COMHISMA7 Report: A Fresh Wind from Morocco

Schoolcraft College Department of Mathematics The Right Angle Vol. 10, No. 1 September 2002 COMHISMA7 Report A Fresh Wind from Morocco by Prof. Randy Schwartz hen I attended a conference in Marrakech, Morocco this past summer, I had a number of Dorothy and Toto moments (“We’re not in Kansas anymore!”). At one of the conference dinners, we were seated in groups of six people around circular tables in the lunchroom of the host college. As we chatted and served ourselves from a platter of Moroccan lamb, it suddenly occurred to me that people from four of the world’s continents were sitting together at our one little table! Besides myself from North America, there was a retired science professor from Córdoba, Argentina; a woman who teaches middle-school mathematics in a suburb of Lisbon, Portugal; a young African mathematics teacher educator from Nouakchott, the capital of Mauritania; and two Moroccan math education students from the host campus. W One of many banners greeting the Marrakech conferees works published in the U.S. were the result of international collaboration, an increase from 19.7% during 1986-8 (National Science Board, Science and Engineering Indicators 2000). At our dinner table that evening, we were all speaking French as a sort of “lowest common denominator.” This we combined with the international language of smiles and gestures. There were also private conversations breaking out in English, Spanish, Portuguese and Arabic. Navigating across differences of language and culture was part of the excitement of attending the Seventh Maghrebian Colloquium on the History of Arabic Mathematics (COMHISMA7), held May 30 – June 1 at the École Normale Supérieure (ENS), a leading teacher’s college in Marrakech. The official colloquium languages were French, Arabic and English, although I was one of only four or five native speakers of English who attended. The 60–70 participants came from 17 different nations on five continents, with a majority from France and the countries of the Maghreb (northwest Africa). A New Route to Europe A number of the presentations at COMHISMA7 made substantial new contributions to our understanding of the invention of mathematical methods in the Arab world, and their interaction with scientific developments elsewhere in Africa, Europe and Asia. It’s been known for a long time that our numeration system, as well as many of our techniques in arithmetic, algebra and other branches of mathematics, were brought to Europe from the Arab world during the Middle Ages. But what were the actual pathways of this transmission? The traditional understanding has been that Islamic mathematics gained only one significant entryway into Christian Europe, via the trade routes linking Algeria with Italy. Most famously, in the late 1100s a young Italian, Leonardo Fibonacci, learned Arabic numerals and techniques while working at his father’s counting house in Bejaia, a port on the Algerian coast. He later spread the knowledge to Europe. This particular history conference was somewhat unusual in having a focus on the teaching of mathematics. The conference was funded and convened by the Moroccan Ministry of Education, and was organized by the college’s Research Group on Teaching Mathematics and Computer Science (GREDIM). Many of the presentations, including my own, specifically addressed the integration of mathematics history into course instruction as a way to enhance learning and to provide students a more multicultural perspective. However, at the Marrakech conference, Jens Høyrup from Denmark presented evidence that other transmission routes must also have been important. Speaking in French on “The Algebra of Jacopo of Florence: A Challenge to the Historiography of Near-Modern Algebra,” Høyrup drew attention to trade and other possible contact between Egypt, Catalonia and Provence in the 14th Century (the latter two coastal regions were politically unified at the time). His evidence was based on a close reading of algebraic treatises now held in the Vatican. The mathematics that we were discussing at COMHISMA7 is now many centuries old, yet the international aspect of the gathering and its focus on education pointed very much toward the future. I think that as globalization knits the nations of the world ever more closely together, we can expect various forms of crossborder communication (face-to-face as well as high-tech) to become an increasingly routine and essential resource for math educators and mathematicians. Check this out: a study found that already in 1995-97, 26.8% of the mathematical Another speaker, J. Lennart Berggren from Vancouver, 6 Schoolcraft College Department of Mathematics The Right Angle Vol. 10, No. 1 September 2002 British Columbia, has been busy editing the known surviving works of Abu al-Jud, a late-10th-Century mathematician who tackled many difficult geometry problems. An example: given line segment BG and a point A not on it, find a point D on BG such that AD BG + BD2 = BG2. Abu al-Jud not only discovered a method to construct such a point D, he also discovered the diorism or condition on point A that is necessary to ensure that a solution exists. Some of these problems were motivated by practical questions involved in the construction of the astrolabe, an astronomical instrument. It was amazing to hear Berggren announce that, based on the number of references made by other mathematicians to Abu al-Jud’s writings, he suspects that the great majority of these writings haven’t even been uncovered yet, much less studied thoroughly. That gave us a sense of how much work remains to be done in bringing to light, and more deeply understanding, the role played by the Arab world in the history of mathematics. Other presentations explored how mathematics was applied in such fields as physics, astronomy, military science and jurisprudence, the latter especially in implementing the Qur’anic rules for inheritance of wealth. A talk by Kheira Megri, an Algerian living in Paris, posed the question “How Did ibn al-Haytham and Kamal al-din al-Farissi Revolutionize Optics?” She detailed how geometry was used by these Persian-born scholars to explain the focusing of light rays and the formation of rainbows. These were aspects of their comprehensive study of optics, whose greatest significance was that it represented the earliest use of the scientific method that still prevails today. Yvonne Dold-Samplonius of Heidelberg, Germany treated us to the first public presentation of her new 10minute video, “Magic of Muqarnas,” which is suitable for classroom use. The video, which has colorful images and haunting music but no narration, depicts the geometry that Jamshid al-Kashi used in the 15th Century to design faceted vaults called muqarnas, a key architectural element of the intricate domed interiors that can still be seen today in structures throughout the Muslim world. Her earlier video “Qubba for al-Kashi” (1996), distributed by the American Mathematical Society, shows how the same mathematician calculated the volume and surface area of domes. Al-Kashi, who worked at the royal observatory in Samarqand (in modern Uzbekistan), is best known for making prodigious calculations useful to astronomers. He was able to approximateto 16 decimal places, and he solved a cubic equation in order to find a value for sin(1º). When SC English Prof. Gordon Wilson was traveling in Italy and Spain this summer, he found a statue of Leonardo Fibonacci in his home town of Pisa, Italy and snapped some th photos. This year is the 800 anniversary of Fibonacci’s completion of his most famous book, Liber Abaci, which played a huge role in popularizing algebra and other Arab mathematical techniques in Europe. Conferences celebrating the anniversary are being held in Bejaia, Algeria (June 24-26) and Pisa, Italy (Nov. 20-23). Springer-Verlag is publishing the first complete English translation of the book. how should teachers be prepared to use such strategies? She outlined a spiral four-step process, in which the instructor analyzes the needs of the class, plans an activity, brings it to fruition, and then evaluates its effectiveness. To create a new course activity, she browses textbooks on the history of math, singles out relevant passages and authors, goes back to the primary sources, and finally prepares new instructional materials, taking into account the means, aims and context of the activity. Lessons for the Classroom Among the presentations dealing more directly with instructional methods in mathematics was that by Fulvia Furinghetti, whose students include both education and science majors at the University of Genoa, Italy. Her talk, “History as a Tool for Mathematics Education and for Research in Mathematics,” focused on three questions: what are the purposes and benefits of using history in mathematics instruction; which teaching strategies should be selected; and Furinghetti made the point that teaching a mathematical topic in its historical and cultural context can provide the kind of fresh perspective that provokes students to think more deeply. At the same time, she said, requiring the students to re-enact the exact historical paths of discovery is continued on next page 7 Schoolcraft College Department of Mathematics The Right Angle COMHISMA7 REPORT (cont’d from previous page) Vol. 10, No. 1  seldom the most effective way for them to assimilate historical material. She also mentioned a number of reference works useful to instructors seeking to incorporate historical material into mathematics coursework.    Omar Rouan of the ENS outlined the “Evolution of Moroccan Programs in Probability and Statistics at the Secondary and College Level,” from their inception in French in 1967-69, through their reform in 1971-74, to their Arabization in 1985 and more recent developments. In Morocco, as in many other countries around the world, decisions about course curricula and teaching materials are made not by individual instructors or departments, but at the ministerial level. Based on what Omar reviewed for us, it appeared to me that the high degree of centralization in Morocco has been a factor that might have slowed the process of reshaping the mathematics curriculum in order to make it more effective and modern. Nevertheless, in the national curriculum for probability and statistics there has been a gradual evolution toward many of the same changes that have occurred in the United States, including a trend away from the abstract and formal toward the more concrete and contextual; away from a dense and ambitious toward a more lean and limited selection of topics; away from passive lecturing toward constructive and other more active pedagogical approaches; and away from an emphasis on statistical laws and toward greater emphasis on graphical presentations and interpretation.   September 2002 explore each concept from a number of different perspectives, including verbal, numerical, graphical, and algebraic guide the students from easier to more difficult problems streamline the discussion and modernize the notation select applications that are appropriate and interesting to these students show the relation between Arab historical techniques and those from China, India, and Europe. And I summarized the major benefits of these efforts:      to begin to break down the Eurocentric bias of the standard mathematics curriculum in the United States to encourage in students a respect for, and interest in, other cultures than their own to enhance the interest and appeal of mathematics lessons for the students to foster the understanding that mathematics isn’t simply a collection of abstract ideas, but is a product of the thinking and work of real people trying to solve practical problems to help strengthen certain skills and attitudes that are important for functioning in an increasingly interdependent world to round out the technical skills of the students by teaching them some mathematical methods that are still useful even though centuries old. COMHISMA7 showed that mathematics instructors from different nations and cultures can get together and exchange ideas about the content and history of mathematics, and how to integrate them into classroom instruction and other forms of education. I was thrilled to be able to travel to Morocco and learn from the experiences of colleagues from across the planet. I’m especially grateful to the Moroccan Ministry of Education and the various organizers of the colloquium, as well as to Schoolcraft College for making my participation possible.  My own presentation was entitled, “Introducing Historical Arab Mathematics to a Two-Year College in the United States.” I spoke in English, but used transparencies written in French so as to increase my access to this international audience. Since the “community college” phenomenon has little or no parallel in most other countries, I devoted the first section of my talk to making sure the listeners understood the role of such a college and the characteristics of our students. Then I related my experiences incorporating selected mathematical contributions of Arab and other cultures into my classes, as crystallized in a series of self-paced written activities that I developed for several of my courses (for more details, see my article “Bringing Other Cultures into the Math Classroom,” The Right Angle March 2002). Naturally, the current state of heightened tension between America and the Arab world made my audience that much more interested in what I was sharing. The U.S. is perhaps the country where the scientific contributions of Arab and other non-European peoples have been least recognized. But even more, my colleagues were interested to learn how I’ve made good use of the history of mathematics in courses that aren’t mainly aimed at math or history majors. I shared with them the methods that I’ve found useful in making such materials accessible to modern, mostly young, English-only students focusing on careers in business and engineering: 8