GIRLS' USE OF SPATIAL PRECEPTORS AND VISUAL SEGMENTION IN THE SOLUTION OF
VISUALLY ORIENTATED PROBLEMS.

Stephen Sproule
The Florida State University, Tallahassee, Florida
and
Ajetta Zietsman
University of the Witwatersrand, Johannesburg

In 'Sylvie and Bruno', Lewis Carroll writes "He thought he saw a banker's clerk
descending from the bus. He looked again and found that it was a hippotamus." The
world is full of visual experiences which we interpret through the lenses of experience
and expectation. The man in 'Sylvie and Bruno' had never seen a hippopotamus on a
bus before, and was not expecting to see one! His mistake was inevitable. In the same
way the twentieth century mathematics classroom requires the learner to of space and
consequently they may perceive the representation in a manner not intended by the
representor. The interpretation of visual stimuli in turn influences the mental imagery as
well as the mathematical activity of the learner.

RESEARCH AIMS AND QUESTIONS
In the paper, facets of girls' visual understanding that may be useful to teachers in
their classroom practice are identified. In particular, central components in how girls
come to perceive three dimensional diagrams are identified. The specific research
Questions addressed are:
1.   How do girls visually segment diagrams?
2.   Does this suggest a qualitative gender difference in perception and visualization ?
3.   What visual methods of solution do girls employ in the solution of visually
orientated questions ?

LITERATURE REVIEW

1. Theories of mental imagery
Two distinct modes of thought, verbal-logical and visual-pictorial, have been described
(Krutetskii, 1976; McGee, 1979). Verbal-logical thought is characterized by left
hemispherical activity in the brain and is the dominant from of thought when dealing
with spoken and written language (Battista, 1990). In contrast Gestalt processing is the
key cognitive component of visual-pictorial thought (Bishop, 1980; Dallisal, Wheatley
and Talsma, 1982, Talre, 1990). Assuming the existence of a visual-pictoral mode of
thought, the nature of such a thought process becomes important where two
contradictory theories exist. A picture-in-the-mind view emanates from a picture
analogy theory suggested by Kosslyn (1976). Mental images are replicas of previously
sensed objects and consist of icons in the mind. Paivio has suggested a dual-code theory
of mental imagery (Clements, 1981), where memory has dual nature involving a linguistic
and an imagery coding system. Pyiyshyn (1977) saw the major weakness of the
'picture-in-the-mind' theories as lack of explanations as to how verbal information can
be transferred into visual information in the mind. Pytyshyn's prepositional theory
claims that mental representations were sets of propositions that enables visual and
verbal information to be generated as required (Pyiyshyn, 1977). However, prepositional
theorists fail to explain how the mind transfers information from a prepositional state
into words or mental images (Clements, 1981).

We will consider the dual-code theory as the most viable explanation of the cognitive
processes generating the visual and verbal modes of thought. Embedded in a dual-code
theory the idea of a mental picture offers the most useful explanation of imagery for the
researcher and the mental picture acts as a useful metaphor in interpreting the
perceptions of children. It allows us to categorise, analyze and interpret children's
actions, perceptions and visual creativity in a manner that is useful for the improvement
of teaching visual mathematics.

2. Characteristics of spatial ability
In mathematics education three dimensional diagrams are common in the study of
mensuration, conic sections, trigonometry, problem solving, and in the communication of
mathematical ideas in the classroom. Segal et al. (1986; 94) suggest that the use of
perspectives in the drawing of three dimensional diagrams is a convention of society and
a product of their traditions. These conventions are learned and not innate (Bishop,
1979; Gregory, 1970). Authors in the field of spatial perception suggest that the
perception of the learner is influenced by 'cues' contained in the diagram (Deregowski,
1980; Segall et al., 1966). These cues may be general or idiosyncratic characteristics of a
representation. General cues are less important in the visual segmention of a three
dimensional diagram when the diagram is rich in idiosyncratic cues. Deregowski (1976a)
illustrates this by showing that Kenyan children will focus on the handle of a tea cup
rather than the general shape when identifying the diagram. Furthermore children from a
less 'carpentered world' are more likely to focus on the idiosyncratic cues in the
diagram (Segall et al., 1966).

We argue that the children's perception is primarily influenced by what Sproule has
called preceptors (Sproule, 1994). The role attributed to these preceptors replaces the
role of the visual cue used by other authors (Deregowski, 1980; Gregory, 1970; Segall et
al., 1966). Preceptors act as initiators of the mental images and are therefore mental
processes that influence how children perceive the conventions illustrated in a diagram.
We would therefore call those aspects of the diagram that children focus their attention
on, the dominant convention, which is only dominant because of the visual preceptor
used by the learner. Therefore three dimensional diagrams experienced by a student in
the mathematics classroom are not perceived on the basis of spatial conventions
contained in the diagram but rather on the basis of the visual preceptors used by the
learner in the interpretation of the diagram.

3. Factors infuencing spatial ability
There is consensus among mathematics educators that education and the child's
environment are important influences in the development of children's spatial ability
(Bishop, 1979, 1980; Jahoda, 1979; Battista, 1990; Presmeg, 1986b). The role of general
education in the development of spatial visualization remains undetermined
(Ben-Chaim, Lappan and Houang, 1988), but studies of ethnic differences indicate that
education does positively influence visual communication through the learning of
conventions of representation (Bishop, 1979; Mitchelmore, 1983).

The living environment of the child is a second factor in the development of spatial
ability (Bishop, 1986). Deregowski (1980) suggests that in the carpentered world, where
orthogonal constructions are prevalent, visual-pictorial development is superior.
However it is the child's experience with and not the presence of orthogonal
constructions that is of benefit in visual-pictorial development (Jahoda, 1980). The
influence of the environment includes the exposure to art and the characteristics of the
art (Deregowski, 1976b). Therefore the child's visual-pictorial experience is influenced
by the social experience and not only the physical environment (Battista, 1990; McGee,
1979; Mitchelmore, 1980; Presmeg, 1986b). Jahoda (1980) illustrates how British
children are more accustomed to pencil and paper activities and therefore more adept at
drawing than Zambian children. In contrast Zambian children make functional model
vehicles that require a great degree of mechanical and spatial understanding.

4. Ethnic and gender differences in spatial ability
Numerous studies show that children educated in developed nations have a more
developed spatial ability than those children living in underdeveloped or developing
nations (Bishop, 1979; Deregowski, 1980; Jahoda, 1979, 1980; Mitchelmore, 1980,1983).
The interpreter's knowledge of the conventions of distant representations used in
mathematics is an integral part of their interpretation of the representation, Bishop
(1979) claims that improvement in the knowledge of conventions by children from
developing nations leads to an improvement in visualization. The slow chronological
improvement in spatial ability among African children suggests that education and the
social environment rather than the lack of some innate ability accounts for the
discrepancy in spatial ability (Mitchelmore, 1980). Therefore visualization of children in
developing nations may not be inferior but the conventions used in mathematics
classrooms to create three dimensional diagrams may be incongruent with the
experiences of the learner. The ability of African children to create complex wire models
of vehicles corroborates the view expressed (Jahoda, 1980).

Boys are on average more advanced than girls in spatial visualization and spatial
odentation tasks (Jahoda, 1979; Ben-Chaim et al, 1988; Battista, 1990). The difference
in ability is established after puberty with studies conducted with prepubescent children
showing no significant difference in spatial ability between boys and girls (McGee, 1979;
Ben-Chaim et al., 1988). The gender differences appear to be more prominent in the
higher level spatial abilities such as spatial visualization and orientation (Fennama and
Tatre, 1985). Gender differences in spatial achievement are declaring (AAUW, 1992)
and parity in mathematical achievement tests suggests that spatial ability is not a
sufficient condition for mathematical achievement. The influence of spatial abilities to be
more influential in mathematical problem solving (Fennama and Tatre, 1985; Battista,
1990) and since curricula are becoming more inclined towards problem solving so issues
relating to gender differences in spatial ability will become more relevant.

Explanations for gender differences in high order spatial ability abound and are
subsumed in the broader nature-nurture argument. An interactionist view argues that
activity with concrete objects at a young age assists the process of visual-pictorial
development (Deregowski, 1978; Jahoda, 1980; Mitchelmore, 1980; Bishop, 1986). The
interactionist view is questioned by Jahoda (1979) who argues that the similarity in
gender differences in Scotland and Ghana cannot be explained by social factors.
Consequently, he suggests a difference in left-right hemisphere mental activity as a
possible explanation for gender differences. The cultural nature of pictorial conventions
precludes this argument as a sufficient explanation for gender differences. 

We propose that spatial ability is influenced by a great diversity of factors, contributing
to the nature of the gestalten processes that dominate the visual-pictorial thought
process.

4. Consequences of spatial abilities for the learning of mathematics

The visual-pictorial development of the learner can not be considered in isolation since it
is one of the cognitive abilities useful in the learning of mathematics. Spatial orientation
and visualization have been identified as contributing factors in various mathematical
activities. These include problem solving (Presmeg, 1986a; Fennama and Tatre, 1988),
geometry (Battista, 1990; Parazysz, 1991) and mathematical reasoning (Wheatley,
1990).

The most elemental influence of spatial perception is the visual communication between
teacher and learner (Bishop, 1986; Parazysz, 1991). This is particularly important when
the diagram is a distant representation used to illustrate or represent an object. Should a
common 'geometrical culture' not exist in the group, the diagram may result in
miscommunication of mathematical ideas. Then it is no longer only a spatial issue but
becomes a mathematical problem because the learners are unable to complete the
required task. In South Africa's muticultural classrooms the onus is on the teacher to
make explicit his/her use of the conventions used in representations, especially in three
dimensional diagrams.

The teaching strategies employed cannot take for granted the girls' visual segmentation
and good visual communication skills to spatially orientate mental images. The teacher
may consider ways of facilitating the child's ability to communicate visually and the
development of imagery. This is particularly important during primary school where the
learner actively participates in object play, acting as a representor rather than an
interpreter. Passive three dimensional viewing is detrimental to the development of high
order mental manipulations (Mitchelmore, 1980).

5. Concluding Remarks

We consider spatial ability as not innate but as learned through visual experiences and
social interactions thus creating a common geometrical culture. This view also assumes
that the interpreter brings as much to the visual experience as the representor. We will
argue that the interpretation of a diagram is an active constructive process influenced
more by the learner than the diagram. The intention is to offer the teacher interpretive
lenses to better understand the spatial ability of the learner and thus facilitate better
visual communication in the mathematics classroom.

RESEARCH METHODOLOGY

Research methodology in the field of spatial ability is as much debated as the subject
itself (Clements, 1981; Bishop, 1980).

1. Methodology

Within a qualitative framework the reassured can be disabused in a symbollo
interactionst tradition. The primary assumption in the tradition is that the actions of the
learner are purposeful and meaningful to the learner at the time. The learner's meaning
and purpose has arisen through social interactions and previous experiences. In this
study the learners had to perceive and interpret a number of visual stimuli presented to
them. It is assumed that the learners will perceive on the basis of their previous visual
experiences and visual communications during interaction with others. However the
perception and interpretation constructed by the learner may not conform to the socially
accepted interpretation of the stimuli.

The research was undertaken by means of a short test and then an extensive interview.
The initial evaluation of the participants was by means of a short test. The results of the
test were studied to determine the children to be interviewed. The interviews did not
involve all the test participants but rather those who displayed an atypical perception.
Atypical perception means that the child's perception of three dimensional diagrams is
either different to the mathematically accepted perception or different to the perception
of his/her peers.

The interviews used verbal self-reporting as the primary source of information in an
attempt to understand the spatial perception of children. In this study verbal reporting
is considered suitable for two reasons. Firstly, the verbal self-reporting was only
undertaken after the completion of the task. Secondly, the research has as its primary
focus the visual use of the mental image in spatial perception and visualization. This as
well as the assumption that the actions of the children are always rational (Whoaticy
and Cobb, 1990), given their understanding and their intention, suggests that the learners
would be able to verbalize their use of the mental image, even if they were unable to
describe the image accurately.

The responses were not categorized for statistical analysis but rather classified for
evaluation. The classification involved a search for interesting perceptions that
expressed diversity in imagery and visualization and allowed me to evaluate them in
terms of their viability and usefulness for the leamer.

2. Implementation of methodology

2.1 Participants
To facilitate an understating of the spatial perception of girls, boys were considered in
the study as well, to allow for a richer understating of the girls' visual segmentation.
Gender comparisons are only employed as a tool to gain greater insight into the nature of
girls' spatial ability. What will further enhance the quality of teaching and the
development of girls' spatial ability is a more comprehensive understanding of girls'
spatial processes. Thus references to spatial abilities of learners in general include both
boys' and girls' abilities.

The children were as far as possible of average scholastic ability and were chosen from
two educational levels. Firstly, children at the beginning of their standard six year,
ranging in age from 12-14. Secondly, children at the end of standard eight, ranging in age
from 15-17. Standard eight children should have been exposed to mensuration and
geometry and therefore many of the diagrams used in the tests. This would facilitate the
evaluation of the children's spatial ability after specific spatial aspects have been
considered in the mathematics classroom. The purpose of considering standard six
children is an attempt to understand more intuitive spatial perception and visualization.

The participlants are:

1.   White urban children from Model C schools situated in middle class
socio-economic communities. All the learners were mathematics students.

2.   Black urban pupils from middle to lower socio-economic backgrounds. All the
learners were mathematics students. The standard eight children were interviewed
while attending a Saturday school. During the week these children were students at
the Department of Education and Training schools in Soweto. The standard six children
were full time students attending the same school but had completed their primary
education at D.E.T. schools in the townships.

3. Initial Testing

The initial test (Appendix A) included diagrammatically presented visual stimull. The
visual stimuli were primarily two dimensional diagrams representing three dimensional
objects. The tests included diagrams common to classroom mathematics as well as
diagrams that would most likely not have been experienced by South African high school
students.

4. Interviews

The focus of the interviews was not to determine the perception or visualization of the
participant. In the interviews we attempted to understand how the interviewee came to
perceive in the way that he/she did. We also attempted to establish how the
interviewee used his/her interpretation of the diagram in the solution of the problem
presented. The interview involved discussing the written answers of the participant. The
interviewees were primarily asked to elaborate on their interpretation of the diagram as
given explanations for their use of the diagram. This required self-reporting from the
participant.

The interviews were undertaken individually with three or more participants from each
group of children. In the interviews any inhibiting bias observed was noted and the
interview discarded. The bias referred to included the inability of the participant to
express him/herself clearly as well as those participants who attempted to give 'right'
answers thus 'romancing' the interviewer rather than expressing their personal
interpretations (Plaget, 1929).

Interviews were conducted in English with all the participants. A translator was
available during the interviews with participants who did not have English as first
language.

DISCUSSION AND ANALYSIS

The analysis of each question offered a number of insights on the nature of the
perception and visualization of, in general, the learners and, in particular, the girls. It is
the identification of common visual and perceptual activity that offers useful analytic
tools to both teacher and researcher in their attempt to better understand girls' visual
problem solving methods. In the analysis the role of visual segmentation in the visual
activity undertaken by the girls was identified.

The visual segmentation of a representation undertaken by the learners was an influence
in the methods employed by the learner in solving the questions. The role of visual
segmentation in the solution process by making reference to protocols is considered. The
discussion will focus on the factors influencing the learners' visual segmentation, the
characteristics of the segmentation and the consequences of the segmentation for the
learners' visualization.

1. Factors influencing visual segmentation

The diagram as well as the learners' cognitive processes were involved in the visual
segmentation of the representation. Firstly the role fo the question and diagram is
considered and then the role of the learner in the visual segmentation of the
representation is discussed.

The nature of the question and the learners' interpretation of the question influenced the
activity of the learner. In question 4b on page one the learner was required to find an
error (appendix A). This resulted in many respondents visually isolating the cylindrical
rods, particularly the center rod. In contrast, on page two, question nine asked for a
description of the learner's perception given the same diagram (appendix A). Typical
responses included three pronged apparatus, wood joint and building block. These
perceptions illustrate that many of the learners considered the diagram as a whole rather
than segmenting the representation into visually manipulable elements. The learners'
attempt to answer the two questions resulted in differing visual segmentations of the
diagram.

The diagram and key elements in the diagram also facilitated the visual segmentation of
the representation. This is illustrated by the learners' segmenting E as the key element in
the question. In the protocol below Kirsten describes how she uses the E to solve the
question.
 

Figure 1.  Question 2:  Which letter is on the opposite side of H and of V?
781: How do yo get that N is opposite H?
K: Okay, em, I look at the E
801: Why did you look at the E?
K: Because the E is below the H, and I look at the other block, I see the
E here and then belows it is the N...

Kirsten has focused her attention on the E. The E occurs in all the cubes and particularly
in the two cubes used by Kirsten (Line 81). She then considered the orientation of the H
and the N in relation to the change in orientation of the E from the first to the second
cube (line 81). Kirsten, as is the case with other interviewees, has used E as a key
element in the diagram with which to direct her manual manipulation of the cube. The
visual segmentation influenced the problem solving approach used by Kirsten.

The learners' knowledge and past experience of spatial conventions and their ability to
create and manipulate visual images, influences the visual segmentation of the
representation. This is illustrated by the following protocols.


Figure 2. Question 1: How many small boxes are there in big box?

1:    In this first question here, you said you got 27, How did you get 27?
R:    I counted by three
1:    Okay, show me how you counted.
R:   {Rose counts in ones from 1 to 27, pointing to each square as she
counted.}
5 1:  How many sides does this small box have?
R:    Four
1:    Show me the four.
R:    1,2,3,4 {Rose points with her hand around the side to the left vertical
face of the small cube.}

Rose interprets the diagram of the 'small box' (figure 4.1) as a three dimensional
diagram (line 8). Her ability to establish a viable visual image of the object is poor the
results in a perception of four sides to the cube. She is then unable to visually isolate a
small cube within the large cube. She consequently segmented the cube into small squares
on each of the visible faces and counted the number of these squares in her solution
process (line 4). No visual image of the unseen faces has been established nor has she
been able to visually segment the plane faces of the cube. Her lack of knowledge and
experience of the spatial conventions used in figure two inhibited her visual imagery
which negatively influenced her solution process.

In contrast Raymond displays a competent knowledge of the spatial conventions of
three dimensional diagrams. This allows him to create a more useful visual segmentation
in his attempt to solve the problem.

1 I: Raymond describe to me how you got 19 boxes here.
R: Firstly I look at this small box and then I count all the box on top and I
count that box.
I: Okay, do that for me.
R: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15. Um. I really made a mistake because I
counted them twice, both sides. {He counts the 9 cubes on the top of the
large cube, then the seen parts of the middle and bottom rows of the large
cube. He miscounts to get 15}
5 I: Now explain that to me, how you made that mistake?
R: I count this side like I count on top. {Counting top 9 cubes}. And I
didn't count here and I didn't count here. {Pointing to the top row of the 2
vertical faces of the large cube.} I count this side and this side, both sides to
get nineteen boxes. Because I took the size of this box {the small cube} and
compared it to the side of this. {Points to one of the small cubes  in the large
cube}

Raymond has clearly identified the small cubes within the large cube (line 2 and 6). The
error in line 4 is a careless mistake and not incongruent perceptual activity. He has
visually segmented the large cube into small cubes which allows him to count those cubes
which can be seen in the diagram (line 4). His method of solution relies on counting
rather than arithmetic or analytic processing. His knowledge of the spatial conventions
used in three dimensional diagrams is superior to the knowledge displayed by Rose in
the first protocol. The difference in visualization abilities and knowledge of spatial
conventions of the two learners influenced their visual segmentation of the
representation.

As argued in the review of relevant literature, the knowledge of the conventions used in
distant representations is learned. The learning occurs through exposure to and
experience with three dmensional diagrams. The experience of the learner influences the
use of relevant preceptors. Therefore any significant gender difference is untenable and
most likely the result of inqualities in schooling. There is a substantially greater
intragender variation than inter-gender difference in the knowledge of spatial
conventions. The differences do not occur in the knowledge but rather in the application
of the knowledge, through preceptors, to the visual segmentation of the diagrams.

3. Viability and visual segmentation

Visual segmentation is characterized by a search for viability and therefore the use of
preceptors to visually focus attention on different aspects in the diagram. We will
outline the search for viability as the primary characteristic of visual segmentation.

The search for a viable method of solution is the primary concern of an individual
attempting to solve a problem (von Glasersfeld, 1987). This is particularly evident in the
visual segmentation of a representation where the learner is attempting to interpret the
diagram, as intended by the representor, as well as to make the diagram a useful element
in the solution of the given question. Thus the learners will visually segment the diagram
in a way that is believed to be the most viable. Different individuals may consider
different segmentations viable for solving the question. The visual segmentation
employed by the learner may not be considered viable by a broader group of perceivers
and may not result in a viable solution.

The individual nature of viablility is illustrated in the differences in visual segmentation
of question three undertaken by Bongani and Heidi.



Figure 3. Question 3b: How many sides do each of these objects have?

1301: And this one here? {question 3b}
H: It's a soccer ball so I mean its got to be these sides here. {Referring
to each plane face of the dodecahedron.}
I: Alright, how did you work out that there was 12?
H: It's 1,2,3,4,5,6,7 then there's ones at the back. {Counts the faces
created by solid lines and one face  enclosed by broken lines.}
I: Now how do you know there's ones at the back?
135H: Because it's a soccer ball.
1421: And it's because of the soccer ball that tells you there's got to be
something at the back? 
H: And because of those lines. {the broken lines}

In question 3b Heidi perceives the representation as a three dimensional diagram (line
131). She has used an environmental preceptor to attribute the properties of an object,
the soccer ball, to the representation (line 131)> Heidi has segmented the diagram on the
basis of the solid and broken lines (line 143) in order to establish a visual image of an
object. Consequently she used an experiential object as a means of solving the question.
The certainty with which she uses the visual image of the soccer ball (line 135) illustrates
the viability (for her) of establishing an image of an object from the perception of a
diagram.

In contrast the number of sides may be counted. Although this process does not appear
as visually sophisticated it does offer a viable means of solution. Bongani adopted this
procedure.

1001: Okay, and this one how did you get 12? {question 3b}
B: I counted all the dotted lines and this black lines.
I: Why did you count the dotted lines?
B: Because the dotted lines show the other side, in the picture.  

Bongani distinguishes between the solid and broken lines (line 101 and 103.) His concem
with the spatial properties of the diagram suggests the use of a spatial perceptor. The
perception facilitates the counting of faces rather than utilizing a more complex
visualization process in his solution (line 101). The process relies more on the spatial
conventions employed in the diagram than the idiosyncratic elements attributed to the
diagram.

Both learners correctly determined the number of faces in 
question 3b yet they used different visual segmentation of the diagram. Bongani found a
viable solution in visually segmenting the diagram into pentagonal faces and then
counting the number of faces. In contrast Heidi counted the front pentagonal faces and
multiplied by two because of the symmetrical nature of the representation. The protocol
is evidence that girls are more likely to use environmental perceptors resulting in greater
emphasis being placed on key elements or idiosyncratic characteristics of the diagram.
The girls' knowledge of the conventions is not in question but how this is applied on the
basis of perceptors may well illstrate gendar differences. No value judgment should be
imposed on the perceived differences in girls and boys knowledge of spatial conventions
and perceptor application.

4. Consequences of visual segmentation

The search for a viable visual segmentation influences the learner's visualization of the
representation and the method of solution employed in the problem. In the analysis of
question one a number of district visual segmentations were identified. These
segmentations resulted in different method of solution being employed by the learner.

Candy and Heidi both described figure three as the Rubic's cube. This suggests that they
were employing an environmental perceptor to make sense of the diagram. Candy
segmented the diagram in figure two on the basis of the squares and the plane faces. She
counted the number of squares on one face and then multiplied by the perceived number
of faces. This procedure involved an application of arithmetic but resulted in an
inappropriate solution. Ultimately, Heidi visual segmented the small cubes in the large
cube. This resulted in a more visual solution requiring a greater visualization ability.
However Heidi was unable to visualize the unseen faces of the cube resulting in an
answer of 19.

In an attempt to force the two girls to undertake a different, and possibly more viable,
segmentation of the cube, a number of direct and sometimes leading questions were
asked. These questions led to perturbation and demanded increased spatial orientation.
A number of responses were obtained:
a)  The girls took longer to respond, sometimes as long as 51 seconds.
b)  Comments like "I don't know" and "I can't see it" during their efforts to answer
my question.
c)   In resignation comments like "It's difficult."

This suggests that a change in visual segmentation was difficult, if not impossible for
Heidi and Candy. Consequently altering their method of solution was also difficult, if
not impossible for the two girls. One would guess that this would apply to many of the
other respondents who used similar methods.


The two district segmentations resulted in different approaches to the solution of the
problem. There was no noticeable gender influence in the implementation of the two
segmentations. Therefore although the two girls used environmental perceptors this did
not result in a particularly unique solution process for gifts.  This view
offers evidence that visual segmentation is not a primary influence in determining a
method of solution for girls. However a lack of knowledge of spatial conventions and an
inappropriate visual segmentation may restrict the method of solution. This was
evidenced in their inability to alter the method first employed.

Another useful example illustrating the influence of perceptors of the perception and
visual segmentation occurs in question 4b. The creation of a mental image of a house
may not lead to attention to environmental features of the house but the aesthetic
characteristics of the diagram.


Figure 4. Question 4b: Circle any mistake in the picture (if you think there is a mistake).

1481:  Why do you say that's wrong? {She has marked both windows as incorrect}
H:  Because this one is a little bit shorter than that door and you can't see out the windows.
1501:   So then, what's wrong with these windows?
H:   They're too high.
I:   Okay, is there anything else wrong with that picture?
H:   Not that I can see.

Heidi has visually compared the diagram as a representation of a house to her image of
a house as a physical structure (line 149). Although she has considered the
environmental features of the house her attention remains focused on the feasibility of
the representation rather than the feasibility of the house. Heidi did not find the error in
perspective (line 153) but in the aesthetic or artistic nature of the               
representation (line 149). Within the context created by Heidi the spatial conventions of
three dimensional diagrams are not perceived as significant. Therefore the use of an
artistic perceptor resulted in an ultimate visual segmentation of the diagram of the
house. In turn this visual segmentation influenced the solution offered by Heidi.

Furthermore the visual segmentations of Candy and Heidi illustrated above are not
considered to be equally useful of viabie. The visual segmentation of the large cube into
plane faces is considered to be ineffective since it does not facilitate counting spatial
orientation of the cube or algebraic methods of solution. Thus the visual segmentation of
a representation may isolate elements of the diagram that do not facilitate the solution
of the problem. In addition the aesthetic segmentation undertaken bu Heidi in question
4b was inflexible (line 153). The advantages of visual segmentation are derived from the
learner's ability to perform flexible visual segmentations that can be focused on the most
useful information given in the diagram. In the mathematics classroom it is the role of the
teacher to be aware of the learner's visual segmentation and to facilitate the development
of an ablitity to perform flexible visual segmentations.


CONCLUSION

In concluding this paper the results discussed are summarized and placed within a
constructivist paradigm. In addition insights that may assist the teacher in the
mathematics classroom are suggested. 

1. Summation
In a constructivist framework the individual's search for viable methods of solution and
answers is recognized as a central characteristic of the conceptualization and cognitive
interpretation of problems (vov Glasersfeld, 1987). This applies irrespective of gender. In
the same way the perception diagrammatic interpretation and mental imagery of the
learner must be seen in terms of the viablility for the learner. The spatial activity of the
learner in this research is characterized by the search for useful and reliable methods of
solution, that is to say viable methods.

It is noted that the segmentation of the diagram is undertaken by the learner on the basis
of the perceived viability of the segmentation. The visual segmentation of the diagram
appears to be a means employed by the learner to simplify the spatial activity required
in the solution of the problem. In essence the use of perceptors influenced the visual
segmentation which in turn influenced the method of solution.

No qualitative gender differences in the knowledge or spatial conventions or in the
selection of viable methods of solving visually orientated problems were identified.
However it appears that girls are more inclined to use environmental and artistic preceptors in their
perceptions and visual sequmentation of diagrams. Evidence is offered in the protocols
of Heidi, Candy and Kisten. The less familiar the spatial conventions used in the
diagram the more likely the use of these perceptors. The result of the learner's use of
environmental and artistic perceptors is the interpretation of the diagram as an object or
a concern with the aesthetic elements in the diagram (Sproule. 1994) Although this did
occur across gender lines it was noticeable that the girls used these perceptors more
readily than the boys. The boys were more likely to impose spatial or geometrical
perceptors on the diagram. This is evidenced by the protocols of Bogani's and Raymond's
interviews. The rich diversity of perceptions discussed in the report justifies the
argument that perception is a construction of the learner, rather than the transfer of the
properties of the diagram from the visual stimuli to the mental image.

2. Suggestions for classroom practice
The recommendations offered here are based on the perceived value of the research and
interpretations but may not have been fully elaborated in the discussion. These insights
are intended as a useful means for teachers to analyze the spatial abilities of the girls in
their classrooms. Awareness of the role of visual communication and attempts to
facilitate more viable visual segmentation are suggested.

It is suggested that the teacher should not assume that the learners interpret the diagram
in the intended manner. The use of a diagram may offer effective communication but it
could also result in miscomunication if the intentions of the
representor are not concomitant with the perceptions of the interpreter. It must be
remembered that the perception and interpretation of the diagram is a consequence of
the learner's construction and not of the cues in the diagram. One of us (Sproule) has
experienced this first hand in the teaching of circle geometry where children appear to
visually isolate lines rather than angles. This is not a result of any dominant elements in
the diagram but of the learner's perception. Thus it is suggested to teachers not to
assume that the diagram offers effective visual communication or that the learner's
visual segmentation of the diagram is appropriate.

In addition the visual segmentation used by the learner may not be the segmentation
necessary for the effective solution of the problem, as was the case with Rose.
Awareness of the learner's visual segmentation of diagrams offers a powerful diagnostic
tool for the teacher. Suggesting that girls are inclined to use environmental and artistic
perceptors in their perceptions and visual segmentation of the diagrams should be seen
as an advantage by the teacher. The teacher may consider adding idiosyncratic
environmental elements to improve communication. This should not result in the teacher
neglecting to teach the learners the spatial conventions used in three dimensional
diagrams. However, the idiosyncratic elements may facilitate perception as well as
visualization. As teachers our awareness of the nature of the girls' perception may
encourage us to facilitate communication by transforming our knowledge into action.


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