Prospective Teachers’ Knowledge of Addition and Division of Fractions

1. Prospective Teachers’ Knowledge of Addition and Division of Fractions R. Elaine Carbone, Ed. D. Clarion University, USA Patricia A. Eaton, Ph. D. Stranmillis University College, Belfast, Northern Ireland

2. Research supports that prospective teachers need to revisit concepts of rational numbers. • Examples from three decades • of research show that • prospective teachers exhibit difficulties • with rational numbers.

3. Focus on fractions • Parker, M. 1996 • Prospective Teachers Knowledge of Referent in Two Complex Areas: Addition of Fractions and Percent Greater Than 100, Annual Meeting of AERA, New York, NY. • Uses Problem posing • To develop students’ understanding of concepts rather than assessing memorized procedures

4. Our research focus • Addition of fractions • Do prospective teachers use a referent whole when the sum of two fractions is greater than one? • Division of fractions • Can prospective teachers pose a valid real life problem for division of fractions?

5. The Population • 34 from the United States • Elementary education students in final weeks of first content course in mathematics • 2 freshmen • 20 sophomores • 7 juniors • 1 senior • 4 post graduates • 6 from Northern Ireland • Elementary education students with a concentration in mathematics • All seniors

6. The Task Write a story problem where students in the elementary grades would add ¾ + ½ to complete the problem. • Write a story problem that shows the meaning of 2 ½ ÷ ½ . Problems were classified as acceptable or unacceptable.

7. Results for Addition problemWrite a story problem that shows the meaning of ½ + ¾ • Acceptable responses • 82% of US students • 28 out of 34 • 20% on pretest (8) • 0% of Northern Ireland students • 0 out of 6

8. Addition of FractionsBasic Difficulties • If established measurement units were not used in the problem, students did not clearly indicate that there were two wholes exactly the same size and shape.

9. Acceptable Response US You are helping your grandma make cupcakes. She tells you to add ½ cup of flour, then add ¾ cup more. How much flour did you put in total?

10. Acceptable Response US John ran ½ mile in the race for life on Saturday. Sue ran ¾ mile, how many miles did the(y) run all together?

11. Unacceptable response US One student ate ½ of a pizza and another student ate ¾ of another pizza. How much pizza did they eat total?

12. Unacceptable response (NI) Susan ate ½ of a strawberry cake and ¾ of a chocolate cake. How much cake did she eat altogether?

13. Students did not consider that the sum of the two fractions is greater than one when writing the problems. Another difficulty

14. Unacceptable response US All the students went to lunch and their was a pizza there for them to eat. If the girls in the 1st grade class ate ½ of the pizza and the boys in the 1st grade class ate ¾ of the pizza. How much pizza did the whole class eat together?

15. Unacceptable response US In Mrs. C’s class ½ of her students got A’s on the test, and ¾ of her student’s got B’s. How many student’s got A’s and B’s.

16. What we learned • US students improved • 8 students proficient prior to instruction (20%) • 28 students proficient after instruction (82%) • Discussing the referent whole is important in teaching addition of fractions especially when the result is greater than one. • NI will address the referent whole in their future instruction of addition of fractions.

17. Results for Division Problem Write a story problem that shows the meaning of 2 ½ ÷ ½ Acceptable responses • 62% of US students • 21 out of 34 • no pretest • 83% of Northern Ireland students • 5 out of 6

18. Majority of students in both countries could write an acceptable division problem.

19. Acceptable responses NI If there are 2 ½ pizzas and each child received ½ pizza. How many children share it? Tony buys 2 ½ pizzas. He divides it out so that each person gets ½ a pizza each. How many people does he share his pizza with?

20. Acceptable response (NI) Simon has two and half pizzas. He wants to share his pizzas with his friends and he wants to give each person a half of a pizza. How many friends can he share with?

21. Acceptable response NI There are two and a half apples, half an apple is going to be given to each child, how many children get a piece of apple for their Health Break?

22. A difficulty regarding division of fractions was the misconception that division meant to take half of 2 ½.

23. Unacceptable response US Mrs. Smith has a collection of 2 ½ pencils. If she were to give ½ of her collection away How many pencils would she have left?

24. Unacceptable response US You are making bows for a wedding. If the amount of ribbon is 2 ½ yards, how manybows can you make with ½ yard of ribbon? ½ yard all of the Acceptable

25. Acceptable US You have 2 whole pizzas of the same size and exactly ½ of a 3rd pizza that was also the same size. How many halves do you have altogether?

26. Acceptable US If you were selecting a team for a 2 ½ mile relay race, and each member is only permitted to run ½ of a mile. How many runners do you need to select?

27. What we learned:To improve student understanding we want to improve our teaching. • Northern Ireland • Will be addressing referent wholes when teaching addition of fractions. • US • Will stress that dividing by ½ is not the same as taking half of a group.