A Learning Trajectory in 6-Year-Olds' Thinking About Generalizing Functional Relationships
Abstract
The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.
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Using algebraic notation would enable students to visualize generalization in fuller detail.
“In line with Blanton et al. (2015), the present authors venture that using algebraic notation would enable students to visualize generalization in fuller detail.”
From Generalization in fifth graders within a functional approach (2018) · cites this paper as Blanton et al. (2015)
Cites
Attneave, F. (1974) · American Psychologist · doi:10.1037/h0036779 · Cited by 137 · see citing works
Comments on the Use of Learning Trajectories in Curriculum Development and Research
Baroody, A. J., Cibulskis, M., Lai, M.-L., & Xia, L. (2004) · Mathematical Thinking and Learning · doi:10.1207/s15327833mtl0602_8 · Cited by 69 · see citing works
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A Learning Progression for Elementary Students’ Functional Thinking
Stephens, A. C., Fonger, N., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Murphy Gardiner, A. (2017) · Mathematical Thinking and Learning · doi:10.1080/10986065.2017.1328636 · Cited by 86 · see citing works
“The ordering of levels of sophistication in a progression can be informed by both the shifts observed in student thinking over time as well as the types of thinking that might be viewed as more sophisticated from a disciplinary perspective (e.g., Battista, 2004; Blanton et al., 2015).”
BackgroundSelf-citationKieran, C. (2022) · ZDM – Mathematics Education · doi:10.1007/s11858-022-01435-6 · Cited by 68 · see citing works
Blanton, M., Isler-Baykal, I., Stroud, R., Stephens, A., Knuth, E., & Gardiner, A. M. (2019) · Educational Studies in Mathematics · doi:10.1007/s10649-019-09894-7 · Cited by 46 · see citing works
Self-citationFonger, N. L., Stephens, A., Blanton, M., Isler, I., Knuth, E., & Gardiner, A. M. (2018) · Cognition and Instruction · doi:10.1080/07370008.2017.1392965 · Cited by 46 · see citing works
Self-citationGeneralizations of third and fifth graders within a functional approach to early algebra
Pinto, E., & Cañadas, M. C. (2021) · Mathematics Education Research Journal · doi:10.1007/s13394-019-00300-2 · Cited by 42 · see citing works
“Further, most studies that focus on the functional relationships that students demonstrate do so in instructional contexts (e.g., Blanton et al. 2015; Carraher et al. 2008a; Radford 2018) We focus on what type of functional relationships students evidence when not directly instructed in…”
BackgroundResultInfluentialPittalis, M., Pitta-Pantazi, D., & Christou, C. (2020) · Journal for Research in Mathematics Education · doi:10.5951/jresematheduc-2020-0164 · Cited by 32 · see citing works
Generalization in fifth graders within a functional approach
Pinto, E., & Cañadas, M. C. (2018) · PNA. Revista de Investigación en Didáctica de la Matemática · doi:10.30827/pna.v12i3.6643 · Cited by 18 · see citing works
“In line with Blanton et al. (2015), the present authors venture that using algebraic notation would enable students to visualize generalization in fuller detail.”
Investigating a learning progression of functional thinking for elementary students
Deng, X., Ding, R., & Huang, R. (2025) · Educational Studies in Mathematics · doi:10.1007/s10649-024-10376-8 · Cited by 4 · see citing works
- Authors
- Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K., & Newman-Owens, A.
- Affiliations
- 1 TERC
- Tufts University (US)
- Corresponding author
- Blanton, Maria
- Newman-Owens, Ashley
- Venue
- Journal for Research in Mathematics Education
- Published
- 2015-11
- Volume
- 46
- Issue
- 5
- Pages
- 511-558
- ISSN
- 0021-8251 (ISSN-L)
- 1945-2306
- OpenAlex
- W2284297282
- Semantic Scholar
- dd283e44c82dd8561be1fadaac763f385a35acae
- Cited by
- 132
- References
- 2
- Citations by year
- 2026: 3 · 2025: 9 · 2024: 14 · 2023: 19 · 2022: 19 · 2021: 14 · 2020: 8 · 2019: 17 · 2018: 5 · 2017: 17 · 2016: 4 · 2015: 3
- FWCI
- 47.84
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal for Research in Mathematics Education (publishedVersion)
- Topics
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
- Physical Sciences › Mathematics › Statistics and Probability › Statistics Education and Methodologies
- Keywords
- functions, learning trajectories, early algebra, first grade students, children's mathematical thinking, algebraic generalization, elementary mathematics education
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