CONCLUSION
6.2 The role and function of Sketchpad as a mathematical modeling tool and its potential role in mathematics education
given diagram of a quadrilateral, perpendicular bisectors had to be used. They were adamant that this was the only way to have done it, thus indicating that it was an obvious conclusion. The new knowledge they discovered became so empowering that they felt strongly about their hypothesis that in any quadrilateral the perpendicular bisectors would be concurrent. Of course, it is this very same idea that lead to the surprise that the learners experienced when they found that their hypothesis was not always true. Perhaps, this type of teaching should be explored in more classrooms in order to establish a definite theory regarding teaching by surprise and contradiction.
6.2 The role and function of Sketchpad as a mathematical modeling tool
The general usefulness of Sketchpad can be summarized as follows:
1. The ease with which the diagrams were constructed and manipulated. This allowed the researcher and the learners the freedom to drag, change diagrams and manipulate the figures as and when required. This may have been impossible to achieve if other mathematical materials were used. This is in agreement with the NCTM STANDARDS 2000 document (1998 : 42) : "Dynamic geometry programs enhance students' experience of two-dimensional and three-dimensional geometry.
Such programs make it easy to generate a large set of instances, and to make conjectures" .
2. The use of buttons saved a lot oftime and allowed the learners to see changes at the simple click of a button. Tedious constructions were avoided by the creation of these buttons.
3. The diagrams constructed using Sketchpad were clear and made misinterpretation less likely. Pencil and paper sketches might have resulted in many errors such as those created by using instruments incorrectly or by using a thick pencil lead.
4. Measurements were easily obtained when they were required. In many instances, this was fundamental to their understanding. It would have been quite time consuming if learners had to print the diagrams on paper and measured using a ruler. The possibility of them making incorrect measurements was also removed.
5. The manipulation of the diagrams on the screen allowed the learners to grasp properties of figures easily and it was evident that they understood the basic properties after a few minutes.
6. The fact that Sketchpad allowed them to develop high levels of conviction within a few minutes meant that more work could be done in a shorter period of time. But, to some extent, the learners expressed a desire for some explanation because they could instantly see that some of their conjectures were incorrect. This immediate feedback surprised them and created in them a need to know why the result was different.
7. The fact that certain relationships could be visualized helped the learners to accept that the concepts were true ("if I did not see it I would not have believed it").
These aspects of Sketchpad's usefulness show that the software has tremendous potential in mathematics education. Geometry teaching could become much easier because of the simple construction and manipulation of sketches. Simple and complex ideas can be understood within a few minutes affording educators greater freedom to work with more challenging problems and allowing the educator more time to work with individual learners.
Carefully planned worksheets enable learners to work comfortably on their own and if learners are au fait with the different tools available in Sketchpad, constructing and measurement becomes a simple activity. There can be very little doubt that Sketchpad is a tremendous mediating artifact in a mathematics classroom and its potential has not been adequately investigated. Off course, the constraints to using Sketchpad in a classroom are real (for example the affordability of computers and the program itself) but this must not allow us to detract from the fact that Sketchpad is a useful mathematics software.
It must also be stated that Sketchpad allowed the learners to gain sudden insight. An example of such an instance was when the learners realized that the perpendicular bisectors of all triangles were concurrent as compared to their initial conjecture. This is exactly the reason why the learners were surprised with some of the results they experienced. The experience of being surprised was in fact more than just that. It was the point at which the learners discovered relationships that were unfolding on the computer screen. It is this capacity to suddenly gain new insights that makes learning through the use of dynamic computer software all the more worthwhile.
In connecting ideas and concepts the learners were personally constructing their own meanings. Battista (2002 : 333) states that "true understanding of mathematics arises as students progress through phases of action (physical and mental manipulations), abstraction (process by which actions become mentally solidified so that students can reflect and act on them), and reflection (conscious analysis of one's thinkingJ". With the use of Sketchpad the learners were able to manipulate (physically) the diagrams and from what they observed, construct meaningful mental responses that aided in the understanding of the concepts that they were discovering and developing.
According to Battista (2002 : 339) "working in this environment (Sketchpad or other dynamic software) helps students build increasingly sophisticated mental models for thinking about shapes, models that form the foundation on which genuine understanding of geometry must be constructed'. He goes on further to state that "such work supports and encourages students' development and understanding of the property-based conceptual system used in geometry to analyze shapes. It encourages students to move to higher levels of thinking instead of having to memorize a laundry list of shape properties. The
environment involves students as conceptualizing participants, not spectators, in the process of doing geometry". This is a significant departure from the mundane pencil and paper geometry. The level of conviction attained here is significant and this together with the time frames within which conviction is achieved makes Sketchpad a powerful tool in the dynamic geometry context.
Over and above all the useful aspects of Sketchpad as a modeling tool, Sketchpad contributed most significantly to the third type of model application, namely creative application ( refer to section 3.2.1). Sketchpad allowed for the effective use of modeling introducing the concept of "equidistance".
FAEEZA RESEARCHER
FAEEZA
RESEARCHER FAEEZA
RESEARCHER FAEEZA
You can locate it around there (pointing), somewhere where it would be equal distance apart.
Do that construction .... (after a point and the segments were constructed) Okay how can we check whether it is equidistant apart?
You measure the distance from the village to the point.
But they are not, what must you do now?
We move it ... I'll drag this point around.
How long should we move it around?
Till we get the.. till the distances are equal.
Thereafter, because of the nature of the software, the learners were able to easily see the link between the idea of equidistance and the concept of perpendicular bisector.
PRAVANIE
RESEARCHER PRAVANIE RESEARCHER PRAVANIE RESEARCHER
And I can measure the distance again, from I to the new point and from J to the new point, and you can move it along until it is equal distant.
How many points do you think we can find like this?
A lot.
A lot? Is there anyway we can find a generalization? Were can we find the other points.
On the perpendicular to the line I J
So you are saying that it is going to be perpendicular to the line I
& J. Maybe we should draw that.... Construct that perpendicular line. (after a while) There we go you are right, it passes through all the points. So what have we determined .. .if we had just two
PRAVANIE RESEARCHER PRAVANIE RESEARCHER PRAVANIE
Just construct the perpendicular line.
Is it any perpendicular line? What does this perpendicular line do?
I mean I could draw a perpendicular line there isn't that so?
It divides the line.
What do we call that line?
Perpendicular bisector.
The fact that the learners could 'see' the points arranged in a linear fashion enabled them to draw an instant conjecture that these points lay on the straight line. Furthermore, they joined these points to immediately confirm their conjecture. Although by looking at the line they constructed they could tell that it was perpendicular to the line joining the two villages, it was a simple task for them to measure the angle formed in order to reinforce the fact that the line was perpendicular. The link between the equidistant points and the perpendicular bisectors was thus achieved.
The ease with which the learners formed the relationship between the concurrent perpendicular bisectors and the quadrilateral being cyclic contributed to their understanding ofthe problem.
RESEARCHER
NIGEL
RESEARCHER NIGEL
RESEARCHER
NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER
Can you construct a circle with the point of concurrency as the centre and move it slowly upwards to this vertex here.( after awhile) What do you notice?
(after a while) The four edges of the quadrilateral meet the circumference of the circle.
Edge? ... which are you referring to as the edge? (after learner points) ... oh we call that a vertex ... all of these are vertices.
Ok.
So the vertices of this quadrilateral lie on the circumference of the circle. Do you think that if I had any kind of quadrilateral the result would be the same?
Yes (very confidently)
So no matter what kind of quadrilateral, the perpendicular bisectors will always be concurrent?
Yes (emphatically)
Perhaps we should check. Change the quadrilateral and see what happens to the circle?
It went away from the vert ... vertices.
Oh and what else do you notice about the perpendicular bisectors?
NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
It does not meet at all.
Change it again, and let see if we can make them meet again. Yes they are meeting again but what do you observe?
The perpendicular bisectors are concurrent ....
What can you say about the vertices ofthe quad.
It touches. . .. It lies on the circle
Then can you draw a conclusion from that?
The perpendicular bisectors are concurrent when it lies on the circumference of the circle.
Do you think that this is always true?
Definitely!