To refine search, see subtopic History of Education. Laterally related topics: Religion, Time and Space, Mathematics in Recreation, Art, Language and Literature, Music, Measurement, Arithmetic, Mathematics and Mysticism, Geometry, Discrete Mathematics, Optimization, Philosophy, Calculus, Statistics, Social Science, Logic, Computation, Probability, Applied Mathematics (General), Algebra, Number Theory, Optics, Archaeology, Medicine, Creativity, Business, Fractals, and Science.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Bidwell, James K. Maya Arithmetic. Mathematics Teacher 74 (1967), 762--68.
A discussion of the base 20 Mayan number system. It will be especially useful to those teaching mathematics at the elementary level. It does not discuss the Mayan calendrical system in detail, which is uses a mixed base of 20 and 360. As the author points out, his versions of Maya arithmetic may not be historically accurate---The main source on the subject, Father Diego de Landa (1524--1579), burned many of the existing Mayan manuscripts because he considered them heretical. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Maya and Number Systems.
D'Ambrosio, Ubiratan. On ethnomathematics. Philos. Math. (2) 4 (1989), no. 1, 3--14. (Reviewer: M. P. Closs.) SC: 01A07 (00A30 01A80), MR: 91e:01005.
The author sees ethnomathematics very broadly. (A different notion is in Ascher, Marcia and Ascher, Robert, Ethnomathematics.) To the author, the term "ethno" includes all people, but implies a focus on social and cultural factors, and the term "mathematics" includes many ways of understanding the social and physical environments. Stresses the importance to education of understanding the ethnomathematics of different groups. Closely related topic: Ethnomathematics General.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
Gerdes, Paulus and Bulafo, Gildo. Sipatsi. Technology, art and geometry in Inhambane. Translated from the Portuguese by Arthur B. Powell and Gerdes. Instituto Superior Pedagógico, Ethnomathematics Research Project, Maputo, 1994. 102 pp. (Reviewer: J. S. Joel.) SC: 01A07 (00A08 00A69 01A13 51M20), MR: 95f:01002.
The authors discuss the construction and mathematical properties of the Mozambican sipatsi, which are essentially woven handbags. They are generally decorated with strip or frieze patterns, and in fact all 7 possible types of strip patterns occur in the sipatsi from Inhambane province in Mozambique. This book includes a description of the processes used to create the sipatsi, a catalog of the strip patterns found, and a chapter designed for people using the sipatsi to teach mathematics. The authors also give just a few examples of strip patterns on wooden spoons (also from Inhambane province) and on vases and pots (from Maputo). Closely related topics: Mozambique, Basket Making, and Frieze Patterns.
Gupta, R. C. Why study history of mathematics? Ga\d nita-Bh\=arat\={\i} 17 (1995), no. 1-4, 10--28. (Reviewer: C. J. Scriba.) SC: 01A99 (00A30 01-01), MR: 97e:01025.
Discusses a wide variety of reasons for studying the history of mathematics. We will only give a few short excerpts, with the goal of suggesting the breadth of the article. Quotes the saying "History makes a man wise" (p. 10). Mathematics is "a great cultural heritage. A study of the history of mathematics will help us to understand this cultural heritance. It will also help us in knowing the relation of mathematics to other varied elements of culture such as art, architecture, crafts, religion, philosophy, etc." (p. 11). Mathematics "is called the mirror of civilization" (p. 11). "History of Mathematics can humanize Mathematics" (p. 13). Quotes George Sarton as saying "the main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men" (p. 14). "We can know as to why and how new branches of mathematics were born. One can find as to how the unified approaches often integrate various apparently divergent branches. Mathematical revolutions sometimes revolutionized not only mathematics, but other sciences and even society" (p. 15). The author also notes the advantages of knowledge of the history of mathematics in education. He notes a parallel with the biological maxim "Ontogeny parallels phylogeny" (p. 17). It can also "grab the attention of students and spark their interest" (p. 17). With regard to problems once solved or found to be unsolvable, notes "those who are ignorant of history are condemned to repeat it" (p. 20). He concludes with the note about mathematics that "Its international appeal illustrates that mathematics has been a unifier of human experience and a vehicle for better global understanding. It thus helps to cultivate the noble idea of a world family (Vasudhaiva Kutumbakam)" (pp. 22-23). There is much more as well--an article to recommend to anyone interested in teaching at any level. The bibliography includes a number of sources that may not be widely known; S. Buchman's book Poetry and Mathematics may be a good example. There appear to be quite a few useful references to articles on mathematics education as well. Closely related topic: Why Study History Of Math.
Hildebrandt, Stefan and Tromba, Anthony. The parsimonious universe. Shape and form in the natural world. Copernicus, New York, 1996. xiv+330 pp. ISBN: 0-387-97991-3. SC: 00A05 (01A99 49Q15), MR: 97c:00001.
This book has many interesting examples of how problems in optimization have been important both historically and in the world around us. For our purposes, we focus on Chapter 2, The Heritage of Ancient Science. The authors start here with a survey the history of some of the mathematics and applied mathematics of the Babylonians, Egyptians, and Greeks. They consider aspects such as astronomy, burning mirrors, and the discovery of the irrationals (they include a modulo 10 proof that the square root of two is irrational). Of course, this part of the book is not intended to be authoritative; the reader should beware of comments about the Egyptians and the Pythagorean theorem. The book continues with discussions of the Ptolemaic system (which they said was once thought to have been handed down from above) and of the heliocentric system. One of the more appealing parts of Chapter 2 is a discussion of the problem where Queen Dido of Carthage obtained the largest possible area that can be enclosed by the hide of an ox. She supposedly cut the hide into strips and formed it into a semicircle bounded by the sea. Elsewhere in the book there is quite a bit of discussion on optical shortest path problems. There are many fine illustrations both here and elsewhere. Example from Chapter 2 include the music of the spheres as imagined by Kepler, an illustration of Dido's minimization problem from the 1630s, pictures of medieval towns built with an optimization principle à la Dido, and a fronticepiece of a treatise on optics from the 1200s where refraction and burning mirrors are clearly illustrated. This book can be a fine educational resource for teachers trying to motivate ideas such as minimization problems in Calculus. Closely related topics: Optimization, Optics, Astronomy, Irrationals, The Circle, and Carthage.
Jones, Phillip S. The history of mathematics---new sources and uses. Southeast Asian Bull. Math. 4 (1980), no. 1, 1--5. (Reviewer: C. R. Fletcher.) SC: 01A15, MR: 83m:01002.
The author gives a few brief examples of how problems in the Ahmes papyrus could be used for pedagogical purposes. Closely related topic: The Rhind/Ahmes Papyrus.
Kapur, J. N. Encounters of a working mathematician with history of mathematics. Ga\d nita Bh\=arat\=\i 11 (1989), no. 1-4, 30--37. SC: 01A99 (01A32), MR: 91i:01150.
In the process of describing his own encounters with the history of mathematics, the author makes a strong argument for its importance, particularly in mathematics education. He notes that mathematicians are too often unaware even of the history of their own research areas. For example, he mentions "a student who had written a Ph.D. thesis on Banach spaces had no idea who Banach was, to which century he belonged and of what country he was a citizen and why this concept was necessary." As the author notes, such ignorance inevitably weakens mathematics, since it separates mathematics from the applied problems that often motivated it. He discusses the quantity of research currently taking place in India in various fields of mathematics, and in the history of mathematics (and Indian mathematics) in particular. He finds room for improvement, and closes with some some recommendations for correction. Closely related topics: Why Study History Of Math, Applied Mathematics (General), and India.
Knight, Gordon. The geometry of Maori art---spirals. New Zealand Math. Mag. 22 (1985), no. 1, 4--7. (Reviewer: H. S. M. Coxeter.) SC: 51N20 (01A10), MR: 87m:51060.
The Maoris frequently use spirals in their tattoos and wood carvings. These appear very much like the spirals of Archimedes, but often interlace two or more such spirals. Although the easiest way to construct a spiral similar to the spiral of Archimedes may be to use sets of concentric semicircles (or other segments of circles) offset with respect to one another, the author believes that the Maoris didn't use this technique. "In Spirals of Archimedes, and, it seems, in Maori spirals, there is a gradual, rather than an abrupt, change in curvature." The author gives several examples from Maori artwork; there are examples with 2, 3, and 4 interlaced spirals. The author notes that the 3 spiral form is more common in tattooing patterns than in carving. Apparently there was once a 6 spiral pattern on one of the figures guarding the gateway of Papawai Pa. The center of the spiral can be varied somewhat; for example, two spirals can come together in an S-curve. In one case, "the plain ridges, which form an S-curve, are made to cross over the notched spirals, giving a woven effect. According to Phillips this was chiefly an Arawa modification." The author concludes with a note that the spiral of Archimedes should perhaps have a Maori name instead. He suggests that an investigation of these spirals might be useful in mathematics education (when polar coordinates are studied). Closely related topics: Spirals, The Maori, Tattoos, Wood Carving, and Archimedes.
Loeb, A. L. The magic of the pentangle: dynamic symmetry from Merlin to Penrose. Symmetry 2: unifying human understanding, Part 1. Comput. Math. Appl. 17 (1989), no. 1-3, 33--48. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 52-03), MR: 91a:01058a.
In this interesting and entertaining article, Merlin the magician assists Arthur and Key in exploring the secrets of dynamic symmetry (in a problem with four beetles in a square always walking towards each other), in the logarithmic spiral (the curve generated by the beetles), the golden rectangle (and its own associated spiral), and the Fibonacci numbers. The article closes with a discussion of the pentangle, which the author says "is central to the late fourteenth-century 'Sir Gawain and the Green Knight', to medieval sign theory as well as to recent research in quasi-periodic alloy crystals. The Socratic discussions here could be turned used as active learning exercises for talented students. Highly recommended. Closely related topics: England in the Middle Ages, Dynamic Symmetry, Spirals, Proportion and the Golden Ratio, Leonardo of Pisa (Fibonacci), and The Pentagram.
Nagy, Dénes. Symmet-origami (symmetry and origami) in art, science, and technology. Symmetry Cult. Sci. 5 (1994), no. 1, 3--12. SC: 00A69 (01A99), MR: 1 309 239.
Discusses the history and philosophy of origami and then (in a little more depth) discusses some of its applications. The author discusses applications in math and science education, and also in art, design, and technology. A particularly interesting application of paper-folding and the theory of polyhedra is in music education, where one researcher devised "a 'tower' of five octahedra, to illustrate some basic concepts in musicology. His inspiration was from a work by Möbius written in 1861. Ganter's compound polyhedron illustrates geometrically the following concepts and their connections: the vertices correspond to the notes of the chromatic scale, the edges corresponds to the thirds and fifths, and the triangular faces correspond to the triads." He mentions that M. C. Escher was interesting in construction paper models (though it is not really clear how deep that interest lay). It is interesting that the well-known book by T. Sundara Row entitled Geometric Exercises in Paper Folding seems to be independent from the Japanese traditions. Closely related topics: Origami, Symmetry, Japan, Music, M. C. Escher, and August Ferdinand Möbius (1790-1868).
Riese, Tara A. and Chen, Yong Zhuo. Crop circles and Euclidean geometry. Internat. J. Math. Ed. Sci. Tech. 25 (1994), no. 3, 343--346. (Reviewer: E. J. F. Primrose.) SC: 51M04 (01A99), MR: 95b:51018.
This article can be viewed as a supplement to an article by I. Peterson (Science News 141 (1992), no. 5, 76--77) in which the author discusses "Gerald S. Hawkins, a retired astronomer, who was fascinated by the intriguing configuration of crop circles near Stonehenge in southern England. After a systematic study of the crop formations, he discovered five geometric theorems which cannot be found in any Euclidean geometry textbooks and references. Four of them were stated in that article. The fifth, he left to the reader to figure out." These theorems turn out to be quite elementary, but might still be of some interest to an introductory geometry class; when Riese and Yong-Zhou Chen used Peterson's article in their geometry class they had "an exciting discussion on Hawkins's theorems", and the class was able to develop its own version of the fifth theorem. The class's theorem is given in the paper, together with three other simple theorems describing the relationships between circles and n-gons. Closely related topics: The Stone Builders and The Circle.
Ritter, James. Prime Numbers. Unesco Courier (November 1989), 12--17.
The title is a bit misleading. Discusses the work of Babylonian and Egyptian scribes and how they fit into society. Although neither society had a word for a mathematician, the ability to do mathematics was highly valued. One Mesopotamian king boasted of his academic achievements by stating proudly "I am perfectly able to subtract and add, [clever in] counting and accounting", and another says "I can find the difficult reciprocals and products which are not in the tables." In Babylonia and Egypt, mathematics was taught by creating a "network of typical examples in which a new problem can be related---by a form of interpolation---to those already known." An edited version appears in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians and Ancient Egypt.
Swetz, Frank J. Seeking Relevance? Try the History of Mathematics. Mathematics Teacher 77 (1984), 54--62.
Focuses on how the history of mathematics can be used to improve mathematics education. It can not only breath new life into the subject, but also allow students to better understand mathematics as a mode of inquiry. If students see mathematical ideas in other times [and in other cultures], they can appreciate the ideas better in our own. Swetz gives examples from the development of algorithms for arithmetic (including square roots). Ancient demonstrations of mathematical ideas, such as the "husan-thu" proof of the Pythagorean theorem from China can be conceptually more suitable for students than more synthetic modern ones. Ancient "homework problems" from Babylonia, China, and Medieval Italy can be more interesting than the more dry and formulaic modern equivalents. (See Swetz, Was Pythagoras Chinese? for many interesting examples from China.) Although the author doesn't discuss this, the Chinese problems in surveying led to interesting questions in algebra, with fourth and higher degree equations. Swetz discusses how Descartes' idea of a coordinate grid was earlier used by Renaissance artists, ancient Egyptian tomb painters, and various cartographers. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Arithmetic, Computation, China, Algebra, Analytic Geometry, Renaissance Art, Ancient Egypt, and Cartography.
Vitrac, Bernard. The Odyssey of Reason. UNESCO Courier (1989), 29--35.
The development of Greek schools, the role of mathematics in Greek thought, "pure" and "applied" mathematics, the mathematical community that existed in the Hellenistic era. Includes a passage by Proclus on Geminus' classification of mathemata (the root mathema originally meant "that which is taught", so included all branches of knowledge). Reprinted in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: History of Education, Greece, and Applied Mathematics (General).