Mathematics Education at the University level in the West

1. Mathematics Education at the University level in the West A personal perspective. Saleh Tanveer The Ohio State University

2. Mathematics for different audiences • Service courses for engineering and physical sciences—basic skills in calculus, linear algebra and differential equations. • Undergraduate Math Majors—usually geared towards building a combination of skills and abstract thinking. • Graduate Mathematics Training. Most programs in the US have a unified core requirements, followed by specialized training.

3. Service Courses • Generally, in the US, while teaching calculus, most proofs in calculus classes are skipped since the emphasis is on computational skills. • In linear algebra, materials like vector spaces are presented abstractly even to a non-math audience with plenty of examples. • Differential equations is also taught the same way, though basic understanding in applying existence and uniqueness theorems, though not proofs, is usually required of all students.

4. Undergraduate Math Major Training • The course requirements is partly geared to the kind of careers math majors usually have. Note only a small fraction of majors pursue post-graduate training. • Besides graduate school, mathematicians find careers in diverse fields. Teaching at the secondary level is a popular choice. At the same time, one of the top ten jobs every year is actuary, where a lot of probability and statistics are used. Mathematicians also work in operations research, computer science, cryptography, biotechnology, and more. • Even MBA , law and medicine programs prefer students with math and physical science background these days because of the rigor in their training and ability to abstract.

5. Different tracks in Undergraduate Math Training • Traditional track--a healthy dose of core abstract classes in algebra and analysis and other ``pure-math” classes. Many programs have a course on “introduction to proofs” before these serious courses begin. • Applied Track--fewer core course requirements instead differential equations, probability, statistics and numerical analysis forms part of the curriculum. • In UK, traditionally, they have separated pure and applied options early in their training. This is not the case in most US programs.

6. Graduate (post-graduate) Training • Many Ph.D program in the US like ours require students to pass qualifying exams that include real analysis and abstract algebra, even when some pursue an applied option. • Most strong programs is the US usually have a well-integrated pure and applied program. • This is very different from the UK, which has a tradition of separate pure and applied math.

7. Graduate Program in Mathematics • In France, what is called applied mathematics is actually quite pure, for instance, theoretical PDE. Germany also has a strong core mathematics bent. What is ``applied math” in the British sense is typically done by Physicists and Engineers in France and Germany. • The US graduate programs has been influenced more by mainland Europe rather than Britain.

8. Flexibility and openness in the US • US system is quite flexible. Double major options are quite possible. Even a Ph.D program allows minors in other areas. • Also, the faculty for the most part, are quite open. It is not uncommon for students to challenge a faculty member about the correctness of a proof or a procedure and it is not usual for faculty to admit that they overlooked something or were incorrect. Students also do not assume that their teachers know everything. This creates a good learning atmosphere.

9. Pure versus Applied Dichotomy • Few universities In the US have separate pure and applied departments. Most have just one department with both pure and applied groups and people in between. • A common culture is essential. Many initial developments in applied math, which resulted from mere manipulations motivated by physics, have now matured into areas that require pure mathematics. Generally, as a field matures, it tends to become more and more abstract. A good example is statistical physics. People in probability and combinatorics regularly prove theorems in statistical physics. Another good example is integrable systems. Many recent contributions come from algebraic geometry, group theory, topology and combinatorics. Fluid Mechanics, Quantum Mechanics, General relativity are other examples. In String Theory, the reverse is true. Esoteric geometry and topology concepts with no previous physical applications have now become relevant to String theory. • It is my opinion that students are better served to catch this new trend if they are broadly trained with a good dose of core mathematics--real and functional analysis, abstract algebra and differential equations. Specialized topics course can wait till graduate school so that they can use more sophisticated tools.

10. Bridging Resource Gap • Improving mathematics education and opening new avenues for students requires bridging resource gaps—institutional inadequacies, inadequacy of teaching resources, etc. • In the UK, most universities do not have resources to offer regular graduate level courses. Some of them are bridging this resource gap by pooling with other universities and offering on-line lecture. Here is an example: • http://maths-magic.ac.uk/index.php

11. Bridging Resource-gap • There are also other ways of overcoming teacher shortage or limitations. For instance, MIT has made all their lectures available on videos free of cost http://ocw.mit.edu/courses/mathematics/ • Another excellent resource, though at a lower level (up to differential equation) and somewhat easier to understand is http://www.khanacademy.org Bangla voice-over of this material is also available. http://khanacademybangla.com/ • http://learning.agami.org/ • Additionally, wikipedia is usually a reliable on-line resource if you should google any subject. • Also, many professors make available their lecture notes freely to any one.

12. Conclusion and Thoughts on improving Bangladesh Math Education • I presented a brief overview of math education in the west—primarily in the US. I have tried to present some modern trends. • I believe on-line resources is the way to go for dramatically raising standards of education in BD. • Further, I think resource sharing between different universities will be helpful. BMS can take a lead in organizing such cooperation. • I think core mathematics classes should be part of the curriculum for all students-- pure or applied. Abstract thinking has to be part of effective mathematics training. • Also, we should think of less hierarchy in BD mathematics departments. With rare exceptions, none of us can claim to be in good control of more than just one or two areas of our discipline. By recognizing this in our attitudes, more collaborations are possible and it contributes to improved learning atmosphere.