INVESTIGATING THE PROBLEM SOLVING COMPETENCY OF
PRE SERVICE TEACHERS IN DYNAMIC GEOMETRY
ENVIRONMENT
Shajahan Haja
This study investigated the problem-solving competency of four secondary pre service
teachers (PSTs) of University of London as they explored geometry problems in
dynamic geometry environment (DGE) in 2004. A constructivist experiment was
designed in which dynamic software Cabri-Géomètre II (hereafter Cabri) was used
as an interactive medium. Through participating in the study, the PSTs demonstrated
their competency exemplified by using their subject knowledge to construct, to make
conjectures and verify their conjectures on their way to solve the given geometric
problems.
INTRODUCTION
Being a mathematics teacher educator I felt the gap between the technology infused
learning experiences in geometry class and the current status of teacher preparation
programs that in general pay little attention to this aspect. For a solid prescription, I
needed an empirical data involving PSTs and DGE. The first issue that stroke me
was: How competent are the PSTs in using subject knowledge to solve problems in
DGE? I decided to pursue my investigation to find answer for this question.
Related Studies
There exists very few experiential data to inform us about the competency of PSTs in
using DGE. Saads and Davis (1997) investigated the van Hiele levels in threedimensional geometry and the spatial abilities of a group of secondary PSTs and
found that PSTs’ language depends on a combination of the PSTs' general geometric
level, their spatial ability and their ability to express the properties of the shape using
language. Pandiscio (2001) studied how secondary mathematics PSTs perceive the
need for and the benefits of formal proof when given geometric tasks in the context
of dynamic geometry software. Jiang (2002) observed the actual learning processes
of two PSTs as they explored geometry problems with dynamic geometry software
and the effects of using the software on developing their mathematical reasoning and
proof abilities.
‘Knowledge-in –Action’ Design
The aim of the study is to find out the PSTs’ problem solving competency in DGE. I
developed a ‘Knowledge–in–Action’ Design (Fig 1) based on geometrical
constructions in Cabri using dragging and other functions to illustrate the problem
solving process of PSTs. The process of converting the conceptual understanding of
the given geometric problems into action demands the PSTs to apply their content
2005. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International
Group for the Psychology of Mathematics Education, Vol. 3, pp. 81-87. Melbourne: PME.
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knowledge to understand the given problem, construct the dynamic figures, make
conjectures, verify the conjectures, and solve similar problems. Accordingly the
competencies are defined as follows:
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Competency 1: PSTs construct figures in DGE.
Competency 2: PSTs make conjectures in DGE.
Competency 3: PSTs verify their conjectures in DGE.
It is been hypothesized that if the PSTs are able to solve the given problem using
their subject knowledge by constructing, conjecturing and verifying the properties in
DGE then their competency is testified. The study builds on the assumption that
learning involves the active construction of knowledge through engagement and
personal experience (Ernest, 1994; von Glasersfeld, 1995). The Van Hiele theory of
learning geometry also provided a useful conceptual framework for this study.
Problem
Subject
Knowledge
Competency 1
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Figure 1: Knowledge-in-Action Design.
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METHODOLOGY
The subjects of this experiment were four secondary PGCE mathematics students (3
males PST1, PST2, PST3 and 1 female PST4) of University of London. None of the
PSTs had used Cabri before taking part in this experiment but they had studied
geometry at the high school/O level. The experiment was conducted on one- to-one
basis on different days in January 2004 at the Institute of Education, London. The
experiment was recorded.
Constructive Experiment: Discussion
The constructive experiment designed for the study is an open-ended problem
involving Varignon’s quadrilateral and medial triangle. I analyze the Varignon’s
quadrilateral using non - conventional quadrilaterals (crossed/ bow tie) in DGE
background, which is otherwise not possible with pencil and paper. Consider a
quadrilateral ABCD (figure 2a) with the midpoints of its sides E, F, G and H. In the
experiment, PSTs conjectured that EFGH is a parallelogram. By dragging the vertices
of the quadrilateral ABCD they observed that EFGH remained a parallelogram no
matter what type of quadrilateral ABCD was. And this result is valid for a crossed
quadrilateral too (figure 2b). Also the area of the Varignon parallelogram EFGH is
exactly half of the area of the original quadrilateral ABCD; even in the case of the
crossed quadrilateral.
(2a)
(2b)
Figure 2: Varignon’s Quadrilateral.
Now let us consider three-sided polygon (n=3) in the above example; the
quadrilateral is then replaced by a triangle. I see that the medial triangle formed by
joining the adjacent midpoints is similar to the original triangle. PSTs explored this
task with special cases: equilateral triangle, right triangle and isosceles triangle. Now,
the midpoints of the sides of a triangle form a triangle DEF (see Fig 3) that is exactly
one quarter of the area of the original triangle ABC.
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Figure 3: Medial Triangle.
Also the medial triangle is not only congruent to the other three small triangles (ADF,
BDE, EFC) but also similar to ABC and the line segment connecting the midpoints
of two sides of a triangle is parallel to the third side and is congruent to one half of
the third side.
Now, how did the dynamic figures control the PSTs exploration? In paper-and-pencil
geometry, all entities are static, but in dynamic geometry, they behave in a specific
way to maintain their geometric properties. For instance, the vertices of quadrilateral
ABCD are completely arbitrary and can be moved freely, while E, F, G and H are
fixed as midpoints that cannot be altered. This is the same with triangles as well. For
instance, in the right triangle the perpendicularity of two lines controlled the activities
of PSTs.
Exploration by PSTs as Replay Construction – The Road to the Solution
In this experiment, the Replay Construction function in the edit menu of Cabri
describes the construction process when a convincing solution was achieved by a set
of physical manipulations by the PSTs. The following two illustrations describe stepby-step how each PST works.
Illustration 1: Medians of Isosceles Triangle
All the four PSTs were able to construct the isosceles triangle using different
conjectures. PST 1 & PST3 used angle bisectors to construct the triangle while PST2
used the conjecture (perpendicular bisector of a chord passes through the centre of
the circle) & PST4 used (isosceles triangle has two equal sides) circles to construct
the triangles. PST2 & PST4 used the radii of circles as the two equal sides of the right
triangle (Fig 4).
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PST1 -Isosceles Triangle
PST2 Isosceles Triangle
PST1 Isosceles Triangle
PST3 -Isosceles Triangle
Point
Point
Segment point, point
Midpoint segment
Perpendicular Line C, segment
Point on Object line
Segment point, C
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
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Midpoint segment
Midpoint segment
Segment A, B
Segment B, C
Segment C,A
Angle C, A, B
Comments
Angle A, B, C
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Angle point, point, point
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Angle point, point, point
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PST2 Icoceless Triangle
Point
Circle point
Point on Object circle
Point on Object circle
Segment point, point
Midpoint segment
Ray point, point
Intersection Points ray, circle
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
Comments
Midpoint segment
Midpoint segment
Segment point, point
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
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Angle point, point, point
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PST4 - Isosceles Triangle
PST4 Isosceles Triangle
PST3 Isosceles Triangle
Point
Point
Segment B, C
Perpendicular line B, segment
Perpendicular line C, segment
Point on Object Line
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Angle bisector point, C, point
Intersection Points line, line
Intersection Points line, segment
Intersection Points line, segment
Segment A, C
Segment B, A
Angle A, B, C
Comments
Angle A, C, B
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Midpoint segment
Midpoint segment
Point on Object segment
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Segment D, E
Segment E, F
Segment F, D
Angle F, D, E
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Segment point, point
Circle point, point
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Segment point, point
Segment point, point
Midpoint segment
Midpoint segment
Midpointsegment
Intersection Points segment,circle, point
Angle point, point, point
Comments
Angle point, point, point
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Angle point, point, point
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Segment point, point
Segment point, point
Segment point, point
Angle point, point, point
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Angle point, point, point
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Parallelsegment, segment
Parallelsegment, segment
Parallelsegment, segment
Figure 4: Medians of Isosceles Triangle.
Illustration 2: Medians of Quadrilateral
PST2 tried non-convex quadrilateral while others tried the convex. In either case
PSTs found that the inner quadrilateral turned out to be a parallelogram (Fig 5). PST1
Used the conjecture diagonals of a parallelogram bisect each other. Others used the
conjecture the opposite sides of a parallelogram are parallel and opposite angles are
congruent. PST1 tried to compare the areas of inner and outer quadrilaterals by
drawing the diagonals. He found that the sum of the areas of the four triangles was
one-half of the area of the entire quadrilateral.
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PST2- Quadrilateral
PST1 -Quadrilateral
PST1 Quadrilateral
Point
Point
Point
Point
Polygon point, point, point, point
Mid Point polygon
Mid Point polygon
Mid Point polygon
Mid Point polygon
Segment a,b
Segment b,c
Segment c,d
Segment d,a
Parallel ab, cd
Parallel ad, bc
Intersection Points ac,bd
Distance and Length eb
Comments
Distance and Length ed
Comments
Distance and Length ec
Comments
Distance and Length ea
Comments
Area polygon
Comments
Polygon a,b,c,d
Area polygon
Comments
Triangle a,b,e
Triangle b,e,c
Triangle e,c,d
Triangle e,d,a
Area triangle
Comments
PST2 Quadrilateral
Point
Point
Point
Point
Segment point, point
Segment point, point
Segment point, point
Segment point, point
Midpoint segment
Midpoint segment
Midpointsegment
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Midpointsegment
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Area triangle
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Area triangle
Comments
Area triangle
Comments
Comments
PST3 -Quadrilateral
PST4 -Quadrilateral
PST4 Quadrilateral
Point
Point
Point
Point
Polygon point, point, point, point
Mid Point polygon
Mid Point polygon
Mid Point polygon
Mid Point polygon
Segment point, point
Segment point, point
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Parallel segment, segment
Parallel segment, segment
PST3 Quadrilateral
Point
Point
Point
Point
Polygonpoint, point, point,point
Midpoint polygon
Midpoint polygon
Midpoint polygon
Midpoint polygon
Segment point, point
Segment point, point
Segment point, point
Segment point, point
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Angle point, point, point
Comments
Figure 5: Medians of Quadrilateral.
CONCLUSION
In this experiment, geometrical facts about medial triangles and mid point
quadrilaterals are examined by the PSTs. The emphasis was on tracing a solution to
the given problem not a formal proof. As PSTs manipulated the dynamic figures,
they could make conjectures about the properties of the shapes. Also, PSTs used onscreen measurements for side lengths and angles to transform their intuitive notions
into more-precise formal ideas about geometric properties (Battista, 1998). For
instance, PSTs used distance and angle measurement to check the geometric
properties of medial triangles. I observed that PSTs are motivated to search for
explanations driven by ‘dragging’ that guided them towards the solution. This is
substantiated by PSTs’ opinion about the experiential learning; to put it in their own
words, ‘This is a new experience and I thoroughly enjoyed it (PST1)’, ‘It is wonderful
that I could create my own way of finding a solution (PST4)’, ‘I learned (PST2)’ and
‘It is useful but time consuming (PST3)’. Analysis of the PSTs solution process
shows that learning took place in the form of construction, conjectures, and
verification and thus validates the assumption of the study (Ernest, 1994; von
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Glasersfeld, 1995). However the interaction with dynamic software can’t be accepted
as a formal proof as argued by Saads & Davis (1997). The problem solving behaviour
exhibited by the four PSTs correlates with the Knowledge-in-Action design (Fig1) in
terms of the components (construction, conjecture, and verification) described in the
design.
The construction process showed that PSTs demonstrated adequate geometric
reasoning in accordance with the hierarchical levels of Van Hiele viz., Visualization,
Analysis, Abstraction, Deduction, and Rigor on the selected tasks. However, this
can’t be generalized to the PSTs understanding of other geometrical concepts
(Mayberry, 1983). Observation of the problem solving processes of the four PSTs
while they attempted to solve geometric problems with the help of Cabri suggests
that:
a) PSTs' content knowledge are adequate to understand the given problems
are competent enough to apply the content knowledge to construct the
figures c) PSTs are competent enough to apply the content knowledge
conjectures d) PSTs are competent enough to apply the content knowledge
the conjectures and e) PSTs are able to use DGE to justify their solution.
b) PSTs
dynamic
to make
to verify
References
Battista, T. (1998), Shape Makers: Developing Geometric Reasoning with the Geometer's
Sketchpad. Berkeley, Calif.: Key Curriculum Press.
Ernest, P. (1995), The One and the Many. In Steffe, L. & Gale, J. (Eds.). Constructivism in
Education (pp.459-486). New Jersey: Lawrence Erlbaum Associates Inc.
Jiang, Z. (2002), Developing Pre-service Teachers' Mathematical Reasoning and Proof
Abilities in the Geometer's Sketchpad Environment. PME-NA XXIV: Proceedings of the
24th Annual Meeting of the North American Chapter of the International Group for the
Psychology of Mathematics Education (Columbus, Ohio: ERIC, 2002), 717-729.
Mayberry, J. (1983), ‘The van Hiele Levels of Thought in Undergraduate Pre-service
Teachers’, Journal for Research in Mathematics Education, 14: pp. 58–69.
Pandiscio E. A. (2001), Exploring the Link Between Pre Service Teachers' Conception of
Proof and the Use of Dynamic Geometry Software, PME XXV, Vol. 1, pp. 353.
Saads, S and Davis, G. (1997), Spatial Abilities, van Hiele Levels, and Language Use in
Three Dimensional Geometry. PME 21, Finland.
Von Glasersfeld, E. (1995), ‘A Constructivist Approach to Teaching’, In Steffe, L. & Gale.
J. (Eds.). Constructivism in Education, (pp.3-16). New Jersey: Lawrence Erlbaum
Associates, Inc.
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