Download EXPLORING THE FUNCTIONALITY OF VISUAL AND NON

NÚRIA GORGORIÓ
EXPLORING THE FUNCTIONALITY OF VISUAL AND
NON-VISUAL STRATEGIES IN SOLVING ROTATION PROBLEMS
ABSTRACT. This article deals with the solving of rotation problems, and shows that there
is an alternative to using mental rotations or their encoding into verbal terms: namely
using geometrical properties. The idea is consistent with the theory which distinguishes
between visual and analytical individuals, but uses the construct strategies instead of the
construct preferred processing mode. Moreover, contrary to many researchers who refer
to this distinction, but who often use it to classify students, this researcher introduces
a new parameter, namely the nature of the task. The article presents the analysis of the
functionality and effectiveness of the different kind of strategies as a function of the task’s
characteristics. The research, dealing not with individual traits but with solving strategies,
offers information that could be helpful for the improvement of geometry teaching.
1. VISUAL PROCESSING AND GEOMETRY
When mathematics educators consider geometry from a theoretical perspective, the key role of spatial abilities is universally accepted, even
though spatial knowledge is not thought of as a synonym for geometric
knowledge. There is also growing evidence that students’ spatial abilities
can be developed by different teaching methods. Therefore, better knowledge about what kinds of strategies students use, and which difficulties
they encounter, when solving geometrical tasks, can contribute not only to
the enlargement of theory but also to the solution of the actual problems
of teaching mathematics.
This kind of knowledge is also significant from a didactical perspective:
it awakens teachers’ awareness of the fact that students may use problemsolving strategies which are different from their own, and that a particular
teaching style can be a learning obstacle for people who use problemsolving processes which are different from those of their teachers, their
manuals, their-textbooks and so on.
Lean and Clements (1981) listed topics where students with poorly
developed spatial abilities might experience difficulties, and these include
geometrical transformations such as translations, reflections, rotations,
dilations, and expansions. All these topics are part of the curricular content
Educational Studies in Mathematics 35: 207–231, 1998.
c 1998 Kluwer Academic Publishers. Printed in the Netherlands.
GR: 201007293, Pipsnr.: 149853 HUMNKAP
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of the compulsory secondary schools in Spain after the recent educational reform. The fact that different children respond to the same tasks in
different ways raises a number of questions which are of interest to the
classroom teacher and the educational researcher. Among these questions
are the ones on which this study tried to throw light.
There are two different ways of viewing a person’s various mathematical
abilities. One way is to consider the level of accomplishment in some given
tasks, which have some common characteristics determined in advance.
The second way is to consider the individual’s cognitive traits that facilitate
the solving processes for those tasks.
Both these theoretical perspectives have generated a number of studies
dealing with spatial abilities. Although these studies have, for the most part,
been reviewed in the past (see Bishop, 1980a, b and 1983; Clements, 1981
and 1982; Clements and Battista, 1992; Eliot and Smith, 1982; Lohman,
1979a, b), it is important to refer to the terms and constructs that have been
used. For instance, constructs like visualisation appear in almost all of the
studies which refer to spatial abilities, independent of the conceptualization
of spatial ability underlying the study. Very often a unique concept has been
given different names, and different concepts appear with a similar name.
One instance of this is found by comparing the disparate meanings attached
to the world ‘visualisation’ by different authors such as Dion et al. (1985),
Bishop (1983), Tartre (1990), and Senechal (1991). This reaffirms the need
to clearly state the conceptualizations on which the research is based, in
order not to lead to misinterpretations of the findings.
Bishop (1980b, 1983), taking as a starting point the idea that it is
impossible to establish a single definition of spatial ability, and trying to
focus attention on the significant learning processes, suggests we consider
two different abilities: ‘the ability of interpreting figural information (IFI)’,
and ‘the ability of visual processing (VP)’ (1983, p. 184). The importance
of VP ability lies in the fact that Bishop emphasizes those aspects related
to processes over those related to the stimuli form.
Bishop refers to visual processing in the mathematical context, in its
broadest sense, and therefore in a context where visual stimuli are not
always needed. Therefore, one could distinguish two different aspects of
VP ability. Firstly, in the broadest context of mathematics, where abstract
relationships do not necessarily have a visual origin, VP ability could be
considered as the visualisation and transformation of non-figural information into visual terms. Secondly, when referring to geometry, one could
restrict VP ability to the ability to mentally manipulate and transform visual
representations and visual imagery.
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This is particularly important in the domain of geometry, where visualisation and visual processing have to be considered carefully. Very often, if
not always, in primary and secondary geometry classes, geometric objects
are introduced to students through a drawing or a model, and through geometric tasks, which easily lead to using, or needing, drawings and models.
In primary and secondary geometry courses, we have to deal with a representation of space which concerns both form and content, and that duality
throws up different kind of obstacles (Bishop, 1986; Presmeg, 1986a). The
objects under study are always associated, to some extent, with objects
having a physical or visual entity. The fact that geometrical concepts are
associated, up to a certain point, with physical objects or drawings makes
the relationship between geometry and visualisation much more complicated than it appears.
In fact, most of the studies that analyze cognitive processes in relation
to visualisation, are interested in the solving processes of mathematical
problems in general. The early research was generally done by people with
a strong interest in visualisation. For a long time, the focus was on the
contribution of visual abilities to the learning processes, a situation that
has contributed to the neglect not only of the analysis of the non-visual
strategies but also of the strategies not tied to the individual’s processing
mode. However, if the research is about geometry and not just about visualisation, one has to remember that geometrical activity involves both visual
and non-visual abilities. Little has been published regarding the analysis of
the strategies present in the solving processes of geometrical tasks taking
into account the possibility of using, or not using visual processing, and
other aspects beyond the processing mode. Therefore we need to know
more about when particular strategies are important and for what tasks
they are important.
2. STRATEGIES INVOLVED IN THE SOLVING PROCESSES OF GEOMETRICAL
TASKS
A review of the existing literature shows that the tendency of researchers concerned with mathematical abilities, and in particular, with spatial
abilities, has been to classify students according to their efficiency of, and
preference for, using particular processing modes, rather than to analyze
strategies. Moreover, the results of their studies have to be interpreted
within a cognitive style frame, which is difficult for the teacher to use.
Therefore, as the interest of this researcher is with teaching and geometry,
the study reported here deals with strategies as a more useful construct
than cognitive style.
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This study deliberately proposes the use of the construct spatial processing ability instead of the construct visual processing ability, in order
to state clearly the difference between the ability to solve any situation by
means of a visual processing strategy, and the ability to cope with a spatial
task, already having visual roots, using any kind of strategy.
In the present research, spatial processing ability is understood as the
ability needed to fulfill the combined mental operations required to solve
a spatial task. It includes not only the ability to imagine spatial objects,
relationships and transformations and to decode them visually, but also the
ability to encode them into verbal or mixed terms. Furthermore, it includes
not only the ability to manipulate the visual images 1 of spatial facts, but
also the ability to solve the tasks using processes that are not merely visual.
Obviously, the spatial processing ability so defined, even if described with
a singular term, has plural meanings. Spatial processing ability includes:
at least as many different abilities as there are different spatial transformations: rotation, cross-sections, unfolding and so on.
the ability to interpret spatial information: the ability needed to understand not only graphical descriptions and modeling of spatial facts
and relationships, but also the verbal or mixed ones, and the specific
vocabulary used in geometrical work.
the ability to communicate spatial information: the ability needed to
produce descriptions of spatial objects, relationships and transformations, being the content of those descriptions, figural2, verbal or mixed.
Existing studies of spatial abilities mostly emphasized the individuals’
characterization, ignoring a defining remark made by Krutekskii (1976),
that ability is always an ability for a concrete type of activity and, therefore,
it only shows itself in the analysis of a specific activity. Some authors
(Guay et al., 1978; Lean and Clements, 1981; and Paivio, 1971) pointed
out the need for considering the amount of observed variance in spatial
tests attributable to the task variable.
Paivio’s work, even though being based on a psycholinguistic model
and dealing, mainly, with memory processes, points out several aspects
that could be paralleled in the field of mathematics education, and offers
us with a suitable model for the purposes of this research. Paivio points to
three variables which influence the amount of visual imagery a person uses
when performing a task, namely, the characteristics of the task; the extent
to which the type of thinking is specified in the definition of the task; and
the different processing modes used by individuals. In particular, Paivio
asserts that ‘effects of imagery are less predictable when this process
is defined by individual difference measures than when it is defined by
stimulus characteristics or instructional sets’ (p. 524). This author proposes
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that more research is needed which incorporates relevant item attributes,
instructional sets and individual differences in a simple design.
Lean and Clements said that ‘research has not thrown much light, for
example, on the question of whether persons who prefer to use visual
imagery, with little verbal coding, when processing mathematical information are likely to do better on certain mathematical tasks than persons who
prefer a verbal-logical processing mode’ (p. 278).
Thus, the research reported here did not aim at characterizing individuals from the analysis of their behaviour while solving different activities,
but at the analysis of how individuals deal with some given activities taking
into account the activities’ characteristics. The goal was to analyze how
the distinct strategies function and, in particular, to determine the conditions under which visual or non-visual strategies are used, and to analyze
their effectiveness for solving geometrical tasks. Although the emphasis is
on the different functions of visual and non-visual strategies, and on their
contribution to spatial processing ability, it is likely that the two kinds of
strategies interact continually in tasks related to geometry that are assumed
to involve imagery.
Given that the field of geometrical transformations is very broad, the
present study focused on one of its aspects, namely spatial rotations. The
research analyzed and characterized the strategies used by a sample of students, aged 12–16, when dealing with geometric tasks that required spatial
rotations. The main goals were: to analyze how the task characteristics
influence students’ strategies, and to search for any relationship between
achievement and the strategies used during the solving processes. Collateral goals were to compare, as sample groups and not as individuals, the
strategies and not just the achievements of boys and girls, and of different
age groups.
Taking as a starting point Burden and Coulson’s study (1981), and
modifying it to fit the present research goals (for further details see Gorgori ó
1995), students’ strategies were analyzed from three different standpoints:
Structuring strategy, the origins and the organizing of the information
used,
processing strategy, the mental representation mode,
approaching strategy, the focus of attention.
These, for every subject and for every task, were not three different kinds
of cognitive strategies, but three different aspects of the student’s solving
strategy.
For the study of structuring strategies, the student’s cognitive strategy
was considered from the standpoint of the different mental ways of facing
the task, the mental organization, and source of the information used to cope
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with the task. When analyzing structuring strategies, they were organized
according to the answers to different aspects:
Does the student use only the information given in the task statement,
or is information resulting from previous experience or knowledge
explicitly used?
Does the student involve him/herself in the context, and how? Does
the student give meaning to the task by creating a real context, and
how?
How does the student organize the given information? For instance,
when the task requires different options to be compared, does s/he
select one as a model and compare the others against it, or does s/he
make all possible comparisons?
Structuring strategies so established are clearly not restricted to the
domain of solving processes of spatial tasks, and this construct could be
used to analyze the solving processes of any mathematical problem.
When analyzing processing strategies, the student’s cognitive strategy
was considered from the standpoint of its form of mental representation.
Following Krutekskii’s (1976) model, the premise was made that all mathematical problems require reasoning or logic in their solving processes.
Furthermore, all the tasks presented in the present research had a figural
support in their presentation. Therefore, what determined the kind of processing strategy used by the student was, as in Presmeg’s (1985) study,
whether or not the student made use of visual images as an essential part
of the solving process, a fact that could only be elicited from student’s
explanations and from observations.
Obviously, mental images and verbal processes do not function independently of each other in any thinking task. Not only do they interact,
they also have supportive functions (Fischbein, 1993; Lean and Clements,
1981; Lohman, 1979b; Mariotti, 1996; and Paivio, 1971). However, if the
purpose is to analyze their functions, it is important to consider their characterizations separately, especially since many of the relevant studies were
done by researchers with a strong interest in visualisation. This means
that non-visual strategies have been all considered within a broad category
without further analysis.
As the present research aimed at typifying the strategies used by students on every task, as a group, and not as individuals, only two kinds of
processing strategies were considered, a priori, visual and non-visual:
A student’s processing strategy was characterized as being visual
when, from the student’s explanation and observation, one could elicit
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that s/he had used visual images as an essential part of the method of
solution.
A non-visual strategy was one where the student made explicit that
s/he had used an argument, and had not relied on visual images when
solving the task.
It is important to note that both visual and non-visual strategies need
logical reasoning, and are used to solve spatial tasks having visual support.
The purpose of that ‘quite simple’ a priori characterization was to allow a
deeper analysis of both kinds of processing strategies. That is, to analyze
not only the different kind of visual images students used but also the
different kind of arguments they elicited.
The analysis of the students’ approaching strategy considered the student’s focus of attention. The construct of ‘approaching strategy’ has a
meaning close to the construct of ‘focus’ which Leinhardt et al. (1990)
developed, even if the latter was defined in the domain of graphs of functions. The student’s approaching strategy was determined by studying
where s/he focused his/her attention, regarding either the geometric object
or the situation, when coping with the task.
Approaching strategies were characterized as global or partial according to the attention focus of the mental strategy over the geometric object.
A student was considered to be using a global approaching strategy when
his/her cognitive strategy was focused on the object considered as a whole.
A student was considered to be using a partial approaching strategy when
his/her cognitive strategy was focused only on some parts of the object.
One of the study goals was to analyze how a task’s characteristics might
affect the students’ strategies. According to Paivio (1971), the characteristics of the task that may affect the processes individuals use when solving
it include: attributes of stimulus (imagery concreteness, meaningfulness
in the sense of familiarity, complexity, the possibility of unitization of
the objects) and explicit instructions to use imagery. Wattanawaha (1977)
also took into account the characteristics of stimulus and answer form, but
argued that the explicitation of the processing mode was also likely to have
an effect.
In the present study, among the characteristics that were considered as
being liable to modify or influence students strategies, the most significant
turned out to be the required action. Required action is the action to be
done by the subject in order to solve the task, in the sense established by
Leinhardt et al. (1990). According to these authors, the required action can
be one of interpretation or of construction.
‘The required action is considered one of interpretation when the
student has to gain meaning or to obtain information from an object or a
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representation’ (p. 8). A task was considered to be one of interpretation
when it required the student to react in the face of a geometric action
presented as accomplished. For instance, multiple choice items where
the subject has to select the drawing which represents the result of a
certain geometric transformation are tasks of interpretation.
‘The required action is one of construction when the student has to
generate a new object, constructing or representing it’ (p. 12). In the
present case, when given the initial object the student had to generate
the final one, which means that s/he had to carry out the geometric
action, mentally or through manipulation, on the object to generate a
new one, real not imagined.
In terms of their relationship to each other, these authors make us note that
‘whereas interpretation does not require any construction, construction
often builds on some kind of interpretation’ (pp. 12, 13).
3. METHOD
Qualitative data obtained through clinical interviews were used in the
analysis of the students’ solving processes, despite an awareness of the
limitations of both qualitative and quantitative methodology. Quantitative
analysis was also used, in order to achieve the other goals of the study.
Qualitative and quantitative analysis, being complementary, generated the
research results and contributed to the study’s validity.
Nine tasks were presented to the sample of students to be solved during
the interviews. The geometric content of all the tasks was a spatial rotation.
All the tasks were presented with visual support, using both real objects
and 2-D representations of 3-D objects. Those representations used either
the usual codes, or introduced new ones to students which were easy for
them to understand. The students’ understanding of those codes had been
tested in previous sessions.
The objects involved in all the tasks were chosen in order to allow either
kind of approaching strategy: even if they did not have any distinctive part,
they had a shape and a structure that, by suggesting some kind of relational
organization or of unitization, allowed the student to consider different
parts on it when wanted or needed.
As in the case of presentation form, the answer form took into account
not only the kind of language used, visual or verbal, but also the representation code required for the answer. Therefore, two similar tasks demanding,
for instance, to construct and to draw an object, were considered as being
different.
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Figure 1.
Among the tasks, there were 4 whose required action was of interpretation. Three of the tasks of interpretation had the form of a multiple choice
question, where students had to decide which was the correct answer by
identifying objects as being the same, or not, through rotation. Task 2–I,
presented in Figure 1, is one such example.
In the fourth interpretation task, 4–I, given a part of a map of a town,
the student had to describe the route from a given point on the upper part
of the map to another given point on its lower part.
There were five tasks whose required action was one of construction. In
those tasks, students had either to draw or to construct, with wooden cubes,
an object satisfying the geometric requirements. Tasks 2–A, presented in
Figure 2, is an example of such a task. The object referred to in task 2–A
is presented in Figure 3.
Some of the tasks presented a situation within a context with real
meaning, and some did not. For instance, task 3–I presented the drawing
of 4 pairs of false dice which were not numbered in the usual way, and
which had dots in place of numbers in order not to give any orientation
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Figure 2.
Figure 3.
clues, and then asked in which pair both drawings could be the same. Dice
were known to be familiar to students, therefore, this task was considered
to be presented within a real meaningful context.
The tasks had different level of complexity, understood as a function of
the difficulty to obtain, from the given object, the information necessary to
deal with the geometric requirements. A complex object here was defined
as one that had several parts, units or details connected with each other.
However, the task complexity depended not only on the complexity of
the object but also on how difficult it was to obtain information from it.
From that point of view, among the previous examples, tasks 2–I and 4–I
had a low level of complexity, and tasks 2–A and 3–I had a high level of
complexity.
Although in all tasks the geometric transformation was a rotation, the
statement of the various tasks presented the request in different forms:
suggesting a change in the object’s position, ‘as it would look like after
rotating it’, or a change in the subject’s position, suggesting a change
of the point of view, ‘as it would look like when seen from behind’.
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However, explicit instruction to form a mental image was avoided in all
tasks, for Guay et al. (1978) and Paivio (1971) assert that simply instructing
subjects to use imagery on the statement of the task can affect their solving
processes.
The tasks were administered to a sample of 24 students, aged 12 to 16,
selected from a broader sample of 645, from different types of schools, who
had been administered a test, created and validated in a previous study (see
Gorgorió, 1995, for further details). The test included some 3-D geometry
items related to particular curricular content, and some other items to test
the performance of students in spatial tasks in general.
When selecting the 24 students to whom the tasks were going to be
administered, the student’s characteristics were diversified, taking into
account theoretical conditions, e.g. gender, age and performance on the
spatial test. The selection was made as follows: first, 4 different class
groups were selected, 2 of 12 year-old students, and 2 of 16 year-old
students, representing different types of schooling. Within each group,
3 boys and 3 girls were selected to ensure that among students of the
same gender within each class-group, there was one with a high level of
performance in the test, one with an average level, and one with a low
level.
For every task, the interviews were prepared beforehand, based on the
results and observations of a pilot experiment. The interviewer was, for all
tasks and students, this researcher herself. In this way, she could ensure
that the information collected through the interviews with different students
could be confidently compared.
The interviews were tape-recorded, and drawings and objects made by
the students were kept. Students’ processes of drawing and construction
were recorded through codified notes. During the interviews, the researcher
also noted actions, movements and gestures made by the students that
were considered to be hints of the strategies being used. The students were
also asked for their own description of their solving processes once the
task was accomplished. The transcripts of all interviews, drawings and
objects produced by students during the interview, and researcher’s notes
comprised the initial data. The consistency of the data and the comparison
of the subjective reports with the observed behaviour suggested that the
reports were valid indicators of the strategies actually used by the subjects.
Systemic networks, (see Bliss and Ogborn, 1979; Bliss et al., 1983)
were used to unfold, structure and reduce the data. Partial and recurrent
systemic networks were created to achieve a general form that enabled the
analysis, for every task and for every student, of the student’s solving pro-
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Figure 4. Systemic network (Task 2–A).
cess. Transcriptions and systemic networks were reviewed by two external
judges apart from the researcher herself.
In figure 4, there is, as an example, a part of the systemic network
dealing with the analysis of the students’ cognitive strategies from the
standpoint of their mental representation form, corresponding to task 2–A
presented before. The initials stand for the names of the students whose
answers are transcribed afterwards as examples.
By means of the systemic network, students were organized into two
initial categories, according to their mental representation form: those for
whom it was clear that they had used visual images during the solving
processes, and those who used an argument. The individual’s report that
s/he ‘saw an image’ was the major criterion to decide wether s/he used
mental images. In addition, when questioned about the strategies they used,
students apparently had no difficulty in distinguishing between images and
verbal mediators, and, in some cases, the student even described the image.
When saying that it was clear that a student had used visual images, it means
that either the student had explicitly said s/he had imagined a particular
situation, or said that s/he imagined performing an action – a physical
action – which actually did not take place. This was the case with E.S.,
who said I went behind the object, when actually she was sitting still all
the time!
Differentiating visual and non-visual strategies empirically is an interesting puzzle in its own right. The quotations here are only given to show
the kinds of verbal arguments or geometrical properties students relied on.
To decide whether the student relied on visual images, on verbal arguments, or used geometrical properties to solve the tasks the researcher and
the judges considered the whole protocols.
In task 2–A, some of those who used visual images, had imagined
changes in the object’s position:
S.F.: I’ve imagined, more or less..., how the object is when turning it round.
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and the others had imagined changes in the subject’s position:
E.S.: ... I’ve imagined... the object, well, from behind, I’ve imagined it... I
went behind the object.
Among students who had not used visual images but who gave an
argument, one could distinguish between those who had used a verbal
argument and those who had used a geometrical argument.
Those who had used a verbal argument described the appearance of
the final object under different conditions; some of them referring to the
changes produced in the appearance of the object when rotating it:
J.N.: ...because, if I turned it round... this part would go to the left.
some referring to what one would observe when seeing it from another
point of view:
H.G.: it goes up to the other side, because if you see it from behind...
As a tentative explanation for these verbal arguments I would suggest
that they are the result of a verbal encoding of a previous experience of
visual processing. The student would have used verbal coding devices to
organize the information s/he had received from a previous perception or
from a previous visual processing task.
On task 2–A, those who used a geometrical argument relied on right/left
inversion properties:
R.V.: ... then there are three (cubes) over them, but instead of being on the
left, they have to be on the right, because as you see it from behind,
you see it has changed...
Comparing the data relating to all the tasks, structured through networks, allowed the characterization and description of the different kind
of strategies.
Other research goals required a quantitative analysis; for example, to
know if the different kind of strategies within each category (structuring,
processing and approaching) led or not to correct answers in different ways.
For each task, and for each category of strategies a table was created which
summarized the number of correct answers and errors collected for each
kind of strategy. From the tables, one could observe, in some cases, the
existence of some tendencies. Further statistical analysis of the tables was
used to decide which tendencies were significant enough to be considered.
Broader results were achieved comparing the evidence obtained through
parallel processes done for each task and for each kind of strategy. For
instance, Table I concerns the relationship between the kind of processing
strategy followed in the process and the answer’s correctness. Table I
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TABLE I
Processing strategies used and answers’ correctness
Task
Complexity
Tendency
Significant
Strategy
correct
Strategy
error
1–I
2–I
3–I
4–I
1–A
2–A
3–A
1–B
3-B
low
low
high
low
high
high
low
low
low
yes
no
no
yes
yes
yes
no
yes
no
–
–
–
95%
–
–
–
90%
–
visual
–
–
visual
vis/non-vis
non-visual
–
visual
–
non-visual
–
–
non-visual
visual
visual
–
non-visual
–
presents, for each task, the existence or not of tendencies relating processing strategies to geometric errors, significance percentage, and which
kind of strategies led to correct answers and errors. From the table arose
some results that are presented in the next section.
A similar process of analysis was used to obtain the results concerning
the differences among strategies used by boys and girls.
4. RESULTS
For each task, students’ cognitive strategies were characterized as being
structuring, processing and approaching strategies. Below, the reader will
find the description and some examples of the different types of strategies
within each category. In all cases, the transcribed phrases are part of the
answers students gave when asked how they had known which one was the
right answer, or how they had arrived at their conclusions. The transcribed
sentences are only illustrative of the different categories, and should be
considered just as examples. Giving a full account of the descriptors of the
categories is beyond the scope of this paper.
The data suggest a differential use of the structuring strategies as a
function of the tasks’ characteristics of context and formulation. When
the task was presented within a context with real meaning, the structuring
strategies observed implied the subject’s involvement in the context, in the
sense of pulling out information which was not essential to fulfill the task’s
geometric request. For instance, in task 4–I , where students were given a
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map and were asked to give a route, some of them referred to the names of
the streets, as indispensable elements, to describe the way:
A.F.: ...if there are no names... I cannot tell him. If I had time, I would lead
him there.
When the task was presented within a context with no real meaning,
the structuring strategies observed implied that the subject made use of
information obtained from previous experiences or knowledge which could
explain the situation or which helped to solve the task. For instance, in task
2–A, some students said, without being asked, and just before initiating
the solving of the task, that they should take into account what changes
take place when turning an object 180 around its base.
A.B.: I have to build up the object, say... the part behind has to be in front,
and the right has to go to the left ... when building it up.
When tasks were of interpretation, the structuring strategies depended
on the task’s formulation, and consisted of simplifying the task’s structure
which relates the given information to the required action. In task 2–I, for
instance, where students had to compare four options, some of them took
one of the options as a model, and compared the others with that one.
Processing strategies were characterized as being visual or non-visual.
The students who used visual processing strategies imagined some of the
following aspects: the task’s context, a rotation or a change in position of
either the subject or the object.
Examples of imagining the context of the task are found in some students
saying that they imagined a real die, when they had to compare the drawings
of false dice:
R.U.: I imagine a real die, with its dots, and I compare it with these here...
Instances of students imagining a rotation or a position’s change had been
given in the previous section.
Among non-visual strategies one could distinguish the verbal strategies
and the geometrical strategies. In the previous section there are some
examples of verbal strategies where the students described the final object,
interpreted as the result of verbally encoding some previous visual experiences or processing. Other verbal strategies are those where the student
used information belonging to the context. One example of this is found
when some students, when explaining their solving processes of task 3–I,
where they had to compare the drawings of false dice, said that they knew
where a real die had its dots, and checked if the drawn dice had the dots in
the same position:
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L.S.: because I know, more or less the positions of the numbers in..., the
dots in real dice..., and I checked here.
Regarding the geometrical strategies recorded, the students relied on
facts related to properties of 180 rotations, symmetry, and congruence,
with the most frequent being those where the subject used 180 rotation
arguments. In the previous section, there are some instances of how students
solved task 2–A, by explicitly relying on properties of 180 rotations.
Examples of students relying on symmetry facts, are found in some
students giving answers similar to the following to task 2–A:
M.L.: When I see it from behind, it is the same... well, as from, as... in a
mirror,... it is symmetrical.
Instances of using a congruence argument are found in some students
giving answers, in task 2–I, similar to the following:
M.C.: I look at its form... this part has to be this part... it is clear, those
three (referring to the figures) have to be the same, this part on them is
longer,... they have the same, ... the same... shape, the other is different.
Functional distinctions between visual and non-visual processing strategies were found depending on the task’s characteristics. The appearance of one of the different processing strategies depended on the task’s
complexity, and the required action. When the required action was of
interpretation, students tended to use visual processing strategies when the
given object was simple, and to use non-visual processing strategies when
it was complex. When the required action was of construction, students
tended to use visual processing strategies when the given object was complex and manipulation was not suggested (a drawing was required), and
to use non-visual processing strategies when the given object was simple
or manipulation was required (e.g. students had to build an object). This
would be, for instance, the case of tasks 1–A and 2–A, both of them dealing
with the object presented in figure 3. For those tasks, some of the students
explained explicitly, without being asked, that when the task required
manipulation it was not necessary to imagine the position changes:
E.S.: Now, (referring to task 2–A) as I can do it, I did not need to imagine
anything... I’ve just... put the cubes where they have to be...
Approaching strategies were characterized as being global or partial.
The students who used global approaching strategies either compared the
object or the situation with a real life object or situation, or referred to the
objects’ congruence. For instance, in task 2–I, some students referred to
the object by comparing it with an everyday object:
R.V.: Those three have a shape, like, like a gun...
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Here is an example of a student who considered the object as a whole
when analyzing the congruence in task 2–I:
H.G.: Because, these two have the same structure, ... changing the position
of the second one, ... they fit into each other.
Regarding the partial approaching strategies recorded, the students took
into account some of the following aspects: the existence of distinctive
parts, their characteristics, their relative position, or the elements resulting
from splitting up the object and which cannot be considered distinctive
parts. Examples of considering the existence of distinctive parts are found
in some students focusing their attention on the upper part of the object,
when solving task 2–A, and completing afterwards its building up:
D.P.: ... I have seen that the apex here had to go there ... and then, from
here, I have constructed it, three, two, one and after, next to them, the
one, two, two.
Here is an example of a student considering the characteristics of distinctive parts when answering to task 2–I:
M.C.: I look at its form... this part has to be this part... it’s clear, those
three (referring to the figures) have to be the same, this part on them is
longer,...they have the same, ... the same... shape, the other is different.
An example of a student focusing her attention on the relative position of
distinctive parts, when she considered the relative position of the numerals
of the dice in task 3–I is:
S.R.: ... in option C, they would fit, isn’t that right? So that when rising,...
when rising the 3 (referring to the dots on the dice) to the upper part,
then, well, the 3 would be over, the 5 on this side, ... one wouldn’t see
the 1, and this would be the 6.
An example of a student focusing his attention on the elements resulting
from the splitting up the object into parts when solving task 2–A is the
following:
O.M.: ... I have split the object into two parts... I removed these two slices
in front, I’ve turned it round, I’ve built it up, then I’ve fitted them in
again ...
Once again, the data raised evidence of the fact that the task’s characteristics influenced the strategies which students used. The appearance of
one or another approaching strategy depended on some of the task’s characteristics: the required action, the similarity among the objects involved
in the task, and their complexity. For instance, when the required action
was of interpretation, students tended to use global approaching strategies
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when the objects appearing in the task were clearly different. Also, in tasks
of interpretation, most of those who used partial approaching strategies
focused their attention on the relative position of the objects’ parts. When
the required action was of construction, students tended to use partial
approaching strategies when tasks allowed manipulation (to build up an
object), or when the given object was simple.
Therefore, for each category of strategy, the appearance of one type or
another depended on one or more characteristics of the task. In particular,
the task led to different cognitive strategies depending on its required
action, being either interpretation or construction.
Note that one of the quotations above (student M.C.) appears both
as an instance of a geometrical processing strategy, using a congruence
argument, and as an instance of a partial approaching strategy, taking
into account the characteristics of significant parts. This quotation appears
twice deliberately, to illustrate that, for every student, to refer to his/her
structuring, processing and approaching strategies does not imply that s/he
used three different cognitive strategies, but that his/her solving strategy
has been considered from three different perspectives. In a similar way, the
quotation of O.M. presented above, as an instance of a partial approaching
strategy, could have also been presented as an example of a visual processing strategy, stated by the student to be based on performing a physical
action, which he actually did not do.
Students’ errors observed during the interviews were: misinterpretation
of the task’s statement, errors when communicating the answer or explaining the solving process, and geometric errors. Among interpretation errors,
the most frequent were those related to the use of 2-D representations of
3-D objects, errors similar to those presented in Baldy (1988).
Referring to communication, the most frequent errors were related to
the use of both verbal and graphical codes. Errors related to the use of
verbal codes when giving the answer or explaining the solving process,
consisted of naming the objects’ parts using words both from everyday
life, or corresponding to 2–D geometry (e.g. side instead of face), and
using ambiguous or incorrect expressions when referring to the position
or the movement of objects. Referring to the use of graphical codes when
presenting the answer, the errors were similar to those in Ben-Chaim et al.
(1989). In general, difficulties and errors relating to communication, both
verbally and graphically, were similar to those presented by Hershkowitz
(1990). The most frequent consisted of not knowing how to draw a part of
an object which was behind, in front of, over or under another one which
has already been drawn.
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Geometric errors showed the absence of right/left inversion or the
absence of front/back inversion when rotating objects through 180 . Thus,
the only geometric error that appeared was to mistake a 180 rotation for
a symmetry.
As has been said, the different uses which students made of the different
kinds of strategies is a function of the task’s characteristics. Moreover,
the data also show that the usefulness of the strategies, in the sense of
effectiveness, also differs. For every category of strategies, its different
types led to correct or incorrect answers depending on some of the task’s
characteristics.
The type of structuring strategy used led, in some cases, to wrong
answers. For instance, when the required action was of interpretation,
and the task was presented within a real meaning context, there was a
structuring strategy that led significantly to geometric errors: the one in
which the subject got involved in the context of the situation, by attaching
a real meaning to it. For instance, in task 3–I, all the students who relied
on ‘real dice’, imagining them or not, instead of considering only the
information given on the task, got wrong answers.
In general, the type of processing strategy or of approaching strategy
used did not lead significantly to any difference between the number of correct answers or geometric errors. Concerning processing strategies, only
in two tasks (see table I), where the given object was a simple one, was a
significant difference found between the number of correct answers and the
number of errors, with the visual processing strategies leading to correct
answers. However, while not being statistically significant, some tendencies appeared relating errors to the processing strategies used, depending
on the task’s complexity. When the given object was a simple one, visual
processing strategies tended to lead to correct answers, while non-visual
processing strategies tended to lead to errors. When the given object was a
complex one, the tendencies were just the opposite.
Concerning approaching strategies, only in one task, where the given object was a simple one, was a significant difference found between
the number of correct answers and errors, with the global approaching
strategies leading to correct answers. Considering all the tendencies which
appeared, even those which were not statistically significant, only in two
tasks, where the given object was a complex one, did partial approaching
strategies lead to correct answers and global approaching strategies lead to
geometric errors.
The present research supports and extends the findings of Guay et al.
(1978), Lean and Clements (1981) and Paivio (1971) concerning the influence of tasks characteristics on the variation in the solving processes of
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spatial tasks. According to the results, the task characteristics which influence the use and effectiveness of strategies are: the context, having or
not a real meaning; the task’s formulation; the complexity of the objects
involved; and, most significantly of all, the required action.
Moreover, the results also add evidence to the argument which some
authors have previously suggested (Guay et al., 1978; Lean and Clements,
1981; and Lohman, 1979a and b) that one should not identify spatial ability
with visual processing for, as we have seen, rotation tasks (at least) can be
correctly solved using both visual and non-visual processing strategies.
In terms of gender differences, some qualitative differences were
observed among the structuring and processing strategies used by boys
and girls, but no differences appeared in their approaching strategies3.
However, the results of this study confirm the fact that gender is not a
significant differentiating variable when analyzing the solving processes
for spatial tasks, because the differences between genders are less than the
differences within genders, at least with tasks whose geometric demand
is a rotation. In particular, the author would agree here with Guay et al.
(1978) in saying that the difference between males and females may not be
so much a function of the strategies used, as a function of the effectiveness
of using the strategies.
In terms of age differences, no significant differences were found among
the strategies used by the different age groups, nor among their effectiveness. When interpreting that fact, however, one should take into account
that age was a variable interfering with the type of school, and therefore
it may still be as important as one might think, a priori. Furthermore there
may also be a school teaching effect being masked by the age variable.
5. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH
The present research has demonstrated that the construct of strategies is
useful for explaining students’ solving processes in geometric tasks. As a
construct for analysis, strategies clearly has a different character from the
constructs of level of ability or preferred processing mode: one can analyze
both the differential strategies available and their effectiveness in terms of
task characteristics.
From the point of view of teaching, strategies can be shared and therefore taught, while preferred processing mode is an individual’s trait. If it is
the teacher’s belief, as it is of this writer, that education should achieve the
best for each individual without having him/her to renounce their personal
traits, the results of research dealing with the characterization of strategies
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are much easier to handle and more useful than those related to cognitive
style.
From the results of this study some general conclusions arise concerning
the spatial orientation ability, understood as the spatial processing ability
needed to solve rotation tasks. The research suggests that other kinds
of strategies apart from those tied to the processing mode are relevant
to geometrical activities. In particular, the individual’s spatial orientation
ability depends on his/her capacity to make successful use of structuring,
processing and approaching strategies.
For each category of strategies, the appearance of one type or another
and its effectiveness depends on one or more characteristics of the task
proposed, with the required action of the task being the most relevant
characteristic. Therefore, the demonstration of one’s spatial orientation
ability depends not only on his/her capacities or potentialities, but also on
the task’s characteristics s/he is facing. In particular, any differences in the
observed results obtained by students by means of visual or non-visual
strategies are potentially attributable either to the relative availability or
relative effectiveness of those strategies, or both, depending on the nature
and demands of the task.
Furthermore, the individual’s spatial orientation ability depends on a
combination of factors including not only abilities understood as characteristics of an individual’s mental processes, but also abilities understood as
capacities that allow those processes to take place, and allow their results
to be revealed. During the research, some students’ difficulties and errors
have been observed relating to their interpretation of 2-D representations
of 3-D objects, to the use of 2D-drawings to represent 3-D objects, and to
the use of verbal codes which refer to spatial facts. All those difficulties
and errors obstructed or hindered the students’ solving processes, because
interpreting and communicating spatial information are elements necessary
for understanding the demands of tasks and for expressing their results.
Therefore, one may conclude that the individuals’ demonstration of their
spatial orientation ability depends also on their abilities for interpreting
and communicating spatial information.
From the results one may conclude also that, in general, the kind of
processing strategy or of approaching strategy used, does not lead to significant differences in the number of correct answers and of geometric
errors. In particular, any task may be correctly solved using both visual or
non-visual processing strategies. Therefore it is inappropriate to continue
to identify spatial ability with visual processing as some authors, tacitly or
explicitly, have done.
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We need more research that contributes to the evidence that visual
and non-visual strategies, and in particular, geometrical strategies, are
distinct and, more importantly, that the difference has both theoretical and
practical significance. This calls for a clear demonstration that the different
kinds of processing strategies are functionally different rather than merely
being names for a kind of strategy or for different kinds of strategies that
nonetheless are indistinguishable in the solving processes of geometrical
tasks.
The results confirm the fact that gender is not enough of a differentiating variable when analyzing solving processes of spatial tasks, at least
when students face tasks whose geometric request is a spatial rotation.
Therefore, this author joins Clements and Battista (1992) in their conclusions: ‘because there is obviously much more variability in performance
and processing within genders than between them, we should eventually
be able to move beyond studying gender differences to the study of different cognitive profiles that underlie successful performance in geometry’
(p. 458).
The methodology used in a research project always affects the nature
of the results arrived at. By large sample testing one may assert general
conclusions, which however only give information about achievements and
not processes. On the other hand, with qualitative analysis of data, obtained
through interviewing a reduced sample, the results one may obtain about
students’ solving processes are of a descriptive nature, and can be used
only to explain students behaviour on analogous situations. Therefore,
more research is needed to complete what has been done until now.
Spatial orientation ability is only a particular kind of spatial processing
ability. It would be interesting to broaden the range of geometric transformations presented to students, symmetry, cross-sections, unfolding and
so on, in order to obtain information to establish conclusions concerning spatial processing ability in general. Also, more research is needed to
analyze students’ cognitive strategies when varying tasks’ characteristics;
particularly, on how the required action controls students’ strategies and
their effectiveness.
At the beginning of this paper, and for the purpose of the present
research, the author suggested that there are two aspects of VP ability
defined by Bishop which should be considered further. Until now, the published studies have focused, in the main, on only one of the two aspects:
either on the visualisation of abstract relationships, or on the mental manipulation of visual representations and imagery. However, considering VP
as a unitary construct, it would be interesting to know more about which
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features visual processing has in common when considering both abstract
relationships and spatial imagery.
The above discussion gives only a sketch of the great deal of work
which this study suggests is still to be done. This author would however
just insist on one final recommendation: given that researchers should keep
on working on visualisation, visual processing, visual thinking, visual
reasoning... and on all kind of ‘visual plus suffixes’, should we not all
make the effort always to state clearly in our research reports the meaning
attached to such words in order that we may all improve our understanding?
NOTES
1 Throughout
this paper, visual image should be understood in the sense
defined by Presmeg (1986b): ‘visual image as mental scheme depicting
visual or spatial information’ (p. 297).
2 All through this paper, figure and figural will refer to drawings and graphical representations.
3 To refer to them on this paper, surpasses its purposes and its length. For
a full account see Gorgorió (1996).
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