1. Kaylee McDowell Mathematics Specialization Children’s Development of Mental Representations for Fractions

2. Original Research Question How can the use of manipulatives in conjunction with stories assist 4 th graders’ development of rich mental representations of fractions?

3. Literature Review Many Students Lack Adequate Knowledge of Fractions (Charalambous & Pitta-Pantazi, 2007; Butler et al., 2003) Conceptual v. Procedural Knowledge (NRC, 2001; Ploger and Rooney, 2005) Whole Number Bias (National Research Council (NRC), 2001; Ni & Zhou, 2005) Rich Mental Representations (Cramer & Wyberg, 2009) 0 3/4

4. Research Question How do 4th graders use strategies to represent and solve problems involving fractions following a unit on fractions? How do these strategies compare between students who frequently used physical manipulatives and stories and those who did not?

5. Participants Control Group (CG) 18 students Experimental Curriculum 3 ½ weeks 10 students Investigations Curriculum No stories Fewer physical manipulatives 3 ½ weeks Experimental Group (EG)

6. Manipulative Models (Cramer & Wyberg, 2009; Cramer et al., 2002; NRC, 2001) Area Paper Folding Fraction Circles Length Student Created Fraction Tiles Number Line Set Unifix Cubes

7. Stories Role of Context and Connection to Stories (Whiten & Wilde, 1995; Butler et al., 2003)

8. Data Collection Surveys Attitudes: Beginning and End Stories Pretest & Posttest Concept, Equivalence, Order, Estimation, Operations (Cramer & Wyberg, 2009; Cramer et al., 2002) Interviews 3 students from each group Recorded

9. Survey Results

11. Test Results

12. Strategy CG Times Used Percent Correct Percent in Error EG Times Used Percent Correct Percent in Error Long Line 2090%10%3100%-- Grid 3*100%--1753%47% Pictorial Representation560%40%3776%24% Other 1--100%333.5%66.5% Long Line Strategy Grid Strategy Pictorial Representation 1 2 3 1 2 3 2 3 5 3 6 9 2 4 6 6 6 6 += 4 2 6 8 8 8 += Students Use of Strategies

13. Interview Results Based on Denominator Based on Denominator and Numerator 1 4 1 10 8 4 2 3 7 4 Benchmarks/ Equivalence Fraction Relationships Grid Strategy Long Line Strategy CategoryQuestion EG Percent Correct CG Percent Correct Concept1100% 233% 666% Order/Equivalence3100% 466%100% Estimation533%100% 7 Operation866%33% COMMON THEMES Percent of Correct Responses

14. Conclusions Strategies Connected to Understanding Long-Line and Grid Student-created comparison Time to Build Conceptual Knowledge. Manipulatives / Pictures Multiple experiences Number sense Emphasizing the multiplicative nature Relationships among fractions Knowledge of multiples

15. References Bray, W. S. & Abreu-Snachez, L. (2010). Using number sense to compare fractions. Teaching Children Mathematics 17(2), 90-97. Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education 19(3), 215-232. Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research and Practice 18(2) 99-111. Charalambous, C., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293-316. Cramer, K., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth- and fifth- grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education 33(2), 111-144. Cramer, K. & Wyberg, T. (2009). Efficacy of different concrete models for teaching part-whole construct for fractions. Mathematical Thinking & Learning 11(4), 226-257. McElligott, M. (2009). The lion’s share: A tale about halving cake and eating it too. New York: Walker. Myller, R. (1991). How big is a foot? New York: Yearling.

16. References Cont. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press. 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), 27-52. Ploger, D. & Rooney, M. (2005) Teaching fractions: Rules and research. Teaching Children Mathematics 12(1), 12-17. Russel, S. J. & Economopoulos, K. (Eds.). (2012). Investigations in number, data, and space: Grade four fraction cards and decimal squares. (Vol. 6). Glenview, IL: Pearson. Siebert, D. & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics. 17(2), 394-400. Smith, D. (2011). If the world were a village: A book about the world’s people. Toronto: Kids Can Press. Van de Walle, J. Karp, K. S. & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston: Allyn & Bacon. Watanabe, T. (2007). Initial treatment of fractions in Japanese textbooks. Focus on Learning Problems in Mathematics. 29(2), 41-60. Whitin, D. J. & Wilde, S. (1995). It’s the story that counts: More children’s books for mathematical learning, K- 6. Portsmouth, NH: Heinemann.

17. Questions?