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

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### 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

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

- 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.

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?

- 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.

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