first box of the
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first line is to be read as "for a quadrilateral, it is enough to have a pair of parallel sides to
be a parallelogram" (which is of course wrong). The table (Table 9) is shown below, filled
with the percentages of right and wrong answers given by the pupils.
Teachers observed that in the previous experience with version 0 the terms "necessary" and
"sufficient" had not seemed very difficult to pupils. These terms were interpreted smoothly,
in a natural way, taking on only a communicative function. But later on difficulties arose,
particularly with the term "sufficient". So teachers decided to leave the term "necessary"
and to substitute the term "sufficient" with "it is enough to". In Exercise 5.2 a summary of
questions on several properties (parallel opposite sides, equal angles, equal diagonals, etc.)
is proposed. Pupils must only check if a certain figure has or does not have a selected
property. For instance, ticking an “X” in the first box of the first line is to be read as "a
parallelogram is a quadrilateral with two pairs of parallel sides" (which is right). The aim of
exercises 5.1 and 5.2 is to see if pupils shifted from symbols to signals, and to introduce
definitions through the subsequent discussion.
Worksheet 5 - Reflections on results. Pupils seem to have acquired the "necessary"
properties (they have only to check that the property is in the "list" of properties of the
known symbol), except, for some pupils, in the questions concerning the equality of the
diagonals.
Pupils show more difficulties in handling the "sufficient" conditions: they meet a proof-
situation/context openly, and the springboard for a definition-situation/context.
We filled in the tables of exercises 5.1 and 5.2 (Tables 9 and 10) with the percentages of
pupils who gave the right answer (R) or the wrong answer (W). Of course pupils
sometimes omitted the answer, or gave more than one answer even in lines that require
21
only one correct answer. The discussion went on and made reference to these general
results and to different behavior in the classes.
Table 9
Table 10
It was necessary to point out again that we could not accept the answer "trapezium" in the
first line of the first exercise (inclusive definition of a trapezium) and in the second line of
exercise 5.2 (exclusive definition of a trapezium) as simultaneously correct, as all the
teachers had done in spite of what was said in the previous meeting. This lack of
consistency has to be resolved by choosing appropriate definitions (see section 1).
Teachers realized that they did not reflect enough on this subject. They proposed (for the
future) to remove the last column relating to the trapezium, and for the time being to
discuss the question in the classroom again.
Four teachers realized that they never talked about the kite, and the sheets refer to the angle
between diagonals only in the case of the square and of the rectangle. This is probably why
their pupils completed the last line of exercise 5.1 with a rhombus; hence, the teachers
decided to consider this a more acceptable answer. Similarly, the high number of wrong
answers to question five of exercise 5.2 led us to notice that in the worksheets we had
failed to include an exercise on the length of the diagonals of the parallelogram.
Mariolina noticed that she had not considered the answer "parallelogram" in row 5 of
exercise 5.1 (equal diagonals) as a mistake, and she did not know the reason for this. The
question is objectively difficult, since there is no correct answer to it, as in line 7. The
teachers underlined that at this stage they had begun to speak to the pupils in language
appropriate to the third level. It is as if this worksheet supplied a sort of impulse for
22
pushing the teachers and the pupils into this new level. Some teachers affirmed they did not
change the language in the classroom in a conscious way, but this worksheet requires the
passage to a mathematical language. The teachers' tolerance of a non-mathematical
structured language during the previous work diminishes. From a cognitive point of view
this situation could be interpreted, as in van Hiele (1986, p. 49), as the presence of
discontinuities in a learning process.
Teachers remarked that with worksheet 5, a greater linguistic awareness is required. Now
explicit definitions arise consciously. In exercise 5.1, all the lines beginning with "it is
necessary" may contain more than one right answer. But the lines beginning with "it is
enough" have at most one right answer. This is the significant signal, the definition.
(O) Some pupils discovered this in an autonomous way. Some other pupils still display
some difficulties: these seem to be substantially linguistic difficulties; in fact a further
sentence reformulation by the teacher (in terms of "if – then", or "it is always true
that…?",…) leads to the correct response.
(A) The discussion on exercise 5.2 goes on with a precise question: which of the listed
properties are also definitions for the corresponding figures? In each line there is now, at
most, only one right answer. Pupils know this: there are no mistakes at all, even not in the
last line, where there is now no correct answer (of course in the meantime the teachers have
at least mentioned the kite).
Teachers believed that the discussion and the framing of the teacher are important,
particularly for those pupils who have not autonomously recognized the definitions, but
Vanna and Mariolina also noticed that it is not appropriate to indulge too much in
discussion, or to exaggerate the logical aspect, thus taking away natural significance from
the language.
23
3.5. Episode 5: Evaluation Sheets – "Conscious" Constructions.
The project unit ends with two evaluation sheets. All the evaluation items ask for the
construction of figures with prescribed conditions and a description of the procedure
chosen.
Looking at the pupils' work together with the teachers, we could deduce - from the frequent
use of the rubber for erasing - that the solutions were corrected and refined more than once.
(O) In each class there were three or four pupils who still operated at a level of perception
of the figure: they "organized" the figure in their mind superposing it onto the given
segment. At a higher level, pupils performed the construction by selecting some properties.
The major difficulty appeared in the construction of figures with given diagonals,
particularly in 7.2 (Table 11), where some pupils tried to draw a square with its sides
parallel to those of the worksheet.
Table 11
All pupils were without doubt at the level which allows them to recognize a figure with its
properties at a global level. The majority was operating at the level of description of a
given and already constructed figure. They need a little bit more to describe the
construction of a non-given figure. They need to understand how some properties may be
derived from a particular given one. According to teachers, this does not mean that the
pupils who coped with the task reached the level of deduction. Many pupils can be at an
intermediate stage, in which the understanding of implication is not yet of the logical type.
But for the teachers this is sufficient: it is exactly this intermediate step that permits the
start of logical deduction.
24
Teachers whose pupils had learned to use Cabri software in an autonomous way proposed
that they verified their procedure using this tool. Cabri software prevents pupils from
adjusting, through perception, the given elements to the image of the final figure. Teachers
observed that some pupils use a “perceptive” or incomplete construction even when having
a ruler and a compass or a square in their hands. Similarly to all Dynamic Geometry
Software, such as Geometer’s Sketchpad, Cabri requires instead a complete Euclidean
construction. For example, to solve exercise 7. 2, a pupil has to remember the properties of
the diagonals of a square and has to perform a series of actions strictly in the stated
succession: trace two points, A and C (creation-menu); create the segment through the two
given points; trace the middle point M of the segment (construction-menu); construct the
perpendicular line to the given segment through the middle point M; create the circle
centered in M and passing through A or C; construct the intersection of the circle with the
perpendicular line obtaining A, B, C, D; create the segments AB, BC, CD, DA.
That is, Cabri pushes pupils a step away from a perceptive procedure and a step toward a
deductive procedure.
4. Conclusions
Teachers, even our expert teachers, forget from time to time the process of adding to and
refining elementary mathematical knowledge. In particular, there is unconscious didactic
planning to anticipate knowledge and transmit it to the pupil in order to reach the targets
too quickly. Set inclusion on elementary figures seems so simple that it is used as a didactic
tool with which to clarify. The trend is to give definitions, which should only be used with
more careful attention paid to semantics and syntax. After our working sessions on van
25
Hiele, teachers in our group realized that their teaching habits, and the fact that set
inclusion is so "simple", were hiding important developmental steps.
Teachers reconsidered worksheets as a didactical tool: the pupil is able to reflect on them,
developing a form of personal mathematical understanding. This reinforces the efficiency
of the discussion phase: symbols, which a pupil has acquired through perception, acquire a
verbal content not only through an individual process, but also through the interaction with
the teacher and with other pupils.
Undoubtedly, it is not easy for the teachers to avoid intervening immediately in pupils'
work, to accept pupils' assimilation time, and to delay the conclusions and re-arrangement
of the topic. All this requires attention and special energies in order to control one’s
interactive behavior with each pupil and with the class as a whole. Certainly the knowledge
teachers had about van Hiele’s levels led to a deeper awareness of pupils’ development,
played a predictive role in their discourse in the classroom and had a strong influence on
the way in which they were guiding classroom discussion. They admitted moreover to
having improved their way of being present silently while the pupils were working on the
worksheets.
Making the first levels of van Hiele explicit allowed teachers to clarify their teaching aims.
In particular the development of the passage from exclusive to inclusive definitions, and
the role of the latter in starting deduction, became clear to them.
Van Hiele's theory is a framework on which teachers can recompose different experiences,
like the fact that a pupil's knowledge acquisition is local and temporary, that definitions
may be inclusive for some figures and exclusive for others, or, analogously, that a pupil
may be at a certain level for some points and at a different level for others.
26
All teachers wanted to try to use this framework regularly in their didactic practice, either
for the worksheets or, more generally, for methodology.
Some aspects not yet sufficiently explored emerged in the workgroup: the way in which
generic images (prototypes) are constructed in order to characterize figures, the role of
definitions in overcoming standard visual images (for instance a square whose sides are
parallel to those of the worksheet), i.e. the role of definitions in "helping the perception"
and generating other symbols. The relationship with conscious language and terms of logic
like "every" and "all" is connected to this.
Finally we want to stress that the chosen itinerary seems to permit and confirm the passage
through the various levels, but its optimal ‘fitting’ with van Hiele's theory is due to the
teachers' awareness of the stages to be followed.
References
Cannizzaro L., & Menghini, M. (2001). From geometrical figures to linguistic rigor: Van Hiele's model and the growing of teachers awareness, in van den Heuvel-Panhuizen M. (ed.) Proceedings of the 25th conference of PME, Utrecht: Freudenthal Institute, I-291.
Cannizzaro, L., & Menghini, M. (2004). Geometric Figures from Middle to Secondary School: Mediating Theory and Practice. In Mariotti M. A. (ed.) Proceedings of CERME 3 (2003), TG11, CD, Pisa: Ed. Plus.
Castelnuovo, E., & Barra, M. (2000/1976). Matematica nella realtà, Torino: Boringhieri. Clements, D., H., & Battista M. T. (1992). Geometry and spatial reasoning. In Grouws D. (ed.)
Handbook of Research on Teaching and Learning Mathematics, New York: Mc Millan, 420-464. DeMarois, P., & Tall, D. (1999). Function: Organizing Principle or cognitive root? In Zaslavsky O.
(ed.) Proceedings of the 23rd conference of PME, 2, Haifa: Israel Institute of Technology, 257–264.
De Santis, C., & Percario, Z. (1997). Dalle costruzioni geometriche al rigore linguistico: un'esperienza didattica all'inizio del biennio. In Micale B. & Pluchino S. (eds.) Atti del 18° Convegno Nazionale UMI–CIIM, Notiziario U.M.I, supplemento al n.7, Bologna: Edizioni U.M.I., 129-132.
Dubinsky, E. (1991). Reflective abstraction in Advanced Mathematical Thinking. In Tall David (ed.) Advanced Mathematical Thinking, Reidel: Dordrecht, 95–126.
Duval, R. (1988) Approche cognitive des problemes de geometrie en terms de congruence. Annales de Didactique et de Sciences Cognitives 1, 57-74: Basic Books.
Freudenthal, H. (1973). The case of Geometry. In Mathematics as an Educational Task, Dordrecht: Reidel Publishing Company.
Heath, T. L. (1908). The thirteen Books of Euclid's Elements: Cambridge University Press.
27
Krainer, K. (1994). Integrating research and teacher in-service education as a means of mediating theory and practice in mathematics education. In Luciana Bazzini (ed.), Theory and practice in mathematical education. Pavia: ISDAF, 121-132.
Jacobson, C., & Lehrer, R. (2000). Teacher appropriation and student learning of geometry through design. Journal for Research in Mathematics Education, 31, 71-88.
Love, E.(1994). Mathematics Teachers’ Accounts seen as Narratives. In L. Bazzini (ed.), Theory and practice in mathematical education. Pavia: ISDAF, 143-155.
Malara, N., Menghini, M., & Reggiani, M. (1996). General Aspects of the Italian Research in Mathematics Education. In Malara N, Menghini M, & Reggiani M. (ed.): Italian Research in Mathematics Education 1988-1995, Roma: Seminario Nazionale di didattica della Matematica e CNR, 7-23
Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics 34, 220-248.
Monaghan, F. (2000). What difference does it make? Children's views of the differences between some quadrilaterals. Educational Studies in Mathematics 42, 179-196.
Pinto, M, M. F., & Tall, D. (1999). Student construction of formal theory: giving and extracting meaning. In Zaslavsky O. (ed.) Proceedings of the 23rd conference of PME, 4, Haifa: Israel Institute of Technology, 65-72.
Pirie, S., Martin, L., & Kieren, T. (1994). Growth in Mathematical Understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165-190.
Skemp R. R. (1979). Intelligence, Learning, and Action, Chichester: John Wiley. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with special
reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169 Usiskin, Z. (1982). van Hiele levels and achievement in secondary school geometry. Chicago,
University of Chicago (ERIC Document Reproduction Service N° ED 220 288). van Hiele, P. M.: (1986). Structure and Insight. New York: Accademy Press. van Hiele, P. M., & van Hiele-Geldof, D. (1958). A method of initiation into geometry at secondary
schools. In H. Freudenthal (ed.) Report on methods of initiation into geometry, Groningen: J. B. Wolters, 67-80.
de Villiers, M. (1994). The Role and Function of a Hierarchical Classification of Quadrilaterals. For the Learning of Mathematics, 14, 1, 11–18.
de Villiers, M. (1998). To teach Definitions in Geometry or teach to define? In Olivier A. & Newstead K. (eds.) Proceedings of the 22nd conference of PME, 2, Stellenbosch: University of Stellenbosch, 248 –255.
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Table 1
Table 2
Table 3
1.2 a) Classification of triangles with regard to their sides:
A triangle is isosceles when it has at least two equal sides. Is a triangle with three equal sides (in other words, an equilateral triangle) isosceles ? YES / NO A triangle is not isosceles if …………………………………………………... b) Classification of triangles with regard to their angles: […] 2.2 Where possible, draw triangles with the following characteristics: a) Isosceles and acute-angled […] d) Equilateral and rectangular […]
Definition XX - "Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal"; Definition XXII - "Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia." (In the translation of Heath, 1908).
Episode 1: Worksheets 1 and 2
Triangles. "Rough" Constructions and Verbalizations.
Episode 2: Worksheet 3 Rectangles, Squares and Rhombi. Comparing properties.
Episode 3: Worksheet 4 Rectangles, Parallelograms and Trapezia. Approaching Definitions
Episode 4: Worksheet 5 Quadrilaterals. The Significant Signal: Inclusive Definitions
Episode 5: Evaluation Sheets "Conscious" Constructions.
29
Table 4
2. 1 Observe the isosceles triangle ABC: If you cut out and fold it so that C does not move and A is placed on B you will find that the parts coincide. The straight line between points C and H resulting from the folding is an axis of symmetry. The isosceles triangle, therefore, as well as having equal sides CA and CB, also has equal angles CAH and CBH. Angles BCH and HCA can also be seen to be………………………….. Hence CH is a bisecting line. […]
Table 5
2.3 The triangle illustrated is not isosceles Draw the height and median for side AB (base); Do the height and median coincide? YES / NO Now rotate triangle ABC and consider AC as the base; Draw the height and median for side AC (base);
2.4 Is the illustrated triangle, ABC, isosceles? YES / NO For which base? ………………… Draw the heights. How many are there?…………… Draw the medians. How many are there? In an isosceles triangle, how many segments do you have to draw in order to draw all the heights and all the medians? ………………
1
Table 6
Table 4
Table 4
Table 7
4.1. Consider the rectangle ABCD constructed with the pieces of the "meccano" (fig.1).
The opposite sides are……………. and they are… The angles are… The diagonals are… At vertex A push bar AB in the direction of BA. You obtain a parallelogram as in fig. 2. On the new shape: Are the opposite sides equal? yes/no/ it depends Are the opposite sides parallel? yes/no/it depends Are the angles all equal? yes/no/it depends Are the diagonals equal? yes/no/it depends More generally: Is a rectangle a parallelogram? Yes/no/it depends Is a parallelogram a rectangle? Yes/no/ it depends Is a square a parallelogram? Yes/no/ it depends Is a parallelogram a square? Yes/no/ it depends Is a rhombus a parallelogram? Yes/no/ it depends Is a parallelogram a rhombus? Yes/no/ it depends
3.6. You can see a square ABCD made by the pieces of meccano:
Using the same pressure, push the square in from the vertexes A and C in direction AC, the
segments AO and OC will shorten, but will remain equal (figure on the right).
Are the diagonals AC and BD still equal? Yes/no.
The sides of the figure on the right are equal / not equal.
Is the figure on the right a rhombus? Yes/no.
Can you say: "The rhombus is a quadrilateral where all the sides are equal"? Yes/no.
The square has all its sides equal: can we state that it is always a rhombus? Yes/no.
Can we say that the rhombus is always a square? Yes/no.
From the construction can you deduce that the diagonals are perpendicular to each other? Yes/no.
Can we say that the rhombus is always a square? Yes/no.
From the construction can you deduce that the diagonals are perpendicular to each other? Yes/no.
Are the diagonals AC and BD still equal? Yes/no.
The sides of the figure on the right are equal / not equal.
Is the figure on the right a rhombus? Yes/no.
Can you say: "The rhombus is a quadrilateral where all the sides are equal"? Yes/no.
The square has all its sides equal: can we state that it is always a rhombus? Yes/no.
2
Table 8
4. 3. As you remember, the image on the right represents a trapezium. It has two parallel sides. Draw the diagonals. Evaluate the following statements: A trapezium is a quadrilateral with a pair of parallel sides Yes/no Is a trapezium a parallelogram? Yes/no Is a parallelogram a trapezium? Yes/no If AD = BC the trapezium is said to be isosceles: draw an isosceles trapezium with its diagonals. If AD is perpendicular to AB the trapezium is said to be right-angled: draw a right-angled trapezium with its diagonals. In the trapezium in the figure above, do the diagonals cut each other in half? YES / NO Check on your drawings whether or not the diagonals of an isosceles trapezium and of a rectangular trapezium cut each other in half. In the parallelogram, do the diagonals cut each other in half ? YES / NO
Table 9 Exercise 5.1: For a Quadrilateral Parallelogram Rectangle Rhombus Trapezium it is enough to have a pair of parallel sides to be a…
W: 17%
W: 2%
W: 2%
R (?): 100%
it is necessary to have a pair of parallel sides to be a…
R: 90%
R: 90%
R: 81%
R: 75%
it is enough to have two pairs of parallel sides to be a…
R: 95%
W:17%
W: 19%
W: 2%
it is necessary to have two pairs of parallel sides to be a…
R:88%
R: 98%
R: 86%
W: 4%
it is enough to have equal diagonals to be a …
W: 40%
W: 21%
W:7%
W: 2%
it is necessary to have equal diagonals to be a…
W: 29%
R: 88%
W: 4%
R:14% 'if it is isosceles'
it is enough to have perpendicular diagonals to be a…
W: 0%
W: 0%
W: 37%
W: 0%
3
Table 10 Exercise 5.2: It is a Quadrilateral with Parallelogram Rectangle Rhombus Trapezium two pairs of opposite parallel sides.
R: 100% R: 96% R: 98% W: 0%
only one pair of opposite parallel sides.
W: 4% W: 2% W: 4% R (?): 44
all equal angles. W: 2% R: 99% W: 9% W: 0% opposite angles equal. R: 96% R: 96% R: 96% equal diagonals. W: 33% R: 93% W: 2% R: 11% "if it is
isosceles" perpendicular diagonals. W: 0% W: 0% R: 98% W: 0%
Table 11
7.2. Construct a square ABCD in which AC is a diagonal. Give a description of the procedure employed. A C [...]