Preliminary results of a study of students’ attempts to write fraction story problems.

1. Preliminary results of a study of students’ attempts to write fraction story problems. Cheryl J. McAllister Southeast Missouri State University November 2007

2. Contact information • A Power Point version of these slides is available at http://cstl-csm.semo.edu/mcallister/mainpage/ • Email: cjmcallister@semo.edu • Phone: 573-651-2778

3. Teachers need a deep conceptual understanding of the mathematics they will teach. • In 2001 the National Research Council published Adding it up: Helping Children Learn Mathematics. • being able to model a mathematical situation with concrete manipulatives, drawing a diagram representative of the situation, creating a word problem from which the mathematical relationship would develop, or correctly representing the situation symbolically (p.118).

4. How proficient are teachers with procedural and conceptual understanding of fractions? • Post, Harel, Behr, and Lesh (1988) • Liping Ma (1999) • Tirosh, (2000)

5. What happens when students attempt to create story problems for fraction operations? • Do identifiable error patterns occur in the problems students write? • How can instruction be improved to help students increase their conceptual understanding, evidenced by the ability to write logical, meaningful fraction story problems?

6. Problems students were given

7. Non-operation Specific Misunderstanding #1 • In writing the story problem, the writer asks a question that calls for a whole number response. • Jill has 2/3 lb of jelly beans. Her mother gives her another 4/5 lb of jelly beans. How many jelly beans does Jill have altogether? (for 2/3 + 4/5)

8. Non-operation Specific Misunderstanding #1 • About 5% of all problems written by 73 preservice teachers contained this type of error. • Happened most often in addition problems, but was found in problems written for all operations.

9. Related to previous • Instead of writing a problem for a/b, the problem is written for a/b x N, where N is an unknown natural number. • Jake ate 2/3 of his animal crackers, while Dani ate 4/5 of his. How many crackers in all did they eat? • ( written for 2/3 + 4/5) • This type of error occurred in over 7% of the samples.

10. Non-operation specific misunderstanding #2 • The writer has one or more ambiguous units or wholes in the problem. • If Mary has 4/5 piece of ribbon and adds 2/3 to it to make a bow, how much ribbon is there for the bow? • ( written for 2/3 + 4/5) • This type of error occurred in about 6% of the samples.

11. Non-operation specific misunderstanding #3 • Errors in logic • For 2 1/5 ÷ 2/3, the following problem was written: 2 1/5 pizzas were available when the class had to split it by 2/3. How many pizzas did they have to have? • What does the student mean by ‘split’?

12. Non-operation specific misunderstanding #3 • Eight out of 9 people at a movie had 3/4 a bag of popcorn left at intermission. What would the fraction number be to represent all the remaining popcorn for all the people in the movie? (This was a problem written for 3/4 x 8/9) • Logical errors occurred in over 10% of the problems and were most common in multiplication and division problems.

13. Subtraction error #1 • Instead of writing a word problem for a-b, the student writes a problem for a-(axb). • Sam has 2/3 of a lb of jelly beans. He gave 2/9 of his share of jelly beans away. How many pounds of jelly beans did he have left? (For 2/3 – 2/9) • This error occurred in over 22% of the subtraction problems written by the students in the study.

14. Multiplication error #1 • Instead of writing a problem for a x b, wrote a problem for a + b • On Monday, Bill ate 3/4 of a pizza. On Tuesday he ate 8/9 of a pizza of the same size. How much pizza did he eat in those 2 days? (For 3/4 x 8/9) • This error occurred in about 9% of the samples.

15. Division error #1 • Note: About 50% of the students could either write no division word problem or wrote something such as “What do you get when you divide 2/5 by 1/3?” • Inappropriate use of sharing concept of division (partitive division). • For 1/4 ÷ 7/9 the student wrote: You have 1/4 of a cake. You want to divide it into 7/9 of a group. How many in each group?

16. What concepts are not being mastered by students? • Fractions are relative amounts. In story problems the ‘whole’ or ‘unit’ for each fraction needs to be clear. • In addition problems the addends and in subtraction problems the minuend and the subtrahend must reference the same whole. • In multiplication problems either both fractions are the dimensions of some rectangle or one fraction factor becomes the whole for the other fraction factor.

17. What concepts are not being mastered by students? • Students don’t understand how the two context types of division problems translate to fractional situations. • Students don’t understand the mathematical difference between ‘how many ?’ and ‘how much ?’ • Students often use discrete set models in inappropriate situations: 2/3 of 7 dogs.

18. What can be done? • Combine the use of manipulative models with student attempts to write a story problem related to what they just did concretely. • Actively teach language skills related to mathematical ideas – what does it mean to take half of something? • Present students with examples of good and bad story problems to evaluate and edit.

19. Future research questions • Will these same error patterns occur in other populations? • Middle school students • Calculus students • How much of difficulty is mathematical and how much is underdeveloped language skills? • Will specific teaching techniques and methods reduce certain types of errors? • Will writing word problems help students improve their ability to solve word problems?