Mathematics and Panay Bukidnon Women in Alan Cabalfin’s Paintings
Rosemarie G. Galvez
Abstract
This case study analyzes of Alan Cabalfin’s paintings from a mathematical point of view. Specifically, the subjects of the paintings are the women of Panay Bukidnon, an indigenous group living in the island of Panay in central Philippines. Purposive sampling, mathematical investigation, and interview complemented the use of mathematical software in the analysis. Mathematical concepts such as square, perspectivity, golden rectangle, and ratio served as the mathematical lens through which the paintings were interpreted.
Recommendations on the use of paintings for arts integration in mathematics instruction are discussed.
Keywords: art, ethnic women, mathematics, Panay Bukidnon
Funded by the Department of Science and Technology – Science Education Institute
©WVSU Research and Development Center
“Let no man who is not a mathematician read the principles of my work”
(Da Vinci, 1906, p.143). The audacity of Da Vinci’s statement shows that the seemingly separate worlds of mathematics and art are not mutually exclusive.
Griffin (2003), Jensen (2002), and O’Connor and Robertson (2003) would also agree that art and mathematics are interrelated in many ways.
However, the inclusion of art in mathematics should not be limited to Western perspectives. This is supported by the principles of ethnomathematics which was popularized in the 1960s by Ubiratan D’Ambrosio (Scott, 2012).
Ethnomathematics supports the need to connect mathematics to other subject areas such as art and to the local culture of the learners by incorporating local mathematics (Shirley, 2001). It has pedagogical implications and leads to the major goals of creativity and citizenship. The prefix, ethno, is not limited to indigenous groups or racial groups. Rather, it refers to a group of people who share a common culture (D’Ambrosio in François & Kerkhove, 2010).
In the Philippines, the Department of Education (2013) has recognized the need to package mathematics in the context of the learners’ culture. The new K to 12 Curriculum includes the learners’ context that encompasses beliefs, environment, language, culture, prior knowledge, and experiences.
With the significance of integrating local culture in mind, this study was conducted to analyze the mathematical concepts found in the paintings by an artist in Iloilo, a province in the island of Panay in central Philippines.
Anchored on constructionism and symbolic interactionism (Crotty, 1998), this paper aimed to determine the mathematical concepts and meanings found in the artworks of the Ilonggo artist, Alan Cabalfin. Specifically, these artworks focused on women of the indigenous group, Panay Bukidnon.
Alan Cabalfin
Known as a potter, painter, and film maker, Alan Cabalfin resides in one of the rustic communities in Panay–an island in the Philippines. He keeps an art studio in his residence in Leon, Iloilo (see Figure 1).
Figure 1. Leon, Iloilo in Panay Island in the Philippines. The zoomed-in maps were taken from Google Maps (2011).
Influences. In the island of Panay, Cabalfin was associated with some experts who may have influenced his art. He had been a close associate of Dr.
Alicia Magos and Dr. Christine Muyco whose research endeavors focused on Panay Bukidnon. He has also worked with writer Alex de los Santos who extended help in organizing and promoting exhibits. Foremost of all Cabalfin’s mentors is Nelfa Querubin, a renowned ceramics artist.
Cabalfin had several travels abroad and these may have widened his perspective on art making. When he went to Malaysia, he was exposed to batik, which he applied as a technique in painting. In 2010, he held an exhibit entitled Binanog: Panay Bukidnon on the Rise in the Philippine Center in New York City. His exhibit featured panubok and binanog, which are Panay Bukidnon’s embroidery and dance, respectively (De los Santos, 2011). In 2012, he also stayed in Virginia where he captured the landscape and spring blossoms through watercolor.
Despite his exposure abroad, he remained true to his roots. He brought Filipino culture outside Philippine borders and showed to the world the beauty of Philippines and the talents of the Filipino people.
Recognition. Cabalfin may not have a roster of national and international awards to his name, but he does not exist as an artist for the sake of prestige and money. Instead, what he has is a genuine concern for Ilonggo art development and does art for art’s sake. His passion and his consistent performance in the field of visual arts did not go unnoticed. He was recognized not only for a single artwork but because of his consistent involvement in the art community that contributed much to the development of Ilonggo culture.
Two reputable institutions recognized Cabalfin for his significant achievements. Western Visayas College of Science and Technology (WVCST), now named as Iloilo Science and Technology University, recognized him for his contribution to the enrichment of Ilonggo culture. He was named outstanding alumnus during WVCST’s centennial celebration. The Provincial Government of Iloilo dubbed him as Distinguished Ilonggo in the field of Visual Arts. A film produced by the Province of Iloilo lauded him for his pottery and exhibits abroad.
Aside from the plaques that he has received, Cabalfin was also recognized in the website of the National Commission for Culture and the Arts as one of the authorities in pottery. Ramirez (2011) mentioned Cabalfin as one of the few active Filipino artists who have consistently produced works on pottery.
Panay Bukidnon
In his own way, Cabalfin had preserved and promoted the beauty of Ilonggo culture, in particular, his advocacy on Panay Bukidnon women.
Muyco (2008) referred to Panay Bukidnon as a group of people living in the highlands of Panay in Western Visayas, Philippines. Panay Bukidnon people
live in some parts of Capiz, Iloilo, Antique, and Aklan.
Cabalfin has coordinated with Dr. Magos, the known researcher on Panay Bukidnon, to join immersions in the Panay Bukidnon locale. He would stay in the mountains to observe the everyday life of the indigenous people whom he captured through his art. He was mostly interested with the Binanog, the hawk-eagle dance. “Parang feeling ko kung indi ka makasaot sina indi ka na-belong. [It seems that when you could not dance Binanog, you don’t belong.],” Cabalfin said.
Since 2010, Panay Bukidnon has been the central theme in Cabalfin’s work. This indigenous group has been depicted through oil, clay, batik, and acrylic for his solo exhibits in New York and the University of the Philippines (UP) Visayas Art Gallery. It was also the reverberating theme in his contributions to group exhibits.
Cabalfin’s solo exhibit at UP-Visayas on the 11 November 2011, displayed numerous artworks on Panay Bukidnon. Entitled Inday-Gen-News:
Recentering the culture bearers, it featured Panay Bukidnon women garbed in traditional attire. The combination of “Inday”, “Gen”, and “News” is based on the word indigenous. Women, referred to as Inday, were Cabalfin’s preferred subject because of their colorful attire well-suited for a work of art. Gen may refer to generation or degenerate. News was added to mean how Inday would cope with current events. Cabalfin’s pieces shown in Figure 2, are apt for the subtitle, Recentering the culture bearers. “This is my way of promoting Panay Bukidnon,” Cabalfin said during the opening of the exhibit.
Figure 2. Women as the main subject of Cabalfin's Inday-Gen-News exhibit.
(Clockwise from top left) The Gathering, Liberty, Harmony, Water, Wind, and Fire are square paintings each featuring a woman with arms outstretched, but with the hands no longer included within the frame. These photos were taken by the researcher.
In February 2012, Cabalfin continued to promote Panay Bukidnon through his participation in Balik Hubon exhibit that gathered great names in Ilonggo art. He joined his fellow Hubon Madiaas members, namely: Joe Amora, Benji Belgica, Erwin Chiongson, Ed Defensor, Martin Genodepa, Edgar Gonzales, Roman Sanares, Dodjie Tan, and PG Zoluaga for a reunion exhibit. There was a varied display of artworks from Hubon and Cabalfin showed his signature work, another Panay-Bukidnon inspired painting.
In March 2012, several Ilonggo artists joined in the exhibit, Small Works at the Museum at Museo Iloilo. Cabalfin, that time, did not waste the opportunity to display a little of his advocacy. He showed two watercolor paintings that artistically highlighted the grace and magic of Binanog.
Methodology Methodological Perspective
Case study of a single case was the method used (Yin, 2003) to provide detailed explanations on the mathematical meanings in Cabalfin’s artworks. The three characteristics of case study as expounded by Merriam (1998) were adapted: particularistic, descriptive, and heuristic. This research is particularistic for it focuses on a particular situation, the mathematics in Cabalfin’s Panay Bukidnon artworks. The rich and detailed explanations, which include mathematical descriptions, verbatim accounts, and photos of artifacts make this case study descriptive. The heuristic quality of this research is manifested by the illumination that a reader may obtain from the results that show a new understanding or the artist’s work.
Procedure
Purposive sampling techniques espoused by Miles and Huberman (1994) and Yin (2003) served as the basis for choosing the informant, an artist who has a significant artistic contribution in the field of cultural preservation among indigenous people. Multiple sources of evidence that include artifacts, interviews, and observations provide data for triangulation (Hammersley &
Atkinson, 2007). The informant reviewed the research manuscript an intensive case study database was developed by the researcher to establish construct validity and reliability, respectively (Yin, 2003).
To determine the mathematical concepts, the nature and some of the steps of mathematical investigation as explained by Bastow, Hughes, Kissane, and Mortlock (1984), and Ronda (2011) were considered. Embedded in the mathematical analysis is the study of signs or symbols, or semiotic analysis.
“Sign” or “symbol” is something that stands for something else or it may be an image that stands for other images or encapsulate ideas (Esterberg, 2002;
Kleiner & Mamiya, 2009). As analyzed through semiotics, geometric figures found in the artists’ paintings were taken as signs.
For the software, Geometer’s Sketchpad (Jackiw, 1997) and Atrise Golden Section (Revvo, 2003-2011) were utilized, to ensure some degree of accuracy and ease in the mathematical description of the artworks.
The computer analysis also backed up the data obtained through manual measurements. The Geometer’s Sketchpad allows one to import a photo of a painting for further analysis, which could include determining the equations of the lines, measures of the angles, coordinates of points, and lengths of segments. The Atrise Golden Section features a golden rectangle that could
hover on top of other system windows. The semi-transparent rectangle could have different orientations and it could also show different kinds of grid and division based on the golden ratio.
Sampling
Not all possible mathematical concepts could be considered in this study. The main limitation is the mathematical expertise of the investigator.
Basic branches of mathematics such as geometry, analytic geometry, and algebra were mainly involved in the analysis. Though wavelets and Fourier transform used by Daubechies, Wang, and Postma for the authentication and classification of Van Gogh’s paintings (Sipics, 2009) could be elegant mathematical analysis techniques, these were set aside to give more focus on what the researcher could essentially do.
Artworks included in this study were chosen purposively based on Cabalfin’s preferences. They were the best representation of his artistic works during the period of September 2011 to June 2013.
Results and Discussion
Cabalfin, with his expertise in drawing figures, stated that sometimes an artist would show a standard face after a series of paintings. He did not disclose any formula in portraiture because he believed in the spontaneity of the painting process. What he said about the standard face opened the door for the mathematical analysis of Water and Fire, artworks shown in the Inday- Gen-News exhibit. Mathematical concepts such as square, perspectivity, golden rectangle, and ratio were used to interpret the artworks. These mathematical concepts may have also served as one of the bases in creating a Panay Bukidnon painting that reflected Cabalfin’s artistry.
The Composition for Square
A rectangle with four congruent sides is a square. According to Cabalfin, this shape is less preferred by some artists because of the difficulty in balancing the painting elements, but he deviated from the norm when he presented a number of square paintings in his exhibit, Inday-Gen-News, at UPV Art Gallery. In each painting, he featured a Panay Bukidnon woman with arms outstretched. Such composition would require a horizontal space for the outstretched limbs.
To deal with the space management predicament, Cabalfin did not include the hands. He shared,
Actually, during sa creation, Siling ko kun may kamot gid sya ya I have to have a format nga palaba. Pagwaon mo gid ang kamot mong. Kun amo na ang frame ang gamiton ko tapos gusto ko pagwaon ko ang kamot, magamay na naman ang figure. [Actually during creation, I said I needed to have a horizontal, rectangular frame. That is because I have to include the hands. But if I were to use such frame, I would have to make the figure smaller.]
The outstretched arms could remind the viewer of the famous sculpture of Guillermo Tolentino, The Oblation, which stands in every campus of the University of the Philippines. Such a gesture of outstretched arms is an offering (U.P. Oblation History, 2005). The Panay Bukidnon woman’s outstretched arms somewhat offers a rich culture to the world while the background depicts the turmoil of the world outside the borders of the mountain. The action might also be metaphorical to crucifixion that is a reminder of how Christ suffered in the cross to save humanity from sins. The severed hands could be interpreted as a sacrifice.
For Cabalfin, cutting a part of the subject somewhat gives more story, mystery, and drama to the picture. He compared his technique to film making, which involves the editing and cutting of some captured scenes. For his paintings, he cut the arms to place the subject at the center and to let it occupy a big area in the canvas.
His choice of using a square frame did not spring from an art gallery’s or aesthetics’ restrictions. It was basically an optimizing issue. He divided commercially available rectangular plywood into squares without waste of material. One plywood has standard size of 4 ft by 8 ft. Cabalfin obtained 2 squares by using the dimensions 4 ft by 4 ft. He could have used other dimensions such as 5 ft by 4 ft and 3 ft by 4 ft; however, his choice could have been influenced by his desire to create more drama and meaning in drawing up an “incomplete” picture of a woman.
Perspectivity
To create the illusion of depth in painting, an artist may apply perspective (Smart, 1998). In linear perspective, parallel lines meet at a vanishing point. That is, the lines on the same plane appear to get closer as they recede into the distance to meet at the vanishing point (Hartt, 1989;
Rodwell, 2001; Clouse, 2002). Linear perspective was used by Renaissance
artists to create more realistic paintings than those of Middle Ages
(Sharp, 2007). With its widespread application in the field of arts, perspective may be used in analyzing the painting such as Cabalfin’s Water and Fire.
In Water, Cabalfin showed the devastating effects of people’s negligence of nature. The murky flood at the background sweeps away houses and trees and the Panay Bukidnon woman in the foreground has her arms outstretched that could signal her intent to stop the destruction. She is a reminder that it is not yet too late to take action.
With the point of origin in between the eyes (see Figure 3), the lines that determine the eyebrows, eyes, tip of the nose, and lips all have a slope of 0.05. The equality of slopes indicates the parallelism of the lines. With parallelism, Cabalfin managed to show that the face is somewhat directly facing the viewer; that is, the facial plane is parallel to the picture plane. At the outset, this is not what a viewer would expect because only the left ear of the woman is shown and this position suggests that the woman has her face turned a little bit to her right.
Figure 3. Cabalfin’s Water (top) and Fire (below) analyzed through Geometer’s Sketchpad.
On the other hand, the Panay Bukidnon woman in Fire is featured with the blazing forest in the background. Just like in Water, the subject’s left ear is shown to the viewer and the right ear is hidden from view. In this painting, the inclination of her head to her right is more defined. This could be attributed to the technique of showing only a fraction of the right side of the face to the viewer.
Furthermore, the perspective suggested by the lines that determine the facial parts tells the viewer that the left face is closer to the audience than the right face (see Figure 3). With the Cartesian plane’s origin between the eyes, the lines that suggest the eyebrows and eyes all have a slope of -0.16; the lines for the tip of the nose, lips and chin have a slope of -0.21. The line that suggests the eyes and the line that suggests the tip of nose intersect at (-8.19, 1.32), which can be considered as the vanishing point.
The lines that determine the facial parts that are observed to be parallel in actuality are expected to intersect at a vanishing point in the painting.
However, the face of the Panay Bukidnon woman when analyzed through the use of perspective principles does not yield the desired results. Nevertheless, the lack of clear perspectivity in the face may be attributed to the short distance between the facial parts and the lack of well-defined leading lines on the painting. The use of perspectivity may be well appreciated in landscape paintings where the distance between objects need to be clearly established.
When the face is perceived to be fully facing the viewer, the facial plane should be parallel to the picture plane. On the picture plane, the lines that determine the eyes and tip of the nose, respectively, are parallel. This was applied in Cabalfin’s Water. When the face is perceived to turn at an angle, just like the subject’s face in Fire, the facial plane is not parallel to the picture plane. Some lines, such as the ones that determine the tip of the nose and eyes, would intersect at a vanishing point. The nearness of the parts of the face (e.g., one eye and the other eye) would usually place the vanishing point outside the boundaries of the frame. Also, the nonparallelism of the lines cannot be easily noticed.
Golden Rectangle
A rectangle is golden if it has the following special property: when a square is a cut away from it, a rectangle of a similar shape is left so that cutting off squares may continue indefinitely. In a golden rectangle, the ratio of the length to the width is called the golden ratio. This ratio is also referred to as phi, ϕ = (1+√5)/2 (Smart, 1998).
The golden ratio comes in various names. Gullberg (1997) tagged the golden ratio as the golden section. It is also referred to as the divine section after the Latin appellation sectio divina, which was first used by Luca Pacioli in his work De divina proportione. Leonardo Da Vinci called it the sectio aurea (Hvidsten, 2005). The name phi is attributed to Greek sculptor Phidias, who used it in the design of the Parthenon (Henry, 2007). Gardner (1959) stated that phi has an aesthetic appeal, a stable, yet never static effect, which has led artists of varying periods and cultures to employ it in determining basic measurements.
Both paintings, Water and Fire, exhibit the integration of the golden rectangle. In particular, the golden rectangle frames the Panay Bukidnon woman’s head (see Figure 4). When the height is divided by the width, the result is 1.62, approximately equal to (√5+1)/2, the golden ratio. Only estimated values could be attained through verification since the golden ratio is irrational.
Even though Cabalfin did not explicitly calculate the dimensions of the rectangle that frames the head, his standard for beauty aligns with the universal standard, the golden ratio. He has also tried using gadgets that could guarantee accuracy. For example, he has used a projector in order to enlarge a picture. By tracing the projected image, he can ensure that the original and the traced projected image are both similar. However, he said that spontaneity still reigns in art making. He would still prefer to use his own artistic inner compass to guide his hands. It turned out his spontaneous artistry leads to a beautiful figure framed by a mathematical standard, the golden rectangle.
Figure 4. The golden rectangle frames the head of Panay Bukidnon women in Cabalfin’s paintings. Rectangles are drawn based on the analysis with the Atrise Golden Section.
Figure 5. The ratio used in dividing the facial plane as shown in Cabalfin's paintings.
The ratios found in the two paintings have corresponding small differences. The average of the percentages may be used as a basis in dividing the facial plane in order to approximate Cabalfin’s depiction of a Panay Bukidnon woman. Furthermore, such ratios may be used to verify the division of the facial plane for other Panay Bukidnon paintings of Cabalfin.
Conclusions and Recommendations
Even with his impressive portfolio, Cabalfin continues to strive to perfect his art. Sometimes, he would muse about his clay works as a metaphor for finding perfection. “Kun may deperensya ang clay, magwa siya sa throwing nga may deperensya. May mga lumps of hard material magwa gid na da siya ya. Ang refinement. Siguro amo na nga kun ngaa nga gina-compare kita sa clay kay kun indi kita refined magwa gid na”. [Imperfections of clay such as lumps of hard material are revealed through throwing, a process of refinement.
That is why sometimes we are compared with clay. If we are not refined, our imperfection will surely show.]
It is an apt metaphor. Revealing and eliminating the imperfections of clay in order to make a beautiful artwork is pottery’s basic principle.
The Ratio for the Standard Face
The ratio of the distance between the lines (that determine the facial parts) to the whole (the distance from the forehead’s hairline and the chin) was determined. The analysis of the ratio rather than the distance is appropriate because ratio is not dependent on the units of measure used. The difference in the size of the digital photo used in the computer analysis and the actual size of the painting will not significantly affect the values of the ratio. The ratios expressed in percentage are shown in Figure 5.
Similarly, research in mathematics and local art may still continue to flourish by sifting out the poor sense of Filipino identity and doubt in the talents of the local artists. In this way, artists and mathematics enthusiasts may continue to bring out the beauty of Panay Bukidnon, the splendor of Iloilo, and the innovativeness of the Ilonggo.
In particular, the education sector could serve as a medium in the preservation of the local culture as manifested in the principles of ethnomathematics (D’Ambrosio, 2004; Shirley, 2011) and arts integration (Burnaford, Brown, Doherty, & McLaughlin, 2007). By contextualizing mathematics through the use of local artworks, mathematics teachers could provide learners the necessary connections for meaningful learning.
In an attempt to define arts integration based on existing literature, Burnaford et al. (2007) mentioned that arts integration may be considered as learning through and with the arts, as a curricular connection process, and as collaborative engagement. For the case of Cabalfin’s artworks, Ilonggo mathematics teachers may use Panay Bukidnon paintings as a context in teaching mathematics concepts such as square, perspectivity, golden rectangle, and ratio. Arts integration, however, should underscore both content and art objectives. That is, art integration should not compromise mathematical content. Some objectives in the Mathematics Curriculum of the Philippines (DepEd, 2013) that may be used in arts integration are solving the area and perimeter of a square, constructing polygons, and proving theorems on different kinds of parallelograms.
As objects of inquiry, paintings such as Water and Fire may be used for students to investigate and inquire in order to form conjectures on mathematical concepts other than the ones investigated in this paper. These paintings may serve as a basis for project-based learning where students may write mathematical investigation papers or construct their own artworks through mathematical construction techniques and application of theorems.
Such projects may be teacher-facilitated or may be accomplished through collaboration with artists.
The paintings could also serve as a gateway through which Ilonggo mathematics learners could be creative, and be proud of their cultural identity and heritage. Cabalfin’s choice of subjects is an appropriate context in which cultural awareness and preservation may be promoted. It is with the recognition of how cultural heritage and art permeate in Ilonggo daily life that would serve as overarching framework for arts integration.
This study is just a scratch in the surface; hence, researchers are encouraged to further explore mathematics in art and how it could be used in the development of sound pedagogy for mathematics. Other researchers may also probe the role of local art and ethnomathematics in the promotion of social justice in the teaching-learning process. Further research on these areas may pave the way for art to thrive not just in museums but in the classrooms as well.
References
Bastow, B., Hughes, J., Kissane, B., & Mortlock, R. (1984). 40 Mathematical Investigations. The Mathematical Association of Australia.
Burnaford, G., Brown, S., Doherty J., & McLaughlin, H. J. (2007). Arts integration frameworks, research, and practice: A literature review.
Retrieved from http://arts.lausd.net/sites/arts.lausd.net/files/
ARTS_INTEGRATION_BOOK_FINAL(1)_0.pdf Clouse, W. (2002).
Step by step art school: Acrylics. London: Octopus Publishing.
Crotty, M. (1998). The foundations of social research: Meaning and perspective in the research process. London: Sage.D’Ambrosio, U. (2004). Preface.
In F. Favilli (Ed.), Ethnomathematics and mathematics education:
Proceedings of the 10th International Congress of Mathematics Education in Copehagen. Pisa: Tipografia Editrice Pisana.
Da Vinci, L. (1906). Thoughts on art and life (M. Baring, Trans.). Retrieved from http://www.gutenberg.org/2/9/9/0/29904/
De los Santos, A. (Datu Lubay). (2011, March 15). The multi-media art of Alan Cabalfin [Web log post]. Retrieved from http://datulubay.
blogspot.com/2011/03/multi-media-art-of-alan-cabalfin.html
Department of Education, Republic of the Philippines. (2013). K-12 curriculum guide: Mathematics (Grade 1 to Grade 10). Pasig City: Author.
Esterberg, K. (2002). Qualitative methods in social research. Boston:
McGraw-Hill.
François, K., & Kerkhove, B. V. (2010). Ethnomathematics and the philosophy of mathematics (education). In B. Löwe, & T. Müller ( E d s . ) . Philosophy of mathematics: Sociological aspects and mathematical practice (pp. 121-154). London: College Publications.
Gardner, H. (1959). Art through the ages (4th ed). USA: Harcourt Brace &
Griffin, W. (2003). The perfect perspective: A mathematical analysis of Co.
perspective using tools available to middle school students. The Journal of Mathematics and Science: Collaborative Explorations, 6, pp. 217-239
Hammersley, M., & Atkinson, P. (2007). Ethnography: Principles in practice (3rd ed.). New York: Routledge.
Hartt, F. (1989). Art: A history of painting, sculpture, architecture (3rd ed.).
New York: Harry N. Abrams.
Henry, R. (2007). Pattern, cognition and contemplation: Exploring the geometric art of Iran. Journal of the Iran Society, 2(6), pp. 24-36.
Hvidsten, M. (2005). Geometry with Geometry explorer. New York:
McGraw-Hill Companies, Inc.
Jackiw, N. (1997). The Geometer’s sketchpad (Version 3.10) [Computer software]. Key Curriculum Press.
Jensen, H. J. (2002). Mathematics and painting. Interdisciplinary Science Reviews, 27 (1). 45-49. doi: 10.1179/030801802225000000
Kleiner, F., & Mamiya, C. (2009). Gardner's art through the ages: Non- western perspectives (13th ed.). Retrieved from http://books.google.
com/books
Merriam, S. B. (1998). Qualitative research and case study applications in education. San Francisco, CA: Jossey-Bass.
Miles, M., & Huberman, M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Thousand Oaks, CA: Sage.
Muyco, M. C. M. (2008). (Re)opening a cultural memory: The bamboo instruments of the Panay Bukidnon. Humanities Diliman, 5(1-2).
Retrieved from http://journals.upd.edu.ph/index.php/humanities diliman/article/view/1485/1440
O’Connor, J. J., & Robertson, E. F. (2003). Mathematics and art – perspective.
Unpublished manuscript, School of Mathematics and Statistics, University of St. Andrews, Scotland. Retrieved from http://
wwwhistory.mcs.st-andrews.ac.uk/HistTopics/Art.html Ramirez, E. L. (2011). Cross-breed and emigre: Visual art in a flux.
Retrieved from National Commission for Culture and the Arts website: http://www.ncca.gov.ph/about-culture-and-arts/articles-on- c-n-a/article.php?i=172&igm=1
Revvo, A. (2003-2011). Atrise golden section (Version 4.2.0) [Software].
Retrieved from http://www.atrise.com
Rodwell, J. (2001). Step by step art school: Drawing. London: Octopus Publishing.
Ronda, E. (2011). What is mathematics investigation? [Web log post].
Retrieved from http://math4teaching.com/2010/03/09/what-is- mathematical-investigation/
Scott, P. (2012). The intellectual contributions of Ubiratan D'Ambrosio to ethnomathematics. Cuadernos de Investigacion y Formacion en Educacion Matematica. 7 (10), pp. 241-246.
Sharp, B. D. (2007). Making the most of digital imagery. Mathematics Teacher, 100(9).
Shirley, L. (2001). Ethnomathematics as a fundamental to instructional methodology. Springerlink, 33(3). doi:10.1007/BF02655699
Sipics, M. (2009). The Van Gogh project: Art meets mathematics in ongoing international study. SIAM News, 42(4).
Smart, J. R. (1998). Modern Geometries (5th ed.). Pacific Grove, CA: Brooks/
Cole.
U.P. Oblation history. (2005). In University of the Philippines Alumni Association of Greater Chicago. Retrieved from http://upaagc.org/
index.php?option=com_content&view=article&id=70&Itemid=98 Yin, R. (2003). Case study research: Design and methods (3rd ed.). Thousand
Oaks, CA: Sage.
