Photography for the Mathematical Culture of the Student. Research

Transcription

International Journal of Pedagogy,
Innovation and New Technologies
journal homepage: www.ijpint.com
ISSN: 2392-0092, Vol. 1, No. 1, 2014
Photography for the Mathematical Culture of the Student.
Research Report
Małgorzata Makiewicz
CONTACT: Professor Małgorzata Makiewicz, PhD, Institute of Mathematics of Szczecin University, Poland,
E-mail: [email protected]
mathematical culture, cognitive
photography, mathematical
photoeducation, pedagogical
experiment, the concept of
teaching mathematics
Abstract:
The concept discussed in the article is based on the constructivist paradigm. It stipulates the
generation of the child’s system of knowledge from independent discoveries on the basis of
cognitive conflicts (discrepancy between mental imagery and actual observations), as well as
the pursuit towards construction of Piagetian cognitive schemata.
The article presents an outline of the author’s concept of mathematical photoeducation
and experimental justification of its didactic application (the influence of photography on the
development of the mathematical culture of the student). It manifests the power of photography as a universal tool for development of all components of the mathematical culture of the
student: mathematical competence, language, imagination, creativity and elegance.
Introduction. Towards Mathematical Culture
Of all the humanities
mathematics is the most humanistic.
Hans Freudental
Mathematical culture, a phenomenon remaining in the shadow of the struggle for measurable and spectacular
teaching results and achievements focusing on the fastest pace of education and the greatest amount of knowledge
acquired, reflects in its wealth the essence of the humanistic significance of mathematics. It incorporates both the
basic mathematical competence and the specific language of mathematics, as well as creativity, imagination, and
the so-called “elegance of reasoning” (the article comprises fragments of the following thesis: Makiewicz 2013).
After all, mathematics is not limited to computations, but rather it is the omnipresent language of the universe and
a system of reasoning by means of which it is possible to define many natural phenomena and practical correlations.
Clifford Pickover places an emphasis on the special significance of mathematics in the service of other sciences.
The invaluable support of mathematics for biology, physics, chemistry, economics, sociology, and engineering is shown
in every area. Mathematics may be very helpful in explaining the colour palette of the setting sun and the architecture
of the human brain. It is used in construction of supersonic aircrafts and roller coasters, simulation of exploitation
of natural resources, examination of elementary particles, and presentation of remote galaxies. Mathematics offers
a wide selection of tools and knowledge required for an alternative view of the universe. […] Mathematics is remarkably useful: it is used in construction of space crafts and in exploration of the geometry of the universe, while numbers
may be used as the very first means of mass communication with extra-terrestrial civilizations (Pickover 2014: 10).
On the other hand, however, the concept of the paramount role of applications of mathematics constitutes an
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This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Keywords:
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M. Makiewicz • Photography for the Mathematical Culture of the Student. Research Report
interesting topic of discussion. Mathematics is not practiced only for pragmatic reasons, even though it undoubtedly
finds its origins in pragmatic needs. One of the driving forces behind exploration of various mathematical problems is
curiosity. Later on, solutions of majority of problems find a variety of applications. Applications, on the other hand,
constitute a source of new problems, and so on … (Ciesielski, Pogoda 2013: 44). This latter approach emphasizes the
cognitive objective of teaching of mathematics. The concept of photoeducation, which is the focus of this article,
is inscribed into a popular among didactitians of mathematics pursuit of ways in which students’ creativity could
be developed while preserving their intellectual autonomy (Hejny 2012: 41).
The humanistic values of mathematical culture reside primarily in the cognitive development and in the
quest to overcome difficulty and to obtain cognizance. Is mathematics not a creation of human reason? Renaissance humanists successfully obtained mathematical results. Ancient philosophers (…) were equally successful.
Following in the footsteps of their predecessors, great humanists should have a versatile mind that is open to
knowledge and ready for challenges (Ciesielski, Pogoda 2013: 48).
Mathematics establishes various sets of formal models. Realization that there is no single and unique
foundation of mathematics, just as there is no single correct way of understanding the world (Lakoff 2011: 361),
inevitably leads to expansion of perception and a new and broader view of meanings of the reality and solutions of problems. Mathematical objects and absolute entities, which cannot be experienced with the senses
and which exist beyond our will are fairly difficult to understand, especially for a child. Therefore, in spite of
the fact that among modern mathematicians proponents of Plato’s ideas dominate over formalists and constructivists (Życiński 2013: 100), it is worth to assume (for the purpose of education) that knowledge is not
given to us a priori, but rather generated from human experience, while cognitive semantics mechanisms may
be used for description of the reasoning discussed in traditional logic (Lakoff 2011: 366). This approach ensures
strong foundations for the development of photoeducation – the concept of the development of mathematical
culture on the basis of photography.
Tapping into the achievements of Greek culture, mathematical culture construes an intellectual attitude
that expresses comprehension of concepts, phenomena, and processes with the help of the conceptual apparatus of mathematics. It incorporates the ability to select the most appropriate general problem solution methods. It consists in the ability to identify in the problem under consideration non-existent mathematical objects,
which, however, offer an unexpected and successful solution (Kordos 2009: 5). Mathematical culture incorporates also geometric creativity and imagination, a good understanding of mathematical concepts, as well as
awareness of the beauty of mathematics. Mathematical culture consists also in the ability to perceive certain
mathematical ideas, problems, and even theorems in the surrounding world, in still life, in nature, and in
human craftwork (Makiewicz 2009: 9).
The possibility to form and develop mathematical culture of the student was already pointed out by Hans
Freudenthal, who highlighted the so-called “constructability” of mathematical culture: culture is creative (…).
The young generation creates a new culture in which they will live, growing into it, while inventing new forms
of, for example, knowledge. Knowledge is not passed on. It is either recreated or created (Freudenthal 1963: 14).
Frantisek Kurina, stressing the deep significance of the development of mathematical culture from early
childhood, formulated the following decalogue:
“1. Acquisition of mathematical competence; 2. Comprehension of the continuity of individual disciplines
of mathematics between mathematics as a science and mathematics as an educational subject; 3. Comprehension of the language of mathematics; 4. The ability to select the correct problem solving method; 5. Possession
of good geometric imagination; 6. Mastering computation techniques; 7. Mastering the ability to carry out
the process of mathematical proof; 8. Mastering the ability to introduce concepts; 9. The ability to practice, to
a certain degree, mathematical creativity; 10. Perception of beauty of mathematics (Kurina 1991: 30)”.
Zofia Krygowska looked at creativity as an extremely important component of the child’s mathematical
culture: Mathematical culture may be acquired while actively listening to a musical composition created for others; it is possible to apprehend the beauty and the essence of music without composing even a single chord. On
the other hand, it is impossible to comprehend the beauty and the essence of mathematics without mathematical
creativity. Fortunately, mathematical creativity is sufficiently available to children at the beginning of their school
education, as long as we know how it can be awakened (Krygowska 1977: 13-14).
Mathematical culture, therefore, is readily available to children. However its formation does not follow an
algorithmic pattern and is not accomplished during 21, 32 or 47 lessons of mathematics. The process of the
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creation of mathematical culture is associated with daily existence in the mathematical world. Mathematical
culture comes to light in various circumstances. It deals its cards when we are looking at an incomprehensible writing on the school blackboard or at a snail spiral, when we are reading an advertising message about
a bank deposit, when we are admiring an ornamental jewellery motif or the golden ration in a painting.
Mathematical culture is beginning to play a universal game. The winner is not someone who is able to recite
Thales’ theorem by heart or to provide the definition of interest rate, but someone who is able to usefully apply
and interpret acquired knowledge (Makiewicz 2011: 21). Therefore, apart from its elite, symbolic and festive
character, culture may be ascribed also with a casual and egalitarian dimension. Dorota Klus-Stańska discerns
culture in everyday habitual actions, conventionality, and mechanical ascription of meaning to our surroundings. It paves the way to the understanding of our surroundings, but, at the same time, also limits the horizon of
meanings and interpretations available to us. Invigorative for learning about and understanding the world, it is
also co-created in these processes (Klus-Stańska 2010: 301). According to Helena Siwek, an essential factor in
the development of mathematical culture is the use of the language of mathematics along with the awareness of
how it differs from the logical structure of the natural language, application of the laws and rules of reasoning in
order to better understand concepts, mathematization and interpretation, as well as uninhibited use of abstract
objects (Siwek 2005: 170).
Taking into account the contents of the mathematical curriculum, the shortage of general and detailed
knowledge within the scope of mathematics as a scientific discipline among junior high school students, the
priority role of geometric imagination in plane geometry and spatial geometry problems, as well as the need
to adapt the model to the structure of the cognitive development of a junior high school student, a two-stage
model of the mathematical culture of the student was specified. In view of its clear structure and the clarity of
the occurring relationships, the logical and graphic examination of the mathematical culture of the student
was based on the model of directional abilities developed by Wiesława Limont (Limont 2010: 50). The model
of the mathematical culture of the student adopted in the research (Makiewicz 2013: 31-59) accounted for
two spheres: the sphere of essential conditions and the sphere of conducive conditions. The sphere of essential
conditions is comprised of two components required for undertaking mathematical activity: the fluency in the
language of mathematics and the mastery of the basic mathematical competence. The conducive conditions
include the following: imagination (Paivo 1978: 380) (on the figurative, verbal, and constructional level), creativity on the “mini-t” level, which, according to Wiesława Limont, has a special role in the process of assimilation, accommodation, and construction of new knowledge (Limont 2010), as well as elegance, which defines
the simplicity of reasoning and the ability to capture a problem in a uniform structure. An elegant solution
of a problem resembles a good figure (in the language of Gestalt), which combines simplicity, symmetry, and
regularity, the cognitive aesthetics of which is simply captivating. As a result, mediocre solutions are superseded, just as good figures are remembered and associated much better than flawed figures (Garner 1974 ).
Why photography?
The invention of photography opened a new dimension of association of physical phenomena and chemical
reactions with issues of sociological nature, and later with artistic and cognitive concepts. The idea of painting
with light may be traced back probably to Aristotle watching a partial sun eclipse. Optics owes its beginnings
to Thales, who measured the height of the pyramid with its shadow (Chmielewski 2009: 9), Euclid, who introduced the notion of radius of visibility, Ptolemy who focused his work, among others, on the phenomenon
of reflection and refraction of light, as well as the father of the modern optics and ophthalmology – Alhazen,
who, at the turn of the 10th and the 11th century, studied the effects of light on human eye and described the
prototype of a camera. Among the precursors of photography are Leonardo da Vinci, Galileo, Johannes Kepler,
Pierre Fermat, and Isaac Newton (Wade, Finger 2001: 1165). Nevertheless, the world had to wait for the first
publication of a photographic image until 1839. Małgorzata Gąbczewska, in her reference to the linguistic
dimension of the nineteenth century’s beginnings of photography, mentions the self-realization of the inventors of the ancient methods (heliography, daguerreotype, and calotype): Nicéphore, Niépce, Louis JacquesMande Daguerre, and William Henry Fox Talbot (Watson 2008: 1376-1379). Remaining under the impression of the power of “miraculous peculiarity”, all three of them searched for names that would best reflect
the nature of the phenomenon. For Talbot, the photographic image, slightly blurry, vibrating like a shadow
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(Skiagraphy) was “almost a perfectly true and exact copy of nature”. Elements of nature: leaves, flowers, trees
(as well as) small objects placed by Talbot on a piece of paper, showed their spirit in a photographic impression (Gąbczewska 2013: 13-14). Niépce ascribed “the copies of nature” and their “true impressions” with the
name of “heliography”. He viewed the sun as “an active actor and a participant in the creation of the image”
(Gąbczewska 2013: 15). Daguerre, was searching within his passion, which boarded on insanity (Siegler 2013:
63-64), for a term that would capture the perfect image (reproduction of nature) and introduced the term
“daguerreotype”. The modern term photography refers to the Greek notion of drawing with the rays of light.
From the moment of its invention, photography has been developing both on the technological plane,
which may be considered as a derivative of the dynamic development of computerization, and on the artistic
plane, associated with its inclusion into the high arts, as well as on the sociological and the communicative
plane. Photography, as a medium combining all the above aspects, was recognized as a universal language spoken by people all over the world (Rowe 2011: 8). Jean van Dijk depicts the evolution of human communication
by placing a special emphasis on the milestones leading to the creation of new media (Dijk 2010: 14). Digitalization was an extremely important breakthrough in the history of the media. Nearly simultaneous inventions
of Daguerre, of cinematograph by Lumière brothers, and of computing machines contributed to all the media
becoming numerical media comprehensible for computers. As a result, graphics, moving pictures, sounds, shapes
(…), and texts are transformed into computer data, which may be used for computation. In a word – the media
are becoming new media (Manovich 2006: 90).
It is difficult today to imagine modern life without photography. Already before a child is born, parentsto-be receive an image of the expected child taken with a USG 3D machine at a doctor’s office. The first minutes of everyone’s life are recorded and immediately published on the Internet for the enjoyment of friends
and family. We are taking photographs of everything, more and more frequently and with the help of continuously advancing technology. Piotr Sztompka emphasizes the relationship between acceleration of technical
developments and the increasing popularity of digital photography, especially among young people – It is
difficult today to find anyone who would not have any personal experience with taking photographs. This results
in the extraordinary attractiveness of photographic images (Sztompka 2012: 36). It is true that the development
and marketing of increasingly advanced technologically personal multitasking devices resulted in increased
accessibility of a photographic camera. Thanks to the simplicity of its operation, even small children are able
to take photographs. We take photographs in all kinds of daily life situations in order to preserve everything
that we find touching, important, beautiful, and interesting. We take photos of bus schedules, lecture notes,
sporting events, cultural events, and family reunions. As the photographic camera is available at all times,
we photograph ephemeral phenomena (such as rainbows, atmospheric discharges, traffic accidents, or civil
engineering disasters). We photograph a lot and with high frequency and without consideration, as we do
not have to worry, as before, about any additional expenses of developing the photographic film and printing
enlargements. Photography has become completely free, maybe except of the exceptional situations when we
are photographed by a traffic camera for speeding or for parking the car in a wrong spot…
In spite of the efforts undertaken by Alfred Stieglitz, who successfully implanted his objective of creating a system of signs based on symbolic and media-based communication of the aesthetic experiences of the
viewer (Stiegler 2009: 72), photography is still struggling for the high art status. Pierre Bourdieu distances
himself from classifying photography (as well as cinema, jazz music, and song) as one of the high arts. Instead
Bourdieu awards photography the status of a validated medium (Bourdieu 2012: 262). A careful approach to
the status of photography is also assumed, among others, by Roland Barthes (Barthes 2012: 409) and Anna
Matuchniak-Krasuska (Matuchniak-Krasuska 2010: 124).
Notwithstanding the opinions that digital photography has shifted radically in the direction of computer
graphics (A. Rouille’ [w:] Łukaszewicz-Alcaraz 2014: 218), the digital image must be associated with technological developments and it is a natural result of a leap from the analogue age to the bit era. The essence of
photography understood as painting with light has remained unchanged. Even though it is not easy to discuss
the spirit of digital photography, modifications affect, first and foremost, technological conditions rather than
perception of our surroundings.
Remarkably, the creation of photography was inspired by the longing to share a given message – information or even an emotion. Why should not this longing be used in education? Apart from mere images, photography carries a certain (additional) message (Sekula 2010: 12). It is associated with the context (of creation,
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exhibition), the process of reading, commenting, and reviewing. Photography reflects the experience and the
condition of the spirit of the photographer, as well as his or her subjective viewpoints, thoughts, and sentiments. When the work of art thrives with its author’s expressiveness, the documentary role is superseded in
the eyes and in the mind of the recipients by an aesthetic experience (Makiewicz 2010: 18). Photography is
emerging as the most momentous event in the history of art. By virtue of its duality as artistic liberation and
achievement, photography freed Western painting from its obsession with realism and inspired it towards aesthetic autonomy (Bazin 2012: 426).
Allan Sekula, in his reference to the subject already undertaken by August Sander concerning the global
sense of visual communication, emphasized a new role of photography applicable also to the cognitive process: an appropriate application of photographic media may give rise to liberal, enlightened, and even critical
pedagogy (Sekula 2010: 82). Photography teaches about perception of multiple aspects of the image, as well as its
depth and levels. It inspires a series of questions, as well as curiosity generated by the willingness to learn. Literal
representation and faithful reflection of the existing scenery by means of photography gives way to peculiar events
takings place in the mind of: the artist and the recipient, the student and the teacher. Photography owes its power
to transform reality to the interplay of metaphors (Stiegler 2009: 9). The photographic image exposes, shows the
Other, things so far unknown, in a predictable way, in a mathematically computable perspective, taming the mysterious areas of reality which so far have remained concealed (Łukaszewicz-Alcaraz 2014: 102). In the service of
education, photography does not have any excessive requirements as regards equipment. On the other hand,
it requires cognitive curiosity, ingenuity, and the interest in the surrounding environment. Posing questions
and searching for answers. The power of modern digital photography is related, among others, to its extensive
availability and easy operation. Decreasing prices, as well as the economic reasoning of the consumer capitalism
intensify the society’s saturation with visuality (Sztompka 2012: 22-24).
Photographic objectivism bestows upon the image the power of credibility, which is non-existent in other
visual arts. Our critical mind may suggest various reservations, however we are compelled to believe in the
existence of the object represented in a photograph, which means an object that is present in time and space.
Photography takes advantage of the fact that the reality of an object is transferred into its reproduction (Bazin
2012: 424-425). At the same time it should be noted that the way in which photography affects a teenager
tends to be more powerful than the impact exerted by all other electronic media. This is related to the
experience gained while using animations and self-modelling interactive applications. The photographic
image (read or created), which has a strong reference to an actual situation, seems to be more real than
a computer visualization. Piotr Sztompka emphasizes the extremely important cognitive role of photography with a didactic propensity: Similarly as in the case of pure sociological photography, it enables application
of a variety of cognitive filters, allowing differentiation of things that matter from things that are insignificant
in the chaos of daily life (Sztompka 2012: 29). Leaving out insignificant properties (such as the colour or the
scale) of an abstract mathematical object proves to be an excellent method of development of the mathematical culture of the student.
Photoeducation
Mathematical photoeducation consists of a set of carefully considered and planned actions undertaken by the
teacher and students and aimed at mathematical education by means of photography. The focus is placed on
cognitive, aesthetic, and creative values, while educational results are achieved through a variety of activities
undertaken by students and teachers. For example, through finding (Jeffrey 2009) concepts in photographs,
recognizing correlations, solving problems, interpreting symbols, or creating own visualizations of concepts
and theorems, constructing models, formulating own problems and tasks, as well as creating metaphors with
the help of images and language.
The model of photoeducation in teaching mathematics was constructed in reference to the two primary areas of student’s activity: reading finished visual materials and creating new visual materials. Such an
approach invokes the two classical paths in sociological photography: decoding with the help of categories or
models, which consist in imposing a conceptual framework and network on a photograph, as well as independent
artistic activity undertaken with the intention to assign meaning (Sztompka 2012: 28).
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Mathematical photoeducation relies on the cognitive concept of human nature (Kozielecki 1976). The
student, through two mutually supplemental processes: interiorization and exteriorization, creates own
cognitive structures, reaching beyond the information provided from an external source and pursuing
higher levels of generalization by means of a gradual detachment from the reality of the surrounding environment. The process of interiorization leads from actual actions towards imaginary actions and incorporates the following: elementary activity of the teacher associated with the organization of a didactic
situation, reading of a photograph (from scanning photographic dominants of the image through reconstruction of the formal composition to interpretation) and formulating problems. The role of the teacher
at the level of reading a photograph is to organize the perception field, as well as to intervene by means of
directing the interpretation towards knowledge. The student is elevated to a higher level on which he or
she begins to discern and name mathematical objects presented in a photograph and is ready to depart
from realistic interpretation towards formal interpretation. Interpretation of a photograph incorporates
the analysis of the content through formal representation of mathematical objects, while commenting
consists in determination of relationships and interplay between concepts. John Berger shows the path to
understanding photography: we learn how to read photography in the same way as we learn how to read
cardiograms or how to trace footprints (Berger 2011: 206). The success of photoeducation is associated, first
and foremost, with the visual sensitivity of the student, as well as the ability to read photographic images.
However, the element of interiorization that students find most challenging is the recognition and the
formulation of problems. The difficulty encountered here is associated with the creation of a semiotic situation that provokes a holistic system of visualization (Lewandowska – Tomaszczyk 2009: 22). Therefore the
teacher, in spite of the natural temptation to offer prompts and assistance, should allow the child to engage
in an unassisted activity. The teacher may only stimulate and organize observations, while surrounding
the student with positive emotions (Gruszczyk – Kolczyńska 2012: 128-130). It is extremely important for
the teacher to give the priority to the student in order to encourage attempts to independently overcome
a cognitive challenge, while proposing active negotiation of meaning instead of assimilation of concepts
(Klus – Stańska 2010: 303).
As opposed to interiorization, which prioritizes actual actions (manipulations) affecting physical objects or
their figurative and symbolic representations over actions taking place on the imaginary plane at first and then on
the verbal – conceptual plane, the child at the same time possesses the ability to externalize iconic signs or verbal
symbols in practical activities or in play (Przetacznik – Gierowska 1993). The process of exteriorization consists
in a transition from what is thought about (for example, learned during a mathematics lesson) to what is to
be seen. Mathematical objects cannot be shown, as they are invisible. The senses merely offer a representation
of concepts, correlations, and theorems. The variety of representation of objects of mathematics, which has
an abstract nature, offers extraordinary wealth to didactics. In the process of exteriorization, it is the teacher’s
role to create a problem situation and to direct students towards visualization of previously discussed concepts
and theorems. Students search for photographic associations during a school trip, a walk, while doing their
homework, etc. They try to recognize such elements of their surroundings as previously discussed trapezoids,
rhomboids, cones, pyramids, functions, theorems, and properties.
The cognitive development of the student relies on assimilation and accommodation. Photoeducation,
combining the process of interiorization and exteriorization, implements the idea of combination of interpretation of the surroundings in accordance with available meanings with a change of cognitive structures as a result of
the influx of new information, as postulated by Dorota Klus-Stańska (Klus – Stańska 2004: 24).
The conception of photoeducation was introduced into an experimental research program in the
geometry lesson series in grade 2 of junior high school as an independent variable. The auxiliary didactic
set provided to teachers participating in the research incorporated 16 lesson scenarios along with attached
materials in the form of computer presentations (16 copies), auxiliary presentation materials (3 copies), as
well as problem sets (5 copies). Overall the didactic package included over 200 photographs and 50 references to photographs published on the Internet. Due to a limited size of this article, in order to present
a problem set, a few lesson scenarios were selected. The selection of descriptions depended on the following factors: understanding mathematical concepts, the ability to recognize mathematical regularities,
recognition, formulation, and solution of problems, as well as experiencing mathematics through creation
of metaphors.
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Description of selected lessons from packages:
Example 1. The topic of a mathematics lesson: The symmetry of a line segment. A circle circumscribed
about a triangle.
Didactic package: Lesson scenario, Presentation consisting of 32 photographs. The definition of the perpendicular bisector of a line segment is introduced to students during a lesson in a natural way by reference
to a photograph of a suspension cable-stayed bridge (Phot. 1).
At first, students engage in unassisted exploration of the properties of the pylon (vertical structural element) and of the cables (steel suspension wires), which is followed by the formulation of the definition of perpendicular bisector of a line segment. The problem of finding the midpoint of the segment is solved in practice
with the help of a piece of string. With the help of the photograph and the experiment with a piece of string,
students discover the properties of perpendicular bisector and proceed to formulate the theorem about the
distance of a locus of points on the perpendicular bisector from the endpoints of the segment (in the language
of a civil engineer, these distances are the length of the cables).
Students proceed to discover the theorem about the perpendicular bisectors of a triangle without any
help on the basis of a computer presentation. The presentation includes a step-by-step series of photographic
images showing how to fold paper triangles along the lines of symmetry. Paper triangles include acute triangle, right triangle, and obtuse triangle.
In the first scenario, all three bisectors intersect at one point located inside the triangle, i.e. the centre
of the circumscribed circle of the triangle. Having mastered the properties of the bisector, students prove
the theorem without assistance. In case of the right triangle, the locus of intersection of all three bisectors is
located in the midpoint of the hypotenuse and is indeed equidistant from all the vertices of the triangle. As
for the obtuse triangle, in order to find the locus of intersection of bisectors it is necessary to consider the area
outside of the triangle itself. Therefore, the midpoint of the circle circumscribed about the triangle is located
by placing the triangle on a larger piece of paper and extending the folded segments by drawing lines. The
proof of the thesis is carried out analogously.
Example 2. The topic of a mathematics lesson: The area of polygons.
Didactic package: Lesson scenario, Presentation consisting of 15 photographic images.
The objective of the lesson is to activate students, awaken their cognitive curiosity, analytical perception,
and the ability to solve problems, develop the ability of logical reasoning, develop their intuition, develop
their spatial imagination, as well as the ability to mathematize real life situations. The lesson should, first and
foremost, encourage students to formulate and solve problems associated with the area of planar figures. For
example, given the dimensions of bricks, students are asked to determine how many square meters of plastic
sheet are required to cover the openings in a barn (Phot. 2).
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Phot. 1 A rectangular bridge. Author: Alicja Piotrowska
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Phot. 2 Repeatability. Author: Rita Witkowska
Student’s ideas how to calculate the area of polygons (by means of division and by completing the rectangle) should be immediately applied in solving problems. For example, in calculation of the area of the trapezoid situated between wooden beams by detaching the right triangle from the rectangle (Phot. 3)
Phot. 3 Masonry polygons Author: Janusz Gąszczak
or by deducting the area of the flower tub to be covered with a protective layer of tree bark (Phot. 4).
Phot. 4 Regular hexagon with the midpoint of a circle. Author: Mateusz Michalak
The ability to convert the area units is developed by means of an instruction enriched with photographic
images and included in the presentation. By noticing regularities in photographic images of windows, scaffolding, skylights, fencing grids, and ceramic floor, students formulate problems which they later solve in
groups. As a result, they learn how to make use of the properties of angles and diagonals in rectangles, parallelograms, rhomboids, and trapezoids also in order to calculate their areas, as well as develop the ability to
solve word problems associated with calculation of the area of polygons on a plane. For example, calculation
of the area of a rhomboid and trapezoid by dividing them into two or three triangles (Phot. 5).
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Phot. 5 Polygons in architecture. Author: Karolina Zachariasz
Phot. 6 Regular octagon. Author: Jadwiga Bosacka
Example 3. The topic of a mathematics lesson: Angle bisectors in a triangle inscribed in a circle.
Didactic package: Lesson scenario, Presentation consisting of 16 photographic images, Photographic
instruction of folding a triangle along angle bisectors.
The lesson introduces new content. Its primary objective is to introduce students to the definition of the
angle bisector, the triangle angle bisector theorem, and to the correct angle bisector construction. The role of
the introductory chat illustrated with photographs and captions is to review the concepts utilized in the following parts of the lesson (angles and their classification, the vertex and an arm of an angle). The angle bisector and its properties are defined in reference to Phot. 7.
Phot. 7 A triangle inscribed in a circle. Author: Małgorzata Rarata
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The development of the ability to recognize polygons in student’s immediate surroundings is ensured by
a homework assignment that requires taking photographs, classifying, and recognizing polygons, while placing a special emphasis on road signs, ornaments, decorative motifs in ceramic floors, textiles, architectural
façades, and fragments of building structures (Phot. 6).
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The angle bisector construction is demonstrated in reference to practical situations (folding a piece of
fabric or paper). An important part of the lesson is experimental determination of the midpoint of a circle
inscribed in the triangle on the basis of a photographic instruction of a series of steps to fold a paper triangle
along its angle bisectors. This task is performed by students without any assistance and its goal is to develop
a sense of certainty about the truthfulness of the theorem (the emphasis is place not on a formal proof, but
rather on inner confirmation of observations).
The lesson summary is presented on the basis of a computer presentation and its purpose is to consolidate
the newly acquired information and to indicate, with the help of photographic images, the applications of the
angle bisector: in optics, geodesy, and carthography.
Example 4. The topic of a mathematics lesson: Calculation of arc length and of circular sector area.
Didactic package: Lesson scenario, Presentation consisting of 80 photographic images.
The proposed scenario of the lesson assumes the pre-existing familiarity with the following concepts: disk,
circle, radius, diameter, chord, round angle, straight angle, acute angle, central angle, formula for the area of
disk and for the length of circle.
In order to fulfil the abovementioned objectives (theoretical and practical mastery of the formulas for the
length of arc and the area of circular sector), students need not only be familiar with elementary concepts and
formulas, but also need to be able to successfully apply them. Therefore, the initial part of the lesson is comprised of a review and a restatement of all the concepts necessary to understand the content of the lesson. For
example, the concept of the chord and the diameter of a circle is consolidated during a discussion about the
changing position of the shadow depending on the time of the day in Phot. 8.
Phot. 8 Apparent diameter. Author: Wojciech Gołębiowski
What follows is a presentation of photographic images of circular sectors captured in: pizza slices, cake
wedges, elements of interior architecture (contemporary shower cabins and shower tubs, hotel and office
receptions), famous buildings (Theatre of Dionysus, Barbican), useful mechanisms (fly-wheel, drive wheel,
steering wheel, sextants, car wipers, merry-go-rounds, windmills), road signs, warning signs, pictograms,
graphic symbols of business entities – for example vehicle manufacturers such as Mercedes, BMW, toys and
puzzles, toy bricks, furniture, flower beds, lawns, sidewalks, park paths, kitchenware, binders, and citrus fruits
cross section.
The arc length formula and the circular section area formula are introduced with the help of a schematic
drawing with a background photographic image of a merry-go-round (Phot. 9).
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71
Phot. 9 Ideal symmetry in Belfast. Author: Ewa Lesiak
Example 5. The topic of a mathematics lesson: Quiz – Recognition of mathematical concepts and regularities.
Didactic package: Lesson scenario, A graphic file including 14 photographic images, Auxiliary printouts.
The lesson is a summary of a cycle of mathematics classes with the application of photoeducation. Its
focus is to demonstrate the interdisciplinary nature of mathematics and to develop the sense of aesthetics
and the sensitivity to beauty in students. The primary objective of the lesson is to review and consolidate the
knowledge within the scope of geometry with the aid of visual photographic materials. The proposed format
of the lesson combines group and individual work and focuses on the development of active listening skills,
perceptiveness, and critical thinking.
During group work the teacher reads the instructions and points to the relevant photographic image. Students are instructed to formulate responses to questions posed by the teacher. Students respond individually
and every response is followed by discussion, the purposes of which is to determine the completeness of the
provided solution. The problem set includes 14 questions accompanied with photographic images with visual
reference to the questions. The question cover the following concepts: perpendicularity and parallelism; line
segments, straight lines; circle and circular arc, disk, circular ring and circular sector; central angle; relative
position of straight lines, circles, and a straight line and a circle; angles and angle types; angle bisector of perpendicular bisector of a line segment; triangles and triangle divisions; polygons; planar and space congruent
figures and similar figures; axial symmetry and central symmetry. Photographic images present: road signs,
mirror reflections in water, scattered necklace beads, a sailboat, a skylight in the Louvre Museum, structural
bridge spans, condensation trails in the sky, regular fragments of interior design, telephone posts, as well as
examples of construction machinery, works of art, and graphic installations and signs.
Following the quiz summary, students engage in individual work on finding the greatest possible number
of mathematical concepts in a photographic image of a Ferris wheel in an amusement park selected by the
teacher. Students demonstrate in writing the ability to name and interpret details presented in the photograph
such as concentric circles, isosceles triangles and equilateral triangles, rhomboids, similar and congruent figures, adjacent and vertical angles pairs, straight angle and round angle, central angles, circular segments and
sectors, and tangent segments to a circle.
Lessons not included in the synopsis were focused on: relative position of straight lines and circles on
the plane; symmetry in architecture and art, analysis of the symmetry of the face, axial symmetry and central
symmetry in letters, numbers, signs, and ornaments; the converse of the Pythagorean theorem, shadows and
cross-sections, as well as an educational excursion.
Research plan
The role of photographic images in teaching mathematics and the development of mathematical culture in
students constitutes a prominent research in the Laboratory of Didactics of Mathematics of the Institute of
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The above photograph is presented in a few anonymous slides of the attached presentation, while stated
problems refer to a practical situation.
72
Mathematics of the University of Szczecin. The experimental research discussed here was preceded by many
years of hermeneutic analysis (discussed in: Makiewicz 2012: 27-39) of verbal and visual materials collected
during the author’s work as a teacher of mathematics and in relation with a series of international photographic competitions organized under the name of MATHEMATICS IN FOCUS, (www.mwo.usz.edu.pl).
The procedure of the pedagogical experiment was prepared with the help of consultations as part of the SelfStudy Task Force of Doctors at the Sciences Committee the Polish Academy of Sciences supervised by prof.
zw. dr hab. Maria Dudzikowa.
The primary objective of the undertaken research was to expand the theoretical knowledge within the
scope of mathematical culture of students as part of mathematical education supported with photoeducation.
The utilitarian task involved laying the foundations of the design of innovative textbooks on mathematics and
didactic materials for teachers who utilize photoeducation.
The main problem in the undertaken research procedure concerned the impact of the application of
photographic images in teaching mathematics (photoeducation) to students of the second year of junior high
school on the development of their mathematical culture. More specific problems were related to the influence of photographic images on individual components of mathematical culture of the student: mathematical
competence, fluency in the language of mathematics, the ability to select proper (i.e. elegant) problem solving
methods, spatial imagination, and mathematical creativity.
The selected research model was embedded in the tradition of critical rationalism (Popper 2002) with
acknowledgment of Imre Lakatos’ position as regards a search for a compromise between the revolutionary
shifts of Thomas Kuhn and the evolutionary process of changes advocated by Karl Popper. The need to explain
the relationships and the conditioning, as well as the interdependencies between variables (Krüger 2007: 175)
and the longing for empirical verification of a research hypothesis about a beneficial influence of photoeducation introduced into the process of teaching mathematics to second year students of junior high schools
on the development of their mathematical culture resulted in application of quantitative methods and in the
choice of the experimental path (Brzeziński 1997: 282). Therefore, the main research method used here was
a natural pedagogical experiment carried out by means of a parallel group study in the natural school environment of junior high school students. In spite of the difficulties in controlling intervening variables and their
dependencies, it was decided to carry out a natural experiment, as, from the point of view of the development
of a mathematics teaching theory, the priority was given to the opportunity to transfer the results into realworld practice. The experiment included diagnostic and psychological tests. The objective of the achievement
test was to determine the overall mathematical competence (considered one of the components of the nonindependent variable) of the respondents.
Another tool used in order to determine the parameters of the non-independent variable was the Test for
Creative Thinking developed by K.K. Urban and H.G. Jellen (Matczak A., Jaworowska A., Stańczak J 2000). This
tool was used twice; at the beginning and at the end of the study, in order to capture the dynamics of the level
of creative thinking. Other non-independent variables were examined by means of the author’s Mathematical
Creativity Test, which was developed on the basis of Lew Wygotski’s concept of the zone of proximal development of the child. The problems incorporated in the abovementioned test pertained to the ongoing and incomplete development cycles, had a high level of difficulty (compare A. Brzezińska 2000) and solutions that required
imagination, divergent thinking, and elegance. The problems were formulated by means of colloquial phrases,
as early maturation of scientific concepts is related to propedeutic paving of the way (Wygotski 1989: 167) with
colloquial terms. The test was developed taking into account the problems specified by Zofia Krygowska as
problems that activate students without relying on standard formulas and procedures (Krygowska 1972: 201).
The test was comprised of 5 problems presented graphically on two A4-format pages. Students were given 40
minutes to freely choose any number of problems and provide their own solutions in the response sheet, along
with drawings and comments. The problems included in the test were not related directly to the didactic material presented to teachers for implementation in experimental groups. They did not originate in school textbooks
or in teachers’ methodological materials. The test set was developed on the basis of didactic literature, as well as
reflections resulting from the author’s ten years of teaching experience. Following Kazimierz Skurzyński’s postulate of teaching mathematics on the basis of independently modified and generated problems (Skurzyński 2012:
14), the research set included a problem verifying the fluency and the flexibility of students’ cognitive processing.
Students had to pose new questions without the need to answer them. The test included problems that allowed
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73
for two ways in which the language of mathematics is used, i.e. from the text to the execution of the instructions
and from own idea to its successful presentation to others. The above selection may be justified with the receptive and the productive role of the language in communication with one’s surroundings, as defined by David
Wood (Wood 2006). The comparison of the thematic areas of problems and the areas verified with problems was
published in M. Makiewicz (2013), O fotografii w edukacji matematycznej. Jak kształtować kulturę matematyczną
uczniów (About photography in mathematics education. How to develop the mathematical culture of students.)
Szczecin SKNMDM US. The attendance of students and the monotonicity of mathematics grades while carrying out the experiment were determined by means of an analysis of school documentation was performed,
which resulted in creation of an attendance sheet and a grade sheet. In order to find out about students’ attitude
towards mathematics, as well as to determine the value of the confounders associated with the need to engage in
additional mathematical activity outside of the classroom, a complementary questionnaire based study was carried out. To determine the predominant learning style and the dominant hemisphere, all students participated in
psychological tests upon completion of the experiment (Linksman 2001).
In the first stage of the study, the experimental group included 130 and the control group 98 participants. As
a result of the application of a 74% classroom attendance filter, the number of group participants was reduced
to 82 participants in the experimental group and 76 participants in the control group. The hypotheses were
verified at the statistical significance level of .05.
In order to determine the parallelism of both groups (E – experimental group and C – control group),
apart from the intentional stratified sampling (Frankfort-Nachmias, Nachmias 2001: 199), which accounted
for the demographic data of Zachodniopomorskie Province, teaching at least two second-year classes by the
same teacher, consent was obtained to carry out the study, the study was carried out for 14 additional factors,
which could potentially differentiate both groups (predominant learning style, dominant brain hemisphere,
parents’ education level, 3 components of student’s approach to mathematics: emotional, intellectual, and
operational, the grade obtained from mathematics from the first semester (directly prior to the experiment),
student’s participation in activities of a mathematics student club or any other extracurricular activities or
tutoring, as well as student’s opinion about the content of home library books on mathematics and scientific
interests of parents and other people in his or her surroundings. The Pearson test with Yate’s correction demonstrated a lack of statistical difference between group E and group C as regards the abovementioned factors
at the statistical significance level of .05.
As for the factor of the mathematics grade received prior to the experiment by students, an additional
median test was carried out, which confirmed that there was no .01 level statistical difference between group
E and group C [χ2 = 0.003; p = .955].
Groups E and C were also analysed from the perspective of the value of partial non-independent variables
prior to commencing the experiment. The study demonstrated that there was no statistical dependence at
a statistical significance level of .05 between group E and group C for the following factors: fluency in the language of mathematics, elegance, verbal imagination and the ability to engage in creative mathematical thinking. As for mental imagery, the Pearson test demonstrated a statistical difference between group E and group
C. In pretesting, group C obtained better scores within the scope of constructive imagination. Mathematical
competence of group E and group C was compared in pretesting by means of student’s t-test for independent
samples. The test value: t = –1.21; p = .227, which demonstrated that there were no differences between the
two groups at a statistical significance level of .05. The average value of the creative thinking index MT-W of
group E and group C was compared in pretesting by means of student’s t-test for independent samples. The
test value: t = –0.71; p = .479, which demonstrated that there were no difference between the two groups at
a statistical significance level of .05.
The significance of the difference in the development of mathematical competence between group E and
group C was compared by means of student’s t-test for independent samples. The test value: t = –2.68; p = .01,
which demonstrated dependence at a statistical significance level of .05. As the average development in group
E was higher than the average development in group C, it was concluded that photoeducation has a positive
impact on improvement of mathematical competence of students from the experimental group.
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Experiment results
74
Pearson test with Yate’s correction carried out in reference to studies focusing on individual components
of mathematical culture demonstrated the following χ 2 and p values (Table 1):
Table 1. Analysis of the impact of photoeducation on the individual components of mathematical culture
Demonstration
of statistical
dependence
benefitting
group E
p
Component of the mathematical
culture of the student
χ2
Fluency in the language of mathematics
18.88
.001
+
.01
Mathematical elegance
10.45
.001
+
.01
Mental imagery
36.85
.001
+
.01
4.17
.04
+
.01
Constructive imagination
14.29
.001
+
.01
Mathematical creativity
20.04
.001
+
.01
Verbal imagination
p
The above results demonstrated statistical dependence between the application of photoeducation in
teaching mathematics and the improvement of all components of the mathematical culture of the student.
The above study, which is described in a greater detail in M. Makiewicz (2013), O fotografii w edukacji matematycznej. Jak kształtować kulturę matematyczną uczniów (About photography in mathematics education. How
to develop the mathematical culture of students.) Szczecin SKNMDM US, demonstrates that there is no differentiation in the development of students’ mathematical culture due to the dominance of the learning style, the
dominance of brain hemisphere, the level of education of parents, the dynamics of student’s attitudes towards
mathematics, as well as the monotony of mathematics grades.
Final conclusions
Photoeducation defined on the foundations of pedagogical constructivism constitutes an alternative proposition
for facilitation of aesthetic and cognitive development, as opposed to transmission-style education. Photoeducation ensures variety and variability of topics as regards the various levels of student’s advancement and a broad
selection of didactic tools available to the teacher. It encourages spontaneous activity while preserving student’s
freedom and independence and inspires joy of independent exploration of the world, sincere delight in solving
a difficult problem, which constitutes one of the fundamental principles of the Hejny method of teaching mathematics (www.h-mat.cz/principy). It pushes aside the danger of mathematical thoughtlessness (Klus-Stańska,
Kalinowska 2004: 19). In a child-friendly environment, without unnecessary prescripts and procedures, it enables the development of cognitive curiosity. It encourages admiration of beauty and wisdom of nature, perception of regularities in craftwork, formulation and solution of self-stated problems, and individual expression of
own thoughts, both verbally and by means of photographic images. Photoeducation opposes the phenomenon
of “school bulimia” defined by Marzena Żylińska as stuffing students over a short period of time with an enormous
amount of unnecessary and disconnected information, which gets forgotten just as quickly (Żylińska 2013: 226).
Whatever students observe and discover on their own tends to be remembered for a long time.
The results of experimental studies of the impact of photoeducation on development of mathematical
culture of the student discussed here may be used in creation of didactic tools that apply photoeducation
at all levels of education. Teaching mathematics on the basis of photography is focused on the student, who
reads and interprets photographic images, formulates problems, and independently visualizes mathematical concepts. Photoeducation is centred on double coding of information (image – photography along with
its description, caption, and commentary). Presumably, in view of the fact that the information acquisition
process is based on two strategies: verbal and imagerial (Hankała 2009: 169-174), teaching mathematics with
the aid of photographic images will prove to be an excellent way to stimulate memorization processes and
cognitive activation of students regardless of the dominant learning style or the brain hemisphere dominance
(Makiewicz 2013: 25), as it continuously synchronizes the function of both brain hemispheres.
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75
Photoeducation activates creative and re-creative mental processes, as well as zooms in on abstract concepts and regularities both by means of visual and verbal metaphors (Makiewicz 2013: 205). As stated by
Wiesława Limont, metaphoric expressions evoke imagery that sheds more light on and ensures a better understanding of specific phenomena (Limont 2003: 168). This explains the effectiveness of photoeducation in the
development of creativity, the formation of mathematical imagination and elegance, as well as in the enrichment of the language of mathematics and general mathematical competence.
Photoeducation is also instrumental in social development, which has not been researched just yet. By
participating in discussion, developing their argumentation skills, and commenting the work of others, students reinforce and expand their contacts with the society at large. Photoeducation provides an excellent
opportunity to practice concentration skills and to develop sensitivity to a variety of visual aspects of social
life. Through photographic images, photoeducation enables access to hidden and invisible to the eye meanings and structures. This is also a chance for us to determine the area of our ignorance and ask pertinent questions about what remains hidden and invisible (Sztompka 2012: 36-37). Mathematical thinking begins with
asking a question.
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DOI: 10.5604/23920092.1134801
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Photography for the Mathematical Culture of the Student. Research

Transcription

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