# Teachers' Development of Didactic and Algebraic Praxeologies through Bi-Institutional Lesson Study

Derya Diana Cosan

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# Teachers' Development of Didactic and Algebraic Praxeologies through Bi-Institutional Lesson Study

Derya Diana Cosan

University of Copenhagen, Department of Science Education

## Abstract

Transition problems in algebra between Danish lower and upper secondary school point to a need for teachers from different institutions to develop shared didactic and mathematical knowledge. The article presents *bi-institutional lesson study*, based on the anthropological theory of the didactic, and examines its potential for supporting teachers' development of shared knowledge of these transition problems. Despite structural asymmetries, the collaboration provided learning opportunities for both groups, helping teachers identify common challenges, such as students' use of guess-and-try methods when solving first-degree equations and difficulties with algebraic rules. Designing tasks requiring algebraic manipulation proved more difficult than expected. Teachers emphasized the importance of designing algebra tasks accessible to students with varying prior knowledge. The findings indicate a need for external inputs and further cycles of bi-institutional lesson study to achieve more substantial development of shared praxeologies, highlighting the limitations of what bi-institutional lesson study can achieve on its own.

**Keywords:** Algebra, lesson study, bi-institutional lesson study, institutional transition, the Anthropological Theory of the Didactic

## Résumé

Les problèmes de transition en algèbre entre l'enseignement secondaire inférieur et supérieur danois mettent en évidence la nécessité pour des enseignants issus de différentes institutions de développer des connaissances didactiques et mathématiques partagées. L'article présente la lesson study bi-institutionnelle, fondée sur la théorie anthropologique du didactique, et examine son potentiel pour soutenir le développement, chez les enseignants, de connaissances partagées concernant ces problèmes de transition. Malgré des asymétries structurelles, la collaboration a offert des opportunités d'apprentissage aux deux groupes, aidant les enseignants à identifier des défis communs, tels que le recours des élèves à des méthodes d'essais-erreurs pour résoudre des équations du premier degré et les difficultés liées aux règles algébriques. La conception de tâches nécessitant des manipulations algébriques s'est révélée plus difficile que prévu. Les enseignants ont souligné l'importance de concevoir des tâches algébriques accessibles à des élèves ayant des connaissances préalables variées. Les résultats indiquent la nécessité d'apports externes et de cycles supplémentaires de lesson study bi-institutionnelle afin d'aboutir à un développement plus substantiel de praxéologies partagées, mettant en évidence les limites de ce que la lesson study bi-institutionnelle peut accomplir à elle seule.
# 1. Introduction

Could collaboration between teachers from two different and neighboring institutions be organized as an efficient means to work with concrete and well-understood transition problems for students as they move between the institutions? This study works with this question based on a specific form of teacher collaboration known as lesson study (Takahashi & McDougal, 2016), where teachers plan, conduct, observe, and reflect on a lesson together to enhance their shared knowledge about teaching and student learning. While lesson study has been widely used in different institutions and grades, it usually happens within a single institution. This leads to a curiosity about how this approach could be used to facilitate shared knowledge and practices across institutions, and particularly to facilitate the transition between them.

In the Danish context, research has recently identified and examined the relatively serious problems that algebra causes in the transition from lower to upper secondary school (Cosan, 2024). Gueudet (2016) emphasizes that algebra “has long been the “transition topic” par excellence, marking the frontier between elementary and secondary education” (p. 18). Cosan (2024) shows that the transition problem in algebra for the Danish students is related to rewriting algebraic expressions based on explicitly known algebraic rules such as the distributive property, and algebraic definitions like $a \cdot a = a^2$ – for instance, in the process of solving equations through algebraic rewriting. These algebraic challenges form the “concrete and well-understood transition problems” addressed in this study.

Gueudet (2016) also notes that many initiatives aiming to ease the transitions often focus on strengthening the relationships between primary and secondary teachers (p. 18). This motivates the present study to experiment with an adaptation of lesson study. We propose a new variant of lesson study, called *bi-institutional lesson study*, in which teachers from two neighboring institutions (here, Danish lower and upper secondary school) collaborate to plan, conduct, observe, and reflect on a lesson in the first institution. By engaging in a bi-institutional lesson study, the hypothesis is that teachers from both institutions can develop new and shared knowledge about how to address transition problems.

The following sections provide a more detailed presentation of a bi-institutional lesson study, based on what is already known about ordinary lesson study. We then introduce the Anthropological Theory of the Didactic, which is used in this study to model and analyse the mathematical and didactic practices involved in our experiment, and their institutional embedding; and, naturally, to formulate precise research questions about how they are affected by bi-institutional lesson study. Finally, the experiment and (elements of) its outcomes are presented and analysed in view of answering the research questions and generating new hypotheses.

# 2. Background on Lesson Study

Lesson study (also called *jugyou kenkyuu*), developed in Japan, is a practice-oriented and collaborative approach to teachers’ professional development aimed at improving teaching and student learning. The most common form of lesson study occurs within a single school as a school-based teacher development program and typically lasts more than five weeks
(Takahashi & McDougal, 2016). By connecting individual teachers' daily instruction with the school's vision for student learning, lesson study fosters both enhanced teaching quality and a stronger professional culture within the school or institution (Lewis et al., 2019).

According to Huang and Shimizu (2016), lesson study provides teachers with the opportunity to develop their professional competencies and gain a deeper understanding of student learning: “Several studies revealed teachers’ development of their competence (including knowledge, skills, beliefs, self-efficacy, and disposition) ... researchers documented teachers’ changes in their mathematics knowledge and skills... understanding of student learning” (p. 399).

The lesson study process can strengthen professional collaboration among teachers, creating a framework for collective knowledge development. This collaborative structure serves as a systematic tool for knowledge sharing across schools or institutions. This structure enables teachers to share their experiences and reflections, thereby creating a collective culture of learning (Lewis et al., 2019). Moreover, Takahashi & McDougal (2016) point out that lesson study is often embedded in a “highly structured, school-wide project” (p. 514) that involves all teachers, ensuring that knowledge about teaching is developed and shared institutionally.

According to Takahashi & McDougal (2016), lesson study is organized into three phases that together form a lesson study cycle.

In the initial phase, *kyouzai kenkyuu*, a team of teachers studies the teaching materials, reviews curricula and textbooks, and identifies common student misunderstandings (Takahashi & McDougal, 2016). Takahashi & McDougal (2016) emphasize that this phase is crucial to the success of lesson study, as it requires “significant time” (p. 516) to provide teachers with deep insights into both subject matter content and students’ thinking. They emphasize that omitting this “crucial phase of lesson study” (p. 514) can lead to superficial planning, thereby undermining the effectiveness of the subsequent phases in the lesson study process.

Seleznyov (2018) highlights that 63% of lesson study implementation outside Japan did not include *kyouzai kenkyuu*. As Fujii (2018) notes, this is an important part of the Japanese lesson study, and he emphasizes that teachers outside Japan who wish to implement lesson study must “both understand and successfully implement kyozaikenkyu” (p. 16). This makes *kyouzai kenkyuu* an essential prerequisite for maximizing the overall benefits of lesson study.

In the next phase, the team of teachers collaboratively designs a lesson addressing a specific, challenging mathematical topic, based on the thorough preparatory work conducted during the *kyozai kenkyuu*. This lesson, referred to as the “research lesson” or the “open lesson” (*kenkyuu jugyou*), is conducted and taught by one teacher from the team, while the other team members observe the lesson with a particular focus on students’ thinking, learning, and behavior (Lewis, 2002). Additional educators may also participate in the observation of the research lesson (Takahashi & McDougal, 2016).

The process concludes with *kenkyuu kyougikai*, a so-called post-lesson meeting, where teachers engage in a structured discussion to share and analyse observations with a focus on
how students responded to the lesson. The goal is to gain insights into the teaching and
learning processes that occurred and to improve the lesson based on these discussions and
reflections. Finally, the lesson may be taught again, and the process is documented in a report
that includes the lesson plan (Takahashi & McDougal, 2016). As pointed out by Lewis et al.
(2019), using a facilitator outside the team in the post-lesson discussions helps keep observers
on track with their collected observations from the open lesson and the goals for the lesson.

A key figure in the lesson study process is the *knowledgeable other*, an external expert with expertise in both the academic content of the lesson as well as expertise in teaching and lesson study. This role is crucial, as the knowledgeable other contributes to the post-lesson discussion by highlighting key observations and linking the lesson to relevant research, thereby supporting the teachers’ reflections and enhancing the growth of professional and didactic knowledge (Murata, 2011; Takahashi & McDougal, 2016). However, as emphasized by Seleznyov (2018), 55% of lesson study implementations outside Japan did not involve a knowledgeable other (p. 221), highlighting the lack of external expertise internationally.

Research has demonstrated that lesson study contributes significantly to improving teachers’ content knowledge, particularly in specific mathematical topics such as fractions (as noted in Lewis et al., 2019), while also shifting their expectations for students’ performance. Moreover, the collaborative nature of lesson study can change teachers’ perspectives on the value of addressing mistakes and provide teachers with deeper insights into how students engage with the content of the teaching materials, thereby improving their professional practice (Lewis et al., 2019).

Huang & Shimizu (2016) emphasize that lesson study can lead to improvements in both classroom teaching and student learning outcomes: “These studies demonstrated three main types of evidence to document improvement of student learning … the third type of evidence is established on the basis of standardized tests that indicate gain of student’s academic achievement” (p. 399).

Several studies document the positive impact of lesson study on student performance. For example, a study on lesson study focusing on fractions demonstrated a notable improvement in students’ mathematical skills (Lewis et al., 2019). Similarly, a long-term case study of a school implementing a school-wide lesson study in mathematics found that the school’s mathematics scores increased nearly three times more than the average in the district (Lewis et al., 2019).

**3. Theoretical framework – The Anthropological Theory of the Didactic**

Our work takes place within The Anthropological Theory of the Didactic (ATD), developed
by Yves Chevallard (Chevallard & Bosch, 2020). This theory allows us to describe and
analyze any human activity as a combination of practice and the corresponding knowledge
and discourse about that practice (Barbé et al, 2005). ATD introduces the concept of
praxeology to model human activities (Barbé et al., 2005).

A praxeology consists of two parts: praxis and logos. Praxis consists of a type of task (T) and a corresponding technique (τ). Praxis can be understood as various types of task, where
"task" can be broadly interpreted to include mathematical tasks, such as solving an equation, and more general actions, such as brushing teeth or opening a door (Chevallard & Bosch, 2020). Logos consists of technology (θ) and a theory (Θ), where the former refers to discourse and knowledge that explain and justify the techniques, while theory is the knowledge and discourse that justifies and clarifies the technology (Miyakawa & Winsløw, 2019).

A mathematical praxeology thus includes mathematical types of task and corresponding techniques, as well as a logos about this practice, consisting of explanations of the techniques used, definitions of terms, rules, theorems, proofs, etc., that justify the technique applied (Miyakawa & Winsløw, 2019).

The focus here is on the mathematical activity – both what is done (praxis) and why it is done (logos). In contrast, didactic praxeology deals with the activities and the knowledge that teachers use to teach and convey mathematical praxeologies. Praxis in didactic praxeology might include planning lessons, presenting a task, or evaluating students’ understanding, while logos contain theoretical knowledge about methods for teaching specific mathematical praxeologies and for evaluating students’ praxeologies (Miyakawa and Winsløw, 2019).

The relation between these two praxeologies is central. Mathematics teachers’ knowledge consists of mathematical and associated didactic praxeologies; the theoretical level of the latter includes, of course, also general ideas about teaching and learning.

As Miyakawa & Winsløw (2019) point out: “Mathematics teacher knowledge consists thus, foremost, of *didactic praxeologies*, but these are naturally inseparable from the mathematical praxeologies whose teaching they bear on, and can also be enriched (especially at the logos level) by related mathematical praxeologies” (p. 284). This emphasizes that although teachers’ core knowledge is often didactic, this knowledge is closely connected to and depends on the mathematical praxeologies.

In ATD, teaching and learning mathematics are modeled as human activity conditioned and constrained by institutional settings. Teachers’ praxeologies are shaped by and developed in the institutions they participate in. Thus, teacher knowledge needs to be developed beyond individual skills, in view of and by the institution as a whole (Bosch & Gascón, 2014).

**4. Bi-institutional lesson study**

This section introduces a new variant of lesson study: *bi-institutional lesson study*. It involves teachers from two connected and neighboring institutions, referred to as I₁ and I₂, collaborating to improve the teaching in I₁, particularly to ease students’ transition to I₂. The collaboration is asymmetrical: I₁ teachers have in-depth knowledge of the students’ current mathematical praxeologies, while I₂ teachers are familiar with the mathematical praxeologies expected when entering I₂. This difference provides an opportunity to develop shared knowledge among the teachers that can support students’ transition.

Bi-institutional lesson study provides a framework for addressing these differences in expectations and for developing new and shared knowledge about students’ learning and thinking. The collaboration allows teachers to explore each other’s mathematical and didactic
praxeologies, which are unique to each institution. Through this collaboration, teachers can gain an in-depth view of how these praxeologies differ and how such differences can impact students' learning and transition to the neighboring institutions.

$I_2$ teachers participate in all three phases of the lesson study: *kyouzai kenkyuu* (preparation and planning), *kenkyuu jugyou* (research lesson), and *kenkyuu kyougikai* (post-lesson discussion). They contribute with their experiences with the students from $I_2$, but it is important to note that there are no fixed guidelines regarding the role teachers from $I_2$ should take on. They do not initially function as facilitators or as knowledgeable others, but these roles may fall to them if they have the necessary expertise.

As in Japanese lesson study, a *knowledgeable other* also plays a central role in the post-lesson discussion in the bi-institutional lesson study. This person must have in-depth subject knowledge about the content of the lesson, as well as knowledge about the didactic and mathematical praxeologies of both institutions.

The facilitator role can be taken on by teachers from $I_2$ or an external person with knowledge of the didactic and mathematical praxeologies in both institutions. The facilitator's role is to promote a productive discussion among the participants.

What makes a bi-institutional lesson study unique is its cross-institutional nature, where collaboration aims not only to improve teaching in $I_1$, but also to build shared knowledge of students' knowledge, difficulties, learning, and transition problems among teachers from $I_1$ and $I_2$.

**5. Research questions**

Given the identified transition problems in algebra between Danish lower and upper secondary schools, and the potential of lesson study to develop shared professional knowledge, it is relevant to investigate how a bi-institutional lesson study can contribute to the development of teachers' didactic and mathematical praxeologies related to the transition problems. This paper, therefore, seeks to address the following research questions:

How can participation in a bi-institutional lesson study between teachers from Danish lower secondary and upper secondary schools affect their didactic and mathematical praxeologies? What are the specific potentials of this bi-institutional collaboration in relation to supporting students' learning of algebra?

**6. Context and methodology**

The implementation of the bi-institutional lesson study is primarily placed in a lower secondary school. Since there are known problems in students' transition from lower secondary to upper secondary school, particularly in algebra, it can be seen as an important task for lower secondary school to prepare students as well as possible. Therefore, the focus should be on the lower secondary level, where the initial difficulties with algebra arise and where the transition to upper secondary school needs to be prepared. This study focuses
specifically on the collaboration between one lower secondary school (I₁) and one upper secondary school (I₂) in addressing these challenges. In Denmark, lower secondary school teachers are educated with a strong emphasis on pedagogy and didactics, whereas upper secondary teachers are educated with a substantially deeper specialization in mathematics and less emphasis on pedagogy and didactics.

The experiment (or invention) of this study was conducted in two Danish 9th-grade classes from the same school and involved six lower secondary school teachers (Leo, Julia, Anna, Mia, David, and Molly) and two upper secondary school teachers (Emma and Alex) from the same municipality. All names are pseudo-names. The teachers were divided into two teams: each consisting of three lower secondary school teachers and one upper secondary school teacher.

Each group planned and conducted three lesson studies on algebra in lower secondary school and observed the open lessons of the opposite group during the 2023/2024 school year. This meant that each teacher participated in a total of six lesson studies, which will be referred to as LS1, LS2, LS3, LS4, LS5, and LS6. All six lesson studies focused on algebra, specifically the setting up of algebraic models (algebraic expressions) in view of solving a given problem, as well as using the model to solve the problem through rewriting.

The lower secondary school involved in this collaboration performs significantly below average, both nationally and within its municipality, in terms of the number of students progressing to upper secondary school. This can create motivation among both teachers and school leaders to enhance (and promote) students' chances of successfully making the transition to upper secondary school.

The lower and upper secondary school teachers participated in all phases of the bi-institutional lesson studies, including:

1. Study of the curriculum

2. Formulation of goals for student learning

3. Lesson planning

4. First trials with a research lesson followed by a post-lesson discussion

5. Revision of the lesson plan for improvement

6. Open lesson followed by a post-lesson discussion

During the planning meetings, the author participated, while invited facilitators and knowledgeable others were present during the research lesson and post-lesson discussions. As part of the planning meetings, the author managed the meetings by keeping track of time and advancing with the purposes related to lesson study – for example, by ensuring that the lesson plan was completed on time.

In addition, during the planning phases, which encompass the study of the curriculum, the formulation of goals, and lesson planning, teachers not only developed the lesson plans but also collaboratively solved the tasks themselves. They discussed possible student solutions,
anticipated common mistakes, and considered how these could be addressed during the lesson.

During the first trials, only the team conducting the lesson participated alongside the author. During the open lesson, one of the lower secondary school teachers taught the lesson while the others, including the teachers from the team and the other team, observed.

After each open lesson, a facilitator, an external guest from the university, leads the subsequent post-lesson discussion by ensuring that all teachers share their observations and select key points for deeper discussions. The knowledgeable other, also an external guest from the university, offered final comments by connecting teachers' observations from the open lesson and linking them to research. These comments related, among other things, to the lesson plan and the observed lesson, including how students worked with the tasks and shared their solutions.

## 6.1 Data collection

All planning meetings, research lessons, and post-lesson discussions were audio-recorded and transcribed in Danish. After the completion of the bi-institutional lesson studies, semi-structured individual interviews were conducted with all participating teachers. These interviews aimed to examine the teachers' own assessment of learning outcomes and the challenges they experienced in participating in the bi-institutional lesson study, both with colleagues from their own institution and with teachers from the other institution. During the interviews, the teachers were asked to recall concrete episodes from the entire bi-institutional lesson study. To support this, they were asked questions such as: "Can you describe an episode where you felt you really learned something (that was beneficial)?", "Can you mention a situation where the collaboration was particularly fruitful or challenging?", and "How has it been to work with algebra in this project? Please mention specific episodes or situations where this became evident". Particular attention was given to episodes from the planning and reflection meetings, as these are the main situations where the teachers collaborated across institutions and reflected together on their teaching practice.

In addition to the audio-recorded planning meetings, research lessons, post-lesson discussions, and interviews, the lesson plans were also used as data material. These were included to identify the teachers' didactic intentions and goals for the open lessons, and to relate these to their reflections during the post-lesson discussions.

## 6.2 Data Analysis

To answer the research questions, the data analysis was carried out in three phases, guided by the Anthropological Theory of the Didactic, with particular focus on identifying and relating didactic and mathematical praxeologies.

In the first phase, all the data material was examined and sorted based on its relevance to the transition problem in algebra or to the bi-institutional lesson study. Specifically, data that did not address how the collaboration could contribute (or not) to the teachers' development of didactic or mathematical praxeologies related to this transition problem were filtered out. The selection criteria focused on whether the episode included either (a) references to specific
algebraic content and its teaching across the two school institutions, or (b) indications or expressions of development of didactic or mathematical praxeologies among the teachers.

In the second phase, the interviews were examined to identify instances in which teachers explicitly described the bi-institutional collaboration as either difficult, challenging, or beneficial.

In the third phase, transcripts from the planning and reflection meetings were analyzed in relation to the interview data. The purpose was to identify episodes in which teachers' discussions either supported or challenged their reflections from the interviews. In this phase, the lesson plans developed for the research lessons were also included in the analysis, as they reflected the teachers' didactic intentions and goals for the lessons. By comparing the planned didactic intentions and goals for the research lesson with the reflections and discussions in the meetings, it was possible to explore how specific didactic choices in the lesson plans aligned (or not) with the teachers' expressed praxeologies, and how the outcomes of the lessons influenced further development or challenging of their mathematical and didactic praxeologies.

To illustrate the data analysis process, one example was selected: during a planning meeting, the teachers collaboratively designed an algebraic task that could be solved by setting up a linear equation and solving it. They discussed how they expected students to approach the problem, as reflected in the lesson plan, which emphasized a guess-and-try method and the use of algebraic rewriting. However, in the subsequent reflection meeting, it became evident that the students did not solve the task as anticipated. This led to a discussion among the teachers about how to better support students in algebraic rewriting. In the interview, one teacher reflected on the differences between teachers' expectations and students' approaches, noting: "Perhaps we should look more into what makes sense for the students". This example shows how the analysis combined planning and reflection meetings with the interview data to illustrate the development of the teachers' didactic praxeologies.

# 7. Analysis and results

The following three sections examine how participation in a bi-institutional lesson study contributed to the development of Danish lower and upper secondary school teachers' didactic and mathematical praxeologies.

## 7.1 How differences in expectations led to changes in teachers' didactic praxeologies

Our data, presented below, highlighted how differences in expectations between lower and upper secondary school teachers became visible during the planning and reflection meetings in the bi-institutional lesson study, particularly in relation to how students solve first-degree equations. These differences contributed to the development of teachers' didactic praxeologies.

As mentioned, the teachers collaboratively designed six lesson studies (cf. Context). In LS5, they developed the following task *T* related to solving first-degree equations:
T: Lotte, Emil, and Daniella receive pocket money from their parents. Emil gets 5 DKK more than Lotte. Daniella receives three times as much as Emil. Daniella gets 125 DKK. How can we determine exactly how much Lotte receives in pocket money?

For the introduction of the task (the devolution in the sense of Brousseau, 1997, p. 31), the teachers planned to have a preparatory activity, where together with the students they formulate three algebraic expressions for Emil's (E), Lotte's (L) and Daniella's (D) pocket money:

$$ 
\begin{array}{l}
\text{Lotte's pocket money: } L \\
\text{Emil's pocket money: } L + 5 \\
\text{Daniella's pocket money: } 3 \cdot (L + 5)
\end{array}
 $$

The students were then asked to investigate: "How can we determine exactly how much Lotte receives in pocket money when Daniella gets 125 DKK?".

The learning goal was to enable the students to "multiply into parentheses and solve an equation using inverse operation, and for the students to realize that the guess-and-try method has limitations".

The motivation and background for focusing on this topic were explicitly stated in the lesson plan:

Many students feel that equations are a difficult topic, and this is something we [teachers] recognize from teaching, both in lower and upper secondary school. Many students have a limited understanding or perception of what an equation is. Many use the guess-and-try method and become discouraged when this approach is insufficient.

During the planning meeting, the teachers agreed on the equation $3 \cdot (L + 5) = 125$ for students to work with. These discussions revealed differences in expectations about how students approach equation solving. While all teachers acknowledged that students often use a guess-and-try method, the lower secondary school teachers considered it as a sufficient technique, whereas the upper secondary school teachers emphasized the importance and difficulty for students to solve first-degree equations by rewriting them. To address this, in these discussions, the teachers shared a common didactic task: to design an algebraic task aimed at guiding students away from the guess-and-try method and toward using algebraic techniques like the distributive property.

The lesson plan for LS5 stated: "We [the teachers] deliberately constructed the task so that the solution to the equation was not an integer and not a number less than 10, which makes the guess-and-try method weaker".

During the planning, when the teachers discussed their expectations for how the students would solve the equation $3(L + 5) = 125$, the lower secondary school teacher expressed that she did not believe the students would be able to guess the solution directly ($L = \frac{125}{3} - 5$), since it is a non-integer less than 10, it makes it more difficult for students to guess the
solution. Although the teachers want to design a task where the solution is not an integer and, in this way, make the guess-and-try method weaker, they still expect that the students will start with values like $L = 10, L = 20$, and $L = 30$ and work their way toward the solution through a guess-and-try method. This reveals a contradiction: while the lesson aims to promote algebraic rewriting, the lower secondary school teacher still expects students to use guess-and-try, even though the solution is a non-integer. In contrast, the upper secondary school teacher assumes that the non-integer solution will make the guess-and-try method unlikely.

During both implementations of LS5, most of the students used a guess-and-try method to find an approximate solution, while only three students solved it orally and expressed the solution as a decimal number (e.g., “What must L be for $(L + 5) = \frac{125}{3} \approx 41.66?”). So only three students approached an algebraic solution to the task, although their reasoning was expressed orally. In the subsequent reflection meeting, the teachers, supported by the knowledgeable other, agreed that this low number indicated a lack of didactic support for most of the students. The issue is not the few who already possess some knowledge of algebraic techniques, but the many who do not. This led to a discussion on the need for didactic techniques to better scaffold the remaining students.

As Mia, highlighted in her interview:

> It has been rewarding to gain insights into my expectations and theirs [upper secondary school teachers], which were not exactly the same ... Their expectations of what the students can do... at the upper secondary school, they expected the students to know more... they had higher expectations than I do.

In the first implementation of LS5, the task used 100 DKK as Daniella's pocket money. When students solved it easily by using a guess-and-try method, the teachers changed it to 125 DKK for the open lesson, aiming to reduce the guessing method. However, since both solutions are non-integer solutions, the changes made by the teachers were unnecessary. In the reflection meeting, the upper secondary school teacher noted that although students solved the revised task more slowly, they still did not use algebraic manipulation.

This outcome highlighted a didactic challenge: although the task was intended to promote algebraic rewriting, its structure allowed students to avoid it entirely. In the open lesson, students worked with equations of the form $A(x + b) = C$. The structure of this equation allows it to be solved by working backwards, which we observed students doing in the above-mentioned example $3(L + 5) = 125$. This issue arose despite the teachers having anticipated possible students' strategies and prepared the task introduction. However, because the task did not start explicitly with algebra and could be solved by working backwards ($\frac{125}{3} - 5$), students saw little reason to engage in algebraic manipulation. The teachers had assumed it was an algebra task but they had not fully considered how students would actually approach the task, or whether algebra would appear meaningful or necessary from the students' perspective.
Although the teachers discussed several possible equations during their planning, they chose an equation of the form $A(x + b) = C$, as their experience suggested that equations like $A(x + b) = Cx + d$ or $A(x + b) = C(x + d)$ would be too difficult for students to solve. But if the goal is to bring the distributive property into focus, as the teachers intended, then equations of the form $A(x + b) = Cx + d$ or $A(x + b) = C(x + d)$ are likely necessary to address.

Through the open lesson, however, the teachers realized that such equations are necessary if they want to make the algebraic manipulation required for solving the equation. This realization marks a development in their didactic praxeologies.

They also recognized that designing tasks that merely allow algebra is not enough. If students are to engage with algebra meaningfully, the task must be structured so that algebraic manipulation is not just possible, but advantageous or even necessary. This might be achieved through equations of the form $A(x + b) = Cx + d$ or $A(x + b) = C(x + d)$. Otherwise, the natural emergence of a need for algebra will not occur. This reflection also revealed that both lower and upper secondary teachers had limited experience in designing algebra tasks that truly necessitate the use of specific algebraic techniques, such as the distributive property. Designing such tasks turned out to be more difficult than anticipated. Many students left the lesson confused and uncertain about the teacher's intentions and would likely continue to solve similar tasks without algebra, unless the task explicitly prevents the guess-and-try technique.

The pocket money task is a task that the teachers designed themselves. In comparison, the tasks in Figures 2 and 3 were more directly inspired by external sources. These externally sourced tasks also address rewriting algebraic expressions, but they function more clearly as algebra tasks that focus explicitly on algebraic techniques. This difference revealed that while the teachers collaborated effectively, the self-designed task lacked the algebraic clarity of the externally derived tasks, highlighting the challenge of designing such tasks from scratch.

The upper secondary school teacher noted that equation solving is not taught as a separate unit in upper secondary school but rather addressed when needed. This probably reflects an expectation that the subject is essentially dealt with in lower secondary school; as it transpired, this is not presently true, even if the curriculum and final exam could give that impression.

In his interview, Alex reflected on this collaboration:

> I think it has been interesting to discuss with the lower secondary school teachers what they believe the challenges are, and what I think ... I have probably become clearer about what exactly students struggle with. Much of it, we probably knew beforehand, but we have discussed it and perhaps become more focused on what students struggle with in the transition from lower to upper secondary school ... which is about mathematizing. That is when it becomes difficult for them [the students], as they move from concrete examples to working with letters. We experience again and again that many students just get left behind.
Alex's reflection highlights how the bi-institutional lesson study helped clarify the challenges students face when transitioning between the two institutions. While the collaboration led to shared assumptions, such as the idea that using non-integer solutions would reduce guess-and-try methods, these assumptions proved insufficient. Although the teachers had not yet succeeded in designing a task that necessitated algebraic manipulation, they became more aware of the complexity involved in doing so. In this way, their reflections marked a modest but meaningful step in the ongoing development of their didactic praxeologies.

## 7.2 Development of teachers’ didactic praxeologies about lesson structuring

During the reflection meetings, teachers discussed how tasks can be presented to engage as many students as possible. A development in lower secondary school teachers’ didactic praxeologies is their increasing emphasis on how tasks are presented (the devolution in the sense of Brousseau, 1997, p. 31) within a lesson. In particular, teachers reflect on how the structure of the constructed task and the lesson can support students’ development of their mathematical praxeologies related to algebraic modeling.

For example, Leo says:

> Perhaps there is also something about starting [the lesson] with the very simple. I think we already do that, but I have definitely been reinforced in the idea that it is okay to make it concrete ... starting with tasks that have a low entry point and then gradually building up. But perhaps being even more aware that there are students who are left behind, and I know they did not grasp it.

This reflection highlights how lower secondary school teachers recognize the importance of designing tasks that all students can at least undertake at some level or to some extent, regardless of their varying levels of prior knowledge ('low entry point'). This point has the potential to lead to a shift in the teachers' didactic practice related to task design, but at the risk of evacuating the need for students to advance into algebra. Specifically, the teachers recognize the importance of designing tasks that enable most or all students to develop their own methods, and the non-trivial character of this didactic task. This shift may also lead to changes in the teachers' view of their own role in teaching. At the same reflection meeting, Anna noted the importance and difficulty of "letting go so that they [the students] have to investigate and explore and come up with methods and ideas to reach an answer". These meetings, where such technical challenges are identified and discussed, also facilitate change in the teachers' didactic technology ($\theta'$), where the focus moves from what teachers should explain to how they could facilitate student activity, while keeping some control of its direction. As Mia reflected in her interview: "it is a challenge to let go of the control and see where it leads ... how can you keep it controlled while still allowing the students to actually contribute knowledge and participate in the lesson".

An analysis of the three lesson plans for LS1, LS2, and LS3, involving Emma, Leo, Julia, and Anna, indicates a development in the teachers’ didactic praxeologies, particularly regarding how they structure the lessons to support the students’ algebraic modeling.
In the main problem for LS1, the students had to determine how many edges existed in a

<figure><img src="image_3.png" /></figure>

Problem 1: How many edges are there on a bridge with 15 bridge spans?

Problem 2: How many edges are there on a bridge with an arbitrary number of bridge spans?

structure of connected rectangular bridge segments (Figure 1).

Figure 1: The main problem for LS1

While students successfully drew the bridge with 15 bridge spans and counted the edges (where an edge corresponds to a side of the rectangle) in problem 1, many struggled to generalize algebraically in problem 2. During the research lesson, many students said: “I can count, but I simply cannot figure out a formula for it”. This illustrates a core challenge in students’ mathematical praxeology related to algebra – moving from counting to an algebraic expression.

Emma emphasized in the post-lesson discussion for LS1 that their hypothesis regarding the development of this lesson was that: “if we do this enough times, we will generally strengthen the students’ sense of algebra and the sense of abstracting from the concrete [numbers] to the abstract letters.” However, the reflections from LS1 challenged this assumption. The students’ difficulties with algebra tasks were not due to a lack of practice but were related to the specific way the lesson was organized. In particular, the transition from a concrete counting task to the formulation of an algebraic expression posed difficulties for many students. This highlighted the need to reconsider the lesson structure, including the didactic techniques used, especially how task design and teacher guidance could better support students in developing algebraic techniques.

In the main problem for LS2, the goal for the students was to gain greater familiarity with translating written instructions into algebraic expressions and rewriting these using the distributive property. In the problem named “Think of a number” (see Figure 2), students performed a sequence of arithmetic operations before being asked to express the process algebraically and then rewrite the algebraic expression.
1. Think of a number and follow the commands. Do this three times with three different numbers.

<table><thead><tr><td>Command</td><td>Number</td></tr></thead><tbody><tr><td>A: Think of a number</td><td></td></tr><tr><td>B: Multiply the number from A by 2.</td><td></td></tr><tr><td>C: Add 4 to the number from B.</td><td></td></tr><tr><td>D: Multiply the number from C by 5.</td><td></td></tr><tr><td>E: Divide the number from D by 10.</td><td></td></tr></tbody></table>

2. Write an expression that corresponds to the commands in the table, where n is the number chosen in command A. Rewrite the expression to make it as short as possible.

<caption>Figure 2: The main problem for LS2</caption>

Leo refers to this in the reflection meeting, stating that “rewriting and simplification of the expressions posed significant challenges, which we anticipated. However, what I have gained is a clearer understanding of where we need to focus our efforts in order to move forward”. This shows an initial awareness of the students’ mathematical praxeology. However, the reflection remains relatively vague and does not yet lead to concrete didactic conclusions. This demonstrates how ongoing engagement in the bi-institutional lesson study might be necessary before teachers can develop more detailed analyses of students’ mathematical praxeologies and how an outside perspective could be valuable in refining such insights.

Emma, on the other hand, noted that students seemed hesitant when it comes to algebraic rewriting and simplification: “In part three [in the research lesson], they [the students] were really unsure, ‘what are we supposed to do?’ ... Maybe this group needed more instructions, but we had no way of knowing in advance”.

This reflection shows how her didactic praxeology evolved towards more deliberate support of students’ algebraic rewriting. Initially, Emma observed that students were uncertain about how to proceed with the task of rewriting the equation step-by-step, as instructed by the teacher. Based on this observation, Emma suggested that, before having students engage with the task, it would have been beneficial to first show the step-by-step process of solving a first-degree equation. This reflects a shift in her didactic approach, from assuming that students would grasp the process on their own, to recognizing the need for more explicit guidance before they were expected to solve the task independently, pointing to an emerging awareness of students’ algebraic praxeologies while leaving the form and timing of such guidance as an open didactic question.

In the main problem for LS3, the students had to identify a number pattern from a geometric situation and then set up an expression based on that pattern, which they had to solve as a first-degree equation. Concretely, the task required students to determine the number of marked squares along the perimeter of a square grid (see Figure 3). Initially, students worked with a specific numerical case (10 marked squares on each side of the square) and were then asked to generalize using algebraic expressions. Finally, they were given a total number of
marked squares (352, see Figure 3) and asked to determine the side length of the square by solving a first-degree equation.

<figure><img src="image_4.png" /><figcaption>Figure 3: The main problem for LS3</figcaption></figure>

By using tasks with a low entry point that can be solved in various ways, teachers can include more students in the teaching. This focus on differentiated and open tasks is further nuanced through the bi-institutional collaboration, where upper secondary school teachers set expectations and contribute ideas on task devolution.

During the planning, the teachers discussed how to structure the task. For example, in LS3, the main problem began with question 1, which was designed to be solved at different levels. Some students counted the marked squares along the perimeter, while others formulated arithmetical expressions for the number of marked squares along the perimeter.

Anna noted in the reflection: "It was great to see that all [the students] were engaged at the first step [question 1]. They just got started. Some counted the marked squares, while others came up with different methods, like '8+8+10+10'".

Emma emphasized in the reflection meeting

The students engage at the low entry point. They are quicker at setting up the expressions than they have been before. I also noticed that we had slightly changed the introduction regarding discussing 'what is n', and the way it was introduced today – as the number of squares in the side length – made the students grasp it extremely quickly compared to what they had previously learned.

Thus, both lower and upper secondary school teachers began to develop shared didactic knowledge about how to design lessons and tasks that enable most students to develop their algebraic knowledge.

## 7.3 Challenges for teachers in probing students' mathematical praxeologies

During the planning meetings for all lessons, the teachers identified a shared didactic type of task: how to support students in describing and justifying the techniques they used (individually) to solve an algebraic task. This didactic task was also emphasized in the lesson
plans, such as in LS1, where one of the focal points for observation was whether students shared their thoughts, described and justified their techniques, and listened to each other's descriptions.

During these lessons, the teachers observed that students often struggled to both present their solutions orally and to justify their techniques. This highlighted the challenge of implementing didactic techniques for probing: how to ask questions that guide students toward describing and justifying their techniques.

The teachers' discussions during the reflection meetings show the development of a didactic technology about the need to moderate students' presentation of techniques involving probing students to make mathematical technology.

During the reflection meeting for LS1, the facilitator commented on that:

> One thing is that students need to say something, another is what they say. They need to speak mathematics using mathematical language. Today, we observed some students ... say something, but it was not entirely mathematical. Is there anything that can be done to help them in this process, not just to come up [to the board] and say something, but rather, how can we support them in achieving a more mathematically precise language?

The facilitator's comment reflected an emerging theoretical view that probing students requires both attention to what students say and how they say it.

The difficulty in probing students became a theme in the reflection meetings. While the lower secondary school teacher found it challenging to facilitate the probing process during the open lessons, the other teachers struggled in the reflection meetings to provide concrete suggestions that could support students in explaining and justifying their techniques. While they acknowledged the importance of having students share their thoughts, whether in groups or at the board, determining how best to support the students' discussions remained a challenge. This difficulty was not limited to lower secondary school teachers; upper secondary school teachers also expressed similar concerns, emphasizing that the issue persisted across institutions. This shared experience indicates limits in the teachers' didactic techniques, prompting a need to expand their praxeologies. Consequently, external input, such as a knowledgeable other or a facilitator, may be necessary to support the development of their didactic praxeologies related to probing.

Despite these challenges, the reflection meetings also help teachers realize that fostering students' reasoning requires not only effective probing but also a focus on summarizing.

As Emma noted during the reflection meeting for LS2:

> Sharing thoughts is also something we have discussed a lot about in upper secondary school. Reasoning is essential, so how do we emphasize it in a product that they [the students] have to create? ... If the focus is to be on reasoning, then it is about valuing that and letting something else take a back seat.
At the same time, teachers realized that summarizing had often been deprioritized in their teaching: As David remarked in an interview:

> Paying attention to wrapping up and following up - you can not spend too much time on it. It is always easy to tack on something afterward, but it is really difficult to actually do it, and you cannot do it in the next lesson. You have to do it right then and there.

In this way, the teachers recognized the importance of the didactic technique of summarizing to help students articulate their descriptions and justifications of the techniques they use to solve tasks more explicitly, although they still found it challenging to implement effectively.

## 8. Discussion

As shown in the analysis, both Danish lower and upper secondary school teachers participating in a bi-institutional lesson study became aware of how difficult it is to design tasks (for instance, word problems in algebra) that necessitate algebraic rewriting. The findings related to LS5 show that although the designed task for this lesson aimed at engaging students in algebraic rewriting, such activity did not occur among the students. If students can solve these word problems without algebraic rewriting, they tend to do so. These experiences were valuable for the teachers from both institutions in relation to how difficult it is to design tasks that require algebraic rewriting.

Designing algebra tasks involving word problems can be difficult for teachers for various reasons. Teachers often have limited experience in task design and are frequently inspired by textbook material, exam tasks, and similar resources. The findings of this study confirm this, as teachers from both institutions ended up designing what was intended to be an algebra task, which in the end had very little to do with algebra at all. This may also be attributed to the fact that teachers, especially those from $I_1$, often design tasks based on their knowledge of students' existing knowledge and misconceptions. Their expectations about how students will solve a task influence the design process, often resulting in tasks that can be solved without algebraic techniques.

Despite LS5 being a lesson that, in many ways, proved somewhat challenging and even, at times, unsuccessful, it still serves as a valuable example of how teachers can sometimes learn more about the students and teaching algebra when the lesson does not go as planned.

A central aspect of the bi-institutional lesson study is its asymmetrical nature. This asymmetry relates not only to the different roles and the individual knowledge of the participating teachers but also to the institutional praxeologies they are part of. Teachers from $I_1$ have experiential knowledge about their students' current algebraic praxeologies and difficulties, while teachers from $I_2$ have experiential knowledge of the algebraic requirements and students' degree of honoring them when entering $I_2$. These differences make the collaboration asymmetrical as the shared goal is to improve teaching in $I_1$ to prepare students for success in $I_2$.

In the Danish context, we could expect this asymmetry to be perceived as hierarchical, since the upper secondary school teachers come from a different and higher-level institution and
enter the lower secondary school context. We could hypothesize that they might act as relative "experts" in mathematical praxeologies, which could empower them to instruct or even correct teachers from lower secondary school. Also, the fact that there are three lower secondary and one upper secondary school teacher collaborating could reinforce this asymmetry. Such an asymmetry could potentially weaken the collaboration if it leads to lower secondary school teachers feeling evaluated or if upper secondary school teachers assume a dominant role. The whole enterprise could also be seen by the lower secondary school teachers as focusing overly on the needs of the rough half of their students who eventually enter upper secondary school. What we see in the findings regarding the collaboration between teachers from the two institutions, however, is that this hierarchy did not materialize; both institutions contributed meaningfully to the collaboration. One might even say that the lower secondary school teachers were regarded as experts, as it was both their institution and their students, while the upper secondary school teachers were, so to speak, on unfamiliar ground, and thus had a greater potential for developing their didactic praxeologies. The findings also revealed that teachers from both institutions struggled with designing tasks in algebra, which provided both groups with an opportunity to develop new and shared didactic praxeologies.

An additional limitation of the bi-institutional lesson study can be the difference in the teachers' educational backgrounds. In Denmark, which is the case we are studying, lower secondary school teachers often have a broad pedagogical education with less specialization in mathematics, while upper secondary school teachers have a higher degree of subject-specific knowledge and specialization in mathematics. This difference can lead to challenges in collaboration, especially if the teachers fail to utilize their complementary backgrounds. For example, upper secondary school teachers' deep mathematical knowledge may come into conflict with the lower secondary school teacher's more pedagogical approach, which can create challenges in collaboration, such as disagreements about teaching areas or focus points regarding the teaching of algebra. None of these aspects was observed in the collaboration we studied. The empirical data and analysis from this study show that although the asymmetry exists, it did not result in a one-sided transfer of knowledge from upper secondary to lower secondary school teachers. The upper secondary school teachers developed their didactic praxeologies through collaboration, particularly by collaborating with the teachers from lower secondary, and the difficulties the students from lower secondary face. For instance, in LS1, the teachers became aware that students struggle to move from a concrete counting task to algebraic generalization, and that this transition requires deliberate task structuring and teachers' guidance. In LS3, both teachers saw how tasks with low entry points and with multiple strategies to solve them can engage all students. In LS5, they realized, through students' continued use of guess-and-try, how difficult it is to design an algebraic task that students cannot solve with a guess-and-try method, which they tend to do if the task can be solved with this technique.

The upper secondary school teachers involved in the bi-institutional collaboration expressed an interest in developing knowledge about the algebraic praxeologies in lower secondary schools, particularly in the 9th grade. This is evident from the interviews, where an upper secondary school teacher reported gaining knowledge about the difficulties that lower
secondary school students face when solving first-degree equations through algebraic rewriting. Given these difficulties and the lack of algebraic knowledge among the students, teachers from both institutions shared a common goal: to support students' learning in algebra.

One limitation of this bi-institutional lesson study is that it took place in 9th grade (final year of lower secondary school), while addressing the transition problems earlier, in 7th or 8th grade, could be more effective, as it would give teachers more time to address the challenges before students face the transition problem. However, this was a choice, as there are also reasons for placing the bi-institutional lesson study in 9th grade — for example, students are more mature, and upper secondary school and algebra are closer at that point than in earlier grades. Still, recognizing a problem is not enough if there is limited time to act on it, which an earlier bi-institutional lesson study might have provided.

We also observed in the analysis that teachers from upper secondary school suggested and used tasks from lower secondary schools' final exams as a starting point for their lesson studies, rather than introducing tasks from I₂. Had the latter happened, it might have reinforced the asymmetry in an unproductive manner. This leads to a broader hypothesis: For bi-institutional lesson study to be effective, it appears reasonable to suggest that I₂ teachers could engage with and utilize resources from I₁ before introducing their own. This approach can help limit the structurally embedded asymmetry and can serve as a motivator for collaboration among institutions if a shared transition problem is acknowledged.

It is also worth reflecting on the specific design of the bi-institutional lesson study. In this case, I₂ teachers observed lessons in I₁, given by its teachers. This choice gave them knowledge about teaching and students' mathematical praxeologies in I₁. However, bi-institutional lesson study could also be designed differently, for instance, by inviting I₁ teachers to observe an open lesson in I₂. One potential benefit of this alternative design is that it could reinforce both groups' view of the transition problem as a shared responsibility rather than solely their own. This could reduce any feeling of asymmetry in the current model, where primarily I₂ teachers observe I₁. Although our findings indicate that no clear dominance of upper secondary school teachers over lower secondary school teachers occurred in the collaboration, it is important to recognize that the underlying assumption of the bi-institutional study could itself be seen as a form of asymmetry.

Moreover, by observing lessons in I₂, I₁ teachers could witness the existence of algebraic transition problems, confirming that these challenges are not just perceived by the upper secondary school teachers. This might increase the motivation and relevance of the collaboration. On the other hand, a challenge might be the difference in mathematical background, which could limit I₁ teachers' contribution to the lesson planning and reflection. Nevertheless, observing the open lessons remains valuable, especially if scheduled early in the first year of upper secondary school, where the connection to lower secondary school and the transition problem are most apparent.

Our study has obvious limitations. First, it is small-scale and exploratory, involving a limited number of teachers in a specific context over one school year. It is carried out in a specific
(Danish) context; at best, our observations there may function as evidentially supported hypotheses on the potentials of bi-institutional lesson study across neighboring institutions, which could be helpful for experiments in other, similar, contexts.

## 9. Conclusion

This study has examined how participation in a bi-institutional lesson study involving Danish teachers from lower and upper secondary school affected their didactic and mathematical praxeologies, and what specific potentials such collaboration may have in addressing the transition problem in algebra between the institutions.

Through the planning and reflection meetings in the bi-institutional lesson studies, teachers from both institutions developed shared didactic praxeologies regarding students' approaches to modelling non-algebraic problems with algebra and solving first-degree equations. Lower secondary school teachers observed that many students relied on a guess-and-try method despite prior teaching in algebraic manipulation. This observation was collaboratively discussed, leading to shared knowledge about students' algebraic techniques and how they solve algebraic tasks.

A challenge identified across both institutions concerned the design of algebra tasks, particularly those meant to involve the distributive property. The study highlighted that designing tasks that genuinely support students in reaching learning goals related to algebraic manipulation is a shared difficulty. When tasks do not explicitly begin with algebra, it becomes important for teachers to consider in advance whether the task actually requires algebra, in what ways, and whether students are likely to construct one or more algebraic models themselves. The bi-institutional lesson study provided an opportunity to address this challenge by exploring various ways to structure and design such algebraic tasks. As the examples show, success in the first trials is not guaranteed, and it is apparent the teachers from both institutions need more external input to advance from recognizing that it is a difficult problem.

Another outcome of the teachers' reflections was the recognition of the importance of designing algebra tasks that are accessible to all students, regardless of their prior knowledge. Lower secondary school teachers, in particular, emphasized the need for tasks that can be approached at different levels of student prerequisites, a concern that was reinforced through the bi-institutional collaboration.

Finally, the difficulty of encouraging students to describe and justify their techniques for solving algebraic tasks emerged as a theme in the reflection meetings. Although teachers from both institutions recognized the importance of encouraging students to share their mathematical techniques in groups or classroom discussions, they struggled to prepare and implement efficient didactic techniques for supporting this practice, and probably would need several more trials before reaching stable and efficient techniques for the didactic type of task. And again, more external input seems to be needed for them to collectively advance with this challenge.
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