COURSE OBJECTIVES

The objectives of the history of mathematics course are to provide mathematicians and teachers of mathematics with:

---1. An understanding of mathematics both as a science and as an art (mathematics as a deductive science is emphasized in most mathematics courses; as an art, mathematics is a creative subject that includes the application of inductive insights and intellectual curiosity to the solution of problems and the formulation of theorems);

---2. The ability to develop a broad concept of the mathematical sciences as approachable from several points of view, including:
-------problem solving, as a basis for the initial development of many concepts;
-------mathematics as a human endeavor, the role of individuals of both genders with their insights and idiosyncrasies;
-------mathematics as a cultural heritage, the evolving role of mathematics in cultures throughout the world;
-------the impact of social, economic, and cultural forces on mathematical study and creativity;
-------interrelations among the various branches of mathematics, especially their role in the solution of significant problems and in extending the horizons of mathematics; and
-------the dynamic nature of mathematics, including recent developments in pure and applied math and the increasing role of technology;

---3. Resources for developing the empirical and mathematical origins of each area of school mathematics, including the notations, terminology, and major topics of algebra, geometry, trigonometry, calculus, number theory, probability, statistics, computer science, and scientific applications of mathematics. Such developments should be recognized as useful at all levels for organizing knowledge in historical perspective and appropriate in more detail as enrichment.


TEXTS (available in the University Store)

--Katz, Victor, History of Mathematics: Brief Version, Addison-Wesley, 2003   [ISBN 0-321-161939]
                                                                                                           
--Shirley, Lawrence (compiler), History of Mathematics Supplementary Materials, Towson University, 2005  [no ISBN]

(see also the course bibliography  and links)

TOPIC OUTLINE AND NOTES (Brief summaries of material covered in class; class numbers assume 75-minute classes; often the course is arranged for 150-minute classes.) (see the Notes in Wordle)

   Mathematics history and culture (classes 1,2):    Most standard histories of mathematics focus on achievements in Europe. However, mathematics is done and has been done by peoples all over the world. Hence, before launching into the chronological history, it is worthwhile to look at ethnomathematics, the mathematics of cultural groups. If we broaden our view of mathematics from the formal-pure academic to include technical, everyday, and recreational mathematics also, we can see mathematics being done by everyone, from Polynesian sailors navigating to Akan weavers designing kente and Australian Aborigines developing kinship structures in abstract algebraic group structures. All of these really need to be recognized as achievements in the history of mathematics. Where can you find mathematics in your own cultural heritage?

   Early civilizations (classes 3-5) People started counting long before recorded history. The significant development was going from subitizing to having number names, especially as number names developed into a hierarchical system of groupings, usually in groups of 5, 10, or 20. Symbols followed and continued the organization of groups--Egyptian in an additive system of base ten groups; Babylonian in a place-value sexasgesimal (base sixty) system, and later, Mayans in a base twenty system with a similar structure to the Babylonian system.  Within these systems, these people handled sophisticated arithmetic: a complex unit-fraction system of the Egyptians, Babylonian sexagesimal fractions, and an interesting doubling-halving way of multiplying by the Egyptians. Beyond that, they calculated square roots and handled some forms of quadratic equations. In geometry, they found some interesting areas and volumes, dealt with complex engineering, architectural, and astronomical mathematics, and discovered the "Pythagorean" property of right-triangles.

   Greek mathematics (classes 6-9) The Greeks of the first millennium BCE were prosperous and encouraged discussion and debate. This led to the first significant work in pure mathematics, notably the idea that mathematical statements should be backed by deductive proofs. One early area of work was in number theory.  Pythagoras extended this into a number-worshipping cult, though it led to a crisis when root-two was found to be irrational. Problems in comprehending irrationals led the Greeks to shift more of their work to non-measurement geometry.  Meanwhile, Pythagoras became more famous for "his" theorem about right triangles. The "golden age" of Greek mathematics was in the second and third centuries BCE, largely launched at the Alexandria Library/Museum. Deductive mathematics was organized in an axiomatic structure of "The Elements" of Euclid, followed by the many achievements in pure and applied mathematics by Archimedes, the greatest of the ancient mathematicians. Even after the Roman Empire conquered the Greeks, Greek-based mathematics at Alexandria continued into the fifth century CE, with further commentaries and expansions on older works.

   Mathematics of the eastern world and medieval & Renaissance Europe (classes 11-14) China and India have similar stories of long civilizations, mostly out of contact with the West. Both had mathematical work dating back into the second millennium BCE. Chinese writings from the classical period of 250-1200 CE show similar triangles, surveying, and much algebraic work, often far ahead of similar work in Europe. Indian mathematics covers similar topics, but the most important development from India was the beginnings of the Hindu-Arabic numeration system that used cipher symbols in addition to the decimal, place-value structure.  Islamic culture honored scholarship and contributed much to mathematical development--partly by translating and preserving Greek and Indian mathematics, but also expanding on the Hindu numeration with new algorithms, developing algebra to the quadratic formula and beyond, and expanding trigonometry. The Islamic art of tessellations led to new geometric understandings.
European intellectual activity dropped between the end of the Roman Empire and the early centuries after the year 1000. But then trade and war introduced Europeans to the work of the Islamic mathematicians and then more translations of Greek work appeared, all helping to restart mathematics in Europe. The invention of the printing press in the late 1400s greatly contributed to the spread of mathematical ideas and methods to the common people.  The period 1450 - 1600 saw the needs for mathematics applications grow in art, astronomy, navigation, etc., and the techniques of mathematics became more powerful with better notation and algorithms and higher algebra techniques.  The groundwork was laid for calculus.

   Calculus and the beginnings of modern mathematics (classes 15-17) The early 1600s primed the mathematical world for calculus. Coordinate geometry offered a contextual setting to allow the necessary graph work of calculus. A few techniques were developed to differentiate functions. Finally, Newton and Leibniz (at about the same time, working independently) developed the fundamentals of calculus. Newton is considered one of the three greatest mathematicians in history, for his calculus work and also his work in binomials and applied mathematics in physics. The logic of calculus was problematic, as the meaning of delta-x going to zero was not clear.  Even with some logical problems, calculus offered a powerful tool for applications in astronomy, engineering, and mechanics, giving a big push to modern science and the Industrial Revolution.  Governments offered support to mathematics in the royal courts and as support for national development. Mathematics education grew, both in higher levels and in the common schools. In the Americas, mathematics was seen in mining books in Mexico, courses in the new universities, interests of the "founding fathers", and some impressive work from self-taught mathematicians.

   The 19th Century abstractions (classes 19-22) The 1800s was a time of revolution in mathematics, especially as a new "golden age" of pure mathematics. Algebra and geometry both were opened up with new abstract ideas. Gauss, the third of the three greatest mathematicians, worked with algebra on the complex plane, number theory, probability, and in applied areas of astronomy, geodesy, and magnetism.   But there were many others who "liberated" algebra and geometry, offering several new algebras and non-Euclidean geometries, which violated the old standard axioms and yet were internally consistent and "worked".   The revolution continued with the development of analysis, which looked at the formal structure of calculus and the concept of limit to explain the delta-x/dx idea.  The generalized abstraction led to topology, the study of invariants under transformations in algebra, geometry, and analysis.  In the second half of the 19th century, mathematicians were studying some weird things. Why should geometry be limited to three dimensions? Can there be several different infinities (and, specifically, is there an infinity between aleph and c)? The combination of "arithmetization" of operations and the completed link of number sets from integers to complex numbers allowed the study of all mathematical foundations to be done by studying the arithmetic of the integers. Next, the task was to put the arithmetic of the integers on a sound axiomatic structure.

   The explosive 20th century (classes 23-27)  Principia Mathematica wrote out the axiomatic structure of the arithmetic of the integers, but the project crashed when it was shown that the axiomatic system cannot be complete (often considered the third crisis of mathematics). Meanwhile, work in pure mathematics continued at an ever-faster pace, partly in response to the "homework assignment" of unsolved problems given in 1900.  Some major areas of pure mathematics work include abstract algebra, number theory, functional analysis, and topology.  There were many mechanical devices through history to aid calculation--abacus, special written formats, geared arithmetic machines, and especially from the late 1800s into the 1970s, slide rules working on the principle of logarithms. The modern idea of computers began from theoretical work in the 1930s, which grew into the first machines of the 1940s, programming languages in the 1950s, and then the miniaturization and interactivity of the 1970s which led to today's widespread use in communications, business, science, ...and even pure mathematics!  The twentieth century also saw impressive achievements in applied mathematics: physics/cosmology and all kinds of engineering; math and statistics for economics, business, and biology; math to deal with complexities and chaos.  Meanwhile, mathematics education emerged as a field of study, looking at issues of curriculum, instruction, and assessment.

TESTS AND EXAM

There are two tests during the semester (classes 10 and 18) and a Final Exam (class 28)

--Assessment literacy components: As a content course on mathematics history, this course does not address assessment literacy.
--Signature assessments: pre-test, two term papers (one with an oral presentation) midterm exam, final exam
--Reference to standards: National Council of Teachers of Mathematics: Principles and Standards of School Mathematics (2000): History of Mathematics relates to the Connections Standard, and to some extent, the Communications Standard; also historical development of mathematical ideas provides background to all of the content Standards
--Reference to Voluntary School Curriculum and Maryland Core Learning Goals:   The Voluntary School Curriculum and Maryland Core Learning Goals spell out details of content in Algebra, Data Processing, and Geometry for high school curricula and well as integrated mathematics content for lower grades. History of mathematics shows the development of these ideas over time and helps to integrate the students knowledge of the content, thus helping pre-service teachers see mathematics from a deeper and more sophisticated point of view, and broadening and strengthening their instructional ability for all mathematical topics.  
--Towson University's Conceptual Framework for Professional Education: MATH 301 fits into the Mission to "inspire, educate and prepare facilitators of active learning for diverse and inclusive communities of learner in environments that are technologically advanced" by satisfying several of integrated themes of the Vision: The historical view of mathematics helps ensure academic mastery; the world historical background of mathematics helps prepare educators for diverse and inclusive classrooms; learning of the thinking of past mathematicians helps develop professional conscience and encourages students to provide leadership through scholarly endeavors.

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