Art Duval and Helmut Knaust

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Art Duval and Helmut Knaust
The most exciting phrase to hear in science, the
one that heralds the most discoveries, is not
Eureka! (I found it!) but Thats funny Isaac
Asimov
Department of Mathematical Sciences The
University of Texas at El Paso
March 6, 2004
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• The Problem
• Math majors take 4-5 courses geared for science
  and engineering majors, heavy on computation,
  then show up for their first classes for math
  majors, and encounter proofs for the first time. 
  They do not know what a proof is, why you'd want
  one, or how to construct one.

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• The Problem (contd)
• Others, unsure whether to major in math or
  something else, base their decision on those same
  4-5 more applied courses, never discovering that
  math is primarily about proofs.
• An existing course focuses on how to write a
  proof, once you have an idea of what you want to
  prove, but not on how to find the idea of the
  proof.  "Nuts and bolts" of proof-writing (sets,
  functions, relation), not enough time for big
  ideas and exploration, not very motivating.

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• The Intervention
• New course "Introduction to Higher Mathematics"
  (Math 2325)
• Sophomore level
• Only co-requisite Calculus I
• 6-7 labs on a variety of mathematical topics
  (two weeks each)
• Goal See why youd want or need a proof, and how
  you might discover it.

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The Laboratories



test conjecture
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• The Laboratories (contd)
• Intriguing open-ended problems
• Short exposition by instructor
• Prove conjecture
• Write report (students can resubmit for a new
  grade once per lab)
• Instructor provides guidance and feedback all
  along

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• Group work
• Students work in teams of 2-3 on the
  experimentation part, while
• writing up reports individually.
• Ways in which this interaction takes place
• One student at computer, other thinks of
  experiments to try
• One student experiments, while other looks for
  patterns
• Both students look for patterns on computer
• Students work separately, but check in with each
  other every so often

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Great course Im very happy it was here this
semester. I recommend it to all students who
are unsure of majoring in Math. The best part
(personally) is that I can actually read a Math
book and understand.

• Student comments
• 
  Course Evaluation, Math 3325, Spring
  2003

Challenging but very effective not
overwhelming. Instructors excitement for math is
contagious.
Cognitive thinking is better. Mathematical
interest is at a peak level for me. Comprehension
is strengthened and reinforced. Mathematical
inspiration!!
This class helped to seal the holes in my
mathematical knowledge base! I was exposed to
many forms of math which I had not been aware of
additionally, terminology and abstract concepts
were clarified.
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• Student comments
• 
  Course Evaluation, Math 3325, Fall
  2003

I have been very pleased all semester. ...
Overall very informative and rewarding. I have
enjoyed this class immensely.
I enjoyed this class. I learned a lot of new
material, which was quite interesting.
I liked the way we explore new mathematical
concepts. I like the way the instructor points
out the mistakes and encouraged us to turn in a
revision.
The textbook is really good and correlates
exactly to the class.
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All Questions Answered, All Answers
Questioned Borrowed from Donald Knuth
? Art Duval artduval_at_math.utep.edu ?
Helmut Knaust helmut_at_math.utep.edu