Archived: The Educational System in the United States: Case Study Findings

Teaching at Hamilton High
The classes of the teachers that were observed in this school differ in profound
ways from those at South Central Vocational High School. Most students were
present, and the teachers were able to deal with subject matter that was more
complex, to present material in a more rigorous fashion, and to employ teaching
techniques that were interactive. The example that follows is that of an experi-
enced, popular teacher teaching an ‘‘average’’ geometry class.
In Mr. G.’s second-period geometry class there are 25 students in the class,
13 females and 12 males. He has been teaching over 20 years, and he contin-
ues to express enthusiasm for math and for teaching. Answers to the previous
night’s homework are projected onto a screen in one corner of the room. Stu-
dents check to see if their homework is correct for the first 5 minutes. The
class is fast-paced. Over the course of the 42-minute period, Mr. G. reviews
the homework as well as hands out—and has the students work through—
three worksheets. What follows highlights an extended segment of class time.

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203
Teacher: Draw some pictures on the bottom. Draw the triangles and write the
values, using the triangles. Do not just punch a button on your calculator. That
is not the objective. I want to get the meaning, not the numerical value, at
this point.
The teacher gives them about 2 minutes to do what he requests, and all are
working intently.
Teacher: OK, let’s take a look at these here. For a sign of 30 degrees, what
triangle should I draw?
Student: 30–60–90.
Teacher: 30–60–90. How do I know it has to be 30–60–90?
Student: Because one of the angles is 30.
Teacher: Just because one of the angles is 30?
Student: Because these are right triangles.
Teacher: OK. What we’re doing only applies to right triangles, so of course
it’s a 30–60–90. So what are the lengths?
Student: [can’t hear.]
Teacher: There’s an idea, What shall we call them then? What would this be?
X?
Student: [can’t hear.]
Teacher: Uh, I don’t think so. The leg is X2 x 3, and this is X x 2. So what
is the sign of a 30-degree angle? How would that go? (Repeats student’s re-
sponse.) Opposite leg would be X over hypotenuse, X over 2X?
Student: One half.
Teacher: One half. Wait a minute. Wait a minute. What if we knew the sides?
In other words, what if I knew this side was like 6? Then what would happen?
[Repeats student’s explanation.] Let’s see, this would be 62 x 3, and this
would be?
Student: 12
Teacher: And then what would the sign of 30 degrees be? It would be 6 over
12 which is? . . . . one half. [Students are feeding him the answers, but his
voice is loud and carrying the thrust of the lesson.] Wait a minute, wait a
minute, what if this were 48? The hypotenuse would be?
Student: 96.
Teacher: 96, and the sign of 30 degrees would be? 48 over 96? That’s . . . .
Student: One half.

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Teacher: You mean it doesn’t matter how big the triangle is? It’s always the
same?
Student: Yes.
Teacher: Ah, why? Why do they have to be the same every time?
Student: Because the hypotenuse is a ratio.
Teacher: Ah, it’s a ratio. Well, why do ratios always have to be the same here?
Say again.
Student: They are all similar triangles.
Teacher: All similar triangles. Of 
course 
the ratios, the corresponding ratios,
are the same. Of course they are. They’re similar triangles. So what does that
mean? Which size triangle can I use to get the sign? Shall I pick this one or
this one or this one?
Student: Any one.
Teacher: 
Any 
one. So could I even make it real easy, and make this length
one? Would that do?
Students are shaking their heads in the affirmative.
Teacher: Sure, if that’s one, this would be square to 3, and this would be 2,
and the sign would be a half. Of course it doesn’t matter. The ratios are the
same. That’s the whole idea right there. They’re similar triangles. The ratios
are the same.
This teacher orchestrates the class so that the logic of the procedures is apparent.
Students provide answers but also explanations. However, not insignificantly, the
ones who answer his questions follow that logic and generally provide the correct
responses. The interaction works for both teacher and students (or, at the least,
for the ones responding).