Abstract
In this chapter , we discus s the algebra framework that guides our work and how this framework was enacted in the design of a curricular approach for systematically developing elementary-aged students’ algebraic thinking . We provide evidence that, using this approach, students in elementary grades can engage in sophisticated practices of algebraic thinking based on generalizing, representing, justifying, and reasoning with mathematical structure and relationships. Moreover, they can engage in these practices across a broad set of content areas involving generalized arithmetic; concepts associated with equivalence, expressions, equations, and inequalities; and functional thinking.
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Notes
- 1.
By early algebra we mean algebraic thinking in the elementary grades (i.e., Grades Kindergarten–5).
- 2.
Technically, such properties are axioms and assumed to be true without proof. However, it is productive for children to think about why such properties are reasonable.
- 3.
We use the term “LEAP” (Learning through an Early Algebra Progression) here in reference to our Grades 3–5 suite of projects that focused on understanding the impact of a systematic, multi-year approach to teaching and learning algebra in the elementary grades.
- 4.
We elaborate on this curricular approach in Fonger et al. (in press).
- 5.
Ultimately, our aim is to develop a Grades K–5 sequence. Our decision to focus initially on Grades 3–5 was guided largely by the more extensive early algebra research base available in upper elementary grades.
- 6.
The LEAP Grades 3–5 instructional sequence and associated assessments are available upon request to Maria_Blanton@terc.edu.
- 7.
See Blanton et al. (2017b) for a more detailed account of this study.
- 8.
Nine items were common across all Grades 3–5 assessments.
- 9.
Adapted from Carraher et al. (2008).
- 10.
We recognize that a child might give a response such as n, m, and n + m, for parts a, b, and c, respectively. In a further analysis of strategy, we considered such responses. However, for overall correctness, we considered only the most stringent case in which students accounted for the fact that Angela and Tim had the same number of pennies in their banks in their representations.
- 11.
It should be noted that the analysis for Grade 6 data was for all items on the assessment (not just items common with the Grades 3–5 assessments) and that it included new, more difficult items.
- 12.
For our analysis of teachers’ fidelity of implementation, see Cassidy et al. (to appear).
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Acknowledgements
The research reported here was supported in part by the National Science Foundation under DRK-12 Awards #1207945, 1219605, 1219606, 1154355 and 1415509 and by the Institute of Education Sciences, U.S. Department of Education, through Grant R305A140092. Any opinions, findings, and conclusions or recommendations expressed are those of the authors and do not represent the views of the National Science Foundation or of the Institute of Education Sciences or the U.S. Department of Education.
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Blanton, M. et al. (2018). Implementing a Framework for Early Algebra. In: Kieran, C. (eds) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68351-5_2
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