Mr. Goldberg has been teaching a unit on fractions. He was pleased that all of his students seemed to quickly
master adding and subtracting two fractions. However, when he began teaching the students how to multiply
fractions, a small number of them did not readily learn the content. But after a quick mini-lesson, it appears that
all but three students seem to understand how to solve the problems. One of these students, Wyatt, seems to be
really struggling. Mr. Goldberg determines that he needs to collect some data to help him decide what type of error
Wyatt is making so that he can provide appropriate instruction to help Wyatt be successful. To do so, he decides to
evaluate Wyatt’s most recent independent classroom assignment.
! Assignment
- Read the Introduction.
- Read the STAR Sheets.
- Score Wyatt’s classroom assignment below by marking each incorrect digit.
- Review Wyatt’s scored assignment sheet.
a. Describe Wyatt’s error pattern.
b. Discuss any exceptions to this error pattern. What might these indicate?
- Based on Wyatt’s error pattern, which of the two strategies described in the Addressing Error
Patterns STAR Sheet would you recommend that Mr. Goldberg use to remediate this error?
Explain your response.
Full Answer Section
4. Review Wyatt’s Scored Assignment Sheet:
a. Describe Wyatt’s Error Pattern:
Based on the hypothetical assignment above, Wyatt's primary error pattern appears to be adding the numerators together instead of multiplying them. He correctly multiplies the denominators.
For example:
- In problem 2 (32×51), he gets 3×52+1=153.
- In problem 3 (53×72), he gets 5×73+2=355.
- This pattern continues in most of the incorrect answers.
b. Discuss any exceptions to this error pattern. What might these indicate?
In the hypothetical assignment, problem 1 (21×43=83) is correct. This could indicate a few possibilities:
- Lucky Guess: Wyatt might have simply guessed correctly on this one.
- Isolated Understanding: He might have, by chance, applied the correct rule in this specific instance but doesn't have a consistent understanding.
- Simpler Numbers: The numbers in this problem are relatively small and straightforward, which might have inadvertently led him to the correct procedure.
The fact that there is at least one correct answer suggests that Wyatt might have some fleeting understanding of the correct procedure or that his misconception isn't absolute. It's important not to dismiss this correct answer but to consider why it might be an exception.
5. Based on Wyatt’s error pattern, which of the two strategies described in the Addressing Error Patterns STAR Sheet would you recommend that Mr. Goldberg use to remediate this error? Explain your response.
Without knowing the exact strategies described in the "Addressing Error Patterns STAR Sheet," I can suggest two general approaches that would likely align with effective remediation for this type of error:
Strategy Recommendation 1: Conceptual Understanding through Visual Models
- Explanation: Wyatt seems to be operating on a procedural level without a strong conceptual understanding of what it means to multiply fractions. Using visual models can help him internalize the concept. Mr. Goldberg could use:
- Area Models: Representing each fraction as a part of a whole and then showing the overlap when multiplying to visualize the product. For example, to multiply 21×43, draw a rectangle, shade half of it horizontally, and then shade three-quarters of it vertically. The area where the shading overlaps represents 83. This visually demonstrates that we are finding a "fraction of a fraction."
- Number Lines: While perhaps less intuitive for multiplication than area models, number lines can still show repeated addition of a fractional amount (though this is more directly related to multiplication by a whole number).
- Manipulatives: Using fraction bars or circles to physically represent the fractions and then combine them to model multiplication.
Reasoning: This strategy directly addresses the likely lack of conceptual understanding. By visualizing what fraction multiplication represents, Wyatt can move beyond simply applying an incorrect rule (adding numerators) and understand why we multiply numerators and denominators.
Strategy Recommendation 2: Explicitly Comparing and Contrasting Addition and Multiplication of Fractions
-
Explanation: Since Mr. Goldberg noted that Wyatt seemed to master adding fractions, it's possible that Wyatt is confusing the rules for the two operations. Mr. Goldberg should explicitly compare and contrast the procedures for adding and multiplying fractions, focusing on:
- When to find a common denominator: Emphasize that a common denominator is necessary for addition and subtraction but not for multiplication.
- What to do with the numerators: Clearly state that in addition/subtraction, we add/subtract the numerators (once a common denominator is established), while in multiplication, we multiply the numerators directly.
- What to do with the denominators: Clearly state that in addition/subtraction, the denominator remains the same (after finding a common denominator), while in multiplication, we multiply the denominators directly.
-
Activity: Mr. Goldberg could give Wyatt pairs of problems (one addition, one multiplication with the same fractions) and ask him to solve both, explaining his steps for each. This would force Wyatt to consciously differentiate between the two operations.
Reasoning: This strategy directly targets the potential confusion between the rules for adding and multiplying fractions. By explicitly highlighting the differences and providing opportunities for comparison, Wyatt can learn to apply the correct procedure for each operation.
The best approach might involve a combination of both strategies. Starting with visual models to build conceptual understanding and then explicitly comparing and contrasting the rules for addition and multiplication could provide a strong foundation for Wyatt to overcome his error pattern. Mr. Goldberg should also provide immediate and specific feedback as Wyatt practices the correct procedure.