Two Scenarios

Classroom One:

Teacher: (drawing a right triangle on the board) Who can tell me the name of this triangle?

Student 1: A right triangle.

Teacher: That is right, it is a right triangle because it has one right angle. We have been talking about finding the area of rectangles, and parallelograms and now I am going to show you how to find the area of a triangle. Any questions?

(No response)

Teacher: Great, so let’s move on and dig right in. Can anyone remember the formula to finding the area of a parallelogram?

Student 2: Length times width.

Teacher: Not quite, that is the area of a rectangle. What particular words am I looking for to describe the area of a parallelogram? Baaa times heee…

Student 3: Is it base times height?

Teacher: (feeling rather pleased) You are correct. Now let’s look at finding the area of a triangle. This is the base of a triangle (pointing to the particular line that represents the base) and this is the height, (indicating the imaginary line from the peak of the triangle to the base). To find the area you simply multiply the base and the height and then divide it by two. Any questions?

Student 1: Miss, so all you have to do is get the base and multiply it by the height and then half it?

Teacher: (again feeling rather pleased that the student ‘got it’) You are absolutely right. Let’s try an example and I’ll show you how it is done.

Student 3: Why do you divide it by two?

Teacher: Just watch and I will show you how to do it.

The teacher then continues to draw several examples of different triangles on the board and finds the area by applying the formula. Students are quietly copying examples in their note books. Once the teacher feels “all students” have got it, she assigns questions 1-17 from the next lesson entitled “Area of Triangles” in the student math book. She announces that any unfinished work is to be completed for homework.

The classroom described above, unfortunately, can be from any classroom or school around the country. This type of teaching and learning can be defined as ‘traditional’, where the teacher’s role is to transmit information to the student who in turn memorizes the procedure and applies the skill to repeating tasks of similar nature. It is no secret that this method of teaching and learning has proved disadvantageous and researchers of today are suggesting a different way, a more constructivist way, of teaching and learning math.

Let’s consider the next classroom.

Classroom Two:

Teacher: O.k, class, over the past few days we have been exploring how to find the area of different rectangles and parallelograms using various methods. Using what you have learned previously, I want you to create and find the area of any rectangle or parallelogram.

(Students begin working and can be seen discussing their creations with other students. During this time the teacher can be seen roaming around the room talking with individual and pairs of students about their shapes and asking them to explain how there were able to find the area, where any types of misconceptions were noted).

Teacher: (after about 15 minutes have passed) Who would like to come to the front of the class to show your shape and explain how you went about finding the area?

Student 1: (in front of the class) Well, the first thing I did was made a parallelogram on the geoboard (using the overhead goeboard, he reproduces the parallelogram Then I counted how many blocks the length was, or the base. It was five (again demonstrating how this is done). I then counted how many blocks high the parallelogram was (demonstrating how this was counted) and it is 6. I know a parallelogram is the same as a rectangle only squished a little, so I made this rectangle, a 5 x 6, out of snap cubes and then counted how many cubes I used. I used 30 snap cubes so I know the area of my parallelogram is going to be 30 units squared.

Teacher: Are there any questions or comments for Student 1? Would someone like to explain in their own words what Student 1 did to find the area?”

Student 3: Well, basically all he did was made a parallelogram on the geoboard, found the base and the height and then multiplied them together.

Student 4: Yeah, but he did not really use multiplication. He made a rectangle out of snap cubes and then counted how many blocks he used.

Student 3: Yes, but really he used multiplication to find the area because when you make a rectangle like he did out of cubes you are really making an array of 5 x 6. And 5 groups of 6 is 5, 10, 15, 20, 25, 30. See?

Student 4: Oh yeah, you are right. I can see that now. I never looked at doing this in that way before.

Student 1: Me either. I now see how you can use multiplication to find the area of the parallelogram. The parallelogram is the same as a rectangle, only it is slanted. So if you can find the area of a rectangle by multiplying length by width, than it would work the same for a parallelogram, only the names of length and width are different. Cool! Thanks Student 3.

Student 3: No problem!

Teacher: So now that we have seen a couple of different ways of getting the area of a parallelogram, I want you to check Student 1’s answer by using your method to find the area of a parallelogram, check to see if you get the same answer.

Students apply their own strategies to check the answer as the teacher walks around the room to check other people’s methods and understanding.

Teacher: So, what can we say about student 1’s answer? How can you be sure?

Student 7: Well, I drew the same parallelogram on graph paper and counted the complete units and got 30 units squared, just like student 1.

Teacher: So, we have all had a chance to come up with different ways to find the area of a rectangle and a parallelogram, all of which a perfectly right. There were a lot of great discussions happening in your groups and I am glad to see that you are helping each other. You are all becoming very great problem solvers. So, I now want you to put your thinking skills to the test. I want you to explore and find out as much as you can about the area of a triangle. I will give you about a half an hour and then we will come back as a whole class to discuss your findings.

Students began exploring the idea of trying to find the area of a triangle by using different manipulatives and through discussions with other students and the teacher. The teacher met with different students, asking questions to help them through their thinking. By the end of the half an hour, there were some pretty different discoveries made about triangles whereby all students were able to find out something about the area of a triangle and most had discovered strategies that worked for them to find the area.

Teacher: Now, that we have all had some time exploring the area of a triangle, let’s talk about what we have found out.

Student 7: Well, I found out that if I drew the triangle on a piece of grid paper, it was a little hard to count the number of squares because there are a lot of little pieces here and there.

Student 10: Yeah, I did the same thing but then I decided to make a triangle on the geoboard. I looked at it for a while and thought that maybe you would multiply the base and height but realized this would not work because that is how you get the area of a rectangle. Then I kept thinking about the triangle on my geoboard which was a right triangle and seen that if I made another triangle just like it, it would make a rectangle. Knowing how to find the area of the rectangle, I knew that if two triangles made one rectangle, then one half of the rectangle would be equal to the area of the triangle. So, I found the area of the rectangle and then cut it in half.

Student 9: Hey, that is almost the same way I did it, but I started off with the parallelogram I used from the last question that was still on my geoboard and cut it in half and realized I could get two equal triangles out of it. Knowing the area of the parallelogram I just divided it by 2 and got my answer.

Student 7: I can see that. I would have never of thought about doing it that way. I was too caught up in trying to figure out some other, more harder way. Hey let me try doing it that way.

Student 10: I’ll give you a hand, here let me show you.

Teacher: Why don’t you (Student 10) come up to the board and show the class.

Student 10: Sure.

(Student 10 continues to explain the process she went through to figure out how to find the area of the triangle on her geoboard.)

Teacher: That is wonderful. I want the rest of you to make a note of your strategy and when we come back tomorrow we will begin there. Before class is over I want you to think about this question and write a few things about it in your math journals – how is a parallelogram and a triangle alike? Different?

Students were given about five minutes to jot down a few ideas and then preceded on to the next class.

So, what is happening in Classroom Two? Well, the students are talking more, the teacher is talking and telling less, the students are constructing meaning through problem solving and coming up with their own personal strategies and methods to solve the problem. Students are learning from one another and they are not expected to simply memorize the formula or procedure. The focus of this classroom is on meaning and understanding, whereas in Classroom One, the focus was on getting the right answer and applying the correct formula. Classroom Two illustrates a reform approach to teaching and learning math. It is here that students learn the intended mathematical concepts through solving problems and discussions. The teacher’s role is to use appropriate questioning to guide student thinking to bring them to an appropriate understanding of concepts taught and to make meaningful connections among the various mathematical strands.