The University of Georgia
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 the TI-92 calculator and the Casio Cassiopeia hand-held computer).
Geometry Expert (GEX, a new computer-expert system from China, Gao, Zhang & Chou, 1998).
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 realised 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 Geometerís 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.
Implications for Secondary Teaching and Learning
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 & Sutherland (1994) point out, the teacher needs to understand the "domain of epistemological validity" of a dynamic geometry environment (or microworld). This can be characterised 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 Honours Geometry class (grade 9) he provided students with definitions of the eight basic quadrilaterals and some basic parts (eg. 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 2 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 2, 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 3).
Figure 2: Plotting base against area of fixed perimeter rectangle
Figure 3: Rectangle with maximum area
The fact that the measures of base and height are not exactly the same in Figure 3 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.
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 generalisation is indeed true. There is a danger here that students may regard this "convincing evidence" as a proof. Michael de Villiers (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, systematisation, 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 centres satisfy the minimum sum requirement (eg. 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 4).
Figure 4: 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 5).
Figure 5: 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 centre 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 (eg. 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.
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. 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 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.
Arzarello, F., Micheletti, C., Olivero, F., Robutti, O.& Paola, D.. (1998b). A model for analysing the transition to formal proofs in geometry. In A. Olivier & K. Newstead (Eds.) Proceedings of the 22nd 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.
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. (1999). Rethinking proof with the Geometer's Sketchpad. Emeryville, CA: Key Curriculum Press.
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.
Gao, X. S., Zhang, J. Z. & Chou, S. C. (1998). Geometry Expert (computer program). Taiwan: Nine Chapter Pub. (in Chinese).
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.
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. & 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.
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.
van Hiele, P. M. (1986). Structure
and insight: A theory of mathematics education. Orlando, FL: Academic