Social constructivism in practice: case study of an elementary school's mathematics program

Focus on Learning Problems in Mathematics, Wntr-Spring, 2005 by Roland Pourdavood, Lawrence V. Svec, Lynn M. Cowen

Student F: First, the answer is $14.00.

Principal: Does everyone agree with this answer. [Most students agree, Student G does not agree.]

Student G: I got $10.00.

Principal: Any other solutions?

Student H: I got $11.00.

Principal: Other answers? [There are no other answers.]

Principal: Ok, so what do we know? Can all the answers be correct?

Student A: No, because this is math and there is usually only one right answer.

Principal: [laughing], Not always but maybe in this case. We will have to find out if we have more than one 'right' answer. You guys know how I feel about 'one right answer' in math. It isn't about the answer but the way you think about the answer.

Students: Yeah, we remember.

Principal: Ok, [calls student F's name], share your thinking with us. How did you arrive at your solution?

Student F: Well, Jason makes $4.00 an hour--just like your picture on the board. He works for 3 and 1/2 hours. Three hours times 4 is 12.

Principal: Why did you multiply by four?

Student F: [Drawing three circles on the board] These are hours. For every hour, Jason earns four dollars [student puts $4.00 inside each circle.] You can add 4 4 4, or you could just multiply 3 times 4 and get the answer which is 12.

Student B: But you said the answer was 14.

Student F: I am not finished. Then, Jason works more than 3 hours. He works 30 minutes more-until 4:00. So, then thirty minutes is 1/2 of an hour, so I halved the four dollars. That is two dollars. I added the two dollars to 12 and got 14. So that's that. [Student F returns to place in the circle. There is a silence as students mull over the information.]

Principal: Well, what are you thinking? I smell smoke, so some of you are really thinking about [Student F's name] solution.

Student G: Well, I am changing my answer. I agree with [student F's name]. I forgot to add the money for the half hour.

Principal: OK, you changed your answer. You found a mistake, good. What about you, [calls student H's name]? What do you think?

Student H: Well, now I am not sure. When I did it, I thought that I had the answer, now, I don't know.

Principal: Why don't you show us your solution? [Student H moves to the easel.]

Student H: I think Jason earned $10.00 because he worked 2 1/2 hours, not 3 1/2.

Principal: How did you figure it was 2 1/2 hours? [Student H reaches for the clock and sets the time for 1:30.]

Student H: [Moving hands of the clock] 1:30 to 2:30 is one hour. 2:30 to 3:30 is another hour. 3:30 to 4:00 is thirty minutes or 1/2 hour. So Jason earned $10.00. I figured the money just like [student F's name] except he had Jason working one more hour. I am not sure if I counted the hours right.

Principal: What do all of you think? We have a difference of one hour. How should we solve this difference of hours Jason worked?

Student I: [Calls student F], show me how you counted the hours.

Student C: Yeah, use the clock, like [student H]

Student F: Ok, watch. [Student F takes the clock and moves the hands.] Begin at 1:30-that's one hour, 2:30, that's 2 hours, 3:30-3 hours, 4:00-thirty minutes more. 3 1/2 hours. [Again, students are silent.]