**Jonah Mutua**

*Jonah Mutua is a Ph.D. Candidate in the Developmental Education program at Texas State University with a specialization in developmental math. He earned his M.S. from the University of Texas at Dallas and taught mathematics for Dallas Community Colleges (2011-2012) and Huston-Tillotson University in Austin (2012-present). His research interests involve finding better and practical ways to teach fractions and quadratic equations to college algebra students.*

**Introduction**

Students enrolled in developmental mathematic courses experience various challenges in learning mathematics in general, and, in particular, they struggle in solving fractions problems. The American National Mathematics Advisory report (2008) states that “difficulty with the learning of fractions is pervasive and is a major *obstacle* to further progress in mathematics and other domains dependent on mathematics, including algebra” (p. 28).

Seigler and Pyke (2012) state that some of the properties of positive whole numbers arithmetic operations–which are taught before fractions are introduced to students–are not transferable in solving fraction problems, which is a major source of confusion among students. For example, positive whole numbers never increase with division, never decrease with multiplication, and they have unique successors. None of these properties are applicable when solving fraction problems. Students often find it difficult to learn and apply new properties required to handle fractions. Instead, they apply positive whole number properties in dealing with fractions and vice versa (Ni & Zhuo, 2005; Vamvakoussi & Vosniadou, 2010).

Another hurdle witnessed among children and adults as they struggle to master fraction concepts and procedures is the realization that the numerator and denominator should be handled together as a single unit and not as two separate and unrelated whole numbers. A study found that “students process the natural number parts of the fractions separately” (Kallai & Tzelgov, 2012). For example, when students solve this problem 2/3 + 4/5 , they often add 2+4=6 and 3+5=8, and they obtain 6/8 (which is wrong) as their final answer. According to Siegler & Lortie-Forgues (2015), there is a positive and significant relation between students’ ability to accurately place a fraction or decimal on the number line and their general arithmetic ability.

**Theoretical Framework**

Procedural and conceptual knowledge theories guided this study. According to Lin (2010), procedural knowledge can be defined as the necessary steps required to solve problems. The use of procedures helps students to solve fractions without necessarily understanding the concepts behind the procedures. Siegler & Lortie-Forgues (2015) found that procedures are easy to memorize and apply, but they are not as flexible as conceptual understanding.

Fazioand, Lisa & Siegler (2011) defined conceptual knowledge as knowing why–showing understanding and the ability to sequentially explain why rules or procedures applied in solving fractions result to correct solutions. Conceptual knowledge can be generalized to a class of problems and is not tied to a specific problem, (Fazioand, Lisa, & Siegler, 2011).

**Research Questions**

- What are the common errors committed by developmental mathematics students when solving fractions?
- Which fractions concepts do developmental mathematics students struggling with the most?

**Importance of the study**

Learning fractions has remained a challenging domain for students in developmental mathematics. This study seek to identify specific area in which students struggle while solving fractions. The study findings can be used to inform lesson planning so that “problematic” can be allotted more emphasis in terms of time and resources.

**Research Method**

**Participant and Procedures**

Three students were identified for this study based on the averages of their pretest and examination one scores, willingness to participate in the study and their availability. Benson is struggling with the basics of solving fractions. His pretest score was below 50%. Mary is an average student compared with students Benson and Loice. Her pretest score was between 50-60 %. Loice is the best of the three students in the study. His pretest score was 65-75 %. Each student solve three problems. All students in the study are freshmen, and they had never enrolled in this course before. This requirement was necessary to ensure a fair comparison among students at the end of the study.

** Research Findings**

**QUESTION 1:** Which is greater: 5/12 or 9/16?

**Benson:** He converted both fractions into decimal and then compared the two before choosing is final answer. Benson was able to solve this problem and explain his solution logically.

**Mary: **She compared 5 with 9; 9 was greater, then she compared 12 with 16; 16 was greater. Therefore, 9/16 was *greater than*. Mary got the correct answer, but her explanation was wrong.

**Loice: **She cited her previous instructor: “fractions are the reverse, the bigger the denominator, the smaller the fraction.” She compared 12 with 16 and decided that 5/12 was *greater than* 9/16 because 12 is smaller than 16. Loice’s answer was wrong.

**QUESTION 2: **Solve x+ 1/6 = 5/8.

**Benson:** He isolated x by subtracting on both fractions. Benson converted the fractions on the right side into decimal numbers, and then he subtracted 0.167 from 0.625 to get his answer (0.458). Benson struggled in converting his final answer (0.458) into a fraction, but he did great work.

**Mary:** She isolated x by subtracting on both fractions. Mary calculated the Least Common Multiple (LCM) of 6 and 8, then performed the subtraction on the fractions. Why find LCM? **“**We find the LCM because I thought we needed to have it, but I don’t know why.”

**Loice:** She isolated x by subtracting 1/6 on both fractions. Mary calculated the Least Common Multiple (LCM) of 6 and 8, then performed the subtraction on the fractions. Why find LCM? “We find LCM because you can’t solve a problem without the denominators being the same. It’s like comparing apples to apples.”

**QUESTION 3:** Divide 5-1/8 by 2-1/6 , write your answer as a mixed number.

**Benson:** He was not able to proceed beyond changing the mixed fractions into improper fractions. “I have to stop because I do not know. I have forgotten how to divide mixed numbers.”

**Mary:** She converted the mixed fractions into improper fractions she verbalized this acronym “keep change flip, (KCF),” which meant keeping the first fraction unchanged, change the division sign to multiplication, and find the reciprocal of the second fraction. Why do we find the reciprocal of the right side fraction? “I do the KCF because it’s what I was taught; it’s the rule when you divide fractions.”

**Loice:** She identified fractions which were divisible by a common factor. Loice simplified the fractions and she obtained her answer. She used fewer steps.Why do we find the reciprocal of the right side fraction? “I’m not sure why we get the reciprocal because I’ve just always done it this way.”

**Conclusions & Discussion**

This study found that participants were struggling with both the procedural methods and conceptual understanding of solving fractions. Participants committed errors in multiplying and simplifying fractions. This study found that students’ procedural skills in solving fractions were better than their conceptual skills. Participants fared relatively well in addition and subtraction of fractions problems, but experienced challenges in dividing and comparing the magnitudes of two or more fractions. Participants who were not proficient with fraction concepts were unable to solve problems when they could not recall the associated acronym or rules. For example, students who could not recall how to find the least common multiple (LCM) of two fractions failed to answer question one. Division of fractions posed a major challenge to almost all participants. Although students tried to apply division of whole numbers rules to solve division of fractions, problems most of the rules were not applicable. The use of acronyms helped some students while others could not go beyond the first step. For example, students knew they were supposed to find the reciprocal of one of the fractions when dividing two fractions, but they were not certain whether to find the reciprocal of the left or right side fraction. There was a strong relationship between students’ abilities in explaining the necessary steps required in solving a problem and obtaining the correct answer.

**References**

Bettinger, E., & Long, B. T. (2005). Remediation at the community college: Student participation and outcomes. *New Directions for Community Colleges, 129*, 17-26.

Fazioand, Lisa & Siegler. (2011). Teaching fractions. The International Academy of Education. http://unesdoc.unesco.org/images/0021/002127/212781e.pdf

Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: the origins and implications of whole number bias. *Educational Psychologist, 40*(1), 27e52. http://dx.doi.org/10.1207/s15326985ep4001_3.

Kallai, A. Y., & Tzelgov, J. (2012). When meaningful components interrupt the processing of the whole: the case of fractions. Acta Psychologica, 139(2012), 358e369. http://dx.doi.org/10.1016/j.actpsy.2011.11.009.

Lin, C. (2010). Web-based instruction on preservice teachers’ knowledge of fraction operations. *School Science and Mathematics,* *110*(2), 59-70.

Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. *Cognition and Instruction, 28*(2), 181e209. http:// dx.doi.org/10.1080/07370001003676603.

Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic.* Journal of Educational Psychology.*

Siegler, R. S., & Pyke, A. A. (2012). Developmental and Individual Differences in Understanding of Fractions. *Developmental Psychology*. Advance online publication. doi: 10.1037/a0031200.