Finally I would like to linger upon the envisioned value of digital images in mathematical investigations of students. What learning advantages does measurement and manipulation of images offer? When answering this question I do not only think of egg investigations, but more on mathematical modeling of concrete objects taken from real world situations. Below I mention in random order some educational benefits found in my own classroom work with still images and video clips (Heck & Uylings, 2006; Heck, 2007), and mentioned in papers of other educational researchers and teachers ((Huylebrouck, 2007; Oldknow, 2003a; Oldknow, 2003b; Pierce et al., 2004; Schumann, 2004; Sharp et al., 2004):
The benefits mentioned above focus on increased engagement of students, appreciation of the usefulness of mathematics, training of their "mathematical eye", training of the use of ICT, and on experimental exploration and analysis of real world phenomena that can be modeled mathematically. The benefits in the process of acquiring procedural and conceptual knowledge in mathematics come less to the foreground. The main reason is that these benefits cannot really be clearly separated from advantages of using a dynamic mathematical software in education in terms of complexity, authenticity, versatility, ease of communcation, and so. A traditional modeling approach in the mathematics classroom also has great potential to enhance the students' mathematical knowledge and enrich their knowledge and skills in applying mathematics. But ICT offers students a greater opportunity to work directly with high-quality real data in much the same way as practicing professionals would do, including the use of the same methods and techniques. In other words, ICT and real-life contexts contribute to the realization of authentic tasks for students,
I cannot resist the tempation to show a nice example of such an authentic task that comes from crime science photography. When you want to measure in a digital image of a crime scene the evidence within it, you are immediately confronted with the problem of a perspective view of the scene. There can be no measuring in perspective in the strict sense of the term (i.e., the establishment of a two-way metrical relationship between a given object and its representation on the picture plane). In order to be able to measure in the digital image distances between objects in the crime scene, the photographer usually places a perspective grid of known size in the scene. In Figure 15 you see a picture of a fake crime scene that contains a square perspective grid consisting of four tiles. I have used the tools of the GeoGebra package to extend the grid on the ground floor in all directions. You can replay the GeoGebra construction to see how it was actually done. The procedure followed is described in detail in (Robinson, 2007) Items of evidence are now located within imaginary tiles constructed, which can be used to get an indication of distances between the objects. By the process of grid reduction one can increase precision of the determination of location and distance.
Figure 15. Perspective grid photogrammetry: grid extension.
The process of grid extension is actually rooted in the one-point perspective constructions of checkerboard pavements that were used for the first time in Renaissance art, such as the method of Leon Battista Alberti (1407-1472), which he called "modo ottimo" (best way) in his work entitled Il Tratato della Pittura e I Cinque Ordini Archittonici ("On Painting") and which is better known under the Italian name "costruzione legittima" (legitimate construction). As a matter of fact, the grid extension method resembles more the so-called distance point construction, found and published by Jean Pélerin, also known as the Viator, in his treatise on perspective (De Artificiali Perspectiva, 1505). It can be proven that Pélerin's method, which is shown in Figure 16 [taken from (Pélerin, 1521)] and which can be replayed in the GeoGebra-supported outline shown in Figure 17, produces the same results as Alberti's method.
Figure 16. The distance point method in Pélerin's book.
Figure 17. The steps in the distance point method.
The steps in drawing the square-tiles pavement following the distance point method are:
This modern application of mathematics brings the students in a natural way in contact with the mathematics in art and architecture, the development of which started in the Renaissance. Nowadays these methods have been incorporated in computer programs so that the forensic scientist does have to go through the details of the construction process time after time. Projective geometry is also a major theoretical background in computer vision and underpins image rectification procedures that are built into software programs like image processing packages and, with educational purposes in mind, Coach. Figure 18 is a screen shot of a Coach activity in which the upper left window contains the perspective corrected image of the picture shown in the upper right window.
Figure 18. Crime scene photogrammetry in Coach.
For measurements on video clips it is worth mentioning that research (Ellermeijer & Heck, 2003; plus references in this paper) indicates that
Studying motion with a dynamic geometry package may seem impossible, but via stroboscopic photographs it can be done. In Figure 19 you see a screen shot of a GeoGebra activity in which the motion of a falling ball is studied with the help of a stroboscopic photograph. The time between successive images is 1/30 second. The vertical position of the falling ball at consecutive timings has been recorded by clicking on the digital image. In the coordinates system we have mentally changed the role of the x-axis as time axis, because GeoGebra does not allow us to use another variable than x. and we have displayed the y-t diagram. The collected data point can be modelled by a parabolic curve, which can be fitted with the help of slider bars. Admittedly, the data collection is a bit time-consuming, but the results make it worth the effort.
Figure 19. Modeling the motion of a falling ball via a stroboscopic photograph.
But also in the context of describing eggs mathematically, acquirement of mathematical knowledge and skills can be promoted by the use of digital images. For example the shape of the partridge egg (Figure 12) can also be described well with a circle and a parabola. But how can a student construct the circle by clicking on three points on the egg image? And how can (s)he construct a parabola by clicking on three points and giving the direction of the point at infinity on it? The first construction is a technique built into GeoGebra; for the second construction a method must be discovered and implemented as a macro definition. This brings the student immediately into the problem of constructing a particular geometrical object (hint: for the parabola, you can use a special case of Steiner’s construction of a conic through five points).
In short, also when the main teaching goal is application and enhancement of mathematical knowledge, working with still images and video clips in a dynamic geometry package offers a lot of opportunities in mathematical modeling activities. What seems to me most important here is that the use of digital images and video clips in education provides a student an excellent opportunity to be engaged in an attractive and more personalized manner in doing mathematics and to express oneselve in the language of mathematics.