John Olive

The University of Georgia

Athens, Georgia, USA

Paper for Conference on Teaching and Learning Problems in Geometry

Fundão, Portugal

May 6-9, 2000

In this talk I would like to explore the implications of using Dynamic
Geometry technology for teaching and learning geometry at different levels
of education. Through example explorations and problems using the *Geometer's
Sketchpad* I hope to provoke questions concerning how children might
learn geometry with such a tool, and the implications for teaching geometry
with such a tool. I shall draw on my own experiences and the experiences
of other teachers and researchers using dynamic geometry technology with
young children, adolescents, and college students.

What is Dynamic Geometry Technology?

This question is best addressed through demonstration. I include any technological medium (both hand-held and desktop computing devices) that provides the user with tools for creating the basic elements of Euclidean geometry (points, lines, line segments, rays, and circles) through direct motion via a pointing device (mouse, touch pad, stylus or arrow keys), and the means to construct geometric relations among these objects. Once constructed, the objects are transformable simply by dragging any one of their constituent parts. Examples of dynamic geometry technology include, but are not limited to the following:

__The Geometer's Sketchpad__ (Version 3 for both Windows and Macintosh
computers, and the new implementation for TI-92 and 93 calculators and
the Casio Cassiopeia hand-held computer).

__The Geometry Inventor__ (Computer software)

__Geometry Expert__ (GEX, a new computer-expert system from China).

* TesselMania®* (dynamic tessellation software).

Implications for Elementary Teaching and Learning

Nathan's use of the dynamic drag feature of this type of computer tool illustrates how such dynamic manipulations of geometric shapes can help young children abstract the essence of a shape from seeing what remains the same as they change the shape. In the case of the triangle, Nathan had abstracted the basic definition: a closed figure with three straight sides. Length and orientation of those sides was irrelevant as the shape remained a triangle no matter how he changed these aspects of the figure. Such dynamic manipulations help in the transition from the first to the second van Hiele level: from "looks like" to an awareness of the properties of a shape (Fuys, Geddes & Tischler, 1988).

What Nathan did during the next 15 minutes with *Sketchpad* also
indicates how such a tool can be used to explore transformational geometry
at a very young age. I showed him how he could designate a line segment
as a "mirror" using the TRANSFORM menu. We then selected his triangle and
reflected it about the mirror segment. Nathan was delighted with the way
the image triangle moved in concert with his manipulations of the original
triangle. He quickly realized that movement toward the mirror segment brought
the two triangles closer together and movement away from the "mirror" resulted
in greater separation. I decided to add a second line approximately perpendicular
to the first mirror segment and designate this second line as a mirror.
We then reflected both the original triangle and its reflected image across
this line, resulting in four congruent triangles. Nathan then experimented
by dragging a vertex of the original triangle around the screen (see Figure
1). He was fascinated by the movements of the corresponding vertices of
the three image triangles. He was soon challenging himself to predict the
path of a particular image vertex given a movement of an original vertex.
At one point he went to the chalkboard, sketched the mirror lines and triangles,
and indicated with an arrow where he thought an image vertex would move.
He then carried out the movement of the original vertex on the screen and
was delighted to find his prediction correct. Note also that Nathan was
not constrained by physical mirrors. He had no hesitation in crossing over
the mirror lines! Goldenberg and Cuoco (1998) challenge us to think seriously
about the educational consequences for children working in an environment
in which such mental reasoning with spatial relationships can be provoked.

Figure 1: Double reflection of a triangle from *Geometer’s
Sketchpad*

Lehrer, Jenkins and Osana (1998) found that children in early elementary school often used "mental morphing" as a justification of similarity between geometric figures. For instance a concave quadrilateral ("chevron") was seen as similar to a triangle because "if you pull the bottom [of the chevron] down, you make it into this [the triangle]." (p. 142) That these researchers found such "natural" occurrences of mental transformations of figures by young children suggests that providing children with a medium in which they can actually carry out these dynamic transformations would be powerfully enabling (as it was for Nathan). It also suggests that young children naturally reason dynamically with spatial configurations as well as making static comparisons of similarity or congruence. The van Hiele (1986) research focussed primarily on the static ("looks like") comparisons of young children and did not take into account such dynamic transformations.

__Logo or Dynamic Geometry in the Elementary School?__

I was one of the many Logo enthusiasts who embraced
this computer programming language as a tool for children's mathematical
explorations. Papert (1980) made a very strong case for how the "turtle
geometry" (accessible through simple Logo computer commands) was closely
related to children's own movements in space (walking forward or backwards,
turning right or left). Balacheff and Sutherland (1994) point out critical
differences between the "learning milieu" (Brousseau, 1988) that may be
created using the Logo computer programming language and dynamic geometry
software (*Cabri-géomètre*). While children can enact
Logo-like commands themselves to walk the pathways they might want to create
on the computer screen, the programming interface is symbolic, requiring
the child to encode their movements (or the turtle's movements on the computer
screen) using words and numbers. This encoding by the children is a crucial
aspect of the Logo learning milieu. It requires quantification and formalization
of the geometric constructs. The direct manipulation of screen objects
through motion of a pointing device in dynamic geometry environments does
not require a priori formalization.

Papert (1980) has argued that the possibility for children to experiment with Logo commands in an interactive way, producing movements of the screen turtle, without having to first write, compile and then run a program, reduces the demand for formalized thought. Children would construct their own notions of a "turtle step" as a unit of linear measure, and their own notions of angle measure through experimenting with the Forward and Back, and the Right and Left turning commands in Logo. Early research in the Logo community, however, indicated that the notions that children constructed of angle measure especially, were very limited and often misleading (Hoyles & Sutherland, 1986).

Balacheff & Sunderland (1994) make the point
that "the interface [of computer software] cannot be strictly separated
from the so-called internal representation, it is not a mere superficial
layer." They go on to say that "What the learner explores is at the same
time the structure of the objects, and their relations and the representation
which make them accessible. In this sense, direct manipulation does not
only make the use of the microworld more friendly, it is an integral part
of it." (p. 7) For young children, then, it would appear that the direct
manipulation interface of dynamic geometry software would bring the children
in direct contact (through action) with the "structure of the objects,
and their relations". Nathan's initial explorations with *Sketchpad*
would certainly bare this out. Elementary teachers can take advantage of
the direct manipulation interface and of the dynamic transformational properties
of the software to introduce young children to rotation (and angle measure)
as an amount of turning from one ray to another, translation as a "slide"
in a given direction that does not change the orientation of a figure,
and reflection as "mirror motion". An example activity developed by a third
grade teacher in Project LIMTUS illustrates these possibilities.

__The Paper Doll Caper__. Children draw a free-hand
stick figure in *Sketchpad *using the segment and circle tools. The
challenge is to construct a row of "paper dolls" similar to the paper dolls
one would get when cutting out a stick figure from a strip of paper that
had been folded many times. In Figure 2, the stick figure has been translated
by the vector RS and its translated image has also been translated by this
same vector.

Figure 2: Translated "Paper Dolls"

While "translation by a vector" may be far too abstract a description of the transformation for young children, they can make sense of the notion of "sliding" a given distance in a given direction. In the sketch shown in Figure 2, the children can explore this sliding motion by moving point S very close to point R and then slowly moving point S away from R. As point S gets close to R the 3 figures will merge into one figure. As point S is moved away from R, the 2 image figures will "slide" away from the original figure. The children enjoyed moving parts of the original stick figure to make the row of figures "dance" together. The teacher who developed this activity used it to also investigate measures of corresponding segments and corresponding angles formed by the elements of the stick figures. These measures change dynamically as the segments are manipulated, thus, the children could see that these measures were the same for each of the stick figures, even when they changed position and length of parts of the original figure.

Some children decided to reflect their stick figure using a vertical segment as a "mirror" as in Figure 3.

Figure 3: "Paper Doll" Reflection

By manipulating elements of their original stick figure, they were able to make them "dance" with one another. The teacher also encouraged measurement of angles and segments in this situation. Some of the children were able to "simulate" their paper folded dolls by creating another vertical mirror to the right of the reflected stick figure and reflecting both figures (and the initial mirror segment) about this new mirror (see Figure 4).

Figure4: Double reflection of "Paper Dolls"

From this double reflection of their original stick figure the children were able to relate the effects of reflection with their paper folding and cutting activity.

As Ballacheff & Sunderland (1994) point out, the learning milieu that one can create in a Logo environment and that one can create in a dynamic geometry environment are essentially different because of the different objects and actions available in each, and the different modes of interaction within each. It is not to say that one is necessarily "better" than the other but that there is a "different complexity in each environment" and that "learning as the result of the interaction with these environments is likely to lead to the construction of quite different meanings." (p. 10)

Implications for Teaching and Learning in the Middle Grades

One of the constraints that educators have found
when using dynamic geometry software with young children is the level of
geometric knowledge needed in order to construct the most common geometric
figures, such as equilateral triangles, squares, rectangles and parallelograms.
Young children can easily DRAW such figures using the segment tool, but
their figures do not maintain their specific configuration under direct
manipulation (Olive, 1998). In order for a square to remain a square whenever
one of its vertices or sides are dragged, the square has to be CONSTRUCTED
using the available geometric construction tools (such as rotation of a
segment about a point, or constructing a line perpendicular to a given
segment through an end-point of the segment). Finzer and Bennet (1995)
have pointed out the necessity for students to make this transition from
drawing to construction when first encountering dynamic geometry software.
But, for young children, this transition is very difficult, as it requires
knowledge of geometric properties and relations they are yet to construct.
Battista (1998a) developed a microworld within *Sketchpad* that provided
young children with ready-made shapes that they could manipulate directly
without having to construct them. According to Battista (1998b):

The new *Standards* in the USA also recommend
that middle grades students develop and apply their understanding of spatial
transformations and symmetry to investigate and interpret geometric figures.
The *Standards *also recommend students investigate composition of
transformations, forming and testing conjectures as a prelude to proving
geometric relations in the high school. Dynamic geometry environments provide
a medium in which the making and testing of conjectures becomes a laboratory
science. The following example is an illustration of how students might
investigate composition of transformations in the plane.

__Finding a single transformation that does the
same as 2 reflections__. In Figure 4 above, students could try to find
a single transformation that would move the first stick figure onto the
third figure. They could reason that the figures are facing the same direction,
so a translation might work. They could draw a segment and designate it
as a translation vector. They could then translate the first stick figure
by this vector and test to see if its image coincides with the third stick
figure. If it does not, they could manipulate their vector segment until
the two images did coincide. They might then make a conjecture concerning
the two mirror lines and the translation vector. They could then test this
conjecture by changing the position of the two mirror lines and making
the necessary adjustment in their translation vector. Measurements of distances
between mirrors and length of the vector could also be taken to quantify
their conjecture. A similar investigation with non-parallel (intersecting)
mirror lines could lead to conjectures relating rotation about a point
to reflection across two intersecting mirrors.

__Creating Escher-like Tessellations__. The transformational
symmetries of figures can be used to produce tiling patterns and artistic
tessellations in the style of M.C. Escher. Middle grades students find
this artistic application of mathematics particularly interesting and motivating.
Dynamic software such as *TesselMania!®* (Lee, 1997) provide students
with a means of directly altering geometric shapes so that they will tessellate
under specific transformations. The altered shape is then automatically
replicated and transformed through an animation process that results in
the tessellation. Properties of the tessellation can be explored and the
original shape can be decorated using a paint program built into the software.
Figure 5 illustrates a tessellation of a quadrilateral using "rotation
about the midpoints of each side" to form the altered shape (a man's head).
This altered quadrilateral is then tessellated using rotations about the
midpoints of each side, combined with translations along each diagonal.

Figure 5: Tessellation of a man's head using *TesselMania!*

Implications for Teaching and Learning in the Secondary School

At the secondary level dynamic geometry environments can (and should) completely transform the teaching and learning of mathematics. Dynamic geometry turns mathematics into a laboratory science rather than the game of mental gymnastics, dominated by computation and symbolic manipulation, that it has become in many of our secondary schools. As a laboratory science, mathematics becomes an investigation of interesting phenomena, and the role of the mathematics student becomes that of the scientist: observing, recording, manipulating, predicting, conjecturing and testing, and developing theory as explanations for the phenomena.

The teacher intending to take advantage of this software,
and change mathematics into a laboratory science for her students, faces
many challenges. As Balacheff & Sunderland (1994) point out, the teacher
needs to understand the "domain of epistemological validity" of a dynamic
geometry environment (or microworld). This can be characterized by "the
set of problems which can be posed in a reasonable way, the nature of the
possible solutions it permits and the ones it excludes, the nature of its
phenomenological interface and the related feedback, and the possible implication
on the resulting students' conceptions." (p. 13) Such knowledge can only
be obtained over a long period of time working with the software both as
a tool for one's own learning as well as a tool for teaching mathematics.
There are resources, however, that teachers can turn to. The publication
"Geometry Turned On" (King and Schattschneider, 1997) provides several
examples of successful attempts by classroom teachers to integrate dynamic
geometry software in their mathematics teaching. Michael Keyton (1997)
provides an example that comes closest to that of learning mathematics
as a laboratory science. In his Honors Geometry class (grade 9) he provided
students with definitions of the eight basic quadrilaterals and some basic
parts (e.g. diagonals and medians). He then gave them three weeks to explore
these quadrilaterals using *Sketchpad.* Students were encouraged to
define new parts using their own terms and to develop theorems concerning
these quadrilaterals and their parts. Keyton had used this activity with
previous classes without the aid of dynamic geometry software. He states:

Keyton's activity with quadrilaterals stays within the bounds of the traditional geometry curriculum but affords students the opportunity to create their own mathematics within those bounds. Other educators have used dynamic geometry as a catalyst for reshaping the traditional curriculum. Cuoco and Goldenberg (1997) see dynamic geometry as a bridge from Euclidean Geometry to Analysis. They advocate an approach to Euclidean geometry that relates back to the "Euclidean tradition of using proportional reasoning to think about real numbers in a way that developed intuitions about continuously changing phenomena." (p. 35) This approach involves locus problems, experiments with conic sections and mechanical devices (linkages, pin and string constructions) that give students experience with "moving points" and their paths.

Both Cabri II and *Sketchpad* have the facility
to trace the path of points, straight objects and circles. The locus of
a moving point can be constructed as an object. Points can move freely
on such constructed loci. These features enable the user to create direct
links between a geometrical representation of a changing phenomenon and
a representation of the varying quantities involved as a coordinate graph.
As an example, consider the investigation of the area of a rectangle with
fixed perimeter. Figure 6 shows a *Sketchpad* sketch in which the
perimeter, length AB, height BC and area of the rectangle ABCD have been
measured. A point has been plotted on *Sketchpad's* coordinate axes
using the measures of AB and area of ABCD as (x, y). The trace of this
point can be investigated as the length AB of the rectangle is changed
by direct manipulation of vertex A. In Figure 6, the locus of point (x,
y) as the length of base AB changes has been constructed. The rectangle
with maximum area can be found by experiment to be when length and height
are approximately equal (see Figure 7). The fact that the measures of base
and height are not exactly the same in Figure 7 should lead to an interesting
discussion, and a need to "prove" by algebraic means that the maximum area
will, in fact, be when AB = BC.

Figure 6: Plotting base against area of fixed perimeter rectangle

Figure 7: Rectangle with maximum area

__Proof in Dynamic Geometry__

The above example of finding a solution in dynamic
geometry by experiment is analogous to finding roots of a polynomial using
a graphing calculator. The solution can be found but the students still
have a need to prove that the solution is valid. In the case of the rectangle
with maximum area there is a need to prove that the conjecture (or hypothesis)
that, for any rectangle with fixed perimeter, the maximum area will be
achieved when the rectangle becomes a square. Manipulating rectangle ABCD
in the sketch (moving vertex B will change the perimeter) can give convincing
evidence that the generalization is indeed true. There is a danger here
that students may regard this "convincing evidence" as a proof. Michael
de Villiers (1997, 1998, 1999) has addressed this concern through a thorough
analysis of the role and function of proof in a dynamic geometry environment.
De Villiers expands the role and function of proof beyond that of mere
verification. If students see proof only as a means of verifying something
that is "obviously" true then they will have little incentive to generate
any kind of logical proof once they have verified through their own experimentation
that something is always so. De Villiers suggests that there are at least
five other roles that proof can play in the practice of mathematics: explanation,
discovery, systematization, communication, and intellectual challenge.
He points out that the conviction that something is true most often comes
*before* a formal proof has been obtained. It is this conviction that
propels mathematicians to seek a logical *explanation* in the form
of a formal proof. Having convinced themselves that something must be true
through many examples and counter examples, they want to know *why*
it must be true. De Villiers (1999) suggests that it is this role of *explanation*
that can motivate students to generate a proof:

__The Power Plant Problem__. A power plant is
to be built to serve the needs of three cities. Where should the power
plant be located in order to use the least amount of high-voltage cable
that will feed electricity to the three cities? If the three cities are
represented by the vertices of a triangle, ABC, then this problem can be
solved by finding a point with minimum sum of distances to all three cities.
In exploring this situation in *Sketchpad* students can measure the
three distances from an arbitrary point P and the three vertices, A, B
and C of the triangle. They can then sum these distances and move P around
to find a location with minimum sum. When such a location appears to have
been found, students can make conjectures concerning relations among P
and the three vertices. Many students conjectures have been to see if any
of the known triangle centers satisfy the minimum sum requirement (e.g.
incenter, centroid or cirumcenter). Some of these may well appear to work
for certain triangles, but not for others. Eventually, some students will
notice that the angles formed by the point P and each pair of vertices
appear to be the same. Measurements would indicate that they are all close
to 120 degrees.

Having discovered a possible invariant in the situation, students then look for a way to construct a point that subtends 120 degrees with each pair of vertices. Various construction methods arise. One way is to construct an equilateral triangle on two sides of the triangle ABC and then construct the circumcircles of these equilateral triangles. Where the circumcircles intersect will subtend 120 degrees with each side of the triangle (see Figure 8).

Figure 8: Constructing the location of the power plant, P.

After the students have successfully located the
position of the power plant and found a way of constructing that position,
I ask them to explain *why* this point provides the minimum sum of
distances to each vertex of the triangle formed by the three cities. This
question challenges them to find a way of proving that their constructed
point P must be the minimum point (at least for triangles with no angle
greater than 120!). This proves to be a difficult challenge for my students
but one they are willing and eager to engage. The proof that I find the
most satisfying and explanatory is one that makes use of the fact that
the shortest distance between two points is along a straight path between
the two points. The proof involves a rotation of segments AP, AB and BP
about vertex A by 60 degrees (see Figure 9).

Figure 9: Rotation of ABP about A by 60û to form AB'P'

As rotation preserves length, AP' is congruent to AP and B'P' is congruent to BP. Thus AP'P is an isosceles triangle with an angle of 60û between the congruent sides. As base angles of an isosceles triangle are equal, and angle sum of any triangle is 180, the base angles must also be 60û. Thus triangle AP'P is equiangular, and, therefore, equilateral. Thus P'P is congruent to AP.

Thus by the above reasoning, the path B'P'PC is equal in length to the sum of the distances BP, AP and PC. The path B'P'PC will have a minimum length if, and only if, the path is a straight path, as the shortest distance between two points (B' and C) lies on a straight path. Rotation preserves the shapes of figures. Thus the angle relationships within triangle AP'B' are the same as in triangle APB. In particular, when angle APB = 120û, angle AP'B' = 120û and thus B'P'P will be 180û -- a straight angle. Also, when angle APC = 120û, angle P'PC will also be a straight angle. Thus B'P'PC will lie on a straight path when (and only when) P subtends angles of 120û with each side of the triangle ABC. Thus, this is the condition that provides the minimum sum of distances from P to the three vertices of the triangle ABC.

The above argument appears logical and rigorous as
well as explanatory. It, therefore comes as a surprise to most students
to learn that the power plant should be built in the center of city B (rather
than at P) when the angle ABC is greater than 120û! This "exception"
to what they have just proved leads to an investigation of the implicit
assumptions in their proof (e.g. that P is in the interior of triangle
ABC). The rotation used in the above proof also gives rise to an alternative
construction for finding the location of P: Draw lines connecting the outer
vertex of each equilateral triangle to the opposite vertex of the original
triangle. Where these lines intersect will be the location of point P,
also known as the *Fermat* point.

Implications for Teaching and Learning at the College Level

While the above example was taken from my course with pre-service secondary teachers of mathematics, I regard it as an appropriate problem to pose to secondary students working with dynamic geometry technology. Geometry teaching and learning at the college level can also be enriched through the use of this technology. Hampson (1997) describes a beginning course in geometry at college for entering freshmen in a British teacher education institution. He used dynamic geometry software as a medium of exploration throughout his course. He students were able to explore topics such as conics and projective geometry as dynamic phenomena to be investigated, generating their own theorems as well as discovering classic ones such as the theorems of Pappus and Desargues.

Parks (1997) suggests an alternate classification
of isometries in the plane made possible in a dynamic geometry medium.
All isometries can be uniquely identified by their *orientation preserving*
property and the nature of any *fixed points* under the transformation
(points that transform to themselves). Any isometry transformation in the
plane can then be identified through an investigation of *the orbit*
of the transformation under iteration. For instance, if the orbit of two
rotations about different points in the plane results in a circular path
then there is one fixed point: the center of that circular path, and the
isometry is a rotation (as two rotations would be orientation preserving).
If the orbit resulted in a straight path then there would be no fixed point
and the isometry would be a translation.

King (1997) provides examples of how an exploration of similarity transformations using dynamic geometry software can lead to alternative methods of construction and proof of classic theorems. Menelaus's theorem, concerning the product of the ratios of parts of each side of a triangle formed by a line intersecting all three sides (extended), is one such example. By composing similarity transformations with isometries, spiral growth can be investigated.

In Olive (1997) I describe how *Sketchpad* can
be used to investigate the Joukowski transformation of points in the complex
number plane. I apply this transformation to a circle in the complex plane
to produce an airfoil shape (see Figure 10).

Figure 10: The Joukowski Airfoil

The Joukowski transformation uses *inversion in
a circle* followed by reflection in the real axis to produce the point
1/z from point z (see Olive, 1997, pp. 169-176 for the mathematical details).
The ability to actually construct an inversion transformation opens up
many new avenues of exploration for the college student. Non-Euclidean
geometries can be modeled and explored visually as well as theoretically.
The Poincaré disc model for hyperbolic geometry can be constructed
in *Sketchpad,* along with script tools for drawing hyperbolic segments,
rays, lines and circles. Hyperbolic angles also can be measured using a
script tool. Figure 11 illustrates some properties of hyperbolic geometry
that would be very difficult to visualize in any other medium.

Figure 11: The Poincaré Disk model of Hyperbolic Geometry

In Figure 11, three hyperbolic lines have been constructed
through point A parallel to the hyperbolic line through points B and C,
thus demonstrating the fundamental difference between hyperbolic and Euclidean
geometry. Two hyperbolic triangles have been constructed on the same chord
DF of the hyperbolic circle with center O. Triangle DOF connects the chord
with the center of the circle. Triangle DEF has all three vertices on the
circumference of the circle. The measures of hyperbolic angles DOF and
DEF illustrate how the angle at the circumference is NOT half the angle
at the center of a hyperbolic circle, as it is in Euclidean geometry. See
Dwyer and Pfiefer (1999) for example explorations with the Poincaré
disk using *Sketchpad.*

Schumann and Green (1994) provide alternative approaches
to geometry using the *Cabri-Geometre* software that can be used at
both the high-school and college levels. They emphasize *theorem-finding*
through the activity of varying geometrical figures dynamically. They also
look at angle theorems as *invariance properties, *and make extensive
use of the loci of varying points under manipulation as alternative approaches
to difficult construction tasks. The set of 20 problems they present at
the end of their book would challenge many college mathematics students.
The following exploration is based on one of these problems concerning
the *Arbelos *of Archimedes (see Figure 12).

Figure 12: The Arbelos of Archimedes

In Figure 12, the two shaded semi-circles (with centers at E and F) are constructed on the diameter of the largest semi-circle with center at A. The two shaded semicircles are tangent to one another at C and tangent to the outer semi-circle at D and B. The Arbelos is the unshaded region bounded by all three semi-circles, so named because of its resemblance to the shoemaker's knife of ancient Greece. The problem is to construct a circle inside the Arbelos that is tangent to all three given semi-circles.

Using *Sketchpad* I started my exploration of
this problem by drawing a circle inside the Arbelos so that it *appeared*
tangent to all three semi-circles. I then constructed tangent lines at
the apparent points of tangency and lines through the centers of each of
the semi-circles (see Figure 13). My strategy was to vary the point C that
defined the diameter of the two inner semi-circles, readjust the tangent
circle and see if I could find any relations that were invariant. I made
several measurements within the various triangles that are formed in Figure
13. I looked at ratios of sides, especially in triangles that should be
similar. I found a surprising relation between two ratios that did NOT
appear to belong to similar triangles: the ratios AB/AC and FM/FB appeared
to remain approximately equal under variance of point C and the readjustment
of the tangent circle. Using this possibility, I CONSTRUCTED point M by
dilating point B about center F by the ratio AB/AC, as FM=FB.AB/AC if the
equality of ratios holds true.

Figure 13: Tangent lines and center lines constructed on the Arbelos

Having constructed the location of point M (the foot of the common tangent to the outer semi-circle and my inner circle) I constructed a circle with diameter AM to find the point of tangency, T in Figure 13. Using the line through A and T, I constructed the tangent at T as the perpendicular line to AT at T. This tangent line (of course) passed through point M. In order to construct the center of my inner circle, I needed one more tangent line. Figure 13 indicates that the tangent line at point B intersects tangent TM at a point (N) coincident with the intersection of the common tangent at point J with semi-circle center F and my inner circle. [This observation gives rise to an interesting property that turns out to be true for any three circles that are tangent to one another: The three tangents at the three points of tangency of the three circles will intersect in a single point. I leave the proof of this relation as an exercise for the reader!] Using this relation among the three possible tangents, I constructed a circle with diameter FN to find the point of common tangency, J. Where the line FJ intersects AT gives me the center of my inner circle, O (see Figure 14).

Figure 14: Construction of the Circle Tangent to all three Semi-Circles.

Moving the arbitrary point C confirms that the above
construction works for all points C between A and B. I was now faced with
the challenge of *explaining* why this construction works. I am close
to a solution using triangle similarities but have not yet found a complete
proof that does not make assumptions based on the given construction. In
my explorations I did discover another surprising identity: The diameter
of the inner tangent circle is equal to the distance of its center from
the diameter of the outer semi-circle! This identity holds for all C between
A and B.

The above exploration is an example of how dynamic
geometry technology can lead to *Theorem Finding* and motivate a journey
of *Theorem Proving* at the college level. Hofstadter (1997) tells
a compelling story of theorem finding and proving with *Sketchpad*
from a research mathematician's point of view, in his *discovery and
dissection of a geometric gem.*

Concluding Comments: The Need for More Research

While there have been many personal accounts of the powerful learning that can take place when students of all ages work with dynamic geometry technology (my own included), there have been very few, well designed research projects to study the effects on learning in such environments. A group in Italy headed by Ferdinando Arzarello (Arzarello et al, 1998a)has conducted investigations of students' transitions from exploring to conjecturing and proving when working with Cabri. They applied a theoretical model that they had developed to analyse the transition to formal proofs in geometry (Arzello et al., 1998b). They found that different modalities of dragging in Cabri were crucial for determining a shift from exploration to a more formal approach. Their findings are consistent with the examples given in previous sections of this paper. The different modalities of dragging that they classified are described as:

The group at the University of Grenoble in France have been conducting research studies on the use of Cabri for many years (Laborde, 1992, 1993, 1995, 1998). They have focussed both on what students are learning when working with Cabri and the constraints both students and teachers face when teaching and learning with Cabri. Laborde (1992 & 1993) and Balacheff (1994) conclude that the observation of what varies and what remains invariant when dragging elements of a figure in Cabri, helped break down the separation of deduction and construction that Schoenfeld (1988) found in his study of geometry teaching and learning. Laborde (1998) points out that it takes a long time for teachers to adapt their teaching to take advantage of the technology. She reports three typical reactions that teachers have to the perturbations caused by the introduction of dynamic geometry software into the teaching-learning situation:

- reaction alpha: ignoring the perturbation
- reaction beta: integrating the perturbation into the system by means of partial changes
- reaction gamma: the perturbation is overcome and looses its perturbing character. (p. 2)

The efforts at researching the effects of technology use on students' learning has been hampered by the prevalence of reactions alpha and beta. As more teachers achieve reaction gamma we have both the opportunity and the responsibility to carefully research the effects of integration of dynamic geometry technology into the teaching and learning of geometry and mathematics in general.

References

Arzarello, F., Micheletti, C., Olivero, F., Robutti,
O., Paola, D. & Gallino, G. (1998a). Dragging in Cabri and Modalities
of Transition from Conjectures to Proofs in Geometry. In A. Olivier &
K. Newstead (Eds.) *Proceedings of the 22 ^{nd} Annual Conference
of the International Group for the Psychology of Mathematics Education
(PME 22).* Stellenbosch, South Africa, Volume 2, 32-39.

Arzarello, F., Micheletti, C., Olivero, F., Robutti,
O.& Paola, D.. (1998b). A model for analyzing the transition to formal
proofs in geometry. In A. Olivier & K. Newstead (Eds.) *Proceedings
of the 22 ^{nd} Annual Conference of the International Group for
the Psychology of Mathematics Education (PME 22).* Stellenbosch, South
Africa, Volume 2.

Balacheff, N. (1994). Artificial Intelligence and
Real Teaching. In C. Keitel & K. Ruthven (eds.), *Learning through
computers: Mathematics and Educational Technology, *131-158*. *Berlin:
Springer Verlag.

Balacheff, N. and Sutherland, R. (1994) *Epistemological
domain of validity* *of microworlds. The case of Logo and Cabri-géomètre*.
Paper to IFIP, The Netherlands.

Battista, M. T. (1998a). SHAPE MAKERS: Developing
Geometric Reasoning with *The Geometer’s Sketchpad*. Berkeley, CA:
Key Curriculum Press.

Battista, M. T. (1998b). Computer Environments that Engender Students’ Construction of Mathematical Ideas and Reasoning: A Constructivist Perspective. Paper presented at the ENC Technology and NCTM Standards 2000 Conference. Arlington VA, June 5-6.

Brousseau, G. (1988). Le contrat didactique: le milieu.
*Recherches en didactique des mathématiques*, 9(3), 309-336.

Cuoco, Albert A., and E. Paul Goldenberg. (1997).
Dynamic Geometry as a Bridge from Euclidean Geometry to Analysis. In James
R. King and Doris Schattschneider (Eds.) *Geometry Turned On!: Dynamic
Software in Learning, Teaching, and Research. *Washington, D.C.: The
Mathematical Association of America, 33-46.

de Villiers, M. (1994). The role and function of
a hierarchical classification of quadrilaterals. *For the Learning of
Mathematics,* 14(1), 11-18.

de Villiers, M. (1997). The role of proof in investigative,
computer-based geometry: Some personal reflections. In James R. King and
Doris Schattschneider (Eds.) *Geometry Turned On!: Dynamic Software in
Learning, Teaching, and Research. *Washington, D.C.: The Mathematical
Association of America, 15-24.

de Villiers, M. (1998). An alternative approach to
proof in dynamic geometry. ? In R. Lehrer & D. Chazan (Eds.) *Designing
Learning Environments for Developing Understanding of Geometry and Space.*
Mahwah, NJ: Lawrence Erlbaum Associates, 369-394.

de Villiers, M. (1999). *Rethinking proof with
the Geometer's Sketchpad.* Emeryville, CA: Key Curriculum Press.

Dywer, M & Pfiefer, R. E. (1999). Exploring Hyperbolic
Geometry with The Geometer's Sketchpad. *The Mathematics Teacher*
92, 7, 632-637.

Finzer, Bill, and Dan Bennett (1995). From Drawing
to Construction with The Geometer's Sketchpad. *The Mathematics Teacher*
88, 5, 428-431.

Fuys, D., Geddes, D. & Tischler, R. (1988). The
van Hiele model of thinking in geometry among adolescents. *Journal for
Research in Mathematics Education Monograph, *No. 3.

Goldenberg, E. P. & Cuoco, A. A. (1998). What
is Dynamic Geometry? In R. Lehrer & D. Chazan (Eds.) *Designing Learning
Environments for Developing Understanding of Geometry and Space.* Mahwah,
NJ: Lawrence Erlbaum Associates, 351-368.

Goldenberg, E. P., Cuoco, A. A. & Mark, J. (1998).
A role for geometry in general education. In R. Lehrer & D. Chazan
(Eds.) *Designing Learning Environments for Developing Understanding
of Geometry and Space.* Mahwah, NJ: Lawrence Erlbaum Associates, 3-44.

Hampson, T. (1997). Beginning geometry at college.
. In J. King & D. Schattschneider (Eds.) *Geometry turned on! Dynamic
software in learning, teaching, and research.* Washington, D.C.: The
Mathematical Association of America, 95-104.

Hofstadter, D. R. (1997). *Discovery and dissection
of a geometric gem. *In J. King & D. Schattschneider (Eds.) *Geometry
turned on! Dynamic software in learning, teaching, and research.* Washington,
D.C.: The Mathematical Association of America, 3-14.

Hoyles, C. & Sutherland, R. (1986). When 45 equals
60. In C. Hoyles, R. Noss & R Sutherland (Eds.)Proceedings of the Second
International Conference for *Logo and Mathematics Education.* London:
Department of Mathematics Statistics and Computing, University of London
Institute of Education.

Keyton, M. (1997). Students discovering geometry
using dynamic geometry software. In J. King & D. Schattschneider (Eds.)
*Geometry turned on! Dynamic software in learning, teaching, and research.*
Washington, D.C.: The Mathematical Association of America, 63-68.

King, J. (1997). An eye for similarity transformations.
. In J. King & D. Schattschneider (Eds.) *Geometry turned on! Dynamic
software in learning, teaching, and research.* Washington, D.C.: The
Mathematical Association of America, 109-120.

King, J. & Schattschneider, D. (Eds.) (1997).
*Geometry turned on! Dynamic software in learning, teaching, and research.*
Washington, D.C.: The Mathematical Association of America.

Laborde C. (1992) Solving problems in computer based
geometry environments: the influence of the features of the software. *Zentrablatt
für Didactik des Mathematik*, 92(4) 128-135.

Laborde, C. (1993). The computer as part of the learning
environment: The case of geometry. In: C. Keitel and K. Ruthven (Eds.),
*Learning from Computers: Mathematics Education and Technology*, 48-67.
Berlin: Springer-Verlag.

Laborde, C. (1995). Designing tasks for learning
geometry in a computer-based environment. In L. Burton & B. Jaworski
(Eds.)*Technology in Mathematics Learning - a bridge between teaching
and learning, *35-68. London: Chartwell-Bratt.

Laborde, C. (1998). Factors of integration of dynamic geometry software in the teaching of mathematics. Paper presented at the ENC Technology and NCTM Standards 2000 Conference. Arlington VA, June 5-6.

Lee, K. (1997). *Tesselmania!* (computer program).
Minneapolis, Minnesota: MECC.

Lehrer, R., Jenkins, M. & Osana, H. (1998). Longitudinal
study of children's reasoning about space and geometry. In R. Lehrer &
D. Chazan (Eds.) *Designing Learning Environments for Developing Understanding
of Geometry and Space.* Mahwah, NJ: Lawrence Erlbaum Associates, 137-168.

NCTM (2000). *Principles and Standards for School
Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

Olive, J. (1997). Creating airfoils from circles:
The Joukowski transformation. . In J. King & D. Schattschneider (Eds.)
*Geometry turned on! Dynamic software in learning, teaching, and research.*
Washington, D.C.: The Mathematical Association of America, 169-176.

Olive, J. (1998). Opportunities to explore and integrate
mathematics with the Geometer's Sketchpad. In R. Lehrer & D. Chazan
(Eds.) *Designing Learning Environments for Developing Understanding
of Geometry and Space.* Mahwah, NJ: Lawrence Erlbaum Associates, 395-418.

Papert, S. (1980). *Mindstorm: Children, computers
and powerful ideas.* Sussex, England: Harvester.

Parks, J. M. (1997). Identifying transformations
by their orbits. . In J. King & D. Schattschneider (Eds.) *Geometry
turned on! Dynamic software in learning, teaching, and research.* Washington,
D.C.: The Mathematical Association of America, 105-108.

Schumann, H. & Green, D. *Discovering geometry
with a computer - using Cabri-géomètre.* Bromley, Kent,
England: Chartwell-Bratt.

van Hiele, P. M. (1986). *Structure and insight:
A theory of mathematics education.* Orlando, FL: Academic Press.