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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.

The question call for a whole number response.

- 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.

Ambiguous wholes in problem (similar 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? (2/3 + 4/5)

Ambiguous units or wholes

- 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)

Logic errors

- 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)

Subtraction specific error

- 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.

Multiplication specific errors

- 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)

Division specific errors

- 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)

Mixed numeral specific error

- 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)

Overarching issues

- 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.

How can we change things?

- 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.

Contact information

- 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]

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