### Fraction Symbols and their Relation to Conceptual Knowledge [thesis]

Heather Douglas
How do students make sense of fractions? Formal fraction knowledge begins when students start mapping fractions shown visually (e.g., area models), with symbols (e.g., ¾) and with words (e.g., three-quarters). Students were recruited from three schools that service rural and smalltown communities. Participating students in grade 4 (N=64) and grade 6 (N = 66) completed measures of cognition, language, and three novel measures developed for this study: mathematical vocabulary, orthography (i.e.,
more » ... orthography (i.e., the conventions for writing symbolic math), and fraction mapping. Five months later, their conceptual fraction skills (i.e., mapping, word problems and number line) were measured. I used two analytical approaches to examine the role of fraction mapping as students acquire conceptual fraction knowledge. In Study 1 (Chapters 4 and 5), I tested a path model in which mathematical vocabulary and orthography predicted fraction mapping, and fraction mapping predicted conceptual fraction skills. The model was largely supported for both grade 4 and grade 6. Moreover, mathematical vocabulary also predicted conceptual fraction skills for sixth graders. Thus, once students have sufficient knowledge of fraction mappings, other skills such as mathematical vocabulary may contribute more strongly to students' knowledge of fraction concepts. In Study 2 (Chapter 6), I used latent profile analysis to group students based on their fraction number line estimation. Three groups emerged. Relational estimators had the most advanced fraction concepts because they viewed the fraction as a unit. Compared to the other groups, relational estimators were more likely to be in sixth grade, have better mapping skills and more accurate whole number line estimation. Whole-component and denominator estimators, respectively, interpreted the fraction based on the magnitudes of both components (i.e., the numerator and denominator) or just the denominator. Only fraction mapping skills iii differentiated whole-component estimators from denominator estimators. Thus, students' knowledge of fraction mappings is a precursor to interpreting fractions as units rather than as composites, and therefore necessary for successfully placing those fractions on a number line. In summary, this research shows that students who struggle to acquire fraction concepts in grades 4 and 6 have not mastered fraction mappings. Knowing how fraction symbols are connected to magnitude is foundational knowledge for fraction learning. iv