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Mathematics and the inquiring mind

September  05, 2012

Assistant math professor Brian Katz is the first subject of the new recurring
"Augie Minute" video series, which will highlight the interesting work
of Augustana faculty and staff.

Distilling Ideas: An Introduction to Mathematics Through Inquiry is the title of a new eBook coauthored by Dr. Brian Katz, assistant professor of mathematics at Augustana, and his colleague Dr. Michael Starbird of the University of Texas-Austin. The book is the second in a new series of "inquiry" books to be published by the Mathematical Association of America, which publishes high-quality mathematics textbooks for undergraduates.

According to the MAA editorial board, they are "especially interested in innovative manuscripts." So what makes Distilling Ideas different from the usual textbook? It's that concept of inquiry.

"Most textbooks are big reference repositories of information," said Dr. Katz. "They contain results." Instead of proposing questions for readers to solve and then providing answers, this book is an introduction to what he calls "the habits of being a mathematician."

"We (mathematicians) have a constellation of habits in response to a mathematical problem," he explained. "So, rather than impose a definition, we propose a mathematical problem and ask the 'reader' to pin down the essential ingredients to create their own definition." 

Experiencing mathematics — or, crossing that bridge when you get there

In the book, the authors present mathematical exercises as scenarios that include characters and their conundrums. Users share in the experience and create their own questions and connections, to try to make meaning of the situation. The skills they gain can help them explore mathematical ideas and prove theorems.

For example, in presenting the Königsberg Bridge Problem at the beginning of the book, a certain Otto challenges his friend Friedrich to leave the café where they met, walk over all seven bridges in their city of Königsberg, and return to their café without crossing the same bridge twice.

To solve this problem, a student would begin by asking questions: What situation or problem is similar to this problem? What makes them similar? Which of those similar features are needed to solve this problem? Such questions are the first of the "essential ingredients" needed to create the next steps.

"The next phase would be experimentation, and building upon that," Dr. Katz continued. "Eventually, the student arrives at a simple statement—like, 'Yes, I can do this if these two other things happen.' Then the student does the work of creating a rigorous argument to support that statement."

This "rigorous argument" is what we would call a mathematical proof. (And the solving of the Königsberg Bridge Problem by Leonhard Euler in 1736 led to modern-day graph theory.)

Sounds complex? It may just be simpler than what we expect. The introduction to Distilling Ideas begins "All mathematical ideas originate from human experience." As children we learn by experience, and over time we become used to being taught.

In Dr. Brian Katz's inquiry-based math courses, students
ask questions and create a "rigorous argument" or proof
to support a statement.

A transition from being taught to learning

Dr. Katz describes inquiry-based learning as similar to the Socratic method, after Socrates and his pupils, in which teachers and learners consider questions in the form of a debate, ruling out and reformulating questions through an aggressive discussion. But while learning mathematics through inquiry relies upon discussion and formulating questions, he points out that "inquiry-based learning is much more collaborative."

So what really happens in the classroom?

"Students prepare solutions to the questions for the day in advance of class," Dr. Katz explained, "and then our time together is spent with them presenting their solutions, which we then complete, hone, polish and connect.

"The fact that class is filled with student presentations is often the hallmark of inquiry-based learning, and is the most surprising aspect for many people," he said.

Dr. Katz points out, "mathematics courses labeled 'inquiry' tend to shift and change, as there are more questions, and then more different kinds of questions." He is excited about the findings of a recent study by the Ethnography & Evaluation Research group at the University of Colorado at Boulder, which show the method to be particularly effective in engaging women and students who previously have struggled in learning mathematics.

Last spring, one of his students wrote in her class evaluation, "It was a great transition from being taught to learning." Another said she had learned to take responsibility for her own learning.

Inquiry and Senior Inquiry

Dr. Katz has used inquiry especially in teaching his courses in discrete mathematics, real analysis and modern geometry. He has team-taught the geometry class with associate professor of mathematics Dr. Stacey Rodman, who then used inquiry-based learning in her linear algebra course. They and others in Augustana's department are thinking that exposure to at least one inquiry-based course could better prepare mathematics majors for Senior Inquiry.

In the inquiry-based classroom, students process the results of their discussion through some explicit activity. Sometimes it may be the corrections to their earlier drafts; but in the geometry course, the students have used their questions and results to put together a WikiTextbook. Two recent geometry classes also led workshops on inquiry as part of an in-service program for faculty at Western Illinois University.

It's the kind of work that makes sense for Senior Inquiry, in which students use their learning from across the liberal arts to create and complete a capstone project—research, a publication or other creative project—that informs their major field and benefits a community.

Like learning math through inquiry and experience, Senior Inquiry is not just about the project or product. It's about the process: questioning, discussing, re-formulating, connecting, honing and polishing. Inquiry asks students to take responsibility for their learning, just as Dr. Katz's student said.