Students with learning disabilities who struggle in mathematics have memory deficits and conceptual gaps that interfere with procedural competence (Geary, 2004). Although explicit and strategy instruction seem generally effective for struggling learners (Gersten et al., 2009), surprisingly little is known about effective fraction instruction for these students (Misquitta, 2011). One promising approach is to design instruction that specifically addresses known errors (Kelly, Gersten & Carnine, 1990); hence, it is important to better understand the error patterns of struggling students learning to solve problems with fractions.

Conceptual Perspective

Prior research on difficulties with fractions suggests that errors often involve misapplications of prior knowledge (Kelly et al., 1990; Ni & Zhou, 2005). For example, adding across numerators and denominators (e.g. 2/5 + 3/4 = 5/9) is a persistent error among upper elementary students (Byrnes & Wasik, 1991). Additionally, students sometimes find/keep equal denominators when multiplying two fractions. This error has been found among proficient middle school students (Siegler, 2011), struggling high school students (Kelly et al., 1990) and preservice elementary teachers (Author, 2008).

Recent research suggests the frequency of these errors could depend on problem characteristics. When adding fractions, students are less likely to add across when the denominators were equal, but findings are inconsistent for the impact of equal denominators on multiplying and dividing (Author, 2008; Siegler, 2011). The impact of equal denominators on errors for struggling learners is unknown.

In one of the few studies examining errors across all four operations with both simple fractions and mixed numbers, Author (2008) found that using like denominators when it was not appropriate was the most common error for preservice teachers. Other examples of misapplying fraction knowledge were also present, as were skill-related errors such as errors changing an improper fraction to a mixed number. The current study examined fraction error patterns of struggling sixth grade students.

We sought to answer the following questions: (1) Within a conceptually oriented classroom, what are the error patterns found among struggling learners engaged fraction computation? (2) How do like and unlike denominators influence these error patterns? 

Methods

Participants/ Setting

The participants were 11 students in a sixth grade classroom, in a private school for students with language-based learning differences. Four of the eleven students were considered by the school to also have learning differences in mathematics. Of the other seven, three were considered to be significantly impacted in mathematics by their language-based differences.

There was approximately 40 minutes each day of math instruction taught by a veteran teacher with over 17 years teaching experience in elementary education. Consistent with Gersten et al. (2009), the teacher blended explicit instruction with other effective practices, such as asking students to verbalize their thinking and using contexts and visual representations to help build and reinforce ideas.

Data Collection

Data for the current study was collected at the end of a unit on fractions. Students were given two separate assessments, both focusing on computational skill using fractions and mixed numbers and all four operations. One of the assessments was created by the school; the other was based on prior research designed to capture particular strategies and errors (Author, 2008). For analyses, the tests were combined. One item was removed because of redundancy both in problem type and students' solution methods, leaving 18 items.

            Two doctoral students independently coded the test items for errors. Coding was guided by the work of Author (2008), but new error codes were created as needed.  The coders met together with a third researcher to discuss and resolve all disagreements.

Results

Of the 198 answers examined, 121 (61%) were correct. Of the 77 incorrect answers, 14 were categorized as skill errors (e.g., errors multiplying whole numbers or renaming fractions), 12 were categorized miscellaneous because only one person made the error, 9 were problems left blank or incomplete, and 42 were categorized as conceptually based errors (i.e., errors based on misconceptions), described in more detail below.

Conceptually Based Error Patterns

For addition problems, one major error existed. Four students added across numerators and denominators (e.g., 4/15 + 2/3 = 6/18).

Two subtraction errors were identified. Four students subtracted the smaller numerator from the larger numerator incorrectly (e.g., 6-2/5 – 2-4/5 = 4-2/5), as if subtraction is commutative. Two students incorrectly subtracted a fraction from a whole number by treating the whole number as if it were a numerator (e.g., 5 – 3/8 = 2/8).

Students made two types of multiplication errors. Five students found/kept a common denominator and then multiplied the numerators  (e.g., 2/9 x 7/9 = 14/9 = 1 5/9).  Seven students multiplied the "like" parts of mixed numbers. Specifically, they multiplied the whole numbers and then the fractions (e.g., 3-5/7 x 4-3/8 = 12-15/56).

One conceptual division error was made. Namely, eight students found/kept a common denominator and then divided the numerators (e.g., 9/10 ÷ 3/10 = 3/10).   

Like/ Unlike Denominators

Because the subtraction errors involved only numerators, we examined how the nature of the denominators influenced errors for addition, multiplication, and division.  Of the four students who added across for addition, three did so only when denominators were not equal. Of the five students who inappropriately found/kept denominators, four did so for both like and unlike denominators. Of the eight students who found/kept a denominator for division, seven did so only when denominators were equal.

Discussion

Findings from this study showed specific error patterns in struggling learners’ fraction computation. These errors were similar in type to preservice elementary teachers' errors (Author, 2008), but they differed in frequency. Consistent with higher performing 6th graders (Siegler, 2011), these students were more likely to add across when denominators were not equal. However, they were also more likely to keep the denominator and divide the numerators when dividing with equal denominators. Given that instruction can impact error patterns (Kelly et al., 1990), one implication of the current study is that teachers may need to explicitly address these errors in the classroom.

References

Author (2008).

Byrnes, J. P. & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27(5), 777-786.

Geary, D. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37(1), 4-15.

Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79(3), 1202 -1242.

Misquitta, R. (2011). A review of the literature: Fraction instruction for struggling learners in mathematics. Learning Disabilities Research & Practice, 26(2), 109-119.

Kelly, B., Gersten, R., & Carnine, D. (1990). Student error patterns as a function of curriculum design: Teaching fractions to remedial high school students and high school students with learning disabilities. Journal of Learning Disabilities, 23(1), 23-29.

Ni, Y. & Zhou, Y. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52.

Siegler, R. S., Thompson, C. A., Sneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273-296.