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)

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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

StoriesRole of Context and Connection to

Stories (Whiten & Wilde, 1995; Butler et al., 2003)

Data Collection

SurveysAttitudes: Beginning and EndStories

Pretest & PosttestConcept, Equivalence, Order, Estimation,

Operations (Cramer & Wyberg, 2009; Cramer et al., 2002)

Interviews3 students from each groupRecorded

Survey Results

Survey Results

Test Results

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.

Questions?