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Kaylee McDowellMathematics Specialization
Children’s Development of Mental Representations for Fractions
Original Research Question
How can the use of manipulatives in conjunction with stories assist 4th graders’ development of rich mental representations of fractions?
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
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?
Participants
Control Group (CG)18 studentsExperimental
Curriculum3 ½ weeks
10 studentsInvestigations
CurriculumNo storiesFewer physical
manipulatives
3 ½ weeks
Experimental Group (EG)
Manipulative Models
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
Data Collection
SurveysAttitudes: Beginning and EndStories
Pretest & PosttestConcept, Equivalence, Order, Estimation,
Operations (Cramer & Wyberg, 2009; Cramer et al., 2002)
Interviews3 students from each groupRecorded
Test Results
StrategyCG
Times
Used
Percent
Correct
Percent in Error
EGTime
s Used
Percent
Correct
Percent in Error
Long Line20 90% 10% 3 100% --
Grid3* 100% -- 17 53% 47%
Pictorial Representati
on5 60% 40% 37 76% 24%
Other1 -- 100% 3 33.5
%66.5
%
Long Line Strategy
Grid Strategy
Pictorial Representation
1 2 3 1 2 3 2 3 53 6 9 2 4 6 6 6 6
+ =
4 2 6 8 8 8
+ =
Students Use of Strategies
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
Category Question
EGPercent Correct
CGPercent Correct
Concept 1 100% 100%
2 33% 33%
6 66% 66%
Order/Equivalence
3 100% 100%
4 66% 100%
Estimation 5 33% 100%
7 100% 100%
Operation 8 66% 33%
COMMON THEMES
Percent of Correct Responses
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
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.
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.