# The Developmental Progression of Early Algebraic Thinking of Elementary School Students

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (i).
- Do distinct groups emerge, characterized by different developmental states, among students based on their performance in the early algebraic thinking test?
- (ii).
- What specific characteristics of early algebraic thinking can be observed among these various student groups?
- (iii).
- Do these observed differences suggest a developmental trajectory in early algebraic thinking, transitioning from more intuitive forms to more sophisticated ones?

## 2. Theoretical Framework

#### 2.1. The Key Components of Early Algebraic Thinking

#### 2.2. Levels of Students’ Thinking about Early Algebraic Concepts

## 3. Materials and Methods

#### 3.1. Procedure and Participants

#### 3.2. Instruments

#### 3.3. Coding Scheme

#### 3.4. Data Analysis

## 4. Results

#### 4.1. Identification of Different Groups of Students Based on Their Performance

#### 4.2. The Characteristics of the Responses Provided by Students in Each Group in Specific Tasks

#### 4.2.1. Arithmetic Thinking Students

#### 4.2.2. Concrete Algebraic Thinking Students

- 1
- R: Do you have any idea about Item 8 (in Figure 5)?
- 2
- S: I couldn’t figure it out.
- 3
- R: Is it because the questions involve variables that you encounter difficulties?
- 4
- S: I usually tend to calculate specific numerical values… (the student re-read the question) So, what is this question asking for?
- 5
- R: So, if there are no specific numbers, you wouldn’t know what it’s asking? If we substitute the letters with actual numbers, do you have an idea?
- 6
- S: If m equals 10… (student started to draw the number line), then The number of chicks represents one unit, and the number of ducks is four times fewer, let’s say it’s 2 fewer. So, the ducks are 10 + 2 = 12, and 12 × 4 = 48 in total.
- 7
- R: So, if substitute it with specific numbers, you know how to calculate, but you’re not sure with the letters?
- 8
- S: If it’s with letters, I don’t understand what the question is asking.
- 9
- R: If the number of chicks is m, and ducks are n, then ducks n = 4 × m − 2. Do you understand this equation? Do you think this equation is correct?
- 10
- S: Firstly, you can move the 2 to the left side of the equation; I also don’t know why it can be done this way.
- 11
- R: You can understand that this equation is valid, but not being able to solve for this letter makes you feel quite frustrated, right?
- 12
- S: Of course, it’s really annoying when you can’t figure out the actual numbers.

#### 4.2.3. Generalized Algebraic Thinking Students

- 1
- R: How did you figure out how many people should be at Table 20?
- 2
- S: I looked at the chart and saw that the number goes up by 3 each time. So I kept adding 3 and got to 62!
- 3
- R: So what if it’s the nth table? Do you get what I mean?
- 4
- S: Does that mean like endless tables?
- 5
- R: If I tell you any table number, can you figure out how many kids are there?
- 6
- S: If it’s a lot of tables, and I keep adding 3, I won’t know how many people there are. Is it n people?
- 7
- R: Why is it n people?
- 8
- S: Because n tables means a lot of people, and endless tables mean endless people!
- 9
- R: ”So, who do you think is bigger, n or “n − 1”?
- 10
- S: If a number is super big, like endless, then you can’t really compare it anymore.

#### 4.2.4. Symbolic Algebraic Thinking Students

#### 4.3. The Developmental Trend in Early Algebraic Thinking

## 5. Discussion

## 6. Limitations and Implications

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Attewell, Paul, and Thurston Domina. 2008. Raising the bar: Curricular intensity and academic Performance. Educational Evaluation and Policy Analysis 30: 51–71. [Google Scholar] [CrossRef]
- Blanton, Maria L. 2013. Equation Structure and the Meaning of the Equal Sign: The Impact of Task Selection in Eliciting Elementary Students’ Understandings. Journal of Mathematical Behavior 32: 173–82. [Google Scholar] [CrossRef]
- Blanton, Maria L., and James J. Kaput. 2011. Functional thinking as a route into algebra in the elementary grades. In Early Algebraization. Edited by Jinfa Cai and Eric Knuth. New York: Springer, pp. 5–23. [Google Scholar]
- Blanton, Maria, Ana Stephens, Eric Knuth, Angela Murphy Gardiner, Isil Isler, and Jee-Seon Kim. 2015a. 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: 39–87. [Google Scholar] [CrossRef]
- Blanton, Maria, Barbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey, and Ashley Newman-Owens. 2015b. A Learning Trajectory in 6-Year-Olds’ Thinking About Generalizing Functional Relationships. Journal for Research in Mathematics Education 46: 511–58. [Google Scholar] [CrossRef]
- Blanton, Maria, Barbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey, and Ashley Newman-Owens. 2017. A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics 95: 181–202. [Google Scholar] [CrossRef]
- Brizuela, Bárbara M., Maria Blanton, Katharine Sawrey, Ashley Newman-Owens, and Angela Murphy Gardiner. 2015. Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas. Mathematical Thinking and Learning 17: 34–63. [Google Scholar] [CrossRef]
- Cai, Jinfa. 2004a. Developing algebraic thinking in the earlier grades: A case study of the Chinese elementary school curriculum. The Mathematics Educator 8: 107–30. [Google Scholar]
- Cai, Jinfa. 2004b. Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs. Journal of Mathematical Behavior 2: 135–67. [Google Scholar] [CrossRef]
- Cai, Jinfa, Swee F. Ng, and John C. Moyer. 2011. Developing Students’ Algebraic Thinking in Earlier Grades: Lessons from China and Singapore. In Early Algebraization. Edited by Jinfa Cai and Eric Knuth. New York: Springer, pp. 25–41. [Google Scholar]
- Carpenter, Thomas P., Linda Levi, Megan Loef Franke, and Julie Koehler Zeringue. 2005. Algebra in elementary school: Developing relational thinking. ZDM: The International Journal on Mathematics Education 37: 53–59. [Google Scholar] [CrossRef]
- Carpenter, Thomas P., Megan L. Franke, and Linda Levi. 2003. Thinking Mathematically. Portsmouth: Heinemann. [Google Scholar]
- Carraher, David W., Analúcia D. Schliemann, Bárbara M. Brizuela, and Darrell Earnest. 2006. Arithmetic and Algebra in Early Mathematics Education. Journal for Research in Mathematics Education 37: 87–115. [Google Scholar] [CrossRef]
- Chimoni, Maria, D. Pitta-Pantazi, and Constantinos Christou. 2018. Examining early algebraic thinking: Insights from empirical data. Educational Studies in Mathematics 98: 57–76. [Google Scholar] [CrossRef]
- Cohen, Louis, Lawrence Manion, and Keith Morrison. 2007. Research Methods in Education, 6th ed. New York: Routledge, p. 213. [Google Scholar]
- Collins, Linda M., and Stephanie T. Lanza. 2010. Latent Class and Latent Transition Analysis: With Applications in the Social, Behavioral, and Health Sciences. New York: Wiley. [Google Scholar]
- Ding, Rui, Rongjin Huang, and Xixi Deng. 2023. Multiple pathways for developing functional thinking in elementary mathematics textbooks: A case study in China. Educational Studies in Mathematics 2: 223–48. [Google Scholar] [CrossRef]
- Dougherty, Barbara. 2008. Measure up: A quantitative view of early algebra. In Algebra in the Early Grades. Edited by James J. Kaput, David W. Carraher and Maria L. Blanton. New York: Taylor & Francis Group, pp. 389–412. [Google Scholar]
- Drouhard, Jean-Philippe, and Anne R. Teppo. 2004. Symbol and Language. In The Future of Teaching and Learning of Algebra. Edited by Kaya Stacey, Helen Chick and Margaret Kendal. Boston: Kluwer Academic Publisher, pp. 227–64. [Google Scholar]
- Greens, Carole E., and Rheta Rubenstein. 2008. Algebra and Algebraic Thinking in School Mathematics, 70th Year-Book. Reston: National Council of Teachers of Mathematics. [Google Scholar]
- Irwin, Kathryn C., and Murray S. Britt. 2005. The Algebraic Nature of Students’ Numerical Manipulation in the New Zealand Numeracy Project. Educational Studies in Mathematics 58: 169–88. [Google Scholar] [CrossRef]
- Kaput, James J. 2008. What is algebra? What is algebraic reasoning? In Algebra in the Early Grades. Edited by James J. Kaput, David W. Carraher and Maria L. Blanton. New York: Taylor & Francis Group, pp. 5–17. [Google Scholar]
- Kieran, Carolyn, JeongSuk Pang, Deborah Schifter, and Swee F. Ng. 2016. Early Algebra: Research into Its Nature, Its Learning, Its Teaching. New York: Springer, p. 10. [Google Scholar]
- Kolovou, Angeliki, Marja van den Heuvel-Panhuizen, and Olaf Köller. 2013. An Intervention Including an Online Game to Improve Grade 6 Students’ Performance in Early Algebra. Journal for Research in Mathematics Education 44: 510–49. [Google Scholar] [CrossRef]
- Lannin, John K., David Barker, and Brian Townsend. 2006. Algebraic generalization strategies: Factors influencing student strategy selection. Mathematics Education Research Journal 18: 3–28. [Google Scholar] [CrossRef]
- Lee, Ji-Eun. 2002. An Analysis of Difficulties Encountered in Teaching Davydov’s Mathematics Curriculum to Students in a U.S. Setting and Measures Found to be Effective in Addressing Them. Ph.D. thesis, State University of New York at Binghamton, Binghamton, NY, USA. [Google Scholar]
- Lo, Yungtai, Nancy R. Mendell, and Donald B. Rubin. 2001. Testing the number of components in a normal mixture. Biometrika 88: 767–78. [Google Scholar] [CrossRef]
- Molina, Marta, Rebecca Ambrose, and Aurora D. Rio. 2018. First encounter with variables by first and third grade spanish students. In Teaching and Learning Algebraic Thinking with 5-to 12-Years Olds. Edited by Carolyn Kieran. New York: Springer, pp. 261–80. [Google Scholar]
- Moses, Robert P., and Charles E. Cobb. 2002. Radical Equations: Civil Rights from Mississippi to the Algebra Project. Boston: Beacon Press. [Google Scholar]
- Muthén, Linda K., and Bengt O. Muthén. 2010. Mplus Users Guide. Los Angeles: Muthén & Muthén. First published 1998. [Google Scholar]
- National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston: NCTM, p. 36. [Google Scholar]
- OECD. 2020. Benchmarking the Performance of China’s Education System PISA. Pairs: OECD Publishing. [Google Scholar] [CrossRef]
- Radford, Luis. 2014. The Progressive Development of Early Embodied Algebraic Thinking. Mathematics Education Research Journal 26: 257–77. [Google Scholar] [CrossRef]
- Radford, Luis. 2018. The emergence of symbolic algebraic thinking in primary school. In Teaching and Learning Algebraic Thinking with 5-to 12-Years Olds. Edited by Carolyn Kieran. New York: Springer, pp. 1–25. [Google Scholar]
- Ralston, Nicole C., Li Min, and Taylor Catherine. 2018. The Development and Initial Validation of an Assessment of Algebraic Thinking for Students in the Elementary Grades. Educational Assessment 23: 211–27. [Google Scholar] [CrossRef]
- Reisch, Christopher P. 2008. College Students’ Understanding of Variable in a Traditional, Developmental, Elementary Algebra Curriculum. Ph.D. thesis, State University of New York, New York, NY, USA. [Google Scholar]
- Rittle-Johnson, Bethany, Percival G. Matthews, Roger S. Taylor, and Katherine L. McEldoon. 2011. Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology 103: 85–104. [Google Scholar] [CrossRef]
- Schifter, Deborah. 1999. Reasoning about operations: Early algebraic thinking in grades K–6. In Developing Mathematical Reasoning in Grades K–12. Edited by Lee V. Stiff and Frances R. Curio. Reston: National Council of Teachers of Mathematics, pp. 62–81. [Google Scholar]
- Schifter, Deborah. 2009. Representation-based proof in the elementary grades. In Teaching and Learning Proof across the Grades: A K-16 Perspective. Edited by Despina A. Stylianou, Maria L. Blanton and Eric J. Knuth. New York: Taylor & Francis, pp. 71–86. [Google Scholar]
- Schoenfeld, Alan H. 1995. Report of working group 1. In The Algebra Initiative Colloquium; Edited by Carole B. Lacampagne, William Blair and Jim Kaput. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement, National Institute on Student Achievement, Curriculum, and Assessment, vol. 2, pp. 11–18. [Google Scholar]
- Stacey, Kaye, Helen Chick, and Margaret Kendal. 2004. The Future of the Teaching and Learning of Algebra: The 12th ICMI Study. Dordrecht: Springer. [Google Scholar] [CrossRef]
- Stephens, Ana C., Amy B. Ellis, Maria Blanton, and Barbara M. Brizuela. 2017a. Algebraic thinking in the elementary and middle grades. In Compendium for Research in Mathematics Education. Edited by Jinfa Cai. Reston: National Council of Teachers of Mathematics, pp. 386–420. [Google Scholar]
- Stephens, Ana C., Nicole Fonger, Susanne Strachota, Isil Isler, Maria Blanton, Eric Knuth, and Angela Murphy Gardiner. 2017b. A Learning Progression for Elementary Students’ Functional Thinking. Mathematical Thinking and Learning 19: 143–66. [Google Scholar] [CrossRef]
- Stephens, Max. 2006. Describing and exploring the power of relational thinking. In Identities, Cultures and Learning Spaces, Paper presented at 29th Annual Conference of the Mathematics Education Research Group of Australasia, Canberra, ACT, Australia, July 1–5. Edited by Peter Grootenboer, Robyn Zevenbergen and Mohan Chinnappan. Sydney: MERGA, pp. 479–86. [Google Scholar]
- Tannisli, Dilek. 2011. Functional thinking ways in relation to linear function tables of elementary school students. The Journal of Mathematical Behavior 30: 206–23. [Google Scholar] [CrossRef]
- Thompson, Patrick W., and Marilyn P. Carlson. 2017. Variation, covariation, and functions: Foundational ways of thinking mathematically. In Compendium for Research in Mathematics Education. Edited by Jinfa Cai. Reston: National Council of Teachers of Mathematics, pp. 421–56. [Google Scholar]
- Ursini, Sonia, and Maria Trigueros. 2001. A model for the uses of variable in elementary algebra. Paper presented at 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands, July 12–17; Edited by Marja Van den Heuvel-Panhuizen. Utrecht: Freudenthal Institute, vol. 4, pp. 327–34. [Google Scholar]
- Usiskin, Zalman. 1988. Conceptions of school algebra and uses of variables. In The Ideas of Algebra, K-12. Edited by Arthur F. Coxford. Reston: National Council of Teachers of Mathematics, pp. 8–19. [Google Scholar]
- Veith, Joaquin M., and Philipp Bitzenbauer. 2022. What group theory can do for you: From magmas to abstract thinking in school mathematics. Mathematics 5: 703. [Google Scholar] [CrossRef]
- Veith, Joaquin M., Meeri-Liisa Beste, Marco Kindervater, Michel Krause, Michael Straulino, Franziska Greinert, and Philipp Bitzenbauer. 2023. Mathematics education research on algebra over the last two decades: Quo vadis? Frontiers in Education 8: 1211920. [Google Scholar] [CrossRef]
- Warren, Elizabeth A., Tom J. Cooper, and Janeen T. Lamb. 2006. Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning. Journal of Mathematical Behavior 25: 208–23. [Google Scholar] [CrossRef]
- Weller, Bridget E., Natasha K. Bowen, and Sarah J. Faubert. 2020. Latent class analysis: A guide to best practice. Journal of Black Psychology 4: 287–311. [Google Scholar] [CrossRef]
- Wilkie, Karina J., and Doug M. Clarke. 2016. Developing students’ functional thinking in algebra through different visualizations of a growing pattern’s structure. Mathematics Education Research Journal 28: 223–43. [Google Scholar] [CrossRef]
- Yu, Rongjie, Xuesong Wang, and Mohamed Abdel-Aty. 2017. A hybrid latent class analysis modeling approach to analyze urban expressway crash risk. Accident Analysis & Prevention 101: 37–43. [Google Scholar] [CrossRef]
- Zhu, Liming. 2017. The Developmental Level of Mathematical Symbol Awareness among Students in the Compulsory Education Stage. Ph.D. thesis, Northeast Normal University, Changchun, China. [Google Scholar]

**Figure 2.**The average scores of four groups of students on 12 mathematical tasks. For the 12th item, a full score is 4, while Items 8 to 10 have a full score of 1, and all other items have a full score of 3.

**Figure 4.**Three examples of commonly employed strategies employed by students with concrete algebraic thinking.

Content Strand | Concept | Task |
---|---|---|

Generalized arithmetic (GA) | Equality, Equal sign, Equivalence, Equations | GA: 1, 2, 3, 4 |

Simple operation | GA: 5 | |

Properties of operations | GA: 6 | |

Functional thinking (FT) | Letter as variable | FT: 11 |

Covariable and correspondence | FT: 7, 12 | |

Quantitative reasoning (QR) | Generalizing the quantitative relationship | QR: 8, 9, 10 |

Model | K | Log(L) | AIC | BIC | aBIC | Entropy | LMR | BLRT |
---|---|---|---|---|---|---|---|---|

1 Class | 31 | −6230.14 | 12,522.28 | 12,654.51 | 12,556.11 | — | — | — |

2 Class | 63 | −5720.43 | 11,566.86 | 11,835.57 | 11,635.60 | 0.82 | ** | ** |

3 Class | 95 | −5618.78 | 11,427.56 | 11,832.77 | 11,531.21 | 0.78 | 0.52 | ** |

4 Class | 127 | −5516.74 | 11,377.47 | 11,919.17 | 11,516.04 | 0.83 | 0.59 | ** |

5 Class | 159 | −5518.23 | 11,354.47 | 12,032.65 | 11,527.94 | 0.82 | 0.79 | ** |

6 Class | 191 | −5483.39 | 11,348.77 | 12,163.44 | 11,557.16 | 0.85 | 0.79 | 1.00 |

Thinking Type | Generalized Arithmetic | Functional Thinking | Quantitative Reasoning |
---|---|---|---|

Arithmetic thinking | Able to judge the equality of equations by calculation. Can only perform calculations on known specific numbers and cannot perform calculations on unknown quantities. Able to identify the existing laws within calculations and to extend examples but unable to generalize. | Able to find recursive rules in patterns (e.g., the number of people increased by 3), but cannot make a generalization. Cannot understand the letters to represent variables. | Inability to understand mathematical problems without specific numbers. |

Concrete algebraic thinking | Able to use relational thinking to determine the equality of an equation. Able to solve the unknowns in the equation by “guess and check”. Able to identify patterns within calculations and extend examples, but unable to generalize them. | Able to discern covariant rules in patterns (e.g., For every additional desk, three more people are added.), but unable to generalize. Cannot understand the letters to represent variables. | Inability to understand mathematical problems without specific numbers. |

Generalized algebraic thinking | Able to use relational thinking to determine the equality of an equation. Can solve for the unknown in the equation directly through inverse operations or the basic properties of the equation. Able to identify patterns in operations and extend them to generalizations but unable to use formal symbols for a comprehensive demonstration. | Able to discern covariant rules in patterns (e.g., For every additional desk, three more people are added.), but unable to generalize. Cannot understand the letters to represent variables. | Able to employ specific numerical values in place of abstract quantities in problems, or to use line graphs for quantitative reasoning. |

Symbolic algebraic thinking | Able to use relational thinking to determine the equality of an equation. The unknown quantity in the equation can be solved directly through the inverse operation or the basic properties of the equation. Able to discover the laws in operations and generalize them and use formal symbols to represent and demonstrate correctly. | Able to use covariation thinking or correspondence thinking to find functional relationships and use formal symbols to correctly express general terms. Able to understand that letters represent variables and can correctly represent the correspondence in functional tasks using algebraic symbols. | Able to directly conduct symbolic algebraic reasoning on the quantitative relationships in mathematical situations. |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sun, S.; Sun, D.; Xu, T.
The Developmental Progression of Early Algebraic Thinking of Elementary School Students. *J. Intell.* **2023**, *11*, 222.
https://doi.org/10.3390/jintelligence11120222

**AMA Style**

Sun S, Sun D, Xu T.
The Developmental Progression of Early Algebraic Thinking of Elementary School Students. *Journal of Intelligence*. 2023; 11(12):222.
https://doi.org/10.3390/jintelligence11120222

**Chicago/Turabian Style**

Sun, Siyu, Dandan Sun, and Tianshu Xu.
2023. "The Developmental Progression of Early Algebraic Thinking of Elementary School Students" *Journal of Intelligence* 11, no. 12: 222.
https://doi.org/10.3390/jintelligence11120222