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Common Error Patterns in Pre-service Teachers’ Attempts at Writing Fraction Word Problems

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Common Error Patterns in Pre-service Teachers’ Attempts at Writing Fraction Word Problems

Cheryl J. McAllister

Southeast Missouri State University

Cheryl Beaver

Western Oregon University

Mathfest 2009, Portland, OR

- Premise: Teachers need a deep conceptual understanding of the mathematics they will teach.
- Question: How proficient are pre-service teachers with procedural and conceptual understanding of fractions?
- Do identifiable error patterns occur when students attempt to write story problems for fraction operations?
- How can instruction be improved to increase students conceptual understanding?

- For each of the following fraction problems
- Solve the problem showing your steps.
- Write a word problem that would be solved by doing the problem you just completed. Be sure you use correct grammar and punctuation.

- Samples collected from Preservice teachers at Western Oregon University and Southeast Missouri State.
- Independently analyzed samples and developed a list of common errors.
- Incorrectly written word problems did not seem to be correlated to computational errors.
- Errors were identified as general or related to a specific operations.

- 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)
- Most often an error for addition problems, but occurred in problems for all operations.

- 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? (2/3 + 4/5)

- 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?
(2/3 + 4/5)

- Jack has 1⅔ of his garden planted with various types of flowers that he bought at Pete’s Flowers. He has 2⅜ of a garden left to plant with flowers. What is the product of his entire garden after he finishes planting all of his flowers in the garden? (1⅔ ∙ 2⅜ )
- Eight out of 9 people at a movie had 6/7 of 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? (8/9 ∙ 6/7)

- 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?(2/3 – 2/9)

- This error occurred in over 22% of the subtraction problems written by the students in the study.

- 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? (3/4 x 8/9)

- Instead of writing a problem for a x b, wrote a problem for a + (a x b)
- Eliza has 6/7 of $100 saved up. She found a job and increased her savings by 8/9. How much money does Eliza have? (6/7 ∙8/9)

- Inappropriate use of sharing concept of division (partitive division).
- You have 1/4 of a cake. You want to divide it into 7/9 of a group. How many in each group? (1/4 ÷ 7/9)

- Instead of writing a problem for a/b ÷ c/d, student wrote a problem for a/b ÷ c.
- You have ¼ of a cake and 7 out of 9 people want cake. How much does each person get?(1/4 ÷ 7/9)

- Instead of using the mixed numeral A b/c, the student writes a word problem using A ∙ b/c.
- If I had 1 whole pie and you had 2 whole pies, how much would we have if you gave me 2/3 of yours and I gave you 3/5 of mine? (1 2/3 ∙ 2 3/5)

- In a set of samples gathered from 73 students, about 50% either wrote no division word problem or wrote “What do you get when you divide a by b?”
- Students don’t understand the difference between how many and how much.
- There are common interpretations of language different from mathematical meaning.
- Students don’t think logically about what their word problems are trying to say.

- Focus on units and wholes.
- Emphasize conceptual understanding of fractional models.
- Actively teach language skills related to mathematical ideas.
- Combine the use of manipulative models with student attempts to write a story problem.
- Ask students to write and think critically about good and bad examples of story problems.

- A Power Point version of these slides is available at
http://cstl-csm.semo.edu/mcallister/mainpage/

- Email Cheryl McAllister: [email protected]
- Email Cheryl Beaver: [email protected]