Growth in children’s understanding of generalizing and representing mathematical structure and relationships


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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  1. 1.

    By early algebra, we mean algebraic thinking in the elementary grades (i.e., grades K–5)

  2. 2.

    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).

  3. 3.

    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

  4. 4.

    For more recent reviews of early algebra research, see Cai and Knuth (2011); Carraher and Schliemann (2007), Kieran (2018), and Stephen, Ellis, Blanton, and Brizuela (2017).

  5. 5.

    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.

  6. 6.

    Grade 2 teachers were administered the survey in year 1 of the study.

  7. 7.

    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

  8. 8.

    Adapted from Carraher, Schliemann, and Schwartz (2008).

  9. 9.

    This item is not included in the longitudinal analysis here because it is not a common item.

  10. 10.

    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.


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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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Correspondence to Maria Blanton.

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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).

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  • Algebraic thinking
  • Early algebra
  • Instructional intervention
  • Quantitative methods
  • Elementary grades