Seminar on Mathematics Education at the University level in the West

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Seminar onMathematics Education at the
University level in the West  A personal
perspective

• Professor Saleh Tanveer
• The Ohio State University, Ohio, USA
• tanveer_at_math.ohio-state.edu
• Presented on December 28, 2012

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Mathematics for different audiences

• Service courses for engineering and physical
  sciencesbasic skills in calculus, linear algebra
  and differential equations.
• Undergraduate Math Majorsusually 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.

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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.

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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, mathematiciansfindcareers
  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.

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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.

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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, evenwhen some
  pursue anapplied 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.

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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.

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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.

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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.

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Bridging Resource Gap

• Improving mathematics education and opening new
  avenues for students requires bridging resource
  gapsinstitutional 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

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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.

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Conclusion and Thoughts on improving Bangladesh
Math Education

• I presented a brief overview of math education in
  the westprimarily 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.