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A progression in first-grade children’s thinking about variable and variable notation in functional relationships

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Abstract

Recent research suggests that children in elementary grades have some facility with variable and variable notation in ways that warrant closer attention. We report here on an empirically developed progression in first-grade children’s thinking about these concepts in functional relationships. Using learning trajectories research as a framework for the study, we developed and implemented an instructional sequence designed to foster children’s understanding of functional relationships. Findings suggest that young children can learn to think in sophisticated ways about variable quantities and variable notation. This challenges assumptions that young children are not “ready” for a study of such concepts and raises the question of whether difficulties adolescents exhibit might be ameliorated by an earlier introduction to these ideas.

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Notes

  1. By variable notation, we refer here to literal symbols, or letters, used to represent variable quantities.

  2. We refer to variable or variable quantity interchangeably throughout this paper.

  3. We expected students to represent a function of the type y = mx additively, if at all (e.g., y = 2x might be represented as y = x + x). A function of the type y = mx + b (m ≠ 1, 0; b ≠ 0) was not addressed until the post-interviews.

  4. “Upper 30%” should not be construed to mean interviewees represented high-performing students overall. One of the two participating schools was designated as low performing based on students’ performance on the state’s standardized assessment.

  5. For more details on the instructional sequence and interview protocols, see Blanton, Brizuela, et al. (2015).

  6. Videotapes of CTE lessons were reviewed as a secondary data source for confirming or disconfirming evidence, but not formally analyzed for this study. The design and implementation of CTE lessons is, however, discussed at length here because the concepts addressed in our CTE are not taught in Grade 1 in the US.

  7. Although W is the 23rd (not 21st) letter of the alphabet, correcting this in Rebecca’s thinking was not critical to us in the interview.

References

  • Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the Learning of Mathematics, 25(2), 42–47.

    Google Scholar 

  • Barrett, J., & Battista, M. (2014). Two approaches to describing the development of students’ reasoning about length: A case study for coordinated related trajectories. In A. Maloney, J. Confrey, & K. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 97–124). Charlotte: Information Age.

    Google Scholar 

  • Battista, M. (2004). Applying cognition-based assessment to elementary school students’ development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185–204. doi:10.1207/s15327833mtl0602_6

    Article  Google Scholar 

  • Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in six-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511–558. doi:10.5951/jresematheduc.46.5.0511

    Article  Google Scholar 

  • Blanton, M., Stephens, A., Knuth, E., Gardiner, A., Isler, I., & Kim, J. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87. doi:10.5951/jresematheduc.46.1.0039

    Article  Google Scholar 

  • Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford & A. P. Schulte (Eds.), The ideas of algebra, K–12 (pp. 20–32). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Braddon, K. L., Hall, N. J., & Taylor, D. (1993). Math through children’s literature: Making the NCTM standards come alive. Englewood: Teacher Ideas Press.

    Google Scholar 

  • Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Gardiner, A. (2015). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17, 1–30. doi:10.1080/10986065.2015.981939

    Article  Google Scholar 

  • Brizuela, B. M., & Earnest, D. (2008). Multiple notational systems and algebraic understandings: The case of the “best deal” problem. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 273–301). Mahwah: Lawrence Erlbaum and Associates.

    Google Scholar 

  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.

    Google Scholar 

  • Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669–705). Charlotte: Information Age Publishing.

    Google Scholar 

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

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 235–272). Mahwah, NJ: Lawrence Erlbaum Associates.

  • Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89. doi:10.1207/s15327833mtl0602

    Article  Google Scholar 

  • Clements, D. H., & Sarama, J. (2014). Learning trajectories: Foundations for effective, research-based education. In A. P. Maloney, J. Confrey & K. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 1–30). Charlotte: Information Age.

    Google Scholar 

  • Cobb, P., & Steffe, L. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83–94.

    Article  Google Scholar 

  • Confrey, J., Maloney, A., & Nguyen, K. (2014). Introduction: Learning trajectories in mathematics. In A. Maloney, J. Confrey, & K. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. xi–xxii). Charlotte: Information Age.

    Google Scholar 

  • Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86. doi:10.2307/749228

    Article  Google Scholar 

  • Cooper, T., & 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. doi:10.1007/978-3-642-17735-4_12

    Chapter  Google Scholar 

  • Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction (Research Report No. 68). Philadelphia: Consortium for Policy Research in Education.

  • DiSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10(2–3), 105–225. doi:10.1080/07370008.1985.9649008

    Article  Google Scholar 

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

    Google Scholar 

  • Filloy, E., Rojano, T., & Puig, L. (2008). Educational algebra: A theoretical and empirical approach. New York, NY: Springer. doi:10.1007/978-0-387-71254-3

    Book  Google Scholar 

  • Freudenthal, H. (1982). Variables and functions. In G. V. Barneveld & H. Krabbendam (Eds.), Proceedings of conference on functions (pp. 7–20). Enschede: National Institute for Curriculum Development.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 127–140). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Glaser, B. G. (1998). Doing grounded theory: Issues and discussions. Mill Valley: Sociology Press.

    Google Scholar 

  • Glaser, B. G. (2002). Conceptualization: On theory and theorizing using grounded theory. International Journal of Qualitative Methods, 1(2), 23–38.

    Article  Google Scholar 

  • Hackenberg, A., & Lee, M. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196–243. doi:10.5951/jresematheduc.46.2.0196

    Article  Google Scholar 

  • Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kaput, J., Blanton, M., & Moreno, L. (2008). Algebra from a symbolization point of view. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 19–55). New York: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kieran, C. (2007). Learning and teaching 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). Greenwich: Information Age Publishing.

    Google Scholar 

  • Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.

    Google Scholar 

  • Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students’ understanding of core algebraic concepts: Equality and variable. International Reviews on Mathematical Education, 37, 1–9.

    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 and variable. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives. Advances in Mathematics Education Monograph Series (pp. 259–276). New York: Springer. doi:10.1007/978-3-642-17735-4_15

    Chapter  Google Scholar 

  • Küchemann, D. E. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics (pp. 102–119). London: Murray.

    Google Scholar 

  • Lucariello, J., & Tine, M. (2011). Algebraic misconceptions: A teacher use test for diagnosing student misconceptions of the variable. In N. L. Stein & S. Raudenbush (Eds.), Developmental science goes to school (pp. 250–266). New York: Taylor & Francis.

    Google Scholar 

  • MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19. doi:10.1023/A:1002970913563

    Article  Google Scholar 

  • Maloney, A., Confrey, J., & Nguyen, K. (2014). Learning over time: Learning trajectories in mathematics education. Charlotte: Information Age.

    Google Scholar 

  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 65–86). Netherlands: Springer. doi:10.1007/978-94-009-1732-3_5

    Chapter  Google Scholar 

  • McNeil, N., Weinberg, A., Hattikudar, S., Stephens, A. C., Asquith, P., Knuth, E., & Alibali, M. W. (2010). A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102(3), 625–634. doi:10.1037/a0019105

  • Moss, J., & McNab, S. (2011). An approach to geometric and numeric patterning that fosters second grade students’ reasoning and generalizing about functions and co-variation. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 277–298). New York, NY: Springer. doi:10.1007/978-3-642-17735-4_16

    Chapter  Google Scholar 

  • National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

  • Piaget, J. (1964). Developing and learning: Conference on cognitive studies and curriculum development. Ithaca: Cornell University Press.

    Google Scholar 

  • Radford, L. (2001). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237–268. doi:10.1023/A:1017530828058

    Article  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. doi:10.1007/978-3-642-17735-4_17

    Chapter  Google Scholar 

  • Resnick, L. B. (1982). Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 136–155). Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Schliemann, A. D., Carraher, D., & Brizuela, B. M. (2007). Bringing Out the algebraic character of arithmetic: From Children’s ideas to classroom practice. Mahwah: Lawrence Erlbaum and Associates.

    Google Scholar 

  • Schwartz, J. (1990). Getting students to function in and with algebra. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 261–289). Washington, DC: Mathematics Associations of America.

    Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. doi:10.1007/BF00302715

    Article  Google Scholar 

  • Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205

    Article  Google Scholar 

  • Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for research in mathematics education. National Council of Teachers of Mathematics: Reston. In press.

  • Strauss, A. L., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Los Angeles: Sage.

    Google Scholar 

  • Szilágyi, J., Clements, D., & Sarama, J. (2013). Young children’s understandings of length measurement: Evaluating a learning trajectory. Journal for Research in Mathematics Education, 44(3), 581–620. doi:10.5951/jresematheduc.44.3.0581

    Article  Google Scholar 

  • Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49(3), 341–359. doi:10.1023/A:1020229023965

    Article  Google Scholar 

  • Vygotsky, L. (1978). Mind in society (M. Cole, S. Scribner, V. John-Steiner, & E. Souberman, Trans.). Cambridge, MA: Harvard University. (Original work published 1934)

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Acknowledgements

The research reported here was supported in part by the National Science Foundation under DRL 1154355. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. TERC, 2067 Massachusetts Avenue, Cambridge, MA, 02140, USA

    Maria Blanton & Angela Murphy Gardiner

  2. Department of Education, Paige Hall, Tufts University, Medford, MA, 02155, USA

    Bárbara M. Brizuela, Katie Sawrey & Ashley Newman-Owens

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  1. Maria Blanton
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  2. Bárbara M. Brizuela
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  3. Angela Murphy Gardiner
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  4. Katie Sawrey
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  5. Ashley Newman-Owens
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Correspondence to Maria Blanton.

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Blanton, M., Brizuela, B.M., Gardiner, A.M. et al. A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educ Stud Math 95, 181–202 (2017). https://doi.org/10.1007/s10649-016-9745-0

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