We consider algebraic thinking as the ways of doing, thinking, and expressing algebra. Until a few decades ago, algebraic thinking was considered inevitably associated to algebraic symbolism (Usiskin, 1988). Today, referring to algebraic thinking does not involve only algebraic symbolism, although it may include it.

Algebraic thinking is “an approach to quantitative situations that emphasizes the aspects of the general relation with expressions that are not necessarily specific to algebraic symbolism” (Kieran, 1996, p. 275). Kaput (2008) characterized algebraic thinking through two key aspects. One of them has to do with generalization and its expression through representations which gain formalism (until they become formal algebraic expressions). The second aspect has to do with reasoning and handling symbolic expressions. Along this same line, other authors (e.g., Soares, Blanton and Kaput, 2006) considered that algebraic thinking is a cognitive process that allows students to establish and build general mathematics relations, which can be expressed in different ways and evolve over time, one of them being algebraic symbolism.

We are referring here to school algebra, pertaining to the levels of compulsory education in Spain. We adopted a multidimensional and therefore, broad approach to school algebra, which is currently shared by the research community. Kaput et al. (2008) argued that this view was essential when attempting to identify the ways in which algebra learning can be based on students’ prior skills and knowledge or can be included in the Primary Education curriculum. These authors insisted that a deep understanding of algebra, its relevance, its structure, and the meanings of fundamental algebraic concepts can only emerge from considering the multiple situations that make them meaningful. Likewise, these cannot be radically separated in education practice since a situation or context often triggers algebraic activities corresponding to different views of algebra (Drijvers and Hendrikus, 2003).

The multiple conceptions of algebra include generalized arithmetic, the study of functional relations and patterns, and the study of structures.

For the construction of knowledge in general and mathematical knowledge in particular, generalization is essential (Castro, Cañadas and Molina, 2010). Mason et al. (1985)
among other authors, highlighted this as the root of algebra. Through different approaches, most authors have pointed out that generalizing is recognizing a regularity that goes beyond the particular cases given and their representation, whether specifying the present similarity or broadening reasoning towards patterns, procedures, structures and relations between them (Kaput, 1999). Stephens, Ellis, Blanton and Brizuela (2017) differentiated generalization as process and product. The product is derived from three processes: (a) identify the regularity of a set of elements, (b) reason beyond the cases at hand, and (c) broaden the results beyond the particular cases. The product is the result of those processes.

In our study, we assumed generalization as the quintessential cross-cutting aspect in most notions of algebra. It can refer to the generalization of arithmetic relations, of patterns, of functional relations, of structures, of methods to solve problems, or rather to solving generalization problems or the representations used in generalization (with algebraic language being one of them).

Generalization allows more flexible mathematical thinking among students, helping them (a) dismiss irrelevant information, (b) adapt, adjust and reorganize prior experiences, (c) pay attention to certain ideas, skills and properties involving different situations, and (d) solve problems and understand different mathematical situations (Carpenter & Levi 2000; Carraher & Schliemann, 2015; Warren, 2005).

The notion of structure is present in most conceptions or approaches recognized in school algebra (Molina and Cañadas, 2018). Within the framework of generalized arithmetic, the term structure refers to a system comprised of a set of mathematical objects, one or more operations, as well as certain properties and relations of and between said objects and operations (Castro, Rico and Romero, 1997).

In the functional conception of algebra, the term structure is equal to regularity or pattern of the values of the two related variables (Torres, Cañadas, Moreno & Gómez, under review). In the study of patterns, this term is used less precisely which makes it difficult at times to distinguish between the notion of pattern and the notion of structure. We consider the former one as broader and containing the latter. It is our understanding that the notion of structure refers to identifying the pattern through particular cases, and the expression or application of the same for general cases.

Based on the conception of algebra as a tool to express general methods that solve classes of problems, the term structure is used to refer to the semantic structure of the problem, which allows distinguishing meanings assigned to the operations in the verbal problems (Molina and Cañadas, 2018).

Functional thinking is considered a part of algebraic thinking. It may be characterized as the process of constructing, describing, and reasoning with and about functions (Blanton, Levi, Crites & Dougherty, 2011), and it is composed of topics, procedures and relations involving functions (Rico, 2006). Therefore, functions are the core mathematical contents of functional thinking. Functions are considered a powerful tool in mathematics (as well as in other fields) to understand, model and predict the behaviour of many real-world phenomena (Blanton, 2008) and they can serve as the theme connecting and even unifying different curriculum contents (Freudenthal, 1982; Schwartz, 1990; Teixidor, Schliemann & Carraher, 2013).

Variables, their relations, recursiveness, the correspondence between the values of the variables, generalization of the functional relation or the use of different representation systems in problem solving with functions, are key elements for the development of functional thinking from the earliest school grades. Understanding the functional thinking capabilities of students in grades lower than Secondary Education can help develop knowledge related to other mathematical contents in these grades, and to functions and other mathematical contents in higher grades.

Given the mathematical contents involving functional thinking (functions), patterns or regularities observed between the covariant quantities have a specific name: functional relations. Studies addressing functions from a mathematical perspective, distinguish between three types of functional relations: (a) recurrence, which means identifying the pattern in a sequence of values of a variable; (b) correspondence, which refers to the relation between corresponding pairs of the dependent and the independent variable; and (c) covariance, where the focus is provided by the analysis of how two quantities vary at the same time and how the changes that occur in one affect the other (Confrey and Smith, 1991; Smith, 2008).

Kieran (2007) pointed out problem solving as an approach to school algebra, which connects to the origin of algebra. Problem solving provides a context in which students can express their own reasoning and strategies as well as use different systems of representation. Taking into account that it is not advisable to involve algebra artificially in tasks that can be solved using common sense and informal strategies ((Drijvers, Dekker & Wijers, 2011), in this project we addressed the challenge of designing or finding tasks related to the students’ environment that allowed working with school algebra in a meaningful manner.

We focused on what Castro & Castro (1997) called external representations. There are different representations coexisting in algebra that help make abstract mathematical objects present (Molina, 2014). Among them are primarily verbal language, algebraic symbolism and representation systems with tables, charts and numbers. There may also be image and manipulative representations.

- In algebraic symbolism, letters represent variables with the meaning of changing indeterminate quantities, and algebraic expressions describe the relations between the quantities. Symbolic representation is very useful to synthetically and precisely represent the relations between indeterminate quantities. However, other representations highlight further elements of mathematical contents and their relations, and they are of particular interest when introduced in the early years of education.

- A tabular representation provides a structure to visibly present related quantities. As children record values on a table, they begin to work with the ideas of correspondence between quantities because they need to pay attention to where the numbers go on the table. They learn the meaning behind the position of numbers distinguishing both by rows and columns. In the early grades, students can begin to recognize where and how to use tabular representation to organize data. As this representation takes on meaning, they can address the functional relations existing between the data.

- Graphic representations, whether conventional or invented, are important tools for students to visually approach information on relations. During primary education, the proposal is to introduce construction and reading of visual tools such as graphs by representing only dots (Blanton, 2008; Blanton et al., 2011).

- Pictorial representations are drawings that help express relations. Children in the early grades of primary education find a familiar language in these representations that support their reasoning and make mathematical ideas more concrete. The use of these representations reduces the cognitive load of the task for the students by helping them make and keep certain information visible and reflect upon the ideas involved. (Blanton, 2008).

- Manipulative representations play the same role as pictorial representations but in this case, they are physical manipulable materials.

The use of different representations mediates and supports the development of algebraic thinking, allowing students to solve the ambiguity of certain representations and generating more flexible thinking (Brizuela and Earnest, 2008).

Balacheff (2000) identified explanation, testing and demonstration (by increasing order of formalization) as validation processes that allow conjecture to become conviction. In an explanation, the speaking subject establishes and ensures the validity. The step from explanation to testing refers to the social process by which a discourse that ensures the validity of a proposal changes position and is accepted by the community. A test is an informal justification that does not have sufficient generality to be considered a formal demonstration.

We are interested in analysing the explanations and tests carried out by students in preschool and primary education. At an early age, there are students who are capable of conducting empirical tests (e.g., Maher and Martino, 1996), but, in general, students in middle grades do not usually conduct formal validation processes to justify their own conjectures (Cañadas, 2007).