CLARIFYING LEVEL DESCRIPTORS
FOR CHILDREN'S UNDERSTANDING OF SOME· BASIC
2-D GEOMETRIC SHAPES
John Pegg
University of New England
Brisba~e
Geoff Davey
College of Advanced Education
ABSTRACT
The purpose of the study, in general terms, was to compare the level descriptors of the van
Hiele Theory and those of the SOLO Taxonomy. More specifically. the study was designed to
clarify inconsistencies between the two theories as they relate to the understandings of
primary and early secondary school pupils.. Th'e study involved pupils from classes in Years
3-7 in New South Wales and Queensland, who were asked to describe a number of common 2D geometric shapes. The results showed that the descriptors associated with the Solo Taxonomy more accurately described the 'level"· of student thinking.
In addition support was found for:
1. the proposal that pupils often function within two modes of reasoning. i.e.. many students need to support their answers with reasoning from more than on~ mode:
2. the categorization of students' responses within a given SOLO 'level' i.e .. there appeared
to be strong evidence to suggest that within some levels of understandings, at least. stages of development could be identified.
INTRODUCTION
The eighties have been marked by a substantial increase in the interest shown by
mathematics educators to children's· understandings of mathematics. Notable catalysts
have been the C.S.M.S. study based at Chelsea
College and the subsequent book edited by
Hart (1981) and several theories of learning
that have focussed on identifying 'levels' of
thinking (e.g" Biggs and Collis. 1982; Halford, 1982; Case, 1985: van Hiele, 1986).
The incentive for research that attempts to
probe children's understandings is that the
findings have the potential to help:
1. curriculum
planners
better
organize
content in a way that can optimize pupil
thinking,
2. teachers better focus their instructional
activities to the students' level of thinking.
Of the above theories of learning two stand
out and will be the focus of the
tion described later in this paper. The
theories are the -van Hie1e Theory and
SOLO Taxonomy (developed by Biggs and
lis) .
Both these theories· have been the subject
many recent investigations. Indicative
the type of studies that have recently been
completed:
1.
for the van Hie1e Theory (which is primarily concerned with student understandings in Geometry). are three major
studies in the U.S. under the direction of
Usiskin (1982). Geddes,Fuys and Tischier
(1985), and Burger and
(1986);
16
2. for the SOLO Taxonomy (which is directed
at a number of subjects across the curriculum), are papers by Collis, Romberg
. and jurdak (1986), Chick, Watson and
Collis (1987), and Watson Chick and Collis, (1988).
In addition to this, earlier
work by Collis is acknowledged as influencing the directions taken by the Chelsea study.
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Of particular interest, are the similarities
between the two theories. For example, both
theories were developed because of identified inadequacies and anomalies with some
of the assumptions underpinning Piaget's
ideas. They were developed through considering students' understandings which see
stUdent growth in terms of identifiable 'levels'. Both theories are also applicable within
the classroom situation. However, apart
from these similarities and some informal
hypothesizing (Pegg, 1987), little has been
done to explore relationships between the
two theories. Yet it is clear that comparisons
between the two theories would be valuable.
Not only would such investigations help
clarify both theories further, but a greater
insight into the nature of learning would
result.
Following is a brief summary of aspects of
both theories as they relate to their 'level'
descriptors. This leads, in turn, to a comparison-of the level descriptors.
The ideas of van Hiele have been summarized many times in the last few years (two
examples readily available for Australian
·re.aders are Pegg (1985) and Davey (1988)).
In 1986 van Hiele released a book, written in
English, which encapsulate~ his current
thinking. The reader is directed to these
sources for a fuLler account. However, in essenee the van Hiele Theory identifies the
main purpose of teaching mathematics as
the development of insigh t in students. Insight is achieved when a person can act in a
new situation adequately (j.e., avoids a cumbersome approach ) and intentionally (j.e.,
can choose a suitable method). Associated
with this purpose are two aspects. The first,
Psychological, which involves a series of
. levels that describe student understandings,
and the seond, pedagogical, which involves
a series of teaching phases that provide a
means of transition from one level to the
next. It is the five level descriptors ( named
Level 1 to Level 5) which are of interest in
this study and brief descriptions of each
level are given below.
Level 1 Geometrical figures are recognized by their shape; a student is familiar with the external form of the figure.
Level 2 Geometrical figures are the bearers of properties; a stu.dent is familiar
with the properties of a figure but these
properties are seen to be independent of
one , another.
Level ,3 The properties of the geometrical
figures are seen as inter-related not independent. Figures also are seen to be
related.
Level 4 The student works formally within a system of theorems, axioms, etc.
Necessary
and
sufficient
conditions
have meaning.
Level .5 The student is able to study different geometries by modifying axioms,
e.g., projective geometry.
In the study by Usiskin (1982), who tested
approximately 2700 students in schools
throughout the U.S. on their geometry
knowledge and van Biele 'level assessment,
it was found necessary to include a further
level. The extra level was created to cater
for those students who failed to reach cri~
tedon on Level 1 tasks - such students
were assigned "Level '0".
..
It is not clear from the writings of van
Hiele whether his work involved students
who were younger than Year 7 (12 year
01ds). If such is the case it would help explain the absence of levels below Level 1 in
his theory. It is interesting that some rethinking at the lower end of the scale has
been acknowledged by van Hiele in his
book (1986). This has resulted in a change
in the labelling of his levels from his earlier work. In the 1950s van Hiele referred to
his lowest level as the Base Level, the next
Level 1 etc. However, he has now abandoned this earlier stance and refers to the
Base Level as Level 1, the old Level 1 as Level 2, .... He stated the reason for the change
was "caused by our not having seen the importance of the visual level (which is now
called the first level) at that time" (p.4
This paper adopts the descriptor Level 0 to
identify those students who fail to identify
n
17
or recognize a figure by its external form.
The SOLO Taxonomy was first described in
Biggs and Collis (1982) and later modifications can be found in Collis (1988). The theory is based-on the analysis of children's responses to questions in a number of subject
areas. Two phenomena are identified as determining the level of a student's response,
namely, the mode of functioning and a series of levels which describe the growth
within each mode. The mode of functioning
is closely related to Piaget's 'stages of cognitive development, i.e.,
Sensorimotor
[ 4 months - 2 years]
Ikonic
(I)
[ 2 - 6 years]
\=oncrete Symbolic (C.S,)
[7 - 15 yearsl
Formal Operational(F 1 )
16+ yearsl
Formal Operational(F 2)
[age parameters
notclearl
The lower age bound is probably the earliest
at which such behaviour can be recognized
in a person. Associated with each of the
modes are a series of levels. These levels are
repeated for each mode of functioning and
the descriptors (adapted from Biggs and Collis, 1982, pp. 24-25) are:
Unistructural: able to generalize on only
one aspect. No felt need for consistency,
closes too quickly.
Multistructural: able to generalize only
in terms of a few limited and independent aspects. Can be inconsistent because
closes too soon.
Relational:
able to generalize within a
given or experienced content using related aspects. No inconsistency within a
given
system.
Inconsistencies
occur
when a person goes outside the system.
Extended
Abstract: able to generalize to
situations not experienced. Inconsistencies resolved. No felt need to give closed
decision - conclusions held open or are
qualified to allow logical possible alternatives.
18
The cyclic nature of these levels manifests
itself by the extended ab stract level of one
mode (say ikonic) being equivalent to the
unistructural level in the next mode ( in
this case the concrete symbolic).
It has been gene'rally accepted that stu-
dents in primary and secondary schools
usually function at concrete symbolic or
formal 1 modes. However recent developments( see Collis 1988) tend to suggest that
it may pe more useful not to consider the
progression through the modes as strictly
linear ..That is, instead of a person's growth
being described as passing from the ikonic
mode and moving into the concrete symbolic mode, it is possible that in the right
circumstances the lower mode (ikonic in
this case) may proceed to grow in its own
right at the same time as the. higher mode
of functioning. An example would be,
when mathematics students who are familiar. with graphing quadratics (concrete
symbolic mode) are able to visualize the
parabola and its position on the cartesian
number plane in their minds' eye (ikonic
mode).
Comparison of Level
Descriptors
When comparing the two theories in terms
of their level descriptors two possibilities
emerge and these are detailed in Table 1
and Table 2.
Van Hiele
Level
0
1
2
3
4
5
SOLO Taxonomy
Level
Relational
Unistructural
Multistructural
Relational
Unistructural
Multistructural
(Mode)
(I)
(C.S.)
(C.S.)
(C.S.)
(F 1)
, (F 1)
Table 1: Hypothesized comparison
between van Hiele levels and the
levels of the SOLO Taxonomy
direct
relationship
,..........,...~= F""~~=~~~-""'~~"~~'=-'
Van Hiele
Level
SOLO Taxonomy
Level
,
(Mode)
""""=
0
1
-
below Relational
Relational
Unistructural
Multistr uet ur al
Relational
(I)
(I)
(C.S.)
(C.S.)
(C.S.)
other end of
This proced ure
mon shapes,
parallelogram,
No. of
Students
a telephone conversation".
was repeated for four comnamely, square, rectangle,
and rhombus.
YEAR
4 5 6
7
TESTED
59 44 55 38
7
Table 2: Hypothesized comparison between van Hiele levels and the levels
of the SOLO taxonomy :
indirect
relationship
INTER VIEWED
16 9
Table 1 suggests a direct one-to-one relationship between both theories and Table 2
suggests some discrepancies between the
theories at both ends of the comparison. It
is clear that both van Hiele's Levels 2 and 3
and the SOLO levels m ultistructural and relational, respectively, describe the same
phenomena. The same cannot be said for
the other levels.
Students who were not familiar with the
shape (i.e. could not draw/sketch the figure) were shown a drawing of the figure.
2
3
4
5
-
(F
3
TOTAL
1)
(F 2)
274
~
Of use would be two studies, one to examine
more closely the functioning of students
below Level 2/m ultistructural (C.S.) and one
to examine the functioning of students
above Level 3/relational (C.S.). While a
studY: '(Pegg, 1989) is currently under development which will focus on the latter investigation it is the former which forms the
basis of the study described below. It revolves around the research question:
Doe'S the closure by students on one aspect
[i.e. unistructural response - described in
SOLO] - relate to a figure being recognized
in its'totality [i,e. Level 1 - described by van
Hiele] or has the closure on one aspect not
been identified by van Hiele and is the van
Hiele Level 1 better compared with a functioning in a lower mode, namely, a relational level in the ikonic mode?
Design
15 11 29
80
,-,
Table 3: Number of students tested
and interviewed by year level
To lessen the impact of rote learnt responses the study was carried out early in the
school year I.e. before Geometry topics had
been introduced that year, and 80 students
(approximately
30%)
were
interviewed
about their written answers.
The interviewing procedure was designed to
avoid prompting and consisted of questions
such as having the students explain what
was written, clarify certain words, and allowing them to change their answer by
adding or removing information. if they
wished.
Interviews were taped for later
analysis.
This questioning procedure, and lack of
prompting, differed from other studies (e.g.,
Geddes et ai., 1985). in that there was not a
teaching role associated with the investigation. It was of concern that prompting
would have a ' level reducing' quality arid,
if it occurred, many students may have appeared, during the interview, to function at
a much higher level than they could
achieve on their own or which their written responses had indicated.
of Study
To examine the research question 274 pupils
from ungraded classes in Year 3 to Year 7 in
New South Wales and Queensland were tested (see Table 3). The students were asked to
draw a specific geometrical shape and then
to describe it in writing as best they could "
as they would to a friend who was on the
RESULTS AND DISCUSSION
As the analysis of the results proceeded it
became quite clear that Table 2 better described the student responses to the questions posed. In addition several findings became evident. These are listed below.
19
1.
Many students focussed on only 0 n e
property when describing plane figures
- this property was invariably the sides
of the figure.
2. The SOLO Taxonomy better explained students' understandings of the geometrical figures.
3.
4.
There was variety in students' responses
within a year level. It was not uncommon for a student in Year 4 to respond
at a m ultistructural level (C.S. mode), although the majority of responses were
either unistructural (C.S. mode) or relational (I mode). In year 7 the majority of
students gave either unistructural or
multistructural (C.S. mode) responses,
although a few students continued to express their descriptions only in terms of
imagery; that is relational (I) responses.
Many students needed to support their
descriptions in the concrete symbolic
mode by using ikonic mode references.
9.
There often was not a uniform level description for each of the four shapes, although the differences between reponses for an individual was small in the vast
majority of cases. The responses to all
figures could be described by no more
than two consecutive SOLO levels.
10. In cases where two levels were needed to
describe a students' responses the higher tevel was associated with the more
'common' shapes; for example square
and rectangle. This probably reflects the
fact that these students have had more
experience with these shapes.
To illustrate the results described above, and
to indicate the type of decisions made in the
analysis, samples of student written descriptions ate given below in each of the
categories identified. The spelling and
grammar used by the students has be.en reproduced, corrections have been written as
[ I when the meaning of what has been
written may be unclear.
S. There appeared ample evidence to suggest that the unistructural level (C.$.
mode) could be sub divided into distinct
stages.
6.
Few students gave detailed responses at
the multistructural level (C.S. mode), and
only one student ( a 12 year old girO was
identified at the relational level (C.S.
mode), as would be expected given the
spread of students across these year levels (see Table 3).
7.
Several words (e.g., 'even', 'corners',
'side',) and negative phrases (e.g., 'not
equal to') that are not generally used by
teachers were very common in students'
language _ and this language transcends
state and national borders.
8. Some students who seemed used to working at a given level, but did not have
sufficient experience with a given
shape
at that level, often gave a description which had errors. For exampIe, students who gave multistructural
answers to questions about a square and
a rectangle, often included in their description of a rhombus comments about
right angles.
20
Ikonic Mode: Relational (Biggs and
Collis) and Level 1 (Van Hiele)
In general the responses given at this level
refer to images. Students know the figure's
global shape, or, if not, when given an example can describe it in terms of some other
object (or image). Students were much more
likely to give ikonic mode responses for the
less familiar shapes, parallelogram and
rhombus, although the number of such responses from students in a class decreased
with school year.
Students at this level were usually able to
make reasonable attempts at drawing the
figures - this was especially true for the
square and the rectangle. However even
though the students made use of such ideas,
say for a square, as equal sides and right
angles in their drawing, these concepts did
not emerge in either their written or verb al descriptions.
The language used by the students in this
mode was consistent across state borders
and usually differed quite markedly from
the language teachers would use themselves
in formal mathematics lessons. Two words
stand out and will serve to emphasise the
point, they are 'even' and 'corner',
its tip to
'can form into a diamond - like a
square and out of shape'
"Even' appears to be a lower form of the
word 'equal'. It was rare for students in the
ikonic mode to use the word equal on their
own initiative, However, it was also not uncommon for students in the concrete symbolic mode also to perservere with 'even'
when they meant equal.
'Corner' was a vague term which appeared
to include, vertex (or 'point'), sides, and the
included angle
an undefined region
around which two lines meet. Students seem
to liken the word to the corner in a room.
Some stuuents felt that, in common with
most rooms, the included angle needed to be
aright angle, i.e.,· a rectangle could have
'corners' but a triangle could not. Such students were unable to clarify precisely what
they meant by the word and often gave conflicting descriptions within the interviews.
In· the concrete symbolic mode it was also
common, especially in a unistructural resppnse, to use the word 'corner' as a form of
ikonic support for the answer given. When
functioning at a higher level some students
continued to use the term 'corner' but in
this case it lost its vagueness of meaning
and it was used as a synonym for either angle or vertex.
one side'
'is a squar on a lean'
'it looks like two triangles drawn
together'
Concrete Symbolic Mode: Unistructural level (Biggs and Co111s) -no
equivalent Van Hiele level
The analysis of student responses which resulted in students focussing on only one issue resulted in several findings. They were:
1. The single issue that students focussed on
was the side(s) of the figure.
2.
The students clearly moved from seeing
the figure in global terms and instead
were beginning to identify properties,
albeit, one property.
Below are examples of students' written descriptions.
3. The responses of students who focussed
on one property could be categorized
into three stages, namely, those students
who by their own initiative could: focl,ls
on one property; try to qualify (but unsuccessfully) the one property; qualify
correctly the property.
.
Typical
4.
Many students added an ikonic type reference to their answer, i.e., they felt the
need to support their answer by using
the ideas of the lower mode of functioning.
S.
The word 'side' became part of the descriptions but often it meant (a) a pair
of lines and (b) lines other than the top
or the bottom, Le., the lines on the side.
6.
It became common for negative phrases
to appear in the descriptions, e.g., 'sides
not even'.
Square:
Ikonic
Responses
'it as to [be] even on each corner'
Rectangle: 'can come in all shape and sizes
even so small
that you cant see it'
'its got 4 corners'
'is almost the same as the square but
its longer'
'has 4 cones [cornersl and it is long'
Parallelogram:
'is like a rectangle
that has been pushed to one side'
'looks like a rectangle but the ends
are crocked'
Rhombus:
'is simi11ar to a square but
the lines are on a slant'
7. The descriptions given were often based
on a limited view of the figure given,
e.g., many students used descriptions for
a square or rectangle which included
the words vertical and horizontal.
Rectangle:
'It
has
four
sides,
two on
21
Below are examples of students' written descriptions that have been classified at each
of the three stages. Associated with each
stage are examples of the ikonic support
many students felt they needed to provide.
the side are longer then the two on
'the top and buttom [ bottom J.
'allways has uneven sides'
'it has four sides, of unequal length'
Typical Unistructural
sponses
Stage
1 Re-
Square:
'stright lines its a flat shape,
4 sides'
Rectangle:
'4 straight' sides'
'it has two sides the same size and the
other two are half the size'
'4 different sides'
'2 strat [ straight
sides'
sides and 2 little
Parallelogram: 'it has four sides'
Parallelogram: 'Its a bit wobbly
four sides of equal length'
'4 sides and 4 points'
Rhombus:
Rhombus:
Unistructural:
with Ikonic
Stage
1
support
'looks like a cube it has four corners
and it have two faces [ pairs of line ]
and it is good to look at'
'has 4 cones [ corners ] and 4 lies [
lines ]'
'4 sides, 4 corners, long
'it has 4 sides it is deffret [ different]
to a square'
Parallelogram: 'It is like a rectangle but
two of the sides are on an. angle'
Rhombus:
'This shap has 4 sids but in
the shap of a diamond'
'is exactly like a square only turned
or tiped on a side it has four sides
which are also joined as a corner'
'4 sides but they are diagonal'
Typical Unistructural:
sponses
Square:
Rectangle:
22
- no response recorded
Responses
Square:
'It has 4 line joining points in
the shap of a box but it only has one
sid of a box'
Rectangle:
shape'
has
Stage
2
Re-
- no response recorded
'is simuler to the square but
Typical Unistructural: Stage 2
Responses with Ikonic support
Square:
'it is about 4cm long, has four
sides some corners go over a bit'
[ The phrase 'over a bit' refers to the
lines in the students' diagram not
finishing exactly at the vertex ]
Rectangle:
'Two sides' are longer than
the other two, four corners, smooth
sides'
'2 long sids and 2 short sids, 4 corners.'
'is just like
the
lines
longer than
facing each
a square except that 2 of
facing
each other
are
the other pair of lines
other'
Parallelogram: 'is like a rectangle except tilted to one side a bit and the
two end sides face the same way'
Rhombus:
'-You draw one triangle but
don't put a bottom on it and draw one
upside down. Don't put a bottom on it
as well. Theres your diament.
Typical Unistructural:
sponses*
Square:
Stage
3
Re-
'4 even lines'
Rectangle:
'...it has lines Jommg but
the lines a [are] not all the same size.
The to [two] side lines are the same size
and the top and bottom one is the same
size'
'has two sides that is the same length
two is 8cm and two are 4 ems'
'the top and bottom are the same
amount of centermeters [centimetres] and the two sides are the same
amount of centermeters'
Parallelogram: 'is a four sided figer. It
has two sides the same and two different. Its a sort of rectangle.'
Rhombus:
'4 sides the same'
* It is worth noting that while many students, who were functioning at unistructural stage 3 for a parallelogram and
rhombus, gave correct mathematical definitions as their descriptions of the figures, i.e., their descriptions comprised in.formation
that
was
necessary
and
sufficient.
Unfortunately,
the
significance of their answers was beyond their
comprehension and did not reflect some
special insight into the figures concerned.
Typical Unistructural: Stage 3
Responses with Ikon-ic support
Square:
'it has 4 corners, 4 sides the
same measurement'
'it has equal centimetres all around
and four corners'
'four even sides, four corners,
smooth sides'
Rectangle:
'two identicale lines and
two identicale [lines], it is flat shape,
four identicale corners'
'four corner and they are equal'
'the two sides are the same and the
top and the bottom are the same. But
the two sides are diagonal [on a slant]'
Par alle logr am: 'it has sloping sides and
it has two straight sides. It looks like
the top of a cube, it is a plain [plane]
shape'
'has sloping sides (3 em) and it has 2
straight sides (6 em). It looks like the
t.op of a cube it is a plain [plane]
shape'
Rhombus:
'it looks just like a diamond,
it has eq u'al sides'
'is like a half squashed square all
sides are equal'
Concrete Symbolic Mode: Multistructural level (Biggs and Collis) and
Level 2 (Van Hiele)
Such responses focussed on more than one
property. The most common second property to be selected was right angle for square
and rectangle and parallel lines for parallelogram and rhombus. Very few students
made mention of axes of symmetry and
there was a complete absence of any mention of properties related to diagonals.
The sequential nature of the levels also became evident through the various responses made by students at this level. In most
cases, either in written descriptions or in
the interview situation, students who gave
more than one property O.e., a multistructural response), and who were correct, had
also correctly given in their answer the appropriate response at a unistructural level stage 3.
The language of the pupils was improving,
although words such as 'even' and 'corner'
were still evident.
As with the previous level it was clear that
there were stages which indicated growth
through the level. However, the number of
students who responded at this level was
considerably less than for the other categories and hence the findings are more tentative. It did appear, though, that there were
students who:
1. felt the need to incorporate another
property or properties into their description but had trouble qualifying
their answer. Often these students made
errors in their descriptions, or
2. could clarify correctly the subsequent
property or properties they used.
The following student responses are organized according to this dichotomy.
23
Typical Multistructural level
2 (van Hiele) Responses:
Without qualification of the
and further
properties.
Level
second
Square:
'has 4 equal sides. First you draw
1 straight line. You then draw another
line straight at right angles from no.!
line. Then you draw another line, no.!I
line then another line.'
Rectangle:
'the length is longer than
the width. There is 2 sets of parell (parallel) lines'
'has 2 parial (parallel) lines going sideways and 2 parial lines going upwars.
All the sides arent the same'
Parallelogram: 'has four sides two Of them
are equal to each other the other two are
equal to each other. If you run the lines
forever it will never meet.'
.
'has 4 sides. The longer sides are 4 cm
long and the shorter ones are 2 cm
long, it does not have right angles.'
Rhombus:
'4 equal sides, no right angles'
'all sides equal, 2 acute, 2 obtuse angles'
Without qualification of the second
and further properties and using
Ikonic
support
Square:
'is 5 cm all around and it is not
very big and it is parallel'
Rectangle:
'is a 4 sided figuie made up of
squares. It has two sides parallel, the
length and the width. Every point or
corner measures 90 degres.'
Parallelogram: 'its sides are parallel and
angles about 60 degrees, it is like a rectangle - it has two short sides and to long
sides.'
'it has two sides the length of 7 cm
and two sides the length of 2 cm. The
two corners down the bottom have an
acute angle in the left side and an ob-
24
tuse angle in the right its theopposite on the top, its like a rectangle on
a lean.'
'is a sort of like a rectangle pushed
out of shape - the two sides facing
each other are equal size, but the angles aren't 90 degrees all the way
along two sides there is an equal distance.'
Rhombus:
'All angles and sides are
even it is like a square on slant'
'has 4 equal sides and 4 equal angles
is also called a diamond' ['4 equal angles' meant equal in pairs]
'is just like a sqliare pushed to about
60 degrees'
'like a square bent over so it does not
have a right angle'
With qualification of,
second
property.
at
least,
the
Square:
'is a slope that has 4 sides and 4
angles. The angles are all 90 degrees. All
sides are even and equal to each other.'
'it has 4 equal sides and each has a
right angle so there should be 4
right angles.'
Rectangle:
'has all parallel sides. The
angles are all 90 degrees. Two lines are
the same but are longer than the other
two lines.'
'has 2 sides that are equal and small
and 2 that are equal and large. Each
corner has a right angle in it. It has
2 horizontal and 2 vertical lines.'
Parallelogram: 'has two long sides and two
short sides but all angles aren't equal.
But two are same and other two are the
same.'
'is a slope of which there are two sets
of equal, parallel lines.'
Rhombus:
'Is a shape that has 4 equal
sides. 2 are horizontal and 2 diagonal
sides that are parallel.'
With qualification of, at least, a secondary property and using Ikonic
Su p port
Square:
'plane figure, 4 sides measuring
the same length. 2 are horizontal,i 2are
vertical. hi each corner, the angles are
all 90 degrees (right angles). Each side is
parallel to one another.'
Rectangle: 'a shape with 2 long sides and 2
short sides similar to a square. 4 right
angles. 2 long sides are opposite. 2 .short
sides are opposite. Short sides· are the
same measurement etc long sides,are the
same measurement.'
Parallelogram: 'Is just like a rec;tangle tilted over. The lines are paralleLandtlle
angles are different to a rec;tangle. It is
similar to a rhombus.'
Rhombus:
'Has four sides of whicllare
in pairs. Each side of a pair is facing the
other so they are parallel. It looks li1<.e a
square pushed over a bit to 1liak~<apair
of lines slope. If you draw a .diamond
you have the right shape.'
Concrete Symbolic Mode: Relational
level (Biggs and Collis) and Level 3
('van
Hiele)
Only one student - a Year 7 gi~l gave what
was considered to be a response at this level.
Her comments for the four shapes are . given
below.
Square:
'Is a flat shape with 4 equal sides
and 4 right angles. This on.e has sides S
ems long. A square is a special tYPc;lof
rectangle. '
Rectangle:
'has four sides with opposite
sides equal. It has 4 right angles and this
particular one is 6cm x 3cm. A rec;tangle
C;an be a square as well.'
Parallelogram: 'Has 4 sides opposite sides
equal and opposite angles equal. All 4
angles are not equal. This parallelogram
is 6cm x 3cm.'
Rhomb us:
'is a parallelogram wi~h all
sides equal. These are Scm x Scm.
Of interest here was how the student Was
beginning to see relationships between
various figures which were not based on
'the look' of the figures. Also, the student
was not able to give the properties of the
figures associated with the diagonal. It
would seem that there could weLl be a hierarchy of properties and the ab sence of
knowledge of some properties may not exclude growth to a higher thought level.
COJ\fCL US I O:N
The findings of the investigation were
many and .extended beyond the original research questions posed.
First, a.ssociated with the research question,
it was clear that a great number of students
between Year 3 and Year 7 describe a geometrical figure in terms of one property,
namely, the number/equality of sides. This
occurred despite students drawings indicating, where appropriate, equal side (by
measurement), angles of 90 degrees and
parallel lines. It appears that the ability to
draw correct diagrams stems from images
that students possess and often these images
do not reflect student understandings in
terms of the properties of a given figure.
The SOLO Taxonomy identifies the concentration on 'one aspect" as unistructural
where the van Hiele theory does not. It is
possible that van Hiele saw the identifying
of one property and of many properties as
'horizontal' growth - hence not deserving
of separate level status. However, the numbers of students who responded with only
one property and the inability (unless directly prompted) to expand their description gave the clear impression that there
was a difference in' thought and that there
was a 'vertical' growth associated with the
move from identifying one property to that
of identifying many properties of a figure.
Further it became clear that there are in
general three stages within the Unistructural level. These stages were: beginning to
focus on a property; attempting (but not
successfully) to qualify the property; being
able to qualify the property. However, for
such figures as the 'square' and the 'rhombus' where the qualification of the property
simply means using the word 'equal' or
'even',it is possible that the intermediate
stage cannot be identified in practice.
25
Other findings that arOSe from the investigation included -
able to integrate these ideas in the descriptions they gave of the geometric figures.
van Hiele's Level 1 is equivalent to the relational level (ikonic mode) of Biggs and Collis. This means that this level of functioning is in a qualitatively different mode
than that of van Hiele's Level 2 or Level 3.
Unfortunately there is little information in
the literature about ikonic mode of functioning. Students who respond within this
mode see figures as shapes related to other.
figures (i.e., 'a pushed over square') or to
common objects (i.e., 'a side of a box'). It
seems that at the relational level within the
ikonic mode the imagining process is consistent (i.e., a student's image of a shape and
reality are similar).
Also associated with the range of responses
was the language used by students. Words
such as flat, smooth, top and bottom, corners, and even, -were constantly used by
students despite the fact that few teachers
in the upper grades appear to use such language. It is of interest that the language of
students transcended state borders and can
be compared directly with student responses documented in the U.S. (see for example
student interviews in' Geddes et aJ.). This
supports van Hiele's proposition that people
on a certain level share a similar language.
It was also noted that as students grew older
their ansWers often became more longwinded but that qualitatively, in terms of
the level descriptors, there was little
change.
Multi-mode
functioning
(i.e.,
responses
from both the ikonic and concrete symbolic
mode) was illustrated by many of the students who were questioned. Here they gave
a property or a few properties and at the
same time felt the need to support their answers with some form of imagery.
Of surprise was that a number of students
in Year 7 were only able to give ikonic descriptions of the figures. One can only
guess at the type of geometrical work that
these students have been involved in over
the years. Although as a balance to these
comments it was not uncommon for students in Year 4 to list at least two properties
for the shapes, square and rectangle. This
enormous diversity once again highlights
problems teachers face in working meaningfully in Mathematics at the level at
which a particular student is functioning.
Associated with this is the v.alue of the interview situation to help correctly identify
a student's level of functioning. For exampIe, a comment such as 'like a square' does
not provide sufficient information. In this
case the student may be relying on imagery
or alternatively the student may be thinking in terms of 'equal sides' and 'opposite
ends being parallel.'
Equally surprising was how rarely the vast
majority of students questioned employed
such concepts as symmetry, parallel lines,
diagonals, and angles to describe the different figures. This is not because students
were unaware of m any of these concepts.
When students were probed, they were often able to describe and talk about the concepts. Their inability arose from not being
26
Finally, despite the fact that the questions
asked of the students were relatively unsophisticated, it was surprising the amount of
information that was 'generated. Although
not a major objective of the study, the questions asked were intended to be of a nature
that practising teachers could make use of
in their teaching. What became clear during the investigation was the great value
that existed for teachers in using such
'open' questions. Students would benefit
from practice in expressing themselves,
writing grammatical sentences, spelling
mathematical words.
More importantly,
however, the questions asked often became
an instrument in student growth. That is, by
the very act of reflecting and having to express an opinion that has not been rote
learnt, students were forced to bring their
ideas together - laying the groundwork for
growth to the next level.
As if this was not enough, the consideration
of the answers within a cognitive developmental framework allowed the students'
level of understanding to be identified. This
provides a direct and realistic means of
identifying the most appropriate type of
content that the teacher should pursue. The
net result of such action in the classroom
would mean the effectiveness of teaching
would be increased and that the professionalism of teaching could be enhanced as the
ad hoc nature of much of the current practice was replaced by procedures which can
take into account student understandings.
REFERENCES
Pegg,].E. (1987)"An Approach to the Year 7/8
Syllab us" Reflections ,12 (3), 1-20.
Biggs,].B. & Collis, K..F. (1982).Evaluating
the Quality of Learning: The SOLO
Taxonomy. NY: Academic Press.
Burger, W.F. & Shaughnessy, ].M. (1986).
"Characterizing the van Hiele levels
of development in geometry", Jour-
nal of Research in Mathematics Education, 17 (1), 31':'48
Case, R. (1985).Intellectual
Birth to Adulthood.
Development:
NY.: Academic
Pegg,j.E. (1989) "Students' Understanding of
Geometric Concepts in the Senior Secondary School" (in preparation).
(1982).
Van Biela Levels and
Achievement in Secondary School Geometry. University of Chicago.
Usiskin,Z.
Van Hiele,P. (1986). Structure and Insight: A
Theory of Mathematics Education. NY:
Academic Press.
Press.
Chick, H.L., Watson, ].M. & Collis, K.F.
(1988). "Using the SOLO Taxonomy
for error analysis in Mathematics",
Research in Mathematics Education
in Australia, May, 34-47.
Watson,].M., Chick, H.L., & Collis, K..F. (1988).
"Applying the SOLO Taxonom.y to Errors
in area problems" in Pegg,j.E. (Ed.)
Mathematical
Interfaces.
(pp.260-281).
Adelaide, AAMT.
Col1is,K.F. (1988). "The "Add up' or 'Take
away' Syndrome" in Pegg ].E. (Ed.)
Nlathematical Interfaces (pp.66-75).
Adelaide, AAMT.
Col1is,K.F., Romberg, T.A. & jurdak, M.E.
(1986). "A technique for assessing
mathematical problem-solving ability," Journal for Research In Mathematics Education 17 (3),206-221.
Davey,G. (1988). "Primary School Geometry - The van Hiele challenge" in
Pegg.].E. (Ed.) Mathematical Interfaces (pp.149-153). Adelaide, AAMT.
Geddes,D., Fuys,D. S, Tischler, R. (1985).
An Investigation of the van Iliele
Model of Thinking itl Geometry
among Adolescents. Brooklyn College, C.U.N.Y.
Halford,G.S. (1982). The Development of
Thought, Hillsdale,. Nj: Erlbaum.
Pegg,J.E. (1985). "How children learn geometry: The van Hiele Theory", The
A ustralian Mathematics Teacher, 14
(2), 5-8.
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