I had a really good class the other day with my grade 11 precalculus students; but it didn't start that way.
We're working on analytic geometry, the link between algebra and geometry. Anyway, in this particular class I was teaching them how to find the shortest distance between a point and a line in the cartesian plane. (ASIDE: I also told them the cartesian plane is named after Rene "I Think Therefore I Am" Descartes and how he died because Queen Christina of Sweden wanted to learn calculus at 5am .... math is full of these interesting little anecdotes .... more on this in a future post. ;-))
So there we were; looking at the point P (0, 0) sitting at the origin; the line y = 2x - 10; and a little green line I called d trying to figure out how to calculate the shortest distance between the two -- the length of d. I asked the class for suggestions on how to proceed and was met with 28 blank stares.
This happens to me a lot. I ask for student input and I get the "wall of silence." At times like this I pull out an idea I learned years ago from reading this book. I was also inspired by an in-service our school had last week with Caren Cameron. She gave us a number of suggestions similar to this:
I ask all the students to take a piece of paper, fold and tear it into four pieces, and pile them up on their desks. I then ask them questions throughout the class which they have to repond to by writing their responses. A response is required, even if it's "Mr. K. I have no bloody idea what you're talking about!"
In this class I asked them to reply to two questions:
- »Is it possible to solve this problem? Do we have enough information? Yes or no.
- »If you could change or add one thing about this problem to make the question easy to answer, what would it be?
I told them they had 90 seconds to write their answers and gave them 60. I also told them I didn't want anybody's name on the papers -- while a response was required they could remain anonymous.
I collected the papers, shuffled them and gave a pile to each of two other students who picked out three or four at random while I did the same. Wow! From about 10 randomly selected replys I learned that some students thought the problem was solvable but they weren't sure how to proceed; some thought there was not enough information to answer the question; some wrote (what seemed to me) random scratches on the paper; some said "Mr. K. I have no bloody idea what you're talking about!" and a few of them said: "Well, if we know the coordinates of the point where d meets the line we could use the distance formula."
Fantastic! I now had enough feedback from the class to clear up some misunderstandings and the students came up with the key to the solution to the problem.
More than that, after using this written reply technique a couple of more times the students started volunteering verbal suggestions to my questions and one of them came up with a novel solution, using trigonometry, that I hadn't thought of!
More and more I find that the main obstacle to students' learning is shyness. Many times I've addressed this specific concern to them. I found that blogging about it helps them open up a bit, nonetheless, "the wall of silence" quickly pops up again.
After the tremendous success I had using this style of questioning I'm committed to using it more. However, I'm concerned that it will get old fast. I need more tools like this in my toolbox. I'm open to suggestions folks.
There's a deeper issue here too. How many of our students are capable of doing well in math, even going on to advanced math, but are hindered by shyness? How many students have failed math courses because of the paralyzing fear they have of asking questions, fearful of looking "stupid" in front of their classmates? How many of our students who get up the courage to ask their teachers questions, during or after class, end up nodding unknowingly because they can't maintain the inertia necessary to get the answers they need? What's the failure rate due to shyness and what can we do to overcome it?