In Spring term 2012, I offered Complex Analysis presented in a modified emporium model at Western Oregon University.  Supporting material was available online for the students, and class time was reserved for working on exercises and proofs.  The course was self-paced.  The layout of the course will be presented, and I will offer an analysis of the efficacy of this model in the setting of Complex Analysis.

In the spring of 2011, I made my first headlong foray into IBL. The course was Methods of Proof, which is our bridge course from lower-division, computational mathematics to upper-division, theoretical mathematics. In this talk I will describe the evolution of my experience, from inception and development through implementation and results. In the end, qualitative and quantitative data indicate that the IBL version of this course was superior to my past more traditional offerings of the same course.

``Automath (automating mathematics), a formal language, was devised by Nicolaas Govert de Bruijn in 1968, for expressing complete mathematical theories in such a way that an included automated proof checker can verify their correctness'' (Wikipedia). De Bruijn's  Automath eventually checked every proposition in a primer on the construction of real numbers that another mathematician,  Edmund Landau, had written for his young daughter.
``Proof checkers'' are used as an educational tool to teach theorem proving. A popular mathematics and physics  monthly  Delta published in Poland printed each month a short tutorial based on the proof checker MIZAR  intended to form an elementary logic course  aimed at high school students.  Currently MIZAR has the largest library (MML) of electronically checked proofs, including large parts of Engelking's General Topology.
R. L. Moore paid particular attention to formulating proofs with a proof-checker precision. Just as in the Moore method, students working with MIZAR are expected to derive proof from axioms. The only lemmas and theorems that can be used are those already proved in MIZAR and included in the MML.

Students enter their intro proofs course accustomed to being able to check their final answers with others and in the back of the book.  In my opinion, one of the greatest difficulties encountered in teaching proofs is helping students adapt to the idea that there are many correct answers.  While in computation-based courses, most students can memorize algorithms and do satisfactorily on tests, a certain level of understanding is required to create a correct proof.  While teaching my first intro proofs course and my first IBL course, I often fought with when to assist students and when to let them struggle just a bit longer on a proof.  The line between frustration and giving up can be hard to see until your students have crossed it.  In IBL classes it can be especially hard to figure out how to give input without positioning yourself as the authority on the subject.  I will discuss my observations on the issue and what worked for me in my class. 

I just finished teaching second semester of real analysis course using a hybrid version of IBL  method.  I will discuss my experience of this method while teaching this course and the reactions of students with the topics, concepts, proving theorems and finding examples and counter-examples.

 

We will discuss university students’ mathematical and non-mathematical proving difficulties. We have observed over forty different proving difficulties and have organized them into nine categories. We will describe these categories and briefly give several examples of difficulties coming under  each category. Our observations come from several years of teaching an experimental proving course to beginning graduate and advanced undergraduate mathematics students and from teaching an experimental voluntary proving supplement to an undergraduate real analysis course. We believe that discussing and categorizing these difficulties will lead to a greater understanding of students’ thinking with regard to proof and proving.