April 17, 2009 Colorado School of Mines

1. AN INVESTIGATION INTO THE USE OF CASE STUDIES IN THE TRAINING OF IN-SERVICE AND PRE-SERVICE MIDDLE SCHOOL MATH TEACHERS.GARY HARRIS, JERRY DWYER, ZENAIDA AGUIRRE-MUNOZ, TARA STEVENS, WARREN KOEPP,  JULI D'ANN RATHEAL April 17, 2009 Colorado School of Mines

2. Acknowledgements • A project of the West Texas Middle School Math Partnership • Funded by the National Science Foundation MSP grant # 0831420

3. Background • A small school in a rural community. The school has a history of being underperforming the state English tests and marginal on the mathematics tests. • The student body is 75% Hispanic, 5% African American, and 20% Caucasian. • Ninety percent of the students qualify for the free, or reduced, lunch program.

4. Case Study Scenario At the beginning of her third period class Ms. Hammel tells the class how pleased she was that the class did so well on their last unit test. She also tells them that she looks forward to them doing even better on next Friday’s unit test over fractions.

5. Scenario Cont’d She then pairs the students up, trying to make each pair have one student she thinks should be able to do things pretty well and the other being a student that might need some help. She then hands out a set of problems for the pairs to work on, and encourages them to make sure that they both understand each problem before moving on to the next one.

6. Scenario Cont’d As the students work on the problems Ms. Hammel walks around the class, looking over the students’ shoulders to see how they are doing, and to offer help and answer questions. She purposely stops by Johnny and Jose’s table to provide encouragement as both of them typically struggle with new content and Jose had not done well on the previous unit test. Ms. Hammel notices that they have completed the first problem correctly and praises them by saying “I like the work you two have done. Great work!”

7. Scenario Cont’d • After a few minutes, she observes that the majority of the class has successfully completed the first problem and comments to the class, “It looks like everyone is working hard on these problems.”

8. Problem In problem 3 the students have to compute . Ms. Hammel notices that Miguel and Susie have gotten the answer . She asks them how they got their answer. Miguel immediately responds that he added the two top numbers and the two bottom numbers. Of course, Ms. Hammel knew this is what had been done. What should she do next?

9. Problem Cont’d Ms. Hammel asks Susie what she thinks of that answer. Susie responds that she first thought it was supposed to be done differently but Miguel had convinced her that this was the right answer. So Ms. Hammel asks Miguel to explain how he arrived at his answer.

10. Explanation • Miguel tells her that he is a big basketball fan. This came as no surprise since Miquel is a starting guard on the school’s basketball team, and even as a sixth grader has gotten the attention of several area high school coaches. Miguel reckoned that if he hit 1 out of 2 three point shot attempts in the first half and 2 our of 3 in the second half, then he would be 3 out of 5 for the game, which is 60% for the game and that is a good percentage. • How should Ms. Hammel react to this explanation?

11. Reaction • Ms. Hammel paused for a second and then replied “Yes I see, you are absolutely correct if you are computing your 3-point shooting percentage for the whole game.” • How can she convince Miguel and Susie that is not correct and that the correct answer is ?