Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements

Teachers’ in-depth diagnostic thinking has been shown to be crucial for student-centered teaching as they need to perceive and interpret students’ understanding for well-informed decision-making on adaptive teaching practices. The paper presents a content-related approach to analyzing diagnostic thinking processes with respect to the mathematical knowledge elements that prospective teachers identify as students’ resources and obstacles. Prospective teachers’ challenge is that some relevant knowledge elements first have to be unpacked, because compact concepts (such as the place value concept) or procedures (such as for multi-digit multiplication) comprise several smaller knowledge elements (such as the positional property) that have to be made explicit for students to foster their learning processes adequately. Our study examines what knowledge elements prospective teachers perceive and interpret in a transcript vignettes on multi-digit multiplication (of decimal and natural numbers) and its underlying basic arithmetic concepts (place value understanding and meaning of multiplication) in written diagnostic judgments on students’ resources and obstacles (N = 196). A comparative design within the vignette is used to investigate how far the process of perceiving can be supported by thematic cues. The analysis reveals that those knowledge elements cued in the vignette by being already unpacked and explicitly addressed are perceived and interpreted more often (but with lower correctness) than those that are uncued and therefore have to be unpacked by the prospective teachers themselves. This confirms the need to prepare prospective teachers for unpacking mathematical concepts themselves.

ers' judgments from different perspectives (see overviews in Leuders et al. 2018 andStahnke et al. 2016). Whereas early approaches focused mainly on teachers' accuracy of judgments predicting achievements in standardized tests (e.g., Schrader and Helmke 1987), the research shifted increasingly to in-depth investigations of teachers' diagnostic thinking, for instance, in rich vignettes of student thinking (Stahnke et al. 2016). When capturing diagnostic thinking, defined as "the cognitive (thinking) process during the genesis of a diagnostic judgment, ... understood as internal information processing" (Loibl et al. 2020, p. 3), three internal cognitive processes are often studied: perceiving (i.e., noticing or problem identification) and interpreting (i.e., making sense of the information and generate hypotheses), with both in preparation for decision making (on teaching actions such as asking questions and referring to artefacts; Codreanu et al. 2021;Loibl et al. 2020;Stahnke et al. 2016;Wildgans-Lang et al. 2020). These diagnostic thinking processes (perceiving, interpreting, and decision making) have been investigated with respect to their interplay with situational characteristics and personal characteristics (Leuders et al. 2018;Loibl et al. 2020;Südkamp et al. 2012).
Although one of these situational characteristics is the content in view (Wildgans-Lang et al. 2020), content so far has rarely been in explicit focus: Stahnke et al. (2016) showed in their survey that all 60 surveyed studies on diagnostic thinking implicitly used a particular mathematical content area for investigating diagnostic thinking (e.g., problem solving, algebra, modelling, and arithmetic strategies), but most of them did not include the content itself in their operationalizations or research questions. The rare studies that have used content-related operationalizations of the diagnostic processes for analyzing how teachers perceive and interpret knowledge elements in their diagnostic judgments (e.g., by analyzing the degree of correctness or the focus of the diagnostic judgments on students' errors) have provided no explicit reports on what knowledge elements the teachers perceived and interpreted and what they left outside (Busch et al. 2015). However, as prospective teachers' diagnostic judgments and with them their diagnostic thinking processes in particular have been shown to develop mainly with respect to particular areas of content (Hoppe et al. 2020;Prediger 2010), designers of content-related university courses need a sound research base, not only on how prospective teachers perceive and interpret knowledge elements in their diagnostic judgments, but also on what knowledge elements teachers do or do not address for a particular content area and how their attention to relevant knowledge elements can be supported in university courses (Hoppe et al. 2020;Prediger and Zindel 2017).
To decrease this research gap on what knowledge elements prospective teachers perceive and interpret, this paper presents an approach for analyzing diagnostic thinking processes for the content area of arithmetic, more specifically multidigit multiplication procedures. The question concerning what knowledge elements prospective teachers perceive and interpret is of particular importance for this content area as in German Grade 6 (with students aged 11 or 12), the learning of multidigit multiplication requires teachers' attention to the basic German Grade 2 (with students aged 7 or 8) concepts of place value understanding and meaning of multiplication. Therefore, teachers' attention must navigate between knowledge elements from several grades (Prediger 2020) and knowledge elements concerning procedural skills and conceptual understanding (which has been shown to be challenging for prospective teachers; Callejo and Zapatera 2017;Jansen and Spitzer 2009). Thus, we pursue the following overall research question: What do prospective teachers perceive and interpret in their diagnostic judgments on students' understanding of multi-digit multiplication procedures and the underlying basic arithmetic concepts in a given vignette?
In Sect. 2, the theoretical background of our content-related approach is presented, including the conceptualizations of diagnostic thinking processes of perceiving and interpreting the basic arithmetic concepts underlying multi-digit multiplication of decimal and natural numbers and their role in unpacking the knowledge elements of the content. In Sect. 3, the refined research questions, the design of the study and the methods of gathering 196 written diagnostic judgments in a diagnostic activity, and the methods of data analysis are presented. Sect. 4 reveals the empirical findings, which are then discussed in Sect. 5.

State of Research and Existing Conceptualizations of Content-Related
Diagnostic Thinking

Conceptualizing Diagnostic Thinking by Perceiving and Interpreting
Across different conceptualizations and research strategies, teachers' diagnostic competences have repeatedly been identified as necessary for reacting to students' utterances in student-centered and adaptive ways (Ball and Cohen 1999;Empson and Jacobs 2008) and as having a significant impact on students' learning gains (Brunner et al. 2013;Franke et al. 2001;Wilson and Berne 1999). Prospective and practicing teachers' informal or formal formative assessments have been investigated from many perspectives, such as noticing practices and diagnostic thinking processes, examining diagnostic competence (i.e., the knowledge, skills, beliefs) and motivations needed to carry out diagnostic processes (e.g., preparing, organizing, executing, and reflecting diagnostic situations, activities and judgments), and assessing task complexity in order to direct didactical actions (Artelt and Gräsel 2009;Leuders et al. 2018;Stahnke et al. 2016;Weinert et al. 1990). Whereas early studies tended to focus mainly on teachers' accuracy of judgments in predicting students' achievement in standardized tests (e.g., Schrader and Helmke 1987), the research has shifted increasingly to teachers' diagnostic thinking, which has been defined as the "cognitive (thinking) process during the genesis of a diagnostic judgment" from an information-processing perspective (Loibl et al. 2020, p. 3) and captured in in-depth investigations, for example, in rich vignettes of student thinking (Stahnke et al. 2016). Borrowing from the construct of professional vision (Goodwin 1994), Sherin (2007) and colleagues (Sherin et al. 2011) successively developed the distinction of different cognitive diagnostic processes that have been taken up by Loibl et al. (2020) in the DiaCom framework: perceiving (i.e., attending to particular aspects in a complex situation), interpreting (i.e., making sense of the perceived aspects by connecting them to a wider picture/professional knowledge), and decision making (i.e., preparing possible actions such as deciding the next move).
Other researchers have also focused on these diagnostic processes, albeit with different nuances: perceiving is sometimes also conceptualized as noticing or problem identification; interpreting is sometimes divided into describing, evaluating, and explaining (which might lead to generating hypotheses and later on evaluating evidence for them), and decision making is referred to as different decisions on actions (eventually in a consecutive order) such as asking questions or addressing artefacts (Codreanu et al. 2021;Loibl et al. 2020;Stahnke et al. 2016;Wildgans-Lang et al. 2020). In their insightful survey on 60 papers on teachers' diagnostic thinking, Stahnke et al. (2016) elaborated that the surveyed papers differed in their focus on this triad of diagnostic processes, with perceiving in 63% of the papers, interpreting in 78%, and decision making in 53%, with each of them in slightly different diagnostic situations and thereby with other operationalizations. Several studies have not included the process decision making (e.g., Pankow et al. 2016;Weiland et al. 2014), whereas others have focused only on decision making (e.g., Bruckmaier et al. 2016;Jacobs and Empson 2016). As many of the existing papers have done, we reduce complexity by focusing on the two processes of perceiving and interpreting, which are the base for later decision making, in this investigation based on vignettes.
The relationship of the cognitive processes of perceiving and interpreting is still being discussed (see Scheiner 2016, for a systematization): While there are (1) conceptualizations that have assumed the processes to be consecutive cognitive phases or steps (see Dunekacke et al. 2016;Hoth et al. 2016;Pankow et al. 2016), (2) other conceptualizations that have assumed the processes to be cyclical (Santagata and Yeh 2016), and (3) conceptualizations that have regarded them as highly interrelated (Dyer and Sherin 2016;Sherin et al. 2011), we follow (3).
We study perceiving and interpreting, inspired by the DiaCom framework by Loibl et al. (2020), from an information-processing perspective. The DiaCom framework was developed in order to support research that aims at investigating diagnostic thinking processes and synthesizes three components that have often been captured for explaining the processes of diagnostic thinking: the elicited and observed external diagnostic behavior with process indicators (e.g., thinking-aloud protocols) and/or product indicators (e.g., written diagnostic judgments); the external situation characteristics influencing the process, the framing (e.g., time pressure or specific goals), and cues (e.g., tasks and responses); and possible internal person characteristics influencing the process in traits (relatively stable characteristics such as knowledge and beliefs) and/or short-term states (mindsets, affects, and stress).
Research on diagnostic processes, diagnostic behavior, and their relation to situation characteristics and person characteristics for prospective and practicing teachers is multifaceted: Wildgans-Lang et al. (2019) used a model of epistemic-diagnostic activities and elaborated that practicing teachers addressed all the epistemic-diagnostic activities (and with them the cognitive processes of diagnostic thinking) in their judgments, even without explicit knowledge of the model. Meanwhile Jacobs et al. (2010) reported in their comparison of perceiving, interpreting, and decision making of teacher groups with different levels of professionalization that all three cognitive processes were challenging for prospective teachers. Further evidence has been provided "that pre-service teachers had difficulties in perceiving and interpreting students' errors and solutions" (Stahnke et al. 2016, p. 14) by several researchers (e.g., Hines and McMahon 2005;Jakobsen et al. 2014;Son 2013). Focusing on prospective teachers' resources, Wildgans-Lang et al. (2020) showed that prospective teachers were well able to distinguish students who were performing best from poor-performing students based on students' written documents, yet without a more fine-grained analysis of students' competences. In the field of fractions, prospective teachers had difficulties formulating adequate diagnostic judgments on students' solutions that were not similar to their own solutions (Jakobsen et al. 2014).
Personal characteristics, especially knowledge as pedagogical content knowledge (PCK), were examined by half of the studies reviewed by Stahnke et al. (2016): Magiera et al. (2013) showed a strong relationship between prospective teachers' competences in algebraic thinking and their evaluation of oral (but not written) diagnostic situations. Blömeke et al. (2015) revealed that prospective teachers that experienced a positive school climate when they entered their job and had higher mathematical content knowledge (MCK), mathematics pedagogical content knowledge (MPCK), and general pedagogical knowledge (GPK) performed better in perceiving, interpreting, and decision making than other prospective teachers.

Content-Related Research on Diagnostic Thinking for Capturing Teachers' Thematic Focus
While important findings have been achieved on generic aspects of diagnostic thinking (see above), this paper pleads for also deepening the content-related research on diagnostic thinking due to its development with respect to particular areas of content (Hoppe et al. 2020;Prediger 2010). Even though all empirical studies examined by Stahnke et al. (2016) used a particular content area, they differed in the ways they included the content within the research questions and operationalizations, and the DiaCom framework helps to express these differences (see Fig. 1 for these possibilities of location): Some studies left the content completely implicit, focusing on generic research questions (e.g., Wildgans-Lang et al. 2019, on epistemic-diagnostic activities). Some studies treated the content as element of the situation characteristics (e.g., when analyzing differences or commonalities of a teacher's diagnostic processes in situations with different areas of content; Loibl et al. 2020;Wildgans-Lang et al. 2020), and many studies referred to content as an element of the personal characteristics (e.g., when analyzing statistical interactions between diagnostic processes and teachers' content knowledge CK, or PCK as relevant elements of personal characteristics; Förtsch et al. 2018;Loibl et al. 2020). Only a few studies have explicitly integrated content-related aspects in their operationalization of the diagnostic processes themselves. For example, Busch et al. (2015) analyzed teachers' diagnostic processes on functional relationships using a qualitative inventorization of knowledge elements that teachers address. The reported results mainly focused on how teachers perceived and interpreted knowledge elements (e.g., teachers are shown to restrict the diagnostic process of interpreting to surface descriptions rather than deeper analysis, and they vary heavily in correctness). Beyond these findings on how teachers perceive and interpret knowledge elements, the study revealed interesting insights on what knowledge elements they addressed: Many diagnostic judgments are identified as missing concreteness (as teachers mainly mentioned general categories for functional relationships, but not subcategories for more detailed knowledge elements). Also, other studies have indicated that prospective teachers often focus on general aspects rather than focusing on content-related aspects of students' mathematical thinking (Jansen and Spitzer 2009), that they identify procedural obstacles but cannot determine their sources (Cooper 2009), and that they identify conceptual understanding even in procedural utterances (Bartell et al. 2013). These findings call for further research on identifying what knowledge elements prospective teachers address for a particular mathematical content area.
This research is crucial because (prospective) teachers' thematic focus of attention (i.e., the knowledge elements teachers' attend to) is important for effective instruction, as we argue in two steps: (1) In instructional psychology, Renkl (2015) has shown in several studies that students need not only to be cognitively activated (about anything), but the cognitive activation needs to really focus the essential knowledge elements.
(2) Based on the findings on the tight connection between perceiving, interpretation, and decision making (see above), we can assume that teachers can only include in the decision making what they perceive and interpret (Sherin et al. 2011). Consequently, teachers' thematic focus of attention while perceiving and interpreting is crucial for effective instruction (Mason 2002). But so far, studies have indicated that prospective teachers' diagnostic judgments need support in focusing on details of student thinking and in particular on conceptual instead of procedural aspects (Callejo and Zapatera 2017;Jansen and Spitzer 2009;Wildgans-Lang et al. 2019).
Empirical knowledge about prospective teachers' thematic perceiving and interpreting in particular areas of content can thereby substantially inform the design of content-related teacher education courses in mathematics education (Prediger and Zindel 2017).
In particular, thematic cues (e.g., already unpacked knowledge elements) should be investigated for their potential to direct prospective teachers' focus of attention (Mason 2002), as many studies have used prompts for fostering diagnostic processes but have not reflected on the way they might direct teachers' perceiving and interpreting: To investigate diagnostic thinking in the studies presented in Sect. 2.1.1 there was a frequent use of prompts to foster perceiving and interpreting (e.g., Jacobs et al. 2010;Kersting 2008). Meanwhile these studies did not investigate the impact of prompts or explicitly thematic cueing on teachers' thematic focus of attention in diagnostic thinking. Therefore, the current study takes explicit thematic cueing into consideration.

Content-Related Approaches to Diagnostic Thinking as Unpacking Procedural and Conceptual Knowledge Elements
This section introduces our conceptual framework on student understanding and teachers' actions for enhancing student understanding.
Often conceptual understanding, defined as the meaning of an operation or understanding of mathematical terms, has been opposed to procedural skills, defined as procedures of carrying out operations or algorithms, even if we know today that both knowledge types are highly interwoven (Kilpatrick et al. 2001). We draw upon Hiebert and Carpenter's (1992) definition of understanding as network of internal representations, according to which a concept or a procedure is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and the strength of the connections. ... [it] is understood thoroughly if it is linked to existing networks with stronger or more numerous connections (p. 67).
To work with this definition, we conceptualize mathematical knowledge as a network of conceptual and procedural knowledge elements that need to be developed and connected in a network. For further investigating the network, we combine Hiebert and Carpenter's definition with the idea that the network can be compacted and unfolded by students, as established by Aebli (1981) and elaborated by Drollinger-Vetter (2011): Once an individual has constructed and connected the single knowledge elements of a network, the internal representation of the network of connected knowledge elements is often compacted into one overall concept. It is Aebli's epistemic process of compacting that allows the individual to build more complex understanding through connections with further elements. Drollinger-Vetter emphasized that deep understanding is reached when students are able to unfold the concept into its elements, if necessary, while usually working with compacted concepts. In order to do so, students also need learning opportunities characterized as the practice of unfolding, especially concerning how and when to unfold. Based on this conceptualization of student understanding as a network of conceptual and procedural knowledge elements (Hiebert and Carpenter 1992) that can be compacted and unfolded (Aebli 1981; Drollinger-Vetter 2011), we can also articulate what learning has to entail, namely learning opportunities for (a) mentally constructing the knowledge elements, (b) connecting them, then (c) compacting them into higher order concepts or procedures, and (d) undoing the compacting process, here called unfolding (Drollinger-Vetter 2011; Prediger et al. 2022).
Based on this conceptual framework about student understanding and student learning, teaching for understanding can be characterized by three teacher activities: (a) unpacking students' insufficiently learned compacted concepts and procedures in order to (b) identify the lacking knowledge elements or lacking connections in students' mental representations. Then (c) explicating the lacking knowledge elements or connections provides a targeted and adaptive learning opportunity. The last teaching activity of explicating can only be targeted (focused instruction in the sense of Renkl 2015) if teachers' diagnostic thinking was successful.
Within this framework, we can refine our conceptualization of the processes of diagnostic thinking in a content-related way. We conceptualize perceiving as pointing to critical events in teacher-student interaction (e.g., identifying parts of students' products and processes that contain relevant knowledge (not necessarily disentangled as concise elements) or concrete statements of the interaction that are critical in the sense of documenting resources-preknowledge and adequate concepts or procedures-or obstacles-inadequate concepts or procedures and inadequate individual constructions of meaning) and interpreting as explicitly naming knowledge elements and unpacking them by identifying subcategories (e.g., against the background of the content not only identify the general category but also the subcategories into which students might unfold it or teachers explicitly unpack it).
Diagnostic judgments of practicing and prospective teachers about conceptual understanding and procedural skills have been studied in several papers. Concerning prospective teachers' diagnostic judgments, different obstacles in perceiving and interpreting conceptual understanding and procedural skills have been documented in previous studies: Bartell et al. (2013) showed that prospective teachers often identify conceptual understanding correctly in students' statements, but also interpret pure procedural statements as indicating conceptual understanding, when superficial diagnostic processes lead to catching keywords. Similarly, Son's (2013) study on error bases for similarity revealed that prospective teachers mostly identified conceptual obstacles as being procedural. Callejo and Zapatera (2017) emphasized the partly unfulfilled necessity of perceiving the conceptual and cognitive dimensions in students' statements. Taking the conceptual dimension into consideration, Jansen and Spitzer (2009) pointed out that focusing on students' thinking as an important category cannot be taken as synonymous to focusing on students' mathematical thinking. In their study on prospective mathematics teachers' reflective thinking skills, they reported that some prospective teachers described "their students' mathematical thinking ..., [whereas others] generally described their students' thinking" (p. 140) without explicitly linking it to the mathematical learning content, for instance, by simply reporting the correctness of solutions. These different findings indicate obstacles in the thematic focus of the diagnostic behavior, namely, what knowledge elements are perceived and interpreted in diagnostic judgments, which might be connected to obstacles in diagnostic thinking. Therefore, we want to elaborate and investigate prospective teachers' diagnostic thinking with a content-related approach on a particular network of arithmetic concepts.

Content-Related Approach to Diagnostic Thinking About Students'
Understanding of Multi-Digit Multiplication and the Underlying Basic Arithmetic Concepts

Model of Students' Understanding of Basic Arithmetic Concepts Underlying Multi-Digit Multiplication
The content in view of our content-related approach is understanding of basic arithmetic concepts, more concretely, place value understanding and meaning of multiplication. This content area is of central importance not only in Grades 1 to 4, but also in Grades 5 and 6, as difficulties in arithmetic of the Grades 5 and 6 mostly stem from missed learning opportunities in Grades 1 to 4 (Moser Opitz 2007). We explain the relevance of place value understanding and meaning of multiplication in the conceptual framework described above (synthetized from Aebli 1981; Drollinger-

Fig. 2
Understanding as network of connected knowledge elements: General categories of compacted conceptual understanding or procedural skills in the current content or foundation from previous years (elements marked with black margins) can be unfolded into subcategories with more detailed elements (marked with grey margins) K Vetter 2011; Hiebert and Carpenter 1992;Kilpatrick et al. 2001) for an example from Grade 5 (displayed in Fig. 2). The example in Fig. 2 starts from a typical procedural student error in the current learning content (procedure of multi-digit multiplication for decimal numbers), that needs conceptual understanding of place value understanding for decimal numbers and meaning of multiplication for decimal numbers. Those knowledge elements are based on foundations from previous years in place value understanding for natural numbers and meaning of multiplication for natural numbers.
The understanding of the complex basic concept of the place value system requires at least three knowledge elements that have to be connected: the positional property (the place of a digit determines its value), the additive property (two-digit numbers can be decomposed into tens and ones) and the multiplicative property for the place values (the second and third digits count the bundled units of tens and hundreds ;Ross 1989;Van de Walle 2007). The multiplicative property is also a relevant part of the decomposition of the original product into four subproducts.
Similarly, the meaning of multiplication for natural numbers requires different knowledge elements: first, determining the meaning of multiplications in arrays combined with, second, relating representations and, third, identifying the multiplication as counting in bundled units (e.g., Götze and Baiker 2021;Lamon 1996;Steffe 1994).
Also, far beyond the example in Fig. 2, fifth and sixth graders are generally supposed to acquire new knowledge as current learning content (such as the multi-digit multiplication of decimal numbers) that relies on a foundation of highly compacted knowledge elements from previous years. For students with mathematical difficulties whose conceptual development was not successfully accomplished, teachers need to unpack the highly compacted knowledge and provide remedial learning opportunities, for example, for place value understanding and meaning of multiplication, so that students can learn to connect the prior learning content to the current learning content. As the research literature on students with mathematical difficulties has shown (Andersson 2010;Moser Opitz 2007), these students often have learning needs in the following basic arithmetic concepts: place value understanding and its elements, meaning of multiplication and division and its elements, and relating multiple representations (as displayed in Fig. 2).
Teachers in Grades 5 and 6, however, are often not aware of these learning needs in basic arithmetic concepts (Karsenty 2010;Moser Opitz 2007;Prediger et al. 2022), so it is a major task of teacher education to focus on them in content-related teacher education courses (Karsenty 2010).

Content-Related Research Approach for Investigating Diagnostic Thinking on Students' Understanding of Multi-Digit Multiplication and Underlying Basic Concepts
Based on the current state of research, the processes of diagnostic thinking and the conceptualization of understanding (Fig. 3) present a research framework for the current study that adapts the DiaCom framework (Loibl et al. 2020) with a stronger focus on content and the what-questions. For a content-related approach, we frame our research in diagnostic situations occurring when students need to remediate understanding of basic arithmetic concepts (situation characteristics). The prospective teachers receive a transcript vignette of a teacher-student interaction starting with a procedural error in multi-digit multiplication of decimal numbers that is unpacked by the teacher and unfolded by the students into different aspects of understanding in a particular content structure (Fig. 2). The impact of person characteristics is not investigated, so they are simplified in Fig. 3 to the assumption of comparable states, as the study is conducted in courses where all prospective teachers had equal learning opportunities for their pedagogical content knowledge on understanding of basic arithmetic concepts.
The major content focus is set on the observation of diagnostic behavior based on content-related product indicators, which is done by eliciting and analyzing diagnostic categories that the prospective teachers address for unpacking knowledge elements in the vignette. With these external observations, we intend to infer the internal diagnostic thinking processes, which are not measured as independent processes but as highly interwoven, as perceiving critical points in the teacher-student interaction is a precondition for interpreting students' understanding.
In the investigation of the diagnostic thinking, we aim to study content-related information processes and infer from the diagnostic behavior what knowledge elements are perceived in pointing to critical events, for example, the presented error and further obstacles in the teacher-student interaction. We aim to study what knowledge elements are interpreted by scrutinizing the prospective teachers' unpacking of knowledge elements by identifying subcategories exemplified in Fig. 2 by a typical procedural error, for instance, to achieve understanding of what is missing, the procedure and its conceptual background must be increasingly unpacked. The process of decision making is not explicitly captured in the diagnostic behavior in our study (see Sect. 2.1).
The focus on the content structure is also reflected by the analysis of a specific variation of thematic cues: When unpacking the knowledge elements is the key process for teachers, it is important to consider that prospective teachers have been shown to struggle with unpacking relevant knowledge elements (Busch et al. 2015;Morris et al. 2009). These documented challenges raise the question of whether thematic cues can support the prospective teachers. In our research design, we cued some knowledge elements in view by having already unpacked them explicitly in the vignette, whereas others stay compacted and require unpacking by the teachers. By varying the degrees of unpacking of the knowledge elements, we investigate the impact of these thematic cues on teachers' processes of perceiving and interpreting.

Refined Research Questions
Within the outlined conceptual framework and our content-related research approach sketched in Fig. 3, our initial question from the introduction can be refined into two more detailed research questions: RQ1 Which knowledge elements of conceptual understanding and procedural skills in the current content and also in the foundations from previous years (in particular, place value understanding and meaning of multiplication) do prospective teachers perceive and interpret?
RQ2 How far do prospective teachers go in unpacking knowledge elements in their diagnostic judgments and does this vary with the thematic cues for unpacking provided by the diagnostic vignette?

Sample
The sample of the study consisted of German prospective mathematics teachers in a mathematics middle school teacher education program from Grades 5 to 10, (N = 196, with 75% female, which represents the usual gender distribution in German teacher study programs). 47% of the prospective teachers were in their 3rd year of the teacher study program, 53% in 4th or 5th year. The research was conducted in the beginning of their mathematics education course. Before that, all prospective teachers had participated in courses on mathematics and mathematics education, in particular a practicum course for first diagnostic experiences and a topic course on arithmetic concepts and typical student errors. Therefore, we can assume that it was possible for all prospective teachers to acquire the relevant knowledge elements for their diagnostic judgment. However, it is a limitation that the prior knowledge was not controlled as a person characteristic (see research approach in Sect. 2.2.2).

Vignettes as Method for Data Gathering
To pursue the research questions, prospective teachers' diagnostic judgments were captured in a framing of remediating sixth graders' understanding of basic arithmetic concepts. Prospective teachers' written diagnostic judgments were elicited using vignettes with open answer formats. This instrument was chosen because vignettes and representations of practices have already been widely used in PD research for capturing and fostering prospective teachers' competences (overview in Buchbinder and Kuntze 2018), for example, by Sherin (2007) to study teacher noticing and professional vision. Vignettes depict typical teaching situations and teacher-student interaction in written formats, video formats, or picture formats and are optimized to capture not only teachers' knowledge, but also diagnostic thinking processes (as perceiving and interpreting) in simulated practice (Friesen and Mecherlein 2020).
We chose a written vignette format to display a diagnostic situation in which a teacher works with a small group of students to discover the students' resources and obstacles and enhance their understanding by unpacking knowledge elements that students used inadequately or did not know. The construction of the vignette started from a real transcript that was sharpened and enriched according to the particular research interest (e.g., the variation of thematic cues).
The transcript of the vignette (see Fig. 4) started with a typical error in multi-digit multiplication of decimal numbers and expanded to a dialogue on the meaning of multiplication. This focus on a typical error in multi-digit multiplication of decimal numbers was chosen, as it contains the procedure for the current learning content in Grade 6 (multiplication algorithms of decimal numbers) which is based on procedure and understanding for natural numbers with all its underlying knowledge elements (see Fig. 2): procedural elements such as the decomposition in subproducts and automatized multiplication facts and compacted conceptual elements such as place value understanding and meaning of multiplication. The list of knowledge elements to be perceived and interpreted (in the sense of the DiaCom framework) was validated in 10 interviews with experts working in mathematics education research groups and specializing on one or more of the following aspects: decimal numbers, understanding of place value, meaning of operations, and diagnosing and enhancing students' understanding.
To study how the unpacking of knowledge elements can be cued within a diagnostic vignette, we kept the place value understanding on a compacted level (but with the same amount of knowledge elements as the meaning of multiplication) and had the meaning of multiplication already unpacked in the vignette. Therefore, the place value understanding was uncued while the unpacking of the meaning of multiplication was supported by thematic cues within the vignette. This allowed comparison of how the prospective teachers unpacked only the knowledge elements already cued in the transcript or could do this on their own.
In Fig. 4, the general categories for compacted and therefore uncued knowledge elements are marked in black and bold, the subcategories for unpacked and therefore cued knowledge elements in non-bold. Among them, the explicitly cued subcategories are marked in black and the implicit subcategories in grey. By varying the degree of explicitness, the relevance of these cues to prospective teachers' diagnostic thinking processes of perceiving critical points and interpreting cued knowledge elements compared to uncued knowledge elements can be investigated.
The prospective teachers first read the transcript of the teacher-student interaction and were asked to provide written diagnostic judgments on students' resources and obstacles. The prospective teachers were informed that the aim of the diagnostic judgment was to find the best starting point for enhancing and remediating students' understanding.

Methods for Data Analysis
To pursue the research questions, the 196 written texts were investigated with respect to the perceived and interpreted knowledge elements and the unpacking of knowledge elements. For this analysis, we coded the written texts with respect to the observable diagnostic behavior and inferred the underlying diagnostic processes (within the adapted framework from Fig. 3), and then analyzed the results quantitatively, following four steps: Step 1: Coding Prospective Teachers' Diagnostic Categories The diagnostic behavior of the prospective teachers (i.e., 196 written diagnostic judgments) was coded with respect to the knowledge elements included in their diagnostic categories. The first coding scheme was deduced from the analysis of relevant knowledge elements in the content structure of the vignette (see Fig. 4) and inductively enriched by all deviant diagnostic categories articulated by prospective teachers (some general categories such as place value understanding and others unfolded into more specific knowledge elements; see Fig. 4). The categories were distinguished according to what they addressed, procedural skills or conceptual understanding, both in the current learning content (here, multi-digit multiplication of decimal numbers) or in students' basic understanding from previous years (here, place value understanding and meaning of multiplication). They were also coded as correct or incorrect judgment by comparing them to the expert judgments. The written texts were coded by two raters with an interrater reliability of Cohen's κ = 0.88.
Step 2: Inferring the Processes of Perceiving The conceptualization of perceiving as pointing at the critical events (i.e., as a process of diagnostic thinking defined in a content-specific way within the adapted DiaCom framework; see Fig. 3) was based on the coded categories in Step 1 and operationalized as follows: From the coded diagnostic categories, we inferred the diagnostic process of perceiving by deducing which turns (previously regarded as critical events by experts) in the vignette the prospective teachers pointed at, even if the used diagnostic category was not adequate.
Step 3: Inferring the Processes of Interpreting The diagnostic process of interpreting (a process of diagnostic thinking defined in a content-specific way within the adapted DiaCom framework) was conceptualized and operationalized based on the coded turns in Step 2 (as perceiving forms the base for interpreting) by capturing all general compacted categories of knowledge elements the prospective teachers addressed and the unpacked knowledge elements they addressed as subcategories (see Figs. 2 and 4). All categories addressed in the diagnostic judgments were taken into consideration, independent from their correctness. Based on the material gained in the inferring process of interpreting, we inferred the prospective teachers' unpacking as part of interpreting by distinguishing general compacted categories from subcategories for unpacked knowledge elements from Fig. 2. In order to analyze how the unpacking varied with the thematic cues for unpacking, place value understanding and meaning of multiplication were compared.
Steps 1-3 were used to answer RQ1 on perceived and interpreted knowledge elements, and Step 3 was used for RQ2 for support in interpreting through cues for unpacking.

Empirical Findings on Prospective Teachers' Content-Related Diagnostic Processes
In order to substantiate the quantitative analysis, Sect. 4.1 provides three example judgments and documents to transparently show how the diagnostic thinking processes (perceiving and interpreting) were inferred in Steps 1-4. Sect. 4.2 then presents the results of the quantitative analysis relating to perceiving in RQ1 based on Steps 1 and 2, and Sect. 4.3 gives the comparisons for interpreting in Steps 1-3 for RQ1 and Step 3 for RQ2.

Insights into Three Cases of Prospective Teachers' Diagnostic Judgments and Thinking Processes
Even for sixth graders, who are supposed to learn elaborate procedures on decimal numbers, for example, multi-digit multiplication, the prospective teachers' diagnostic focus on students' place value understanding and meaning of multiplication (of natural numbers) is essential for focused content-related decisions. In order to illustrate that this is not self-evident for all prospective teachers, this section provides qualitative insights into three illustrative cases of three prospective teachers' diagnostic judgments about students' obstacles and judgments for the vignette with the students Mia and Ali in Fig. 4. These three cases of diagnostic judgments (of prospective teachers Pjotr, Lena, and Seda) have been chosen to provide maximal contrast concerning their analyzed perceiving and interpreting. Fig. 5 shows Case 1, the diagnostic judgment of the prospective teacher Piotr (in the German original and the English translation) and the three analytic steps. It reveals that Piotr addresses knowledge elements from both the current content and the foundation of previous years, for instance, multiplying as multiplication facts as Ali's resource and meaning of multiplications in arrays as Mia's resource. Piotr therefore unpacks the meaning of multiplication as a compacted knowledge element into the meaning of multiplication in arrays as an underlying basic knowledge element. With respect to the relevance of thematic cues, it is worth mentioning that Piotr's diagnostic judgement is mostly tied to the content of multiplication that has been cued in the vignette.
While case 2, Lena's diagnostic judgment, is not shown in Fig. 5, analytic Step 1 reveals that she addresses the code procedure of multi-digit multiplication four times and multiplication facts one time. From which we derive that she mostly addresses the current learning content of multi-digit multiplication and explicates mostly proce-

Fig. 5 Two cases of prospective teachers' (Piotr's and Seda's) diagnostic judgment and analysis in
Steps 1-3: Focus on current learning content and procedural knowledge elements dural skills. She does not relate these compacted knowledge elements to knowledge elements from the foundation of previous years.
In contrast to Cases 1 and 2, Case 3 (of prospective teacher Seda, shown in Fig. 5) also explicates knowledge elements of place value understanding and unpacks the compacted knowledge element of place value understanding into its subcategories (shown in bold). Those elements have not been cued in the vignette. As a whole, her diagnostic processes are richer and with the most adequate thematic focus, most in line with the experts' focus of attention.
The three cases exemplify differences in focusing on either procedural knowledge elements (Case 2) or conceptual knowledge elements (Cases 1 and 3) and differences in focusing on the current learning content (Case 2) or also the foundation from previous years (Cases 1 and 3). In addition, they reveal differences in either unpacking only cued knowledge elements (Case 1) or also uncued knowledge elements (Case 3). In the next section, these exemplified differences are investigated systematically for the whole sample of 196 diagnostic judgments.

Prospective Teachers' Perceiving of Focus on Students' Understanding of the Current Learning Content and the Foundation from Previous Years
RQ1 asks for the knowledge elements that teachers perceive, comparing procedural skills and conceptual understanding, and the current content and the foundations from previous years (in particular place value understanding and meaning of multiplication).
The coding of diagnostic categories in the teachers' diagnostic judgments reveals that only 17 out of 196 students perceived and interpreted all four aspects: procedural skills in current content and foundations from previous years and conceptual understanding in the current content and from previous years. Table 1 provides a more detailed overview by listing all general diagnostic categories coded as perceived in prospective teachers' diagnostic judgements. Table 1 reveals that 73.2% of the coded diagnostic categories referred to foundations from previous years. Meanwhile, 48.1% of all statements dealt with conceptual understanding within the foundation from previous years, our main focus of the basic concepts place value understanding and meaning of multiplication (marked in italics in the table and focused on the following analyses).

Prospective Teachers' Interpreting with Cues for Unpacking Diagnostic Interpretations and Their Relation to Adequacy
As prospective teachers are known to struggle with unpacking knowledge elements alone, the diagnostic vignette provided many thematic cues by unpacking the rele-  vant knowledge elements for subcategories of multiplication. In contrast, students' resources and obstacles in the place value understanding became apparent, but were not unpacked in the vignette itself. This allows more in-depth study of the interpretation of RQ1 and RQ2, which asks how far prospective teachers went in unpacking knowledge elements in their diagnostic judgements, and whether this varied with the thematic cues for unpacking provided by the diagnostic vignette. Table 2 lists the general categories for compacted knowledge elements in place value understanding and meaning of multiplication and the subcategories for unpacked knowledge elements (see Fig. 2). In italic, we marked those knowledge elements that had not been cued in the vignette so that teachers needed to unpack them themselves, whereas in not italic categories had already been cued in the vignette by unpacking them. The listed percentages of codes among all codes for place value understanding/meaning of multiplication reveals a substantial difference: Although there was a similar number of subcategories applicable for place value understanding and meaning of multiplication, the cued knowledge elements were much more often addressed in diagnostic subcategories than non-cued knowledge elements (significant on a 5% level of the t-test). Whereas 49% of the coded utterances on place value understanding refer to the general category and 51% to a subcategory, the relation for multiplication is 13 to 87%. The diagnostic judgments  are therefore more concrete in the sense of unpacked subcategories when these are cued in the vignette. These differences are confirmed in Table 3, in which the codes are counted for each prospective teacher rather than for all utterances together.
43% of the prospective teachers did not address place value understanding as a diagnostic category and 23% only as a general category, whereas 21% named one and 13% more than one subcategory. In contrast, only 7% did not refer to meaning of multiplication and 3% only as a general category, whereas 20% mentioned one subcategory and 70% more than one. Again, these differences are significant on a 5% level of the t-tests.
However, the cues might also show a limitation when the adequacy of teachers' diagnostic interpretations is considered. Although the overall correctness of teachers' diagnostic judgments is very high (in total, 80% of all interpreted knowledge elements articulated in the prospective teachers' diagnostic statements were coded as correct), Table 4 reveals possible differences: No difference occurs between the two general categories in which correct interpretations of knowledge elements were given in 92.7% (non-cued) and 92.0% (cued) of the utterances. For the subcategories, the correctness seems to covariate with the cues: Whereas cued subcategories for multiplication were used in correct interpretations of knowledge elements in 75.5% of the utterances, the non-cued subcategories for place value understanding where correct in 88.3% of the cases. These findings might show first indications that cued unpacking of subcategories does not necessarily lead to correct interpretations of students' resources and obstacles, whereas those prospective teachers who were able to unpack the non-cued knowledge elements for place value understanding for themselves also did it correctly.

Content-Related Approach and Conceptualization
In this study, we have investigated the effect of the diagnostic thinking processes of perceiving and interpreting of prospective teachers on students' understanding of the current learning content and the underlying basic arithmetic concepts. This content was chosen due to often-documented learning difficulties in arithmetic understanding of decimal numbers and their operations in Grades 5 and 6 and their origin in place value understanding, for example, meaning of multiplication as operation from Grades 2 and 3 (Andersson 2010;Moser Opitz 2007).
Our content-related research approach draws upon an adapted version of the DiaCom framework (Loibl et al. 2020), in which the content structure is not only included as situational or personal characteristics but as a relevant part also of diagnostic processes, here perceiving and interpreting (Sherin et al. 2011;Stahnke et al. 2016).
The operationalization of diagnostic thinking in cognitive processes is combined with a conceptualization of conceptual understanding as a network of connected knowledge elements (Hiebert and Carpenter 1992), which are then compacted into higher order concepts and procedures (Aebli 1981). According to this conceptualization, an adaptive enhancement of understanding requires unpacking relevant knowledge elements by the teacher or unfolding them by students (Drollinger-Vetter 2011; Prediger et al. 2022).
For the current study, we focused on perceiving and interpreting and conceptualized both cognitive processes with reference to the content structure (Figs. 2 and 3): Perceiving as pointing to critical events in teacher-student interaction or students' solutions and interpreting as elaborating knowledge elements either in a compacted form or unpacking knowledge elements by identifying subcategories. By these conceptualizations, we can not only replicate but also refine existing findings on prospective teachers' diagnostic thinking on conceptual and procedural knowledge and their varying concreteness (Bartell et al. 2013;Callejo and Zapatera 2017;Jansen and Spitzer 2009).

Findings on What Knowledge Elements Prospective Teachers Perceive and Interpret with and Without Cues
Beyond the general findings on summarized diagnostic processes presented in the theoretical background of this paper, the content-related approach provides deep insights into the what question (RQ1). First differences have been displayed by giving insights into three cases of prospective teachers' diagnostic judgments and the inferred diagnostic thinking processes (in Sect. 4.1), which illustrate possible differences in what knowledge elements prospective teachers perceive and interpret. Zooming out to the 196 written judgments, Table 1 reveals that unlike an often-problematized perceiving being restricted to procedural skills (Bartell et al. 2013;Empson and Jacobs 2008), our prospective teachers perceived and interpreted a whole range of knowledge elements in conceptual understanding and procedural skills, in actual content and foundations from previous years. This goes along with findings of Callejo and Zapatera (2017), who emphasized that many different categories can be found in prospective teachers' diagnostic judgments. As the range of perceived knowledge elements might be influenced by previous learning opportunities, it is unfortunate that limited testing time hindered empirical control of the PCK of the prospective teachers. In spite of this limitation, the detailed content-focused analytic approach confirmed the great need to work with teachers not only on general categories, but also on unpacking them into the knowledge elements needed to compose the understanding (Jansen and Spitzer 2009 In order to support prospective teachers in this critical part of interpreting by unpacking, the role of thematic cues provided in the diagnostic vignette was investigated through systematic variation depending on the topic (RQ2). Table 2 reveals that for knowledge elements uncued in the vignette, 49% of the statements in the diagnostic judgments only interpret the general category and 51% of the statements also focus on subcategories. In contrast, for cued knowledge elements, only 13% of the statements interpreted the general category, whereas 87% of the statements unpacked the basic understanding into subcategories for more refined knowledge elements. In total, 70% of prospective teachers addressed more than one subcategory for the cued and only 13% for the uncued categories (Table 3). Even if this comparison must be further contextualized, these significant differences provide interesting indications that prospective teachers' diagnostic processes of unpacking might be influenced by thematic cues in a diagnostic vignette, not only by new thematic inputs (Prediger and Zindel 2017) but also by situative cues.
Concerning the correctness of prospective teachers' diagnostic judgments, the general categories of place value understanding and meaning of multiplication were interpreted correctly to an equally high amount. This high number of correct interpretations is in line with Bartell et al.'s (2013) findings. However, for cued subcategories, more incorrect interpretations were found (24.5%) than for non-cued subcategories (11.7%). We interpret this difference as an indication that the cues can also be too strong: When the unpacking is already provided by a diagnostic vignette, the prospective teachers do not necessarily interpret them correctly. But activating the necessary categories through the cues in the vignette might be a starting point for further professionalization. Those prospective teachers who are able to unpack subcategories when not cued in the vignette might have a higher success rate for interpreting them also correctly.
In total, we interpret our findings as a first empirical support that the contentrelated conceptualization of the diagnostic thinking processes can indeed reveal important windows into prospective teachers' thinking. This is of high relevance for continuing a didactical research program that goes beyond investigating how prospective teachers notice knowledge elements to what knowledge elements they notice (Prediger and Zindel 2017;Busch et al. 2015).

Limitations and Future Research
The first contribution to deepening our content-related understanding of diagnostic processes presented in this paper is still constrained by the methodological limitations of the current study, given by the very specific shape of the diagnostic vignette. The vignette was designed based on empirical and theoretical assumptions and validated by experts; however, it is only one vignette that cannot exhaustively capture a complex construct in all generality. Furthermore, the vignette and the role of thematic cues should be further investigated, as the results might stem from very specific design choices in the diagnostic vignette. As the coding system is content related, future research is planned that is aimed at investigating whether comparable results can be generated by a comparable coding system for other mathematical areas of content . Therefore, future studies might use more vignettes in order to provide further insights.
Because this study was methodologically limited by prospective teachers being from only one university, future studies should investigate how different learning opportunities shape diagnostic thinking patterns. Although all of the participating prospective teachers attended university courses in mathematical content and general diagnostic practices, future studies should control for person characteristics more systematically so that they can also determine whether prospective teachers have better pedagogical content knowledge for either of the content areas of meaning of multiplication or place value understanding.
Since the current sample consisted of prospective teachers, the findings cannot be directly transferred to diagnostic thinking of practicing teachers. To overcome this gap, the developed diagnostic vignettes should also be administered with practicing teachers to reveal similarities as well as differences (Prediger et al. 2022).
For both target groups, prospective and practicing teachers, intervention research should investigate how expertise in diagnostic thinking can be systematically developed.
Hence, future studies should aim to overcome these limitations by collecting data from several universities to overcome the possible biases of a specific teacher education program and exploring the identified associations in more depth.

Implications for Developing Prospective Teachers' Diagnostic Thinking in Content-Related Ways
Even if future studies are required to validate and deepen the research, the presented study can already inform the design of preservice professional development courses. The content-related approach to diagnostic thinking suggests making more explicit the underlying ideas about understanding as a compacted network of knowledge elements and the need for prospective teachers to unpack the knowledge elements in particular for students with mathematical difficulties (Jansen and Spitzer 2009). Prospective teachers' professionalization should focus on the epistemological analysis of networks of knowledge elements needed for understanding (as in Fig. 1), the diagnostic judgments of vignettes with a view to the underlying epistemological core, and the need to address not only general categories such as place value understanding but also to unpack them into the subcategories of refined knowledge elements.
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