CONTEMPLATION, INQUIRY, AND CREATION: HOW TO TEACH MATH WHILE KEEPING ONE’S MOUTH SHUT Andrew-David Bjork Siena Heights University 13 th Biennial Colloquium of Dominican Colleges and Universities June , 2014

Introduction What makes the education one receives from a Dominican institution distinctly Dominican?  One starts from a position of contemplation.  The focus of the contemplation leads towards truth.  The search for truth is done as a community.  The fruit of the contemplation is shared.

Introduction (continued…) How do students often experience math classes?  They observe the expert (professor) handing down the knowledge.  They are expected to copy the techniques shown through the examples.  The repetition of the exercises cements the learning.  Technology or manipulatives may be used to help understand the concepts.

Introduction (…end) Goals:  To have the Dominican tradition directly inform how the class is conducted.  To make the students active participants in their learning.  To let the departmental learning outcomes drive the pedagogy in the classroom.

Departmental Learning Outcomes 1. Students will read and understand mathematics, differentiating between correct and incorrect mathematical reasoning. 2. Students will effectively communicate mathematics to others, both in writing and speaking. 3. Students will demonstrate abilities to work independently and in-groups to develop mathematical models using appropriate technologies. 4. Students will demonstrate a mathematical maturity leading to independent investigations, increased responsibility for learning, and participation in the professional mathematics community.

Inquiry-Based Learning After many years of wanting a Dominican approach to teaching mathematics, I discovered Inquiry-Based pedagogy.

Inquiry-Based Learning What does an Inquiry-Based course look like?  There is no textbook.  I almost never lecture.  Students are not given examples to emulate.  The exams don't really matter.

There is no textbook I write notes as the class proceeds. The notes contain:  definitions  axioms  a carefully crafted sequence of problems to be solved  almost no examples

There is no textbook: Advantages  Students like to save money  I control the exact sequencing of the material.  I hand out the notes at the appropriate time  I can change the course according to what the students discover.

There is no textbook: Disadvantages? Students don't have examples in print. I have to write the book for every class. Journal for Inquiry-Based Learning in Mathematics

I almost never lecture A typical class period proceeds like this:  Ask if there are any questions or discussions points from the previous class period.  Ask for any volunteers to present their solutions on the board.  During the presentations, makes notes on the presentation and the questions or comments from the other students.  Give praise to the presenter regardless of the outcome of the presentation.  If enough results were presented, give the next set of notes, and discuss the new ideas, definitions or axioms.  Give some in class time for collaboration.

I almost never lecture: why? Students become responsible for the creation of the content of the course. Students are actively engaged in search for mathematical truths. The search is done as a community. They have no model to emulate: instead, the students create. Creating math is hard. It is frustrating. When the obstacles are overcome, it is so rewarding.

Students are not given examples to emulate Students present their own solutions:  the learning is constructed.  the learning is owned.  Mathematics is the contemplation, discovery and sharing of mathematical ideas: my students become mathematicians.  My class goes from being informative to transformative.

The exams don't really matter  Exams tend to measure imitation of in class examples.  They completely fail to measure any of the departmental learning outcomes.  I can evaluate each one of my students during each class session.  I can keep a journal of in class activities to reflect on individual student progress.

Does this actually work?  Upper level courses  Lower level courses

What happened in my Calculus 2  The class had 1 math major, 7 science majors and 7 high- schoolers.  I covered the same amount of content I usually do.  All the students presented their own solutions.  I lectured about 5 times all semester (we meet every day) for about 20 min each time.  Every week the students surprised me with their insight, creativity, and enthusiasm for the class.  I had never had as much fun teaching Calculus as this past semester.

Were the goals achieved?  One starts from a position of contemplation. Problems are given to students with no hints or examples.  The focus of the contemplation leads towards truth. I give the sequencing, the students are responsible for finding the mathematics.  The search for truth is done as a community. Cooperation, discussion, shared frustrations all build our classroom community  The fruit of the contemplation is shared. The student presentations drive the class. Each student presents dozens of problems through the course of the semester.

Were the goals achieved? 1. Students will read and understand mathematics, differentiating between correct and incorrect mathematical reasoning. 2. Students will effectively communicate mathematics to others, both in writing and speaking. 3. Students will demonstrate abilities to work independently and in-groups to develop mathematical models using appropriate technologies. 4. Students will demonstrate a mathematical maturity leading to independent investigations, increased responsibility for learning, and participation in the professional mathematics community.

Conclusion Dominican Principles Learning Outcomes Inquiry- based Model