Seminar on
Mathematics Education at the
University level in the West
A personal perspective
Professor Saleh Tanveer
The Ohio State University, Ohio, USA
tanveer@math.ohio-state.edu
Presented on December 28, 2012
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.
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.
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.
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.
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.
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.
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.
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.
•
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 online lecture. Here is an example:
• http://maths-magic.ac.uk/index.php
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.orgBangla 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.
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.