Contemplation, Inquiry, and Creation: How to Teach Math While Keeping One’s Mouth Shut

1. Contemplation, Inquiry, and Creation: How to Teach Math While Keeping One’s Mouth Shut Andrew-David Bjork Siena Heights University 13th Biennial Colloquium of Dominican Colleges and Universities June 12 -15, 2014

2. 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.

3. 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.

4. 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.

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

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

7. 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.

8. 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

9. 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.

10. 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

11. 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.

12. 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.

13. 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.

14. 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.

15. Does this actually work? • Upper level courses • Lower level courses

16. 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.

17. 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.

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

19. Conclusion