Kent wrote: <<There are so many lecture topics it would be difficult to name them.
Example: the introduction of triangle trig. You can teach the definitions of the sine, cosine and tangent for common angles between 0 and 90 degrees-- also other angles using the calculator and do a few short applications (my first lecture in trig). Of course this, like all of my lectures is a very give and take situation with the odd irrelevant story tossed in every 10 minutes of so (these are children were dealing with after all).>>
Thanks, Kent,this is a good example. I understand that there are many topics in math for which one can stand up and deliver the current paradigm. I know that there are wonderfully entertaining teachers who can keep students listening too. I had some of those wonderful teachers myself. Much of the math I studied was delivered by lecture and question and answer format, and I kept going with it to complete an undergraduate degree in mathematics and went on into engineering. So I realize that basically the whole curriculum can be told. But I was specifically looking for those concepts and skills which students cannot possibly construct for themselves and so must be told (in as engaging and clear a manner as possible).
Your example of trig is a good one for me to think about. I agree that the students will not be able to construct the terms sine, cosine and tangent and that they must be told those words so that they can communicate within the mathematics community. When should they be told those terms? Up front-- followed by the definitions, examples of how to manipulate them, and then problems that can be solved using them? I have done that during my 30 years of classroom teaching-many times. But I also felt that somehow the students were going through the motions without a real sense of what those words represented and without a sense of control over the concepts involved and an appreciation of the power of these relationships.
As I thought about this I tried to come up with ways that students could understand the relationships between sides and angles in triangles before I delivered the labels. I could direct an investigation by giving them a math laboratory sheet and some right triangles to measure and record the measures of sides and angles and the ratios of the sides taken two at a time. The triangles could be carefully constructed by me in preparation(out of junk cardboard) so that there were 2 or 3 sets of several similar triangles in each lab packet. Then the students in groups could be directed to measure and then compare the values of the ratios of the legs to the hypotenuse or the legs to each other. (I have actually done this lesson.) What would they see? The discussion that followed might be very rich as the students were able to tell what there was about the triangles which wound up with equal ratios. How could this property of similar triangles be used to our advantage? I could even include the task of finding the ratios of certain special angles 45 ,60 and 30 degrees. I could then share the trig tables with them or a calculator with trig functions and say "Look, it's all been done for you already." Then you can have some fun and say, "Does this work with other triangles? Are there relationships between the sides and the angles of non-right triangles? What are they?"
Or I could say- here are some assorted right triangles. What can you find out about the relationships between the sides and the angles of these triangles? --and send them off to cooperatively explore and verify and question and report their findings.
Or I could give them one of the problems surreptitiously taken from the back of the chapter on trig in their text and ask them to do what it takes to solve it.
This is what I'm trying to work out for myself. Although I am no longer in the public school classroom, I still consider myself an "in the trenches teacher" and I am still interested in working to improve math teaching with the teachers I visit in their classrooms.
This discussion has really gotten me hooked. I apologize if I have been writing too much or coming off sounding like a lecturer lecturing about cooperative learning. There's a lot of passion here for me. I'm trying to share my thinking where it is now and really want to engage in a cooperative effort to further construct my own idea of optimal math teaching. I don't think the Standards say it all. I think they are a great beginning-- with a lot of problems still to be worked out.
I'm going home now, honest. Once again, happy Fourth! Lucille
Lucille L. Peterson Math Leadership Program Bank Street Graduate School of Education Tel: 212-875-4665 610 West 112th Street Fax: 212-875-4753 New York, NY 10025 E-mail: firstname.lastname@example.org