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 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 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 dynamic 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 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.