Toward a Modern View of Mathematics

We face many opportunities in the coming years, in our professions of mathematics and mathematics education.  Will we seize the opportunities, or merely survive with the least efforts that avoid the largest problems?

As professionals, we know that mathematics is a collection of sciences dealing with quantities, shapes and relationships.  We have allowed one of these sciences — calculus — to dominate the mathematical experience of our students, and often only have students study other mathematical sciences after a mastery of calculus (even when there is not conceptual connection).

Now, I realize (as we all must) that calculus deserves a prominent location in undergraduate mathematics.  Not only are the concepts and methods of calculus used in a variety of fields, but the study of calculus allows students to experience some of the greatest achievements in science (and see the beauty as well).  I would like more students to learn calculus.

However, we lack balance in our curriculum.  The vast majority of undergraduate mathematics courses are part of the path to calculus, where the content is (loosely) based on what is needed to learn calculus.  The fact that this path is not effective and needs a new design is a related but separate conversation.

Many recent conversations have amounted to “calculus/calculus-path OR statistics”, with the refrain “people can actually use statistics”.  I question the accuracy of that statement in many ways, but more importantly — are there no other areas of mathematics that have a modern practicality?  Do we really believe that life begins after calculus … that study of other areas must be delayed?

Graph theory is ‘hot’; much of our modern technology is related to this work.  Is there a reason not to include a basic understanding of graph theory in undergraduate mathematics?  The work of graph theory seems accessible.  How about basic number theory and ideas of cryptography?  Discrete mathematical ideas? Matrices and numeric method?

Forum 5 of the Conference Board of Mathematical Sciences (October 2014) focused on mathematics in the first two years of college, with a prime motivation coming from the book “Mathematical Sciences in 2025” http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025   As people talked about the vitality of mathematics, my question was (and still is):

Do we integrate any of these topics or concepts into basic college mathematics, or do those courses continue as single-minded diversions into mathematics that nobody cares about?

Many of you have a deeper understanding of the mathematics described in the “2025” book.  What I recognize is that our students are (in general) prevented from seeing any topics or concepts related to current mathematical research until after the first two years.  Perhaps we can not avoid that condition; however, I think we can include multiple mathematical sciences within the basic mathematics courses our students take.

I hoe that mathematical diversity is coming to a math course near you.

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1 Comment

• 

  By schremmer, April 18, 2015 @ 6:21 pm

  By the way, the main problem with calculus is that it does not allow one to … calculate.

  Regards
  –schremmer

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