# Proposing and testing a model to explain traits of algebra preparedness

## Abstract

Early experiences with theoretical thinking and generalization in measurement are hypothesized to develop constructs we name here as logical reasoning and preparedness for algebra. Based on work of V. V. Davydov (1975), the Measure Up (MU) elementary grades experimental mathematics curriculum uses quantities of area, length, volume, and mass to contextualize the relationships among the quantities in, for example, *R* + *C* = *T*. This quasi-experimental study, conducted with 129 fifth- and sixth-grade students, examines MU effects on students’ preparedness for algebra. Structural equation modeling is used to identify a system of relationships among the variables in our proposed model. Findings show significant direct standardized effects from MU to preparedness (0.28, *p* < .05) and from logical reasoning to preparedness (0.89, *p* < .05). Although positive, the effect of MU mediated by logical reasoning was not statistically significant. This suggests that the development of logical reasoning abilities, attributed to theoretical thinking and generalization, lag preparedness for algebra. It also suggests that MU can potentially contribute to algebra preparedness for students who may not have developed strong logical reasoning abilities. The findings are discussed in terms of their theoretical and practical implications for the successful study of algebra.

## Keywords

Davydov Vygotsky Measure Up project Algebra preparedness Logical reasoning Structural equation modeling## References

- Agresti, A. (2013).
*Categorical data analysis*(3rd ed.). NY: Wiley.Google Scholar - Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 5–23). New York: Springer.Google Scholar - Britt, M. S., & Irwin, K. C. (2011). Algebraic thinking with and without algebraic representation: A pathway for learning. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 137–159). New York: Springer.Google Scholar - Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.),
*Testing structural equation models*(pp. 136–162). Newbury Park: Sage.Google Scholar - Cai, J., Ng, S. F., & Moyer, J. C. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 25–41). New York: Springer.Google Scholar - Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in mathematics education.
*Journal for Research in Mathematics Education, 37*(2), 87–115.Google Scholar - Coady, C., & Pegg, I. (1993). An exploration of students’ responses to the more demanding Küchemann test items. In W. Atweh, C. Kanes, M. Carss, & G. Booker (Eds.),
*Proceedings of the Sixteenth Annual Conference of MERGA, Brisbane*(pp. 191-196). Brisbane: MERGA. Retrieved from http://www.merga.net.au/documents/RP_Coady_Pegg_1993.pdf - Cook, T. D., & Campbell, D. T. (1979).
*Quasi-experimentation: Design & analysis issues for field settings*. Boston: Houghton Mifflin.Google Scholar - Cooper, T. J., & Warren, E. (2011). Years 2 to 6 students’ ability to generalize: Models, representations and theory for teaching and learning. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 187–214). New York: Springer.Google Scholar - Cronbach, L. J. (1951). Coefficient alpha and the internal structure of a test.
*Psychometrika, 16*, 297–334.CrossRefGoogle Scholar - Davydov, V. V. (1966). Logical and psychological problems of elementary mathematics as an academic subject (A. Bigelow, Trans.). In D. B. Elkonin & V. V. Davydov (Eds.),
*Learning capacity and age level: Primary grades*(pp. 54–103). Moscow: Prosveshchenie.Google Scholar - Davydov, V. V. (1975). The psychological characteristics of the “prenumerical” period of mathematics instruction. In L. P. Steffe (Ed.),
*Children’s capacity for learning mathematics. Soviet studies in the psychology of learning and teaching mathematics, Vol. VII*(pp. 109–205). Chicago: University of Chicago.Google Scholar - Davydov, V. V. (2008). Problems of developmental instruction: A theoretical and experimental psychological study. In V. Lektorsky & D. Robbins (Eds.), P. Moxhay (Trans.), L. V. Bertsfai. (Comp.).
*International perspectives in non-classical psychology*. New York: Nova Science.Google Scholar - Davydov, V. V., Gorbov, S., Mukulina, T., Savelyeva, M., & Tabachnikova, N. (1999).
*Mathematics*. Moscow: Moscow Press.Google Scholar - 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: Erlbaum.Google Scholar - Dougherty, B., & Venenciano, L. (2007). Measure up for understanding.
*Teaching Children Mathematics, 13*(9), 452–456.Google Scholar - 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.Google Scholar - Flora, D. B., & Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data.
*Psychological Methods, 9*, 466–491.CrossRefGoogle Scholar - Hart, K., Brown, M., Kerslake, D., Küchemann, D., & Ruddock, G. (1985).
*Chelsea diagnostic mathematics tests*[Student assessment]. (Reprinted from by NFER-Nelson), Nottingham: K. Hart, Shell Centre for Mathematical Education, University of Nottingham.Google Scholar - Hu, L. T., & Bentler, P. M. (1999). Cut-off criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives.
*Structural Equation Modeling, 6*, 1–55.CrossRefGoogle Scholar - Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 707–762). Reston: National Council of Teachers of Mathematics.Google Scholar - Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2011). Middle school students’ understanding of core algebraic concepts: Equivalence & variable. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 259–276). New York: Springer.Google Scholar - Krutetskii, V.A. (1976). The psychology of mathematical abilities in school children. In J. Kilpatrick & I. Wirszup (Eds.), J. Teller (Trans.). Chicago: University of Chicago Press.Google Scholar
- Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.),
*Children’s understanding of mathematics: 11–16*(pp. 102–119). Oxford: Alden Press.Google Scholar - Loehlin, J. C. (1992).
*Latent variable models: An introduction to factor, path, and structural analysis*. Mahwah: Erlbaum.Google Scholar - Morris, A., & Sloutsky, V. (1995, October).
*Development of algebraic reasoning in children and adolescents: A cross-cultural and cross-curricular perspective*. Paper presented at the annual meeting of the North American Chapter of the International Group of the Psychology of Mathematics Education, 17th PME-NA, Columbus, OH.Google Scholar - Muthén, L. K., & Muthén, B. O. (1998-2012).
*Mplus user’s guide*. Seventh edition. Los Angeles, CA: Muthén & Muthén.Google Scholar - Pedhazur, E. J., & Schmelkin, L. P. (1991).
*Measurement, design, and analysis: An integrated approach*(pp. 213–215). Hillsdale: Erlbaum.Google Scholar - Radford, L. (2007). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts.
*ZDM Mathematics Education*. doi: 10.1007/s11858-007-0061-0 Google Scholar - Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 303–322). New York: Springer.Google Scholar - Raykov, T. (2007). Reliability if deleted, not ‘alpha if deleted’: Evaluation of scale reliability following component deletion.
*The British Psychological Society, 60*, 201–216.Google Scholar - Schliemann, A. D., Carraher, D. W., Brizuela, B. M., Earnest, D., Goodrow, A., Lara-Roth, S., et al. (2003). Algebra in elementary school. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.),
*Proceedings of the 2003 Joint Meeting of PME and PME-NA: Vol. 4*(pp. 127–134). Honolulu, HI: CRDG, College of Education, University of Hawai‘i.Google Scholar - Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy.
*European Journal of Psychology of Education, 19*(1), 19–43.CrossRefGoogle Scholar - Schmittau, J. (2011). The role of theoretical analysis in developing algebraic thinking: A Vygotskian perspective. In J. Cai & E. Knuth (Eds.),
*Early algebraization, A global dialogue from multiple perspectives*(pp. 71–85). New York: Springer.Google Scholar - Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov.
*The Mathematics Educator, 8*, 60–87.Google Scholar - 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, 4*(pp. 273–280). Morelia, México: Cinvestav-UMSNH.Google Scholar - Smith, J., & Thompson, P. (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., & Izsák, A. (2002). Pre-service middle-school teachers’ construction of linear equation concepts through quantitative reasoning. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Noony (Eds.),
*Proceedings of the Twenty-Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 1163–1172). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar - Tatsuoka, K. K., Corter, J. E., & Tatsuoka, C. (2004). Patterns of diagnosed mathematical content and process skills in TIMSS–R across a sample of 20 countries.
*American Educational Research Journal, 41*, 901–926.CrossRefGoogle Scholar - TIMSS 1999 Mathematics Items: Released Set for Eighth Grade. (2001). International Association for the Evaluation of Educational Achievement (IEA). TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College. [Assessment items, documentation, and results]. Retrieved from http://timssandpirls.bc.edu/timss1999i/pdf/t99math_items.pdf
- Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary grades mathematics.
*For the Learning of Mathematics*, 34(1), 18–24.Google Scholar - Venenciano, L., Dougherty, B., & Slovin, H. (2012).
*The Measure Up program, prior achievement, and logical reasoning as indicators of algebra preparedness*. Paper presented at the 12th International Congress on Mathematical Education (ICME-12), Seoul. Google Scholar - Vygotsky, L. (1978). In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.),
*Mind in society: The development of higher psychological processes*. Cambridge: Harvard University.Google Scholar - Wegmann, K. M., Thompson, A. M., & Bowen, N. K. (2011). A confirmatory factor analysis of home environment and home social behavior data from the elementary school success profile for families.
*Social Work Research, 35*, 117–127.CrossRefGoogle Scholar - Yu, C-Y. (2002).
*Evaluating cutoff criteria of model fit indices for latent variable models with binary and continuous outcomes*(Doctoral dissertation). University of California, Los Angeles. Retrieved from http://www.statmodel.com/download/Yudissertation.pdf - Zukerman, G. (2004). Development of reflection through learning activity.
*European Journal of Psychology of Education, 14*, 9–18.CrossRefGoogle Scholar