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Re: lecture topics
Posted:
Jun 30, 1995 10:39 AM
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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: lpeterson@bnk1.bnkst.edu
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