Introduction

Traditionally, mathematics education follows the approach of teaching arithmetic before algebra. Students first master arithmetic procedures in elementary school and then move on to learning algebra through procedural methods in middle school (Blanton et al., 2007). However, this shift from concrete arithmetic thinking to abstract algebraic reasoning in high school and beyond can lead to difficulties in students’ mathematics development (Carpenter et al., 2000; Knuth et al., 2016). This problem led educators and mathematics education researchers to consider “deep, long-term algebra reform” (Kaput, 1999, p. 134). Kaput (1999) described a route to that reform as “infusing algebra throughout the mathematics curriculum from the very beginning of the school” (p. 134). By starting algebra education in kindergarten, students will develop a stronger algebra background with a deeper understanding and experience, preparing them for middle and high school (NCTM, 2000). This is the rationale behind including early algebra in students’ mathematics education.

Early algebra does not mean teaching common algebraic concepts and procedures addressed in the middle grades to elementary students earlier (Carraher et al., 2008). Besides, early algebra is not an attempt to expand the elementary curriculum (Kaput et al., 2008). Early algebra is a way of thinking that allows students to generalize relationships and mathematical facts by delving into the concepts already in the curriculum to provide a deep and coherent mathematical understanding (Blanton et al., 2007). Although integrating algebra in the elementary grades is a relatively new idea, recent research findings enable us to recognize the capability of kindergarten students (e.g., Stephens et al., 2022) and elementary students (e.g., Blanton et al., 2015b) to think algebraically. We also gain insights into classroom environments for early algebraic thinking (e.g., Bastable & Schifter, 2008). However, the question: “How can prospective teachers be given a good start on developing essential knowledge of algebra for teaching?” (Fey et al., 2007, p. 27) has not been answered sufficiently.

The problem of the theory–practice gap in teacher education is well known. A suggested solution to this challenge is incorporating classroom cases in teacher education, providing a more practical and authentic perspective than a purely theoretical approach (Smith & Friel, 2008). With these considerations, this study aimed to examine the development of prospective elementary teachers’ (PETs) knowledge of teaching early algebra through participation in case discussions. The research question that guided this study was: How do prospective elementary teachers’ knowledge of teaching early algebra change after participating in early algebra lessons including case discussions?

Theoretical background

Teacher knowledge

Ball et al. (2008) created the Mathematical Knowledge for Teaching framework (MKfT), which classifies teacher knowledge into two main domains: subject matter knowledge (SMK) and pedagogical content knowledge (PCK). The SMK domain is divided into three subdomains: common content knowledge (CCK), specialized content knowledge (SCK), and horizon content knowledge. The PCK domain is also divided into three subdomains: knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of content and curriculum.

According to Ball et al. (2008), under SMK, while CCK refers to knowledge and skills that are not specific for teaching and used in other settings, SCK refers to teaching specific knowledge and skills. For instance, expecting and responding to students’ “why” questions might be categorized under SCK. The last component of SMK, horizon content knowledge, is "an awareness of how mathematical topics are related over the span of mathematics included in the curriculum" (p. 403). Regarding PCK, KCS encompasses knowledge of students’ understanding and misunderstandings by merging content and student knowledge. KCT contains knowledge of selecting and ordering mathematical tasks while considering their benefits and drawbacks, facilitating class discussions, and using students’ strategies to enhance mathematical understanding. The final component of PCK, knowledge of content and curriculum, refers to a teacher’s understanding of what students have learned in previous years and what they will learn in the future regarding their current study area.

The current study, which aimed to investigate the development of PETs’ knowledge to teach early algebra, focused on KCS and KCT dimensions regarding (a) students’ conceptions and misconceptions and (b) instructional strategies and representations, respectively, as part of PCK. According to Hohensee (2017), in the learning and teaching process of early algebra, while elementary students need to transition from arithmetic to algebra, PETs need to transition from formal algebra back to early algebra. Following this approach, the knowledge and skills that elementary students should gain, as recommended by early algebra research, were considered CCK for prospective elementary teachers and investigated in this study.

Early algebra

Early algebra refers to integrating algebraic reasoning and instruction for young learners, typically between the ages of 6 and 12, and immersing them in the culture of algebra (Carraher & Schliemann, 2007; Lins & Kaput, 2004). While there is no consensus on what early algebra entails, any content or activity that helps students move beyond arithmetic and computational fluency to understanding mathematical structures can be considered a part of early algebra (Cai & Knuth, 2005; Lins & Kaput, 2004).

According to Blanton et al. (2018), generalizing, representing, justifying, and reasoning with mathematical relationships are four core algebraic practices, which are constructed around three big ideas: (i) equivalence, expressions, equations, and inequalities; (ii) generalized arithmetic; and (iii) functional thinking (Blanton et al., 2019). In the scope of this study, the development of PETs’ SMK and PCK was investigated in those three big ideas. However, the first big idea was limited to equivalence and equations. Since variable notation is a powerful tool for expressing generalizations (Carraher & Schliemann, 2007), PETs were expected to use variables to represent the generalizations related to generalized arithmetic and functional thinking. Thus, we introduced the concept of variables as CCK for PETs.

Equivalence and equations

The big idea of equivalence and equations focuses on students’ development in the relational understanding of the equal sign and solving simple linear equations by analyzing the structure of the equation. The relational view of the equal sign is about recognizing the equivalence or "sameness" between quantities or expressions on both sides of an equation (Stephens et al., 2013). Studies suggest that the notion of equality and relational view of the equal sign become significant for solving equations (Knuth et al., 2005) and required for operating on the structure of equations (i.e., carrying out the same operations on both sides) (Kieran, 1992) and solving equations with an understanding rather than memorizing a series of rules (Falkner et al., 1999). However, it is well documented that students do not view the equal sign as a symbol of equivalence; instead, they think that the equal sign is a signal to "do something" or an announcement of the result of an arithmetic operation (e.g., Knuth et al., 2006; McNeil & Alibali, 2005). Carpenter et al. (2003) describe that students approached solving the equation 8 + 4 = __ + 5 differently based on their view of the equal sign. Those who thought operationally tended to fill in the blank with 12 or 17, while those thinking relationally recognized 7 as the correct answer by understanding the equation’s structure and relationship between its numbers. On the other hand, studies have shown that students’ conceptions of the equal sign can develop from operational to relational (e.g., Blanton et al., 2015b; Warren et al., 2009). For this to happen, teachers need to have a relational view of the equal sign and the required pedagogical content knowledge to support students in developing meaningful understanding and use of the equal sign.

Generalized arithmetic

Generalized arithmetic focuses on students’ development in core algebraic practices in the context of arithmetic relationships (Blanton et al., 2019). It involves "helping children see, describe, and justify patterns and regularities in operations and properties of numbers" to "move beyond arithmetic to algebraic thinking" (Blanton, 2008, p. 12). The relationships related to generalized arithmetic mainly focus on a) the fundamental properties of numbers and operations (e.g., commutative property of addition, additive identity), b) the relationships among operations (e.g., inverse relationship between addition and subtraction), and c) the relationships in a class of numbers and outcomes of calculations (e.g., operations with odd and even numbers).

Research indicates that elementary students have limited experiences related to engaging in generalizations in arithmetic. For instance, studies by Anthony and Walshaw (2002) and Warren (2001) found that while students were confident in the commutative property for addition and multiplication, they faced challenges with commutativity in subtraction and struggled to justify their conjectures with models, often due to incorrect sense-making and misleading teaching materials. On the other hand, studies have also indicated that with suitable instruction, the difficulties elementary students encounter in conjecturing, justifying, and generalizing can be addressed, allowing students to learn to construct and justify arithmetical generalizations, including representation-based arguments (e.g., Hunter, 2010; Isler et al., 2013). Hence, teachers who can design classroom environments that promote generalizing and “algebrafy” their existing mathematical resources to incorporate algebraic thinking practices are needed (Blanton & Kaput, 2003, p. 76).

Functional thinking

Functional thinking focuses on students’ development in core algebraic practices in the context of relationships between covarying quantities (Blanton et al., 2015a). Promoting the concept of generalizing with a focus on functions encourages the use of pattern activities as a means to learn algebra (McAuliffe & Vermeulen, 2018). Research showed students display different functional thinking modes in pattern generalization. According to Stephens et al.’s (2017) framework (see Table 11), which was developed from the original framework outlined in Blanton et al., (2015a), these modes can be categorized as variational thinking, covariational thinking, and correspondence thinking.

Previous research showed that elementary students could identify and generalize relationships among quantities using figural patterns (e.g., Wilkie, 2014) and use tables to represent and reason with the relationships (e.g., Brizuela & Lara-Roth, 2002; Isler et al., 2014). Although students initially tend to focus on recursive patterns, not on the relationships between variables (e.g., Lannin et al., 2006), studies indicated that supportive instruction could help elementary students engage in covariational and correspondence thinking (e.g., Akın & İşler-Baykal, 2024; Blanton et al., 2019). Therefore, teachers need to know how to set up a learning environment that promotes functional thinking to gradually prepare students for studying functions.

Case discussions in teacher education

Teacher education programs are criticized because of the distinction between theory and practice (Ball & Cohen, 1999). An instructional method in teacher education is asserted to resolve this issue: case-based instruction, also referred to as case-study pedagogy (Heitzmann, 2008).

In case-based instruction, students read, analyze, and reflect on the cases of real classroom events (Kowalski, 1999). Shulman (2002) defined a classroom case as "… a piece of controllable reality, more vivid and contextual than textbook discussion, yet more disciplined and manageable than observing or doing work in the world itself" (p. xiv). Cases help teachers create a repertoire of solutions for everyday problems in teaching (Kleinfeld, 1992) and encourage critical thinking and decision-making abilities (Butler et al., 2006).

Merserth (1996) divided case purposes into three categories: (a) cases as exemplars, (b) cases as opportunities to practice analysis and contemplate actions, and (c) cases as stimulants to personal reflection. The forms of cases are also various in the literature and include text-based, video-based, and multimedia cases. This study investigated PETs’ knowledge development through text-based cases and discussions about these cases. The selected cases were content-specific, which focused on the big ideas of early algebra and used as exemplars of ways to guide students to think algebraically. Text-based cases were chosen because creating video or multimedia cases was not feasible during this study.

Method

The study aimed to investigate the development of PETs’ knowledge to teach early algebra in a bounded setting. Therefore, the study was conducted as a case study, which enables a comprehensive examination of complex settings (Merriam & Tisdell, 2016) and helps delving into the how and why questions to explore the distinct aspects of a case (Simons, 2009). Moreover, this study falls under the category of a descriptive case study, as defined by Yin (2003), where the researchers aim to "describe an intervention or phenomenon and the real-life context in which it occurred" (Yin, 2003, p. 15).

Participants

The study included third-year students enrolled in the Elementary School Education program who took the Mathematics Teaching I course in the fall semester of 2020–2021 in Turkey. Mathematics Teaching I is a compulsory course that aims to provide PETs with insight and knowledge related to the principles of mathematics teaching.

Forty-three PETs (38 female and five male) participated in the course and attended early algebra lessons. None of them had prior experience with early algebra. The 18 students who volunteered to participate in the study were grouped as high, medium, and low-achieving according to their GPA and Basic Mathematics course grades. Subsequently, three participants from each group were randomly selected to form a study sample of nine female participants to create a heterogeneous group.

Early algebra (EA) lessons

The final five weeks of the Mathematics Teaching I course focused on introducing early algebra content to PETs, which served as the intervention in this study. The lessons, lasting 2 hours per week, were conducted online by the first author via video conferencing. The learning goals for each big idea were categorized as subject matter knowledge (focused on CCK) and pedagogical content knowledge (focused on KCT and KCS). An additional two goals were created for the concept of variable since variable notation is a powerful tool for expressing generalizations (Carraher & Schliemann, 2007). The CCK goals were adapted from Blanton et al. (2018), which provided a roadmap for implementing a framework for early algebra. The KCT and KCS goals were established by researchers based on Ball et al.’s (2008) MKfT categories. The learning goals for EA lessons are listed in Table 1.

Table 1 The learning goals for the EA lessons
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To align with these goals, each week’s first lesson focused on the PETs’ CCK, and activities were carried out to foster algebraic thinking. The second lesson of each week focused on enhancing their PCK through case discussions, allowing them to consider possible student thinking, instructional strategies, and representations. More specifically, PETs were asked to read, analyze, and discuss the cases from the point of mathematical understanding, students’ thinking, and teachers’ moves. For instance, they were asked to reflect on students’ responses, why the teacher might have posed particular questions, and what instructional strategies they would prefer as a teacher. Moreover, the mathematics curriculum objectives (Ministry of National Education, 2018) that can be associated with each big idea were presented and discussed in the last lesson for each big idea, along with possible classroom implementation strategies. The learning goals guided the selection of the activities and classroom cases from the literature. Table 2 shows the schedule and content of the EA lessons.

Table 2 The schedule and content of the EA Lessons
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As an example of classroom discussion, in the fifth week, PETs were asked to participate in the case discussion through the case of Two of Everything. In this case, the students examine a growth pattern, represented as a linear function rule, a = 2b, record the data on a table, and describe the relationships they see in the table in words and variables. While analyzing that classroom case, PETs were led to think and discuss the possible purposes and advantages of constructing a table, students’ ways of examining patterns in the table, students’ ways of generalizing and describing functional relationships, students’ other possible answers, the purposes and effectiveness of the questions that the teacher asked, and how to use the number patterns to guide elementary students to think functionally. See the Appendix for an example of a case discussion.

Data collection tool and procedure

Individual interviews before and after the EA lessons were used to collect data. The interviews were conducted via video conferencing, occurring five weeks apart and lasting approximately one hour each. An identical semi-structured protocol was used for both pre- and post-interviews. The protocol consisted of three parts; each focused on asking questions about CCK, KCS, and KCT for a big idea. The questions were taken from literature or adapted to align with the learning goals of the EA lessons. They were distinct from the questions covered in the lessons, particularly regarding the problem contexts. Table 3 shows the alignment of the learning goals and the interview questions. Each question will be shared in the findings.

Table 3 The alignment of the learning goals and the interview questions
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Data analysis

The unit of analysis was each PET’s explanation for each item. The analysis was performed to identify participants’ strategies in response to interview questions. After completing the transcriptions of verbal responses, the analysis process started with the development of coding schemes. Except for C2 and C3, we classified the strategies in all items using codes derived from the data. The analysis of items C2 and C3 was conducted using the codes from the framework by Stephens et al. (2017).

The emergent codes were identified from patterns in responses. This process is guided by the learning goals of EA lessons. For example, as one of the learning goals (EE.KCT. in Table 3), participants were expected to develop instructional strategies that would assist students in acquiring a relational understanding of the equal sign during their lessons. Consequently, item A1 was analyzed to determine whether PETs would integrate the relational meaning of equality into their strategies. When more than one participant used a strategy, it became part of the coding scheme. After generating a set of codes for each item, the researchers engaged in discussions to clarify their meanings and evaluate their appropriateness. Necessary adjustments were made, and the codes were then confirmed. An additional code labeled "Other" was introduced for all items to complete the coding scheme. This code was utilized when a participant employed a strategy not covered by the predetermined options or when the strategy used was unclear. All the codes and definitions are provided in detail in the findings section.

Lastly, in the data analysis process, for each item, a second coder coded all of the responses of randomly selected three participants independently to assess the coding reliability. When there was a disagreement between the two coders, the codes were debated, and adjustments were recorded in the analysis until the two coders reached an agreement.

Findings

The findings concerning PETs’ strategies on individual items are presented for each big idea. The frequencies of the Other strategy are noted under each table when relevant and not discussed in the text.

Equivalence and equations

Regarding PETs’ CCK related to equivalence and equations, two true/false items were provided to elicit participants’ understanding. The items, codes, and the frequencies of the codes are shown in Table 4.

Table 4 Participants’ strategies for responding to the true/false question
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The PETs proposed three distinct strategies on why they believe the statement in item A2.1 is either true or false, two of which explain that the statement is mathematically meaningless. The ones who supposed that the statement was meaningless thought either that it was already known, we did not need to write such an expression, or it needed to be included in an operation. In pre-interviews, two of the three PETs thought that the statement was a trivial expression. For example, PET7 said, “I don’t think it makes sense. Because it’s the same number, it’s obviously equal. There’s no point in writing like this.” The remaining participant, PET3, asserted that it needed to be written with an operation by saying, “It did not really make much sense to me. For example, it would make more sense to say 17 equals 15 plus 2 or, say 10 plus 7.” However, after EA lessons, all participants affirmed the correctness of the statement by recognizing it as an expression of equality, which is the third strategy. For example, PET9 said, “It makes sense. It has the same number on both sides, showing they are both equal.”

While examining the trueness of the statement in item A2.2, the participant used the strategies of solving equations or considering the structure of equations by recognizing that the same operation is performed on both sides of the second equation. Besides, some participants thought the equations had different solutions due to their misconceptions about variables. They thought that these two equations were different and the value for x could be different for each equation. In the pre-interviews, five PETs stated that the statement was false. Three of these participants’ incorrect answers were due to variable misconception. For example, PET9 said, “The value we substitute to x in the first expression and the value we substitute to x in the second expression may be different. That is why they do not have the same solution.” In the post-interviews, all participants except one thought the statement was true. PET1, the participant who believed the statement was false, solved the equations but obtained different solutions for them due to an error she made when calculating them. Unlike the pre-interviews, no variable misconceptions were noted in the post-interviews, and four PETs explained the truth of the statement by observing the structure of the equation. PET9, for example, said, “It is true because the same number is added to both sides of the equation.”

These findings highlight the need to support PETs in developing a proper understanding of equivalence and equations. In addition, it can be inferred that examining cases involving examples of different conceptions of equality and the equal sign (relational and operational) and engaging in discussions on them improved PETs’ knowledge related to equivalence.

To assess PETs’ KCS in terms of students’ possible conceptions and misconceptions, they were presented with some student responses for a missing value problem and asked to interpret them. Table 5 shows the students’ sample responses, the codes for participants’ ways of interpretation, and the frequencies of the codes.

Table 5 Participants’ ways of interpretation of the equality item responses
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Examining how participants understand students’ answers revealed that PETs might misunderstand student thinking. It is evident that when explaining students’ thoughts, some PETs solely emphasize the mathematical steps taken by the student, neglecting their understanding of the equal sign. Besides, other responses recognize variations in how students perceive the equal sign in their answers. Before the EA lessons, for both items—A3.1, where the answer is 12 due to operational understanding of the equal sign, and item A3.2, where a student answered 7 by recognizing the equation’s structure—the majority of participants described mathematical procedures without addressing the student’s conception of the equal sign. For example, for item A3.1, PET3 interpreted this student’s response as “[the student] summed up 8 and 4. He may have seen the equality, but not the 5, or he may not have cared.” For item A3.2, PET5 stated, “If there is 4 on one side, I add one, and it becomes 5. Then he might have thought that if I subtract one from 8, it becomes 7.”

After EA lessons, for item A3.1, eight out of nine PETs explained the ways of thinking behind the given answers and identified the student’s conception of the equal sign. For instance, PET3 interpreted the student’s answer this time as follows:

The student looking at 8 + 4 thinks that equality does not require balance on both sides, but it gives the result of the computation. S/he thinks we will write the result directly without looking at the other side when we see equality.

For item A3.2, misinterpretations of the student’s approach were observed. In pre-interviews, four participants, and in post-interviews, two participants misunderstood the student’s thinking. For instance, PET2 believed the student rounded the numbers. Although no participants recognized the student’s understanding of the equal sign in pre-interviews, two did so in post-interviews. One of them, PET3, stated, “I think [this student] understands equality. He understands that both sides are equal. He knows that both sides of the equal sign are equal, keeping both sides balanced.”

The findings suggest that PETs may find it more challenging to understand the student’s answer of 7, which is based on the relational-structural understanding (Stephens et al., 2013), compared to the student’s answer of 12, which relies on operational understanding. In item A3.1, most participants improved their explanations from focusing on procedural steps to considering the student’s conceptions in the post-interview. For item A3.2, only 2 participants showed similar improvements. Nevertheless, it is possible to conclude that engaging in case discussions and exploring reasons for variations in student responses related to equality enhanced PETs’ understanding of student conceptions in this topic.

Regarding PETs’ KCT, they were asked to describe a lesson for a given objective. The objective, codes, and the frequencies of the codes are presented in Table 6.

Table 6 Participants’ ways for describing a lesson regarding the meaning of the equal sign
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This question aimed to evaluate whether PETs would incorporate the relational meaning of equality, as suggested by the early algebra approach, into the lessons they create for the specified objective. The analysis revealed that PETs attributed four distinct meanings to the equal sign in the lessons they described: as a result, as representing the same object, as denoting the same amount, and as a symbol of balance.

Before EA lessons, five of the PETs did not utilize the equal sign to convey relational understanding in their lessons. Two PETs developed tasks that led students to believe the equal sign indicated a command to write a result. For example, PET8 proposed a lesson as follows:

Ali has three pens. Ayşe gave Ali 5 more. How many pens did Ali have? […] Then, when you go to mathematical notation, you prepare the numbers with magnets; you show them by sticking them on the board. You write both 3 and 5. As a result of these, I think we can teach equals as the result.

In post-interviews, six out of nine PETs included activities to demonstrate the balance meaning of the equal sign in their lessons. For instance, PET5 explained her post-interview lesson design as follows:

“I’d bring a pan balance to class. When we put a mass on one side, let’s say it is 10 kilograms; when 2 kilograms are put on the other side, equality is not achieved; one side is outweighed. I could show it as an equation that the scales balance when we put in a value that weighs an equal amount of 10 kilograms.”

Furthermore, in pre-interviews, only one PET considered a pan balance, while in post-interviews, six participants considered using it in their lessons. This result underscores the enhancement of participants’ understanding of instructional strategies for equivalence and equations. We attribute these improvements to discussions about cases where a student demonstrated operational understanding. Delving into questions like “What would you ask next if you were the teacher?” and “How can you support relational understanding?” likely contributed to these positive outcomes.

Finally, when evaluating participants’ responses across all knowledge categories, we noted that the two participants (PST7 and PST8) who did not provide desired responses to both CCK questions in the pre-interview also failed to provide suitable answers in the KCS and KCT categories. This implies that participants lacking a relational understanding of equality interpreted student answers solely procedurally and did not highlight the equal sign’s balance meaning in their lesson descriptions. However, the findings do not conclusively support the reverse statement. Three out of four participants who provided the desired response for CCK questions in the post-interview highlighted the balance meaning of equality in their lessons. While interpreting student responses, although the same three participants accurately identified student conception in item A3.1, only one participant did so in item A3.2.

Generalized arithmetic

In generalized arithmetic, PETs were expected to identify and generalize arithmetic relationships and describe those relationships in words and variables. Accordingly, the PETs were asked to make a conjecture from a set of computations (see Fig. 1).

Fig. 1
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Set of computation for conjecturing

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Participants’ responses showed that PETs might struggle to formulate a conjecture or might suggest mathematically incorrect conjectures. The conjectures made by participants were categorized into three groups, as presented in Table 7 with the frequencies.

Table 7 Participants’ generalizations in the computation task
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Five PETs in the pre-interview and seven in the post-interview proposed correct conjectures. They explained the relationship they identified in calculations by either emphasizing the addition of a number’s inverse to itself and getting zero or performing inverse operations and getting the same number that was started with. For example, in post-interviews, while PET7 stated her conjecture as “when we subtract a number from itself, we get zero,” PET9 said, “If we add the same number to a number and then subtract it, the result will be our first number.” On the other hand, two PETs in the pre-interview and one in the post-interview proposed incorrect conjectures by considering the order of the operations. For example, in the pre-interview, PET2 stated her conjecture as “first, addition operations should be performed, not subtraction operations.”

Then, the PETs who made a correct conjecture in item B2 were expected to write a complete equation corresponding to their conjectures. However, the participants were observed to create equations that did not correspond to the conjecture they stated or were incomplete (they were expressions instead of equations and/or included numbers instead of variables). Therefore, distinctions have been made for these situations in the strategy codes. The codes, explanations, and frequencies are presented in Table 8.

Table 8 Participants’ strategies for representing conjectures in variables
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In the pre-interviews, all five participants who formulated correct conjectures in item B2 constructed incomplete equations. Among them, two participants had expressions corresponding to their earlier conjecture, while three did not. For example, PET9, who made the conjecture focused on performing inverse operations, wrote her conjecture in variables as 100 + x−x. This expression corresponded to her conjecture but was incomplete. PET4, whose conjecture was based on the addition of a number’s inverse to itself, described her conjecture in variables as x + 19 − 19 = x. This response mainly does not correspond to the conjecture and included numbers, so it was “not corresponding—incomplete.”

In the post-interviews, almost all participants had expressions corresponding to their earlier conjecture but most were incomplete equations. However, different from pre-interviews, two PETs could write corresponding—complete equations for their conjectures. One of them, PET9, wrote “x + y − y = x” for the conjecture focused on performing inverse operations and getting the same number that was started with. Based on these findings, it can be said that EA lessons supported the participants’ content knowledge. Nonetheless, it is evident that PETs still require support in formulating conjectures for arithmetical relationships and translating verbal conjectures into expressions in variables. We believe this outcome may be due to the case discussions’ primary emphasis on representing arithmetical generalizations in words and justifying these generalizations. While participants had the chance to discuss conjectures with variables in these case discussions, in our inference, dedicating a separate case discussion to this topic could be beneficial.

Concerning PETs’ KCS about generalized arithmetic, they were provided with student responses to justify an arithmetic generalization related to the sum of three odd numbers and asked to interpret those responses. To analyze item B4, we assessed participants’ comprehension of the responses and their ability to identify the justification approaches. Responses accurately identifying the student’s justification approaches were coded as "JUS," while others were categorized as "Other" since they did not fit into any specific codes. Table 9 shows the presented student response, the explanation, and the frequency of the code.

Table 9 Participants’ way of interpretation of three odd numbers item responses
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For the response in which the student justifies the fact empirically, B4.1, while in the pre-interviews, only 1 PET identified that the student generalized based on an example; in the post-interviews, five out of nine participants identified the student’s thinking. For instance, PET9 stated, “It is interesting that s/he came to this conclusion directly from an example. I would probably ask the question: how do you know it will be the same for all numbers?” Similarly, for the student response that included general arguments, in the pre-interviews, none of the PETs correctly mentioned the student’s way of justification for response B4.2. However, in the post-interviews, two participants correctly identified the students’ justification ways. PET7 stated that,

This student expressed clearly that the sum of an even number and an odd number is an odd number. He said that the sum of two odd numbers is an even number, and when a third odd number is added, it will always be an odd number. This student made a generalization.

During the case discussions on generalized arithmetic, PETs were presented with the ways that students could use for justification, and comparisons between these ways were discussed. Results indicate that these discussions helped PETs understand that students can justify their arguments through empirical arguments to some extent but were inadequate in identifying students’ use of general arguments.

To assess the PETs’ KCT, they were asked to describe a lesson for the objective related to generalized arithmetic. The analysis showed that while describing a lesson for that objective, participants either considered showing students this fact works through examples or leading students to recognize this relationship by themselves. The objective, codes, and frequencies of the codes are presented in Table 10.

Table 10 Participants’ ways of describing a lesson regarding the arithmetic generalization
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In the pre-interviews, all PETs used one or more examples to show that this relationship works rather than enabling students to make an arithmetic generalization. For instance, PET6 reported her lesson description as follows:

I would write it on the board and show it. 2 · 3 · 5. I’d want them to multiply 2 by 3 and then multiply by 5. Then I would have them multiply 2 by 5 and then multiply by 3. They would see that the results were the same.

In the post-interviews, six PETs described a lesson to provide students an environment to make conjectures by thinking, “I wouldn’t give the rule of multiplication directly. I want them to reach it at the end” (PET8). They considered asking questions to lead students to generalize the arithmetic relationship. For example, “Will this be the same for every number?” (PET4), “Why did you do that? How do you think? How do you know it is so? Is it true for all numbers?” (PET8).

These findings may indicate that PETs understood how to create opportunities for students to participate in core algebraic practices. This was one of the purposes of case discussions. Participants were encouraged to develop their teaching strategies for early algebra instruction by discussing examples of mathematics lessons that included core algebraic practices.

When considering all knowledge categories collectively, it becomes evident that PETs require the most support in generalized arithmetic. The findings suggest that the processes of generalizing arithmetic relations, generating different representations, and justifying these relations were novel experiences for the participants. However, while not to the same extent as in other domains, it can be argued that case discussions featuring core algebraic practices within generalized arithmetic enhanced the participants’ understanding to some degree.

Functional thinking

Regarding functional thinking, PETs were expected to identify and describe recursive, covariational, and correspondence relationships. Accordingly, to evaluate their knowledge related to quantities that change together, they were asked to reason on the “Saving for a Bicycle” problem (see Fig. 2; adapted from Blanton, 2008, p. 179).

Fig. 2
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Saving for a bicycle problem

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Participants were asked to describe relationships in words and variables based on the problem. The analysis of their responses followed the framework by Stephens et al. (2017). The question, strategy codes, and frequencies of the codes are introduced in Table 11.

Table 11 Strategies for describing and representing functional relationships in words
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The findings showed that before EA lessons, most participants described the pattern they saw in the table as recursive by focusing on only one quantity in the problem. For example, PET2 stated, “their difference is the same, increasing by three,” by considering only the amount of total money. After the lessons, all PETs described patterns at the correspondence level, which is the most sophisticated level according to the framework. For instance, PET1 expressed, “three times the number of weeks equals the total money each week” by considering the relationship between the two quantities. It is worth noting that despite being identified and discussed in case discussions, none of the participants’ responses demonstrated covariational thinking.

Then, the participants were asked to describe those relationships in variables. The same framework in the previous item guided the analysis. The strategy codes and frequencies are shown in Table 12.

Table 12 Participants’ strategies for representing the relationship in variables
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The analysis revealed that before EA lessons, most participants described the relationship in variables as an incorrect function rule. Obviously, this is connected to how they describe the relationship in words in the preceding item. The ones who described the relationship in variables as n + 3 had seen the relationship in the table as a recursive pattern. Similarly, two participants whose expressions were coded as functional-emergent in pre-interviews were the ones who described the relationship in words at the level of correspondence thinking. Besides, it may be asserted that EA lessons supported PETs in describing the relationship in variables. Unlike the pre-interviews, two participants could describe the relationship as Functional-Condensed in the post-interviews. For instance, PET7 described the relationship as “3x = y,” where x refers to the number of weeks, and y refers to the amount of total money. Nonetheless, the prevalence of functional-emergent responses from most participants after the lessons may be interpreted as that PETs still require guidance regarding the structure of equations.

For PETs’ KCS, they were shown student responses to the Saving for a Bicycle problem and were asked to interpret them. The aim was to observe whether they could decide the students’ levels of generalization of functional relationships. Table 13 shows students’ responses, the participants’ strategies of interpretation, and the frequencies of the strategies.

Table 13 Participants’ ways of interpretation of the functional thinking item
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Like in the equivalence and equations, before EA lessons, most participants explained procedural steps without recognizing the functional thinking approach when interpreting students’ answers. For example, for item C4.1, the student’s answer that describes the relationship using variational thinking, PET5 stated, “Since the total amount of money goes up 3, 6, 9, [the student] might have given such an answer because [s/he] saw that the difference between them was constantly increasing by three.” For item C4.2, in which the student shows covariational thinking, PET4 said, “For every week, three were added. [The student] could have used rhythmic counting by adding three on top of it.” Lastly, for item C4.3, where the student shows correspondence thinking, PET4 responded as, “[the student] saw that it increased by three liras a week. He realized that the sum of the increase in 3,6,9 gives 9 as 3,3,3 and 3 × 3 gives 9.”

After EA lessons, it was noted that PETs who discerned the functional thinking approach behind student answers were in the majority for all items. For instance, for item C4.1, PET8 reported, “This student goes through a single variable. She only considers the total amount of money. She does not coordinate the number of weeks with the total amount of money.” For item C4.2, PET3 said, “[This student] tried to show that the total money is related to the week. He looked at both sides [of the table] and coordinated them.” For the last item, C4.3, PET7 explained the student’s thinking: “[This student] completely coordinated the total amount of money with the number of weeks. S/he said it as a rule.” From these findings, we can deduce that discussions focusing on cases illustrating student thinking across various levels of generalization in functional relationships assist PETs in improving their ability to identify recursive, covariational, and correspondence relationships.

As for the KCT related to functional thinking, they were asked to describe a lesson for an objective related to number patterns to understand how PETs present varying quantities and whether they consider guiding students to investigate relationships between quantities. Table 14 shows the lesson’s objective, the codes of strategies, and the frequencies of the codes.

Table 14 Participants’ ways of describing a lesson regarding functional thinking
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The analysis revealed that before EA lessons, PETs outlined lessons for that objective where students were tasked with either expanding a given pattern or finding the missing term in the pattern. For example, PET4 thought to ask the students to find the next term of the given pattern and described her lesson as:

“I would have 2 students come to the board. Then I would add two more students, and it would be 4. Later, when I added two more students, it would be 6. They would see it increased and expanded as two students were added […]. All students would be on the board, and they would see it increased by two.”

However, after EA lessons, it was observed that several PETs attempted to go beyond the stated objective and used it to guide students toward functional thinking. Among these participants, some considered presenting a problem situation with quantities that vary together rather than directly giving a number pattern to students. For instance, PET6 reported as follows:

I would show this through a problem. There was the question of the chair. If a chair has 4 legs, how many legs do 2 chairs have? We then convert this into a table. We let them see through the table […] I would make them see the amount of increase between them … How many will it increase compared to the first chair and two chairs, or how many will it increase for the other three chairs? They could see the difference between them and express themselves by establishing relationships.

These findings underscore that with adequate support to develop relevant knowledge, PETs can outline lessons centered on patterns to lead students toward algebraic thinking by considering relationships between variables. We believe that case discussions helped PETs in this manner. Thinking about a case that exemplified how patterns can be used for functional thinking and discussing students’ thinking and teacher’s actions might have supported participants’ knowledge of content and teaching.

When all categories of knowledge are taken into account, we observe that after EA lessons, the participants who attempted to guide functional thinking in their lessons were the ones who showed development in their CCK and KCS. Therefore, based on this, it can be inferred that teachers need to develop an understanding of recursive, covariational, and correspondence relationships and how students demonstrate these forms of thinking to adapt their instruction to promote functional thinking effectively.

Discussion

In this study, we found that, in terms of the required knowledge for teaching, elementary teachers may not be ready to foster algebraic thinking in elementary grades to the same extent that elementary students can think algebraically. On the other hand, we also found that this type of knowledge of prospective teachers can be developed through method courses, particularly those that include case discussions.

The lack of necessary knowledge among PETs to promote algebraic thinking may not be unexpected, given that they were educated under the traditional arithmetic-then-algebra approach and did not take any courses focused on early algebra during their teacher training. However, these findings highlight some similarities between elementary students and PETs’ transition to early algebra, which was pointed out by Hohensee (2017). The findings indicated that before the EA lesson, PETs experienced similar difficulties as elementary students who have not received early algebra instruction in intervention studies (e.g., Blanton et al., 2015b; İşler-Baykal et al., 2023). For example, some of the participants stated that the expression 17 = 17 was not mathematically meaningful by thinking the same as Ana, who was a second-grade student and said, "Well, yes, eight equals eight, but you just shouldn’t write it that" (Falkner et al., 1999, p. 235). The finding that PETs lack understanding related to big ideas is consistent with earlier studies. This study confirmed that teachers have difficulty using words and variables to describe arithmetic relationships (Ding et al., 2013), struggle to understand the axiom-based structure of the number system and to apply operational rules in practical calculations (Kim & Kim, 2022), tend to use localized or recursive methods when addressing functional relationships (Oliveira et al., 2021), struggles with representing functional relationships as complete equations (Wilkie, 2016), have difficulty expressing pattern generalizations using variables (Özyıldırım Gümüş, 2021), and face challenges in recognizing errors related to the use of the equal sign in students’ work (Vermeulen & Meyer, 2017). Moreover, this research builds upon previous studies by providing a thorough evaluation of PETs’ considerable development in all aspects of early algebra, including both subject matter and pedagogical content knowledge. In addition to uncovering participants’ understanding of students’ conceptualizations and misconceptions (referred to as KCS), the study found that preservice elementary teachers struggle to design activities or tasks within the current curriculum that encourage algebraic thinking known as KCT, specifically in the realms of three big ideas.

On the other side, this study was not conducted to "simply document teacher weaknesses but to inform the design of teacher education in particular aspects of early algebraic reasoning" (Stephens, 2008, p. 275). We found that the specifically designed method courses can support the development of PETs’ knowledge of teaching early algebra. After EA lessons, PETs’ common content knowledge was found to develop in all domains of big ideas, although relatively less in generalized arithmetic. The less development in the generalized arithmetic may be because building, expressing, and justifying conjectures about mathematical structure is seen as a "habit of mind" (Blanton & Kaput, 2004, p. 142), and the four lessons that focused on generalized arithmetic may not have been enough for developing such a habit of mind. This finding corroborates Hohensee’s (2017) study which pointed out that method courses centered on topics such as generalized arithmetic, functional relationships, and the meaning of the equal sign can offer fresh insights into recognizing the operational and relational understanding of the equal sign, representing quantities which are unknown or variable in informal ways without using algebraic symbols, and representing functional relationships. This confirmation holds true for the study conducted by McAuliffe and Vermeulen (2018), which also demonstrated that an early algebra course helped PETs in enhancing their specialized content knowledge for employing various representations of functions. This study advances previous research by presenting a type of method course that not only improves PETs’ subject matter knowledge but also strengthens their pedagogical content knowledge for teaching early algebra. Different from the previous studies, we believe that providing participants with real-world scenarios as classroom cases in EA lessons might have enabled them to consider theory and practice together and "think like a teacher" (Kleinfeld, 1992, p. 33). Engaging in discussions on particular students, teachers, and learning goals enabled PETs to construct their knowledge concerning students’ conceptions and instructional strategies. In our inference, immersing PETs in specific classroom cases and encouraging them to contemplate questions like “Why does this student think in this way?” or “What other potential answers might students have?” and “How can we address and help correct misconceptions?” as well as “Why did the teacher pose these particular questions?” served as valuable opportunities to enhance their pedagogical content knowledge.

Based on the findings of this study, we recommend including early algebra as a main learning domain in method courses for prospective teachers and using classroom cases and case discussions to facilitate their knowledge to teach early algebra.

Conclusion

Our goal in this study was to investigate the development of prospective elementary teachers’ knowledge of teaching early algebra through participation in case discussions. We examined the PETs subject matter knowledge and pedagogical content knowledge in the domains of equivalence and equations, generalized arithmetic, and functional thinking before and after their participation in early algebra lessons including case discussions. The findings revealed that the prospective teachers need specific support to develop the required knowledge for teaching early algebra. They showed a lack of understanding regarding big ideas and insufficient pedagogical content knowledge before the lessons. In addition, by expanding on earlier research showing that method courses can be effective in enhancing prospective teachers’ understanding of early algebra, we found that case discussions can be a valuable technique for strengthening prospective teachers’ pedagogical content knowledge within these courses.

This study has some limitations. First, the findings rely on a small, non-randomized sample and specific frameworks, which means that the findings may not be applicable to different contexts or larger groups. Additionally, we used the objectives outlined in Blanton et al. (2018) as the basis for common content knowledge for preservice elementary teachers, which could limit the study’s scope. Different studies might define alternative learning goals for teaching early algebra. Thus, additional research is needed to better understand the knowledge required for teaching early algebra and to identify effective strategies to help PETs acquire this knowledge.