published in the International Study Group on Ethnomathematics (ISGEm)
Newsletter, Volume 8, Number 2, May 1993. Located at: http://web.nmsu.edu/~pscott/isgem82.htm.
Article reproduced 2003 with permission of the ISGEm Newsletter editor for use in the Ethnomathematics Digital Library (www.ethnomath.org) developed by Pacific Resources for Education and Learning (www.prel.org).
Joanna O. Masingila
Paper presented at the Annual Business and Program Meeting of ISGEm at the NCTM Annual Meeting, Seattle, Washington, April 1, 1993. It is based on the author's doctoral dissertation completed at Indiana University-Bloomington under the direction of Frank K. Lester, Jr.
The body of literature known as ethnomathematics incorporates research on the mathematics practice of distinct cultures and research on the mathematics practice in everyday situations within cultures. In the first case, researchers have tended to look at the mathematics practice of a whole culture (e.g., Lancy, 1983; Posner, 1982), whereas researchers investigating mathematics practice in everyday situations within cultures have focused on one situation or work context (e.g., grocery shopping, carpentry) within a culture.
Some of these researchers (e.g., Brenner, 1985; Carraher, 1986; Carraher, Carraher & Schliemann, 1985; Ferreira, 1990) have contrasted mathematics practice in school with mathematics practice in everyday situations and noted the gap between the two. Lester (1989) suggested that knowledge gained in out-of-school situations often develops out of activities that occur in a familiar setting, are dilemma driven, are goal directed, use the learner's own natural language, and often occur in an apprenticeship situation. Knowledge acquired in school all too often is formed out of a transmission paradigm of instruction that is largely devoid of meaning.
It is my contention that the gap between doing mathematics in school situations and doing mathematics in out-of-school situations can only be narrowed after more is learned about mathematics practice in the context of everyday life. The majority of researchers who have examined mathematics practice in everyday situations within cultures have investigated situations involving arithmetic and geometry concepts and processes. To extend this inquiry to a measurement situation, I spent a summer with a group of carpet layers to see the mathematics concepts and processes involved in estimating and installing floor coverings (Masingila, 1992a). I was also interested in the process through which novice carpet layers become expert carpet layers. To connect the ethnomathematics of carpet layers with school learning, I analyzed the measurement chapters of six seventh- and eighth-grade mathematics textbooks and had pairs of ninth-grade general mathematics students work some of the problems that had occurred in the floor covering context.
Mathematics Practice in the Carpet Laying Context
I observed four categories of mathematical concepts used by floor covering estimators and/or installers: measurement, computational algorithms, geometry, and ratio and proportion. Measurement concepts and skills were involved in most of the work done by the estimators and installers. In particular, I observed four different categories of measurement usage: finding the perimeter of a region, finding the area of a region, drawing and cutting 45 angles, and drawing and cutting 90 angles.
Although algorithms are processes rather than concepts, I mention computational algorithms in this section because I am interested in the mathematical concept of measurement underlying these algorithms. I observed estimators use computational algorithms in the following measurement situations to determine the quantity of materials needed for an installation job: estimating the amount of carpet, estimating the amount of tile, estimating the amount of hardwood, estimating the amount of base, and converting square feet to square yards.
In addition to the use of measurement concepts and algorithms, I also observed the use of the geometry concepts of a 3 - 4 - 5 right triangle, and constructing a point of tangency on a line and drawing an arc tangent to the line. Floor covering estimators also used ratios and proportion concepts when working with blueprints and drawing sketches detailing the installation work to be done.
Besides the use of mathematical concepts, the estimators and installers made use of two mathematical processes: measuring and problem solving. As would be expected, the process of measuring is widespread in the work done by floor covering estimators and installers. Although being able to read a tape measure is vital, other aspects are equally as important in the measuring process: estimating, visualizing spatial arrangements, knowing what to measure, and using non-standard methods of measuring.
The mathematical process of problem solving is used by floor covering workers every day as they make decisions about estimations and installations. Job situations are problematic because of the numerous constraints inherent in floor covering work. For example: (a) floor covering materials come in specified sizes (e.g., most carpet is 12' wide, most tile is 1' x 1'), (b) carpet in a room (and often throughout a building) must have the nap (the dense, fuzzy surface on carpet formed by fibers from the underlying material) running in the same direction, (c) consideration of seam placement is very important because of traffic patterns and the type of carpet being installed, and (d) tile must be laid to be lengthwise and widthwise symmetrical about the center of the room. The problems that estimators and installers encountered required varying degrees of problem-solving expertise. As the shape of the space being measured moved away from a basic rectangular shape, the problem-solving level increased. To solve problems occurring on the job, I observed estimators and installers use four types of problem solving strategies: using a tool, using an algorithm, using a picture, and checking the possibilities. The following situation illustrates how the strategy of checking the possibilities is used. In this case, an estimator is weighing cost efficiency against seam placement in carpeting a room.
An Estimating Situation
I accompanied an estimator (whom I call Dean) as he took field measurements and figured the estimate to carpet a pentagonal-shaped room in a basement. The maximum length of the room was 26' 2" and the maximum width was 18' 9" (see figure 1). Since carpet pieces are rectangular, every region to be carpeted must be partitioned into rectangular regions. The areas of these regions are then computed by multiplying the length and width. Thus, this room had to be treated as a rectangle rather than a pentagon. Dean figured how much carpet would be needed by checking two possibilities: (a) running the carpet nap in the direction of the maximum length, and (b) turning the carpet 90 so that the carpet nap ran in the direction of the maximum width. [Insert carpet graphics]
In the first case, two pieces of carpet each 12' x 26' 4" would need to be ordered. Note that two inches are always added to the measurements to allow for trimming. After one piece of carpet 12' x 26' 4" was installed, a piece of carpet 6'11" x 26' 4" would be needed for the remaining area. Since only one piece 6'11" wide could be cut from 12' wide carpet, multiple fill pieces could not be used in this situation. Thus, a second piece of carpet 12' x 26' 4" would need to be ordered for a total of 70.22 square yards. The seam for this case is shown by a thin line in the figure.
Turning the carpet 90 would require two pieces 12' x 18'11" and a piece 12' x 4' 9" for fill. The 12' x 4' 9" piece would be cut into four pieces, each 2' 4" x 4' 9". The seams for this case are shown by thick lines in the figure. The total amount of carpet needed for this case would be 56.78 square yards. This second case has more seams than the first, but the fill piece seams are against the back wall, out of the way of the normal traffic pattern. Thus, these seams do not present a large problem. In both cases there would be a seam in the middle of the room. The carpet in the first case would cost at least $200 more than the carpet in the second case. Dean weighed the cost efficiency against the seam placement and decided that the carpet should be installed as described in the second case.
Becoming an Expert
Through my observations of and conversations with the floor covering workers as I examined the apprenticeship process through which novice carpet layers became experts, I made characterizations of both a helper (apprentice installer) and an installer (expert installer). A helper is characterized as becoming an expert by: (a) observing installation work, (b) questioning the installer, (c) participating in the installation process, (d) learning from mistakes, and this culminates in the helper (e) coming to know what the installer knows. I characterized an installer as: (a) maintaining control of the installation process, (b) having a feel for the installation work, (c) determining the progress of the helper, and (d) supporting the helper.
In the School Context
To connect the ethnomathematics of the carpet layers with school learning, I analyzed the measurement chapters in six seventh- and eighth-grade textbooks and observed and talked with pairs of ninth-grade general mathematics students as they solved problems from the floor covering context.
The textbook exercises that I analyzed have some advantages over the problems encountered by the floor covering workers. Whereas the situations encountered in carpet laying are specific to that context and use only customary units of measurement, the textbooks provide students with experiences in both customary and metric units. The textbooks also provide a variety of measurement situations, whereas the floor covering workers encountered the same type of situations on a daily basis. However, the most striking difference between measurement in the floor covering context and its presence in the six textbooks is that the floor covering workers were involved in doing measurement--measuring, making decisions, testing possibilities, and estimating in a natural way as the situation dictated--whereas students using the textbooks would be involved in completing computational exercises placed artificially in everyday situations. The textbook exercises are devoid of the real-life constraints found in the floor covering context and, as a result, do not require students to engage in the type of problem solving required of carpet layers.
The six pairs of ninth-grade general mathematics students I observed and talked with worked on the following problems: (a) find the square footage of a piece of carpet and convert the square footage to square yardage, (b) decide what measurements are necessary to determine the amount of carpet needed for a set of steps with one side exposed, (c) measure a pentagonal room and decide the amount of carpet needed and how to place the carpet considering cost efficiency and seam placement, (d) decide how to install a piece of carpet in a room with a post in the center, and (e) decide how tile should be placed in a kitchen so as to be lengthwise and widthwise symmetrical about the center of the room.
Revisiting the Estimating Situation
The pairs of students who worked the problem concerning the pentagonal-shaped room estimate discussed above all realized that the room needed to be treated as a rectangle, and took the appropriate measurements. The students also understood, with some prompting, that the carpet could be laid two different ways: (a) with the nap running in the direction of the maximum length, or (b) with the nap running in the direction of the maximum width. However, the students seemed to have trouble visualizing how carpet would be laid if the nap ran in the direction of the maximum width, especially how fill pieces could be cut from a carpet piece 12' x 4' 9" and laid to fill the remaining space. This resulted in a lack of ability to compare the amounts of carpet used in the two possible installations: All the pairs decided that both situations used the same amount of carpet since the area of the room did not change.
Contrast this with Dean who, through experience, had gained the ability to visualize how installed carpet would look in an empty room and how fill pieces could be cut so that they filled the remaining space and had their naps running in the same direction as the rest of the carpet. This visualization ability allowed Dean to consider the different possibilities and weigh cost efficiency against seam placement.
Comparing the Students and Carpet Layers
Several differences characterize the gap between the school-based knowledge of the students and the experience-based knowledge of the floor covering workers. The noticeable difference is the lack of a deep understanding of the concept of area on the part of the students. To most of these students, area is a formula determined by the geometric shape (e.g., area of a rectangle = length x width). Because they have not experienced finding area in a real-life manner (at least not in school), these students do not have an understanding of area that can be applied to concrete situations. On the other hand, the estimators and installers, who work with area in concrete ways every day, have a deep and flexible understanding of the concept of area and are able to apply this concept to a variety of floor covering situations.
The second difference between the students and the floor covering workers is that the latter have developed problem-solving skills and strategies that the students lack. If the students have only been exposed to the type of exercises I found in the six textbooks, they have not had sufficient experience with solving problems to develop a repertoire of functional strategies. Related to this, students have often not been exposed to problems with real-life constraints that must be considered and addressed in order to find solutions.
Connecting In-School and Out-of-School Mathematics Practice
This study suggests three key ideas for connecting in-school and out-of-school mathematics practice: (a) Teachers should build upon the mathematical knowledge that students bring to school from their out-of-school situations; (b) Teachers should introduce mathematical ideas through situations that engage students in problem solving; (c) Teachers should establish master - apprentice relationships with their students to guide students in doing mathematics and help initiate them into the mathematics community.
By building upon the mathematical knowledge students' bring to school from their everyday experiences, teachers can encourage students to: (a) make connections between these two worlds in a manner that will help formalize the students' informal mathematical knowledge, and (b) learn mathematics in a more meaningful, relevant way. "Mathematics teaching can be more effective and will yield more equal opportunities, provided it starts from and feeds on the cultural knowledge or cognitive background" of the students (Pinxten, 1989, p. 28).
Introducing mathematical ideas through problem solving means that the mathematical information arises out of the problem-solving activity, along with an understanding of the mathematical concepts and processes involved. In teaching via problem solving, "problems are valued not only as a purpose for learning mathematics, but also as a primary means of doing so. The teaching of a mathematical topic begins with a problem situation that embodies key aspects of the topic, and mathematical techniques are developed as reasonable responses to reasonable problems" (Schroeder & Lester, 1989, p. 33). Teachers can use rich, constraint-filled problems that build upon the mathematical understandings students have from their everyday experiences and engage the students in doing mathematics in ways that are similar to doing mathematics in out-of-school situations.
Teaching via problem solving is consistent with the way in which apprentice floor covering workers learn about estimating and installing. A number of researchers have discussed apprenticeship and its application to the classroom (e.g., Lave, Smith & Butler, 1989; Schoenfeld, 1989) and have found the apprenticeship model to be a viable one for teaching and learning. However, the apprenticeship model that could be used in a classroom is different in two important ways from the apprenticeship model used in work situations, and in particular in the carpet laying context.
The first difference involves the master - apprentice relationship: In the work place, a master and apprentice are working one-on-one; in the classroom, a teacher and possible 30 students or more are working together. In the work place, the apprentice is guided and directed by the master as he or she participates in the work activity; in the classroom, the students are guided by the teacher, but more importantly are guided and challenged by other students as they work cooperatively in doing mathematics. Thus, applying the apprenticeship model to the classroom implies a heavy reliance on cooperative learning. A second difference between the use of the apprenticeship model in the work place and in the mathematics classroom is that apprentices in the work place are constructing situation-specific knowledge; in the mathematics classroom students are constructing mathematics content knowledge and processes that are more general, and hopefully can be applied to a variety of situations.
The end goal of my suggestion that teachers introduce mathematical ideas via rich, constraint-filled problems (e.g., problems from a carpet laying context) is not that students acquire the knowledge necessary to become expert carpet layers. Rather, problems of this type are vehicles for engaging students in doing math and aiding them in developing the mathematical reasoning and problem-solving abilities used by expert problem solvers.
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