# Reinventing fractions and division as they are used in algebra: the power of preformal productions

## Abstract

In this paper, we explore algebra students’ mathematical realities around fractions and division, and the ways in which students reinvented mathematical productions involving fractions and division. We find that algebra students’ initial realities do not include the fraction-as-quotient sub-construct. This can be problematic because in algebra, quotients are almost always represented as fractions. In a design experiment, students progressively reinvented the fraction-as-quotient sub-construct. Analyzing this experiment, we find that a particular type of mathematical production, which we call preformal productions, played two meditational roles: (1) they mediated mathematical activity, and (2) they mediated the reinvention of more formal mathematical productions. We suggest that preformal productions may emerge even when they are not designed for, and we show how preformal productions embody historic classroom activity and social interaction.

## Keywords

Realistic mathematics education RME Fractions Division Algebra Preformal productions## Notes

### Acknowledgments

We are grateful to the students in the class, from whom we learned so much, and to David Webb for his thoughtful comments on our study and on multiple drafts of this article. We also gratefully acknowledge the anonymous reviewers whose comments significantly improved the article.

## References

- Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.),
*Rational numbers: An integration of research*(pp. 157–195). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Behr, M. J., & Post, T. R. (1992). Teaching rational number and decimal concepts. In T. R. Post (Ed.),
*Teaching mathematics in grades K-8: Research-based methods*(2nd ed., pp. 201–248). Boston, MA: Allyn and Bacon.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics: Didactique des mathématiques*. In N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield (Eds.), (pp. 1970–1990). Dordrecht, the Netherlands: Kluwer Academic.Google Scholar - Brousseau, N., & Brousseau, G. (1987).
*Rationnels et décimaux dans la scolarité obligatoire*. Bordeaux, France: IREM de Bordeaux.Google Scholar - Brousseau, G., Brousseau, N., & Warfield, V. (2004). Rationals and decimals as required in the school curriculum part 1: Rationals as measurement.
*The Journal of Mathematical Behavior, 23*(1), 1–20.Google Scholar - Brousseau, G., Brousseau, N., & Warfield, V. (2007). Rationals and decimals as required in the school curriculum part 2: From rationals to decimals.
*The Journal of Mathematical Behavior, 26*(4), 281–300.CrossRefGoogle Scholar - Brousseau, G., Brousseau, N., & Warfield, V. (2008). Rationals and decimals as required in the school curriculum part 3: Rationals and decimals as linear functions.
*The Journal of Mathematical Behavior, 27*(3), 153–176.CrossRefGoogle Scholar - Brousseau, G., Brousseau, N., & Warfield, V. (2009). Rationals and decimals as required in the school curriculum part 4: Problem solving, composed mappings, and division.
*The Journal of Mathematical Behavior, 28*(2–3), 79–118.CrossRefGoogle Scholar - Burris, C. C., & Welner, K. G. (2005). Closing the achievement gap by detracking.
*Phi Delta Kappan, 86*(8), 594–598.CrossRefGoogle Scholar - Charalambous, C. Y., & Pitta-Pantazi, D. (2006). Drawing on a theoretical model to study students’ understandings of fractions.
*Educational Studies in Mathematics, 64*(3), 293–316.CrossRefGoogle Scholar - Charles, K., & Nason, R. (2000). Young children’s partitioning strategies.
*Educational Studies in Mathematics, 43*, 191–221.CrossRefGoogle Scholar - Clark, M. R. (2005). Using multiple-missing-values problems to promote the development of middle-school students’ proportional reasoning. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.),
*Frameworks that support learning.**Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(pp. 1–6). Roanoke, VA: PME-NA.Google Scholar - Clarke, D. M., Roche, A., & Mitchell, A. (2007). Year six fraction understanding: A Part of the whole story. In J. Watson & K. Beswick (Eds.),
*Mathematics: Essential research, essential practice. Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia*(Vol. 1, pp. 207–216). Adelaide: MERGA Inc.Google Scholar - Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. E. Kelly & R. Lesh (Eds.),
*Research design in mathematics and science education*(pp. 307–333). Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Cobb, P., Confrey, J., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13. doi: 10.3102/0013189X032001009.CrossRefGoogle Scholar - Cobb, P., Zhao, Q., & Visnovska, J. (2008). Learning from and adapting the theory of realistic mathematics education.
*Éducation & Didactique, 2*(1), 105–124.CrossRefGoogle Scholar - Cole, M. (2010). What’s culture got to do with it? Educational research as a necessarily interdisciplinary enterprise.
*Educational Researcher, 39*(6), 461–470.CrossRefGoogle Scholar - Cole, M., & Wertsch, J. V. (1996). Beyond the individual-social antimony in discussions of Piaget and Vygotsky.
*Human Development, 39*, 250–256.CrossRefGoogle Scholar - Confrey, J. (2012). Better measurement of higher cognitive processes through learning trajectories and diagnostic assessments in mathematics: The challenge in adolescence. In V. F. Reyna, S. B. Chapman, M. R. Dougherty, & J. Confrey (Eds.),
*The adolescent brain: Learning, reasoning, and decision making*(pp. 155–182). Washington DC: American Psychological Association.CrossRefGoogle Scholar - Confrey, J., Maloney, A., Nguyen, K. H., Mojica, G., & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.),
*Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 345–352). Thessaloniki, Greece: PME.Google Scholar - Connell, M. L., & Peck, D. M. (1993). Report of a conceptual change intervention in elementary mathematics.
*The Journal of Mathematical Behavior, 12*, 329–350.Google Scholar - Cramer, K. (2003). Using a translation model for curriculum development and classroom instruction. In R. Lesh & H. M. Doerr (Eds.),
*Beyond constructivism. Models and modeling perspectives on mathematics problem solving, learning, and teaching*(pp. 449–464). Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Cramer, K., Behr, M. J., Post, T. R., & Lesh, R. (2009).
*Rational Number Project: Initial fraction ideas*. Retrieved from http://www.cehd.umn.edu/ci/rationalnumberproject/RNP1-09/RNP1-09_withBlankPages.pdf - Cramer, K., Bezuk, N., & Behr, M. J. (1989). Proportional relationships and unit rates.
*Mathematics Teacher, 82*(7), 537–544.Google Scholar - Cramer, K., & Post, T. R. (1993). Connecting research to teaching proportional reasoning.
*The Mathematics Teacher, 86*(5), 404–407.Google Scholar - 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.CrossRefGoogle Scholar - Cramer, K., Wyberg, T., & Leavitt, S. (2009).
*Rational Number Project: Fraction operations and initial decimal ideas*. Retrieved from http://www.cehd.umn.edu/ci/rationalnumberproject/RNP2/rnp2.pdf - Davydov, V. V. (1990).
*Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula*(Vol. 2). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom.
*Cognition and Instruction, 17*(3), 283–342.CrossRefGoogle Scholar - Empson, S. B. (2002). Organizing diversity in early fraction thinking. In B. Litwiller & G. Bright (Eds.),
*Making sense of fractions, ratios, and proportions*(pp. 29–40). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2006). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children’s thinking.
*Educational Studies in Mathematics, 63*(1), 1–28. doi: 10.1007/s10649-005-9000-6.CrossRefGoogle Scholar - Falmagne, R. J. (1995). The abstract and the concrete. In L. Martin, K. Nelson, & E. Tobach (Eds.),
*Sociocultural psychology: Theory and practice of doing and knowing*(pp. 205–228). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar - Fosnot, C. T. (2007).
*Field trips and fundraising: Introducing fractions*. Portsmouth, NH: Heinemenn.Google Scholar - Fosnot, C. T., & Dolk, M. (2001).
*Young mathematicians at work: Constructing multiplication and division*. Portsmouth, NH: Heinemenn.Google Scholar - Fosnot, C. T., & Dolk, M. (2002).
*Young mathematicians at work: Constructing fractions, decimals, and percents*. Portsmouth, NH: Heinemenn.Google Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht, the Netherlands: Reidel.Google Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education: China lectures*. Dordrecht, the Netherlands: Kluwer Academic.Google Scholar - Gee, J. P. (2011).
*How to do discourse analysis: A toolkit*. New York: Taylor & Francis.Google Scholar - Gravemeijer, K. (1994). Educational development and developmental research in mathematics education.
*Journal for Research in Mathematics Education, 25*(5), 443–471. doi: 10.2307/749485.CrossRefGoogle Scholar - Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics.
*Mathematical Thinking and Learning, 1*(2), 155–177.CrossRefGoogle Scholar - Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.),
*Educational design research*(pp. 17–51). New York: Routledge.Google Scholar - Gravemeijer, K., & Terwel, J. (2000). Hans Freudenthal: A mathematician on didactics and curriculum theory.
*Journal of Curriculum Studies, 32*(6), 777–796.CrossRefGoogle Scholar - Heritage, J., & Clayman, S. (2010).
*Talk in action: Interactions, identities, and institutions*. Chichester, UK: Wiley-Blackwell.CrossRefGoogle Scholar - Hutchins, E. (1995).
*Cognition in the wild*. Cambridge, MA: MIT Press.Google Scholar - Jaworski, A., & Coupand, N. (2006). Perspecitves on discourse analysis. In A. Jaworski & N. Coupand (Eds.),
*The discourse reader*(2nd ed., pp. 1–37). New York: Routledge.Google Scholar - Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.),
*The development of multiplicative reasoning in the learning of mathematics*(pp. 235–287). Albany, NY: SUNY Press.Google Scholar - Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents’ proportional reasoning on “rate” problems.
*Educational Studies in Mathematics, 14*(3), 219–233.CrossRefGoogle Scholar - Keijzer, R., & Terwel, J. (2001). Audrey’s acquisition of fractions: A case study into the learning of formal mathematics.
*Educational Studies in Mathematics, 47*(1), 53–73.CrossRefGoogle Scholar - Kieren, T. (1980). The rational number construct: Its elements and mechanisms. In T. Kieren (Ed.),
*Recent research on number learning*(pp. 125–149). Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education.Google Scholar - Kindt, M. (2010). Principles of practice. In P. Drijvers (Ed.),
*Secondary algebra education: Revisiting topics and themes and exploring the unknown*(pp. 137–178). Rotterdam: Sense Publishing.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. (2005).
*Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers*(2nd ed.). Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 629-668). Charlotte, NC: Information Age.Google Scholar - Lesh, R., Post, T. R., & Behr, M. J. (1988). Proportional reasoning. In J. Heibert & M. J. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 93–118). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Luria, A. R. (1928). The problem of the cultural behavior of the child.
*Journal of Genetic Psychology, 35*(4), 493–506.Google Scholar - Middleton, J. A., & van den Heuvel‐Panhuizen, M. (1995). The ratio table.
*Mathematics Teaching in the Middle School, 1*(4), 282–288.Google Scholar - Middleton, J. A., Van den Heuvel-Panhuizen, M., & Shew, J. A. (1998). Using bar representations as a model for connecting concepts of rational number.
*Mathematics Teaching in the Middle School, 3*(4), 302–311.Google Scholar - Moss, J., & Case, R. (2011). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum.
*Journal for Research in Mathematics Education, 30*(2), 122–147.CrossRefGoogle Scholar - Nunes, T., Desli, D., & Bell, D. (2003). The development of children’s understanding of intensive quantities.
*International Journal of Educational Research, 39*(7), 651–675.CrossRefGoogle Scholar - Núñez, R. E. (2009). Numbers and arithmetic: Neither hardwired nor out there.
*Biological Theory, 4*(1), 68–83.CrossRefGoogle Scholar - O’Connor, K. (2003). Cummunicative practice, cultural production, and situated learning: Conducting and contesting identities of expertise in a heterogeneous learning context. In S. Wortham & B. Rymes (Eds.),
*Linguistic anthropology of education*(Vol. 37, pp. 61–91). Westport, CT: Praeger.Google Scholar - Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis.
*Mathematical Thinking and Learning, 1*(4), 279–314.CrossRefGoogle Scholar - Olive, J., & Steffe, L. P. (2001). The construction of an iterative fractional scheme: The case of Joe.
*The Journal of Mathematical Behavior, 20*(4), 413–437.CrossRefGoogle Scholar - Paley, V. G. (1986). On listening to what the children say.
*Harvard Educational Review, 56*(2), 122–132.CrossRefGoogle Scholar - Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts.
*Educational Studies in Mathematics, 30*(1), 5–38.CrossRefGoogle Scholar - Post, T. R., Behr, M. J., & Lesh, R. (1988). Proportionality and the development of pre-algebra understanding. In J. Hiebert & M. J. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 93–118). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Rotman, J. (1991).
*Arithemetic: Prerequsite to algebra?*Paper presented at the Annual Convention of the American Mathematical Association of Two-Year Colleges, Seattle, WA.Google Scholar - Schmittau, J. (2003). Cultural-historical theory and mathematics education. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.),
*Vygotsky’s educational theory in cultural context*(pp. 225–245). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar - Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov.
*The Mathematics Educator, 8*(1), 60–87.Google Scholar - Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Heibert & M. J. Behr (Eds.),
*Number concepts and operations in the middle grades*(Vol. 2, pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Schwarz, B., & Hershkowitz, R. (2001). Production and transformation of computer artifacts toward construction of meaning in mathematics.
*Mind, Culture, and Activity, 8*(3), 250–267.CrossRefGoogle Scholar - Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
*Journal for Research in Mathematics Education, 26*(2), 114–145.CrossRefGoogle Scholar - Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. Kelly (Eds.),
*Research design in mathematics and science education*(pp. 267–307). Hillsdale, NJ: Erlbaum.Google Scholar - Streefland, L. (1991).
*Fractions in realistic mathematics education: A paradigm of developmental research*. Dordrecht, the Netherlands: Kluwer Academic.CrossRefGoogle Scholar - Streefland, L. (1993). Fractions: A realistic approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.),
*Rational numbers: An integration of research*(pp. 289–326). Hillsdale, NJ: Erlbaum.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. 179–234). Albany, NY: SUNY Press.Google Scholar - van Reeuwijk, M. (2001). From informal to formal, progressive formalization: An example on “solving systems of equations”. In K. Stacey, H. Chick, & M. Kendal (Eds.),
*The future of the teaching and learning of algebra: The 12th ICMI study*(pp. 613–620). Melbourne, Australia: The University of Melbourne.Google Scholar - Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 141–161). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - von Glasersfeld, E. (1989). Cognition, construction of knowledge, and teaching.
*Synthese, 80*(1), 121–140.CrossRefGoogle Scholar - Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the tip of the iceberg: Using representations to support student understanding.
*Mathematics Teaching in the Middle School, 14*(2), 110–113.Google Scholar - Whitson, T. (1997). Cognition as a semiotic process: From situated mediation to critical reflective transcendence. In D. Kirshner & J. A. Whitson (Eds.),
*Situated cognition: Social, semiotic, and psychological perspectives*(pp. 97–149). Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Wilson, P. H., Myers, M., Edgington, C. P., & Confrey, J. (2012). Fair shares, matey, or walk the plank.
*Teaching Children Mathematics, 18*(8), 482–489.CrossRefGoogle Scholar