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An Approach to Geometric and Numeric Patterning that Fosters Second Grade Students’ Reasoning and Generalizing about Functions and Co-variation

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Early Algebraization

Part of the book series: Advances in Mathematics Education ((AME))

Abstract

In this chapter, we present illustrations of second grade students’ reasoning about patterns and two-part function rules in the context of an early algebra research project that we have been conducting in elementary schools in Toronto and New York City. While the study of patterns is mandated in many countries as part of initiatives to include algebra from K-12, there is a plethora of evidence that suggests that the route from patterns to algebra can be challenging even for older students. Our teaching intervention was designed to foster in students an understanding of linear function and co-variation through the integration of geometric and numeric representations of growing patterns. Six classrooms from diverse urban settings participated in a 10–14-week intervention. Results revealed that the intervention supported students to engage in functional reasoning and to identify and express two-part rules for geometric and numeric patterns. Furthermore, the students, who had not had formal instruction in multiplication prior to the intervention, invented mathematically sound strategies to deconstruct multiplication operations to solve problems. Finally, the results revealed that the experimental curriculum supported students to transfer their understanding of two-part function rules to novel settings.

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References

  • Amit, M., & Neria, D. (2008). “Rising to challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 111–129.

    Article  Google Scholar 

  • Beatty, R., & Moss, J. (2006a). Grade 4 generalizing through online collaborative problem solving. Paper presented at the annual meeting of the American Educational Research Association, April 2006.

    Google Scholar 

  • Beatty, R., & Moss, J. (2006b). Connecting grade 4 students from diverse urban classrooms: virtual collaboration to solve generalizing problems. In C. Hoyles & J. Lagrange (Eds.), Proceedings of the Seventeenth ICMI Study: Technology Revisited. Hanoi, Vietnam.

    Google Scholar 

  • Beatty, R., Moss, J., & Scardamalia (2006). Grade 4 generalizing through online collaborative problem solving. Paper presented at the annual meeting of the American Educational Research (April 2006), San Francisco, CA.

    Google Scholar 

  • Beatty, R., Moss, J., & Bruce, C. (2008). Bridging the research/practice gap: Planning for large-scale dissemination of a new curriculum for patterning and algebra. Paper presented at NCTM Research Presession, Salt Lake City.

    Google Scholar 

  • Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. Jonsen Hoines & A. Hoines (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education.

    Google Scholar 

  • Cai, J., & Knuth, E. (2005). The development of students’ algebraic thinking in earlier grades from curricular, instructional and learning perspectives. ZDM, 37(1), 1–3.

    Article  Google Scholar 

  • Carraher, D. W., & Earnest, D. (2003). Guess my rule revisited. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 173–180). Honolulu: University of Hawaii.

    Google Scholar 

  • Carraher, D., Schlieman, A., Bruzuella, B., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 37(2), 87–115.

    Google Scholar 

  • Carraher, D., Martinez, M., & Schlieman, A. (2008). Early algebra and mathematical generalization. ZDM Mathematics Education, 40, 3–22.

    Article  Google Scholar 

  • Case, R., & Okamoto, Y. (1966). Monographs of the Society for Research in Child Development: Vol. 61. The Role of Central Conceptual Structures in the Development of Children’s Thought. (1-2, Serial No. 246).

    Google Scholar 

  • Cooper, T., & Warren, E. (2008). The effect of different representations on Years 3 to 5 students’ ability to generalise. ZDM Mathematics Education, 40, 23–37.

    Article  Google Scholar 

  • Dorfler, W. (2008). On route from patterns to algebra: comments and reflections. ZDM Mathematics Education, 40, 143–160.

    Article  Google Scholar 

  • English, L. D., & Warren, E. (1998). Introducing the variable through pattern exploration. The Mathematics Teacher, 91(2), 166–171.

    Google Scholar 

  • Ferrini-Mundy, J., Lappan, G., & Phillips, E. (1997). Experiences with patterning. In Teaching Children Mathematics, Algebraic Thinking Focus Issue (pp. 282–288). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Ginsburg, & Seo (1999). Mathematics in children’s thinking. Journal Mathematical Thinking and Learning, 1(2), 113–129.

    Article  Google Scholar 

  • Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through Algebra in Prekindergarten—Grade 2. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Griffin, S., & Case, R. (1997). Re-thinking the primary school math curriculum an approach based on cognitive science. Issues in Education, 3(1), 1–49.

    Google Scholar 

  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.

    Article  Google Scholar 

  • Hewitt, D. (1992). Train spotters’ paradise. Mathematics Teaching, 140, 6–8.

    Google Scholar 

  • Kalchman, M., Moss, J., & Case, R. (2001). Psychological models for the development of mathematical understanding: Rational numbers and functions. In S. Carver & D. Klahr (Eds.), Cognition and Instruction: 25 Years of Progress.

    Google Scholar 

  • Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), The Handbook of Research on Mathematics Teaching and Learning (pp. 390–419). New York: Macmillan.

    Google Scholar 

  • Kuchemann, D. (2008). Looking for structure: A report of the proof materials project. BookPrintingUK.com Peterborough PE29JX UK.

    Google Scholar 

  • Lannin, J. K. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 343–348.

    Google Scholar 

  • Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through pattern activities. Mathematical Thinking and Learning, 7(3), 231–258.

    Article  Google Scholar 

  • Lannin, J., Barker, D., & Townsend, B. (2006). Recursive and explicit rules: How can we build student algebraic understanding. Journal of Mathematical Behavior, 25, 299–317.

    Article  Google Scholar 

  • Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: Perspectives for Research and Teaching (pp. 65–86). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Lee, L., & Wheeler, D. (1987). Algebraic Thinking in High School Students: Their Conceptions of Generalisation and Justification (Research Report). Concordia University, Montreal, Canada.

    Google Scholar 

  • London McNab, S., & Moss, J. (2004). Mathematical modeling through knowledge building: Explorations of algebra, numeric functions and geometric patterns in grade 2 classrooms. In D. McDougall & J.A. Ross (Eds.), Proceedings of the Twenty-Sixth Annual Meeting North American Chapter of the International Group for the Psychology of Mathematics Education, Toronto, Canada, Vol. 2.

    Google Scholar 

  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: Perspectives for Research and Teaching (pp. 65–86). Dordrecht, Netherlands: Kluwer.

    Google Scholar 

  • Moss, J. (2004). Pipes, tubes, and beakers: Teaching rational number. In J. Bransford & S. Donovan (Eds.), How People Learn: A Targeted Report for Teachers. Washington: National Academy Press.

    Google Scholar 

  • Moss, J. (2005). Integrating numeric and geometric patterns: A developmental approach to young students’ learning of patterns and functions. Paper presented at the Canadian mathematics education study group 29 Annual Meeting at the University of Ottawa, May 27–31.

    Google Scholar 

  • Moss, J., & Beatty, R. (2006a). Knowledge building in mathematics: Supporting collaborative learning in pattern problems. International Journal of Computer Supported Collaborative Learning, 1(4), 441–465.

    Article  Google Scholar 

  • Moss, J., & Beatty, R. (2006b). Multiple vs numeric approaches to developing functional understanding through patterns—affordances and limitations for grade 4 students. To be included in the Proceedings of the Twenty-Eight Annual Meeting of the Psychology of Mathematics Education, North American Chapter. Merida, Mexico.

    Google Scholar 

  • Moss, J., & Case, R. (1999). Developing children’s understanding of rational numbers: A new model and experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122–147 119.

    Article  Google Scholar 

  • Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008). What is your theory? What is your rule? Fourth graders build their understanding of patterns. In C. Greenes (Ed.), NCTM Yearbook for Algebraic Reasoning. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.

    Google Scholar 

  • Mulligan, J. T., Prescott, A., & Mitchelmore, M. C. (2004). Children’s development of structure in early mathematics. In M. Høines & A. Hoines (Eds.), Proceedings of the 28th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 393–401). Bergen, Norway: Bergen University College.

    Google Scholar 

  • National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings. Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33(2), 203–233.

    Article  Google Scholar 

  • Ontario Ministry of Training and Education (2005). The Ontario Curriculum, Grades 1–8: Mathematics, Revised. Queen’s Printer for Ontario.

    Google Scholar 

  • Orton, J. (1997). Matchsticks, pattern and generalisation. Education, 3–13.

    Google Scholar 

  • Orton, A., & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics (pp. 104–120). London: Cassell.

    Google Scholar 

  • Orton, J., Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.), Patterns in the Teaching and Learning of Mathematics (pp. 121–136). London: Cassell.

    Google Scholar 

  • Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.

    Article  Google Scholar 

  • Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM Mathematics Education, 40, 83–96.

    Article  Google Scholar 

  • Rivera, F. (2006). Sixth Graders’ ability to generalize patterns in algebra: issues, insights. In J. Novotná, H. Moraová, M. Kratká, & N. Stehlková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 320). Prague, Czech Republic.

    Google Scholar 

  • Rivera, F., & Becker, J. R. (2007). Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 140–155.

    Google Scholar 

  • Rubenstein, R. (2002). Building explicit and recursive forms of patterns with the function game. Mathematics Teaching in the Middle School, 7(8), 426–431.

    Google Scholar 

  • Sasman, M., Olivier, A., & Linchevski, L. (1999). Factors influencing students’ generalization thinking processes. In O. Zaslavski (Ed.), Proceedings of the 23th International Conference for Psychology of Mathematics Education (Vol. 4, pp. 161–168). Haifa, Israel.

    Google Scholar 

  • Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2001). When tables become function tables. In Proceedings of the XXV Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 145–152). Utrecht, The Netherlands.

    Google Scholar 

  • Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspective. Journal of Mathematical Behavior, 12, 353–383.

    Google Scholar 

  • Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.

    Article  Google Scholar 

  • Stacey, K., & MacGregor, M. (1999). Taking the algebraic thinking out of algebra. Mathematics Education Research Journal, 11(1), 25–38.

    Google Scholar 

  • Steele, D., & Johanning, D. (2004). A schematic-theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57(1), 65–90.

    Article  Google Scholar 

  • Warren, E. (2000). Visualisation and the development of early understanding in algebra. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 273–280). Hiroshima.

    Google Scholar 

  • Warren, E. (2006). Learning comparative mathematical language in the elementary school: A longitudinal study. Educational Studies in Mathematics, 62(2), 169–189.

    Article  Google Scholar 

  • Watson, A. (2000). Going across the grain: Generalisation in a group of low attainers. Nordisk Matematikk Didaktikk (Nordic Studies in Mathematics Education), 8(1), 7–22.

    Google Scholar 

  • Willoughby, S. (1997). Functions from kindergarten through sixth grade. Teaching Children Mathematics, 3, 314–318.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Human Development and Applied Psychology, Ontario Institute for Studies in Education, University of Toronto, Toronto, Canada

    Joan Moss

  2. Ontario Institute for Studies in Education, University of Toronto, Toronto, Canada

    Susan London McNab

Authors
  1. Joan Moss
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  2. Susan London McNab
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Correspondence to Joan Moss .

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Editors and Affiliations

  1. Dept. Mathematical Sciences, University of Delaware, Ewing Hall 501, Newark, 19716, Delaware, USA

    Jinfa Cai

  2. University of Wisconsin-Madison, Teacher Education Building, 225 N. Mills Street, Madison, 53706, Wisconsin, USA

    Eric Knuth

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Moss, J., London McNab, S. (2011). An Approach to Geometric and Numeric Patterning that Fosters Second Grade Students’ Reasoning and Generalizing about Functions and Co-variation. In: Cai, J., Knuth, E. (eds) Early Algebraization. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_16

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