# Early Curricular Experiences with Nonnumeric Quantities, Evidence of an Enduring Perspective

## Abstract

In this study we explore possible long-term effects of an adaptation of the El’konin–Davydov elementary grades curriculum, Measure Up or MU. The objectives for the study are to assess how students relate an equation of nonnumeric quantities to a length representation, and if former MU students develop and retain a perspective characteristic of the curriculum. Data were collected from thirteen former MU students and a group of fourteen peers who were instructed together with the MU students in identical middle and high school programs, but did not receive MU instruction. Findings show that former MU students reasoned about lengths as generalized quantities, applied a method for marking and labeling quantities, and justified a representation of relationships given by an equation. Implications are discussed for how a measurement context in elementary mathematics supports meaning making in the later study of algebra, particularly with regard to variables and multiple representations.## References

Curriculum Research & Development Group. (2006, Draft). Measure Up Program, Teacher Notes & Masters and Student Materials. Honolulu, HI: Curriculum Research & Development Group.

Davydov, V.V. (1975). Logical and psychological problems of elementary mathematics as an academic subject. In L. P. Steffe (Ed.), Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII (pp. 55-107). Chicago, IL: University of Chicago.

Davydov, V. V., Gorbov, S., Mukulina, T., Savelyeva, M. & Tabachnikova, N. (1999) Mathematics. Moscow, Russia: Moscow Press.

Dougherty, B. (2008). Measure Up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 389–412). New York, NY: Erlbaum.

Dougherty, B., & Slovin, H. (2004). Generalized diagrams as a tool for young children’s problem solving. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education, Vol. 2. (pp. 295–302). Bergen, Norway: Bergen University College.

Eddy, C. M., Quebec Fuentes, S., Ward, E. K., Parker, Y. A., Cooper, S., Jasper, W. A., Mallam, W. A., Sorto, M. A., & Wilkerson, T. L. (2015). Unifying the algebra for all movement. Journal of Advanced Academics, 26(1), 59–92.

Filloy, E., Puig, L., & Rojano, R. (2010). Educational algebra, a theoretical and empirical approach. New York, NY: Springer.

Hein, V., Smerdon, B., & Sambolt, M. (2013). Predictors of postsecondary success. Washington, D.C.: American Institutes for Research.

Karp, K. S., Bush, S. B., Dougherty, B. J. (2014). 13 rules that expire. Teaching Children Mathematics, 12(1), 18–25.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.

Linchevski, L. & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39–65.

McNeil, N. M., & Alibali, M. W. (2005). Knowledge change as a function of mathematics experience: All contexts are not created equal. Journal of Cognition and Development, 6(2), 285-306.

Merriam, S. B. (2009). Qualitative research. San Francisco, CA: Wiley.

Minskaya, G. I. (1975). Developing the concept of number by means of the relationship of quantities. In L. P. Steffe (Ed.), Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII (pp. 207–261) Chicago, IL: University of Chicago.

National Council of Teachers of Mathematics, Inc. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

Slovin, H., & Venenciano, L. (2008). Success in algebra. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda. (Eds.), Proceedings of the Joint Meeting of PME 32 and PME-NA XXX, Vol. 4. (pp. 273–280). Morelia, México: Cinvestau-UMSNH.

Stake, R. (1995). The art of case study research. Thousand Oaks, CA: Sage.

Tharp, R. G., & Gallimore, R. (1995). Rousing minds to life: Teaching, learning, and schooling in social context. New York, NY: Cambridge University Press.

Wells, G. (1994, September). Learning and teaching “scientific concepts”: Vygotsky’s ideas revisited. Paper presented at the conference of Vygotsky and the Human Sciences, Moscow.

Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary grades mathematics. For the Learning of Mathematics, 34(1), 18–23.

Venenciano, L., & Heck, R. (2015). Proposing and testing a model to explain algebra preparedness. Educational Studies in Mathematics, 92, 21–35.

Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.). Cambridge, MA: Harvard University.

Davydov, V.V. (1975). Logical and psychological problems of elementary mathematics as an academic subject. In L. P. Steffe (Ed.), Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII (pp. 55-107). Chicago, IL: University of Chicago.

Davydov, V. V., Gorbov, S., Mukulina, T., Savelyeva, M. & Tabachnikova, N. (1999) Mathematics. Moscow, Russia: Moscow Press.

Dougherty, B. (2008). Measure Up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 389–412). New York, NY: Erlbaum.

Dougherty, B., & Slovin, H. (2004). Generalized diagrams as a tool for young children’s problem solving. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education, Vol. 2. (pp. 295–302). Bergen, Norway: Bergen University College.

Eddy, C. M., Quebec Fuentes, S., Ward, E. K., Parker, Y. A., Cooper, S., Jasper, W. A., Mallam, W. A., Sorto, M. A., & Wilkerson, T. L. (2015). Unifying the algebra for all movement. Journal of Advanced Academics, 26(1), 59–92.

Filloy, E., Puig, L., & Rojano, R. (2010). Educational algebra, a theoretical and empirical approach. New York, NY: Springer.

Hein, V., Smerdon, B., & Sambolt, M. (2013). Predictors of postsecondary success. Washington, D.C.: American Institutes for Research.

Karp, K. S., Bush, S. B., Dougherty, B. J. (2014). 13 rules that expire. Teaching Children Mathematics, 12(1), 18–25.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.

Linchevski, L. & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39–65.

McNeil, N. M., & Alibali, M. W. (2005). Knowledge change as a function of mathematics experience: All contexts are not created equal. Journal of Cognition and Development, 6(2), 285-306.

Merriam, S. B. (2009). Qualitative research. San Francisco, CA: Wiley.

Minskaya, G. I. (1975). Developing the concept of number by means of the relationship of quantities. In L. P. Steffe (Ed.), Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII (pp. 207–261) Chicago, IL: University of Chicago.

National Council of Teachers of Mathematics, Inc. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

Slovin, H., & Venenciano, L. (2008). Success in algebra. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda. (Eds.), Proceedings of the Joint Meeting of PME 32 and PME-NA XXX, Vol. 4. (pp. 273–280). Morelia, México: Cinvestau-UMSNH.

Stake, R. (1995). The art of case study research. Thousand Oaks, CA: Sage.

Tharp, R. G., & Gallimore, R. (1995). Rousing minds to life: Teaching, learning, and schooling in social context. New York, NY: Cambridge University Press.

Wells, G. (1994, September). Learning and teaching “scientific concepts”: Vygotsky’s ideas revisited. Paper presented at the conference of Vygotsky and the Human Sciences, Moscow.

Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary grades mathematics. For the Learning of Mathematics, 34(1), 18–23.

Venenciano, L., & Heck, R. (2015). Proposing and testing a model to explain algebra preparedness. Educational Studies in Mathematics, 92, 21–35.

Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.). Cambridge, MA: Harvard University.