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

*. I asked the class for suggestions on how to proceed and was met with 28 blank stares.*

**d**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 **un**knowingly 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?

## 10 comments:

Good questions -- and a good strategy. I often ask for hand signals -- but that doesn't allow folks to be anonymous -- I'll be trying your written response idea in my classes.

I use portable whiteboards in my math class. I have the students work in groups of 2 and write down or draw graphs of responses to questions I ask. You are right when you say students are shy. But the resposes I get on the whiteboards are awesome!!!!

I like this a lot. I read an article recently about how the old saws about girls not being as good at math may in fact reflect their unwillingness to speak up in class and the teacher then ceasing to "see" them...

Teaching English in Italy, I had a VERY shy high schooler taking a private lesson from me (beginning level). The lessons were conversational and being a private lesson he (obviously) had to do a lot of talking. I would often ask him a question and he would sit in silence. I would patiently wait wanting to give him time to answer but it got to the point where there were a lot of silences and we were wasting time. (I'd also been trained to NEVER ask "do you understand" so I avoided that question at all costs.) Finally one day I wrote on the board at the end of the class "I don't know," "I don't understand," and "I forgot." I told him to write these sentences down (they were not new to him) and in the next class he was to start using them. It worked. From the next class he started using the phrases (I didn't even have to remind him) and we avoided that "blank wall". I knew when to give him a minute to think and when the information was new to him.

Wonderful tool!! I will have to use that. Having worked in different school settings and now home educating, I pose the question regarding math in general... along with shyness how much could lack of brain maturity be playing in the silence? I have seen many kids (myself included) accused of laziness in learning math, when in fact, I was a late bloomer in this area. When I turned 30, the concepts clicked and now I can do algebra. In high school I really wanted to understand but could not.

What do you think??

I think "brain maturity" (I usually describe this as "readiness") plays a significant role.

Pictorially think of all the math (or any domain of knowledge) you know as lying in a circle. The stuff you don't quite know but are

readyto learn lies on the circumference. (I can't help this, I'm a math teacher. ;-)) As you learn more, the things that were on the circumference of the circle move within, the circle expands and you'rereadyto learn more. The role a teacher plays, through questioning, guided discovery and guided practice, is tolead you down the garden path(I appologise to all the english teachers for the mixed metaphor. ;-)) through your ever expanding circle of knowledge.Of course none of this works if the student doesn't participate. This stuff is

hard work. Students who have a passive approach to their education do poorly if they're gifted and very poorly if they're not. Students who take an active apporach to their education (work hard) do well, and if they're gifted they do very well.Ideally the goal of education should be to simply keep expanding your circle of knowledge. The artifical age cohorts that we have created aren't always in tune with this view of education. Will Richards alluded something like this a while back. You may find it interesting.

This post is linked to from the 11th Week of the Carnival of Education. Check it out.

This is a great idea. I can't wait to give it a try. Thanks for sharing it!

It's spelled "inertia."

You can totally delete this comment after you fix it. :)

btw -- the length of d is around 4.472, yes? Or am I too old to remember this stuff properly?

Thanks for the correction. ;-)

And yes, d is approximately 4.472. We usually write the exact value as 10/sq.rt.(5) or 2*sq.rt.(5).

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