Relationship between Chinese Pre-service Mathematics Teachers’ Knowledge, Pedagogical beliefs, and Noticing

Relationship between Chinese Pre-service Mathematics Teachers’ Knowledge, Pedagogical beliefs, and Noticing | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Relationship between Chinese Pre-service Mathematics Teachers’ Knowledge, Pedagogical beliefs, and Noticing Xinrong Yang, Jun Deng, Johannes König, Gabriele Kaiser This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5925262/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract While the theoretical discourse posits teacher knowledge and beliefs as critical factors influencing teacher noticing, few studies have empirically explored the interrelationship among these constructs within a single investigation, particularly in a non-Western context. This paper examines the relationships among teacher knowledge, pedagogical beliefs, and teacher noticing, with a specific focus on the mediating role of beliefs between teacher knowledge and noticing, based on a study involving 583 pre-service mathematics teachers within the Chinese context. The findings indicate that in contrast to common expectations and earlier results pre-service teachers’ mathematical content knowledge (MCK), rather than their mathematical pedagogical content knowledge (MPCK), exhibits a stronger correlation with teachers’ noticing. However, as expected, transmissive pedagogical beliefs significantly and negatively correlate with noticing, while constructivist pedagogical beliefs demonstrate a significant positive relationship with noticing. Furthermore, the study reveals that teacher knowledge and pedagogical beliefs distinctly influence various facets of teacher noticing confirming theoretically derived assumptions. Notably, pedagogical beliefs serve as a significant mediator between teacher knowledge and noticing. The findings suggest that apparently societal and cultural norms, alongside teaching experience, moderate the relationships among teacher knowledge, beliefs, and noticing. Educational Psychology Mathematics content knowledge mathematics pedagogical content knowledge pedagogical beliefs noticing Figures Figure 1 Figure 2 1. Introduction In the last two decades, the investigation of teacher noticing has emerged as a significant topic in educational research globally, particularly in the field of mathematics education (König et al., 2022 ; Weyers et al., 2024 ). Previous studies have generally accepted teacher noticing as one of the most critical and important professional competencies that profoundly influence the quality of mathematics instruction and students’ learning outcomes (Sherin et al., 2011 ). Given its importance, designing effective professional development programs or interventions to foster teacher noticing is therefore essential for both pre-service and in-service teacher education. However, a comprehensive understanding of the factors that shape and contribute to the development of teacher noticing should be a necessary first step. Such influences are of course complex; empirical studies have identified that teachers’ teaching experience, knowledge, and beliefs are among the most critical factors affecting the development of teacher noticing (e.g., Jacobs et al., 2010 ; Weyers et al., 2024 ). Theoretically, it has been posited that teachers’ noticing is “intimately tied to” their knowledge (Schoenfeld, 2011 , p. 231), with the assertion that “knowledge is a necessary precondition” for the development of teachers’ professional noticing (Blömeke & Kaiser, 2017 , p. 1796). Previous studies indeed support that teacher noticing and teacher knowledge can be empirically separated (e.g., Blömeke et al., 2015 ; Copur-Gencturk &Tolar, 2022 ). Moreover, recent empirical evidence further demonstrates that various types of teachers’ knowledge—such as general pedagogical knowledge (GPK), mathematics content knowledge (MCK), and mathematical pedagogical content knowledge (MPCK)—correlate with teacher noticing to varying degrees (Dreher & Kuntze, 2015 ; König et al., 2014 ; Meschede et al., 2017 ; Yang et al., 2021 ). However, it has also been established that while teacher knowledge is necessary for noticing, it is not sufficient on its own (Callejo & Zaptera, 2017; Yang et al., 2021 ). Instances have arisen where teachers do not automatically enact their knowledge during noticing, leading to a weak or negligible association between teacher knowledge and noticing (Sánchez-Matamoros et al., 2019 ;Steinwachs & Martens, 2024). Additionally, teachers’ beliefs have been suggested as another critical influence, functioning jointly with teacher knowledge to facilitate noticing (Schoenfeld, 2011 ). Indeed, teacher beliefs have been considered as “filters” for teacher noticing (e.g., Lee & Francis, 2018 ; Roose et al., 2019 ). However, the relationship between teacher noticing and beliefs remains “underexplored” (Weyers et al., 2024 , p. 259). More importantly, currently, the majority of existing studies tend to examine the relationships between teacher knowledge and noticing or between teacher beliefs and noticing separately. Very few investigations have integrated these three constructs into a single framework, with only a few exceptions (e.g., Hoth et al., 2022 ; Zeeb et al., 2023 ). In teacher education, however, beliefs have been recognized as critical mediating factors between teacher knowledge and behaviors. Similarly, past research suggests that the effects of teacher knowledge on noticing may be “mediated by beliefs” (Hoth et al., 2022 ). Therefore, a systematic exploration of the interplay between teacher knowledge, beliefs, and professional noticing is necessary for a comprehensive understanding of how knowledge and beliefs function jointly to influence teacher noticing. Furthermore, teaching experience has been identified as a significant factor affecting teacher noticing. The differences in noticing between expert and novice teachers often stem from how teachers with different experiences utilize their knowledge (Bastian et al., 2024 ). Hence, it is reasonable to conjecture that the relationships among teacher knowledge, beliefs, and noticing differ between pre-service and in-service teachers (Schoenfeld, 2011 ). Therefore, a focused investigation into these relationships among pre-service teachers or in-service teachers separately is essential to gain insights specific to this group. Additionally, similar to knowledge and beliefs, teacher noticing is increasingly recognized and commonly accepted as a socially and culturally shaped construct (Louie, 2018). However, most available studies have concentrated on contexts within English-speaking or Western nations (Blömeke & Kaiser, 2017 ). Therefore, research encompassing diverse social and cultural contexts is vital for achieving a holistic understanding of the relationship between the three constructs. In light of these considerations, the present study aims to investigate the relationship between Chinese pre-service mathematics teachers’ professional knowledge (including MCK and MPCK), pedagogical beliefs, and their professional noticing. Specifically, this study seeks to understand how these constructs interact and influence each other within the context of Chinese mathematics education. The findings will provide empirical evidence for a deeper understanding of the interrelations between various aspects of teacher knowledge, beliefs, and professional noticing, particularly from a non-Western social and cultural perspective. Therefore, this study not only addresses existing gaps in the literature but also contributes to the development of targeted professional development programs that enhance teacher noticing within diverse educational contexts. 2. Literature Review, Theoretical Framework and Research Questions 2.1 Mathematics teacher knowledge Teacher knowledge is widely regarded as a multifaceted construct essential to effective teaching and learning (Ball et al., 2008; Shulman, 1987). Researchers have proposed various frameworks to classify components of teacher knowledge basing on the seminal work of Shulman (1986, 1987). Building on this foundation, the Teacher Education and Development Study in Mathematics (TEDS-M) classifies teacher knowledge into three key domains: mathematics content knowledge (MCK), mathematics pedagogical content knowledge (MPCK), and general pedagogical knowledge (GPK) (Tatto et al., 2008). These domains are recognized as critical components of professional competence, significantly influencing instructional quality and student outcomes (Ball et al., 2008; König et al., 2014). This study focuses on MCK and MPCK within the context of Chinese pre-service mathematics teacher education, adopting the TEDS-M framework to guide the analysis. Content knowledge mainly refers to knowledge of the subject and its organizing structure teachers are required to teach (Shulman, 1986). Likewise, Mathematics Content Knowledge (MCK) refers to teachers’ understanding of mathematical concepts, principles, and structures, enabling them to effectively communicate the subject matter (Shulman, 1986; Blömeke & Delaney, 2012). In the context of TEDS-M, three domains of mathematics teachers’ mathematical cognitive skills were specified, namely knowing, applying and reasoning (Tatto et al., 2008). The sub-domain knowing mainly covers teachers’ abilities to recall definitions and properties, recognize mathematical objects, retrieve information from given sources such as graphs and tables, and use measuring instruments. The sub-domain applying refers to teachers’ abilities such as select appropriate methods, represent mathematical information, and generate appropriate model. Finally, the sub-domain reasoning includes teachers’ abilities to prove and reason mathematically, analyze and characterize mathematical relations, and make necessary generalization (Tatto et al., 2008). Pedagogical content knowledge (PCK) refers to subject-specific knowledge for the purpose of teaching so as to make subject matter accessible to students (Shulman, 1986). MPCK is central to bridging the gap between knowing mathematics and teaching it effectively, pertaining to the specialized knowledge required to teach mathematics effectively (Depaepe et al., 2013). In the TEDS-M project, the following two sub-domains of MPCK were differentiated: curricular knowledge and knowledge of planning for mathematics teaching and learning, and knowledge of enacting mathematics for teaching and learning (Tatto et al., 2008). The former mainly refers to knowledge at the pre-active stage, such as establish appropriate learning goals, see connections within the curriculum, plan appropriate activities and methods, and identify approaches for problem solving. The later however refers to knowledge at the interactive stage, including knowledge such as analyze and evaluate students’ mathematical solutions and arguments, provide appropriate feedback, and analyze and diagnose students’ questions (Döhrmann et al., 2012; Tatto et al., 2008). 2.2 Mathematics teachers’ beliefs about mathematics teaching and learning Teachers’ beliefs are among the most extensively studied topics in mathematics teacher education. A common understanding defines beliefs as “psychologically held understandings, premises, or propositions about the world that are thought to be true” (Philipp, 2007, p. 259). Similar to teacher knowledge, beliefs are considered multifaceted (Ernest, 1989). In terms of mathematics teacher’s beliefs, the critical components include beliefs about the nature of mathematics and beliefs about mathematics teaching and learning (Ernest 1989; Thompson,1992). Teachers’ beliefs about mathematics teaching and learning, that is, their pedagogical beliefs, refer to teachers’ views on their preferred ways of mathematics teaching and learning, for example, their conceptions of ideal classroom teaching activities, what behaviors and mental activities are involved in mathematics learning, and what constitutes appropriate and prototypical mathematics learning activities (Chan & Elliott 2004; Ernest 1989; Thompson 1992). Generally speaking, the literature has identified two typical views of mathematics learning and teaching: a knowledge transmission (or “traditional”) view and a constructivist view (Blömeke & Kaiser, 2014). Similarly, in the TEDS-M framework, teachers’ beliefs about mathematics teaching and learning were mainly differentiated between two views on mathematics teaching and learning: 1) a knowledge transmission (or “traditional”) view, where mathematics teaching is seen as a process of knowledge transmission and students receive knowledge from teachers passively, and 2) a constructivist view, where mathematics teaching is seen as facilitating students’ knowledge construction (Blömeke & Kaiser 2014; Tatto et al. 2008). 2.3 Mathematics teacher professional noticing Teachers’ noticing is widely recognized as a key element of teaching expertise, especially in mathematics (Sherin et al., 2011). However, there has been no commonly accepted definition for teacher noticing and teacher noticing has been defined from various perspectives such as cognitive-psychological perspective and socio-cultural perspective (see König et al., 2022 for a detailed discussion). A majority of previous studies referred in their theoretical frameworks to Goodwin’s (1994) concept of professional vision as the theoretical underpinning, which emphasizes that noticing is shaped by social contexts and cultural backgrounds. A few available cross-cultural comparative studies indeed identified different cultural patterns of noticing among teachers from different cultural contexts such as China, Germany and the US (e.g., Ding e al., 2022; Yang et al., 2019). From the cognitive-psychological perspective, teacher noticing has been generally conceptualized as a set of interrelated and cyclical mental processes, including three components: identifying key instructional elements, connecting specific events, and applying contextual knowledge to reasoning (Sherin & van Es, 2009; Sherin et al., 2011). Similarly, Jacobs et al. (2010) refined the definition and proposed that teachers’ professional noticing of students’ mathematics thinking includes: (a) attending to students’ strategies; (b) interpreting students’ understanding; (c) deciding how to respond on the basis of students’ understanding. In follow-up studies of TEDS-M like TEDS-FU and TEDS-Instruct in Germany, teacher noticing was conceptualized as an action-oriented construct (Kaiser et al., 2015; 2017). In the theoretical framework developed in these follow-up studies, teacher noticing was defined as “Perception, Interpretation, and Decision-making” and called this approach the PID model (Kaiser et al., 2015; Kaiser et al., 2017). This model consists of perceiving instructional events, interpreting these actions, and making decisions about student responses or instructional strategies at a broad level which does not limit teachers’ noticing to a specific aspect of teachers’ work. It encompasses all facets of quality mathematics teaching, including pedagogical design, cognitive activation, learning support, and classroom management. The PID model differentiates professional noticing into two domains: general pedagogy (P_PID) and mathematics instruction (M_PID). This study employs the TEDS-Instruct framework to analyze teacher noticing within the Chinese context. 2.4 The relationship between teacher knowledge, beliefs, and noticing There is consensus that teacher knowledge and teacher noticing are theoretically connected yet distinct constructs (Blömeke et al., 2015; Schoenfeld, 2011). Recent empirical studies support the notion that teacher knowledge and teacher noticing are two distinct yet interconnected constructs (Blömeke et al., 2015; Copur-Gencturk & Tolar, 2022). Moreover, recent studies also revealed a few patterns in how various types of teacher knowledge impact noticing. First, different forms of teacher knowledge, such as MCK and MPCK, influence teacher noticing in distinct ways. For instance, Dunekacke et al. (2016) found that in Germany, pre-service preschool teachers’ MPCK significantly predicted their perception, while MCK did not. Similarly, Yang et al. (2021) reported stronger associations between pedagogy-related knowledge (e.g., GPK and MPCK) and noticing among Chinese in-service teachers compared to MCK. Second, teacher knowledge relates more strongly to the interpretation and decision-making aspects of noticing than to perception. For example, Sánchez-Matamoros et al. (2019) observed that pre-service mathematics teachers’ abilities to interpret student understanding and make instructional decisions were influenced by their mathematical knowledge, while their perception of critical incidents was less affected. Similarly, Yang et al. (2019) revealed stronger correlations between GPK, MPCK, and the “interpretation and decision-making” facets of noticing compared to the “perception” facet in a study of Chinese in-service teachers, Additionally, teaching experience appears to moderate the relationship between teacher knowledge and noticing. For example, Dreher and Kuntze (2015) found a weak correlation between pre-service teachers’ MCK and noticing, which diminished for in-service teachers, while MPCK showed strong correlations for in-service but not pre-service teachers. Moreover, the strength of these associations varies across social and cultural contexts. For instance, Yang et al. (2021) observed weaker correlations between knowledge and noticing among Chinese in-service teachers compared to findings in Western cultures, suggesting that cultural and contextual factors may influence these relationships. Concerning the relationship between teacher beliefs and noticing, it has also been argued in literature that teachers’ noticing is “intimately tied to their orientations (including beliefs)” (Schoenfeld, 2011, p. 231). Indeed, in previous research in teacher education, teachers’ beliefs have been argued to act as an important filter for teachers’ decision-making and teaching behaviors as well (Pajares, 1992). Recently, besides within mathematics education field, empirical efforts have been made by researchers from various fields such as inclusive education (Keppens et al., 2021) and science education (Steinwachs & Martens, 2024) to examine the relationship between teacher beliefs and teacher noticing. In the previous studies, it was first found that different types of beliefs tend to influence or correlate to teacher noticing quite differently. For example, Eßling et al. (2023) observed that pre-service science teachers with stronger transmissive beliefs scored lower for their professional vision of instructional support. Similarly, Dunekacke et al. (2016) reported that application-oriented mathematical beliefs, but not static or process-oriented beliefs, significantly predicted preschool teachers’ noticing. Moreover, it has been argued in literature that cultural context will further moderate the association between teacher beliefs and teacher noticing (Eßling et al., 2023; Santagata & Yeh, 2016). For example, in a recent comparative study, it was found that the American mathematics teachers’ noticing aligned more closely with their individual pedagogical beliefs than their perceptions of general cultural pedagogical beliefs, by contrast, the Chinese mathematics teachers’ noticing was closer to their perception of cultural pedagogical beliefs other than their own pedagogical beliefs (Zhu, 2023). Furthermore, in the past years, a few studies have examined the interplay between teacher knowledge, beliefs, and noticing. Findings in these studies suggest a tendency that compared with teacher beliefs, teacher knowledge (or at least one type of knowledge) seems to act as a much stronger influence to teacher noticing. For example, Meschede et al. (2017) found that both pre-service and in-service teachers’ MPCK correlated more strongly with professional vision than their transmissive or constructivist beliefs. Similarly, Zeeb et al. (2023) reported that teachers’ declarative knowledge about learners’ growth mindset, but not their beliefs about growth mindset, was significantly associated with noticing. However, findings in mathematics education are not entirely consistent. For example, Dunekacke et al. (2016) found that even though both application orientation beliefs and MPCK can predict teacher noticing, MPCK has a much stronger prediction power. Notably, MCK had no direct effect on noticing, but only acted as a predictor of MPCK. Interestingly, Hoth et al. (2022) found that only teachers’ beliefs, not teachers’ MCK and MPCK will not significantly predict teacher noticing. These inconsistencies highlight the need for further research, particularly in diverse cultural contexts. 2.4 Research questions As reviewed above, the literature increasingly recognizes that teacher knowledge is not the sole cognitive construct influencing teacher noticing. Other factors, such as teachers’ goals, orientations, and beliefs, play significant roles in shaping their noticing. While prior research has explored the relationships among teacher knowledge, beliefs, and noticing, studies situated in the Chinese context—characterized by its unique cultural and educational traditions in mathematics—remain scarce. Importantly, existing research suggests that the influence of teacher knowledge and beliefs on noticing is moderated by social and cultural contexts (e.g., Santagata & Yeh, 2016). Additionally, teaching experience has been shown to significantly shape the relationships among these constructs (Bastian et al., 2024; Dreher & Kuntze, 2015). Additionally, while a small number of available studies have examined the interplay between teacher knowledge, beliefs, and noticing, the mediating role of beliefs in the relationship between teacher knowledge and teacher noticing has yet to be explicitly addressed. This gap highlights the need for a more comprehensive investigation into these constructs and their interconnections, particularly within culturally specific contexts such as China. To address these gaps, the present study aims to explore the following research questions: 1. What are the relationships between pre-service teachers’ knowledge, pedagogical beliefs, and their noticing at a general level, and how do these relationships vary across the sub-facets of the three constructs? 2. What mediating role do pedagogical beliefs play in the relationship between teacher knowledge and teacher noticing? 3. Methodology 3.1 Research context and participants of the study In Mainland China, pre-service secondary school teachers are predominantly education in a specific type of university, so-called normal university, over a four-year period. Currently, secondary school teacher programs are dominated by a concurrent curriculum model of teacher preparation and still specialized and discipline-based in Mainland China (Wu et al., 2017). That is, the majority of pre-service teachers at secondary school level are trained in a specific academic subject department where the studies of a specific discipline and pedagogy are integrated and taught at the same time within four years (Musset, 2010; Paine, 1990). Normally, the first two years of pre-service training in university will mainly focus on the learning of subject content at tertiary-level and general education content. In the third year, subject-related educational courses will start to be offered but only occupy a very small portion of teaching time (around 10% to 30% of the total credits) (Wu & Huang, 2018). In addition, pre-service teachers are required to participate in a one-semester teaching practicum in school within their pre-service training period. For the present study, 584 pre-service secondary school mathematics teachers in their fourth year of training were selected. Participants included 432 pre-service teachers from one national-level normal university, 99 trainees from two province-level normal universities, and 52 trainees from one local-level normal university. All participants had completed an 18-week teaching practicum. Approximately 40% undertook their practicum in junior secondary schools (Grades 7–9), while the remaining 60% completed it in senior secondary schools (Grades 10–12). 3.2 Instruments 3.2.1 MCK and MPCK The instruments developed in TEDS-M to measure pre-service secondary school mathematics teachers’ MCK and MPCK were used in the study. Based on the consensus of IEA, the traditional Chinese version of the instruments used in Taiwan were slightly modified to make a few expressions closer to the situation in Mainland China and then were directly employed to test pre-service mathematics teachers’ MCK and MPCK (Tatto et al., 2008). In total, the two knowledge instruments comprised 103 items, 76 of which were MCK items and 27 of which were MPCK items. In TEDS-M, those items were allocated to three booklets using a balanced-incomplete-block design to capture adequate domain coverage of teacher knowledge within a reasonable administration time (Blömeke et al., 2014, Tatto et al., 2008). Items assessing MCK covered four content areas, namely number, geometry, algebra and data and were further classified into three cognitive dimensions, namely knowing, applying, and reasoning. The MPCK items included aspects of curricular and planning knowledge and knowledge about how to enact mathematics in the classroom. The majority of the items in both MCK and MPCK tests were complex multiple-choice items with a few multiple choice and open response items. The participants were given 60 minutes to complete both the MCK and MPCK test in paper-and-pencil format. Item examples can be found in Blömeke et al. (2013). 3.2.2 Beliefs about mathematics learning The original Chinese version of the questionnaires designed in TEDS-M to investigate pre-service mathematics teachers’ beliefs about learning mathematics was used in the present study. The beliefs about the learning of mathematics had two dimensions: learning mathematics through following teacher direction (or labeled as “transmissive view” in the study, 6 items sample item like “Pupils need to be taught exact procedures for solving mathematical problems”.), and learning mathematics through active involvement (or labeled as “constructivist beliefs”, 6 items, sample items like “In addition to getting the right answer, it is important to understand why the answer is correct”). Pre-service teachers had to rate the beliefs statements on a five-point Likert scale (1= “strongly disagree” and 5= “strongly agree”). To validate the questionnaire for the purpose of the present study, the CFA showed that the two-factor model with 12 items had a good fit . 3.2.3 Noticing Our previous validated instruments adapted from TEDS-Follow-Up and TEDS-Instruct/Validate (for adaptation details, see Yang et al., 2018, 2019) were used in the study to examine pre-service mathematics teachers’ professional noticing skills. These assessments include three video vignettes designed to evaluate German secondary school mathematics teachers' professional noticing, focusing on aspects of mathematics instruction (M_PID) and general pedagogical issues (P_PID). The items assess three facets of noticing: perception, interpretation, and decision-making, encompassing almost all aspects of a mathematics lesson. To meet the research aim of the present study, we regrouped the items into two categories: 1) Perception, including mathematics instruction and general pedagogy-related perception (labeled as M_Perception and P_Perception respectively); and 2) Interpretation & Decision-making, including mathematics instruction and general pedagogy-related interpretation and decision-making (labeled as M_Interpretation&Decision-making and P_Interpretation&Decision-making respectively). After participants watched each of the three videos, they were given around 15 minutes to answer a few items related to each video (in total not exceeding 60 minutes for the three videos). There are two types of items for this part: 1) 38 items based on Likert scales (four choices from “fully correct” to “not correct”) to assess participants’ noticing mainly related to the perception facet; 2) 36 constructed-response items to assess the participants’ noticing related to interpretation and decision-making facets. In addition, an extensive coding manual was developed to evaluate participants’ answers. Item examples can be found in Kaiser et al. (2015). Although the instrument was originally developed for in-service teachers, a further study could be shown that the instrument can be used for pre-service teachers as well (Weyers et al., 2023). 3.3 Scaling and data analysis The data analysis comprised the following steps. First, the open response items of the MCK and MPCK test and Noticing were coded according to the coding rubrics of the coding manual developed in TEDS-M and TEDS-FU. Four trained postgraduate students in mathematics education worked as independent raters and first coded 80 of the questionnaires together. Good values of Cohen’s Kappa were reached (k>0.78 and K average = 0.86). For all the open response items, items without answers or with incorrect answers were scored 0, and correct answers were scored 1. After the completion of coding, the relative item difficulties for a one-parameter (Rasch model) item response theory (IRT) model were calculated separately on the six dimensions of knowledge, beliefs and noticing. Items with extreme difficulty were removed for the final analysis, as they do not substantially contribute to the measurement of the construct due to weak discrimination (Bond & Fox, 2007). After this, the internal consistency of the remaining items in each of the dimensions and sub-dimensions related was estimated using Cronbach’s alpha reliability coefficient. Most of the reliability scores were above 0.6 indicating acceptable or good reliability. After this, scale scores were created for each of the six dimensions and their sub-dimensions by applying the one-dimensional Rasch model. The second step of the data analysis was to systematically examine the relationship between teacher knowledge, pedagogical beliefs, and noticing. First, using estimated scores for each dimension of teacher knowledge, beliefs and noticing as manifest variables, Pearson correlational analyses were conducted between pre-service mathematics teachers’ knowledge (MCK and MPCK), beliefs (transmissive and constructivist), and noticing (Perception and Interpretation & Decision-making). Afterwards, with the use of estimated scores for each sub-component or sub-aspect of teacher knowledge, beliefs, and noticing as manifest variables, Pearson correlational analyses were again conducted between them. MCK was differentiated according to its cognitive domains: knowing, applying, and reasoning. Finally, path analyses were conducted to examine the mediation effect of beliefs between knowledge on Perception and Interpretation & Decision-making respectively. Mplus was used for the path analyses. 4. Results 4.1 Correlation Analysis To examine how each of the components of knowledge and beliefs were related to each aspect of teacher noticing, Pearson’s r based on the estimated scores of the six aspects was estimated. Table 1 summarizes the results. Table 1 Perception Interpretation & Decision-Making MCK 0.161*** 0.239*** MPCK 0.033 0.135*** Transmissive -0.249*** -0.163*** Constructivist 0.298*** 0.182*** Correlations between teacher knowledge, beliefs and teacher noticing (n = 583) Note: *p < 0.05, **p < 0.01, ***p < 0.001. As illustrated in Table 1 , regarding to the relationship between knowledge and noticing, first of all, statistically significant positive correlations were identified between MCK and Perception (r = 0.161) and Interpretation & Decision-Making (r = 0.239). By contrast, MPCK was only significantly positively related with Interpretation & Decision-Making (r = 0.135). Moreover, out of our expectation, comparatively speaking, MCK was much more strongly related to both Perception and Interpretation & Decision-Making. However, as expected, both MCK and MPCK were found to be much more strongly related to Interpretation & Decision-Making than to Perception. With respect to the relationship between beliefs and noticing, transmissive beliefs were found to significantly but negatively relate to Perception and Interpretation & Decision-Making. By contrast, Constructivist beliefs were found to significantly and positively relate to Perception and Interpretation & Decision-Making. Moreover, both beliefs were found to relatively more strongly relate to Perception. In addition, compared with MCK and MPCK, pedagogical beliefs were found to much more strongly relate with Perception. But for the process of Interpretation & Decision-Making, compared with pedagogical beliefs, MCK was relatively more strongly, but MPCK was relatively weaklier related to Interpretation & Decision-Making. To further examine how each of the sub-domains of mathematics teacher knowledge and beliefs were related to the sub-facets of noticing, Pearson’s r based on the estimated scores of the nine sub-domains of teacher knowledge and beliefs and the four sub-facets of teacher noticing was estimated. Table 2 summarizes the results. As illustrated in Table 2 , the relationships between the five sub-domains of teacher knowledge and the four sub-facets of teacher noticing were all quite weak, suggesting that different sub-domains of teacher knowledge are a factor influencing different phases of teacher noticing with various effect sizes, but not in a very strong manner. Again, transmissive beliefs were found to relate to all facets negatively, but constructivist beliefs were found to be positively related with the four sub-facts of teacher noticing. Table 2 Correlations between Sub-domains of Knowledge, beliefs and noticing (Pearson’s r; n = 583) Perception Interpretation & Decision-Making P_Perception M_Perception P_Interpretation &Decision-Making M_Interpretation & Decision-Making MCK Knowing 0.068 0.088 * 0.073 0.126** Applying 0.140** 0.153*** 0.113** 0.232*** Reasoning 0.170*** 0.101** 0.073 0.160*** MPCK Curriculum -0.029 -0.032 0.024 0.039 Planning and enacting 0.025 0.039 0.017 0.093* Beliefs Transmissive -0.224*** -0.221*** -0.122 -0.116** Constructivist 0.256*** 0.288*** 0.141** 0.127** Note: *p < 0.05, **p < 0.01, ***p < 0.001. Moreover, as shown in Table 2 , for the relationship between MCK and teacher noticing, relatively stronger relationships could be observed between the higher cognitive understanding of mathematics such as Applying and Reasoning and the four sub-facets of teacher noticing. In addition, the relationship between MCK and the sub-facet of M_Interpretation & Decision-Making is relatively stronger than the relationships between MCK with other sub-facts of teacher noticing. However, again, both of the sub-domains of MPCK were found to be quite weakly correlated with the four sub-facts of teacher noticing except for a statistically significant positive correlation between “Planning and enacting” and “M_Interpretation & Decision-Making”. Very interestingly, very weak but negative correlations were identified between “Curriculum” and “P_Perception” and “M_Perception”. 4.2 Path analysis In order to examine the mediating role played by pedagogical beliefs between teacher knowledge and teacher noticing, a path analysis was further conducted. As being reviewed above, because MCK has been generally accepted as a prediction for MPCK and both knowledge and beliefs will influence the process of perception and interpretation and decision making differently, we first performed a path analysis to examine the relationship between knowledge, beliefs and perception. The path model fits the data well \(\:({\chi\:}^{2}\:=\:6.485,\:{\chi\:}^{2}/df\:=\:1.62,\:CFI\:=\:.996,\:TLI\:=\:.985,\:RMSEA\:=\:.033)\) and Fig. 1 represents the overall results . As shown in Fig. 1 , the knowledge domains of MCK and MPCK could not significantly predict the process of teacher noticing of Perception. However, regarding pedagogical beliefs, Constructivist beliefs could significantly positively predict Perception, while Transmissive beliefs could significantly negatively predict Perception. Moreover, both Constructivist and Transmissive beliefs were found to play a significant mediation role between MPCK and Perception. Moreover, to specifically examine the relationship between teacher knowledge, pedagogical beliefs and the process of Interpretation & Decision-Making, a second path analysis was performed. The path model fits the data well \(\:({\chi\:}^{2}\:=7.421,\:{\chi\:}^{2}/df\:=\:1.48,\:CFI\:=\:.995,\:TLI\:=\:.986,\:RMSEA\:=\:.029)\) and Fig. 2 represents the overall results. As shown in Fig. 2 , MCK was found to be able to significantly positively predict teacher noticing of Interpretation & Decision Making. By contrast, the relation between teachers’ MPCK and their Interpretation &Decision Making was again found to be not significant and even negative (β=-0.005). In addition, for pedagogical beliefs, only constructivist beliefs were found to be able to significantly and positively predict pre-service mathematics teachers’ Interpretation & Decision-Making. Moreover, Constructivist beliefs could further mediate the relationship between MPCK and Interpretation & Decision-Making. 5. Summary and Discussion The primary aim of this study was to systematically investigate the relationship between pre-service mathematics teachers’ knowledge (including MCK and MPCK), pedagogical beliefs, and teacher noticing, focusing on the mediating role of pedagogical beliefs between teacher knowledge and teacher noticing. First, out of our expectation, both correlation and path analysis results illustrate that it was Chinese pre-service mathematics teachers’ MCK, not theoretically assumed MPCK, exhibited a stronger influence on teacher noticing. This finding diverges from prior research, which has often identified MPCK as having a more robust relationship with teacher noticing (Dunekacke et al., 2015), even within the Chinese context (Yang et al., 2021 ). Moreover, while the correlation coefficients between MCK, MPCK (including their sub-components), and noticing (including its sub-facts) are generally positive, the overall relationships between Chinese pre-service mathematics teachers’ knowledge and noticing are weaker than the strong empirical relationships identified in some Western contexts (e.g., Meschede et al., 2017 ). These differences may indicate that factors from specific societal and cultural contexts and differences of teaching experience will significantly influence or moderate the relationship between teacher knowledge and noticing. The differing backgrounds and experiences of the participants should be taken into account to provide explanations for the unexpected results, which are in parts inconsistent to the results of other studies. In most previous studies, the participants were all in-service teachers (Yang et al., 2021 ). Consequently, after a number of years of teaching practice, these in-service teachers developed MPCK as usable knowledge that they could readily access to address complex classroom situations, particularly when analyzing observed teaching events and making subsequent decisions (Kersting et al., 2012 ). In contrast, pre-service teachers do not automatically use their knowledge such as mathematical knowledge for teaching to interpret or analyze what they notice (Spitzer & Phelps-Gregory, 2024 ). Similarly, although all participants in the present study had one semester of teaching practicum experience in schools, it may still have presented a challenge for them to flexibly apply their mathematics pedagogical content knowledge to inform their interpretations of what they noticed. Additionally, the tradition of pre-service teacher education within the Chinese context may also influence the weak association between pre-service mathematics teachers’ MPCK and noticing. Traditionally, teacher education in Mainland China has followed an academically oriented preparation model (Liao & Hu, 2017 ; Paine, 1990 ). As a result, most of the education time is spent on content knowledge learning, limiting opportunities for courses related to pedagogical content knowledge (Wu & Huang, 2018 ). Moreover, “direct lecture” is the predominant teaching method during pre-service teacher training, restricting opportunities to link theoretical content with practical teaching applications (Wu et al., 2017 ). Consequently, it is understandable that for Chinese pre-service mathematics teachers, it is challenging to connect their academic knowledge with their classroom observations, which are inherently more context-specific. Furthermore, it was found that Chinese pre-service mathematics teachers did not perceive their school-based teaching practicum experience as beneficial in connecting theory to practice or enhancing their university learning (Yang et al., in press). Therefore, the weak connections between pre-service teachers’ academic knowledge and their practical mathematics teaching experiences likely have a significant impact on the relationship between MPCK and teacher noticing. In contrast, Chinese pre-service teachers generally possess a solid mathematics subject background and have ample opportunities during their training to deepen their cognitive understanding of mathematics. Thus, MCK becomes more accessible for them, which further strengthens the relationship between MCK and teacher noticing. Indeed, this study identified a closer relationship between a higher level of understanding of mathematics and various facets of teacher noticing. Second, regarding to the relationship between teacher beliefs and noticing, the findings indicate that Chinese pre-service mathematics teachers’ transmissive beliefs negatively correlate with their noticing, while constructivist beliefs positively correlate with their noticing. These results are generally consistent with findings from previous studies conducted in diverse cultural contexts (e.g., Eßling et al., 2023 ; Keppens et al., 2021 ; Meschede et al., 2017 ). In earlier studies, constructivist beliefs were similarly found to exert a significant and positive influence on teachers’ noticing, whereas transmissive beliefs were generally associated with a strong negative relationship. These findings suggest that, across various contexts, teachers’ beliefs indeed serve as a primary influence on teacher noticing, as theorized previously (Schoenfeld, 2011 ). However, the strength of the relationships between pre-service mathematics teachers’ beliefs and their noticing identified in this study appears to be comparatively weaker than in contexts such as Germany (e.g., Hoth et al., 2022 ; Meschede et al., 2017 ). These differences may again stem from the variations in societal and cultural settings. As reviewed above, Zhu (2023) found that for Chinese teachers, their noticing is significantly influenced by their perceptions of cultural pedagogical beliefs rather than solely by their own pedagogical beliefs. Thus, the findings of this study may further suggest that, similar to teacher knowledge, social and cultural norms significantly influence the enactment of teacher beliefs during noticing (Eßling et al., 2023 ; Santagata & Yeh, 2016 ). Third, both correlation and path analysis results indicate that the knowledge and pedagogical beliefs of Chinese pre-service mathematics teachers influence various sub-facets of teacher noticing in distinct ways. In general, the relationship between teacher knowledge and noticing is characterized by a stronger association between MCK and MPCK and the sub-fact of interpretation and decision-making. These results are congruent with previous research findings (e.g., König et al., 2014 ; Sánchez-Matamoros et al., 2019 ; Yang et al., 2021 ). This supports the theoretical distinction between different facets of teacher noticing, positing that interpretation and decision-making are more knowledge-driven facets of noticing (Bastian et al., 2024 , Sherin et al., 2011 ). Specifically, the phase of attending will be easier than, or only a prerequisite to, the facet of interpretation and decision-making (Sánchez-Matamoros et al., 2019 ). Thus, teacher knowledge tends to play varied roles within the different facets of noticing. Conversely, it was found that the pedagogical beliefs of pre-service mathematics teachers were more closely related to the facet of perception rather than to interpretation and decision-making. Furthermore, both correlation and path analysis results indicate that the strength of the relationship between pedagogical beliefs and the perception sub-fact is significantly stronger than the relationship between MCK and MPCK and perception. These findings suggest a theoretical differentiation between the perception facet and the afterwards following facet of interpretation and decision-making phases. Comparatively speaking, in complex classroom environments, a teacher’s decision to attend to or disregard specific classroom events is often an intuitive action (Sherin & Star, 2011 ). Consequently, the perception facet is more significantly influenced by teachers’ orientations, goals, and beliefs (Schoenfeld, 2011 ). Although both beliefs and knowledge inform teachers’ interpretation and decision-making, beliefs often serve as “filters” for what teachers initially perceive (Lee & Francis, 2018 ; Roose et al., 2019 ). Fourth, a primary objective of the present study was to examine the mediating role of pedagogical beliefs between teacher knowledge and teacher noticing. Path analysis results corroborate our hypothesis that both transmissive and constructivist beliefs function as mediators between teacher knowledge and noticing. However, the study also found that different types of pedagogical beliefs play distinct mediation roles in relation to the different facets of teacher noticing. Constructivist beliefs were identified as a much stronger positive mediator between teacher knowledge and both perception and interpretation & decision-making. In contrast, transmissive beliefs were found to play a negative mediating role between teacher knowledge and the perception phase of noticing. These findings are consistent with previous research, which has indicated that constructivist beliefs have a more substantial impact on mediating the relationship between teachers' knowledge and their instructional practices (Leder et al., 2002 ; Yang et al., 2020). Furthermore, the results illustrate that teacher beliefs significantly influence their ability to effectively employ their knowledge in the process of noticing. 6. Conclusions and limitations Although it has been theoretically proposed and empirically observed that teacher knowledge and beliefs have a fundamental impact on teachers’ noticing, there remains limited empirical evidence to substantiate the joint effect of teacher knowledge and beliefs, particularly regarding the mediating role of beliefs within a non-Western context. This study aimed to address this gap using standardized testing instruments. The findings reported above first provide empirical evidence that teacher knowledge, beliefs, and noticing are empirically separated constructs. Moreover, findings of the study further suggest that the differences of social and cultural contexts and teaching experience exert significant influence to the relationship between these three constructs. That is, social and cultural norms, along with teaching experience, may play critical roles in moderating the effects of teacher knowledge and beliefs on teacher noticing. While this study contributes to the understanding of the relationship between mathematics teacher knowledge, beliefs, and noticing, particularly within the Chinese context, several limitations should be acknowledged. First, although the study involved a relatively large sample of participants, these participants were selected from only four teacher education universities. As such, the sample may not be sufficiently representative to capture the diversity of pre-service mathematics teachers’ knowledge, beliefs, and noticing across China. Future studies could address this limitation by including a larger and more diverse sample, encompassing pre-service teachers from additional institutions, particularly those at provincial and local levels. Second, the study assessed teachers’ knowledge, beliefs, and noticing in a general manner without focusing on specific branches of mathematics (e.g., geometry or algebra) or particular mathematical topics (e.g., fractions or equations). This general approach may limit the applicability of the findings to specific instructional contexts. Future research could explore these relationships within the context of specific mathematical domains or topics. Additionally, incorporating qualitative methods, such as interviews, could enrich the findings and provide deeper insights into the nuances of teacher noticing. Third, the study employed comprehensive instruments, including the adapted MCK and MPCK tests from TEDS-M and the noticing test from TEDS-FU, which required approximately three hours for each participant to complete, alongside a beliefs survey. This extensive testing burden may have impacted participant engagement and the overall validity of the results. To address this issue, future studies could consider using shorter versions of the assessment instruments or administering the tests across two separate time points. Such approaches could help reduce the testing burden and enhance the validity and reliability of the findings. By addressing these limitations, future research can build upon the current study to deepen our understanding of the complex relationships among teacher knowledge, pedagogical beliefs, and teacher noticing, thereby providing more robust and contextually relevant insights. Declarations Ethics statement: The relevant committees at Southwest University responsible for ethical clearance have approved the study and confirmed that it complies with the ethical standards set by these committees. Internal note: Roma OK'd Declaration of conflict of interest The authors declare no conflict of interest. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5925262","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":408712110,"identity":"8224eda4-fc32-44df-946c-0be384ddd3fb","order_by":0,"name":"Xinrong Yang","email":"","orcid":"https://orcid.org/0000-0003-3520-898X","institution":"University of Macau","correspondingAuthor":false,"prefix":"","firstName":"Xinrong","middleName":"","lastName":"Yang","suffix":""},{"id":408712111,"identity":"b28210cd-ddc0-4fb4-b412-adfa1a338a29","order_by":1,"name":"Jun Deng","email":"","orcid":"","institution":"University of Macau","correspondingAuthor":false,"prefix":"","firstName":"Jun","middleName":"","lastName":"Deng","suffix":""},{"id":408712112,"identity":"64544b71-8697-487d-a46e-9764d451db5e","order_by":2,"name":"Johannes König","email":"","orcid":"https://orcid.org/0000-0003-3374-9408","institution":"University of Cologne","correspondingAuthor":false,"prefix":"","firstName":"Johannes","middleName":"","lastName":"König","suffix":""},{"id":408712113,"identity":"37d2d2c9-11c8-46e9-be9e-1c87aedb8eae","order_by":3,"name":"Gabriele Kaiser","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0002-6239-0169","institution":"University of Hamburg","correspondingAuthor":true,"prefix":"","firstName":"Gabriele","middleName":"","lastName":"Kaiser","suffix":""}],"badges":[],"createdAt":"2025-01-29 15:27:54","currentVersionCode":1,"declarations":{"humanSubjects":true,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":true,"humanSubjectConsent":true,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-5925262/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5925262/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":75486050,"identity":"42cfa608-ad7c-4e1e-8f5b-f0f78a5e39ff","added_by":"auto","created_at":"2025-02-05 06:25:21","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":53453,"visible":true,"origin":"","legend":"\u003cp\u003ePath analysis results for the relationship among teacher knowledge, beliefs and perception\u003c/p\u003e\n\u003cp\u003eNotes: The dotted lines represent non-significant path coefficients\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5925262/v1/daa887e5a0c6831fe1a6b40c.png"},{"id":75485176,"identity":"46ab8391-394d-4742-b3c2-0c108ece07cf","added_by":"auto","created_at":"2025-02-05 06:17:21","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":67034,"visible":true,"origin":"","legend":"\u003cp\u003ePath analysis results for the relationship among teacher knowledge, beliefs and interpretation \u0026amp; decision-making.\u003c/p\u003e\n\u003cp\u003eNotes: The dotted lines represent non-significant path coefficients\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5925262/v1/40c3d2329b544e0b20259540.png"},{"id":75486380,"identity":"2c204429-0b4c-4087-98ee-e1fb3af95285","added_by":"auto","created_at":"2025-02-05 06:33:21","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":875366,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5925262/v1/4aea59b3-df51-4d02-9c71-89a2dfd1f959.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eRelationship between Chinese Pre-service Mathematics Teachers’ Knowledge, Pedagogical beliefs, and Noticing\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn the last two decades, the investigation of teacher noticing has emerged as a significant topic in educational research globally, particularly in the field of mathematics education (K\u0026ouml;nig et al., \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Weyers et al., \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Previous studies have generally accepted teacher noticing as one of the most critical and important professional competencies that profoundly influence the quality of mathematics instruction and students\u0026rsquo; learning outcomes (Sherin et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Given its importance, designing effective professional development programs or interventions to foster teacher noticing is therefore essential for both pre-service and in-service teacher education. However, a comprehensive understanding of the factors that shape and contribute to the development of teacher noticing should be a necessary first step. Such influences are of course complex; empirical studies have identified that teachers\u0026rsquo; teaching experience, knowledge, and beliefs are among the most critical factors affecting the development of teacher noticing (e.g., Jacobs et al., \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Weyers et al., \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eTheoretically, it has been posited that teachers\u0026rsquo; noticing is \u0026ldquo;intimately tied to\u0026rdquo; their knowledge (Schoenfeld, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2011\u003c/span\u003e, p. 231), with the assertion that \u0026ldquo;knowledge is a necessary precondition\u0026rdquo; for the development of teachers\u0026rsquo; professional noticing (Bl\u0026ouml;meke \u0026amp; Kaiser, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2017\u003c/span\u003e, p. 1796). Previous studies indeed support that teacher noticing and teacher knowledge can be empirically separated (e.g., Bl\u0026ouml;meke et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Copur-Gencturk \u0026amp;Tolar, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Moreover, recent empirical evidence further demonstrates that various types of teachers\u0026rsquo; knowledge\u0026mdash;such as general pedagogical knowledge (GPK), mathematics content knowledge (MCK), and mathematical pedagogical content knowledge (MPCK)\u0026mdash;correlate with teacher noticing to varying degrees (Dreher \u0026amp; Kuntze, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; K\u0026ouml;nig et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Meschede et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Yang et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). However, it has also been established that while teacher knowledge is necessary for noticing, it is not sufficient on its own (Callejo \u0026amp; Zaptera, 2017; Yang et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Instances have arisen where teachers do not automatically enact their knowledge during noticing, leading to a weak or negligible association between teacher knowledge and noticing (S\u0026aacute;nchez-Matamoros et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2019\u003c/span\u003e;Steinwachs \u0026amp; Martens, 2024).\u003c/p\u003e \u003cp\u003eAdditionally, teachers\u0026rsquo; beliefs have been suggested as another critical influence, functioning jointly with teacher knowledge to facilitate noticing (Schoenfeld, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Indeed, teacher beliefs have been considered as \u0026ldquo;filters\u0026rdquo; for teacher noticing (e.g., Lee \u0026amp; Francis, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Roose et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). However, the relationship between teacher noticing and beliefs remains \u0026ldquo;underexplored\u0026rdquo; (Weyers et al., \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2024\u003c/span\u003e, p. 259). More importantly, currently, the majority of existing studies tend to examine the relationships between teacher knowledge and noticing or between teacher beliefs and noticing separately. Very few investigations have integrated these three constructs into a single framework, with only a few exceptions (e.g., Hoth et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Zeeb et al., \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). In teacher education, however, beliefs have been recognized as critical mediating factors between teacher knowledge and behaviors. Similarly, past research suggests that the effects of teacher knowledge on noticing may be \u0026ldquo;mediated by beliefs\u0026rdquo; (Hoth et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Therefore, a systematic exploration of the interplay between teacher knowledge, beliefs, and professional noticing is necessary for a comprehensive understanding of how knowledge and beliefs function jointly to influence teacher noticing.\u003c/p\u003e \u003cp\u003eFurthermore, teaching experience has been identified as a significant factor affecting teacher noticing. The differences in noticing between expert and novice teachers often stem from how teachers with different experiences utilize their knowledge (Bastian et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Hence, it is reasonable to conjecture that the relationships among teacher knowledge, beliefs, and noticing differ between pre-service and in-service teachers (Schoenfeld, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Therefore, a focused investigation into these relationships among pre-service teachers or in-service teachers separately is essential to gain insights specific to this group. Additionally, similar to knowledge and beliefs, teacher noticing is increasingly recognized and commonly accepted as a socially and culturally shaped construct (Louie, 2018). However, most available studies have concentrated on contexts within English-speaking or Western nations (Bl\u0026ouml;meke \u0026amp; Kaiser, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Therefore, research encompassing diverse social and cultural contexts is vital for achieving a holistic understanding of the relationship between the three constructs.\u003c/p\u003e \u003cp\u003eIn light of these considerations, the present study aims to investigate the relationship between Chinese pre-service mathematics teachers\u0026rsquo; professional knowledge (including MCK and MPCK), pedagogical beliefs, and their professional noticing. Specifically, this study seeks to understand how these constructs interact and influence each other within the context of Chinese mathematics education. The findings will provide empirical evidence for a deeper understanding of the interrelations between various aspects of teacher knowledge, beliefs, and professional noticing, particularly from a non-Western social and cultural perspective. Therefore, this study not only addresses existing gaps in the literature but also contributes to the development of targeted professional development programs that enhance teacher noticing within diverse educational contexts.\u003c/p\u003e"},{"header":"2. Literature Review, Theoretical Framework and Research Questions","content":"\u003cp\u003e\u003cstrong\u003e2.1 Mathematics teacher knowledge \u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTeacher knowledge is widely regarded as a multifaceted construct essential to effective teaching and learning (Ball et al., 2008; Shulman, 1987). \u0026nbsp; Researchers have proposed various frameworks to classify components of teacher knowledge basing on the seminal work of Shulman (1986, 1987). Building on this foundation, the Teacher Education and Development Study in Mathematics (TEDS-M) classifies teacher knowledge into three key domains: mathematics content knowledge (MCK), mathematics pedagogical content knowledge (MPCK), and general pedagogical knowledge (GPK) (Tatto et al., 2008). These domains are recognized as critical components of professional competence, significantly influencing instructional quality and student outcomes (Ball et al., 2008; König et al., 2014). This study focuses on MCK and MPCK within the context of Chinese pre-service mathematics teacher education, adopting the TEDS-M framework to guide the analysis.\u003c/p\u003e\n\u003cp\u003eContent knowledge mainly refers to knowledge of the subject and its organizing structure teachers are required to teach (Shulman, 1986). Likewise, Mathematics Content Knowledge (MCK) refers to teachers’ understanding of mathematical concepts, principles, and structures, enabling them to effectively communicate the subject matter (Shulman, 1986; Blömeke \u0026amp; Delaney, 2012). In the context of TEDS-M, three domains of mathematics teachers’ mathematical cognitive skills were specified, namely knowing, applying and reasoning (Tatto et al., 2008). The sub-domain knowing mainly covers teachers’ abilities to recall definitions and properties, recognize mathematical objects, retrieve information from given sources such as graphs and tables, and use measuring instruments. The sub-domain applying refers to teachers’ abilities such as select appropriate methods, represent mathematical information, and generate appropriate model. Finally, the sub-domain reasoning includes teachers’ abilities to prove and reason mathematically, analyze and characterize mathematical relations, and make necessary generalization (Tatto et al., 2008).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003ePedagogical content knowledge (PCK) refers to subject-specific knowledge for the purpose of teaching so as to make subject matter accessible to students (Shulman, 1986). \u0026nbsp;MPCK is central to bridging the gap between knowing mathematics and teaching it effectively, pertaining to the specialized knowledge required to teach mathematics effectively (Depaepe et al., 2013). In the TEDS-M project, the following two sub-domains of MPCK were differentiated: curricular knowledge and knowledge of planning for mathematics teaching and learning, and knowledge of enacting mathematics for teaching and learning (Tatto et al., 2008). The former mainly refers to knowledge at the pre-active stage, such as establish appropriate learning goals, see connections within the curriculum, plan appropriate activities and methods, and identify approaches for problem solving. The later however refers to knowledge at the interactive stage, including knowledge such as analyze and evaluate students’ mathematical solutions and arguments, provide appropriate feedback, and analyze and diagnose students’ questions (Döhrmann et al., 2012; Tatto et al., 2008).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.2 Mathematics teachers’ beliefs about mathematics teaching and learning\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTeachers’ beliefs are among the most extensively studied topics in mathematics teacher education. A common understanding defines beliefs as “psychologically held understandings, premises, or propositions about the world that are thought to be true” (Philipp, 2007, p. 259). Similar to teacher knowledge, beliefs are considered multifaceted (Ernest, 1989). In terms of mathematics teacher’s beliefs, the critical components include beliefs about the nature of mathematics and beliefs about mathematics teaching and learning (Ernest 1989; Thompson,1992).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTeachers’ beliefs about mathematics teaching and learning, that is, their pedagogical beliefs, \u0026nbsp;refer to teachers’ views on their preferred ways of mathematics teaching and learning, for example, their conceptions of ideal classroom teaching activities, what behaviors and mental activities are involved in mathematics learning, and what constitutes appropriate and prototypical mathematics learning activities (Chan \u0026amp; Elliott 2004; Ernest 1989; Thompson 1992). Generally speaking, the literature has identified two typical views of mathematics learning and teaching: a knowledge transmission (or “traditional”) view and a constructivist view (Blömeke \u0026amp; Kaiser, 2014). Similarly, in the TEDS-M framework, teachers’ beliefs about mathematics teaching and learning were mainly differentiated between two views on mathematics teaching and learning: 1) a knowledge transmission (or “traditional”) view, where mathematics teaching is seen as a process of knowledge transmission and students receive knowledge from teachers passively, and 2) a constructivist view, where mathematics teaching is seen as facilitating students’ knowledge construction (Blömeke \u0026amp; Kaiser 2014; Tatto et al. 2008).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.3 Mathematics teacher professional noticing\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTeachers’ noticing is widely recognized as a key element of teaching expertise, especially in mathematics (Sherin et al., 2011). However, there has been no commonly accepted definition for teacher noticing and teacher noticing has been defined from various perspectives such as cognitive-psychological perspective and socio-cultural perspective (see König et al., 2022 for a detailed discussion). A majority of previous studies referred in their theoretical frameworks to Goodwin’s (1994) concept of professional vision as the theoretical underpinning, which emphasizes that noticing is shaped by social contexts and cultural backgrounds. \u0026nbsp;A few available cross-cultural comparative studies indeed identified different cultural patterns of noticing among teachers from different cultural contexts such as China, Germany and the US (e.g., Ding e al., 2022; Yang et al., 2019).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFrom the cognitive-psychological perspective, teacher noticing has been generally conceptualized as a set of interrelated and cyclical mental processes, including three components: identifying key instructional elements, connecting specific events, and applying contextual knowledge to reasoning (Sherin \u0026amp; van Es, 2009; Sherin et al., 2011).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSimilarly, Jacobs et al. (2010) refined the definition and proposed that teachers’ professional noticing of students’ mathematics thinking includes: (a) attending to students’ strategies; (b) interpreting students’ understanding; (c) deciding how to respond on the basis of students’ understanding.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn follow-up studies of TEDS-M like TEDS-FU and TEDS-Instruct in Germany, teacher noticing was conceptualized as an action-oriented construct (Kaiser et al., 2015; 2017). In the theoretical framework developed in these follow-up studies, teacher noticing was defined as “Perception, Interpretation, and Decision-making” and called this approach the PID model (Kaiser et al., 2015; Kaiser et al., 2017). This model consists of perceiving instructional events, interpreting these actions, and making decisions about student responses or instructional strategies at a broad level which does not limit teachers’ noticing to a specific aspect of teachers’ work. It encompasses all facets of quality mathematics teaching, including pedagogical design, cognitive activation, learning support, and classroom management. The PID model differentiates professional noticing into two domains: general pedagogy (P_PID) and mathematics instruction (M_PID). This study employs the TEDS-Instruct framework to analyze teacher noticing within the Chinese context.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.4 The relationship between teacher knowledge, beliefs, and noticing\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThere is consensus that teacher knowledge and teacher noticing are theoretically connected yet distinct constructs (Blömeke et al., 2015; Schoenfeld, 2011). Recent empirical studies support the notion that teacher knowledge and teacher noticing are two distinct yet interconnected constructs (Blömeke et al., 2015; Copur-Gencturk \u0026amp; Tolar, 2022). Moreover, recent studies also revealed a few patterns in how various types of teacher knowledge impact noticing. First, different forms of teacher knowledge, such as MCK and MPCK, influence teacher noticing in distinct ways. For instance, Dunekacke et al. (2016) found that in Germany, pre-service preschool teachers’ MPCK significantly predicted their perception, while MCK did not. Similarly, Yang et al. (2021) reported stronger associations between pedagogy-related knowledge (e.g., GPK and MPCK) and noticing among Chinese in-service teachers compared to MCK.\u003c/p\u003e\n\u003cp\u003eSecond, teacher knowledge relates more strongly to the interpretation and decision-making aspects of noticing than to perception. For example, Sánchez-Matamoros et al. (2019) observed that pre-service mathematics teachers’ abilities to interpret student understanding and make instructional decisions were influenced by their mathematical knowledge, while their perception of critical incidents was less affected. Similarly, Yang et al. (2019) revealed stronger correlations between GPK, MPCK, and the “interpretation and decision-making” facets of noticing compared to the “perception” facet in a study of Chinese in-service teachers,\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAdditionally, teaching experience appears to moderate the relationship between teacher knowledge and noticing. For example, Dreher and Kuntze (2015) found a weak correlation between pre-service teachers’ MCK and noticing, which diminished for in-service teachers, while MPCK showed strong correlations for in-service but not pre-service teachers. Moreover, the strength of these associations varies across social and cultural contexts. For instance, Yang et al. (2021) observed weaker correlations between knowledge and noticing among Chinese in-service teachers compared to findings in Western cultures, suggesting that cultural and contextual factors may influence these relationships.\u003c/p\u003e\n\u003cp\u003eConcerning the relationship between teacher beliefs and noticing, it has also been argued in literature that teachers’ noticing is “intimately tied to their orientations (including beliefs)” (Schoenfeld, 2011, p. 231). Indeed, in previous research in teacher education, teachers’ beliefs have been argued to act as an important filter for teachers’ decision-making and teaching behaviors as well (Pajares, 1992). Recently, besides within mathematics education field, empirical efforts have been made by researchers from various fields such as inclusive education (Keppens et al., 2021) and science education (Steinwachs \u0026amp; Martens, 2024) to examine the relationship between teacher beliefs and teacher noticing.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn the previous studies, it was first found that different types of beliefs tend to influence or correlate to teacher noticing quite differently. For example, Eßling et al. (2023) observed that pre-service science teachers with stronger transmissive beliefs scored lower for their professional vision of instructional support. Similarly, Dunekacke et al. (2016) reported that application-oriented mathematical beliefs, but not static or process-oriented beliefs, significantly predicted preschool teachers’ noticing.\u003c/p\u003e\n\u003cp\u003eMoreover, it has been argued in literature that cultural context will further moderate the association between teacher beliefs and teacher noticing (Eßling et al., 2023; Santagata \u0026amp; Yeh, 2016). For example, in a recent comparative study, it was found that the American mathematics teachers’ noticing aligned more closely with their individual pedagogical beliefs than their perceptions of general cultural pedagogical beliefs, by contrast, the Chinese mathematics teachers’ noticing was closer to their perception of cultural pedagogical beliefs other than their own pedagogical beliefs (Zhu, 2023).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFurthermore, in the past years, a few studies have examined the interplay between teacher knowledge, beliefs, and noticing. Findings in these studies suggest a tendency that compared with teacher beliefs, teacher knowledge (or at least one type of knowledge) seems to act as a much stronger influence to teacher noticing. For example, Meschede et al. (2017) found that both pre-service and in-service teachers’ MPCK correlated more strongly with professional vision than their transmissive or constructivist beliefs. Similarly, Zeeb et al. (2023) reported that teachers’ declarative knowledge about learners’ growth mindset, but not their beliefs about growth mindset, was significantly associated with noticing.\u003c/p\u003e\n\u003cp\u003eHowever, findings in mathematics education are not entirely consistent. For example, Dunekacke et al. (2016) found that even though both application orientation beliefs and MPCK can predict teacher noticing, MPCK has a much stronger prediction power. Notably, MCK had no direct effect on noticing, but only acted as a predictor of MPCK. Interestingly, Hoth et al. (2022) found that only teachers’ beliefs, not teachers’ MCK and MPCK will not significantly predict teacher noticing. These inconsistencies highlight the need for further research, particularly in diverse cultural contexts.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.4 Research questions\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAs reviewed above, the literature increasingly recognizes that teacher knowledge is not the sole cognitive construct influencing teacher noticing. Other factors, such as teachers’ goals, orientations, and beliefs, play significant roles in shaping their noticing. While prior research has explored the relationships among teacher knowledge, beliefs, and noticing, studies situated in the Chinese context—characterized by its unique cultural and educational traditions in mathematics—remain scarce. Importantly, existing research suggests that the influence of teacher knowledge and beliefs on noticing is moderated by social and cultural contexts (e.g., Santagata \u0026amp; Yeh, 2016). Additionally, teaching experience has been shown to significantly shape the relationships among these constructs (Bastian et al., 2024; Dreher \u0026amp; Kuntze, 2015).\u003c/p\u003e\n\u003cp\u003eAdditionally, while a small number of available studies have examined the interplay between teacher knowledge, beliefs, and noticing, the mediating role of beliefs in the relationship between teacher knowledge and teacher noticing has yet to be explicitly addressed. This gap highlights the need for a more comprehensive investigation into these constructs and their interconnections, particularly within culturally specific contexts such as China. To address these gaps, the present study aims to explore the following research questions:\u003c/p\u003e\n\u003cp\u003e1. What are the relationships between pre-service teachers’ knowledge, pedagogical beliefs, and their noticing at a general level, and how do these relationships vary across the sub-facets of the three constructs?\u003c/p\u003e\n\u003cp\u003e2. What mediating role do pedagogical beliefs play in the relationship between teacher knowledge and teacher noticing?\u003c/p\u003e"},{"header":"3. Methodology","content":"\u003cp\u003e\u003cstrong\u003e3.1 Research context and participants of the study\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn Mainland China, pre-service secondary school teachers are predominantly education in a specific type of university, so-called normal university, over a four-year period. Currently, secondary school teacher programs are dominated by a concurrent curriculum model of teacher preparation and still specialized and discipline-based in Mainland China (Wu et al., 2017). That is, the majority of pre-service teachers at secondary school level are trained in a specific academic subject department where the studies of a specific discipline and pedagogy are integrated and taught at the same time within four years (Musset, 2010; Paine, 1990). Normally, the first two years of pre-service training in university will mainly focus on the learning of subject content at tertiary-level and general education content. In the third year, subject-related educational courses will start to be offered but only occupy a very small portion of teaching time (around 10% to 30% of the total credits) (Wu \u0026amp; Huang, 2018). In addition, pre-service teachers are required to participate in a one-semester teaching practicum in school within their pre-service training period.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFor the present study, 584 pre-service secondary school mathematics teachers in their fourth year of training were selected. Participants included 432 pre-service teachers from one national-level normal university, 99 trainees from two province-level normal universities, and 52 trainees from one local-level normal university. All participants had completed an 18-week teaching practicum. Approximately 40% undertook their practicum in junior secondary schools (Grades 7\u0026ndash;9), while the remaining 60% completed it in senior secondary schools (Grades 10\u0026ndash;12).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2 Instruments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2.1 MCK and MPCK\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe instruments developed in TEDS-M to measure pre-service secondary school mathematics teachers\u0026rsquo; MCK and MPCK were used in the study. Based on the consensus of IEA, the traditional Chinese version of the instruments used in Taiwan were slightly modified to make a few expressions closer to the situation in Mainland China and then were directly employed to test pre-service mathematics teachers\u0026rsquo; MCK and MPCK (Tatto et al., 2008). In total, the two knowledge instruments comprised 103 items, 76 of which were MCK items and 27 of which were MPCK items. In TEDS-M, those items were allocated to three booklets using a balanced-incomplete-block design to capture adequate domain coverage of teacher knowledge within a reasonable administration time (Bl\u0026ouml;meke et al., 2014, Tatto et al., 2008).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eItems assessing MCK covered four content areas, namely number, geometry, algebra and data and were further classified into three cognitive dimensions, namely knowing, applying, and reasoning. The MPCK items included aspects of curricular and planning knowledge and knowledge about how to enact mathematics in the classroom. The majority of the items in both MCK and MPCK tests were complex multiple-choice items with a few multiple choice and open response items. The participants were given 60 minutes to complete both the MCK and MPCK test in paper-and-pencil format. Item examples can be found in Bl\u0026ouml;meke et al. (2013).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2.2 Beliefs about mathematics learning\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe original Chinese version of the questionnaires designed in TEDS-M to investigate pre-service mathematics teachers\u0026rsquo; beliefs about learning mathematics was used in the present study. The beliefs about the learning of mathematics had two dimensions: learning mathematics through following teacher direction (or labeled as \u0026ldquo;transmissive view\u0026rdquo; in the study, 6 items sample item like \u0026ldquo;Pupils need to be taught exact procedures for solving mathematical problems\u0026rdquo;.), and learning mathematics through active involvement (or labeled as \u0026ldquo;constructivist beliefs\u0026rdquo;, 6 items, sample items like \u0026ldquo;In addition to getting the right answer, it is important to understand why the answer is correct\u0026rdquo;). Pre-service teachers had to rate the beliefs statements on a five-point Likert scale (1= \u0026ldquo;strongly disagree\u0026rdquo; and 5= \u0026ldquo;strongly agree\u0026rdquo;). To validate the questionnaire for the purpose of the present study, the CFA showed that the two-factor model with 12 items had a good fit\u0026nbsp;\u0026nbsp;\u003cimg src=\"data:image/png;base64,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\"\u003e\u0026nbsp;. \u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2.3 Noticing\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOur previous validated instruments adapted from TEDS-Follow-Up and TEDS-Instruct/Validate (for adaptation details, see Yang et al., 2018, 2019) were used in the study to examine pre-service mathematics teachers\u0026rsquo; professional noticing skills. These assessments include three video vignettes designed to evaluate German secondary school mathematics teachers\u0026apos; professional noticing, focusing on aspects of mathematics instruction (M_PID) and general pedagogical issues (P_PID). The items assess three facets of noticing: perception, interpretation, and decision-making, encompassing almost all aspects of a mathematics lesson. To meet the research aim of the present study, we regrouped the items into two categories: 1) Perception, including mathematics instruction and general pedagogy-related perception (labeled as M_Perception and P_Perception respectively); and 2) Interpretation \u0026amp; Decision-making, including mathematics instruction and general pedagogy-related interpretation and decision-making (labeled as M_Interpretation\u0026amp;Decision-making and P_Interpretation\u0026amp;Decision-making respectively).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAfter participants watched each of the three videos, they were given around 15 minutes to answer a few items related to each video (in total not exceeding 60 minutes for the three videos). There are two types of items for this part: 1) 38 items based on Likert scales (four choices from \u0026ldquo;fully correct\u0026rdquo; to \u0026ldquo;not correct\u0026rdquo;) to assess participants\u0026rsquo; noticing mainly related to the perception facet; 2) 36 constructed-response items to assess the participants\u0026rsquo; noticing related to interpretation and decision-making facets. In addition, an extensive coding manual was developed to evaluate participants\u0026rsquo; answers. Item examples can be found in Kaiser et al. (2015). Although the instrument was originally developed for in-service teachers, a further study could be shown that the instrument can be used for pre-service teachers as well (Weyers et al., 2023).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3 Scaling and data analysis\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data analysis comprised the following steps. First, the open response items of the MCK and MPCK test and Noticing were coded according to the coding rubrics of the coding manual developed in TEDS-M and TEDS-FU. Four trained postgraduate students in mathematics education worked as independent raters and first coded 80 of the questionnaires together. Good values of Cohen\u0026rsquo;s Kappa were reached (k\u0026gt;0.78 and K \u003csub\u003eaverage\u003c/sub\u003e= 0.86). For all the open response items, items without answers or with incorrect answers were scored 0, and correct answers were scored 1.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAfter the completion of coding, the relative item difficulties for a one-parameter (Rasch model) item response theory (IRT) model were calculated separately on the six dimensions of knowledge, beliefs and noticing. Items with extreme difficulty were removed for the final analysis, as they do not substantially contribute to the measurement of the construct due to weak discrimination (Bond \u0026amp; Fox, 2007). After this, the internal consistency of the remaining items in each of the dimensions and sub-dimensions related was estimated using Cronbach\u0026rsquo;s alpha reliability coefficient. Most of the reliability scores were above 0.6 indicating acceptable or good reliability. After this, scale scores were created for each of the six dimensions and their sub-dimensions by applying the one-dimensional Rasch model.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe second step of the data analysis was to systematically examine the relationship between teacher knowledge, pedagogical beliefs, and noticing. First, using estimated scores for each dimension of teacher knowledge, beliefs and noticing as manifest variables, Pearson correlational analyses were conducted between pre-service mathematics teachers\u0026rsquo; knowledge (MCK and MPCK), beliefs (transmissive and constructivist), and noticing (Perception and Interpretation \u0026amp; Decision-making). Afterwards, with the use of estimated scores for each sub-component or sub-aspect of teacher knowledge, beliefs, and noticing as manifest variables, Pearson correlational analyses were again conducted between them. MCK was differentiated according to its cognitive domains: knowing, applying, and reasoning. Finally, path analyses were conducted to examine the mediation effect of beliefs between knowledge on Perception and Interpretation \u0026amp; Decision-making respectively. \u0026nbsp; Mplus was used for the path analyses.\u0026nbsp;\u003c/p\u003e"},{"header":"4. Results","content":"\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1 Correlation Analysis\u003c/h2\u003e\n \u003cp\u003eTo examine how each of the components of knowledge and beliefs were related to each aspect of teacher noticing, Pearson\u0026rsquo;s r based on the estimated scores of the six aspects was estimated. Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes the results.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"3\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePerception\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eInterpretation \u0026amp; Decision-Making\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMCK\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.161***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.239***\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMPCK\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.033\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.135***\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTransmissive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.249***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.163***\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConstructivist\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.298***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.182***\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"3\"\u003eCorrelations between teacher knowledge, beliefs and teacher noticing (n\u0026thinsp;=\u0026thinsp;583)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"3\"\u003eNote: *p\u0026thinsp;\u0026lt;\u0026thinsp;0.05, **p\u0026thinsp;\u0026lt;\u0026thinsp;0.01, ***p\u0026thinsp;\u0026lt;\u0026thinsp;0.001.\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eAs illustrated in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, regarding to the relationship between knowledge and noticing, first of all, statistically significant positive correlations were identified between MCK and Perception (r\u0026thinsp;=\u0026thinsp;0.161) and Interpretation \u0026amp; Decision-Making (r\u0026thinsp;=\u0026thinsp;0.239). By contrast, MPCK was only significantly positively related with Interpretation \u0026amp; Decision-Making (r\u0026thinsp;=\u0026thinsp;0.135). Moreover, out of our expectation, comparatively speaking, MCK was much more strongly related to both Perception and Interpretation \u0026amp; Decision-Making. However, as expected, both MCK and MPCK were found to be much more strongly related to Interpretation \u0026amp; Decision-Making than to Perception.\u003c/p\u003e\n \u003cp\u003eWith respect to the relationship between beliefs and noticing, transmissive beliefs were found to significantly but negatively relate to Perception and Interpretation \u0026amp; Decision-Making. By contrast, Constructivist beliefs were found to significantly and positively relate to Perception and Interpretation \u0026amp; Decision-Making. Moreover, both beliefs were found to relatively more strongly relate to Perception. In addition, compared with MCK and MPCK, pedagogical beliefs were found to much more strongly relate with Perception. But for the process of Interpretation \u0026amp; Decision-Making, compared with pedagogical beliefs, MCK was relatively more strongly, but MPCK was relatively weaklier related to Interpretation \u0026amp; Decision-Making.\u003c/p\u003e\n \u003cp\u003eTo further examine how each of the sub-domains of mathematics teacher knowledge and beliefs were related to the sub-facets of noticing, Pearson\u0026rsquo;s r based on the estimated scores of the nine sub-domains of teacher knowledge and beliefs and the four sub-facets of teacher noticing was estimated. Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e summarizes the results. As illustrated in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, the relationships between the five sub-domains of teacher knowledge and the four sub-facets of teacher noticing were all quite weak, suggesting that different sub-domains of teacher knowledge are a factor influencing different phases of teacher noticing with various effect sizes, but not in a very strong manner. Again, transmissive beliefs were found to relate to all facets negatively, but constructivist beliefs were found to be positively related with the four sub-facts of teacher noticing.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eCorrelations between Sub-domains of Knowledge, beliefs and noticing (Pearson\u0026rsquo;s r; n\u0026thinsp;=\u0026thinsp;583)\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" colspan=\"2\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003ePerception\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eInterpretation \u0026amp; Decision-Making\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP_Perception\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eM_Perception\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP_Interpretation \u0026amp;Decision-Making\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eM_Interpretation \u0026amp; Decision-Making\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"3\"\u003e\n \u003cp\u003eMCK\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eKnowing\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.068\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.088\u003csup\u003e*\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.073\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.126**\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eApplying\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.140**\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.153***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.113**\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.232***\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eReasoning\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.170***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.101**\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.073\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.160***\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eMPCK\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCurriculum\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.029\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.032\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.024\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.039\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePlanning and enacting\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.025\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.039\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.017\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.093*\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eBeliefs\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTransmissive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.224***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.221***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.122\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.116**\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConstructivist\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.256***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.288***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.141**\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.127**\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"7\"\u003eNote: *p\u0026thinsp;\u0026lt;\u0026thinsp;0.05, **p\u0026thinsp;\u0026lt;\u0026thinsp;0.01, ***p\u0026thinsp;\u0026lt;\u0026thinsp;0.001.\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eMoreover, as shown in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, for the relationship between MCK and teacher noticing, relatively stronger relationships could be observed between the higher cognitive understanding of mathematics such as Applying and Reasoning and the four sub-facets of teacher noticing. In addition, the relationship between MCK and the sub-facet of M_Interpretation \u0026amp; Decision-Making is relatively stronger than the relationships between MCK with other sub-facts of teacher noticing. However, again, both of the sub-domains of MPCK were found to be quite weakly correlated with the four sub-facts of teacher noticing except for a statistically significant positive correlation between \u0026ldquo;Planning and enacting\u0026rdquo; and \u0026ldquo;M_Interpretation \u0026amp; Decision-Making\u0026rdquo;. Very interestingly, very weak but negative correlations were identified between \u0026ldquo;Curriculum\u0026rdquo; and \u0026ldquo;P_Perception\u0026rdquo; and \u0026ldquo;M_Perception\u0026rdquo;.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2 Path analysis\u003c/h2\u003e\n \u003cp\u003eIn order to examine the mediating role played by pedagogical beliefs between teacher knowledge and teacher noticing, a path analysis was further conducted. As being reviewed above, because MCK has been generally accepted as a prediction for MPCK and both knowledge and beliefs will influence the process of perception and interpretation and decision making differently, we first performed a path analysis to examine the relationship between knowledge, beliefs and perception. The path model fits the data well\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({\\chi\\:}^{2}\\:=\\:6.485,\\:{\\chi\\:}^{2}/df\\:=\\:1.62,\\:CFI\\:=\\:.996,\\:TLI\\:=\\:.985,\\:RMSEA\\:=\\:.033)\\)\u003c/span\u003e\u003c/span\u003eand Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e represents the overall results .\u003c/p\u003e\n \u003cp\u003eAs shown in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, the knowledge domains of MCK and MPCK could not significantly predict the process of teacher noticing of Perception. However, regarding pedagogical beliefs, Constructivist beliefs could significantly positively predict Perception, while Transmissive beliefs could significantly negatively predict Perception. Moreover, both Constructivist and Transmissive beliefs were found to play a significant mediation role between MPCK and Perception.\u003c/p\u003e\n \u003cp\u003eMoreover, to specifically examine the relationship between teacher knowledge, pedagogical beliefs and the process of Interpretation \u0026amp; Decision-Making, a second path analysis was performed. The path model fits the data well\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({\\chi\\:}^{2}\\:=7.421,\\:{\\chi\\:}^{2}/df\\:=\\:1.48,\\:CFI\\:=\\:.995,\\:TLI\\:=\\:.986,\\:RMSEA\\:=\\:.029)\\)\u003c/span\u003e\u003c/span\u003e and Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e represents the overall results.\u003c/p\u003e\n \u003cp\u003eAs shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, MCK was found to be able to significantly positively predict teacher noticing of Interpretation \u0026amp; Decision Making. By contrast, the relation between teachers\u0026rsquo; MPCK and their Interpretation \u0026amp;Decision Making was again found to be not significant and even negative (\u0026beta;=-0.005). In addition, for pedagogical beliefs, only constructivist beliefs were found to be able to significantly and positively predict pre-service mathematics teachers\u0026rsquo; Interpretation \u0026amp; Decision-Making. Moreover, Constructivist beliefs could further mediate the relationship between MPCK and Interpretation \u0026amp; Decision-Making.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"5. Summary and Discussion","content":"\u003cp\u003eThe primary aim of this study was to systematically investigate the relationship between pre-service mathematics teachers\u0026rsquo; knowledge (including MCK and MPCK), pedagogical beliefs, and teacher noticing, focusing on the mediating role of pedagogical beliefs between teacher knowledge and teacher noticing. First, out of our expectation, both correlation and path analysis results illustrate that it was Chinese pre-service mathematics teachers\u0026rsquo; MCK, not theoretically assumed MPCK, exhibited a stronger influence on teacher noticing. This finding diverges from prior research, which has often identified MPCK as having a more robust relationship with teacher noticing (Dunekacke et al., 2015), even within the Chinese context (Yang et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Moreover, while the correlation coefficients between MCK, MPCK (including their sub-components), and noticing (including its sub-facts) are generally positive, the overall relationships between Chinese pre-service mathematics teachers\u0026rsquo; knowledge and noticing are weaker than the strong empirical relationships identified in some Western contexts (e.g., Meschede et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). These differences may indicate that factors from specific societal and cultural contexts and differences of teaching experience will significantly influence or moderate the relationship between teacher knowledge and noticing.\u003c/p\u003e \u003cp\u003eThe differing backgrounds and experiences of the participants should be taken into account to provide explanations for the unexpected results, which are in parts inconsistent to the results of other studies. In most previous studies, the participants were all in-service teachers (Yang et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Consequently, after a number of years of teaching practice, these in-service teachers developed MPCK as usable knowledge that they could readily access to address complex classroom situations, particularly when analyzing observed teaching events and making subsequent decisions (Kersting et al., \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). In contrast, pre-service teachers do not automatically use their knowledge such as mathematical knowledge for teaching to interpret or analyze what they notice (Spitzer \u0026amp; Phelps-Gregory, \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Similarly, although all participants in the present study had one semester of teaching practicum experience in schools, it may still have presented a challenge for them to flexibly apply their mathematics pedagogical content knowledge to inform their interpretations of what they noticed.\u003c/p\u003e \u003cp\u003eAdditionally, the tradition of pre-service teacher education within the Chinese context may also influence the weak association between pre-service mathematics teachers\u0026rsquo; MPCK and noticing. Traditionally, teacher education in Mainland China has followed an academically oriented preparation model (Liao \u0026amp; Hu, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Paine, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e1990\u003c/span\u003e). As a result, most of the education time is spent on content knowledge learning, limiting opportunities for courses related to pedagogical content knowledge (Wu \u0026amp; Huang, \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Moreover, \u0026ldquo;direct lecture\u0026rdquo; is the predominant teaching method during pre-service teacher training, restricting opportunities to link theoretical content with practical teaching applications (Wu et al., \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Consequently, it is understandable that for Chinese pre-service mathematics teachers, it is challenging to connect their academic knowledge with their classroom observations, which are inherently more context-specific. Furthermore, it was found that Chinese pre-service mathematics teachers did not perceive their school-based teaching practicum experience as beneficial in connecting theory to practice or enhancing their university learning (Yang et al., in press). Therefore, the weak connections between pre-service teachers\u0026rsquo; academic knowledge and their practical mathematics teaching experiences likely have a significant impact on the relationship between MPCK and teacher noticing. In contrast, Chinese pre-service teachers generally possess a solid mathematics subject background and have ample opportunities during their training to deepen their cognitive understanding of mathematics. Thus, MCK becomes more accessible for them, which further strengthens the relationship between MCK and teacher noticing. Indeed, this study identified a closer relationship between a higher level of understanding of mathematics and various facets of teacher noticing.\u003c/p\u003e \u003cp\u003eSecond, regarding to the relationship between teacher beliefs and noticing, the findings indicate that Chinese pre-service mathematics teachers\u0026rsquo; transmissive beliefs negatively correlate with their noticing, while constructivist beliefs positively correlate with their noticing. These results are generally consistent with findings from previous studies conducted in diverse cultural contexts (e.g., E\u0026szlig;ling et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Keppens et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Meschede et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). In earlier studies, constructivist beliefs were similarly found to exert a significant and positive influence on teachers\u0026rsquo; noticing, whereas transmissive beliefs were generally associated with a strong negative relationship. These findings suggest that, across various contexts, teachers\u0026rsquo; beliefs indeed serve as a primary influence on teacher noticing, as theorized previously (Schoenfeld, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). However, the strength of the relationships between pre-service mathematics teachers\u0026rsquo; beliefs and their noticing identified in this study appears to be comparatively weaker than in contexts such as Germany (e.g., Hoth et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Meschede et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). These differences may again stem from the variations in societal and cultural settings. As reviewed above, Zhu (2023) found that for Chinese teachers, their noticing is significantly influenced by their perceptions of cultural pedagogical beliefs rather than solely by their own pedagogical beliefs. Thus, the findings of this study may further suggest that, similar to teacher knowledge, social and cultural norms significantly influence the enactment of teacher beliefs during noticing (E\u0026szlig;ling et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Santagata \u0026amp; Yeh, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThird, both correlation and path analysis results indicate that the knowledge and pedagogical beliefs of Chinese pre-service mathematics teachers influence various sub-facets of teacher noticing in distinct ways. In general, the relationship between teacher knowledge and noticing is characterized by a stronger association between MCK and MPCK and the sub-fact of interpretation and decision-making. These results are congruent with previous research findings (e.g., K\u0026ouml;nig et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; S\u0026aacute;nchez-Matamoros et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Yang et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). This supports the theoretical distinction between different facets of teacher noticing, positing that interpretation and decision-making are more knowledge-driven facets of noticing (Bastian et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2024\u003c/span\u003e, Sherin et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Specifically, the phase of attending will be easier than, or only a prerequisite to, the facet of interpretation and decision-making (S\u0026aacute;nchez-Matamoros et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Thus, teacher knowledge tends to play varied roles within the different facets of noticing.\u003c/p\u003e \u003cp\u003eConversely, it was found that the pedagogical beliefs of pre-service mathematics teachers were more closely related to the facet of perception rather than to interpretation and decision-making. Furthermore, both correlation and path analysis results indicate that the strength of the relationship between pedagogical beliefs and the perception sub-fact is significantly stronger than the relationship between MCK and MPCK and perception. These findings suggest a theoretical differentiation between the perception facet and the afterwards following facet of interpretation and decision-making phases. Comparatively speaking, in complex classroom environments, a teacher\u0026rsquo;s decision to attend to or disregard specific classroom events is often an intuitive action (Sherin \u0026amp; Star, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Consequently, the perception facet is more significantly influenced by teachers\u0026rsquo; orientations, goals, and beliefs (Schoenfeld, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Although both beliefs and knowledge inform teachers\u0026rsquo; interpretation and decision-making, beliefs often serve as \u0026ldquo;filters\u0026rdquo; for what teachers initially perceive (Lee \u0026amp; Francis, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Roose et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2019\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFourth, a primary objective of the present study was to examine the mediating role of pedagogical beliefs between teacher knowledge and teacher noticing. Path analysis results corroborate our hypothesis that both transmissive and constructivist beliefs function as mediators between teacher knowledge and noticing. However, the study also found that different types of pedagogical beliefs play distinct mediation roles in relation to the different facets of teacher noticing. Constructivist beliefs were identified as a much stronger positive mediator between teacher knowledge and both perception and interpretation \u0026amp; decision-making. In contrast, transmissive beliefs were found to play a negative mediating role between teacher knowledge and the perception phase of noticing. These findings are consistent with previous research, which has indicated that constructivist beliefs have a more substantial impact on mediating the relationship between teachers' knowledge and their instructional practices (Leder et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Yang et al., 2020). Furthermore, the results illustrate that teacher beliefs significantly influence their ability to effectively employ their knowledge in the process of noticing.\u003c/p\u003e"},{"header":"6. Conclusions and limitations","content":"\u003cp\u003eAlthough it has been theoretically proposed and empirically observed that teacher knowledge and beliefs have a fundamental impact on teachers\u0026rsquo; noticing, there remains limited empirical evidence to substantiate the joint effect of teacher knowledge and beliefs, particularly regarding the mediating role of beliefs within a non-Western context. This study aimed to address this gap using standardized testing instruments. The findings reported above first provide empirical evidence that teacher knowledge, beliefs, and noticing are empirically separated constructs. Moreover, findings of the study further suggest that the differences of social and cultural contexts and teaching experience exert significant influence to the relationship between these three constructs. That is, social and cultural norms, along with teaching experience, may play critical roles in moderating the effects of teacher knowledge and beliefs on teacher noticing.\u003c/p\u003e \u003cp\u003eWhile this study contributes to the understanding of the relationship between mathematics teacher knowledge, beliefs, and noticing, particularly within the Chinese context, several limitations should be acknowledged. First, although the study involved a relatively large sample of participants, these participants were selected from only four teacher education universities. As such, the sample may not be sufficiently representative to capture the diversity of pre-service mathematics teachers\u0026rsquo; knowledge, beliefs, and noticing across China. Future studies could address this limitation by including a larger and more diverse sample, encompassing pre-service teachers from additional institutions, particularly those at provincial and local levels.\u003c/p\u003e \u003cp\u003eSecond, the study assessed teachers\u0026rsquo; knowledge, beliefs, and noticing in a general manner without focusing on specific branches of mathematics (e.g., geometry or algebra) or particular mathematical topics (e.g., fractions or equations). This general approach may limit the applicability of the findings to specific instructional contexts. Future research could explore these relationships within the context of specific mathematical domains or topics. Additionally, incorporating qualitative methods, such as interviews, could enrich the findings and provide deeper insights into the nuances of teacher noticing.\u003c/p\u003e \u003cp\u003eThird, the study employed comprehensive instruments, including the adapted MCK and MPCK tests from TEDS-M and the noticing test from TEDS-FU, which required approximately three hours for each participant to complete, alongside a beliefs survey. This extensive testing burden may have impacted participant engagement and the overall validity of the results. To address this issue, future studies could consider using shorter versions of the assessment instruments or administering the tests across two separate time points. Such approaches could help reduce the testing burden and enhance the validity and reliability of the findings.\u003c/p\u003e \u003cp\u003eBy addressing these limitations, future research can build upon the current study to deepen our understanding of the complex relationships among teacher knowledge, pedagogical beliefs, and teacher noticing, thereby providing more robust and contextually relevant insights.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics statement:\u003c/strong\u003e The relevant committees at Southwest University responsible for ethical clearance have approved the study and confirmed that it complies with the ethical standards set by these committees.\u003c/p\u003e\n\u003cp\u003eInternal note: Roma OK\u0026apos;d\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of conflict of interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no conflict of interest.\u003c/p\u003e\n\u003cp\u003eThis study was supported by the National Office for Education Sciences Planning, Ministry of Education, China (Grant Number: BMA 220225).\u003cstrong\u003e\u003cbr\u003e\u003c/strong\u003e\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eBall, D. 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An exploratory study. \u003cem\u003eTeaching and Teacher Education, 130\u003c/em\u003e, 104170.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[{"identity":"7100698d-b0cf-4c30-9c09-2edf1843afa6","identifier":"10.13039/501100002855","name":"Ministry of Science and Technology of the People's Republic of China","awardNumber":"BMA 220225","order_by":0}],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Mathematics content knowledge, mathematics pedagogical content knowledge, pedagogical beliefs, noticing ","lastPublishedDoi":"10.21203/rs.3.rs-5925262/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5925262/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWhile the theoretical discourse posits teacher knowledge and beliefs as critical factors influencing teacher noticing, few studies have empirically explored the interrelationship among these constructs within a single investigation, particularly in a non-Western context. This paper examines the relationships among teacher knowledge, pedagogical beliefs, and teacher noticing, with a specific focus on the mediating role of beliefs between teacher knowledge and noticing, based on a study involving 583 pre-service mathematics teachers within the Chinese context. The findings indicate that in contrast to common expectations and earlier results pre-service teachers\u0026rsquo; mathematical content knowledge (MCK), rather than their mathematical pedagogical content knowledge (MPCK), exhibits a stronger correlation with teachers\u0026rsquo; noticing. However, as expected, transmissive pedagogical beliefs significantly and negatively correlate with noticing, while constructivist pedagogical beliefs demonstrate a significant positive relationship with noticing. Furthermore, the study reveals that teacher knowledge and pedagogical beliefs distinctly influence various facets of teacher noticing confirming theoretically derived assumptions. Notably, pedagogical beliefs serve as a significant mediator between teacher knowledge and noticing. The findings suggest that apparently societal and cultural norms, alongside teaching experience, moderate the relationships among teacher knowledge, beliefs, and noticing.\u003c/p\u003e","manuscriptTitle":"Relationship between Chinese Pre-service Mathematics Teachers’ Knowledge, Pedagogical beliefs, and Noticing","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-02-05 06:17:16","doi":"10.21203/rs.3.rs-5925262/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5c145390-bd89-4cb8-a854-90d297b76b72","owner":[],"postedDate":"February 5th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":43582587,"name":"Educational Psychology"}],"tags":[],"updatedAt":"2025-02-05T06:17:16+00:00","versionOfRecord":[],"versionCreatedAt":"2025-02-05 06:17:16","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5925262","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5925262","identity":"rs-5925262","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}