Fall 2009

Pushing the Boundaries
Michael S. Palmer, Associate Professor, Assistant Director, Teaching Resource Center; Chemistry

“Je le vois, mais je ne le crois pas.”

Georg Cantor (1845-1918), a German mathematician, penned these words to fellow mathematician Richard Dedekind. Cantor had delved into the depths of infinity, a problem long ignored by academics, to discover a property that even he found shocking. “I see it, but I don’t believe it,” he wrote hastily. What Cantor saw was an elegant set of mathematical arguments proving that the number of points contained in a line segment is identical to the number of points contained in the face of a square. This infinite number of points, known as the continuum, is also identical to the number of points contained in a cube, or for that matter any n-dimensional object.

Though difficult to believe, Cantor’s simple, undeniably creative solution makes understanding this curious aspect of infinity a bit more palatable. At least, as instructor of a University Seminar on infinity, this was my hope. The course, Falling from Infinity, invited 18 first- and second-year students to explore and imagine the infinite. Drawing on a diverse set of perspectives — literary, artistic, mathematical, scientific, religious, philosophical — we spent a semester grappling with uncountable numbers, immeasurable spaces, and unending times. Cantor’s continuum problem was one of many challenges we encountered. While it was not entirely necessary for students to believe all of Cantor’s ideas — after all, even he didn’t believe them initially — I did want students to see, or more precisely, understand the ideas. Telling them, and even showing them, would not be enough. I needed to provide a means for them to think about the concepts, play with them, and discover the mind-bending features of infinity for themselves.

Throughout the course, students kept a reflective journal, where they tracked their voyage through the infinite, posed questions, suggested answers, and documented how their questions and answers changed over time. For some journal entries, students chose the topics they explored as well as the direction they took. For others, they first completed a short learning activity and then reflected on the activity. The reflective activities ranged from conducting interviews to creating a photo essay. I asked students tackling Cantor’s continuum problem to complete a Peer Discovery exercise. Here’s what I asked them to do:

While you may not yet believe it, I hope you are starting to see that Cantor’s ideas completely changed the landscape of infinity. Do you see it? Do you believe it?

As part of your reflective journal entry this week, I’d like you to spend some time trying to help one of your peers — someone not in the class — understand Cantor’s infinites. Show them how two infinite constructs can be equivalent in size, even though they don’t look equivalent on the surface. Don’t try to make them believe; help them understand the arguments. Then reflect on the experience: what was the person’s initial reaction, what parts were hard for you to explain, did the exercise help you understand (or believe)?

Following the exercise, one student wrote in her reflective journal, “I thought I had a handle on Cantor’s ideas — it all seemed so straightforward in class — but I didn’t even know how to begin teaching them. The questions my ‘student’ asked quickly revealed all the holes in my understanding. Working through Cantor’s ideas together helped us both better understand his infinities.”

Embedded in every discipline are similarly beautiful and elegant ideas that belie intuition and common sense. These ideas bring us to the limits of our cognitive space, to the boundaries where our current beliefs and understandings begin to break down. As academicians, we learn to identify, seek out and embrace these constructive ambiguities, these productive contradictions, because we know they lead to learning and new knowledge. Encouraging our students to ignore the barriers they sense and to step beyond the edge of their current understanding, however, is challenging. For students to learn, they must push the boundaries themselves. Reflective learning activities, like the Peer Discovery exercise described above, offer students opportunities to ask, “Do I see it? Do I believe it?” Such exercises allow students to engage with the questions and answer from their own experience, providing both students and instructor meaningful feedback about their learning.