2. Long and Short Term Goals To develop a responsible and positive attitude we chose Respect for Self, Others and Learning for the long term goal. Our students are weak in number sense, numeration and problem solving. We chose Compare unit fractions in the context of solving a real world problem for the short term goal.

3. Questions Asked During the Planning of the Lesson What do students already know about fractions? What materials do we need? What will be the context of the problem? How can we design the lesson so students will discover the concepts we are teaching? How do we think students will answer the question: Which is bigger 1/3 or 1/4?

4. The Lesson Problem Mr. Bateman, the principal, is willing to share a candy bar with me. He said I could either have 1/3 or 1/4 of the candy bar. Seeing that I really like chocolate I am wondering, which is larger, 1/3 or 1/4?

5. The More Difficult Question The Hershey Bar is divided into 12 smaller pieces. Should we ask a second question – How many pieces would you get? When should we ask this question? How should we ask this question – individually or in the small groups?

6. Lesson Booklet A work booklet was designed. The question, “How many pieces would I get?” was inside the booklet on the second page. After children worked independently for about 5 minutes, the teacher would ask them to share their ideas with their group. She would suggest that thinking about the number of pieces you would get in 1/3 or 1/4 may help you decide which is the bigger part.

7. First Implementation Children first worked individually, then in pairs.

8. Students’ Ideas This is the left over. 1414 1/4 is more than 1/3. We picked ¼ because the bottom number is 4 and the other bottom number is smaller so ¼ is bigger. Some students still didn’t understand.

9. Students’ Ideas Me and my partner think the 1/3 is bigger because the less pieces the bigger the pieces are. Some children used multiple representations to show this concept and they could explain it.

10. Sharing Ideas  At the beginning of the lesson 18 children thought 1/4 was larger and 6 children thought 1/3 was larger.  After working in small groups, 4 more children thought 1/3 was larger.  Then children came to the front of the room to share their ideas.  After listening to other children during the large group discussion, 12 children seemed to be convinced that 1/3 was larger than 1/4.

11. Feedback Session Joan’s Question: How do you facilitate children’s understanding when one child in the group says 1/4 is larger and the other says 1/3 is larger and they both think they are correct?

12. Feedback Session Problem: Students did not have enough room on their worksheet. Some got confused when they opened the booklet during the independent work time and saw the second question; for others, seeing the twelve parts of the candy bar opened up their thinking. Solution: Revise the worksheet to one large sheet, keep the image of the candy bar with the twelve sections, but eliminate the second question from the paper.

13. Feedback Session Problem: Not all children used manipulatives during the independent working time. Solution: Remind children that they can use any materials in the basket to help them solve the problem.

14. Feedback Session Problem: Children had a hard time explaining their correct ideas. Solution: Teacher could help the children choose words when necessary so the rest of the class can better understand what the child is saying.

15. Feedback Session Problem: With pairs, little progress is made if both children are confused. Solution: Make the groups larger so there is a better chance that one of the children will have productive ideas.

16. Second Implementation At first, all students said 1/4 was larger than 1/3.

17. Second Implementation Their ideas began to change during small group work.

18. Second Implementation At the beginning of the class discussion, some students were still convinced that 1/4 was larger. 1414 1313 1414 is larger 1313 is smaller 1/4 is bigger or larger because there is 4 pieces. 1/3 is smaller because there is 3 pieces.

19. Second Implementation In this group, all the students agreed that 1/3 was larger, but one child said 1/3 was 9 pieces and another child said it was 4.

20. The Turning Point After this group argued that just because the bottom number is biggest it doesn’t mean you get more, students began to raise their hand to say they changed their mind. 1414 1313 1 whole We think it is 1/3 is because you want as much as you can get. Just because the bottom number is biggest doesn’t mean you get more..

21. The Turning Point At the end of class all except one student agreed that 1/3 was bigger than 1/4. This student said, “If you subtract 1 from 1/3 it would be two big pieces. If you subtract 1 from 4, it would be 3.” He was still thinking about whole numbers.

22. Feedback The larger student work groups seemed to stimulate a better exchange of information and discussion and prevented the “stalled” groups we had observed in the first setting. The elimination of the second page and additional question did away with the confusion that it had caused in the AM session. We found that the paper manipulatives helped some children and confused others.

23. Reflections I learned from watching other teachers teach and watching students learn. I saw the importance of giving enough time for students to think-- whereas teachers often rush to give answers. I saw the value in encouraging students to share among themselves and explain their thinking-- whereas teachers often do most of the talking. These observations made an impression on me that will affect how I approach both the teaching and learning in my classroom.

24. Reflections I really enjoyed watching all of us contributing so many ideas and then being able to pull it all together and focus on our final lesson study plan. It provided an opportunity to encourage and support one another as we tried new instructional methods aligned with the teaching and learning standards from the Michigan Curriculum Framework and NCTM. It is never easy to find the amount of extra time needed to accomplish such a task as we took on but we did.