Algebraic Thinking at Elementary School: Comparison
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 9–10-year-old pupils work with equations, a central aspect of algebraic thinking in early grades and a cornerstone for more formal learning of algebra.

  • arithmetic word problems
  • early algebra
  • elementary education

1. Background

Currently, different curricular guidelines consider algebraic thinking as a transversal topic from the beginning of schooling [1][2][3][4]. These proposals recommend promoting in pupils the identification of general mathematical relationships and structures based on situations appropriate for their age, which are part of their daily experiences and natural intuitions. Despite the presence of algebraic thinking in the different curricula, there are still challenges on how to introduce this type of thinking in elementary education classrooms. Specifically, researchers seek to contribute with ways of approaching algebraic thinking from contents that have traditionally been seen as exclusively arithmetic. Researchers focus on identifying how elementary school pupils represent and refer to indeterminate quantities when they establish relationships between the resolution of arithmetic word problems (AWPs, hereafter) and their translation using algebraic language, and vice versa. This translation process will allow researchers to delve into the paths that pupils have to give meaning to the indeterminate.
Indeterminate quantities constitute a central aspect of algebraic thinking, which can be associated with different meanings depending on the context. They can be interpreted as a generalized number, an unknown quantity, a variable quantity, or a parameter. Getting elementary pupils to generate rich meanings from the indeterminate depends on the learning opportunities and diversity of learning experiences they are faced with [5]. Researchers address how students interact with the indeterminate quantities through the process of problem posing and translating from algebraic to natural language, and vice versa.
A growing body of research has shown that pupils between the ages of 6 and 12 refer to and represent indeterminate quantities using multiple representations [6][7]. Regarding the use of algebraic language, and in particular the use of letters, elementary pupils accept its use and correctly represent variable quantities by generalizing relationships between quantities that covary [5][7][8][9]. It has been evidenced that teaching and learning environments that encourage children to utilize non-numerical symbols to represent indeterminate quantities, such as variable notation, can help them construct an understanding of variables [10]. However, in the transition to using this type of notation correctly, some errors and difficulties evidenced in higher grades are replicated [11][12][13]. For example, it is observed that pupils spontaneously assign values to literal symbols according to their position in the alphabet, or although they recognize that they can represent different values, they attribute specific values chosen at random [13]. On the other hand, the literature recommends giving concrete meaning to mathematical language through familiar contexts and recognizing familiarity as an important factor in the problem-solving process.

2. Algebraic Thinking

The conceptual framework that directs the study considers that algebraic thinking refers to indeterminate quantities, and these quantities are treated analytically, that is, even if the quantities are unknown, they are added, subtracted, multiplied, or divided [7]. More specifically, algebraic thinking can be understood as the four core practices of generalizing, representing, justifying, and reasoning with mathematical structure and relationships [14]. Specifically:
  • Generalize can be interpreted, in a broad way, as the action of recognizing that some attributes of a mathematical situation can change, while others remain invariable [15]. Attending to generalization allows pupils to move away from the particularities associated with arithmetic calculation and, in turn, allows them to identify the structure and mathematical relationships involved in each situation [15].
  • Representing general mathematical ideas can involve different semiotic means, some conventional and others not, such as gestures, the rhythm of speaking, and natural language [7]. The expression of generalization will have different degrees of sophistication depending on the means of representation used.
  • Justifying generalizations requires pupils to determine and explain the truth of a conjecture or claim [7]. This supports a better understanding of the problem, its structure, and its relationships. Promoting justification in the classroom helps to: refine generalization [16]; for pupils to express themselves more clearly; and for teachers to make well-informed pedagogical decisions since they can understand what pupils think based on what they say or the use they make of signs [17].
  • Reasoning involves treating generalizations as objects in themselves [18], which implies that pupils use the generalizations that they have found, represented, and justified in other types of mathematical situations.
The four core practices are embodied in the different approaches to early algebra: (a) generalized arithmetic, which involves generalizing, representing, justifying, and reasoning with arithmetic relationships, including fundamental properties of operations as well as other types of relationships on classes of numbers [16]; (b) equivalence, expressions, equations, and inequalities, which include developing a relational understanding of the equal sign and generalizing, representing, and reasoning with expressions, equations, and inequalities, including in their symbolic forms [14]; and (c) functional thinking, which includes generalizing relationships between co-varying quantities and representing, justifying, and reasoning with these generalizations through natural language, variable notation, drawings, tables, and graphs [18].

3. Linear Equations in Elementary School

Researchers focus on linear equations because these are deemed suitable for the age and their work is suggested in elementary school curricula [1][2][3][4]. They understand a linear equation is a mathematical sentence that involves an equal sign to show that two algebraic or numeric expressions are equivalent [14], with one or more unknowns. Radford [19] pointed out that using an equation to reason about the representation and communication of relationships between quantities is a cornerstone of algebra. In addition, many problems are better solved if the equation is first written to represent the problem statement. He highlighted that developing an understanding of how equations can be written to represent problems at elementary school can build a foundation for later learning of formal algebra.

4. Translation between Verbal Language and Algebraic Language

AWPs contain information that is presented exclusively through natural language, and to solve them and find the value of some unknown quantity it is necessary to apply one or more elementary mathematical operations. Within the framework of school algebra, AWPs encourage pupils to make sense of the indeterminate, which does not appear without support as it is a quantity of something that is not known [20]. In this context, problems can be represented using different representations. Their interpretation and solution can lead to several translations carried out by the solver.
Regarding the translation of natural language to algebraic language, most authors focus mainly on grades after elementary education. These studies have shown that to be successful in translating between natural language and algebraic language, elementary pupils must identify the variables involved, the relationships between them, and the syntax of the symbolic representation. Regarding the difficulties that they face, one of them is understanding the meaning of algebraic language since this type of representation is considered opaque to them. They tend to have difficulty visualizing the advantages of algebraic language [11], so elementary school pupils prefer to use arithmetic-type strategies and representations [21].
The reverse translation, from algebraic language to natural language, can be considered in the context of problem posing. This activity requires pupils to formulate mathematical problems from given situations that may include mathematical expressions or diagrams, or by reformulating existing problems [22]. Stoyanova [23] proposes three categories of problem-posing tasks: (a) free situations, (b) semi-structured situations, and (c) structured situations. Researchers focus on the second category. These tasks are characterized by being based on an open situation, particularly an equation. From this point, researchers invite elementary school pupils to create a problem by applying mathematical procedures, concepts, and relationships from their own experiences. This type of task is associated with high cognitive demand; whoever invents the problems must reflect on the structure of the situation rather than on the procedures for solving the problem [24].

References

  1. Ministerio de Educación de Chile. Bases Curriculares para la Educación Básica (Curricular Bases for Elementary School); Ministerio de Educación de Chile: Santiago, Chile, 2012.
  2. Boletín Oficial del Estado (BOE). Real Decreto 157/2022, de 1 de marzo, por el que se Establecen la Ordenación y las Enseñanzas Mínimas de la Educación Primaria (Royal Decree 157/2022, of March 1, which Establishes the Organization and Minimum Teachings of Primary Education); BOE: Madrid, Spain, 2022.
  3. Common Core State Standards Initiative (CCSSI). Common Core State Standards Mathematics; National Governors Association Center for Best Practices, Council of Chief State School Officers: Washington, DC, USA, 2010.
  4. Ontario Ministry of Education and Training. The Ontario Curriculum Grades 1–8: Mathematics; Ministry of Education: Toronto, ON, Canada, 2020.
  5. Ayala-Altamirano, C.; Molina, M. Meanings Attributed to Letters in Functional Contexts by Primary School Students. Int. J. Sci. Math. Educ. 2020, 18, 1271–1291.
  6. Pinto, E.; Cañadas, M.C. Generalizations of third and fifth graders within a functional approach to early algebra. Math. Educ. Res. J. 2021, 33, 113–134.
  7. Radford, L. The Emergence of Symbolic Algebraic Thinking in Primary School. In Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds; Kieran, C., Ed.; Springer: Dordrecht, The Netherlands, 2018; pp. 3–25.
  8. Blanton, M.L.; Brizuela, B.M.; Gardiner, A.M.; Sawrey, K.; Newman-Owens, A. A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educ. Stud. Math. 2017, 95, 181–202.
  9. Blanton, M.L.; Stephens, A.C.; Knuth, E.J.; Gardiner, A.M.; Isler, I.; Kim, J.S. The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. J. Res. Math. Educ. 2015, 46, 39–87.
  10. Brizuela, B.M. Variables in elementary mathematics education. Elem. Sch. J. 2016, 117, 46–71.
  11. Castro, E.; Cañadas, M.C.; Molina, M.; Rodríguez-Domingo, S. Difficulties in semantically congruent translation of verbally and symbolically represented algebraic statements. Educ. Stud. Math. 2022, 109, 593–609.
  12. Ruano, R.M.; Socas, M.M.; Palarea, M. Secondary Students’ Error Analysis and Classification in Formal Substitution, Generalization and Modelling Process in Algebra (Análisis y clasificación de errores cometidos por alumnos de secundaria en los procesos de sustitución formal, generalización y modelización en algebra). PNA 2008, 2, 61–74.
  13. Brizuela, B.M.; Blanton, M.L.; Gardiner, A.M.; Newman-Owens, A.; Sawrey, K. A first grade student’s exploration of variable and variable notation. Estud. Psicol. 2015, 36, 138–165.
  14. Blanton, M.L.; Levi, L.; Crites, T.; Dougherty, B.J. Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3-5; NCTM: Reston, VA, USA, 2011.
  15. Mason, J. Overcoming the algebra barrier: Being particular about the general, and generally looking beyond the particular, in homage to Mary Boole. In And the Rest is Just Algebra; Stewart, S., Ed.; Springer: Cham, Germany, 2017; pp. 97–117.
  16. Stephens, A.C.; Ellis, A.B.; Blanton, M.L.; Brizuela, B.M. Algebraic thinking in the elementary and middle grade. In Compendium for Research in Mathematics Education, J. Cai, Ed.; NCTM: Reston, VA, USA, 2017; pp. 386–420.
  17. Morgan, C.; Craig, T.; Schuette, M.; Wagner, D. Language and communication in mathematics education: An overview of research in the field. ZDM 2014, 46, 843–853.
  18. Blanton, M.; Brizuela, B.; Stephens, A.; Knuth, E.; Isler, I.; Gardiner, A. Implementing a framework for early algebra. In Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds: The Global Evolution of an Emerging Field of Research and Practice; Kieran, C., Ed.; Springer: Hamburg, Germany, 2018; pp. 27–49.
  19. Radford, L. O ensino-aprendizagem da ágebra na teoria da objetivação. In Pensamento Algébrico nos Anos Iniciais: Diálogos e Complementaridades entre a Teoria da Objetivação e a Teoria Histórico-Cultural; Moretti, V., Radford, L., Eds.; Livraria da Física: Sao Paulo, Brazil, 2021; pp. 171–195.
  20. Janßen, T.; Radford, L. Solving equations: Gestures, (un)allowable hints, and the unsayable matter. In Proceedings on the Ninth Congress of the European Society for Research in Mathematics Education, Prague, Czech Republic, 4–8 February 2015; Krainer, K., Vvondrová, N., Eds.; CERME: Prague, Czech Republic, 2015; pp. 419–425.
  21. Kieran, C. Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In Second Handbook of Research on Mathematics Teaching and Learning; Lester, F.K., Ed.; NCTM: Reston, VA, USA, 2007; pp. 707–762.
  22. Cai, J.; Hwang, S. Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. Int. J. Educ. Res. 2020, 102, 1–8.
  23. Stoyanova, E. Empowering students’ problem solving via problem posing: The art of framing’good’questions. Aust. Math. Teach. 2000, 56, 33–37.
  24. Cai, J.; Moyer, J.C.; Wang, N.; Hwang, S.; Nie, B.; Garber, T. Mathematical problem posing as a measure of curricular effect on students’ learning. Educ. Stud. Math. 2013, 83, 57–69.
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