Specialized Knowledge of Prospective Teachers in a Function TaskABSTRACT
This study explored the level of specialized knowledge of 48 Chilean prospective teachers in solving a task involving the function f(x) = 2x + 6. The qualitative analysis revealed that the basic level predominates, using pictorial and verbal representations. However, intermediate and advanced levels were also observed, with the use of correspondence and covariation strategies, developing functional generalization. It is concluded that it is crucial to further investigate the specialized knowledge of future teachers in order to improve their training.
Keywords
Early Algebra
; Functional Thinking; Specialized Knowledge
RESUMEN
Este estudio exploró el nivel de conocimiento especializado de 48 futuros profesores chilenos de Educación Primaria al resolver una tarea que involucró la función f(x) = 2x + 6. El análisis cualitativo reveló que predomina el nivel básico, usando represent aciones pictóricas y verbales. Sin embargo, también se observó nivel intermedio y avanzado, con uso de estrategias de correspondencia y covariación, alcanzando la generalizando de la función. Se concluye que es crucial investigar más a fondo el conocimiento especializado de los futuros profesores para mejorar su formación.
Palabras-clave
Álgebra Temprana; Pensamiento Funcional; Conocimiento Especializado
Introduction
School algebra has attracted growing interest in both teaching and educational research, being recognized as a key tool for the development of mathematical thinking (Kaput, 2000; Brizuela, 2024). This interest has been driven by research highlighting the importance of integrating algebraic content from the early years of schooling, demonstrating that even young children can develop forms of algebraic thinking when provided with appropriate experiences (Brizuela, 2024). However, its teaching has traditionally been relegated to secondary levels, with a focus on mechanical procedures and symbolic manipulation to solve algebraic tasks (Kaput, 2008; Blanton et al., 2015). This approach has generated serious difficulties for students to understand algebraic concepts, limiting their abilities to apply them in varied contexts (Kieran, 2007). For example, some students perceive solving a first-degree equation as a merely mechanical process, where it is only necessary to move terms from one side to the other of the equal sign and perform the corresponding operations to determine the value of the unknown, without a deep understanding of the meaning of these actions (Shliemann et al., 2008). In the same vein, various researchers, aiming to reverse this situation, have advocated fostering algebraic thinking from the early years of schooling, highlighting the benefits this brings to mathematical learning (Cañadas; Castro, 2007; Strachota, 2020). The literature suggests that stimulating algebraic thinking in the early educational levels enriches knowledge and facilitates progress in learning mathematics (Mejías; Alsina, 2020). Thus arises the curricular proposal known as early algebra, which promotes the "algebrization" of the curriculum (Kaput, 2000), integrating forms of algebraic thinking in the educational levels prior to Secondary Education (Cañadas; Molina, 2016).
This research focuses on functional thinking, one of the early algebra approaches considered central to addressing algebraic thinking in Primary Education. (Kaput, 2000; Blanton, 2010; Cañadas; Molina, 2016). This type of thinking centers on the concept of function, which is understood as a relationship between covarying quantities. Functional thinking seeks to promote key ideas of algebraic thinking, such as the relationships between quantities and their joint variation (Rico, 2006; Smith, 2008; Brizuela, 2024). This approach considers that it is not only about introducing functions as they are addressed in Secondary Education, but about presenting them intuitively, through tasks adapted to the early educational stages, so that students observe relationships between quantities and manage to generalize such relationships (Cañadas; Molina, 2016).
In various curricular updates, functional thinking tasks have been considered key elements for promoting algebraic notions and mathematical skills such as: justifying, representing, reasoning with mathematical structures and relationships, and generalizing (Rico, 2006; Blanton et al., 2011; Strachota, 2020; Stephens et al., 2021). The above reasons have led various countries such as Australia, Canada, China, Chile, South Korea, the United States, Japan, Portugal, and Spain to incorporate objectives associated with this type of thinking into their curricula from the early educational levels (Merino et al., 2013; MINEDUC, 2023). For example, the Chilean Primary Education curriculum suggests that students should identify relationships between quantities by exploring how one quantity changes in relation to another (MINEDUC, 2012; Morales; Parra-Fica, 2022). This is also reflected in the Guiding Standards for teacher training in Chile, which emphasize the mastery of algebraic content for its teaching from the early educational levels (MINEDUC, 2021). Despite the fact that Chilean curricular guidelines incorporate elements of functional thinking, the body of knowledge in prospective Primary Education teachers regarding how they understand and use tasks of this type of thinking in their training is limited (Morales; Parra-Fica, 2022). This gap prevents obtaining evidence that impacts pedagogy programs regarding the development of the specialized mathematical knowledge that prospective teachers require to teach school algebra. From this perspective, our study makes sense, as it seeks to provide evidence in this area, constituting a specific contribution to the national and international context. The future Primary Education teacher must acquire specialized mathematical knowledge in a practical and broad manner, ensuring the utility of what they learn as a way to enhance algebraic learning in their students (Ball et al., 2008; Blanton; Kaput, 2005; Sánchez; Llinares, 2003). Based on the previous arguments, we set as our research objective: to determine the levels of knowledge of specialized content that prospective Primary Education teachers demonstrate when solving a functional thinking task.
Theoretical Elements
Functional Thinking
Functional thinking is considered a type of algebraic thinking that is conducive to introducing algebra from the early educational levels, and whose main mathematical content is the function (Blanton; Kaput, 2011; Cañadas; Molina, 2016). It focuses on the relationship between variables, where the emphasis can be on particular cases or on the general case (Smith, 2008). This type of thinking centralizes its focuses on how quantities vary together and and emphasizes the correspondence between values of the variables involved in the functional relationship, in addition to the use of different systems of representation in a problem-solving context (Cañadas; Molina, 2016). For functional thinking, the generalization of the functional relationship is an important element, as well as its justification and the use of representations that can encompass both natural language and pictorial, tabular, graphical, symbolic, or algebraic forms. In this sense, Ureña et al. (2024) propose that developing functional thinking involves identifying patterns of change, relating different representations to each other, and constructing significant generalizations from a mathematical perspective. Therefore, in a functional thinking task, it is expected that the student can reason fluently through these generalized representations in order to understand and predict the behavior of the function (Blanton et al., 2015).
Blanton (2008) defines three types of relationships that can be established in a functional thinking task, which are: a) recurrence or recursive pattern, b) covariation, and c) correspondence. Recurrence is the relationship defined based on the values of the same set of values (Johnsonbaugh, 2005), that is, it only describes the variation of the quantities of a single variable, such as adding one unit. In this way, this pattern is the most basic and is not considered a functional relationship (Morales; Parra-Fica, 2022). For example, in a problem where the ages of Álvaro and Carmen are related, with Carmen being 5 years older than Álvaro (f[x] = x + 5), a student could find Carmen's age by adding one unit to Carmen's previous age, as shown in Figure 1.
It is important to mention that recurrence hinders generalization because it is necessary to know a previous value of the dependent variable to provide a satisfactory answer (Morales et al., 2018). For this reason, authors like Smith (2008) do not consider it a functional relationship, even though it is within a functional context.
In this regard, various authors (e.g., Blanton et al., 2011; Morales et al., 2018; Smith, 2008) suggest that both covariation and correspondence are key functional relationships for determining functional thinking in a problem solver, as they involve relationships between the values of both variables. From this perspective, covariation focuses on the simultaneous and coordinated variation of the quantities of both variables (Blanton et al., 2011). For example, in Figure 2, it is observed that as Álvaro's age increases, Carmen's age also increases, maintaining the difference between them constant. In this case, both variables increase by one.
Finally, correspondence is a rule that associates each value of the independent variable with a unique value of the dependent variable (Clapham, 1998). Identifying correspondence involves finding the rule that allows one to determine a value of the dependent variable given a value of the independent variable (Blanton et al., 2011). For example, if we consider the same situation regarding ages, we highlight the correspondence when relating the quantities of both variables, such that to find Carmen's age, we must add 5 to Álvaro's age. In Figure 3, we observe a correspondence relationship from the previous example; Carmen's age is obtained by adding 5 to Álvaro's age.
Generalization
Generalization is considered a core aspect of algebra and a starting point for algebraic learning.(Mason, 1996; Strachota, 2016). It is a situated activity in which reasoning emerges not only focused on particular cases but also on the patterns and relationships established in those particular cases and the processes through which such reasoning is communicated (Strachota, 2020). This implies that generalizing requires fulfilling the property for all elements of a given collection (Villa-Ochoa, 2006).
From the context of functional thinking, generalization refers to both an end and a process (Ayala-Altamirano; Molina, 2021). For example, one can achieve generalization through the exploration of particular cases and the discovery of patterns. This means that to achieve generalization, one must link the identified pattern with a general rule that considers not only isolated cases but all cases (Cañadas; Castro, 2007). On the other hand, Mason (1996) argues that generalization can also derive from a single example or particular case with specific characteristics, known as a generic example. Generalization can be expressed through various representations.
Representations
Functional thinking can be identified when students explicitly express the relationship between quantities (functional relationships), which can be through various representations including natural language, symbols, tables, graphs, and drawings (Merino et al., 2013; Morales et al., 2018). Molina (2014) mentions that types of representations can be classified as: verbal, concrete, pictorial, tabular, and symbolic, each of which allows for addressing the understanding of functional relationships from different perspectives, facilitating the gradual transition from the concrete to the abstract. This aligns with Duval's (2016) proposal, which emphasizes that the diversity of representations is fundamental for mathematical understanding, as each system provides a distinct way to approach and understand the relationships between variables. The combination of these representations not only promotes the development of functional thinking but also allows students to identify patterns and generalize relationships, crucial elements in learning algebraic concepts (Brizuela; Earnest, 2019).
In this sense, it is pertinent to detail the characteristics of each type of representation. The verbal representation corresponds to the natural expression through verbal or written language according to the relationship between variables (Morales et al., 2018). In turn, concrete representation refers to the manipulation of objects that activate mathematical relationships, facilitating the understanding of abstract concepts through practical experiences (Clements; Sarama, 2020). Pictorial representation employs visual resources, without symbolic notation, allowing students to visualize patterns and relationships without the intervention of formal codes (Duval, 2016). Additionally, tabular representation organizes information in a rectangular format, with rows and columns, to identify structured relationships, which is useful for comparing values and observing regularity among them (Stein, 2019). Finally, symbolic representation uses alphanumeric expressions, that is, combinations of numbers and letters to express relationships between quantities, a representation that facilitates the generalization of functional relationships through algebraic notations (Rico, 2009). In a broad sense, Kaput et al. (2017) notes that numerical and operational expressions also constitute forms of symbolic representation when used to express relationships and patterns.
Mathematics Teachers’ Knowledge
Mathematical Knowledge for Teaching (MKT) is a theoretical model that refers to the knowledge that every mathematics teacher must possess (Hill et al., 2005). This directly impacts the learning of their students (Lins; Kaput, 2004). In the context of mathematics teaching, the teacher's professional knowledge must encompass both mastery of disciplinary content and the management of pedagogical strategies. Disciplinary content knowledge, referred to as the deep understanding of concepts and structures inherent to mathematics, allows teachers to comprehend and convey mathematical knowledge accurately (Shulman, 1986). Likewise, knowledge of content and teaching involves the ability to present these concepts in an accessible manner for students, adapting strategies and approaches according to the classroom context (Ball et al., 2008). Thus, the teacher's knowledge is structured into different domains and subdomains that encompass both mathematical content and the pedagogical approaches necessary for its teaching (see Figure 4).
This research focuses on the domain of content knowledge in which we distinguish three subdomains: a) common content knowledge, b) specialized content knowledge, and c) horizon content knowledge (Ball et al., 2008). The first subdomain encompasses the mathematical knowledge and skills that any person possesses and that can manifest in contexts outside of teaching. This type of knowledge is reflected in the ability to calculate correct answers or solve problems effectively, skills that are developed through formal education (Ball et al., 2008). Secondly, specialized content knowledge corresponds to the skills and mathematical knowledge that are exclusive to teaching, that is, that specific knowledge that the teacher uses in their daily practice. This knowledge enables the teacher to perform fundamental tasks such as accurately representing mathematical ideas, providing clear explanations for common rules and procedures, and understanding unusual solution methods for various mathematical problems (Ball et al., 2008; Hill et al., 2005). Finally, horizon content knowledge refers to the teacher's ability to relate mathematical topics to the curriculum in an integrated manner. This knowledge allows one to establish connections between mathematical content and other subjects in the curriculum, fostering a holistic understanding of mathematics and its relationship with other areas of knowledge (Shulman, 1986).
This study specifically explores specialized content knowledge in relation to functional thinking specifically aimed at the strategies and justifications that prospective Primary Education teachers make, as well as the representations they use when solving tasks that involve functional relationships. Addressing this type of knowledge seeks to provide a deeper understanding of how teachers may mobilize this knowledge when teaching functional thinking in the classroom (Cañadas; Molina, 2016).
Background
Existing literature suggests that teachers with a high level of specialized knowledge can help students overcome the static view of functions, promoting a more relational understanding of concepts, which is fundamental for the development of functional thinking (Oliveira; Mestre, 2014). In the context of school algebra, authors such as Blanton and Kaput (2011) and Ellis (2011) suggest that teachers guide students towards the use of multiple representations (tables, graphs, and symbolic expressions) for a better understanding of a functional relationship. Similarly, Hauck and Alsina (2021) emphasize the need to deepen the algebra training of prospective Primary Education teachers, given that there are considerable limitations in their ability to interpret and use algebraic representations.
In this context, previous studies address the knowledge of mathematics teachers in Secondary Education focused on school algebra (e.g., Flores et al., 2018). However, specific studies in Primary Education, particularly regarding prospective teachers, remain limited. Available studies indicate that these future teachers face various difficulties and errors in solving algebraic tasks. For example, Morales and Parra-Fica (2022) report that prospective Primary Education teachers have difficulties using symbolic representation in generalizing simple functional relationships, such as f(x)=2x+2. Similarly, Aké (2021) shows that, in a task based on f(x)=4x+2, only 18 out of 40 prospective teachers managed to solve it correctly, primarily relying on verbal representation to formulate a general rule instead of using a symbolic representation. Additionally, 11 prospective teachers achieved a partial solution by addressing only specific cases, and another 11 applied algebraic concepts inadequately. In the same way, Polo-Blanco et al. (2019) reveal that prospective Primary Education teachers in Spain and Portugal experience difficulties in establishing generalizations through correspondence relationships in geometric contexts. Participants tend to use recursive and covariational strategies, which limits their ability to formulate general rules. In a similar vein, Wilkie (2014) found that only 30% of 105 Australian teachers demonstrated an adequate level of functional thinking when extending a geometric pattern, primarily employing recursive strategies; only 2% of the sample used a symbolic representation to generalize the functional relationship.
Given this background, it is suggested that focusing training on specialized content knowledge in the context of functional thinking not only facilitates a more solid understanding of algebraic notions and skills for generalization but also prepares teachers to face the difficulties that students often have when interpreting graphical and symbolic representations involved in algebraic learning (McAuliffe; Vermeulen, 2018; Strand; Mills, 2014).
Method
Research Design
Our research is exploratory and descriptive in nature. Exploratory studies investigate topics that are under-researched and about which there are concerns to open new perspectives on the phenomenon being studied (Hernández et al.). 2014). This research aims to explore the specialized content knowledge of prospective Primary Education teachers in the functional context of school algebra, given that previous research suggests its approach due to the lack of information regarding this topic available in the scientific community. This study seeks to open new perspectives that contribute to obtaining guidelines for the training of prospective teachers and prospective research, through the exploration and description of their functional thinking.
Participants
The subjects of the study consist of 48 prospective teachers who are in their first year of the General Basic Education Pedagogy program at a Chilean university. The selection of subjects was intentional, according to the objectives of the research, the availability of the university center, and the motivation these students had to participate in the study. It is noteworthy that the research subjects have only received instruction in algebra during their Primary and Secondary Education, but not in their pedagogy studies.
Data Collection Instrument
The data collection instrument was a written test that each future teacher answered individually. The test contained a problem adapted from Pinto and Cañadas (2017) that involves a functional relationship of the type (f[x] = 2x + 6). In this problem, the number of white tiles (independent variable) and the number of gray tiles (dependent variable) in a school hallway are related. To elicit evidence of the functional relationship involved in the task, various questions were posed following the inductive reasoning model of Cañadas and Castro (2007). This means that, starting from close cases and then moving to distant cases, the generalization of the functional relationship involved in the problem was sought. The task questions were reviewed by experts in mathematics didactics with experience in the MKT approach, which ensured their coherence and relevance. Before the application of the instrument, each future teacher was asked for informed consent, and they signed a document that outlined the ethical aspects of the research and guaranteed the privacy and confidentiality of the collected data. The application took place in one of the facilities of the aforementioned university and lasted 90 minutes. Figure 5 shows the problem presented to the prospective teachers.
As a way to understand the type of reasoning that prospective teachers employ when solving the presented problem situation, questions are asked to guide the work according to the inductive reasoning model. Table 1 presents the type of question and the example used in the data collection instrument.
Data analysis
For the analysis of the data, we employed the content analysis technique (Fernández, 2002). The units of analysis considered in this research come from two sources of information related to the responses to each of the task questions, which are: a) verbal (written) productions and b) pictorial productions. The responses were analyzed according to the categories described in Table 2. The categories were constructed considering: a) the conceptual framework, b) a prior analysis of the same task, and c) through a priori analysis of the data. Specifically, these categories were constructed based on: a) the Specialized Content Knowledge framework of MKT (Ball et al., 2008), b) previous research (Wilkie, 2014). Additionally, they were adapted and contextualized to functional thinking, focusing on strategies centered on functional relationships and their various forms of representation. Likewise, a peer analysis was conducted within the research team as a strategy to contrast interpretations and enhance information saturation.
For the analysis process, each of the prospective teachers was assigned a code for identification. For example, a specific future teacher was assigned as P(N°). Next, we coded the responses of each of them concerning each of the questions in the task. For this process, we used categories of levels of knowledge of specialized content: basic (B), intermediate (I), advanced (A), no response (SR), and responses not related to the problem (NRP). Finally, the changes in the level of knowledge of mathematical content of each of the prospective teachers was determined when they solved the proposed task.
Results
Next, we present results regarding the levels of knowledge of specialized content demonstrated by the prospective Primary Education teachers in each of the questions of the functional thinking task. In turn, we present representative examples of each of these levels.
Specialized content knowledge in question A
Table 3 shows general results for question A.
In this question, corresponding to a particular consecutive close case, the most evidenced level of specialized knowledge was basic (B), as 26 of the 48 prospective teachers applied it. The most commonly used representation by these prospective teachers was pictorial. The prospective teachers who used this strategy occasionally complemented it with a verbal and even numerical representation. We observed such a response in the production of P7 (see Figure 6) where they drew the design of the tiles considering the 6 white tiles and the 18 gray tiles surrounding them, to which they added numerical representations, thus arriving at the answer to the question (18 tiles).
On the other hand, in this question, the intermediate knowledge level (I) was the second most evidenced, identified in 20 of the analyzed responses. They employed various strategies at this level, particularly highlighting those associated with functional relationships, used by 14 prospective teachers, where covariation was the most predominant. An example of covariation can be observed in the response of P12 (see Figure 7) where the variation between the values of the task variables is evident, such that when they added 1 white tile, the gray tiles increased by 2.
In turn, the correspondence strategy was used by only 4 of the 48 prospective teachers. They determined that to find the number of gray tiles, they needed to multiply the 6 white tiles by 2, and then add the 3 from each side, resulting in a total of 6 gray tiles, ultimately resulting in 18. Figure 8 shows the response of P7, who, in addition to using the correspondence strategy, utilized support from two representations to answer, namely the verbal and pictorial representations, respectively.
Additionally, it is worth mentioning that only 2 prospective teachers employed the recursive pattern strategy, indicating that to find the number of gray tiles, it is necessary to add two tiles to the previous consecutive case. Such a situation can be observed in the response of P3 (see Figure 9), who explained through a pictorial and verbal representation that two gray tiles should be added around.
Finally, it is important to highlight that no prospective teachers were recorded at the advanced level, as no responses were identified that included a generalization or the use of an algebraic representation.
Specialized content knowledge in question B
Table 4 shows general results on the specialized content knowledge for question B, which addressed a particular non-consecutive case.
As observed in Table 4, 25 out of 48 prospective teachers are grouped at the basic level. This is because they only used pictorial and/or verbal representations, as evidenced by the response of P23 (see Figure 10), who justified that when there are eight white tiles, there must be 22 surrounding gray tiles.
Regarding the intermediate level of knowledge, 20 prospective teachers expressed this, of which 12 used a covariation relationship, 7 utilized a correspondence relationship, and 1 applied a recursive pattern.
On the other hand, the advanced level of knowledge was evidenced by 2 prospective teachers, who generalized the functional correspondence relationship, using verbal and symbolic representation. An example of verbal representation of generalization is observed in the response of P11 (see Figure 11), who described the general rule based on a particular case (8 white tiles) and did so by doubling the number of white tiles plus an additional 6.
Finally, in this question, only one future teacher (P43) provided a response unrelated to the problem (see Figure 12), as he indicated that the tiles at the top and bottom of the pictorial representation must each have 10 gray tiles. Given this response, it is not observed that this future teacher is able to establish a relationship between the gray tiles and the white ones.
Knowledge of specialized content in question C
Table 5 presents the overall results regarding the specialized content knowledge for question C, which addressed a particular non-consecutive distant case.
In Table 5, it is observed that the most frequent level of knowledge is the basic level, with 24 prospective teachers who do not manage to demonstrate a functional relationship. A characteristic example of this level of knowledge was the response of P23 (see Figure 13), who used a pictorial representation by drawing the 25 white tiles and surrounding them with 56 gray ones (painted in yellow), following the pictorial configuration given at the beginning of the task, which he ultimately counted by marking each one with a dot to determine the total number of gray tiles.
This was followed by 22 responses classified at the intermediate level. These responses were primarily characterized by the evidence of a functional relationship. In particular, a predominant use of the correspondence strategy was observed, which was employed by 12 prospective teachers. An example of this was the response from P14 (see Figure 14), who used symbolic representation to duplicate the 25 white tiles and then added another 6 (3+3).
Another strategy used by prospective teachers at the intermediate knowledge level, although to a lesser extent, was covariation, present in 8 responses. Such a strategy is illustrated in the response from P21 (see Figure 15), who pictorially, symbolically, and in tabular form indicated that each time the white tiles increase by one, the gray tiles increase by two.
The least used strategy by prospective teachers (2) at the intermediate knowledge level was the use of the recursive pattern. This strategy is exemplified in the response from P20 (see Figure 16), who explained that one must add two more tiles than in the previous case.
Regarding the advanced level, this was the least evidenced in this question, as only 2 prospective teachers expressed it. In the responses of these prospective teachers, the use of symbolic representation stands out as observed in the response from P11 (see Figure 17), who used the letter (X) to represent the number of white tiles within an algebraic expression that allows determining the number of gray tiles.
Knowledge of specialized content in question D
Table 6 shows the overall results related to the specialized knowledge of content in question D, which addressed a particular non-consecutive distant case.
In Table 6, it is observed that in this question the most frequent knowledge level was the intermediate level, with a total of 23 responses. This level groups the responses that demonstrate a functional relationship. At this level, the predominant strategy was matching, with 17 expressing it, while another 5 used covariation and only one used the recursive pattern. The strategies used by prospective teachers in this question are similar to those described in the previous question.
The basic knowledge level in this question was reached by 22 prospective teachers who relied on pictorial and/or verbal representations, without demonstrating a relationship between the quantities involved in the task. Among the responses, there are those that are direct as they only express a correct answer without justification, as seen in the response from P2 (see Figure 18), who indicated that there are 106 gray tiles when there are 50 white ones.
The advanced level of knowledge in this question was achieved by 2 prospective teachers. Their responses demonstrate a generalization and the ability to use a symbolic representation that illustrates the relationship between the quantities of white and gray tiles. Such representation corresponds to the following expression X * 2 + 6 where X represents the number of white tiles and the 6 represents the tiles at the ends of the pictorial configuration of the task.
Knowledge of the specialized content in question E
Table 7 shows the overall results related to the specialized knowledge of the content in question E, which addressed the identification of variables (dependent and independent) present in the task and the relationship between them.
In this question, it is highlighted that the majority of prospective teachers (16) did not respond to it. The second most frequent category was the basic level manifested by 14 prospective teachers. Although they identified the dependent and independent variables, they did not explain what the dependency relationship between them was. An example of the above was P13’s response (see Figure 19), in which the participant did not justify how the white tiles and the total number of tiles are related when both increase.
Next, there is the category of response unrelated to the problem, identified in 12 of the analyzed responses. In them, the prospective teachers provided ambiguous arguments that do not clearly recognize the involved variables or how they relate to each other. An example of this type of response is from P2 (see Figure 20), who, although recognizing the existence of a dependency relationship, does not detail what the independent and dependent variables are, nor does he explain how both relate.
Fourth, there is the intermediate level with three prospective teachers who manifested it. In their responses, they fail to express what the dependent and independent variables are, although they explicitly allude to a relationship between the variables involved in the task. A characteristic example is the response of P26 (see Figure 21), as it explained that there is a dependency relationship between variables, but did not specify which is the independent and dependent variable and how they relate to each other. Rather, it mentioned the pictorial configuration of the task, that is, it stated that there should be as many gray tiles above and below as there are white tiles in the center, and three additional gray tiles on each side.
Finally, three other prospective teachers also expressed advanced-level responses. These responses demonstrate the identification of variables involved in the task and how they relate to each other. A characteristic example is the production of P38 (see Figure 22), who explained the dependency between variables, mentioning that the gray tiles are dependent on the white ones. In his explanation, the way both variables relate is observed, so that if the white tiles increase by one, the gray tiles increase by two, thus evidencing a covariation relationship.
Specialized content knowledge in question F
Table 8 shows the overall results related to specialized content knowledge in question F, which seeks a general rule to calculate the number of gray tiles given an unknown number of white tiles.
In Table 8, it is observed that the category with the highest frequency corresponds to nonsensical responses to the posed question, as 15 prospective teachers expressed this. A representative example of this category was the response of P47 (see Figure 23), who explained that the number of white tiles is obtained by dividing the number of gray tiles, but does not mention the number by which it should be divided. Therefore, this response lacks meaning for addressing the posed question.
Secondly, 12 prospective teachers did not answer the question. On the other hand, 10 prospective teachers present responses at the advanced level, which demonstrate the generalization of the functional relationship implied in the problem. An example of this level is the response of P36 (see Figure 24), who expresses through a symbolic (algebraic) representation the functional relationship and explains what each term of that expression means.
The responses of five prospective teachers are categorized at the intermediate level. They demonstrated functional relationships using specific strategies based on particular cases. For example, of these five, three of them employed the strategy of covariation, as evidenced by the response of P20 (see Figure 25), who explained using a symbolic representation x+1 (with the white tiles) and x +2 (with the gray tiles), noting that when the number of white tiles increases by one, the number of gray tiles increases by two. However, a conceptual error is observed, as both variables are represented with the same algebraic symbolism (X). Subsequently, this future teacher justified with a particular case, that is, when there are 10 white tiles, there are 26 gray tiles, and when there are 11 white tiles, there are 28 gray tiles.
On the other hand, at the same intermediate level, two prospective teachers made use of a recursive pattern, as observed in the response of P6 who explained as follows: "if the number of white tiles is unknown, we can find the relationship […] it follows a pattern of two by two," therefore, this prospective teacher calculates the number of gray tiles through the pattern of adding two to the previous amount.
Finally, the responses of five prospective teachers are categorized at the basic level, characterized by the use of pictorial and/or verbal representations, without evidencing a generalization of a functional relationship. A representative example of this category is the response of P18 (see Figure 26), in which the use of a verbal representation is evident to describe the calculation procedure and thus find the number of gray tiles, based on the pictorial representation of the task.
Knowledge of specialized content in question G
Table 9 shows the overall results related to the specialized knowledge of content in question G, which seeks diverse representations for the generalization of the functional relationship of the task.
Table 9 shows that the category with the highest frequency corresponds to nonsensical responses to the posed problem, recorded in 16 of the analyzed responses. A representative example of this category was the response of P30 (see Figure 25), who developed representations related to the concept of ratio as seen in the left and center images of Figure 27, while in the right image of the same figure, multiplication is used (2 * 3 = 6). Therefore, the representations used by P30 do not consider the functional relationship required in the task.
In this same question, 12 prospective teachers did not respond. This was followed by 3 future teachers whose responses were categorized at the advanced level, as they used diverse representations to generalize the functional relationship implied in the task. An example of the above is the response of P36, who represented the functional relationship through a symbolic (algebraic) expression (see left image, Figure 28), which consists of the letter "X" representing the white tiles (as also shown in Figure 22) multiplied by two plus 3 * 2 (tiles at the ends of the pictorial representation). For its part, P36 verbally represented the pictorial representation of the functional relationship (see right image, Figure 28).
Within the intermediate level, three prospective teachers provided responses that demonstrate a functional relationship using two representations. An example of a response at this level is that expressed by P24 (see Figure 27), who made use of verbal representation (left image, Figure 29) and a pictorial representation that configures the distribution of the two white tiles with the ten gray tiles (right image, Figure 29).
Finally, there was a smaller number of prospective teachers (4) who demonstrated a basic level of specialized knowledge. This was because their responses were limited to the use of verbal or pictorial representations to show a particular case. An example of this level is the response of P15 (see Figure 30), who created a pictorial representation to illustrate the number of white and gray tiles, and their subsequent counting to determine the total number of gray tiles.
Discussion and Conclusions
This study analyzed the specialized knowledge of prospective Primary Education teachers when solving a task of functional thinking, through the strategies and representations used in the seven questions of the proposed task. It was observed that pictorial representations predominated in the initial questions (A and B), associated with particular cases, while in the later questions a significant portion of the prospective teachers employed symbolic representations related to functional correspondence type responses. This result reinforces the importance of inductive reasoning proposed by Cañadas and Castro (2007), which promotes the transition from particular cases to generalizations.
Regarding the representations used, the results show a reduced use of symbolic representations to express the generalization of the functional relationship involved in the task. This result is in line with Aké (2021) and Morales and Parra-Fica (2022), who identified similar difficulties in expressing generalization through symbolic representations. On the other hand, the ability of some participating prospective teachers to use diverse representations suggests the need to continue deepening the development of this type of task, as mentioned by Polo-Blanco et al. (2019) and Wilkie (2014), who emphasize the need to strengthen functional thinking through formative strategies focused on the diversity of representations.
Regarding the different levels of knowledge exhibited by prospective teachers, it can be concluded that the basic level predominates, characterized by the use of pictorial representations for particular cases or direct responses without mentioning the strategies used. However, there was a small number of prospective teachers who reached intermediate and advanced levels, who demonstrated the use of strategies based on correspondence and covariation relationships, showing a deeper understanding of the functional relationship involved in the task. The previous result may be conditioned by the traditional conception of algebra teaching in which these prospective teachers were trained. This implies that the knowledge of these prospective teachers may be supported by a mechanical approach focused on algorithmic procedures and the manipulation of algebraic symbolism, rather than an understanding of structures and mathematical relationships present in algebraic tasks, as well as the appropriate use of increasingly sophisticated representations (Blanton; Kaput, 2011; Cañadas; Molina, 2016; Morales; Parra-Fica, 2022). The previous ideas align with the findings in the literature (e.g., Ellis, 2011; Blanton; Kaput, 2011; Morales and Parra-Fica 2022) that suggest designing formative interventions that allow prospective teachers to transition to more sophisticated levels of mathematical reasoning, promoting the use of generalizations and diverse representations to express it. In turn, those prospective teachers who reached an advanced level of knowledge were characterized by combining different strategies and representations to address the task. This aspect is relevant, as it suggests that the design of this type of activity can favor the development of deeper specialized content knowledge in prospective teachers by promoting the articulated use of various approaches to solve functional problems.
As prospective teachers progressed in the development of the task, it was observed that they managed to mobilize different levels of specialized content knowledge. For example, some began the task showing a basic level of knowledge and finished demonstrating an intermediate and advanced level of knowledge. This result suggests that the task involved in this study can favor the mobilization of specialized knowledge about functional relationships in future teachers. This could be an incentive to incorporate tasks of this type into initial teacher education, which is supported by Oliveira and Mestre (2014) and Hauck and Alsina (2021), who argue that solid specialized knowledge is essential to overcome a static view of functions and algebraic concepts.
This study opens several lines of action for both the initial training of teachers and educational research in school algebra. In the area of teacher training, it is suggested to systematically integrate functional thinking tasks into initial training programs, not only as assessment resources but also as didactic tools that allow prospective teachers to develop a deep and flexible understanding of the function as mathematical content. These tasks should be designed in a way that favors the transition from concrete representations to more abstract forms, thus promoting an articulated use of representations that involves the concept of function. It is also proposed that teacher training incorporate explicit opportunities for the analysis and discussion of conceptual errors, such as those identified in this study (use of algebraic symbolism), and that collaborative work spaces be designed where prospective teachers can justify, contrast, and refine their solution strategies. This not only contributes to the development of specialized content knowledge but also strengthens fundamental metacognitive and pedagogical competencies for teaching algebra.
However, this study presents some limitations. One of them is the small size of the sample, which makes it difficult to generalize the results to a broader population. Likewise, the analysis focused on a single task of functional thinking, which limits the variety of contexts and strategies that could have been explored.
Future research should conduct quantitative or mixed-methods studies with larger samples and diversify the instruments used. It would also be pertinent to develop research that considers a greater variety of tasks in a functional context of school algebra in order to confirm, deepen, and enrich the findings obtained.
Availability of research data
the dataset supporting the results of this study is published in the article itself.
References
-
AKÉ, Lilia Patricia. El carácter algebraico en el conocimiento matemático de maestros en formación. Tecné, Episteme y Didaxis: TED, n. 49, p. 15-34, 2021. Disponible en: https://doi.org/10.17227/ted.num49-9871 Consultado el: 18 de mayo de 2025.
» https://doi.org/10.17227/ted.num49-9871 - AYALA-ALTAMIRANO, Cristina; MOLINA, Marta. Fourth-graders’ justifications in early algebra tasks involving a functional relationship. Educational Studies in Mathematics, v. 107, n. 2, p. 359-382, 2021.
-
BALL, Deborah et al. Conocimiento del contenido para la enseñanza: ¿Qué lo hace especial? Journal of Teacher Education, v. 59, n. 5, p. 389-407, 2008. Disponible en: https://doi.org/10.1177/0022487108324554 Consultado el: 18 mayo 2025.
» https://doi.org/10.1177/0022487108324554 - BLANTON, Maria. Algebra and the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NH: Heinemann, 2008.
- BLANTON, Maria. Early algebra. In: USISKIN, Zalman et al. (Org.). Future curricular trends in school algebra and geometry Dordrecht: Springer, 2010. p. 45–61.
- BLANTON, Maria et al. Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5 Reston, VA: National Council of Teachers of Mathematics, 2011.
- BLANTON, Maria et al. The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, v. 46, n. 1, p. 39-87, 2015.
- BLANTON, Maria; KAPUT, James. Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, v. 36, n. 5, p. 412-446, 2005.
-
BLANTON, Maria; KAPUT, James. Functional thinking as a route into algebra in the elementary grades. In: CAI, Jinfa; KNUTH, Eric (Org.). Early algebraization: a global dialogue from multiple perspectives. Dordrecht: Springer, 2011. p. 5–23. Disponible en: https://doi.org/10.1007/978-3-642-17735-4_2 Consultado el: 18 mayo 2025.
» https://doi.org/10.1007/978-3-642-17735-4_2 - BRIZUELA, Bárbara. Una mirada sobre el aprendizaje del álgebra en niños en niveles parvulario y básico desde los sistemas notacionales. Revista Chilena de Educación Matemática, v. 16, n. 1, p. 3–11, 2024.
- BRIZUELA, Bárbara; EARNEST, Darrell. Mathematical conversations within the practice of mathematics London: Routledge, 2019.
- CAÑADAS, Maria.; CASTRO, Encarnación. A proposal of categorisation for analysing inductive reasoning. PNA, v. 1, n. 2, p. 69-81, 2007.
- CAÑADAS, Maria; MOLINA, Marta. Una aproximación al marco conceptual y principales antecedentes del pensamiento funcional en las primeras edades. In: CASTRO, Encarnación et al (Ed.). Investigación en educación matemática: homenaje a Luis Rico. Granada: Comares, 2016. p. 209–218.
- CLAPHAM, Christopher. Diccionario de matemáticas Madrid: Complutense, 1998.
- CLEMENTS, Douglas; SARAMA, Julie. Learning and teaching early math: The learning trajectories approach. 2. ed. London: Routledge, 2020.
- DUVAL, Raymond. Un análisis cognitivo de problemas de comprensión en el aprendizaje de las matemáticas. In: DUVAL, Raymond; SÁENZ-LUDLOW, Adalira (Org.). Comprensión y aprendizaje en matemáticas: perspectivas semióticas seleccionadas. Bogotá: Universidad Distrital Francisco José de Caldas, 2016. p. 61–94.
- ELLIS, Amy. Generalizing-promoting actions: How classroom collaborations can support students' mathematical generalizations. Journal of Mathematics Teacher Education, v. 42, n. 4, p. 123-145, 2011.
- FERNÁNDEZ, Flory. El análisis de contenido como ayuda metodológica para la investigación. Revista de Ciencias Sociales (Cr), v. 2, n. 96, p. 35-53, 2002.
- FLORES, Ariana rodríguez et al. Conocimiento especializado de los profesores de matemáticas: Un estudio de caso sobre la enseñanza de los conceptos básicos de función. Uniciencia, v. 32, n. 1, p. 89-107, 2018.
- HAUCK, Nataly Pincheira; ALSINA, Ángel. Hacia una caracterización del álgebra temprana a partir del análisis de los currículos contemporáneos de Educación Infantil y Primaria. Educación matemática, v. 33, n. 1, p. 153-180, 2021.
- HERNÁNDEZ, Roberto et al. Metodología de la investigación 6. ed. México, DF: McGraw-Hill, 2014.
- HILL, Heather et al. Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, v. 42, n. 2, p. 371-406, 2005.
- JOHNSONBAUGH, Richard. Matemáticas discretas 6. ed. México: Pearson Educación, 2005.
- KAPUT, James et al (Ed.). Algebra in the early grades London: Routledge, 2017.
- KAPUT, James. Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K–12 curriculum Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science, 2000.
- KAPUT, James. What is algebra? What is algebraic reasoning? In: KAPUT, James; CARRAHER, David; BLANTON, Maria (Org.). Algebra in the early grades London: Routledge, 2008. p. 5–17.
- KIERAN, Carolyn. Developing algebraic reasoning: The role of sequenced tasks and teacher question from the primary to the early secondary school levels. Quadrante, v. 16, n. 1, p. 5-26, 2007.
- LINS, Romulo.; KAPUT, James. The early development of algebraic reasoning: the current state of the field. In: STACEY, Kaye; CHICK, Helen; KENDAL, Margaret (Org.). The teaching and learning of algebra: the 12th ICMI Study. Dordrecht: Kluwer, 2004. p. 47–70.
-
MASON, John. Expressing generality and roots of algebra. In: BEDNARZ, Nadine; KIERAN, Carolyn; LEE, Lesley (Org.). Approaches to algebra Dordrecht: Springer, 1996. p. 65-86. Disponible en: https://doi.org/10.1007/978-94-009-1732-3_5 Consultado el: 18 mayo 2025.
» https://doi.org/10.1007/978-94-009-1732-3_5 - MCAULIFFE, Sharon; VERMEULEN, Cornelis. Preservice teachers’ knowledge to teach functional thinking. In: KIERAN, Carolyn (Org.). Teaching and learning algebraic thinking with 5- to 12-year-olds: the global evolution of an emerging field of research and practice. Cham: Springer, 2018. p. 403–425.
- MEJÍAS, Cristian; ALSINA, Ángel. La incorporación del Early Algebra en el currículo de Educación Primaria. Números. Revista de Didáctica de las Matemáticas, v. 105, p. 81-102, 2020.
- MERINO, Eduardo et al. Uso de representaciones y patrones por alumnos de quinto de Educación Primaria en una tarea de generalización. Edma, v. 2, n. 1, p. 24-40, 2013.
-
MINEDUC. Bases curriculares 1º a 6º básico Santiago: Ministerio de Educación, 2023. Disponible en: https://www.curriculumnacional.cl/614/w3-article-174160.html
» https://www.curriculumnacional.cl/614/w3-article-174160.html - MINEDUC. Bases curriculares para la Educación Primaria Santiago: Ministerio de Educación, 2012.
- MINEDUC. Estándares orientadores para carreras de Pedagogía en Educación Básica: formación inicial y desempeño profesional Santiago: Unidad de Curriculum y Evaluación, 2021.
- MOLINA, Marta. Traducción del simbolismo algebraico al lenguaje verbal: Indagando en la comprensión de estudiantes de diferentes niveles educativos. Gaceta de la RSME, v. 17, n. 3, p. 559-579, 2014.
- MORALES, Rodolfo et al. Relaciones funcionales y estrategias de alumnos de primero de en un contexto funcional. Enseñanza de las Ciencias, v. 36, n. 3, p. 59-78, 2018.
-
MORALES, Rodolfo; PARRA-FICA, Jose. Estrategias que emplean futuros profesores de Educación Primaria en una tarea de relación funcional. Acta Scientiae, v. 24, n. 3, p. 32-62, 2022. https://doi.org/10.17648/acta.scientiae.6959
» https://doi.org/10.17648/acta.scientiae.6959 - OLIVEIRA, Hélia; MESTRE, Célia. Opportunities to develop algebraic thinking in elementary grades throughout the school year in the context of mathematics curriculum changes. In: LI, Yeping; SILVER, Edward; LI, Shiqi (Org.). Transforming mathematics instruction: Multiple approaches and practices. Dordrecht: Springer, 2014. p. 173-197.
- PINTO, Eder; CAÑADAS, María. Generalization in fifth graders within a functional approach. PNA, v. 12, n. 3, p. 173-184, 2017.
- POLO-BLANCO, Irene et al. Portuguese and Spanish prospective teachers’ functional thinking on geometric patterns. In: JANKVIST, Uffe Thomas; HEUVEL-PANHUIZEN, Marja van den; VELDHUIS, Michiel (Eds.). Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education Utrecht: Freudenthal Group; Freudenthal Institute, Utrecht University; ERME, 2019. p. 670-671.
-
RICO, Luis. La competencia matemática en PISA. PNA, v. 1, n. 2, p. 47-66, 2006. Disponible en: https://revistaseug.ugr.es/index.php/pna/article/view/6215 Consultado el: 9 deciembre 2025.
» https://revistaseug.ugr.es/index.php/pna/article/view/6215 -
RICO, Luis. Sobre las nociones de representación y comprensión en la investigación en educación matemática. PNA, v. 4, n. 1, p. 1-14, 2009. Disponible en: https://revistaseug.ugr.es/index.php/pna/article/view/6172 Consultado el: 9 deciembre 2025.
» https://revistaseug.ugr.es/index.php/pna/article/view/6172 - SÁNCHEZ, Victoria; LLINARES, Salvador. Four student teachers' pedagogical reasoning on functions. Journal of Mathematics Teacher Education, v. 6, n. 1, p. 5-25, 2003.
- SHLIEMANN, Analúcia et al. El carácter algebraico de la aritmética: de las ideas de los niños a las actividades en el aula. Buenos Aires: Paidós, 2008.
- SHULMAN, Lee. Those who understand: Knowledge growth in teaching. Educational Researcher, v. 15, n. 2, p. 4-14, 1986.
- SMITH, Erick. Representational thinking as a framework for introducing functions in the elementary curriculum. In: KAPUT, James; CARRAHER, David; BLANTON, Maria (Eds.). Algebra in the early grades London: Routledge, 2008. p. 133-160.
-
STEIN, Itamar. The global dimension of the algebra of the monoid of all partial functions on an n-set as the algebra of the EI-category of epimorphisms between subsets. Journal of Pure and Applied Algebra, v. 223, n. 8, p. 3515-3536, 2019. Disponible en: https://doi.org/10.1016/j.jpaa.2018.11.015 Consultado el: 9 deciembre 2025.
» https://doi.org/10.1016/j.jpaa.2018.11.015 - STRACHOTA, Susanne. Conceptualizing generalization. Open Mathematical Education Notes, v. 6, n. 1, p. 41-55, 2016.
- STRACHOTA, Susanne. Generalizing in the context of an early algebra intervention. Journal for the Study of Education and Development, v. 43, n. 2, p. 347-394, 2020.
- STRAND, Krista; MILLS, Briana. Mathematical content knowledge for teaching elementary mathematics: A focus on algebra. The Mathematics Enthusiast, v. 11, n. 2, p. 385-432, 2014.
- UREÑA, Jason et al. Generalization: strategies and representations used by sixth to eighth graders in a functional context. Mathematics Education Research Journal, v. 36, n. 3, p. 519-545, 2024.
- VILLA-OCHOA, Jhony Alexánder. El proceso de generalización matemática: Algunas reflexiones en torno a su validación. Revista TecnoLógicas, n. 16, p. 139-151, 2006.
- WILKIE, Karina. Upper primary school teachers’ mathematical knowledge for teaching functional thinking in algebra. Journal of Mathematics Teacher Education, v. 17, n. 5, p. 397-428, 2014.
Source: 
Source: 
Source: 
Source: 
Source: Adapted from 
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.
Source: The author.