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In 2005, The Georgia Department of Education introduced the Georgia Performance Standards (GPS) as a means by which Georgia's failing curriculum could be restructured. The new GPS Curriculum was designed to specifically address several concerns, one of which was that "Instruction needs to be student-centered rather than teacher-centered. Educators should focus on what students are learning".  In an effort to address this concern, learning tasks were developed as an instructional resource to address the aforementioned concerns and to accompany the new GPS high school mathematics curriculum. Mandated by The Georgia Department of Education, these learning tasks promote problem-based learning (PBL) and were created to foster a learning environment different from the typical classroom environment that people have generally considered good, where classes are well managed and students get high scores on standardized tests.
In a PBL model, students work in small collaborative groups to solve a task situated in the real world. The teacher functions as a cognitive coach while students develop, test, and refine their mathematical thinking towards goal resolution and concept mastery. In so doing, students are engaged in what is known as situated learning, or, thinking by doing.
For the past two years, I have been teaching ninth-grade math under the GPS curriculum. With learning tasks as the mandated medium for implementation, I quickly observed that the completion of these tasks required a level of critical thinking and problem solving that, unfortunately, my students by-in-large have not possessed. I have been very aware that I should not expect students to be effective problem solvers right away. Since learning to solve problems is one of the main goals of PBL, I know that my students need guidance during the searching and solving process. Instructional strategies such as asking guiding questions and modeling have been marginally helpful in aiding my students to draw conceptual meaning from the learning tasks. However, the strategies in my personal arsenal and those provided at professional development sessions have not provided me any insight into how I can transform my students into situated learners who are efficient, and thus effective, critically thinking problem solvers.
As the study of ethnomathematics pushes us toward a global, multicultural view of mathematics (Ascher, 1), I became interested in the problem-solving and critical thinking processes present and used in other cultures. There are many cultures wherein most, if not all goods are indigenously produced by those who have no formal mathematics education. My goal, then, with this research was to observe the critical thinking skills and problem-solving processes of the artisans of Old Fez, Morocco, within the context of situated learning, in an attempt to discover useful insights and techniques that could be translated into effective instructional strategies.
What is the nature of the problem solving processes used by bronze and embroidery artisans? How are critical-thinking skills used within the context of the two crafts? What kind of instructional strategies can be formed and translated into the mathematics classroom?
Situated Cognition & Learning
Situated cognition posits that knowing is inseparable from doing (Brown, et al 1989) by arguing that all knowledge is situated in activity bound to social, cultural and physical contexts. A fundamental concept of situated learning is that all learning takes place within in a specific context and the context significantly impacts learning. Situated learning involves a practice-based approach which tends to address, and oftentimes erase, the dichotomy between school-based learning and real-world learning. Because situated learning can take place in settings that are culturally and socially diverse, the learning environment can be dynamic and highly effective (Duffy & Cunningham, 1996). Learning can occur naturally as a consequence of the learner recognizing the practical usefulness of knowledge conveyed as well as the need to use it to interpret, analyze and solve real-world problems (Lunce, 2006; Hung & Chen, 2002).
A second key component of situated learning extremely relevant to this research is the assumed presence of tacit knowledge. This is knowledge which experts have developed over a long period of time, but which they may not be able to articulate to a novice (Lunce, 2006; George, 2001). While tacit knowledge can be difficult to define and communicate, it is often a fundamental part of the learning trajectory.
Finally, everyday cognition is an integral part of situated learning and refers to the process of learning to use a tool or artifact in a real-life situation to accomplish a real-world objective (Lunce, 2006; Henning, 1998). Because this type of knowledge is associated with cues from a real-world environment, it can be more readily recalled by the learner when needed (Lunce, 2006; Moore, et. al., 1996). In the classroom context, everyday cognition is usually outmoded by the procedural atmosphere of the traditional mathematics classroom.
According to Lunce (2006), educational simulations are often adopted and used to provide a method for students to check their understanding of the real world by modeling a conceptual system or a real environment. A well-designed simulation can engage the learner in interaction by helping the learner stimulate critical thinking. Educational simulations can also provide the learner with "feedback throughout the learning process" (Lunce, 2006; Granland, et. al., 2000). The authors assert that in so doing, simulations can approach the realities of a situated learning context.
While the instructional use of simulations is a relatively new trend about which research is limited, there are ongoing efforts towards developing and evaluating the use of simulations to facilitate situated learning. The learning tasks that supplement the GPS high school math curriculum are among these efforts.
Problem-Based Learning Tasks
As the result of a 2001 audit of Georgia's Quality Core Curriculum (QCC), it was found that in several areas the decades-old curriculum lacked rigor and was inadequate to guide teaching and to ensure common expectations for all students.  Consequently, writers of the GPS curriculum chose to implement a more holistic approach to the teaching and learning of high school mathematics, using problem-based learning tasks as the conduit for implementation.
Scholars and researchers challenge that in contrast to conventional classroom instructional strategies, problem-based learning tasks provide students with opportunities to develop their abilities to adapt and change methods to fit new situations. Further, students engaged in problem-based learning tasks typically have greater opportunity to learn mathematical processes associated with communication, representation, modeling, and reasoning (Erickson, 1999). Because the design of the learning tasks is guided by modeling principles the type of tasks used attempt to reflect reality. The idea is to teach students to learn how to learn. By having to solve problems, students practice learning rather than merely memorizing. The intended goal is to not only introduce students to facts while solving a problem within a learning task, but to have them remember these facts because they are no longer a collection of random information--rather they are meaningful and relevant to solving actual problems. Students should learn to apply new information to solve problems.
PBL tasks thus require pupils to construct knowledge, self-assess, make their thinking visible, adapt or transfer ideas to other similar situations, and establish relationships between variables in a situation. As such, the PBL model is part of a hypothetical learning trajectory by which "a set of instructional tasks are designed to engender the mental processes or actions hypothesized to move students through a developmental progression of levels of thinking, created with the intent of supporting students' achievement of specific goals in a particular mathematic domain" (Clements & Samara, 2004).
According to Simon & Tzur (2004), "mathematical (learning) tasks provide tools for promoting learning of particular mathematical concepts and are, therefore, a key part of the instructional process" (p. 93). While the GPS curriculum supports this claim, students in classrooms throughout the state are struggling with this particular type of educational simulation as evidenced by the yearly standardized test scores that have been fair, at best. How, then, can the cognitive gaps in Georgia's high school students mathematical development, where operational conceptions should give rise to structural conceptions, be addressed?
Prior to selecting research sites a tour was taken of the medina in Old Fez, Morocco, during which a myriad of shops were visited. Maison du Bronze and an embroidery shop were selected based on the type of work being done, the opportunity for mathematical ideas to be observed, as well as the likelihood for mistakes which may require the artisans to exercise problem-solving and/or critical thinking skills. Arrangements were made to return to the embroidery shop the following day and the bronze shop on the day after. However, when on the first day of research the artisan at the embroidery shop was away on a four-day vacation and was not available, a second site, the embroidery school of the medina, was located and arrangements were made to return the following day. Thus, research began at the bronze shop and concluded at the embroidery school.
The participant observation study began at each site with direct observation of the physical environment. Video footage was taken of the physical environment, as well as of the artisans engaged in their craft. Next, field notes were taken while the artisans were observed. Finally, the semi-formal interview was conducted. During this time questions regarding the artifacts used and thought processes were asked. Questions asked during each interview were:
How do you approach the start of a new piece? What, if anything, do you think about before you begin? Is there any planning involved before the start of a new piece?
What are the origins of the motifs used in the designs created?
How does the motif pattern evolve?
How would you enlarge or shrink a motif?
Are you able to duplicate a design that was given to you or described to you?
What types of mistakes are usually made within this type of work? How do you approach fixing mistakes?
What is the highest level of schooling you have completed?
The research project was designed to investigate how artisans with little to no formal mathematic education approach problem-solving within the context of their craft. In addition, the critical thinking skills used by these artisans were also of interest. Participant observation was conducted over a two-day period within the medina of Old Fez, Morocco. Artisans from two shops within the medina, a bronze shop and an embroidery school, were observed. During the study, direct observation and semi-formal interviews were the primary research methods used.
Data collection Instruments
During the direct observation, videotaping and field notes were used to document the physical environment and artisans engaged in their craft. A worker from each shop served as a translator during the semi-formal interview.
Fez, the oldest city of Morocco, consists of two parts; the Medina and the Ville Nouvelle. Fez differs from other cities by its divided Medina which include New Fez (Fes-el-Djedid) and Old Fes (Fes-el-Bali). New Fez was build in the 14-th century, while Old Fez was founded in the 9th century by the first Muslim dynasty to rule Morocco, the Idrissids. Most people of Fez continue to live in the Medina-city, Fez-el-Bali instead of moving to the Ville Nouvelle, which is more modern and urban.  Within the medina is an awe-inspiring marketplace; a maze of narrowing, cobblestoned streets lined with small shops and street merchants, selling anything from fresh Moroccan spices to hand-made cedar wood mirror frames. One such shop was Maison du Bronze, a one-room shop, larger than most in the medina, that displayed and sold fine bronze, silver, and wood crafts.
The walls of Maison du Bronze were covered from floor to ceiling with metal artwork, bronze and silver mirrors and plates of all sizes. On the four massive bookshelves that covered the remaining wall space were bronze vases, animals, tea kettles, mirrors, jewelry boxes, candleholders, hand-carved wooden trunks, saucers, and shoe horns. Most of these items were made of bronze, while some, such as the tea kettles, were made using nonshoor, a blend of copper, silver, and tin. The two show tables found in the middle of the shop held items made of bronze and patina, a substance used by the artisan to give the items a more antiquated look. Behind the tables was an open case filled with Berber daggers and a Berber hanjar (sword).
The second site, the only embroidery school in the medina, was a two-story space with the showroom and workroom on the bottom floor and the school on the second floor. The showroom was visible upon entrance and displayed neatly folded and hand-made tablecloths, napkins, handkerchiefs, and pillows. To the side of the showroom was the artisans' workspace where more work was displayed on the walls, a large glass table in the middle of the workspace, two wall shelves holding inventory, and an L-shaped bench used by the women who worked on the embroidery.
The artisans of Maison du Bronze were two Arab males. Abdul, a sixty-eight year old widower, has been a bronze artisan for more than fifty years. His craft was passed to him by his father who began teaching him around age seven; he considers his craft the work of his ancestors. Abdul always works alone on his pieces, only works six hours each week, and swims three hours each day to stay in shape. The second artisan, Baasim, not only works on pieces but also deals with customers, as he speaks fluent Arabic, French, English, and Spanish. Unlike Abdul, Baasim has had some years of formal schooling. Baasim does most of his craft work at home "in a special chair".
Both embroidery artisans were female and began embroidery work at the age of seven. Kinza, a thirty-six years old woman, learned the craft from her mother and has no formal schooling. Smehan was celebrating her thirtieth birthday on the day of observation. Unlike Kinza, she began attending school but dropped out so she could learn embroidery from her sister. While Kinza is married, Smehan is not.
Data Analysis & Synthesis
During the observation, both bronze artisans were working on the same large, bronze plate in a small corner of the workshop. With only a chair and a wooden stand that stood about 3 feet tall, Abdul began a design on the perimeter of the circular plate, about 4 inches in width; when Baasim worked, he continued the design covering the remaining area within the perimeter border. The artifacts used by both artisans were a matarakat sakleedia (hammer), needle, and chisel. The needle was to create 'Filikian' work, or designs made by a series of tiny dots; the chisel was used to make groove designs. Both designs were made with the chisel or needle in the left hand, held by the thumb and three fingers. The design is created by using the ring finger and thumb to rotate the chisel or needle, while repeatedly using the hammer to make the desired indentation.
The desired motif for each part of the plate was a series of curves, each connected to the other. As the artisans worked the plate never moved, only the artifacts. Both described that there were two types of patterns usually created for bronze work - Berber and Arab. It was explained that the Arab designs represent nature, as they take on a floral design full of connected curves, many deriving from the Arab Henna designs. The Berber motifs are more geometric in nature and highly symmetric.
Sitting against a small pillow on a cloth-covered bench, both of the embroidery artisans were working on the border of a large tablecloth. The artifacts used by the artisans were the murma (circular, wooden hoop used to hold the white cotton cloth in place), a needle, and ketha demse (thread made of cactus leaves). The fabric usually 80cm wide and 2 meters long are stretched on a special loom, then rolled up so the women actually only see the part which she works on. It was explained that the thread must be twisted by the artisans before the work begins since it is too thin to thread through the needle directly from the spool.
Smehan was creating the main motif of her tablecloth, a design originating from Fez called zushlena. The zushlena motif is a circular design formed by a series of curves with a horizontal line, heavy in stitch, and a vertical line in light stitch intersecting in the middle. The heavier stitch is created by stitching over the same pattern an additional time once it has already been stitched into the cloth. Two diagonal lines with opposite slopes intersect with the other two in the middle of the circle. The collection of motifs, each connected to the other, created a design traveling the perimeter of the tablecloth; only when the needle needed to be re-threaded did stitching cease.
During the semi-formal interview, several notions were revealed regarding the majesty of the artisans' work and the mathematical ideas observed. All artisans stated that there is no planning involved before starting a piece. Each explained that there are no patterns used for their designs, as all work either comes from their imagination or is based on a pattern they have created before. Baasim specifically described his and Abdul's work as "imaginary work". When asked how she decides what patterns or designs to use, Smehan pointed to her head saying, "It's all up here. I know what to do".
Several mathematical ideas emerged from the observations and interviews as well. Baasim continuously stressed the symmetric nature of the designs used in the bronze work, explaining the method of visually sectioning the plate in "eights, sixteens, twenty-foursâ€¦so that the entire design is completely symmetric". In the embroidery shop, attention was drawn to a tablecloth covered with a design more intricate than what Kinza and Smehan were working on. The manager, who was serving as translator, described the design as "highly graphic and geometric". He further explained that Fez embroidery has no pattern whatsoever transferred or drawn onto the actual fabric. Its measured stitches are very small, only a few millimeters. Small girls who are taught the art of embroidery at the school are taught to count each thread; one stitch spans two to four threads of the fabric. Embroidering horizontally, vertically or diagonally, leaving blanks to combine the desired design. This is an extremely time-consuming technique, which requires mathematical precision and a lot of concentration and patience, as there is no reverse side. Usually the women first see the outcome when they are finished all of the intricate work, in the case of Kinza's and Smehan's pieces, approximately 8-9 weeks.
The concept of geometric dilations was discussed, as Abdul, Baasim, and Smehan noted that they never measure or count if there is need to shrink or enlarge a motif. In Smehan's words, "I just know where to go." In addition, all three also exclaimed that they are quite capable of reproducing an image shown to them without first sketching or tracing the design on paper. Baasim stated that as long as he could remember the image, he "can see the image in (his) head and do it."
From the participant study yields three important insights into the practice of the bronze and hand-embroidery work. First, there is an extreme amount of tacit knowledge gained and used by each artisan. Tacit knowledge involves learning a skill but not in a way that can be written down. Each artisan described that they ascertained the knowledge needed for their craft through personal experience. As there are no patterns used, and very little direct instruction given, there is a tacit aspect of their mathematical knowledge as well. With tacit knowledge, people are not often aware of the knowledge they possess or how it can be valuable to others. For example, when Smehan was asked if she was counting when creating the tablecloth motif, she responded, "No"; even though the manager would later explain that counting spaces in the fabric is exactly how young girls are taught when they begin to learn the craft.
The tacit aspect of knowledge of their craft can be directly related to the problem-solving and critical thinking skills used by the artisans. In response to the question regarding common mistakes made and how is fixing them approached, neither of the bronze artisans admitted to making any mistakes at all; both citing the amount of years spent working on their craft, and the fact that their designs are organic, as the reasons for their perfect works. Smehan, on the other hand, confessed that the most often-made mistake in hand-embroidery, specifically when working with rectangular tablecloths, is closing the last corner or edge. She explained that since each motif is connected to the last one, the entire border must be un-stitched until the previous corner is reached, and that side of the border must be re-stitched so that the last stitch can connect with the very first of the border design. A soft conclusion can be made, then, that the amount of tacit knowledge held by the artisans counteracts a need for highly-developed problem-solving techniques.
Second, the cultural connection to the craftwork is undeniable. In both art forms, motifs are used from the Berber and Arab aspects of Moroccan culture. The Berber designs are named for the indigenous people of North Africa west of the Nile Valley.  The history of the Berber people in northern Africa is both extensive and diverse and their connection to the Arab nation dates back to the 7th century. In a Berber motif, any sign has a magical or prophylactic meaning. The real meaning is always hidden by the women who would use them, originally when weaving rugs, in order to protect their female culture that is distinct from the male's one. These secrets are transmitted from the mother to the daughter for hundreds of generations. Within Berber motifs is the art of making symmetrical combination from simple forms in order to put the "world" that is depicted in whatever is created in a certain order.
Many of the Arab motifs are inspired by the designs of Henna tattoos, an art form that started in the Arab culture and remains popular to this day and growing in popularity in other countries. Designs made from henna are completely free form and have no specific meaning and are meant for the sheer beauty of the art with no religious or cultural meaning. Henna designs can consist of nearly any pattern or series of patterns and the more elaborate the design the more erotic and sensual it is. As described by Baasim, the Arab designs used are made primarily of several design styles which includes "flowers, paisley designs, intricate lines, shading and doily designs".
Finally, the connection to and motivation drawn from their spiritual practice is extremely noteworthy. With the exception of a very small Jewish community on the outskirts of Old Fez, all of Fez practices Islam. Thus, it was observed and explained that many aspects of the religious practice influence the artisans with respect to their motivation for excellent work. Abdul made clear that he begins all work sessions by speaking, "Bismillah" (In the name of God). When describing a design inscribed on a mirror frame, Baasim pointed out the hand of Fatima, who was the Prophet Muhammad's daughter, explaining that it is often used in Arab designs for protection against evil. Many of the same designs used in both crafts are also found on the mosques within the city, symbolizing the continuity of faith and its appearance in the everyday lives of followers of Islam. Moreover, the way in which the artisans appeared effortlessly excellent in their craftwork can be explicably tied to the never-ending desire they have to please God with their work.
Several striking observations were made during the participant study of bronze and embroidery artisans in Old fez, Morocco. That learning took place within in a specific context and the context significantly impacted learning, and that a great amount of tacit knowledge and everyday cognition were observed, clearly shows that situated learning and cognition were not only present but were a driving force behind the excellence of the craftwork done by the artisans. It is evident that situated learning is an extremely valuable learning trajectory worth further research and should be used to motivate as many student-centered lessons and learning tasks as possible.
Previously written learning tasks can and should be re-examined to find ways in which more situated learning can truly take place so that students can truly engage and benefit from problem-based learning tasks. Curriculum writers need not be dismayed by perceived cultural limitations or budgetary constraints, as students rely on the determination and imaginations of educational experts to produce engaging and effective curriculum despite these and other issues. Therefore, it is highly recommended that curriculum writers be extremely intentional about exposing learners to learning experiences that are grounded in situated learning as an effort to reach the ultimate goal of producing life-long learners, thinkers, and doers.