Presentation:

Students can learn how to create new concepts and explore ideas. Inquiry-based classes can provide a whole class of students with a quasi-research experience by asking students to tackle questions that are new to them, yet not necessarily new to the world. By carefully designing a sequence of challenge questions, students systematically confront a predictable range of research experiences. Students can develop attitudes of self-confidence and personal reliance. They can develop intellectual stamina and the sense that by taking deliberate steps they can personally solve unknown problems that arise in their academic and personal worlds. The challenge for the instructor is to design a sequence of questions such that the students have the experience of discovering and exploring ideas on their own. This technique is well developed in mathematics courses that teach students how to prove theorems independently, but the technique is generally applicable to most fields, including other liberal arts subjects, fine arts and topics treated in professional schools.
This session began with the participants working on some puzzles in groups of three or four to put them in the frame of mind for discovering ideas on their own. Here are the puzzles, which prompted lively discussion.

A Commuter Fly

A passenger train traveling at a steady 50 miles per hour left Austin, Texas at 12:00 noon bound for Dallas, exactly 210 miles away. At the same instant, a freight train traveling at 20 miles per hour left Dallas headed for Austin on the same track. At this same high noon, a fly leaped from the nose of the passenger train and flew along the track at 100 miles per hour. When the fly touched the nose of the oncoming freight train, she turned and flew back along the track at 100 miles per hour toward the passenger train. When she reached the nose of the passenger train, she instantly turned and flew back toward the freight train. She continued turning and flying until the expected tragedy occurred—she was squashed as the trains collided head on. How far did she fly before being squashed?

A Fair Fare

Three strangers, Bob, Mary, and Ivan, meet at a taxi stand and decide to share a cab to cut down the cost. They have different destinations, but all the destinations are right on the highway leading from the airport, so no circuitous driving is required. Bob’s destination is 10 miles away, Mary’s is 20 miles, and Ivan’s is 30 miles. The taxi costs $1.50 per mile including the tip regardless of the number of passengers. How much should each person pay? (Note: There is more than one way of looking at this situation.)

A Shaky Story

Stacy and Sam Smyth were known for throwing a heck of a good party. At one of their wild gatherings, five couples were present (including the Smyths). The attendees were cordial, and some even shook hands with other guests. Although we have no idea who shook hands with whom, we do know that no one shook hands with themselves and no one shook hands with his or her own spouse. Given these facts, a guest might not shake anyone’s hand or might shake as many as eight other people’s hands. At midnight, Sam Smyth gathered the crowd and asked the nine other people how many hands each of them had shaken. Much to Sam’s amazement, each person gave a different answer.
That is, someone didn’t shake any hands, someone else shook one hand, someone else shook two hands, someone else shook three hands, and so forth, down to the last person, who shook eight hands. Given this outcome, determine the exact number of hands that Stacy Smyth shook.

An Average Square

Imagine a checkerboard that extends forever in each direction. Each square shares a side with four surrounding squares.
Suppose you write a natural number, {1, 2, 3,}, in each square such that the number in each square is the average of the surrounding four. What can you conclude about the numbers written on the infinite checkerboard?

Blindfold Equality

You are blindfolded and are wearing gloves. In front of you on a table are some pennies: some heads up, and some tails up. You are told how many of the pennies are heads up, but you cannot feel which are heads up or tails up. Your challenge is to slide the coins around and turn some of the pennies over in such a way that you create two groups in front of you where each group has the same number of pennies that are heads up—all without looking or being able to determine which side is up for any coin.

Participants suggested several reasonable solutions to the Fair Fare puzzle (and an unreasonable solution or two). The various solutions and attitudes about the solutions reflected a variety of attitudes and perspectives that the participants brought to the discussion.

Discussion:

After the introduction, session participants discussed some of the basic roles of education, including the fundamental goal of teaching students to adopt the habit of thinking for themselves. All classes should include contributions to broad educational objectives. These broad educational goals were discussed in the session, especially through a Hologram Metaphor, a discussion of teaching creativity, and a discussion of how to bring research-like experiences to the students through daily experiences in the classroom.

The Hologram Metaphor

One metaphor for education is the jigsaw puzzle view—the English piece, mathematics piece, science piece, history piece, art piece, and so on fit together to create a whole picture. Another metaphor comes from a feature of holographic film. If you cut out a piece of a holographic film and project it, you will see the whole picture but with less detail than you see when you project the whole film. Each course and educational experience for undergraduates might be thought of as a piece of holographic film. Each course or project can contribute to developing the suite of skills, attitudes, and perspectives that create a whole person. Every part adds detail and a nuance that colors and refines the whole. Mathematics courses should help create better artists; biology courses should help create better historians; and history classes should help create better engineers.

Teaching Creativity

Frequently people perceive the discovery of new, creative ideas as a mysterious result of magical inspiration.  But that concept has two defects: first, it’s wrong; and, second, it’s useless.
Coming up with new creative ideas is not a special gift of a select few. Students can learn habits of effective thinking—how to solve problems that arise in their lives, how to become more creative, and how to see and appreciate their world with greater clarity. Courses can be structured to systematically foster these developments. Inquiry-based learning in courses requires students to meet the challenge of developing the habit of creative thinking. Session leader Starbird presented a list of ten ways to create creativity.

Research in the Classroom

Individual research projects for undergraduates present many challenges for faculty members and for institutions. Frequently finding and directing an individual research project is challenging, time-consuming, and the resulting experiences for the students are variable. Sometimes the experience is life-changing, exciting, and intellectually satisfying; sometimes the project turns out to have some unexpected defect, such as not yielding sufficient results or a sufficient variety of challenges to be of great interest. These problems are inherent in research but present special obstacles to undergraduate experiences. In almost all cases, finding and overseeing an individual research project is a time-consuming effort for the faculty member. Large universities do not have the faculty resources to provide individual undergraduate research experiences for more than a small percentage of their undergraduates.

Recommendations:

• Develop inquiry-based classes that systematically provide an entire class of students with quasi-research experience in which students tackle questions that are new to them, yet not necessarily new to the world. In many ways this experience is often superior to individual research projects because the careful control of the challenge questions gives students a predictable range of research experiences. Students can develop attitudes of self-confidence and personal reliance, intellectual stamina, and the sense that they can personally undertake the task of solving unknown problems that arise in their academic and personal worlds.
• Faculty should design courses that are creative and provide life-changing experiences for all students.