We share here results from a quasi-experimental study that examines growth in students’ algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3–5. Analyses showed that, while there were no significant differences between experimental and control students on a grade 3 pre-assessment measuring students’ capacity for generalizing and representing generalizations, experimental students significantly outperformed control students on post-assessments at each of grades 3–5. Moreover, experimental students were able to more flexibly interpret variable in different roles and were better able to use variable notation in meaningful ways to represent arithmetic properties, expressions and equations, and functional relationships. This study provides important evidence that young children can learn to think algebraically in powerful ways and suggests that the earlier introduction of algebraic concepts and practices is beneficial to students.
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By early algebra, we mean algebraic thinking in the elementary grades (i.e., grades K–5)
In connection to this learning progression approach, the suite of projects in which the study reported here occurred is referred to collectively as Project LEAP (Learning through an Early Algebra Progression). Hereafter, we refer to the components of this learning progression using the LEAP moniker (e.g., LEAP instructional sequence, LEAP assessments).
In our prior work (phase 1), we included variable and proportional reasoning as Big Ideas. However, as noted in Blanton, Stephens et al., (2015), proportional reasoning was not explicitly addressed in our grades 3–5 instructional sequence (its algebraic significance arises more naturally in middle grades), and was subsequently removed as a significant content domain for grades 3–5 in our revised EALP (phase 2). Moreover, because concepts associated with variable were integrated organically throughout the other Big Ideas (GA, EEEI, and FT) in our instructional sequence, we elected not to treat variable as a distinct Big Idea in our revised EALP
Grade 2 was included to assess the extent to which students received instruction in algebraic concepts and practices prior to the start of the study.
Grade 2 teachers were administered the survey in year 1 of the study.
We use variable equation to mean an equation that contains one or more variables, where the role of variable may be that of fixed unknown, varying quantity, or generalized number (Blanton, Levi, Crites, & Dougherty, 2011). Here, for example, the variable equation representing the commutative property of addition could be a + b = b + a, for real numbers a and b
Adapted from Carraher, Schliemann, and Schwartz (2008).
This item is not included in the longitudinal analysis here because it is not a common item.
We have observed this same anomaly in an ongoing follow-up study, where preliminary analysis shows that 58% of experimental students correctly represented the rule with variable notation for this item, while only 33% could do so with words. Differences for control students also favored variable notation (25%) over words (21%), although the margin was quite small.
Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221–247. https://doi.org/10.1080/10986060701360902
Baroody, A. J., & Ginsburg, H. P. (1983). The effects of instruction on children’s understanding of the “equals” sign. The Elementary School Journal, 84, 199–212. https://doi.org/10.1086/461356
Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 165–184). New York, NY: Lawrence Erlbaum Associates.
Blanton, M., Brizuela, B., Gardiner, A., & Sawrey, K. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, 95(2), 181–202. https://doi.org/10.1007/s10649-016-9745-0
Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511–558.
Blanton, M., Brizuela, B., Stephens, A., Knuth, E., Isler, I., Gardiner, A., Stroud, R., Fonger, N., & Stylianou, D. (2018). Implementing a framework for early algebra. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 27–49). Hamburg, Germany: Springer.
Blanton, M., Levi, L., Crites, T., & Dougherty, B. (2011). Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3–5. Essential Understanding Series. Reston, VA: National Council of Teachers of Mathematics.
Blanton, M., Otalora Sevilla, Y., Brizuela, B., Gardiner, A., Sawrey, K., Gibbons, A., & Yangsook, K. (2018). Exploring kindergarten students’ early understandings of the equal sign. Mathematical Thinking and Learning, 20(3), 167–201. https://doi.org/10.1080/10986065.2018.1474534
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. https://doi.org/10.5951/jresematheduc.46.1.0039
Blanton, M., Stroud, R., Stephens, A., Gardiner, A., Stylianou, D., Knuth, E., Isler, I., & Strachota, S. (2019). Does early algebra matter? The effectiveness of an early algebra intervention in grades 3–5. American Educational Research Journal. https://doi.org/10.3102/0002831219832301.
Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization: A global dialogue from multiple perspectives. Heidelberg, Germany: Springer.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
Carraher, D., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669–705). Charlotte: Information Age.
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.
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/Taylor & Francis Group.
Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89.
Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45(2), 443–494. https://doi.org/10.3102/0002831207312908
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). Heidelberg: Springer.
Daro, P., Mosher, F., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. Retrieved from http://www.cpre.org.
Fonger, N. L., Stephens, A., Blanton, M., Isler, I., Knuth, E., & Gardiner, A. (2018). Developing a learning progression for curriculum, instruction, and student learning: An example from early algebra research. Cognition and Instruction, 36(1), 30–55. https://doi.org/10.1080/07370008.2017.1392965
Jones, I., Inglis, M., Gilmore, C., & Dowens, M. (2012). Substitution and sameness: Two components of a relational conception of the equals sign. Journal of Experimental Child Psychology, 113(1), 166–176.
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). Mahwah, NJ: Lawrence Erlbaum Associates/Taylor & Francis Group.
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, NY: Lawrence Erlbaum Associates.
Kieran, C. (Ed.). (2018). Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice. Hamburg, Germany: Springer.
Knuth, E., Stephens, A., McNeil, N., & Alibali, M. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.
Maloney, A. P., Confrey, J., & Nguyen, K. (Eds.). (2011). Learning over time: Learning trajectories in mathematics education. Charlotte, NC: Information Age.
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, the Netherlands: Springer.
Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316–350.
Morris, A. K. (2009). Representations that enable children to engage in deductive arguments. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K–16 perspective (pp. 87–101). Mahwah, NJ: Taylor & Francis Group.
Moses, R. P., & Cobb, C. E. (2001). Radical equations: Math literacy and civil rights. Boston, MA: Beacon Press.
Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008). “What is your theory? What is your rule?”: Fourth graders build an understanding of functions through patterns and generalizing problems. In C. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 70th yearbook (pp. 155–168). Reston, VA: National Council of Teachers of Mathematics.
Museus, S., Palmer, R. T., Davis, R. J., & Maramba, D. C. (2011). Racial and ethnic minority students' success in STEM education. Hoboken, NJ: Jossey-Bass.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Council of Chief State School Officers Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
O’Donnell, C. (2008). Defining, conceptualizing, and measuring fidelity of implementation and its relationship to outcomes in K-12 curriculum intervention research. Review of Educational Research, 78(1), 33–84.
Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37–62.
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, NJ: Lawrence Erlbaum Associates.
Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology, 103, 85–104. https://doi.org/10.1037/a0021334
Russell, S. J., Schifter, D., Kasman, R., Bastable, V., & Higgins, T. (2017). But why does it work?: Mathematical argument in the elementary classroom. Portsmouth, NH: Heinemann.
Schifter, D. (1999). Reasoning about operations: Early algebraic thinking in grades K–6. In L. V. Stiff & F. R. Curio (Eds.), Developing mathematical reasoning in grades K–12: 1999 yearbook (pp. 62–81). Reston, VA: National Council of Teachers of Mathematics.
Schifter, D. (2009). Representation-based proof in the elementary grades. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across grades: A K–16 perspective (pp. 71–86). New York: Routledge.
Schifter, D., Monk, S., Russell, S. J., & Bastable, V. (2008). Early algebra: What does understanding the laws of arithmetic mean in the elementary grades? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 413–447). New York, NY: Lawrence Erlbaum Associates.
Schoenfeld, A. H. (1995). Is thinking about 'Algebra' a misdirection? In C. Lacampagne, W. Blair, & J. Kaput (Eds.), The algebra colloquium. Volume 2: Working group papers (pp. 83–86). Washington, DC: US Department of Education, Office of Educational Research and Improvement.
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. https://doi.org/10.1007/BF00302715
Shin, N., Stevens, S. Y., Short, H., & Krajcik, J. S. (2009). Learning progressions to support coherence curricula in instructional material, instruction, and assessment design. Paper presented at the Learning Progressions in Science (LeaPS) Conference, Iowa City, IA.
Stephens, A., Knuth, E., Blanton, M., Isler, I., Gardiner, A., & Marum, T. (2013). Equation structure and the meaning of the equal sign: The impact of task selection in eliciting elementary students’ understandings. The Journal of Mathematical Behavior, 32(2), 173–182.
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 (pp. 386–420). Reston, VA: National Council of Teachers of Mathematics.
Stephens, A. C., Fonger, N. L., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Gardiner, A. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143–166. https://doi.org/10.1080/10986065.2017.1328636
Stevens, S. Y., Shin, N., & Krajcik, J. S. (2009). Towards a model for the development of an empirically tested learning progression. Paper presented at the learning progressions in science (LeaPS) conference, Iowa City, IA.
The research reported here was supported in part by the National Science Foundation under Awards No. 1219605 and 1219606. 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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Blanton, M., Isler-Baykal, I., Stroud, R. et al. Growth in children’s understanding of generalizing and representing mathematical structure and relationships. Educ Stud Math 102, 193–219 (2019). https://doi.org/10.1007/s10649-019-09894-7