A microgenetic analysis of teachers’ learning through teaching

Background What and how teachers learn through teaching without external guidance has long been of interest to researchers. Yet limited research has been conducted to investigate how learning through teaching occurs. The microgenetic approach (Siegler and Crowley, American Psychologist 46:606–620, 1991) has been useful in identify‑ ing the process of student learning. Using this approach, we investigated the development of teacher knowledge through teaching as well as which factors hinder or promote such development. Results Our findings suggest that teachers developed various components of teacher knowledge through teach‑ ing without external professional guidance. Further, we found that the extent to which teachers gained content‑free or content‑specific knowledge through teaching depended on their robust understanding of the concept being taught (i.e., content knowledge), the cognitive demand of the tasks used in teaching, and the lesson structure chosen (i.e., student centered vs. teacher centered). Conclusions In this study, we explored teacher learning through teaching and identified the sources leading to such learning. Our findings underscore the importance of teachers’ robust understanding of the content being taught, the tasks used in teaching, and a lesson structure that promotes teachers’ learning through teaching on their own.


Introduction
Teaching is a lifelong learning profession.Teachers are surrounded by both formal and informal opportunities to enhance their knowledge and skills.Yet teacher learning has been mainly explored in the form of the external professional support teachers receive from mentors, educators, or researchers (Hoekstra & Korthagen, 2011;Hoekstra et al., 2009;Kyndt et al., 2016).Scholars have acknowledged that the task of teaching itself can be a rich learning resource through which teachers can develop key knowledge and skills on their own (Enthoven et al., 2023;Kang & Cheng, 2014;Lannin et al., 2013;Leikin & Rota, 2006;Leikin & Zazkis, 2010;Russ et al., 2016;Tzur, 2010).Indeed, teachers frequently report their teaching practice as a source of learning (Flores, 2005;Hoekstra et al., 2009).However, systematic investigations of whether and under which conditions the work of teaching contributes to the development of professional knowledge and skills have been scarce (Kyndt et al., 2016;Leikin & Zazkis, 2010).Given that teaching is a major component of teachers' daily activities, exploring the compelling idea of what and under which conditions teachers learn from their daily activities without any explicit support and guidance (Leikin & Rota, 2006;Lloyd, 2008) has important implications for educational research and teacher education programs.
Prior work in this area has documented that the work of teaching has the potential to enhance teachers' knowledge (Copur-Gencturk & Li, 2023;Hiebert et al., 2017; Page 2 of 26 Copur-Gencturk and Atabas International Journal of STEM Education (2024) 11:29 Lannin et al., 2013;Leikin & Rota, 2006;Lloyd, 2008;Remillard, 2000).One line of work has documented that the work of teaching, such as by using instructional materials and certain instructional practices, has the potential to help teachers gain knowledge (Leikin & Rota, 2006;Lloyd, 2008;Remillard, 2000).Yet in these studies, teachers were given these tasks or asked to use these practices; therefore, the extent to which teachers can learn on their own through teaching without external intervention remains unclear.Another line of work has captured learning through teaching by collecting data from teachers months and years apart (Copur-Gencturk & Li, 2023;Hiebert et al., 2017;Lannin et al., 2013).These studies have documented that through teaching, teachers develop pedagogical content knowledge (PCK), a type of knowledge that is key to quality instruction and student learning of a particular subject matter (Copur-Gencturk & Li, 2023;Hibert et al., 2017), but they have fallen short in explaining the process by which such knowledge is developed, leaving this an unexplored research area (Walkoe & Luna, 2020).
In this study, we aimed to contribute to the literature by exploring teacher knowledge development through teaching by zooming in on the work of teaching (i.e., by collecting data before and after teachers taught a mathematics topic in a series of two sequential lessons three times, totaling six teaching events).We conducted structured interviews and collected lesson plans from five elementary and middle school teachers to explore the development of knowledge through teaching (both content-free and content-specific knowledge) as well as the factors contributing to the development of different components of teacher knowledge.

Teacher knowledge
What constitutes teacher learning and how teachers learn have been articulated differently across perspectives (Russ et al., 2016).Here we conceptualize teacher learning as the activation and acquisition of the knowledge teachers use in the work of teaching.Deducing what teachers may or may not know is already a challenging task, and inferring the development of such knowledge through teaching is an even more delicate task.We adapted Izsák's (2008) perspective in our conceptualization of teacher knowledge in that.
if a teacher has but, for whatever reason, does not use a piece of knowledge when responding to a range of situations that arise in the course of teaching-situations where that knowledge could be applied productively-then for that teacher the knowledge is not an instance of knowledge for teaching.(p. 106) We also applied this reasoning to our definition of teacher learning (i.e., teacher knowledge development) such that if a teacher appeared to begin using a piece of knowledge after teaching, then we considered this knowledge as having developed through teaching.Our claims about teacher knowledge and teacher knowledge development were restricted to the knowledge teachers appeared to use or to activate in situations around the work of teaching.With these caveats, by drawing on prior definitions of the knowledge teachers need for teaching (e.g., Ball et al., 2008;Copur-Gencturk & Tolar, 2022;Grosmann, 1990;Shulman, 1986), we conceptualized teacher knowledge as including both the generic components of teacher knowledge (i.e., those that did not depend on a particular subject matter) and the content-specific components of teacher knowledge (i.e., the knowledge and skills needed in teaching mathematics).Adapted from Grossman's (1990) model of teacher knowledge (see Fig. 1), our conceptualization of teacher knowledge had four components: subject matter knowledge (mathematics content knowledge in this study), general pedagogical knowledge, PCK, and knowledge of the context.Being situated in mathematics, our conceptualization of subject matter knowledge was grounded in the knowledge and skills needed for mathematical proficiency (National Research Council, 2001).Specifically, subject matter knowledge encompassed procedural understanding (i.e., knowing the definitions, rules, and procedures), conceptual understanding (i.e., knowing the conceptual underpinnings of these rules and procedures, Copur-Gencturk, 2021), mathematical reasoning (i.e., being able to reason about the relationships among concepts and situations, Copur-Gencturk & Tolar, 2022), and word-problem-solving skills (i.e., being able to perform the tasks and problems found in the school curricula; Copur-Gencturk & Doleck, 2021;National Research Council, 2001).The second component of teacher knowledge was general pedagogical knowledge, content-independent knowledge related to the general principles of instruction (e.g., the wait time after asking a question before providing the answer or asking another question), instructional practices for classroom management (e.g., the choice of small-group or whole-class instruction), and general principles of learning (e.g., creating engaging learning environments for students).
The third component, PCK, was content-dependent pedagogical knowledge.In the context of mathematics education, PCK encompassed teachers' knowledge of how students learn a particular concept (i.e., their knowledge of students' mathematical thinking) and their knowledge of how certain instructional tools and practices could promote or hinder students' understanding of particular concepts (i.e., their knowledge of teaching mathematics), the two most commonly agreed-on components of such knowledge in the literature (Ball et al., 2008;Copur-Gencturk & Tolar, 2022;Copur-Gencturk et al., 2019;Krauss et al., 2008).Knowledge of students' mathematical thinking was composed of teachers' knowledge of students' understanding of specific topics, typical student strategies, and the ideas students would find easier or more difficult when learning a new concept (Ball et al., 2008;Copur-Gencturk & Tolar, 2022;Hill et al., 2008).For example, knowing that students often overgeneralize their knowledge of whole numbers and treat the numerator and denominator of a fraction as separate entities instead of seeing each fraction as representing a single number with a magnitude is an indicator of teachers' knowledge of students' mathematical thinking.Teachers' knowledge of mathematics teaching included their knowledge of the affordances and limitations of particular representations and instructional tools, as well as their knowledge of how to sequence mathematics activities (Ball et al., 2008;Copur-Gencturk & Tolar, 2022;Hill et al., 2008).
The last component of teacher knowledge, knowledge of the context, referred to teachers' knowledge of their own students' background and interests as well as their knowledge of the school and the local environment.Given that teaching occurs in an interactional space between learners and the teacher around the content, teachers' knowledge of their students and the community is an important aspect of teacher knowledge.

Learning through teaching
As illustrated in Fig. 2, as a profession, teaching involves numerous opportunities, ranging from informal learning opportunities, such collaborations with colleagues, to formal opportunities, such as professional development for teachers to develop their knowledge and skills.In this study, we focused on learning through teaching, which we defined as teachers' development of knowledge by engaging in activities that revolved around the task of teaching without systematic external intervention.
The task of teaching has three main components: planning a lesson, implementing the lesson, and reflecting on the lesson taught (Leikin, 2005).Theoretically and empirically, each of these components of teaching provides opportunities for teachers to enhance their knowledge (Leikin, 2005;Russ et al., 2016).Curriculum materials can offer opportunities to learn different approaches and tools (Lloyd, 2008;Remillard, 2000), whereas implementing a lesson can open opportunities for teachers to learn more about their students and their students' mathematical thinking (Lannin et al., 2013;Leikin, 2005).Teachers can also learn by reflecting on their instructional decisions (e.g., how successful the task was in helping students learn a particular idea), particularly when their anticipated strategies did not lead to student learning (Tzur, 2010).Therefore, teachers often enhance their knowledge as they engage in each aspect of teaching.

Factors affecting learning through teaching
While the literature around learning through teaching without external support has not been investigated in depth, from prior literature we identified three key factors that have the potential to influence what teachers can learn through teaching on their own.

Tasks used in teaching
Scholars have highlighted the importance of the tasks used in teaching as it relates to the substance of what students learn and how students think about and make sense of the subject matter (Stein et al., 1996).Tasks with limited cognitive demands leave little room for students to devise various strategies and present different ideas, whereas cognitively demanding tasks are more apt to encourage students to explore different ideas, thereby connecting strategies and concepts to create richer interactions around the content.Given that students could produce different levels of mathematical understanding depending on the tasks, we anticipated that the tasks used in teaching would also influence the kinds of knowledge teachers could develop through teaching.

Lesson structure
The lesson structure could theoretically play a role in teachers' learning through teaching (Stein et al., 2017;Weingarden & Heyd-Metzuyanim, 2023).Although the research is scant regarding the structure of lessons in teacher learning, we hypothesized that teacher-led instruction would provide fewer opportunities than student-centered teaching for teachers to develop knowledge, given that in teacher-led instruction, teachers front-load students with specific strategies for solving problems (e.g., Cazden, 2001).In contrast, in studentcentered instruction, students are allowed to work on problems and share their unique ideas and strategies (Thanheiser & Melhuish, 2023).Thus, we conceptualized that when teachers allow students to work on mathematics tasks individually or as a group before introducing a particular strategy, they expose students to a greater variety of thinking that would not have been available to them if they had introduced a particular strategy first.

Robust understanding of the content
Another factor that potentially plays a role in teachers' knowledge development, especially their PCK, is teachers' own understanding of the content being taught (Copur-Gencturk & Li, 2023;Tröbst et al., 2018).Teachers' content knowledge shapes how they interpret students' solutions and strategies, their understanding of students' thinking, and how they lead class discussions (Copur-Gencturk, 2015;Santagata & Lee, 2021).Therefore, we anticipated that teachers' understanding of the content covered in a particular lesson shapes the kinds of knowledge they could develop through teaching.

Prior research on learning through teaching
Prior literature on teacher learning has focused on how certain elements of teaching have the potential to develop teacher knowledge (e.g., Leikin & Rota, 2006;Remillard, 2000;Remillard & Kim, 2020) or on teachers' overall learning through teaching over time.The first line of work has documented that the curriculum materials as well as the tasks used in teaching and certain pedagogical practices have the potential to create room for teachers to learn from their teaching (Leikin & Rota, 2006;Remillard, 2000).The second line of work has suggested that teachers gain PCK throughout their teaching career (Copur-Gencturk & Li, 2023;Hiebert et al., 2017).Looking across these studies, teachers in general, and novices in particular, seem to gain some knowledge and skills through teaching (Copur-Gencturk & Li, 2023;Hiebert et al., 2017;Kyndt et al., 2016;Leikin & Rota, 2006; McNally, 2016; Santagata & Lee, 2021;Tzur, 2010).These results are not surprising, given that teaching is generally unknown territory for novices as they set about putting into practice their tacit knowledge regarding teaching and reflect on and learn from that experience.In fact, many novices experience reality shock as they enter their own classrooms (Mintz et al., 2020).Discrepancies between anticipated learning outcomes and actual student learning from certain teaching practices can encourage novices to reevaluate and reflect on their decisions, which in turn has the potential to improve their knowledge (Hiebert et al., 2007;Kyndt et al., 2016;Tzur, 2010).
Prior work has focused heavily on learning on the job cumulatively rather than delineating learning through teaching from other forms of learning, such as learning through professional development (e.g Kyndt et al., 2016;Leikin & Zazkis, 2010;Wilson & Demetriou, 2007).One exception to this line of work is the study conducted by Copur-Gencturk and Li (2023).They found, based on an analysis of longitudinal data collected from 207 novice teachers, that teachers developed PCK through teaching even after the authors took into account other formal and informal opportunities the teachers were exposed to during that period.Their findings also underscored the importance of subject matter knowledge in that teachers with strong mathematics content knowledge gained PCK at a higher rate than did teachers with less robust mathematics content knowledge.Although this study provided some empirical evidence of a change in teachers' knowledge through teaching, it did not capture the development of other forms of knowledge, such as general pedagogical content.Furthermore, the data were collected annually, which made it impossible to investigate how teachers developed this knowledge through teaching on their own.Given that teaching is a daily activity for teachers, understanding how teachers learn from teaching on their own and under what conditions they learn has important implications for research and teacher education.
In terms of the kinds of knowledge developed through teaching, prior work has also suggested that different types of knowledge can be developed through teaching.When teachers are asked to use innovative curriculum materials or unfamiliar tasks, they seem to develop general pedagogical knowledge (e.g., how student engagement may change during a particular time of the year; see Leikin, 2006) or PCK (e.g., an understanding of students' mathematical thinking; see Leikin & Zazkis, 2010;Zazkis & Mamolo, 2018).Although these studies have documented the variation in what knowledge teachers develop through teaching, even among those with similar experiences or educational training (e.g., Collopy, 2003;Lannin et al., 2013), precisely which factors lead to different outcomes for teachers has remained unclear.

Present study
Our work aimed to contribute to the literature by systematically investigating teachers' knowledge development through teaching.To do so, we applied an approach that has been successful in exploring the process of children's learning, which Siegler and Crowley (1991) termed the microgeneric approach.Following this approach, we closely monitored teachers' learning through teaching by collecting data before and after teachers completed the task of teaching so that we could detect any changes in their knowledge and confidently attribute those changes to the task of teaching.As shown in Fig. 3, we used lesson plans and preteaching interviews as the means to gauge teachers' baseline knowledge.After teachers taught the first lesson, we conducted another round of structured interviews and continued to use their lesson plans to capture any changes in their knowledge through the first day of teaching. 1We followed the same process after they taught the second day of the lesson.Each participating teacher completed this cycle three times.Thus, this approach allowed us to look closely at the changes in their knowledge through the task of teaching (i.e., planning, enacting, and reflecting).
The microgenetic approach also calls for studying the process of learning by collecting a sample showing the potential for rapid learning.Thus, we collected data on novice teachers' knowledge development given that, as we documented in the review of literature, novice teachers have the potential for rapid learning through teaching (Luft et al., 2015).In addition to the study of learning through teaching, our work also contributes to the literature by exploring three factors (i.e., the quality of tasks used in teaching, student-centered vs. teacher-centered instruction, and a robust knowledge of the mathematics taught in the lesson) in this process of knowledge development.We aimed to answer the following three questions: 1. How did teachers' knowledge change through the task of teaching?2. How were the tasks used in teaching, the way the lesson was structured, and teachers' understanding of the mathematics covered in the lesson related to the development of their knowledge through teaching?3. What did the development of knowledge through teaching look like for individual teachers?

Study context
The data used in this study came from a multifaceted project that was designed to investigate how teachers' content-specific expertise develops through their own teaching (Copur-Gencturk & Li, 2023).We conducted individual meetings with teachers who were also interested in taking part in this smaller study2 to provide information regarding the data collection process for the study.As shown in Table 1, five novice teachers took part in this study.Two of these teachers self-identified as White, two as multiracial, and one as Black.The teachers came from four different states, and they had all graduated from teacher education programs whose duration varied from 4 to 5 years.All the teachers had experience in teaching, and except for one (Hanna), all the teachers had taught mathematics before participating in the study.

Lesson plan tasks
We used lesson plans as the means to capture the extent to which teachers learned through teaching.Lesson plans provide insights into the knowledge teachers draw on while teaching (see Appendix A for the lesson plan template used in the study).Specifically, the template, which we adapted from prior work (Morris & Hiebert, 2017), included sections that allowed us to explore how teachers drew on their knowledge, particularly PCK.Teachers were asked to provide the tasks and representations they would use, along with the struggles they anticipated their students having, what strategies the students would use, and what formative assessment data they planned on collecting from their students.Each teacher was asked to create a 2-day lesson plan 3 times.

Interviews
Another data source we used to gather insights into teachers' knowledge development was interviews (see Appendix B for the interview questions).We conducted structured interviews (a) before teachers started teaching the planned lessons; (b) right after they had implemented the first day of the lesson on the same day or before they taught the second day to capture what they had learned from teaching the first lesson; and (c) after they had implemented the second day of the lesson to capture what they had learned from teaching the second day's lesson.Thus, we interviewed each teacher three times for each concept and asked teachers to undergo this process three times (i.e., for a total of nine interviews per teacher).
The interview questions were informed by prior literature (Smith et al., 2008) and focused on the teachers' knowledge development around the main activity they chose to introduce the concept, the students' strategies and struggles they anticipated, their responses to students' struggles, what they wanted to be on the lookout for to achieve the learning goals of the lesson, and how they assessed student learning.The preteaching interviews were aimed at understanding the baseline knowledge teachers had before teaching these lessons, whereas the postteaching interviews were aimed at capturing shifts or changes in teachers' knowledge.All the interviews were conducted online, videotaped, and transcribed verbatim.

Data collection procedure
Because our purpose was to investigate how teachers developed knowledge through teaching, we collected data from the work of teaching as frequently as possible to detect their learning; therefore, we collected data on a daily basis.Before the data collection began, the research team, led by the first author, held individual meetings with the participating teachers to determine the timeline for data collection and to identify three units in the same content area from which we would collect data.If teachers were teaching at the elementary grade levels (i.e., Grades 3-5), they would select units on fraction concepts, whereas if they were teaching middle school mathematics (i.e., Grades 6-7), they would select units on ratios and proportional relationships.We restricted the content to fractions, ratios, and proportional relationships because of their importance to students' success in algebra and high school mathematics (e.g., Siegler et al., 2012) as well as their role in students' understanding of STEM concepts (e.g., Hoyles et al., 2001;Thompson & Saldanha, 2003).Moreover, given that both teachers' subject matter knowledge and PCK were content specific, focusing on the same content allowed us to investigate the development of knowledge more accurately by reducing other factors affecting the observed patterns.We asked teachers to create a 2-day lesson plan for the first week of each unit in which a new concept was introduced.Our rationale for asking teachers to create a lesson plan for the first week of a lesson was that these lessons would offer more opportunities for teachers to learn and allow more unexpected outcomes to happen, which in turn could lead to more opportunities for them to develop knowledge.
The teachers gave us a tentative schedule of when they would teach these units.We then contacted them before they started to teach the first unit to share their 2-day lesson plan and to schedule a meeting to conduct the preteaching interviews.The preteaching interviews were generally conducted one day before the teachers taught the first day of the lesson.The middle-and last-day interviews were generally conducted on the same day teachers taught the corresponding lesson or the day after they taught before teaching the next day's lesson.

Data coding
We analyzed the data obtained from the lesson plans and interviews in tandem.We separated the lesson plans and transcripts of the interviews into idea units that demonstrated evidence of teachers' knowledge.An idea unit was conceptualized as "a distinct shift in focus or change in topic" (Jacobs et al., 1997, p. 13).Within these idea units, we began coding the data individually with predetermined knowledge codes aligned with the categories of teacher knowledge in Fig. 1 (Ball et al., 2008;Copur-Gencturk & Tolar, 2022;Copur-Gencturk et al., 2019;Grossman, 1990).In the initial coding process, we aimed to capture a more nuanced understanding of their knowledge while also providing room for a variety of theoretical directions to emerge from our reading of the data.The initial codes were removed, extended, or revised through an open and iterative approach informed by the data (Saldaña, 2015).Specifically, our codes attended to nuances in teachers' PCK.For example, we noticed that in some instances, teachers synthesized the two elements of PCK (knowledge of students' mathematical thinking and knowledge of mathematics teaching), whereas in others, teachers displayed the development of either one or the other of the PCK components.Thus, we wanted to capture synthesized PCK because it is qualitatively different; it simultaneously indicates teachers' understanding of the content, students, and teaching.
After some refinement of the codebook, both authors coded the data set individually and engaged in a consensus-building process to reconcile any disagreements and refine the categories (Harry et al., 2005).Through this iterative coding process, we finalized the codebook to capture teachers' learning through teaching and used it to code the data (see Table 2 for the codes and their descriptions, along with sample responses).The level of agreement in our codes was substantial, with a kappa statistic of 0.71 (Landis & Koch, 1977).All the data were double-coded, and any disagreements were resolved by discussion (Harry et al., 2005).In this excerpt, Linda explains why she added a brief discussion about the problem they solved on the first day before she launched into the main problem.Her justification for this additional discussion was presented in connection with the specific mathematical understanding that she aimed to accomplish.
Synthesized PCK Knowledge that indicates teachers can select from a repertoire of various tools and instructional responses and that this selection is informed by their knowledge of students' mathematical thinking.
For example, a teacher might use a particular representation that students are familiar with to help them overcome their struggle to see fractions as numbers with a magnitude.Their use of a particular instructional representation (according to their knowledge of mathematics teaching) is informed by their synchronous knowledge of the subject matter and students' mathematical understanding.The teacher showed knowledge of her students at a personal level related to their behaviors during a particular activity (math journal time).Copur-Gencturk and Atabas International Journal of STEM Education (2024) 11:29 We also coded the lesson plan and interview data to look for evidence of the three factors that could influence teachers' learning: the tasks used in instruction, the lesson structure, and teachers' understanding of the content being taught.To code for the tasks, we examined the main tasks teachers used to teach a lesson by capturing whether the tasks had the potential for students to develop a deeper understanding of a concept or to develop skills only to execute procedures (Stein & Smith, 1998).Thus, we coded each main task as cognitively lowdemanding or cognitively high-demanding.Low-level tasks involve producing previously learned facts, rules, formulae, or definitions, or they require using procedures that are specifically called for.In contrast, high-level tasks require complex, nonalgorithmic thinking with the possibility of providing suggested pathways for students to develop a deeper understanding of the mathematical concepts.To code for lesson structure, we explored how teachers organized their teaching in terms of who was doing the mathematics: teachers who were showing a prescribed strategy or students who tackled mathematics problems to solve them in ways that made sense to them (Sengupta-Irving & Enyedy, 2015).To code for teachers' understanding of the content, we explored instances when teachers showed a robust or limited understanding of the content being taught.Definitions of these coding categories along with sample excerpts can be found in Table 3.

Validity check for capturing knowledge through lesson plans and interviews
The method we used in capturing teachers' knowledge3 development through teaching (i.e., lesson plans and interviews) was novel.To ensure the validity of our approach, we explored the association between teachers' knowledge as measured in this study and teachers' knowledge as captured by an external measure.To do so, we relied on data collected for the larger project of which these teachers were a part.The project was designed to investigate teachers' PCK development quantitatively (Copur-Gencturk & Li, 2023;Li & Copur-Gencturk, 2024); therefore, teachers (including those in this study) completed an external, valid PCK measure annually.Although this external PCK measure was not in perfect alignment with the way we conceptualized and measured PCK in this study, if our approach to capturing teachers' knowledge was valid, we expected to see some alignment between teachers' scores on both measures.We tested this hypothesis in two ways.First, if the way we captured teachers' PCK before they learned through teaching was valid, the average frequencies of the three domains of PCK codes before the teachers began teaching the lessons should be associated with their scores on the external PCK measure before they began teaching that academic year.The correlation was substantial (r = 0.84).Second, if the way we captured learning through teaching was valid, then the average frequencies of the teachers' PCK after teaching should be linked to their PCK scores at the end of the academic year.Again, the correlation was 0.74, suggesting a substantial relationship between these two scores.Taken together, these results give us some confidence that the way knowledge was measured, particularly PCK, was aligned with an external PCK measure.

Data analysis
This study was guided by the three aforementioned research questions.To answer the first question, which was how the teachers developed knowledge through teaching, we reported the frequencies of additional codes assigned for each knowledge category after each teaching day.This is because if teachers developed new knowledge through teaching, we anticipated they would receive an additional code in that knowledge category.If there was no change in teachers' knowledge component after teaching a lesson, then no additional code was assigned to that component.If teachers gained some knowledge through teaching, then the frequency of that knowledge was higher than its frequency before teaching.We also reported the frequencies for subcategories of PCK because this is critical for the quality of teaching the subject matter (e.g., Copur-Gencturk & Han, in press) and students' learning of the subject matter (e.g., Baumert, 2010).
The second research question sought to answer how the tasks used in teaching, the lesson structure, and teachers' understanding of the content related to their development of knowledge through teaching.In response to this question, we categorized the teaching days based on the tasks used on that particular day of teaching (cognitively high vs. cognitively low), the way they structured the lesson (student centered vs. teacher centered), and the content knowledge teachers exhibited for that particular lesson (partial vs. strong understanding).We then reported the total number of additional codes in each knowledge domain aggregated at the teacher level per teaching day for the subgroups of each factor.For example, if the teachers who used cognitively low-demanding tasks on the first day of teaching Unit 1 received a total of five additional codes for the knowledge of students' thinking category, then the frequency of the development of knowledge of students' mathematical thinking would

Tasks
Cognitively low-demanding tasks Tasks require using specific procedures that were introduced previously.These tasks do not require students to understand how the procedure works or explain their thinking; the sole focus is on producing correct answers by using a particular procedure. " The following problem is a regular fraction.Please answer and label the fraction accordingly: 1/4, 1/2, 1/3 1) What number in the above problem is the numerator?
2) What does the numerator represent?
3) What number in the above problem is the denominator?4) What does the denominator represent?5) In the above problem, how many parts would be shaded?6) In the above problem, how many parts does the fraction represent?"(Xavier, Session 2) Cognitively high-demanding tasks Tasks require students to engage in a deeper understanding of the mathematical concepts by not only providing an answer but also explaining their reasoning, making connections to other learned concepts, or developing generalizations.These tasks may require nonalgorithmic thinking to explore the nature of the relationships among mathematical concepts by asking students to provide their thinking and reasoning.
"There are 8 kids and I want to give them each 1/3 of a large pizza.How many pizzas do I need to buy?How much pizza will be left over?Explain."(Hanna, Session 6) "John has 3 7/8 pizzas.His friends just ate 2 1/3 pizzas a. Use benchmarks to estimate the amount of pizza left b.Find the exact amount of pizza left."(Linda, Session 6)

Pedagogical approach
Lesson structure

Teacher centered
The teacher shows the class how to solve a problem without giving students a chance to solve it on their own.In this type of teaching, teachers generally use a particular strategy to teach rather than allowing students to come up with their own methods based on their current level of understanding.
What   [it] or any strategy that they might come up about, and then I will give them some time to try to solve it on their own.And I might tell them they can draw pictures, or they can use equations.It is their choice.And this will be their time when they are just thinking about the problem and how they can solve it, and as they are doing that, I am going to be looking around.And I did not list the common strategies, but the ones that I listed later in the-right below it,… I'm gonna be looking for those strategies and looking for some misconceptions that I could choose to share.And then I am gonna write down the ones, the students who did those strategies I am looking for and then ask them to share with the class at the end."(Hanna, Session 1)

Partial understanding
The teacher demonstrates a superficial understanding of the content being taught.This also includes imprecise language.For instance, an indicator of a superficial understanding of equivalent fractions is using language that implies simplifying a fraction means reducing the value of the fraction.
"… Now one thing we will discuss then will be an idea of simplifying the fraction.2/4, can we make it smaller?Can we make it an equivalent fraction that is smaller?And that' s where we will use the term equivalent fraction.And then they will see that they can change that 2/4 to a half."(Mike, Session 7) [Simplifying a fraction will not make the fraction smaller because equivalent fractions represent the same numerical value.]

Strong understanding
The teacher demonstrates knowledge of the conceptual underpinnings of the content being taught.For instance, an indicator of conceptual understanding is knowing that fractions are equivalent when the two fractions have the same amount and require comparisons based on the same referent whole.be 5 for Day 1 in Unit 1 for the cognitively low-demanding task category.
To answer the third research question, which was what the development of knowledge through teaching looked like for individual teachers, we reported the frequency of codes assigned to teachers individually at each data point (i.e., before teaching, after the first day of teaching, after the second day of teaching) and grouped teachers based on similarities in the aforementioned three factors (the cognitive demand of the tasks, the lesson structure, and a robust understanding of the lesson being taught,) to further investigate their roles in teachers' knowledge development.If a teacher did not gain a particular knowledge component through teaching, then the frequency of the codes for that knowledge would be equal to that of the previous day in that knowledge category.

Knowledge development through teaching
Figure 4 shows the frequency of additional knowledge codes assigned to the teacher knowledge components after teachers taught the mathematics lessons.The highest frequency belonged to the PCK category, whereas GPK came second.These results were consistent across the lessons the teachers taught.Unlike these two knowledge categories, the teachers did not seem to gain subject matter knowledge or knowledge of the context from teaching, as indicated by the low frequency of these categories.In addition, the frequency of the PCK and GPK codes for the second day of each unit was lower than that for the first day of teaching, suggesting that teachers learned the most when they started a new unit.
As we explained in the Data Coding section, we further investigated the nature of PCK developed through teaching.The kinds of PCK developed through teaching seemed to vary across the mathematics lessons taught (see Fig. 5).Similar to the pattern we noticed across the other knowledge components, teachers appeared to gain more of the two subcomponents of PCK (knowledge of students' mathematical thinking and knowledge of mathematics teaching) on the first day of teaching than on the second day (see the steepness of line segments), yet synthesized PCK did not show a clear pattern.In Unit 1, the learning from the first day of teaching was less pronounced than the learning from the second day, whereas in the other units, teachers appeared to develop more synthesized PCK from the first day of teaching than from the second, as indicated by the steepness of the line segments.The role of the tasks used in teaching, the pedagogical approach, and teachers' content knowledge in the development of teacher knowledge through teaching To answer the first research question, we investigated the development of teachers' knowledge through teaching.Here we explored how the three sources (the tasks used in teaching, the pedagogical approach, and teachers' understanding of the content being taught) could explain the variations observed in the development of different knowledge components through teaching.As shown in Figs. 6 and 7, when teachers used cognitively low-demanding tasks, they were developing contentfree knowledge, such as general pedagogical knowledge or knowledge of the context.On the other hand, when teachers used cognitively demanding tasks, they were gaining both PCK and general pedagogical knowledge.When delving into the components of PCK, we noticed that teachers generally developed more knowledge of students' mathematical thinking, followed by synthesized PCK.Unlike the variation in individual teachers' knowledge and the cognitive demands of the tasks used across lessons, when it came to how teachers structured their lesson, Linda and Hanna consistently used a studentcentered approach, whereas the other three teachers used a teacher-centered approach.As shown in Figs. 8 and  9, teachers who used student-centered teaching gained more PCK, whereas teachers in both groups seemed to gain general pedagogical knowledge through teaching.Finally, teachers seemed to gain knowledge of students' mathematical thinking and to develop a more synthesized understanding of their knowledge of students' thinking and mathematics teaching when they allowed the students to engage with the mathematics problems.
Similar to the pattern we observed with cognitive demand of the tasks used in teaching, when teachers had a strong understanding of the content being taught in the lesson they gained more PCK than when they had a less robust understanding of the content covered in the lesson (see Fig. 10).In a closer analysis of the development of the PCK categories, we found when teachers with a robust knowledge of the mathematics, they synthesized their knowledge of students' mathematical thinking and mathematics teaching as they taught (i.e., gained more synthesized PCK; see Fig. 11).When teachers had a robust knowledge of the mathematics, they also seemed to develop a greater understanding of their students' mathematical thinking.Teachers with a less robust understanding of the content being taught seemed to enhance their knowledge of mathematics teaching.

Teacher knowledge development at the individual teacher level
So far, we have investigated the knowledge development pattern across five teachers and have looked at the role of three factors in their knowledge development.Here we zoom in on the patterns of individual teachers.As shown in Table 4, Linda and Hanna demonstrated a robust understanding of the content being taught for each lesson in the study, consistently used cognitively rich tasks, and enacted student-centered teaching methods.They were also teaching at the same grade level.One difference between these two teachers was their experience in teaching.Hanna was a first-year teacher, whereas Linda had some general teaching experience and some mathematics teaching experience in particular.
Both teachers seemed to learn about students' mathematical thinking and synthesized their PCK (see Fig. 12).During the interviews conducted after teaching their lessons, both often talked about student strategies they did not know before teaching the lesson.For example, after teaching the first day of the second unit, Linda identified a strategy the students used that she did not anticipate they would use.In the lesson, students worked on a story problem in which they had to find the difference between 3 1/3 and 1 3/8.While working on the problem, students converted 1/3 to 8/24 and 3/8 to 9/24 so that the two proper fractions would have the same denominator.
As Linda reflected on her experiences at the end of the lesson, she talked about this strategy: Of the remaining three teachers who used a teachercentered approach, Mike and Danielle also had similar characteristics in terms of their understanding of the content being taught and their use of cognitively demanding tasks.In particular, Mike showed a mixed understanding of the lesson being taught in the first unit and partial understanding of the content in the second unit.Yet there was not enough evidence in the data to conclude his understanding of the content covered in the third unit.He used cognitively high-demanding tasks in the first unit, whereas in the other lessons in the following units, he used cognitively low-demanding tasks.Danielle showed partial understanding of the content covered  in the first unit but evidence of her understanding of the content covered in the other two units was insufficient to draw conclusions.She inconsistently used cognitively demanding mathematical tasks throughout the lessons.
As shown in Fig. 13, when both these teachers showed partial understanding of the content covered in the lessons (see Unit 2 for Mike, and see Unit 1 for Danielle), they did not gain much synthesized PCK.However, the use of cognitively demanding tasks in the lessons in Unit 1 seemed to help Danielle gain some understanding of students' mathematical thinking, compared with Mike, who used cognitively low-demanding tasks in Unit 2. The role of a robust understanding of mathematics combined with cognitively demanding tasks in teachers' PCK development was also supported by the learning pattern from Mike's teaching of Unit 1, in which he showed a mixed understanding of the content and used cognitively demanding tasks.Still, the frequency of these knowledge codes was lower compared with those of Linda and Hanna, who showed a robust understanding of the content and used a student-centered teaching approach.Between these two groups of teachers, Xavier's case was unique in that he consistently used a teacher-centered approach, like Danielle and Mike, but he also consistently used cognitively demanding tasks, like Hanna and Linda (Fig. 14).His understanding of the content in the lesson being taught was mixed for the first and last units, whereas he showed a robust understanding of the mathematics covered in Unit 2. Thus, his learning pattern in Unit 2 underscored the importance of having a strong understanding of the content being taught.He gained more understanding of students' mathematical thinking.

Discussion
We investigated novice teachers' knowledge development through teaching when they received no external support.Our findings indicate that the work of teaching holds rich opportunities from which teachers can learn.Further, the cognitive demand of the tasks used in teaching, teachers' robust understanding of the content covered in a lesson, and how they organize their teaching (student centered vs. teacher centered) affects not only the kinds of knowledge they develop through teaching but also the amount.
Before we delve into our findings and their implications for research and education, we would like to draw attention to the study limitations.First, because our findings were based on five teachers, the generalizability of the findings is limited.To understand how teachers learn through teaching, we collected an extensive amount of data from teachers, which led to restricting the number of teachers on which we could focus.Future work with larger samples is needed to replicate our findings.Second, although we provided some evidence for the validity of our approach in terms of gauging teachers' knowledge, particularly their PCK, other approaches that have the potential to collect data at scale are needed.Our purpose for this study was to zoom in on the work of teaching and recognize the nuances in teachers' knowledge development, which required more comprehensive data collection.As our study has now suggested the areas in which teachers seem to learn through teaching, future work could be more streamlined in terms of which knowledge components to investigate and measure.Finally, as mentioned, it is important to take into account our conceptualizations of teachers' knowledge and teachers' knowledge development in interpreting our findings.We could make claims only about the development of knowledge used in the situations that arose in the work of teaching.Therefore, our findings and claims should not be used to infer what teachers may or may not know.
Our findings are in alignment with prior work (e.g., Copur-Gencturk & Li, 2023;Hiebert et al., 2017;Kyndt et al., 2016;Leikin & Rota, 2006;Li & Copur-Gencturk, 2024;McNally, 2015;Tzur, 2010) suggesting that teachers develop some expertise through teaching.By closely examining learning from teaching on a daily basis, we found that the development of teachers' learning could be attributed to the teaching, not to other sources such as professional development, which is a novel contribution to the literature.Our findings also shed light on the variation across teachers' learning from prior work (Santagata & Lee, 2021) by showing that teachers' robust understanding of the content plays a key role in the development of PCK, particularly synthesized PCK.
One of the unique contributions of our study is to uncover which factors influence the development of PCK.As reported previously, although all teachers seemed to develop generic pedagogical knowledge, subject matter knowledge played a substantial role in whether or not teachers developed PCK through teaching.This finding suggests that if teachers do not have a robust understanding of the content they teach, they are less likely to develop the content-specific expertise to make the content accessible to students.Thus, providing opportunities for teachers to develop a more robust understanding of the subject matter being taught could help them continue to learn through teaching on their own.Scholars have emphasized the importance of teachers' subject matter knowledge in teaching and student learning (e.g., Ball et al., 2008;Baumert et al., 2010;Copur-Gencturk, 2015;Copur-Gencturk et al., 2024a, 2024b).Our findings also highlight its vital role in teachers' development of several components of PCK from teaching.One implication of this finding is the need to provide opportunities for teachers to develop a more robust understanding of the subject matter being taught.This is particularly important, given that current research suggests teachers need ample opportunities to learn the conceptual underpinnings of the concepts they are teaching (Copur-Gencturk, 2021).
The second factor influencing whether teachers developed PCK through teaching was the types of problems used in teaching.When teachers use cognitively demanding tasks, they seem to learn more about their students' mathematical thinking and to develop more synthesized PCK.In prior work, curriculum materials have been used as a means of professional development for teachers (Davis & Krajcik, 2005;Moore et al., 2021).Our results indicate that cognitively demanding tasks could be a resource for teachers to develop expertise from their teaching on their own.Given that cognitively challenging tasks require students to use higher reasoning levels, they could reveal more about students' understanding of a given content as well as they could create rich interactions between the teacher and students around the content.This, in turn, could enhance teachers' knowledge of how students learn certain concepts.Our findings underscore the importance of using challenging tasks to develop knowledge and skills, not only for students but also for teachers.Third source of teachers' PCK development is the lesson structure, which indicates who is doing the mathematics in class.When instruction is centered around teachers demonstrating a particular strategy before allowing their students to tackle the problem with their own mathematical ideas, teachers do not gain PCK.This limitation could be because the lesson structure limits the interactions around the content, which in turn limits the learning opportunities for teachers.Our findings emphasize that both high-quality tasks and student-centered pedagogy are beneficial not only for students but also for the teachers themselves.We contend that the identification of these factors has important implications for research and teacher education.Teacher educators should emphasize the benefits of these factors for their own growth in addition to students' growth in their programs.In addition, providing opportunities for teachers to learn how to select and adapt high-quality tasks and creating opportunities for teachers to practice studentcentered teaching would increase their opportunities for learning on the job.

Conclusions
Teaching is considered a rich learning environment through which teachers can grow professionally on the job.Overall, scholars generally agree on the potential for learning through teaching, yet systematic investigations of the extent to which teachers can learn from their teaching experiences on their own have been limited (Kyndt et al., 2016;Leikin & Zazkis, 2010), making it challenging to understand whether and how teaching by itself helps teachers develop expertise.Our study contributes to the literature by showing that teachers can gain knowledge on their own through teaching (without external guidance and support).Further, this study sheds light on the process of learning through teaching by using the microgenetic approach, and it sheds light on the role of a robust understanding of the subject matter, the tasks used in teaching, and the pedagogical approach in what can be learned from teaching.It also underscores the fact that despite the variation in individual teachers' knowledge development, patterns of teacher learning appear similar when the factors affecting the interactions between teachers and students around the content are considered.This means that increasing teachers' use of cognitively demanding tasks in instruction, centering instruction around the students, and enhancing teachers' understanding of the key conceptual underpinnings of the concept being taught will create a continuous learning environment for teachers.

Fig. 2
Fig. 2 Visual Representation of the Learning through Teaching Model

Fig. 3
Fig. 3 Application of the Microgenetic Approach to Study Learning through Teaching

Fig. 4
Fig. 4 The Additional Codes Teachers Received in Each Knowledge Category per Teaching Day.SMK =subject matter knowledge; GPK = general pedagogical knowledge; KOC = knowledge of the content; PCK= pedagogical content knowledge

Fig. 5
Fig. 5 The Cumulative Frequency of Pedagogical Content Knowledge (PCK) Gained through Teaching

Fig. 6 Fig. 7 Fig. 8 Fig. 9
Fig. 6 The Frequency of Additional Codes Teachers Received for Knowledge Categories Per Teaching Day by the Cognitive Demand of the Task Used on That Teaching Day.GPK =general pedagogical knowledge; KOC = knowledge of the content; PCK = pedagogical content knowledge

Fig. 10 Fig. 11
Fig. 10 The Cumulative Frequency of Additional Teacher Knowledge Codes per Teaching Day by Teachers' Understanding of the Content Taught in that Lesson.GPK general pedagogical knowledge; KOC knowledge of the content; PCK = pedagogical content knowledge

Fig. 12
Fig. 12 The Cumulative Frequency of General Pedagogical Knowledge (GPK) and Pedagogical Content Knowledge (PCK) Codes for Linda and Hanna

Fig. 13
Fig. 13 The Cumulative Frequency of General Pedagogical Knowledge (GPK) and Pedagogical Content Knowledge (PCK) Codes for Danielle and Mike

Fig. 14
Fig. 14 The Cumulative Frequency of General Pedagogical Knowledge (GPK) and Pedagogical Content Knowledge (PCK) Codes for Xavier

Table 1
Teachers' Personal and Educational Background Information

Table 2
Teacher Knowledge Codes Used to Capture Their Learning Through Teaching "… I think I'm going to start asking just a few leading questions about 'What did you notice about the answer you thought it was going to be?What did you notice about the answer it actually was?' and see if I can, without telling them, help keep them curious and making those connections that this is going to end up being division and different things like that.Hopefully, someone will break it down and go, 'Oh, we are dividing and multiplying' and 'We can use [the] order of operations, ' and just questions every day that can help them make new connections without me having to tell them." (Linda, Session 7)

Table 2
(continued) "… my students are fully capable of coming back from a holiday weekend and getting focused and getting their work done.For them, behavior can be an issue, especially when it comes to math journal time.They get super chatty typically during math journal time.Today, they were just so on it.I was like, 'Wow.These kids.' I already think they're amazing, and I love them.Just seeing them so focused and on their game, especially after a holiday weekend, was the coolest thing." (Danielle, Session 2)

Table 3
Teacher Knowledge Codes Used to Capture Their Sources of Learning

Table 3
(continued) "Because if they draw it longer, they are not going be able to see the value too….They may see a bigger value.It is kind of like a mini pie and a huge pie.Yeah, they both still have one fourth.But if they have drawn two different sizes, you are not going to be able to notice that."(Xavier,

Table 4
Characteristics of The Teachers By Their Lessons