Volume 8 Number 2, May 1993

Minutes from the ISGEm Meeting in Seattle

Claudia Zaslavsky

The meeting was called to order by President Gloria Gilmer. An audience of about thirty people attended. The secretary, Claudia Zaslavsky, read the minutes of the April 2, 1992 ISGEm business meeting and program in Nashville and the August 18, 1992 meeting in Quebec in connection with the Seventh International Congress on Mathematical Education. The Quebec meeting minutes were amended to state that Sunday Ajose had accepted the chairmanship of the Special Interest Group on Out-of-School Applications. Claudia Zaslavsky also read the financial and membership reports sent by the treasurer, Anna Grosgalvis, showing a December 31, 1992 balance of $1,205.65, and 482 names listed, although not all have paid dues.

Erica Voolich announced the meeting of the History and Pedagogy of Mathematics (HPM) organization, as well as the joint HPM-ISGEm meeting scheduled for Saturday afternoon, April 3, 1993.

Newsletter editor Rick Scott discussed the newsletter, requested contributions for the forthcoming (May) issue, and offered for sale the Compendium of past newsletters at a price of $10 for members. Several people volunteered to work on the editorial board of the newsletter.

New Board member Alverna Champion and retiring member David Davison collected dues while the meeting was in progress.

Lawrence Shirley, chair of the Special Interest Group on School Applications, reported on the meeting of his SIG earlier that day, attended by sixteen people. They discussed contributions on multicultural mathematics to the NCTM journals, as well as the publication of multicultural texts and materials by major textbook publishers. Sunday Ajose's SIG had also met earlier.

Lawrence Shirley reported on the meeting of the NCTM Delegate Assembly. ISGEm had submitted three of the seven resolutions to be considered. They dealt with the inclusion in the program booklet of information about affiliates meeting at the time of the annual NCTM meeting; better handling of the message bulletin board, and the establishment of an international committee of NCTM.

The highlight of the meeting was the talk by Joanna O. Masingila of Syracuse University, dealing with her fascinating study entitled "Connecting the Ethnomathematics of Carpet Layers with School Learning."

The meeting concluded with the announcement that Claudia Zaslavsky was retiring as secretary. She will be ably replaced until the 1996 elections by Maria Reid as recording secretary and Jolene Schillinger as corresponding secretary. The meeting was adjourned, and participants enjoyed a reception hosted as usual by Addison-Wesley.

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Report of the Publicity Committee

Lawrence Shirley

Towson State University

I don't believe this committee has any members except for the chairperson, so my first recommendation is that we should appoint some committee members, preferably people who may have ties to various organizations and/or geographical regions where publicity would be valuable.

Secondly, I would like us to ask all ISGEm members to assist with publicity by mentioning the organization and its address (Gloria's address or Rick's for the newsletter)-anytime they make a presentation in schools, PTAs, university meetings, conferences, radio interviews, etc. I suspect that many members make such presentations at local meetings. I have found much interest even as I find that people are surprised to learn that the mathematics of cultures is even studied, not to speak of having an organization! Just by mentions at such occasions, we can spread the word of ISGEm.

Third, in the same vein, we should urge anyone who writes on ethnomathematical topics for publication (in academic journals, newspapers, etc.) to also mention the organization (and, if possible, the address) in the article or as a note at the end.

Fourth, ISGEm members who are participating in, or (better) helping to organize state or regional NCTM, MAA, SSMA, etc. meetings should see if formal or informal ISGEm gatherings can be added to the program. The Executive Board might assist by writing formal letters to request such inclusion, but the groundwork and details need to be done by the local ISGEm member.

A final suggestion is more for internal publicity. We should urge ISGEm members who have access to e-mail to use the ISGEm e-mail bulletin board more. The few messages I have received were very interesting and have even stimulated further contacts. However, it seems that few use it. Perhaps most members do not know it exists. In the same direction, we should include e-mail addresses when we collect information from members.

Since the lack of members has prevented the committee from meeting, these have been my own suggestions, but I hope they are useful to the overall ISGEm.

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A View of Ethnomathematics

P. Rajagopal

York University

Mathematics is a construct of the human mind. It is knowledge generated by human beings in societies. It arises out of activities such as counting, measuring, locating, designing, playing, and explaining. While not all of these are present or develop to the same extent in all cultures, each culture invents these as it finds the need for them. In course of time the art or technique for these activities link with each other to provide an understanding of the socio-cultural and natural environment. It gets organized in a certain intellectual framework. Thus mathematics is the art or technique of doing these.

Today we understand mathematics to include the mathematics of non-literate people, which require interpretation (see Ascher's book or Zaslawsky's book), the mathematics which requires decoding (Babylonian, Egyptian or Mayan mathematics), the mathematics which requires translating (Indian, Chinese, or Arab mathematics), and our modern Western mathematics (multi-cultural in authorship and international in development).

Since knowledge is developed in cultures what happens when mathematical ideas of one culture are encountered by another culture? Sometimes they meet, shrug, and pass on. Sometimes they meet, and influence each other ("technology transfer" occurs). Sometimes they engage ln a rivalry for imposing one on the other (through trade, colonialism, imperialism or war). Sometimes, when one culture meets another and informs itself of the other it will find that a result or an idea known in one culture was also found or known in the other; but history is written by the victorious about the canonical (or official or authorized) version. The ideas of other cultures may at best rate a footnote.

Any history of mathematics has a problem in that most mathematicians neither know nor care for the mathematics of other cultures. Often they do not care for the culture of their own mathematics either. They treat mathematics as a subject which has no history and is not the product of past human contribution.

What this suggests to me is that while writing about Indian Mathematics forget about parts of it having been done by Indians before the Westerners etc (avoid prioritis); forget that it may be a precursor of later mathematics (Arab or Italian or Western) etc (avoid implying all past mathematics is a prologue to present mathematics). Just write about Indian Mathematics in its own cultural setting. It is the flowering of a culture; it stands on its own and is independent. Say: "this is what it looks like". and write to show the mathematics of another culture. Write it because it is worth knowing for its own sake.

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ISGEm Communications Network Now Operational

The ISGEm Communications Network is now operational and available to anyone with an Internet connection. The network provides a forum for members of ISGEm to pose questions, offer opinions, further discussions, relay research results, and announce meetings and conferences on Ethnomathematics.

The ISGEm Communications Network operates like most electronic bulletin boards. Subscribers may post their message to the ISGEm e-mail address. The message is then relayed to all subscribers to the network. Special interest bulletin boards are a major factor in the rapid dissemination of information in a variety of fields. Ethnomathematics now has one of its own.

To subscribe to the ISGEm Communications Network send an e-mail message to:

isgem@mail.millikin.edu

No subject is needed. Your message should contain the word SUBSCRIBE and your name. Once you are on the network ISGEm messages will be automatically sent to you. You can send messages to the network using the ISGEm address: isgem@mail.millikin.edu

The ISGEm Communications Network is managed by ISGEm member James Rauff (Department of Mathematics, Millikin University, Decatur, IL 62522). Millikin University has donated the network facilities for ISGEm.

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Ethnomath, Cognitive Anthropology & Psychology in Math Ed

Vergani, Teresa (1991). O zeros e os infinitos: Una experiência de antropologia cognitiva e educação matemática intercultural. Lisboa, Portugal: Editorial Minerva. 180 pp. + Bibliography, +XXVIII.

Reviewed by: Luis Ortiz-Franco, Chapman University

One of the four Special Interest Groups (SIGs) of ISGEM is Curriculum and Classroom Application. This book provides a good example of a classroom application of ethnomathematics in teacher education courses.

The book is a report of a project on the mathematical education of prospective elementary school teachers in Setubal, Portugal. The report is divided into six major parts describing the theoretical framework that guided the project, the activities undertaken, and the evaluation of the project.

The purpose of the project was to enhance the mathematical education of 25 prospective elementary school teachers to enable them to improve their teaching of mathematics. The experiences described in this book are very exciting from an ethnomathematics perspective because they vividly demonstrate the links between mathematics, culture, and psychology in the mathematics classroom.

For instance, because of the poor mathematical preparation of the project participants and their negative experiences in their previous attempts at learning mathematics, the classroom interactions between the prospective teachers and the teacher (Professor Vergani who was the director of the project) at times took aspects of therapy sessions. In those sessions, the project participants talked about their mathematics anxieties, their apprehension and fears to attempt to learn mathematics. Professor Vergani discusses how her experiences in those "therapy" sessions led her to the idea of affording the prospective teachers the opportunity to discuss mathematical ideas in a multicultural (ethnomathematics) context. Thus, the project became an experience in the exploration of mathematical thought and intercultural education.

The first few chapters of the book discuss the concepts and methods of cognitive anthropology and intercultural education that provided the theoretical framework for the project. Within this framework, selected mathematical elements in the Mayan and Chinese cultures, and an African culture guided the mathematical activities in the classroom. Descriptions of the respective counting and numbering systems of these cultures and applications of these systems are discussed in a perspective of cognitive anthropology. The concepts of zero, infinity and complementarity are compared and contrasted across these cultures. Vergani infers modes of thought and world views of the Mayans and Chinese based on the mathematical creations of these cultures.

For example, the calendric system developed by the Mayans and the philosophical and religious meaning that they attached to some of the dates are likened to the Chinese world view of complementarity. In particular, Vergani (pg. 80) alludes to the meaning of the Mayan date 4 Ahau as an illustration of the presence in Maya culture of philosophy of complementarity similar to notion of Yin-Yang common in Chinese thought.

With respect to numbers and Yin-Yang, Vergani asserts that the Mayas viewed the positional use of the number zero as signifying both the end of the cycle and the beginning of another cycle. Furthermore, this dynamic view of zero among the Mayas coincided with the changes in the 24-hour day, from darkness to daylight, which in turn fit very well into the Yin-Yang complementarity viewpoint. Thus, the Mayas linked their mathematical creations to their ontology.

In another section of the book, Vergani argues that the thought processes of the Mayas, their convention of writing numbers vertically, and their social activity of building pyramids were in harmony with each other. They all flowed from the bottom up (pg. 79). On the other hand, she points out, Western thought flows in the opposite direction, top-down, we construct buildings from the bottom up, and we write numbers in a horizontal convention. Thus, compared to the Mayas, our thought processes, social activities, and number writing convention are not in harmony with each other.

Concerning the evaluation of the project, students were administered a questionnaire designed to generate quantitative and qualitative data. The quantitative data came from questions asking respondents to place their responses in a Likert-type scale. The qualitative data came from responses to open-ended questions. The results of the evaluation are included in an appendix of the book which includes samples of the open-ended responses.

Although the overall results of the evaluation showed that the project was successful, the most revealing information comes from the open-ended responses. These qualitative data give a glimpse of the exciting intercultural mathematical topics that were explored in the project and how much the prospective teachers liked the intellectual voyage.

In summary, Vergani's experience in using ethnomathematics to further the mathematical education of prospective teachers led her to characterize mathematics as a universal code that can allow us to delve into human cognition. Teacher and students in the project experienced the use of mathematical concepts in various cultures as catalysts in their interpersonal communication sessions to explore the diverse dynamic forms of thought in different cultures. Moreover, the evaluation data illustrates that ethnomathematics is an excellent vehicle to stimulate discussion of abstract concepts found in non-western mathematical systems, and promote interpersonal communication in the classroom in a multicultural context.

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Connecting the Ethnomath of Carpet Layers with School Learning

Joanna O. Masingila

Syracuse University

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

Mathematical Concepts

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.

Mathematical Processes

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.

References

Brenner, M. (1985). The practice of arithmetic in Liberian schools. Anthropology and Education Quarterly, 16 (3),177-186.

Carraher, T. N. (1986). From drawings to buildings: Working with mathematical scales. International Journal of Behavioral Development, 9, 527-544.

Carraher, T. N., Carraher D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21-29.

Ferreira, E. S. (1990). The teaching of mathematics in Brazilian native communities. International Journal of Mathematics Education and Scientific Technology, 21 (4), 545-549.

Lancy, D. F. (1983). Cross-cultural studies in cognition and mathematics. New York: Academic Press.

Lave, J., Smith, S., & Butler, M. (1989). Problem solving as an everyday practice. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 61-81). Hillsdale, NJ: Lawrence Erlbaum Associates.

Lester, F. K., Jr. (1989). Mathematical problem solving in and out of school. Arithmetic Teacher, 37 (3), 33-35.

Masingila, J. O. (1992a). Mathematics practice and apprenticeship in carpet laying: Suggestions for mathematics education. Unpublished doctoral dissertation, Indiana University, Bloomington, Indiana.

Masingila, J. O. (1992b, August). Mathematics practice in carpet laying. In W. Geeslin, & K. Graham (Eds.), Proceedings of the 16th Annual Meeting of the International Group for the Psychology of Mathematics Education, Vol. II (pp. 80-87). Durham, NH: University of New Hampshire.

Masingila, J. O. (in press). Learning from mathematics practice in out-of-school situations. For the Learning of Mathematics.

Pinxten, R. (1989). World view and mathematics teaching. In C. Keitel (Ed.), Mathematics, education, and society (pp. 28-29), (Science and Technology Education Document Series No. 35). Paris: UNESCO.

Posner, J. K. (1982). The development of mathematical knowledge in two West African societies. Child Development, 53, 200-208.

Schoenfeld, A. H. (1989). Problem solving in context(s). In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 82-92). Hillsdale, NJ: Lawrence Erlbaum Associates.

Schroeder, T. L., & Lester, F. K., Jr. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New directions for elementary school mathematics (pp. 31-42). Reston, VA: National Council of Teachers of Mathematics.

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McMath Project

The Department of Mathematics at Grand Valley State University in Grand Rapids, Michigan has been awarded a grant from the National Science Foundation to support the development of a multicultural mathematics course. The project, entitled Multicultural Mathematics (McMath) will develop resource materials and a model syllabus for an undergraduate liberal arts geometry course.

The principal investigator of the McMath project, Associate Professor of Mathematics Alverna Champion, describes the course as unique in both its interdisciplinary and multicultural focus. Faculty from the disciplines of mathematics, computer science, engineering, anthropology, geography, and sociology will work together to construct a course which will attract students to mathematics in ways that many traditionally designed courses have not.

According to Champion, the course will consider the ways in which various groups of people use geometric design principles to construct their houses and to inhabit space. By considering the housing and the communities of diverse cultural groups, students will be encouraged to master mathematical principles and to develop an awareness of history and culture in the process. The course will use the contemporary urban community as a resource, and will also consider the construction of contemporary and historic dwelling places of Native-Americans, African-Americans, Latino-Americans, Asian-Americans, and European-Americans. Funds from the National Science Foundation will be used to construct a computer assisted design program, to develop hands-on mathematics learning tools, and to compile a course package of readings.

NSF awarded the grant on the basis of the course's truly innovative approach to both mathematics and to multicultural education. NSF reviewers note that the course is responsive to the need to make connections between mathematics and other areas of inquiry, and provides a model for curriculum development in this area. In addition to Professor Champion, other Grand Valley team members include, Professor of Sociology, Jacqueline Johnson, co-author; Professors Larry Kottman, Salim Haidar, and Steve Schlicker, Department of Mathematics and Computer Science; Professor Shirley Fleischman, School of Engineering; Professor Janet Brashler, Anthropology; and Professor Ron Poitras, Geography.

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XIXth International Congress of History of Science

The XIXth International Congress of History of Science will be held from August 22-29, 1993 in Zaragoza, Spain. For further information contact:

Prof. Mariano Hormigón
Facultad de Ciencias (Matemáticas)
Ciudad Universitaria
50009 Zaragoza, SPAIN
E-mail: ICHS@cc.UNIZAR.ES

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Have You Seen

"Have You Seen" is a regular feature of the ISGEm Newsletter in which works related to Ethnomathematics can be reviewed. We encourage all those interested to contribute to this column.

Favilli, Franco and Villani, Vinicio. Disegno e Definizione del Cubo: Un'experienza Didattica in Somalia (Drawing and Defining a Cube: A Didactical Experience in Somalia), L'insegnamento della Matematica e delle Scienze Integrate, forthcoming.

The knowledge of a geometric object, like a cube, implies neither the ability to "define" it, nor the ability to "visualize" it with a drawing. A test was given to 19 Somali students. The same test was also given to Italian students. In this article the findings of the test are analyzed and compared.

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Gerdes, Paulus. Survey of Current Work on Ethnomathematics. Invited paper presented at the Annual Meeting of the American Association for the Advancement of Science (AAAS), Boston, February 11-16, 1993.

For the first time in its history the Annual Meeting of the American Association for the Advancement of Science (AAAS) held a session on Ethnomathematics. Paulus Gerdes was invited to present "the first AAAS survey on current work on Ethnomathematics". In his presentation he reflected on the following themes:

Isolated forerunners
Ubiratan D'Ambrosio, the intellectual father of the ethnomathematical program
Gestation of new concepts
Illustrative examples
Ethnomathematics as a field of research
Ethnomathematical movement
Paulo Freire and ethnomathematics

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Indigenous Knowledge and Development Monitor

This new journal "is meant to foster information exchange among the international community of persons and institutions who are interested in the role that Indigenous Knowledge can play in truly participatory approaches to sustainable development". If you wish to be on their mailing list write to:

CIRAN, Centre for International Research & Advisory Networks
P.O. Box 90734
2509 LS The Hague, THE NETHERLANDS

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Claudia Zaslavsky. Multicultural Mathematics: Interdisciplinary Cooperative Learning Activities, J. Weston Walch, Portland, OR, 1993 (call 1-800-341-6094 to order).

The activities in this book emphasize patterns and numbers as used by different people throughout the world. They are appropriate for students in grades 6-9.

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Gilmer, Gloria; Soniat-Thompson, Mary; and Zaslavsky, Claudia. Multiculturalism in Mathematics, Science and Technology: Readings and Activities, Addison-Wesley, 1992.

These readings and activities for the secondary level include the history and accomplishments in science and mathematics of people that have generally been underrepresented in school materials. A wall chart, "A World of Mathematics, Science and Technology", is also available.

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Mathematics Plus: Multicultural Projects, Harcourt Brace Jovanovich, 1-800-544-6678.

A kit of activity cards, a blackline Passport with stamp to keep a record of students' mathematical "Journey", an easel-style World Map to plot their route and a Teacher's Guide with blackline masters and plans.

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Exploring Your Multicultural World, Silver Burdett Ginn, 1-800-848-9500.

These Multicultural Project Booklets, one each for grades K-8, have students participate in projects that reinforce math skills while building multicultural awareness and appreciation.

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ZDM (Zentralblatt für Didatik der Mathematik) - International Review on Mathematical Education, is an abstracting journal with text mostly in English that is now available from Scientific Information Service, 7 Woodland Ave, Larchmont, NY 10538 (914/834-8864).

ISGEm Distributors

The following individuals print and distribute the ISGEm Newsletter in their region. If you would be willing to distribute the ISGEm Newsletter please contact the Editor.

ARGENTINA, María Victoria Ponza, San Juan 195, 5111 Río Ceballos, Provincia de Córdoba

AUSTRALIA, Jan Thomas, Teacher Education, Victoria University of Technology, P.O. Box 64, Footscray, VIC3011

AUSTRALIA, Leigh Wood, PO Box 123, Broadway NSW 2007

BOLIVIA, Eduardo Wismeyer, Consulado de Holanda, Casilla 1243, Cochabamba

BRAZIL, Geraldo Pompeu jr, Depto de Matemática, PUCCAMP, sn 112 km, Rodovia SP 340, 13100 Campinas SP

FRANCE, Frédéric Métin, IREM, Moulin de la Housse, 51100 Reims

GUADALOUPE, Jean Bichara, IREM Antilles - Guyane, BP 588, 97167 Pointe a Pitre, CEDEX

GUATEMALA, Leonel Morales Aldaña, FISICC Universidad Francisco Marroquín, Apartado Postal 632-A, Guatemala

ITALY, Franco Favilli, Dipartimento di Matematica, Universita di Pisa, 56100 Pisa

MEXICO, Elisa Bonilla, San Jerónimo 750-4, México DF 10200

NEW ZEALAND, Andy Begg, Centre for Sicence & Math Ed Research, University ofWaikato, Private Bag 3105, Hamilton

NIGERIA, Caleb Bolaji, Institute of Education, Ahmadu Bello University, Zaria

PERU, Martha Villavicencio, General Varela 598, Depto C, Miraflores, LIMA 18

PORTUGAL, Teresa Vergani, 16 Av. Bombeiros Vol., 2765 Estoril

SOUTH AFRICA, Adele Gordon, Box 32410, Braam Fontein 2017

SOUTH AFRICA, Mathume Bopape, Box 131, SESHESO, 0742 Pietersburg

UNITED KINGDOM, Sharanjeet Shan-Randhawa, 14 Grove Hill Road, Handsworth, (Birmingham B219PA)

UNITED KINGDOM, John Fauvel, Faculty of Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA

VENEZUELA, Julio Mosquera, CENAMEC, Arichuna con Cumaco, Edif. Sociedad Venezolana de Ciencias Naturales, El Marques - Caracas

ZIMBABWE, David Mtetwa, 14 Gotley Close, Marlborough, Harare

ISGEm Advisory Board

Gloria Gilmer, President
Math Tech, Inc.

Ubi D'Ambrosio, 1st Vice President
Universidade Estadual de Campinas

Alverna Champion, 2nd Vice President
Wyoming, MI 49509 USA

Luis Ortiz-Franco, 3rd Vice President
Chapman University

Maria Reid, Secretary
Rosedale, NY 11413 USA

Anna Grosgalvis, Treasurer
Milwaukee Public Schools

Patrick (Rick) Scott, Editor
College of Education, U of New Mexico

Henry A. Gore, Program Assistant
Dept of Mathematics, Morehouse College

David K. Mtetwa, Member-at-Large
Marlborough, Harare, ZIMBABWE

Lawrence Shirley, Member-at-Large
Dept of Mathematics, Towson State U