## 11. Bringing mathematics to life via digital images and vice versa

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):

- Mathematics and the real world are connected with each other in a rather direct way. This does not only increase the attractiveness of the work in the students’ eyes, but it also brings them in touch with applications of mathematics on objects from daily life. Exploratory activities range from measurements, calculations, and figural analysis to functional analysis of models of physical objects on the basis of digital images. The examples on egg mathematics illustrate the possibilities for reconstructing the 2-dimensional geometry of static objects via digital images. In the literature mentioned above, a couple of examples of dynamic models constructed with dynamic geometry packages can be found that allow functional testing and simulation of a dynamic model of a movable object via the dragging tool or animation tool in the software. In this type of activities, you (i) look for an object of interest in the real world, (ii) take a picture of this object with a digital camera or webcam, (iii) import the digital image itno a dynmaic geometry worksheet, (iv) analyze the picture of the object by drawing, measuring and calculating, (v) reconstruct the identified figure with the tools provided by the dynamic geometry package, and (vi), in case of a movable object, simulate the motion by dragging or animation. The main technical restrictions in the geometrical reconstruction process are that only objects can be modeled which can be described in 2-dimensional geometry terms and that care must be taken in central projective photography of objects that the image plane is parallel to the object plane and that the camera is focused on the center of the object in order to minimize perspective distortion. But the real danger lies in the fact that, for example when you are looking at art, architecture, and engineering through a mathematical/geometrical looking glass, you may discover a "hidden" proportions or curves which the artist or architect may not have considered or intended for use, or the use of which he or she even may dispute. Great care must be taken that the use of mathematical overlays of digital images of real objects are purposeful and have a clear relation with practice.
- Doing mathematics with still images and movies offers the opportunity to personalize mathematics, thereby increasing engagement of students. It already makes a study more interesting if it is the student him/herself who is visible in an image or movie, instead of another person or an impersonal picture or video clip of some object of study. Examples of the use of digital images can easily motivate students to find and work on their own examples.
- Students can experience in a playful manner that mathematical functions and geometrical transformations are not just a hobby of mathematics teachers, but that these mappings are really used in many applications of mathematics to real life problems, and in particular that there are used in all software systems that allow image processing. A benefit of using still and moving images that must not be underestimated is that it support the integration of mathematics with many other subjects such as science, technology, physical education, and so on.
- Arithmetic, algebra, and geometry go hand in hand with mathematics on digital images and students are stimulated to or enticed into making geometric constructions that fit with imaged objects or lead to formulas that are as simple as possible. Finding formulas serves a concrete purpose, viz., the computation of derived quantities, which cannot be determined directly in experiments. A visual representation, which is close to the real physical situation, is added to the well-established symbolic, numerical, graphical, and textual representations of mathematical ideas.
- Students can practice useful ICT skills.
- Real measurements on a concrete object can be compared with results obtained through mathematical models. This contributes to reflection about the quality of a mathematical model. ICT makes it possible that students create various mathematical models and compare them with each other and experimental results. In this way, the students’ work can resemble the scientific approach of professionals.
- Measurement on digital images is a modern, much-used research technique that students can apply themselves at high-level in practical investigation tasks and research projects. It facilitates research on real objects that are otherwise difficult to measure. For example, think of the shape of big objects like the main span of a suspension bridge and think of small objects like the shape of plants cells plant, the size of bacteria populations, and so on. But you may also think of the use of digital images in space research, medicine, geography, and forensic. Students can use methods and techniques that are also applied by professionals in the field.

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.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now

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.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now

The steps in drawing the square-tiles pavement following the distance point method are:

1. Draw the base line of the pavement and choose the position of the centric point C.
2. Draw the horizon through C and choose a point D on it such that the distance CD is the distance of the eye from the picture.
3. Divide the base line into the required number of parts and join the points of division to C.
4. Join D to the further end of the ground line.
5. Where this line through D cuts each of the lines through C put a line parallel to the base line.

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

- Students’ abilities to interpret graphs improve with video-based laboratory work in comparison with traditional approaches.
- Video-based laboratory work clarifies the use of a variable as a symbol for a variable object, in addition to the use of a variable as a placeholder and a polyvalent name. The position of a moving object has coordinates that change in time. The dynamic nature of dynamic geometry package, which means that an object may depend on other objects that can be dragged by the user, also supports this concept.

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.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now

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.