Paris Semester Spring 2014 Program Resident

Director Application

Monique Chyba, Professor Mathematics – UH Mānoa

Math 257 – A Mathematical Journey Through the Human History (3 credits)

SAC Course Syllabus Course 2:

1. Description of the purpose or objectives of the course(s) and student learning outcomes (SLOs) (refer to the attached “University of Hawai‘i at Manoa Study Abroad Center Mission Statement and Student Learning Outcomes” with suggestions and samples of how to develop SLOs)

Program Objectives. A historical development of mathematical techniques andideas, including the inter-relationships of mathematics and sciences. Highlights willinclude: Euclidean geometry and number theory including classical constructions,a history of the calculus, foundations for analysis, polynomial equations, and settheory and logic.Part of the course will revolve around Henri Poincaré, one of the most prominent French mathematicians. Henri Poincaré was called the "last universalist" because he was a great contributor to so many fields in mathematics, but his work was so broad that it encompassed physics, philosophy, and psychology.

1. The students will get an appreciation of the historical climate of mathematical research in the 19th century in France, and will study in details the contribution of one French mathematician.

2. 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);

3. The course is not focus solely on computational skills.

Student Learning Outcome. In Mathematics student learning outcome are evlauated in a different way that in any other field I believe. Mathematics is an exact science, and therefor there is no subjectivity, especially in the upper level classes. Typically for 300 level classes we have: Completely Correct (CC), Generally Correct (GC), Partially Correct (PC) andIncorrect (IC).

Assessment of the learning outcomes will be achieved through activities such as class discussion, board work, short non-graded quizzes, selected non-graded homework, and other optional activities. It is important to note that these assessments are for the student learning benefit only and will NOT affect their grade. Class grades will be assigned as described at the end of this document.

Problem solving . Solve problems using historical techniques, including ancient mathematics(for example Egyptian fractions) and more modern mathematics (for example solving cubic equations with Cardano’s formula).Students use calculus–based mathematical techniques to analyze celestial mechanics problems.

Student performance indicators Assessment method

Student solve problems using historical techniques, including ancient mathematics(for example Egyptian fractions) and more modern mathematics (for example solving cubic equations with Cardano’s formula).

On exams, students must demonstrate the ability to apply historical knowledge tyo solve mathematical problems.

The ability to develop a broad concept of the mathematical sciences as approachable from several points of view. The students will learn about mathematics as a human endeavor, the role of individuals of both genders with their insights and idiosyncrasies. They will see mathematics as a cultural heritage, the evolving role of mathematics in cultures throughout the world

Student performance indicators Assessment method

The student will develop skills for problem solving, as a basis for the initial development of many concepts.

The student will stufy the impact of social, economic, and cultural forces on mathematical study and creativity;

The student will develop an appeciation of the 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 student will be exposed to the dynamic nature of mathematics, including recent developments in pure and applied math and

This problem solving ability will be measured by student success at applying techniques to new real–world problems on homework and exams.

The comprehension of historical contect on the developement of mathematics will be measured throughout projects, reports on field trips and presentations given by the students.

the increasing role of technology;

Appreciation of Mathematics. Students appreciate the ways that mathematical techniques and reasoning are applied to physics and astronomy.

Student performance indicators Assessment method

Students work on applications of mathematical techniques to one of the most famous problem of celestial mechanics. Students demonstrate an appreciation for how Mathematics relates to the world around us.

This ability will be measured by student success at applying techniques learned in class to new real–world problems on homework and exams.

Resources for developing the empirical and mathematical origins of each area of school mathematics. This includes the notations, terminology, and major topics of algebra, geometry, trigonometry, calculus, number theory, probability, statistics, computer science, and scientific applications of mathematics.

Student performance indicators Assessment method

Such developments should be recognized as useful at all levels for organizing knowledge in historical perspective and appropriate in more detail as enrichment.

This ability will be measured by students’ success in writing clear, articulate answers to questions involving explanation and interpretation of mathematical concepts.

Model formulation and interpretation. Students will frame quantitative problems and model them mathematically. Model formulation will play a critical role in his class since the questions we will analyse are based on a real-life problem. This is probably one of the major learning outcome that I expect from my students.

Student performance indicators Assessment method

Students translate descriptions of real world phenomena from celestial mechanics into a mathematical models for the 2-body and 3-body problems. After analyzing the mathematical model, the students translate theoretical and numerical answers into real world conclusions.

Student performance evaluation will include setting up and describing a mathematical model for the central force problem and the restricted 3-body problem, interpreting theoretical and numerical answers and writing mathematically reasoned conclusions.

2. Course alpha, number, title(s) and the number of credits.

Math 257 – A Mathematical Journey Through the Human History (3 credits)

3. Detailed course description(s); syllabus/syllabi and student learning outcomes including reading lists.

Syllabus. This Math 257 is a course that is offered here at in Hawai’i. My version differs from the classcical one since it will be heavily inspired by the fact that the course will take place in Paris. A special emphasize will be put on the Mathematics of the 19th and 20th century.

The course is designed based on many manuscripts as well as my own research. No specific textbook will be used but my own notes that will be shared with the students.

Content.

This course may be of interest to any student with a stronginterest in mathematics and the appropriate background. It is particularly recommended for students wishing to teach secondary mathematics. Research shows thatsecondary teachers with a knowledge of history are more likely to believe that investigation is an important part of studying mathematics, that everyone should learnmathematics, and that mathematics is an ongoing, evolving discipline.

Mathematics History and Culture. Most standard histories of mathematics focus on achievements in Europe. However, mathematics is done and has been done by peoples all over the world.

• Look at ethnomathematics, the mathematics of cultural groups• 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.

• Where can you find mathematics in your own cultural heritage?

Early Civilizations. 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. 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. 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.We will therefore focus on its historical aspect and current knowledge.

• 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.The life of Henri Poincaré.

• 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.

Mathematics of the eastern world and medieval & Renaissance Europe. 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 . 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.

• 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.

• 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.

• The French mathematician and engineer Girard Desargues is considered one of the founders of the field of projective geometry, later developed further by Jean Victor Poncelet and Gaspard Monge. Projective geometry considers what happens to shapes when they are projected on to a non-parallel plane.

• René Descartes has been dubbed the "Father of Modern Philosophy", but he was also one of the key figures in the Scientific Revolution of the 17th Century, and is sometimes considered the first of the modern school of mathematics.

• Leonhard Euler was one of the giants of 18th Century mathematics. Like the Bernoulli’s, he was born in Basel, Switzerland, and he studied for a while under Johann Bernoulli at Basel University. But, partly due to the overwhelming dominance of the Bernoulli family in Swiss mathematics, and the difficulty of finding a good position and recognition in his hometown, he spent most of his academic life in Russia and Germany, especially in the burgeoning St. Petersburg of Peter the Great and Catherine the Great.

The 19th Century abstractions. 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.

• "Liberation" of 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 life of Henri Poincaré. Statement of the three-body problem. Historical perspective on Poincaré and the Three-Body Problem. The restricted three-body problem. Application to space mission of the three-body problem.

The explosive 20th century. 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. 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.

• 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!

• Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still discussed.

• The “Hilbert problems” effectively set the agenda for 20th Century mathematics, and laid down the gauntlet for generations of mathematicians to come. Of these original 23 problems, 10 have now been solved, 7 are partially solved, and 2 (the Riemann hypothesis and the Kronecker-Weber theorem on abelian extensions) are still open, with the remaining 4 being too loosely formulated to be stated as solved or not.

TextBook. No specific textbook will be used. Instead, notes and exercises will be made available under various formats (latex, PDF).

Course Objectives and Student Learning Outcomes. The objective of this class is to expose the students to the history of mathematics throughout the ages and immerse them into sphisticated concepts linked to historical development.

Among the student learning outcome associated to this course are:

Understanding mathematics in historical and contemporary contexts; Understandingthe interrelationships within mathematics; Understanding the relationships to other disciplines; Understand the dichotomy/complementarity of mathematics as an abstract area of study versus an applied discipline; Be aware of the role of technology and be able to appropriately use technology for mathematical problems.

4. Number of contact hours.

As for any 3 credits course, the number of contact hours per week is 2 hours and 30 minutes. In addition I will have office hours (2 hours a week). We will also go on field trips, which will add some hours since it cannot be done in 50 minutes.

5. Pre-requisites, if any.

One year of calculus required. Math 311 or 321 recommended.

6. Appropriateness of the course(s) in relation to the overseas setting. For example, please describe how the environment, people, university or college resources may be utilized to maximize the potential outcome of the course(s).

Paris was a great centre for world mathematics towards the end of the 19th Century, and Henri Poincaré was one of its leading lights in almost all fields - geometry, algebra, analysis - for which he is sometimes called the “Last Universalist”. Most emphasize of the course will be around that era.

The course is designed to discuss the history of mathematics and a majpr focus will be on the 19th and 20th century. In partyicular, we will discuss the work of Henri Poincaré, one of the most influential French mathematicians. I am planning to make intensive use of the location as follows. I have a close colleague (Jérémie J. Vaubaillon of the Observatoire de Paris) with whom we work on designing space missions to Near Earth temporarily captured objects (in collaboration with colleagues at the Institute For Astronomy here at University of Hawai’i). This will be a very unique opportunity for my students to not only learn about one of the most famous problems in celestial mechanics but also experiment through visits at the observatory, participation to seminars (frequently given in English on the work of Poincaré) and much more. Additionally, the Institute Henri Poincaré (IHP), created in 1928, is one of the oldest and most active international bodies dedicated to mathematics and theoretical physics. It is located on the campus Pierre et Marie Curie, in the fifth arrondissement of Paris where they regularly organize special activities and events; an example of such activities directly related to the subject of this class can be found at http://www.poincare.fr/. Even though we will be in Paris at a later time, I have no doubt that traces and exhibits from what is currently offered will still be in place and will benefit the students.

In addition, I am planning to expose my Math 257 students to the same field trips than my Math 100 students.

Musée des Arts et Métiers. The Musée des Arts et Métiers presents the fascinating history of the tools and machineries developed by scientists and engineers dating from the 1500s to the present. Its main focus is on Scientific Instruments, exhibiting tools of astronomy. A guided visit of this museum will be part of the Math 499 curriculum.

Eiffel Tower. I am planning a visit to the Eiffel tower with my students to literally touch on some of the concept that will be discussed in class such as congruence and similitude. They are displayed in this magnificent structure, and during our visit the students will learn how to look at it in a mathematical way.

The Pantheon. The pantheon in Paris is home to the Foucault pendulum. It is centrally located in Paris and therefore a place to be! This pendulum measures the rotation of the Earth. On site the students will work on a deep understanding of the functionality of the pendulum.

Le Palais de la Découverte. This fantastic place will excite the brain of our students with many hands-on experience, displays and movies. For the 499 students, we will focus on the astronomy section rather than on the mathematical exhibit.

Sorbonne. The 20th century began with a historic mathematical convention at the Sorbonne in Paris in the summer of 1900 which is largely remembered for a lecture by the young German mathematician David Hilbert in which he set out what he saw as the 23 greatest unsolved mathematical problems of the day.

7. The criteria by which the students will be evaluated (e.g., exams, term papers, attendance, etc.). Indicate percentage for each criteria (totaling 100 per cent)

Exams and Grading. Students will be graded on several aspects. There will be exams, participations, and projects. More precisely, one midterm plus one final will be assigned through the semester. This is very typical of our upper-level mathematics classes here at UH. The students will also work on projects related to the content of the class. Participations will be judged by attendance and engagement in class. My goal is to create a very interactive environment and I would like to value student's motivation and participation.

Homework will be assigned regularly and collected. The homework will be incorporated into their experience in Paris; they will have to visit some places, interview people and turn in reports. Regular mathematical problems to solve will be part of the homework as well.

Final 30%, Midterms 20%, Projects 20%, Homework 20%, Participation 10%