Longitudinal Comparison of Montessori versus Non-Montessori Students’ Place-Value and Arithmetic Knowledge
Keywords:arithmetic, base-10, place-value, Montessori, early childhood mathematics
Base-10 and place value understanding are important foundational math concepts that are associated with higher use of decomposition strategies and higher accuracy on addition problems (Laski, Ermakova, & Vasilyeva, 2014; Fuson, 1990; Fuson & Briars, 1990; National Research Council, 2001). The current study examined base-10 knowledge, place value, and arithmetic accuracy and strategy use for children in early elementary school from Montessori and non-Montessori schools. Children (N = 150) were initially tested in either kindergarten or first grade. We followed up with a subgroup of the sample (N = 53) two years later when the children were in 2nd and 3rd grade. Although Montessori curriculum puts a large emphasis on the base-10 structure of number, we found that children from Montessori schools only showed an advantage on correct use of base-10 canonical representation in kindergarten but not in first grade. Moreover, there were no program differences in place value understanding in 2nd and 3rd grade. Although Montessori children used different strategies to obtain answers to addition problems in 2nd and 3rd grade as compared with non-Montessori children, there were no program differences in addition accuracy at any grade level. Educational implications are discussed.
Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: a test of three verification models. Memory & Cognition, 9(2), 185–196. doi:10.3758/BF03202334
Aunola, K., Leskinen, E., Lerkkanen, M.K., & Nurmi, J.E. (2004). Developmental dynamics of math performance from preschool to grade 2. Journal of Educational Psychology, 96, 699-713.
Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta- analysis of the efficacy of teaching mathematics with con- crete manipulatives. Journal of Educational Psychology, 105, 380-400
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A Longitudinal Study of Invention and Understanding in Children’s Multidigit Addition and Subtraction. Journal for Research in Mathematics Education, 29(1), 3–20.
Carr, M., & Alexeev, N. (2011). Fluency, accuracy, and gender predict developmental trajectories of arithmetic strategies. Journal of Educational Psychology, 103(3), 617–631. doi:10.1037/a0023864
Carr, M., Steiner, H.H., Kyser, B., & Biddlecomb, B. (2008). A comparison of predictors of early emerging gender differences in mathematics competency. Learning and Individual Differences, 18, 61–75.
Cauley, K. M. (1988). Construction of logical knowledge: Study of borrowing in subtraction. Journal of Educational Psychology, 80(2), 202–205.
Cobb, P., & Wheatley, G. (1988). Children's initial understandings of ten. Focus on Learning Problems in Mathematics, 10, 1-28.
Cowan, R., Donlan, C., Shepherd, D.-L., Cole-Fletcher, R., Saxton, M., & Hurry, J. (2011). Basic Calculation Proficiency and Mathematics Achievement in Elementary School Children. Journal of Educational Psychology. doi:10.1037/a0024556
DeLoache, J. S., Peralta de Mendoza, O. A., & Anderson, K. N. (1999). Multiple factors in early symbol use: Instructions, similarity, and age in understanding a symbol-referent relation. Cognitive Development, 14, 299-312.
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., … Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–46. doi:10.1037/0012-16188.8.131.528
Fennema, E., Carpenter, T. P., Jacobs, V. R., Franke, M. L., & Levi, L. (1998). A longitudinal study of gender differences in young children’s mathematical thinking. Educational Researcher, 27, 6–11.
Fuson, K. C. (1986). Roles of representation and verbalization in the teaching of multidigit
addition and subtraction. European Journal of Psychology of Education, 1, 35-56.
Fuson, K. C. (1988). Children’s counting and concepts of number. New York, NY: Springer–Verlag.
Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343–403
Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first-and second-grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21(3), 180–206.
Fuson, K. C., & Li, Y. (2009). Cross-cultural issues in linguistic, visual-quantitative, and written-numeric supports for mathematical thinking. ADM Mathematics Education, 41(6), 793–808. doi:10.1007/s11858-009-0183-7
Fuson, K. C., Smith, S. T., & Lo Cicero, A. (1997). Supporting Latino first graders’ ten-structured thinking in urban classrooms. Journal for Research in Mathematics Education, 28, 738–760.
Geary, D. C., Bow-Thomas, C., Liu, F., & Siegler, R. S. (1996). Development of arithmetical competencies in Chinese and American children: Influence of age, language, and schooling. Child Development, 67,2022–2044.
Geary, D., Fan, L., & Bow-Thomas, C. (1992). Numerical cognition: Loci of ability differ- ences comparing children from China and the US. Psychological Science, 3,180–185.
Geary, D. C., Hoard, M. K., Byrd-Craven, J., & DeSoto, M. C. (2004). Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability. Journal of Experimental Child Psychology, 88(2), 121–51. doi:10.1016/j.jecp.2004.03.002
Ginsburg, H.P. (1989). Children's Arithmetic. (2nd Ed.) Austin, TX: Pro-Ed.
Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Brenwald, S. (2009). Highlights from TIMSS 2007 mathematics and science achievement of U.S. fourth- and eighth-grade students in an international context. Washington, DC: National Center for Education Statistics, Institute of Education Sciences, U.S. Dept. of Education.
Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98-122.
Jordan, N. C., Kaplan, D., Olah, L. N., & Locuniak, M. N. (2006). Number sense growth in kindergarten?: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77(1), 153–175.
Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early Math Matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45(3), 850–867. doi:10.1037/a0014939.
Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1(2), 75–86.
Kilpatrick, J. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Laski, E. V., Ermakova, A., & Vasilyeva, M. (2014). Early use of decomposition for addition and its relation to base-10 knowledge. Journal of Applied Developmental Psychology, 35(5), 444–454. doi:10.1016/j.appdev.2014.07.002
Laski, E. V., Jor’dan, J. R., Daoust, C., & Murray, a. K. (2015). What Makes Mathematics Manipulatives Effective? Lessons From Cognitive Science and Montessori Education. SAGE Open, 5(2). doi:10.1177/2158244015589588
Laski, E. V., & Siegler, R. S. (2014). Learning from number board games: you learn what you encode. Developmental Psychology, 50(3), 853–864.
Lillard, A. S. (2005). Montessori: The science behind the genius. New York: Oxford University Press.
Miura, I. T., & Okamoto, Y. (1989). Comparisons of US and Japanese first graders' cognitive representation of number and understanding of place value. Journal of Educational Psychology, 81(1), 109–114.
Miura, I., Okamoto, Y., Kim, C., Steere, M., & Fayol, M. (1993). First graders' cognitive representation of number and understanding of place value: Cross-national comparisons —France, Japan, Korea, Sweden, and the United States. Journal of Educational Psychology, 1993.
Mix, K., Prather, R., Smith, L. B. & Stockton, J. (2014) Young Children's Interpretation of Multi-Digit Number Names: From Emerging Competence to Mastery. Child Development, 85(3), 1306-1319.
Montessori, M., & Simmonds, F. (1917). The advanced Montessori method: Scientific pedagogy as applied to the education of children from seven to eleven years. London: W. Heinemann.
National Mathematics Advisory Panel. (2008). The Final Report of the National Mathematics Advisory Panel. Foundations, 37(9), 595–601. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Perry, M. (2000). Explanations of mathematical concepts in Japanese, Chinese, and US first- and fifth-grade classrooms. Cognition and Instruction, 18(2), 181–207.
Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In Glaser, R. (Ed.), (1987). Advances in instructional psychology, Vol. 3 (pp. 41e96). Hillsdale, NJ:
Ross, B. H. (1989). Distinguishing types of superficial similarities: Different effects on access and use of earlier problems. Journal of Experimental Psychology: Learning, Memory, and
Cognition, 15, 456-468.
Ross, S. H. (1990). Children’s acquisition of place-value numeration concepts: The roles of cognitive development and instruction. Focus on Learning Problems in Mathematics, 12, 1–17.
Saxton, M., & Towse, J. N. (1998). Linguistic relativity: The case of place value in multi- digit numbers. Journal of Experimental Child Psychology, 69(1), 66–79.
Shrager, J., & Siegler, R. S. (1998). A model of children’s strategy choices and strategy discoveries. Psychological Science, 9(5), 405–410.
Siegler, R. S., & Ramani, G. (2009). Playing linear board games--but not circular ones--improve preschoolers’ numerical understanding. Journal of Educational Psychology. 101, 545– 560. doi:10.1037/a0014239
Stevenson, H. W., & Newman, R. S. (1986). Long-term prediction of achievement and attitudes in mathematics and reading. Child Development, 57(3), 646–659. doi:10.2307/1130343
Torbeyns, J., Verschaffel, L., & Ghesquiere, P. (2004). Strategic aspects of simple addition and subtraction: the influence of mathematical ability. Learning and Instruction, 14(2), 117-195.
U. S. Department of Education (2013). Mathematics 2013: National Assessment of Educational Progress at Grades 4 and 8. Washington, D.C.
Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L. and DeLoache, J. (2009), Dual representation and the linking of concrete and symbolic representations. Child Development Perspectives, 3: 156–159. doi: 10.1111/j.1750-8606.2009.00097.x
Varelas, M., & Becker, J. (1997). Children’s developing understanding of place value: Semiotic aspects. Cognition and Instruction, 15, 265–286.
Vasilyeva, M., Laski, E. V., Ermakova, A., Lai, W.-F., Jeong, Y., & Hachigan, A. (2015). Reexamining the language account of cross-national differences in base-10 number representations. Journal of Experimental Child Psychology, 129, 12–25.
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