Loci (2008)
Mathematical Brooding over an Egg, André Heck

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

Finally I would like to discuss in more depth the potential value of digital images in mathematical investigations by students. What learning advantages does measurement and manipulation of images offer? When answering this question I do not think of egg investigations in particular, but in general of 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 activity, you
1. look for an object of interest in the real world,
2. take a picture of this object with a digital camera or web-cam,
3. import the digital image itno a dynamic geometry worksheet,
4. analyze the picture of the object by drawing, measuring and calculating,
5. reconstruct the identified figure with the tools provided by the dynamic geometry package, and
6. 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 information technology 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. Information technology 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 information technology, 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. It is true that a more 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 information technology 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, information technology and real-life contexts can contribute to the realization of authentic tasks for students, in which they can practice and enhance their mathematics knowledge, understanding, skills, and attitudes.