# RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE AND ALGEBRAIC REASONING: THE CASE OF WILLA

## Abstract

To investigate relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems that required explicit use of unknowns to write equations. As a part of the larger study, the first author conducted a case study of a seventh grade student, Willa. Willa’s fractional knowledge—specifically her reversible iterative fraction scheme and use of fractions as multipliers—influenced how she wrote equations to represent multiplicative relationships between two unknown quantities. The finding indicates that implicit use of powerful fractional knowledge can lead to more explicit use of structures and relationships in algebraic situations. Curricular and instructional implications are explored.

## Key words

algebraic reasoning fractional knowledge fractions as multipliers reciprocal reasoning reversible iterative fraction scheme## Preview

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## References

- Behr, M. J., Harel, G., Post, T. R. & Lesh, R. (1993). Rational numbers: Toward a semantic analysis—emphasis on the operator construct. In T. P. Carpenter, E. Fennema & T. A. Romberg (Eds.),
*Rational numbers: An integration of research*(pp. 13–47). Hillsdale: Lawrence Erlbaum Associates.Google Scholar - Carraher, D. W., Schliemann, A. D. & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 235–272). New York: Lawrence Erlbaum.Google Scholar - Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh & A. E. Kelly (Eds.),
*Handbook of research design in mathematics and science education*(pp. 547–589). Hillsdale: Erlbaum.Google Scholar - Corbin, J. & Strauss, A. (2008).
*Basics of qualitative research*(3rd ed.). Thousand Oaks: Sage Publications.Google Scholar - Driscoll, M. J. (1999).
*Fostering algebraic thinking: A guide for teachers, grades 6–10*. Portsmouth: Heinemann.Google Scholar - Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships.
*Journal for Research in Mathematics Education, 38*(3), 194–229.Google Scholar - Empson, S. B., Levi, L. & Carpenter, T. P. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai & E. J. Knuth (Eds.),
*Early algebraization*(pp. 409–428). Berlin: Springer.CrossRefGoogle Scholar - Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis.
*Journal of Mathematical Behavior, 26*, 27–47.CrossRefGoogle Scholar - Hackenberg, A. J. (2009).
*Relationships between students’ fraction knowledge and equation solving*. Paper presentation at the Research Pre-session of the annual conference of the National Council of Teachers of Mathematics, Washington, D.C.Google Scholar - Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships.
*Cognition and Instruction, 28*(4), 383–432.CrossRefGoogle Scholar - Hackenberg, A. J. & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes.
*Journal of Mathematical Behavior, 28*, 1–18.CrossRefGoogle Scholar - Kaput, J. (2008). What is Algebra? What is Algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 5–17). New York: Lawrence Erlbaum Associates.Google Scholar - Kaput, J. J., Carraher, D. W. & Blanton, M. L. (Eds.). (2008).
*Algebra in the early grades*. New York: Lawrence Erlbaum.Google Scholar - Kieren, T. E. (1995). Creating spaces for learning fractions. In J. T. Sowder & B. P. Schappelle (Eds.),
*Providing a foundation for teaching mathematics in the middle grades*(pp. 31–65). Albany: State University of New York Press.Google Scholar - Lamon, S. J. (1996). The development of unitizing: Its role in children’s partitioning strategies.
*Journal for Research in Mathematics Education, 27*(2), 170–193.CrossRefGoogle Scholar - Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. J. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 629–667). Charlotte: Information Age.Google Scholar - National Mathematics Advisory Panel (2008).
*Report of the task group on conceptual skills and knowledge*. Washington, DC: U.S. Department of Education.Google Scholar - Norton, A. (2008). Josh’s operational conjectures: Abductions of a splitting operation and the construction of new fractional schemes.
*Journal for Research in Mathematics Education, 39*(4), 401–430.Google Scholar - Norton, A. & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes.
*Journal of Mathematical Behavior, 28*, 150–161.CrossRefGoogle Scholar - Russell, S. J., Schifter, D. & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In J. Cai & E. J. Knuth (Eds.),
*Early algebraization: A global dialogue from multiple perspectives*(pp. 43–69). Berlin: Springer.CrossRefGoogle Scholar - Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In M. J. Behr & J. Hiebert (Eds.),
*Number concepts and operations in the middle grades*(pp. 41–52). Reston: National Council of Teachers of Mathematics.Google Scholar - Smith, J. & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 95–132). New York: Erlbaum.Google Scholar - Steffe, L. P. (1991). Operations that generate quantity.
*Learning and Individual Differences, 3*(1), 61–82.CrossRefGoogle Scholar - Steffe, L. P. (1992). Schemes of action and operation involving composite units.
*Learning and Individual Differences, 43*, 259–309.CrossRefGoogle Scholar - Steffe, L. P. (2001, December 9–14).
*What is algebraic about children’s numerical operating?*Paper presented at the Conference on the Future of the Teaching and Learning of Algebra, University of Melbourne, Australia.Google Scholar - Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge.
*Journal of Mathematical Behavior, 20*, 267–307.CrossRefGoogle Scholar - Steffe, L. P. & Olive, J. (2010).
*Children’s fractional knowledge*. New York: Springer.CrossRefGoogle Scholar - Steffe, L. P. & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.),
*Handbook of research design in mathematics and science education*(pp. 267–306). Hillsdale: Erlbaum.Google Scholar - Thomas, G. (2011).
*How to do your case study: A guide for students and researchers*. Thousand Oaks: Sage.Google Scholar - Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.),
*The development of multiplicative reasoning in the learning of mathematics*(pp. 181–234). Albany: SUNY Press.Google Scholar - Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.),
*Providing a foundation for teaching mathematics in the middle grades*(pp. 199–219). Albany: State University of New York Press.Google Scholar - Thompson, P. W. & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.),
*Research companion to the principles and standards for school mathematics*(pp. 95–113). Reston: National Council of Teachers of Mathematics.Google Scholar - von Glasersfeld, E. (1995).
*Radical constructivism: A way of knowing and learning (vol. 6)*. New York: Routledge Falmer.CrossRefGoogle Scholar - Yin, R. K. (2009).
*Case study research: Design and methods*(4th ed.). Thousand Oaks: Sage.Google Scholar