If your math class is snoozing, try exploring the paradoxes that math touches on. One way is to just listen carefully to student’s questions, and see if there are any “big ideas” hidden in the question. Here’s an example of such an experience.
Recently I was tutoring a girl on the concept of rounding off decimals. This might sound dull, but this girl asked a question that, if you think about it, touches on the concept of infinity.
The student was looking at the number line and noticing how her textbook enlarged one small segment on it to display the problem, which was to round 2.72 to the nearest tenth.
The textbook’s number line took a “magnifier-approach,” blowing up only the section from 2.7 to 2.8, but showing all of the hundredth’s places in between.
The girl took a hard look at that and said, “So couldn’t you also take the space between something like 2.73 and 2.74, and blow that up?”
I asked her to explain. She said that if this smaller space were also ‘blown up’ or expanded, the new number line would display even smaller numbers, like 2.731, 2.732, 2.733, with 2.735 in the middle.
I told her that you could do this.
Then she asked, “Couldn’t you go even further?” Meaning, it turns out, can you then take an ever smaller part of the number line, such as the space from 2.731 and 2.732, and blow up that space?
I said you could.
She said, “Can you just keep doing it forever?”
I said you could.
She paused, then said: “I just don’t get that idea of ‘forever.'”
That was the moment …
I said it’s a hard idea to understand forever. But it’s an interesting thing to think about. I didn’t mention it to her, but later I realized that this 4th grader was essentially intrigued by the same question that captivated the mathematician/philosopher Blaise Pascal back about 300 years ago. Namely that humans are surrounded by two different infinities: the infinity of hugeness and the infinity of smallness. For more on Pascal’s words on this amazing matter, see http://www.leibniz-translations.com/pascal.htm
In any case, following this girl’s thought led her and me into a whole discussion about forever and infinity and the edge of the universe, and the conversation would have literally gone on “forever,” but eventually it was time to stop.
In our daily attempts to teach math we sometimes neglect to mention that math touches on the infinite, as an asymptote approaches a curve, you might say. Taking time, once in a while, to explore the infinite can make your class or tutoring session come very much alive.
And if you’d like to see a book that encourages “edgy” math questions by both students and teachers, check this out: http://www.amazon.com/Good-Questions-Math-Teaching-Them/dp/0941355519 It’s a book devoted to generating and listening to startling math questions. And it shows how these questions take a jackhammer to old musty classrooms, letting the light of curisoity and exploration get their day in the sun.