Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School

Citation: Ayala-Altamirano, C.; Pinto, E.; Molina, M.; Cañadas, M.C. Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School. Mathematics 2022,10, 2229. https://doi.org/10.3390/ math10132229 Academic Editor: Jay Jahangiri Received: 29 April 2022 Accepted: 23 June 2022 Published: 25 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 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/). mathematics Article Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School Cristina Ayala-Altamirano 1, Eder Pinto 2,* , Marta Molina 3and María C. Cañadas 4 1Department of Education, University of Almeria, Carretera San Urbano, La Cañada, 04120 Almería, Spain; [email protected] 2Faculty of Education, Universidad del Desarrollo, Av. Plaza 680, Santiago 7610658, Chile 3Departamento de Didáctica de las Matemáticas y de las Ciencias Experimentales, Escuela de Educación y Turismo de Ávila, Universidad de Salamanca, Madrigal de las Altas Torres, 3, 05003 Ávila, Spain; [email protected] 4Department of Mathematics Didactics, Faculty of Education Sciences, University of Granada, 18071 Granada, Spain; [email protected] *Correspondence: [email protected] Abstract: In this study, we analyze how 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. Specifically, we seek: (a) to describe the main characteristics of the tasks that support algebraic thinking through a translation process from arithmetic word problems to algebraic language and vice versa, and (b) to identify how pupils refer to indeterminate quantities in these contexts and what meaning they give to them. The analysis focuses on the semantic congruence of the expressions proposed by them and on the dialogue they held during the translation process. We analyzed the oral discussion in the pools and the written responses to the problem that pupils posed. The results show that arithmetic word problems allow the indeterminate to become an object of thought for pupils, who represent it in multiple ways and refer to it when proposing equations that represent the structure of each problem. Another finding highlights that reflection on the interpretation of the equations supports the identification of two meanings associated with indeterminate quantities, namely, unknown and variable. Keywords: arithmetic word problems; early algebra; indeterminate quantities; elementary education; problem posing; translation MSC: 97H20 1. Introduction Currently, different curricular guidelines consider algebraic thinking as a transversal topic from the beginning of schooling [ 1 – 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, with this study, we seek to contribute with ways of approaching algebraic thinking from contents that have traditionally been seen as exclusively arithmetic. In this paper, we 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 us to delve into the paths that pupils have to give meaning to the indeterminate. Mathematics 2022,10, 2229. https://doi.org/10.3390/math10132229 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 2229 2 of 18 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 ]. In this study, we 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 – 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 – 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. Considering the previous aspects, our proposal is to carry out tasks whose objective is to support the algebraic thinking of elementary school pupils and the correct representation of indeterminate quantities considering familiar contexts for them. 1.1. Algebraic Thinking The conceptual framework that directs our 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, Mathematics 2022,10, 2229 3 of 18 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 ]. In this study, we focus on the equations, i.e., the second content area described previously. 1.2. Linear Equations in Elementary School In this study, we focus on linear equations because these are deemed suitable for the age and their work is suggested in elementary school curricula [ 1 – 4 ]. We 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. 1.3. 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. In this study, we focus on the second category. These tasks are characterized by being based on an open situation, particularly an equation. From this point, we 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 ]. Previous studies have shown that posing problems from given mathematical equations or calculations requires understanding the meaning of the operations [ 25 ]. In addition, in this type of task, pupils usually follow an algorithmic process focused on the operational and non-semantic structure of the problems [26]. Mathematics 2022,10, 2229 4 of 18 Problem posing, from the teaching perspective, is a means of evaluating the conceptions of pupils regarding a particular topic [ 22 , 27 ], and allows their abilities to use their mathematical knowledge to be recognized. 2. Research Objectives In this study, we analyze the work completed by a group of 9–10-year-old pupils. Specifically, we seek to: (a) describe the main characteristics of the tasks that support algebraic thinking through a translation process from AWP to algebraic language and vice versa, and (b) identify how pupils refer to indeterminate quantities in these contexts and what meaning they give to them. 3. Materials and Methods This study is part of a broader Classroom Teaching Experiment (CTE) [ 28 ], and is part of the research-design paradigm [ 29 ]. The general objective of the CTE was to guide pupils ages 9–10 in the expression and justification of general mathematical ideas by working on three approaches to school algebra. 3.1. Context This study was conducted in the context of a summer school for pupils who had just finished 4th grade of the elementary school. The summer school is an activity organized every year by the Faculty of Education of the Universidad del Desarrollo (Santiago, Chile), with the aim of providing effective and fair opportunities to children through the development of thinking and innovation. Both the design and the implementation of this activity contemplated a collaborative work that involved the participation of researchers and teachers from the area of mathematics education in Chile and Spain. Due to the health situation caused by the COVID-19 pandemic, the summer school was developed virtually. Specifically, the pupils accessed the activities from their homes, through different devices connected to the internet: mobile phones, tablets, or computers. Additionally, each pupil used physical materials that were provided by the university: a board with markers of different colors, a folder to record their findings on worksheets, and manipulatives. The students were encouraged to keep the cameras on, so that the teacher–researchers could observe their work. In turn, the students’ families were asked to allow them to work on their own without the help of others. 3.2. Participants This qualitative study involved 21 pupils who were between 9 and 10 years old and who had completed 4th grade online, given the health context. The pupils belonged to two schools that are part of the same Educational Foundation that serves children and young people from low-income sectors. Specifically, nine pupils from one school and 12 from another. Pupils’ anonymity in this paper was ensured by assigning each a code: Si, where i = 1 . . . 21. The pupils were selected with the help of their regular math teachers under the following three criteria: (a) gender parity (10 girls and 11 boys); (b) willingness to work during the summer; and (c) different paces of learning. Regarding the previous knowledge of the pupils, it is important to point out that, although the Chilean curriculum contemplates a thematic topic of algebra and patterns [ 1 ], the learning objectives related to this topic corresponding to 4th grade were not fully addressed due to the country’s health situation [ 30 ]. In the mathematics classes, the learning objectives that refer to the topic of numbers and operations were mainly addressed. Regarding the topic of patterns and algebra, we worked with numerical patterns that involve an operation, which was registered in tables. In the previous grade (3rd grade), pupils solved one-step equations involving addition and subtraction and a geometric symbol representing an unknown number. They employed strategies such as trial and error or the inverse operation. The mathematical representations that they used were Mathematics 2022,10, 2229 5 of 18 numerical–symbolic ones and the emphasis of the classes was focused on promoting fluency in the calculation. 3.3. Design The summer school was organized in 10 sessions, including a pre-test and a posttest, and following the approaches to algebraic thinking [ 14 ]: (a) generalized arithmetic; (b) equivalence, expressions, and equations; and (c) functional thinking (see Figure 1). Mathematics 2022, 10, x FOR PEER REVIEW 5 of 19 [1], the learning objectives related to this topic corresponding to 4th grade were not fully addressed due to the country’s health situation [30]. In the mathematics classes, the learning objectives that refer to the topic of numbers and operations were mainly addressed. Regarding the topic of patterns and algebra, we worked with numerical patterns that involve an operation, which was registered in tables. In the previous grade (3rd grade), pupils solved one-step equations involving addition and subtraction and a geometric symbol representing an unknown number. They employed strategies such as trial and error or the inverse operation. The mathematical representations that they used were numerical–symbolic ones and the emphasis of the classes was focused on promoting fluency in the calculation. 3.3. Design The summer school was organized in 10 sessions, including a pre-test and a post-test, and following the approaches to algebraic thinking [14]: (a) generalized arithmetic; (b) equivalence, expressions, and equations; and (c) functional thinking (see Figure 1). Figure 1. Organization of sessions. In the first and last sessions, the pupils’ responses to different algebraic tasks were assessed. Sessions 2–9 followed a similar structure, organized into three parts: • small groups (4–5 pupils), in which the aim was for the pupils to dialogue and collaborate with each other in the search for regularities, conjectures, and solutions to the problems presented; • whole group, where each group presented their findings and two teachers led the discussion so that the pupils synthesized their ideas; and • medium-sized groups (10–11 pupils), in which the objective was to transfer what had been discussed to another similar situation or to delve into a finding from the previous parts on the problems presented. Each part supported the installation of spaces for cooperation, confrontation, and discussion of ideas. 3.4. Instruction Sequence: Sessions 2–5 In this study, we focus on session 6. However, we also describe, in general terms, what happened previously, without considering the initial assessment. In session 2, pupils expressed their general ideas through natural language by arguing what happens when odd and even numbers are added. Then, in sessions 3 and 4, they discussed the meaning of the equal sign. Here, for the first time, the letter is introduced as a representation for generalizing arithmetic properties (for example, the commutativity of addition). In this first encounter with letters, the pupils concluded that it could represent any number (generalized number). In the fifth and sixth sessions, we focused on the expression and resolution of equations. During these sessions we suggested to the pupils to (a) represent and solve equations using different strategies, (b) use the letter as a representation of an unknown, and (c) provide evidence to validate given explanations. We introduced the ideas of equality, equation, mathematical histories (an idea we use to refer to the Figure 1. Organization of sessions. In the first and last sessions, the pupils’ responses to different algebraic tasks were assessed. Sessions 2–9 followed a similar structure, organized into three parts: • small groups (4–5 pupils), in which the aim was for the pupils to dialogue and collaborate with each other in the search for regularities, conjectures, and solutions to the problems presented; • whole group, where each group presented their findings and two teachers led the discussion so that the pupils synthesized their ideas; and • medium-sized groups (10–11 pupils), in which the objective was to transfer what had been discussed to another similar situation or to delve into a finding from the previous parts on the problems presented. Each part supported the installation of spaces for cooperation, confrontation, and discussion of ideas. 3.4. Instruction Sequence: Sessions 2–5 In this study, we focus on session 6. However, we also describe, in general terms, what happened previously, without considering the initial assessment. In session 2, pupils expressed their general ideas through natural language by arguing what happens when odd and even numbers are added. Then, in sessions 3 and 4, they discussed the meaning of the equal sign. Here, for the first time, the letter is introduced as a representation for generalizing arithmetic properties (for example, the commutativity of addition). In this first encounter with letters, the pupils concluded that it could represent any number (generalized number). In the fifth and sixth sessions, we focused on the expression and resolution of equations. During these sessions we suggested to the pupils to (a) represent and solve equations using different strategies, (b) use the letter as a representation of an unknown, and (c) provide evidence to validate given explanations. We introduced the ideas of equality, equation, mathematical histories (an idea we use to refer to the translation from natural language to algebraic language), and letters as unknowns, among others. In session 5, they proposed an adaptation of the cards and envelopes problem described in [ 20 ] (Figure 2). The pupils were asked to express and solve the equation involved in the problem using manipulatives and pictorial representations. However, it became clear that it was necessary to deepen the approach to equations. Although they were able to represent them with manipulative material, drawings, and even letters, doubts remained about the meaning of the indeterminate. Mathematics 2022,10, 2229 6 of 18 Mathematics 2022, 10, x FOR PEER REVIEW 6 of 19 translation from natural language to algebraic language), and letters as unknowns, among others. In session 5, they proposed an adaptation of the cards and envelopes problem described in [20] (Figure 2). The pupils were asked to express and solve the equation involved in the problem using manipulatives and pictorial representations. However, it became clear that it was necessary to deepen the approach to equations. Although they were able to represent them with manipulative material, drawings, and even letters, doubts remained about the meaning of the indeterminate. Figure 2. Problem session 5. 3.5. Session 6 In the first part of the session, two problems were presented to the pupils: • I bought a box of colored pencils. At home I had pencils, now I have 20 in total. How many pencils are in the box? and • I have a basket of apples. Inside the basket, there are 20 green apples and other red ones. How many apples are in the basket? These AWPs were represented with natural language and the pupils had to translate and represent with algebraic language. The first AWP involves an unknown, has a unique solution, and involves the structure y + 15 = 20. The other AWP implicates two unknowns whose values could not be determined due to the lack of data in the statement, and it involves the structure y = 20 + b. The pupils had to represent the AWPs on their boards. Firstly, the pupils were asked to “tell the story” (i.e., to represent verbal sentences) with mathematical symbols. We were interested in pupils representing indeterminate quantities however they wanted. One by one they explained how they did it and discussed whether what their classmates did was the same as what each one represented. Then the possibility of representing them with letters was mentioned by the teacher and discussed within the group, as had happened in previous sessions. In Figure 3 the AWP and representations that the teacher raised for each situation are presented. The teacher presented two different representations for each problem: the drawing of the box and colored pencils or the basket of apples, and a symbolic–algebraic expression, with the use of letters. It was believed that familiar representations for the pupils, such as drawings, would allow them to understand the statement of the problem and the meaning of each element of the algebraic expression. After reaching an agreement on the representation of the problems in a group discussion, pupils solved each equation and discussed what the value of each unknown was. Figure 3. Problem session 6. Figure 2. Problem session 5. 3.5. Session 6 In the first part of the session, two problems were presented to the pupils: • I bought a box of colored pencils. At home I had pencils, now I have 20 in total. How many pencils are in the box? and • I have a basket of apples. Inside the basket, there are 20 green apples and other red ones. How many apples are in the basket? These AWPs were represented with natural language and the pupils had to translate and represent with algebraic language. The first AWP involves an unknown, has a unique solution, and involves the structure y+ 15 = 20. The other AWP implicates two unknowns whose values could not be determined due to the lack of data in the statement, and it involves the structure y= 20 + b. The pupils had to represent the AWPs on their boards. Firstly, the pupils were asked to “tell the story” (i.e., to represent verbal sentences) with mathematical symbols. We were interested in pupils representing indeterminate quantities however they wanted. One by one they explained how they did it and discussed whether what their classmates did was the same as what each one represented. Then the possibility of representing them with letters was mentioned by the teacher and discussed within the group, as had happened in previous sessions. In Figure 3the AWP and representations that the teacher raised for each situation are presented. The teacher presented two different representations for each problem: the drawing of the box and colored pencils or the basket of apples, and a symbolic– algebraic expression, with the use of letters. It was believed that familiar representations for the pupils, such as drawings, would allow them to understand the statement of the problem and the meaning of each element of the algebraic expression. After reaching an agreement on the representation of the problems in a group discussion, pupils solved each equation and discussed what the value of each unknown was. Mathematics 2022, 10, x FOR PEER REVIEW 6 of 19 translation from natural language to algebraic language), and letters as unknowns, among others. In session 5, they proposed an adaptation of the cards and envelopes problem described in [20] (Figure 2). The pupils were asked to express and solve the equation involved in the problem using manipulatives and pictorial representations. However, it became clear that it was necessary to deepen the approach to equations. Although they were able to represent them with manipulative material, drawings, and even letters, doubts remained about the meaning of the indeterminate. Figure 2. Problem session 5. 3.5. Session 6 In the first part of the session, two problems were presented to the pupils: • I bought a box of colored pencils. At home I had pencils, now I have 20 in total. How many pencils are in the box? and • I have a basket of apples. Inside the basket, there are 20 green apples and other red ones. How many apples are in the basket? These AWPs were represented with natural language and the pupils had to translate and represent with algebraic language. The first AWP involves an unknown, has a unique solution, and involves the structure y + 15 = 20. The other AWP implicates two unknowns whose values could not be determined due to the lack of data in the statement, and it involves the structure y = 20 + b. The pupils had to represent the AWPs on their boards. Firstly, the pupils were asked to “tell the story” (i.e., to represent verbal sentences) with mathematical symbols. We were interested in pupils representing indeterminate quantities however they wanted. One by one they explained how they did it and discussed whether what their classmates did was the same as what each one represented. Then the possibility of representing them with letters was mentioned by the teacher and discussed within the group, as had happened in previous sessions. In Figure 3 the AWP and representations that the teacher raised for each situation are presented. The teacher presented two different representations for each problem: the drawing of the box and colored pencils or the basket of apples, and a symbolic–algebraic expression, with the use of letters. It was believed that familiar representations for the pupils, such as drawings, would allow them to understand the statement of the problem and the meaning of each element of the algebraic expression. After reaching an agreement on the representation of the problems in a group discussion, pupils solved each equation and discussed what the value of each unknown was. Figure 3. Problem session 6. Figure 3. Problem session 6. In the second part of the session, the teacher generated a space for discussion with the pupils about the discoveries obtained in the previous part. The discussion sought: (a) to investigate the use of the letter when expressing the equations, and (b) to collect evidence to determine whether the letter represents an unknown quantity or a variable quantity. The teacher asked questions such as: Does the expression X+ 15 = 20 represent the pencil problem? Does the expression T= 20 + Srepresent the apple problem? Do you know how many pencils are in the box? Can we know how many apples are in the basket? How do you know? What evidence do you have? Could you explain it in a different way to your partner? Mathematics 2022,10, 2229 7 of 18 At the end of this part, pupils were asked to answer a test via a virtual test to assess whether they could transfer what was discussed to other situations with similar characteristics. They were asked to choose the alternative that allowed them to tell each of the stories shown in Figure 4. In the second situation, there were two possible correct answers: 40 + r=jand j+ 40 = r, the objective was to discuss the reasons for why both were correct. Mathematics 2022, 10, x FOR PEER REVIEW 7 of 19 In the second part of the session, the teacher generated a space for discussion with the pupils about the discoveries obtained in the previous part. The discussion sought: (a) to investigate the use of the letter when expressing the equations, and (b) to collect evidence to determine whether the letter represents an unknown quantity or a variable quantity. The teacher asked questions such as: Does the expression X + 15 = 20 represent the pencil problem? Does the expression T = 20 + S represent the apple problem? Do you know how many pencils are in the box? Can we know how many apples are in the basket? How do you know? What evidence do you have? Could you explain it in a different way to your partner? At the end of this part, pupils were asked to answer a test via a virtual test to assess whether they could transfer what was discussed to other situations with similar characteristics. They were asked to choose the alternative that allowed them to tell each of the stories shown in Figure 4. In the second situation, there were two possible correct answers: 40 + r = j and j + 40 = r, the objective was to discuss the reasons for why both were correct. Figure 4. Test part 2. Finally, in the third part, we wanted the pupils to extend what they had learned to other cases and they were asked to carry out the reverse process, that is, to create mathematical stories from the equation 25 + u = 45. Each pupil wrote the problem that each one invented on the board and then discussed the relevance of each situation. 3.6. Analysis We analyzed pupils’ responses to (a) describe the characteristics of the tasks that support algebraic thinking, and (b) identify how they refer to indeterminate quantities. The authors of this study classified the responses concerning the first two parts of the session, and all the written and oral responses of the pupils to the AWPs presented and the invented problems in the third part. Discrepancies were then discussed until an agreement was reached. To analyze the pupils’ translations, we based our method on the ideas of Duval’s [31] proposal. He pointed out that two representations are congruent when the following three conditions are met: (a) semantic correspondence between the significant units that constitute them; (b) semantic univocity, i.e., each initial significant unit of output corresponds to one and only one significant elementary unit of the input record; and (c) the order within the organization of the significant output units is maintained in the arrival representation. When one of these criteria is no longer met, the representations are not congruent with each other. However, this author added that two expressions can be referentially equivalent without being semantically congruent. Semantic congruence allows us to see the degree of transparency of the relationship between two representations. Figure 4. Test part 2. Finally, in the third part, we wanted the pupils to extend what they had learned to other cases and they were asked to carry out the reverse process, that is, to create mathematical stories from the equation 25 + u= 45. Each pupil wrote the problem that each one invented on the board and then discussed the relevance of each situation. 3.6. Analysis We analyzed pupils’ responses to (a) describe the characteristics of the tasks that support algebraic thinking, and (b) identify how they refer to indeterminate quantities. The authors of this study classified the responses concerning the first two parts of the session, and all the written and oral responses of the pupils to the AWPs presented and the invented problems in the third part. Discrepancies were then discussed until an agreement was reached. To analyze the pupils’ translations, we based our method on the ideas of Duval’s [ 31 ] proposal. He pointed out that two representations are congruent when the following three conditions are met: (a) semantic correspondence between the significant units that constitute them; (b) semantic univocity, i.e., each initial significant unit of output corresponds to one and only one significant elementary unit of the input record; and (c) the order within the organization of the significant output units is maintained in the arrival representation. When one of these criteria is no longer met, the representations are not congruent with each other. However, this author added that two expressions can be referentially equivalent without being semantically congruent. Semantic congruence allows us to see the degree of transparency of the relationship between two representations. If only correspondence and semantic uniqueness were observed, the translations were classified as equivalent. Semantically consistent translations and their equivalents were considered correct translations. Regarding the incorrect translations, we identified four types of errors: (a) incomplete translation because some data of the statement are missing; (b) translation related to the problem but that does not refer to indeterminate quantities and includes the answer to the problem (e.g., 15 + 5 = 20); (c) apparently correct translation, but the correspondence between the semantic units poorly relates the terms of the equation to the statement; and (d) translation with invented information or information unrelated to the problem. When translating a given statement into natural language, the pupils’ responses were classified into: (a) pictorial translation if drawings were used; (b) symbolic–arithmetic translation if using numbers and mathematical symbols; and (c) algebraic language translation Mathematics 2022,10, 2229 8 of 18 if it used numbers, mathematical symbols, and some symbols to refer to indeterminate quantities (letters or “?”). To analyze the invented problems, we translated the problems posed into algebraic language, maintaining the structure of the given equation, in addition to a left–right congruence whenever possible. Then we compared the translation obtained with the proposed equation. We also identified if the pupils proposed a new context or adapted the problems proposed in the previous parts. To identify how the pupils referred to indeterminate quantities, we analyzed the oral discussions in the pools and the written records of the problems they invented. We looked for linguistic expressions that used indefinite adjectives (e.g., little, a lot, one, another, too many, same, some, none, any), key phrases that conveyed that the pupils recognized an indeterminate amount in the analyzed contexts. For example: “it is a number that we do not know”, and “it is the number you want”, among others. In previous research, such as [ 5 , 15 ], elementary pupils referred to indeterminate quantities with the keywords “many” or “infinite” or the phrase “the number you want”. On the meaning given to indeterminate quantities, we analyzed the answers given to the questions Can we know the answer to the problem? Can we know what quantity the letter represents? In these cases, if the pupils answered that the problem only had one solution, we interpreted that the meaning associated with the indeterminate quantity was unknown. While if they answered that the answer to the problem could not be known because there was not enough information and that the answer depended on some data of the problem, we interpreted that the meaning given was that of a variable quantity. 4. Results In this section we first present the answers of the pupils to the tasks in which they had to translate verbal statements into algebraic language. Then, the answers given in the inverse process, translation from algebraic language to natural language, will be discussed. 4.1. From Natural Language to Algebraic Language: The Pencil Problem The first problem introduced involved the structure y+ 15 = 20. In this instance, nineteen elementary school pupils participated in its resolution. Initially, the focus was on the translation of the statement into algebraic language. After this, as the last step, the unknown value was discussed. Table 1shows a characterization of the translations proposed by the pupils. Thirteen pupils proposed a translation using algebraic language, one made a symbolic–arithmetic translation, four proposed a pictorial translation, and one pupil did not respond. Table 1. Equations proposed by the pupils in the pencil problem (structure y+ 15 = 20). Translation Pupil Equation Structure Equation Proposed by the Pupils Correct Translation Semantic Consistency SC SU OA Algebraic language S15 and S16 y+ 15 = 20 ? + 15 = 20 Yes Yes Yes Yes S5a+ 15 = 20 Yes Yes Yes Yes S3b+ 15 = 20 Yes Yes Yes Yes S14 15 + y= 20 15 + ? = 20 Yes Yes Yes No S1and S13 15 + a= 20 Yes Yes Yes No S7, S8, S17 and S18 15 + x= 20 Yes Yes Yes No S915 + a= 20 1No No Yes No S615 + y= 35 15 + a= 35 No No Yes No Symbolic–arithmetic S12 15 + 5 = 20 No No Yes No Pictorial S2, S4, S10 and S11 No No Yes No Not responding S19 SC = semantic correspondence; SU = semantic univocity; OA = order of apprehension. Note. 1 The equation is incorrect because the pupil says that he bought 15 pencils and “a” represents the ones he had at home. Mathematics 2022,10, 2229 9 of 18 Initially, pupils were asked to represent the problem statement freely on the board. Eleven of the pupils correctly translated the natural language expression into algebraic language. Four of them formulated an equation consistent with the pencil problem, while another seven proposed referentially equivalent equations, but these were not consistent since they did not meet the order of apprehension criterion. In this case, the structure represented by them was 15 + y= 20, whose verbal statement would have to mention first the number of pencils they have (15) and then the number of pencils in the box (y). In the design of the task, it was considered important that the pupils reflect on the equivalence between the expressions, that is why, in the discussion, the answers that involved the equations in the forms y+ 15 = 20 and 15 + y= 20 were contrasted. They were asked if they represented the same thing, even though the order of the addends was different. In this regard, S 8 pointed out: “if we change their order, the result does not change due to the commutative property”. S 15 also mentioned the commutative property to justify the equivalence between the expressions. Another important aspect considered in the design of the task was to guide the discussion so that the elementary pupils explicitly made the correspondence between the terms of the equation and the verbal statement. Focusing on indeterminate quantities and their representation, 10 pupils used letters (a,b, and x) and another four used the question mark “?” sign. In the discussion, the pupils referred to the letters or to the sign “?” as the number of pencils in the box they do not know. For example, S 3 , who proposed the equation b+ 15 = 20, said: “I thought. He says that he bought a box, but you don’t know how many pencils that box has, and that represents “b”. 15 at home. I used “b”, but any letter can be any number.” This last sentence highlights that he accepted that the representations of his classmates were also correct, even if different letters were used. The reference to indeterminate quantities was not only made in the algebraic language translations, in this problem it was also evidenced in a pictorial translation (see Figure 5). S 2 represented the indeterminate quantity on her drawing using the symbol “?”. She explained that she represented the 15 pencils and a box of mystery pencils. However, her translation is incomplete since she does not refer to or represent the total number of pencils (20). The pictorial representations of other pupils do not refer to indeterminate quantities either. Mathematics 2022, 10, x FOR PEER REVIEW 10 of 19 Figure 5. Translation of the verbal statement using drawings of S 2 . The last aspect considered in the design was the discussion about the value that the indeterminate quantity could take. In this problem, the pupils agreed that there could only be five pencils in the box. They complemented their argument by replacing the letter or the question mark “?” with that number and solving the sum to verify that the result was 20. In this way, they reaffirmed their idea that the indeterminate quantity can have only one value. 4.2. From Natural Language to Algebraic Language: The Apple Problem The second problem discussed was that of apples, whose structure is y = 20 + b. As in the previous problem, the translation was carried out first and then the possible value of each of the indeterminate quantities was discussed. Table 2 shows the characterization of the pupils’ translations. On this occasion, the number of correct translations with algebraic language was less than in the previous problem (9 out of 19); however, the number of pupils who made a translation of this type was the same. One pupil made a symbolic– arithmetic translation, three pupils a pictorial translation, and two pupils did not answer. Table 2. Equations proposed by the pupils in the apple problem (structure y = 20 + b). Translation Pupil Equation Structure Equation Proposed by the Pupils Correct Translation Semantic Consistency SC US OA Algebraic language S 5 b + 20 = y a + 20 = ? Yes Yes Yes No S 13 c + 20 = ? Yes Yes Yes No S 15 20 + b = y 20 + a = ? Yes Yes Yes No S 2 , S 6 and S 12 20 + x = ? Yes Yes Yes No S 3 20 + c = ? Yes Yes Yes No S 8 20 + b = x Yes Yes Yes No S 9 20 + x = a Yes Yes Yes No S 7 and S 17 20 + b = b 20 + x = x No Yes No No S 4 20 + b = 40 20 + n = 40 No No Yes No S 18 20 + x = No No Yes No Symbolic–arithmetic S 16 10 + 10 = 20 No No No No Pictorial S 1 , and S 11 No No Yes No S 10 No No No No Not responding S 14 and S 19 SC = semantic correspondence; SU = semantic Univocity; OA = order of apprehension. Figure 5. Translation of the verbal statement using drawings of S2. The last aspect considered in the design was the discussion about the value that the indeterminate quantity could take. In this problem, the pupils agreed that there could only be five pencils in the box. They complemented their argument by replacing the letter or the question mark “?” with that number and solving the sum to verify that the result was 20. In this way, they reaffirmed their idea that the indeterminate quantity can have only one value. Mathematics 2022,10, 2229 16 of 18 notation. The expressions that we identified in this study could be used in other contexts to highlight the indeterminate quantity and pupils can recognize it in different contexts. In addition, the characteristics of the tasks proposed in this study allowed the elementary pupils to associate the indeterminate quantities with two different meanings. As an unknown quantity with a fixed value when they had enough information in the statement about it, or as a variable number that depends on the values given to some of the unknown quantities. This distinction is of great relevance given the complexity of the polysemy of indeterminate quantities in algebraic contexts. As we have mentioned in our conceptual framework, in previous research with elementary pupils who needed to refer to or give meaning to indeterminate quantities [5,8,9] , pupils showed a tendency to assign specific values to them and make mistakes that are also evident in higher grades [ 11 , 12 ]. In this case, it is important to highlight that this did not occur; we assume that it was because the proposed situations were associated with contexts that were close to the pupils and involved numbers that allowed calculations to be carried out easily. The pupils gave meaning to the indeterminate considering that it was something that is not known in history, an element that they also highlighted in their study [ 18 ]. This helped them focus on the information given in the statement and avoid choosing numbers at random or resorting to the alphabet. It is worth mentioning that this type of error was observed in session 4, in which the letter was presented as a representation for indeterminate quantities for the first time and in a mathematical context when representing some properties of the addition. One limitation of our study is that it did not investigate how the previous difficulties were overcome until reaching the session that we discuss in this work. An open line of work is to analyze the relationship between the different areas of algebra and carry out a global analysis of all the sessions, which will allow the information presented here to be completed, describing how the different tasks proposed cause different or similar ways of referring to indeterminate quantities and give meaning to them. Author Contributions: Conceptualization, C.A.-A., E.P., M.M. and M.C.C.; methodology, C.A.-A., E.P., M.M. and M.C.C.; validation, C.A.-A., E.P., M.M. and M.C.C.; formal analysis, C.A.-A., E.P., M.M. and M.C.C.; investigation, C.A.-A., E.P., M.M. and M.C.C.; resources, C.A.-A., E.P., M.M. and M.C.C.; data curation, C.A.-A., E.P., M.M. and M.C.C.; writing—original draft preparation, C.A.-A., E.P., M.M. and M.C.C.; writing—review and editing, C.A.-A., E.P., M.M. and M.C.C.; visualization, C.A.-A., E.P., M.M. and M.C.C.; supervision, C.A.-A., E.P., M.M. and M.C.C.; project administration, E.P. and M.C.C.; funding acquisition, E.P. and M.C.C. All authors have read and agreed to the published version of the manuscript. Funding: This work has been developed within the project 23.400.173, financed by Universidad del Desarrollo (Chile), and the project with reference PID2020-113601GB-I00, financed by the State Research Agency (SRA) from Spain. Institutional Review Board Statement: The study was conducted in accordance with the Declaration of Helsinki, and approved by the Ethics Committee of Universidad del Desarrollo (CEEII) (date of approval 17 December 2020). Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Data Availability Statement: Data are available upon request due to restrictions. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. References 1. 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