Implications of Using Dynamic Geometry Technology for Teaching and Learning

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:

Cabri-Geometry (Cabri I and Cabri II for desktop computers, and Cabri on the TI-92 calculator).

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).

Goldenberg & Cuoco, (1998) provide an in-depth discussion on the nature of Dynamic Geometry. A common feature of dynamic geometry is that geometric figures can be constructed by connecting their components; thus a triangle can be constructed by connecting three line segments. This triangle, however, is not a single, static instance of a triangle which would be the result of drawing three line segments on paper; it is in essence a prototype for all possible triangles. By grasping a vertex of this triangle and moving it with the mouse, the length and orientation of the two sides of the triangle meeting at that vertex will change continuously. The mathematical implications of even this most simple of operations was brought home to me when my seven-year old son was "playing" with the Geometer's Sketchpad software (hereafter referred to as Sketchpad or GSP). As he moved a vertex around the screen he asked me if the shape was still a triangle. I asked him what he thought. After turning his head and looking at the figure from different orientations he declared that it was. I asked him why and he replied that it still had three sides! He continued to make triangles which varied from squat fat ones to long skinny ones (his terms), that stretched from one corner of the screen to another (and not one side horizontal!). But the real surprise came when he moved one vertex onto the opposite side of the triangle, creating the appearance of a single line segment. He again asked me if this was still a triangle. I again threw the question back to him and his reply was: "Yes. It’s a triangle lying on its side!" I contend that this seven-year old child had constructed for himself during that five minutes of exploration with the Sketchpad a fuller concept of "triangle" than most high-school students ever achieve. His last comment also indicates intuitions about plane figures which few adults ever acquire: That they have no thickness and that they may be oriented perpendicular to the viewing plane. Such intuitions are the result of what Goldenberg, Cuoco, & Mark (1998) refer to as "visual thinking."

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):

This microworld was designed to promote in students the development of mental models that they can use for reasoning about geometric shapes. In it, each class of common quadrilaterals and triangles has a "Shape Maker," a Geometer's Sketchpad construction that can be dynamically transformed in various ways, but only to produce different shapes in the class. For instance, the computer Parallelogram Maker can be used to make any desired parallelogram that fits on the computer screen, no matter what its shape, size, or orientation—but only parallelograms. It is manipulated by using the mouse to drag its control points—small circles that appear at its vertices. Battista has developed a sequence of activities with the Shape Maker microworld that he claims "encourage students to pass through the first three van Hiele levels—from the visual, to the descriptive-analytic, and into the abstract relational." (1998b) He describes this sequence as follows: In initial activities, students use Shape Makers to make their own pictures, then to duplicate given pictures. These activities encourage students to become familiar with the movement possibilities of the Shape Makers viewed as holistic entities. Students are then involved in activities that require more careful analysis of shapes — unmeasured Shape Makers are replaced by Measured Shape Makers that display instantaneously updated measures of angles and side lengths. Students are guided to find and describe properties of shapes. Finally, students are involved in classification by comparing the sets of shapes that can be made by each Shape Maker. (1998b) Providing students with ready-made script-tools (in Sketchpad) or macros (in Cabri) that students can use in the ways described by Battista, is one way of overcoming the constraints of prior knowledge mentioned above. Given such tailor-made microworlds within dynamic geometry environments, teachers in the middle grades can involve their students in activities that could help their progression to the higher levels of thinking in geometry, as recommended by the new Principals and Standards for School Mathematics (NCTM, 2000) in the USA.

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:

In previous years I had obtained an average of about four different theorems per student per day with about eight different theorems per class per day. At the end of the three-week period, students had produced about 125 theorems… In the first year with the use of Sketchpad, the number of theorems increased to almost 20 per day for the class, with more than 300 theorems produced for the whole investigation. (p.65) Goldenberg and Cuoco (1998) offer a possible explanation for the phenomenal increase in theorems generated by Keyton's students when using Sketchpad. Dynamic geometry "allows the students to transgress their own tacit category boundaries without intending to do so, creating a kind of disequilibrium, which they must somehow resolve." (p. 357) They go on to reiterate a point made by de Villiers (1994), that "To learn the importance and purpose of careful definition, students must be afforded explicit opportunities to participate in definition-making themselves." (Goldenberg and Cuoco, 1998, p. 357)

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:

When students have already thoroughly investigated a geometric conjecture through continuous variation with dynamic software like Sketchpad, they have little need for further conviction. So verification serves as little or no motivation for doing a proof. However, I have found it relatively easy to solicit further curiosity by asking students why they think a particular result is true; that is, to challenge them to try and explain it. (p. 8) The following example from my own class of pre-service secondary mathematics teachers illustrates the explanatory function of a proof when solving a problem using Sketchpad.

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:

(i) wandering dragging, that is dragging (more or less) randomly to find some regularity or interesting configurations; (ii) lieu muet dragging, that means a certain locus C is built up empirically by dragging a (dragable) point P, in a way which preserves some regularity of certain figures. (p. 3) They also describe a third modality: dragging test, that is used to test a conjecture over all possible configurations. I used all three modalities in my exploration of the Arbelos problem, above.

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:

It is only in the last stage (reaction gamma) that teachers make an adaptation in their teaching that truly integrates the technology.

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.

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