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# Alternative correct strategies

## Flexibility with Fractions

Research on flexibility with fractions is scarce, but some research suggests that competent students and experts in mathematics are quite flexible with fractions. Smith (1995) interviewed students at the elementary, middle, and secondary levels about several ordering and equivalence tasks, to better understand their strategies and reasoning about fractions. His findings suggested that students skilled with fractions used a variety of strategies for solving problems, not just the traditional ones learned in school. In fact, they tended to use the most general strategy as a last resort, opting for strategies specific to the problem at hand. Their preferred strategies were often ones that made the problem easier in some way (e.g., required less computation).

Newton (2008) found a similar pattern for experts such as mathematicians, engineers, and high school mathematics teachers that took a fraction assessment designed for a study of preservice teachers. When asked to solve fraction computation items, they generally opted for strategies that made the problem easier for themselves. Out of 11 possible points for flexibility, they scored 8.44 points on average, compared to 2.27 for preservice teachers. Although not published as part of the study, one example expert strategy was to solve 6 % - 2 4/s by subtracting the whole numbers and then subtracting the fractions. Doing so resulted in a new subtraction problem, 4 - %, which can be mentally calculated to be 3 - 3/s based on the fact that two-fifths and three-fifths is one whole. A more common strategy was to rename 6 - % as 5 - % and then subtract like parts. In contrast, a strategy used frequently by preservice teachers was to rename both mixed numbers as improper fractions and then subtract numerators, keeping the denominator the same. The preservice teachers in this study generally used the strategy of changing mixed numbers to improper fractions whenever mixed numbers were involved, even when it required more steps.

A case study of a small special education classroom revealed that non-experts are capable of learning and using a variety of strategies for solving fraction problems (Newton et al., 2014). Interestingly, for some problems the more general approach led to errors for these students, whereas an efficient, alternative method led to success. For example, on an end-of-unit fraction assessment, 6 out of 11 students correctly solved 9/i4 У? by dividing the numerators and dividing the denominators. On the other hand, three students attempted to solve the problem by multiplying by the reciprocal (i.e., 9/i4 x 7/i), but they all made errors converting the product, 63/i4, to a mixed number. On the other hand, seven students correctly multiplied by the reciprocal to solve % 4- 3/s, which did not involve simplifying.

Of note is that four students used both methods successfully during the assessment. However, as mentioned previously the teacher in this classroom regularly introduced a new topic by asking students to make conjectures based on their prior knowledge. This atypical practice lends itself to the use of multiple strategies, since not all students will make the same conjectures each time. Based on the work of Bishop et al. (2016), it seems likely that the requests for conjectures about how things might work helped these students to focus on underlying structures and patterns. In the case of dividing across numerators and denominators, students may reason that since division is related to multiplication and fraction multiplication works by multiplying numerators and denominators, then fraction division should work similarly. Research is needed to better understand the role of conjecturing in the development of flexibility.

Representational flexibility with fractions, or an ability to move smoothly between different fraction representations (e.g., circle, rectangle, and number line), has been a focus for some researchers. Deliyianni, Gagatsis, Elia, and Panaoura (2016) examined the relationship between representational flexibility, problem solving with novel tasks, and understanding of fraction addition using confirmatory factor analysis. Findings suggest that problem solving and representational flexibility constitute major components of fraction understanding. Further, representational flexibility with fraction addition is better supported by the number line and rectangle representations than by circles. These results are consistent with standards documents for mathematics education. In particular, the National Council of Teachers of Mathematics (2000) forwarded problem solving and movement between representations as two important processes that students should experience in mathematics classrooms. And as noted above, the Common Core State Standards (National Governors Association Center for Best Practices & the Council of Chief State School Officers, 2010) recently asserted that the number line is a critical representation for learning fractions.

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