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This paper documents efforts to develop an instrument to measure mathematical knowledge for teaching high school geometry (MKT-G). We report on the process of developing and piloting questions that purported to measure various domains of MKT-G. Scores on a piloted set of items had no statistical relationship with total years of experience teaching, but all domain scores were found to have statistically significant correlations with years of experience teaching high school geometry. Other interesting relationships regarding teachers??? MKT-G scores are also reported. We use these results to propose a way of conceptualizing how instruction specific considerations might matter in the design of MKT items. In particular, we propose that the instructional situations that are customary to a course of studies, can be seen as units that organize much of the mathematical knowledge for teaching such course.

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... comprehensively would be a good starting point to unravel the complex relationship between teacher knowledge, teaching quality, and student learning. In the field of mathematics teacher education, there have been substantial efforts to conceptualize key mathematical knowledge that shapes the quality of instruction and measure such knowledge Chapman, 2007;Ellerton & Clements, 2011;Ferrini-Mundy et al., 2005;Hill et al., 2004;Hill et al., 2008;Herbst & Kosko, 2014;Manizade & Mason, 2011;McCrory et al., 2012;Olanoff et al., 2014;Steel, 2013). Compared to teachers' knowledge for teaching numbers, operations, and algebra Chapman, 2007;Ellerton & Clements, 2011;Ferrini-Mundy et al., 2005;Izsák, 2008;McCrory et al., 2012;Newton, 2008;Olanoff et al., 2014;Welder & Simonsen, 2011), investigations into teachers' geometry knowledge is relatively limited and often focused on teaching middle or high school level geometry (Herbst & Kosko, 2014;Herbst et al., 2020;Manizade & Mason, 2011;Martinovic & Manizade, 2018;Steele, 2013;Zambak & Tyminski, 2017). ...

... In the field of mathematics teacher education, there have been substantial efforts to conceptualize key mathematical knowledge that shapes the quality of instruction and measure such knowledge Chapman, 2007;Ellerton & Clements, 2011;Ferrini-Mundy et al., 2005;Hill et al., 2004;Hill et al., 2008;Herbst & Kosko, 2014;Manizade & Mason, 2011;McCrory et al., 2012;Olanoff et al., 2014;Steel, 2013). Compared to teachers' knowledge for teaching numbers, operations, and algebra Chapman, 2007;Ellerton & Clements, 2011;Ferrini-Mundy et al., 2005;Izsák, 2008;McCrory et al., 2012;Newton, 2008;Olanoff et al., 2014;Welder & Simonsen, 2011), investigations into teachers' geometry knowledge is relatively limited and often focused on teaching middle or high school level geometry (Herbst & Kosko, 2014;Herbst et al., 2020;Manizade & Mason, 2011;Martinovic & Manizade, 2018;Steele, 2013;Zambak & Tyminski, 2017). ...

... Developing reliable measures of teachers' mathematical knowledge for teaching has been attempted on the topics of numbers, operations, and algebra at the elementary level (Hill et al., 2004) and geometry at the middle or high school levels (Brakoniecki et al., 2016;Herbst & Kosko, 2014). Such an effort is rare in quantitatively measuring pre-service elementary teachers' geometry knowledge in the existing literature, especially for both content and pedagogical content knowledge for teaching 2D shapes. ...

... One is StoryCircles , a design for practice-based professional learning that attends to the specific demands of teaching a lesson. The other is the MKT-G assessment of mathematical knowledge for teaching high school geometry (Herbst & Kosko, 2014). We briefly review both contributions, then describe how we have used the latter to appraise teacher growth in the former. ...

... Our group developed items that also attempted to tap into the domain definitions proposed by Ball et al. (2008) and where the content addressed by items was specific to the high school geometry (HSG) course in the United States. The items went through several cycles of revision, cognitive interviewing, and piloting, ending with a set that Herbst and Kosko (2014) have described collectively as the MKT-G instrument. Like others, we have used item responses to model a unidimensional latent continuous construct. ...

... Each item in Form A had been initially designed to tap into one of the four MKT domains (3 CCK, 5 SCK, 4 KCS, 4 KCT). Despite the fact that, in a previous study, Herbst and Kosko (2014) had treated MKT-G as a unidimensional construct, in this study we explored the possibility and advantage of separating SMK (subject matter knowledge) from PCK (pedagogical content knowledge), to understand the changes associated with the StoryCircles intervention. In particular, we explored this possibility with two different assumptions on the nature of the MKT-G construct: continuous latent construct and discrete latent construct. ...

We show how we used a national distribution of responses from 416 practicing teachers to items of a test of mathematical knowledge for teaching geometry to estimate changes in knowledge by a group of 11 practicing teachers who participated in a 2-year practice-based professional development programme. To draw a reliable interpretation of the change, we use multiple measurement models under different assumptions on the scale of MKT-G. We demonstrate how Diagnostic Classification Modelling was used to determine whether participants had grown. The participants’ gain (status change in achievement profile) was also examined in relationship with the amount of time participants spent during the PD, which was estimated using log data recorded by the online platform. We discuss the findings from these explorations, in the context of the broader problem of improving conceptualizations of MKT.

... Hill's (2007) study also gave impetus to investigating the MKT of teachers of other levels of schooling. Herbst and Kosko (2014) reported on an effort to develop items that measure MKT for high school geometry. Using a test of MKT whose items reflected the content in U.S. high school geometry, Herbst and Kosko (2014) found evidence that teachers experienced in teaching high school geometry score better than those without such experience, especially on items contextualized in instructional situations that are common in high school geometry courses. ...

... Herbst and Kosko (2014) reported on an effort to develop items that measure MKT for high school geometry. Using a test of MKT whose items reflected the content in U.S. high school geometry, Herbst and Kosko (2014) found evidence that teachers experienced in teaching high school geometry score better than those without such experience, especially on items contextualized in instructional situations that are common in high school geometry courses. Herbst and Kosko (2014) used such findings to suggest that the domains proposed by Ball et al. (2008) may not be single constructs but composites that depend in fundamental ways on the participants' recognition of the instructional situations teachers of different courses of studies need to manage. ...

... Using a test of MKT whose items reflected the content in U.S. high school geometry, Herbst and Kosko (2014) found evidence that teachers experienced in teaching high school geometry score better than those without such experience, especially on items contextualized in instructional situations that are common in high school geometry courses. Herbst and Kosko (2014) used such findings to suggest that the domains proposed by Ball et al. (2008) may not be single constructs but composites that depend in fundamental ways on the participants' recognition of the instructional situations teachers of different courses of studies need to manage. Doyle's (1983) notion of academic task could be used to describe, from the perspective of students, the exchanges of work for grades that students need to manage. ...

... Building on the 2008 work of Ball and colleagues, Herbst and Kosko (2014) focused on identifying aspects of Mathematical Knowledge for Teaching high school Geometry (MKT-G). They used different state curricula to develop survey items that covered definitions, properties, and constructions of plane figures (e.g., transformations), 3-D figures, and coordinate geometry, as well as measurements related to perimeter, area, and volume. ...

... The difference in language shows that in both cases the PCK dimensions were influenced by the dominant rhetoric of the epoch. In addition, since instruments developed in these studies incorporated different geometry tasks, they required different knowledge for teaching them (Herbst and Kosko 2014). To emphasize that teachers' subject matter knowledge (i.e., geometry knowledge related to the area of a trapezoid) is a prerequisite for developing PCK on the same topic, we position it at the top of the second column in Table 1, as well as one of the dimensions of teacher's profile in Fig. 8. ...

... Research teams across the United States have recently worked on developing measures of mathematical knowledge for teaching and related constructs, including: (a) KAT (Ferrini-Mundy et al. 2005;McCrory et al. 2012); (b) LMT and MKT Manizade and Mason 2011); (c) MKT-G (Herbst and Kosko 2014;Steele 2013); and (d) GAST (Bush et al. 2008). Most of these instruments were designed to cover and assess a wide range of mathematics topics. ...

In our research, we focused on the design of assessment instruments for measuring teachers’ mathematical knowledge for teaching geometry. Since 1987 when Lee Shulman conceptualized pedagogical content knowledge, different research groups have expanded on this construct and developed their own measurement instruments. The strengths and limitations of these instruments have been described in the literature. Compared to researchers who have designed instruments for a broad range of mathematical topics, we propose that the measures should be designed as “probes” around specific topics commonly taught by a targeted group of teachers (e.g., middle school). In this paper, we focus on methodological issues of measuring mathematical knowledge for teaching, describe our approach in designing the probe targeting knowledge for teaching the Area of a Trapezoid task and accompanied assessment tools, identify challenges in designing assessments, and discuss possible solutions. In designing the measures and rubrics of teachers’ knowledge, we used the Delphi and Grounded Theory research methods.

... What distinguishes SCK is not whether mathematicians or others who use mathematics might also hold similar understandings (e.g. most mathematicians are aware of multiple proof techniques and appreciate the explanatory role of proof), but that such knowledge is not necessarily critical to their everyday work (Herbst & Kosko, 2014). While I recognize that there is certain overlap and the distinction is not always clear in practice, highlighting proof knowledge unique to teaching allows us to better address the content needs of mathematics teachers and those in training to become teachers (Ball, Lubienski, & Mewborn, 2001). ...

... My focus on CCK, SCK, KCS and KCT is consistent with recent work that seeks to develop measures to assess mathematical knowledge for teaching in other content areas (e.g. Herbst & Kosko, 2014;Hill, Schillings & Ball, 2008). ...

... This limitation is not unique to this study. For example, when using classroom scenarios to measure MKT, or analyzing teachers in-the-moment decision making, it is difficult to discern whether teachers are relying on purely mathematical knowledge (SCK), or knowledge they have garnered from years of experience interacting with students (KCS) (Herbst & Kosko, 2014). However, by detailing how proof knowledge surfaces in PD settings, this work has the potential to inform those designing learning experiences for teachers. ...

Research documenting teachers’ fragile understanding of proof and how it is advanced suggests that enhancing the role of proof in school mathematics will demand substantial teacher learning. To date, there is little research detailing what mathematical knowledge might support the teaching of proof or how professional development (PD) might afford such learning. This paper advances a framework for Mathematical Knowledge for Teaching Proof (MKT for Proof) that specifies proof knowledge across subject matter and pedagogical domains. To explore the utility of the framework, this paper examines data from four different PD settings in which teachers and leaders worked on the same proof-related task. Analysis of discussions across these four settings revealed similarities and differences in knowledge of proof made available in each setting. These findings provide a means to detail the MKT for Proof framework and demonstrate its usefulness as a tool for designing, analyzing and evaluating proof experiences in PD.

... The items represent a variety of tasks of teaching and they were situated in classroom contexts; each item calling the respondent to make a choice on behalf of a teacher involved in a task of teaching. Herbst and Kosko (2014) argued that this instrument captures specific mathematical knowledge for teaching high school geometry on the basis that participants' experience teaching high school geometry significantly correlates with their MKT-G scores, while experience teaching mathematics in general was not significantly correlated with MKT-G scores ). ...

... Also, the hypothesized role of these organizers in capturing differences in mathematical knowledge among the teachers (3.3) is consistent with previous empirical studies. For example, Herbst and Kosko (2014) showed that whether the items are contextualized in an instructional situation or not is related to the difference in the knowledge between novice teachers and experienced teachers. ...

... CG and DP were distinguishable within the task of CGP (5.1.2). The distinction between CGP_CG and CGP_DP is understandable in light of work by Herbst and Kosko (2014), which states that "tasks of teaching could call for different kinds of mathematical work depending on specifics of the work of teaching geometry" (Herbst & Kosko, 2014, p. 7). In their study, conjectures on the specifics were described within the context of designing a problem, which is similar to the category of CGP in my study. ...

This study proposes a way of organizing mathematical knowledge for teaching that permits to reveal its multidimensionality. Scholars concerned with teachers’ mathematical knowledge have traditionally distinguished knowledge dimensions by knowledge types, such as mathematical content knowledge or pedagogical content knowledge (e.g., the MKT framework). This approach has been widely adopted in studies that measure teachers’ knowledge using assessment items. But it remains an open question whether these conceptualizations can lead to precise measures of the different domains, as it is highly likely teachers simultaneously use multiple knowledge types when teaching mathematics. This creates challenges in measuring only mathematical content knowledge not mixed with any pedagogical aspects but still used in the work of teaching. While this way of conceptualizing knowledge dimensions has allowed researchers to develop measures that reflect professional knowledge, it has been less adept to documenting whether and how the knowledge varies depending on the specific teaching assignments teachers have experience with. The challenge in developing distinct measures has motivated me to propose a new way to organize assessment items. I describe this new way in terms of an item blueprint that specifies the correspondence between the organization of the items and the dimensions of the knowledge purported to be measured by the items. The proposed item blueprint is then evaluated regarding its purposes: 1) to capture multiple aspects of teachers’ mathematical knowledge used in teaching; 2) to develop precise multiple measures reflecting the identified dimensions of knowledge. Ultimately, the developed measures were designed to allow a fine-grained description of the knowledge used in the work of teaching secondary mathematics. The proposed item blueprint uses two organizers: task of teaching and instructional situation. Task of teaching alludes to each of the activities that comprise the practice of a mathematics teacher (e.g., understanding students’ work). Instructional situation alludes to each of the types of mathematical work students are assigned within a course of study (e.g., doing proofs in geometry). Following the blueprint, I assigned each set of items to measure one knowledge dimension associated with one task of teaching and one instructional situation. By organizing the knowledge using these two organizers, the item blueprint allows a description of teachers’ knowledge with respect to the characteristics of the components of the work of teaching. With this conceptual rationale, the methodological feasibility of the item blueprint was evaluated by fitting item-factor models to the item responses collected from a nationally distributed sample of 602 U.S. practicing mathematics teachers. The distinctions among the factors were examined using model-comparison tests conducted under three different measurement models: structural equation modeling, item response theory, and diagnostic classification models. The results consistently showed that the majority of the hypothesized dimensions are statistically distinguishable by either or both of the organizers within and across both geometry and algebra courses of study. This distinction was further supported by different relationships with teachers’ educational background and teaching experience across the identified knowledge dimensions. By presenting an innovative item blueprint that is theoretically warranted and methodologically feasible, this study shows great promise for measuring multiple dimensions of teachers’ mathematical knowledge used in the work of teaching. It contributes to developing theory of mathematics teaching and to future item development for measuring knowledge used in professional tasks and instructional situations.

... This study reports on the design and initial validation of an MKT assessment of whole number multiplication and division. Current assessments of MKT have focused on either specific courses, such as Geometry or Algebra I (Herbst & Kosko, 2014;McCrory et al., 2012) or a wide range of content within a single mathematical domain, such as numbers and operations (Hill et al., 2008a). For example, McCroy et al. (2012) developed an instrument to test teachers' mathematics-teachingknowledge of Algebra, constructing items specific to student reasoning of algebra problems. ...

... McCrory et al.'s (2012) definition of mathematics-teaching-knowledge is similar to PCK, as it 765 includes knowing a student's mathematical reasoning and understanding possible misconceptions. Similar to McCrory et al. (2012), Herbst and Kosko (2014) developed items to investigate MKT in Geometry teachers by constructing items based on students reasoning and approach to geometry problems. Hill et al. (2008a) also created items focusing on PCK, but the majority of their assessment is focused on both common and specialized content knowledge for teaching. ...

... Since Hill et al.'s (2008a) writing this statement, items assessing PCK have been successfully written and validated. However, these are typically couched in an overarching assessment of MKT (Depaepe et al., 2015;Herbst & Kosko, 2014). By contrast, this paper focuses explicitly on PCK. ...

Multiplication and division are vital topics in upper level elementary school. A teacher's pedagogical content knowledge (PCK) influences both instruction and students' learning. However, there is currently little research examining teachers' PCK within this domain, particularly regarding professional education of future teachers. To help address this need, the present paper presents an initial validity argument for a survey of preservice teacher's PCK for multiplication and division.

... This paper investigates the mathematical knowledge for teaching of secondary preservice teachers (PSTs) and compares it with that of secondary in-service teachers (ISTs). To the extent that mathematical knowledge for teaching (MKT) is used in practice and that some measures of it show it correlates with experience (Herbst & Kosko, 2014), it is worth asking questions about this construct at the preservice stage: What is the variance of MKT among PSTs, and how much of that variance can be accounted for by components of PSTs' subject matter preparation such as coursework? ...

... While there is debate about how to conceptualize mathematical knowledge for teaching at the secondary level (Thompson, 2015;Speer, King, & Howell, 2015;McCrory et al. 2012), efforts have been made to utilize Ball et al.'s (2008) framework to measure MKT for specific secondary courses. The present study builds on the work of Herbst and Kosko (2014) who developed an instrument for measuring teachers' MKT for secondary geometry (MKT-G) which included items designed to tap into four domains of Ball's MKT framework (CCK, SCK, KCS, KCT). ...

... Similarly, Izak and colleagues (2010) found that experience teaching high school made a significant positive difference for middle school teachers' MKT. Herbst & Kosko (2014) found a positive relationship between secondary in-service teachers' overall MKT-G scores and their years of experience teaching geometry. Yet, they found no correlation between the number of college mathematics courses and teachers' MKT-G scores. ...

This study describes an investigation exploring relationships between pre-service teachers' (PSTs') mathematical knowledge for teaching geometry (MKT-G) and their educational experiences. Our data from 108 pre-service teachers from 6 universities suggest that PSTs' experience in the classroom had the most significant effect on their MKT-G scores. The close examination of correlations between PSTs' field experience and subsets of items reveals that PSTs' field experience has a greater positive correlation with the amount of MKT-G used in the task of formulating problems than the amount of MKT-G used in the task of reviewing students' work.

... pedagogical content knowledge (PCK). These domains, and their subdomains, have been particularly useful in creating multiple-choice quantitative assessments at the elementary (Hill et al., 2004), middle (Hill, 2007), and secondary level (Herbst & Kosko, 2014), as well as assessments based on representations of practice (Kersting, 2008;Kersting, Givvin, Sotelo, & Stigler, 2010). Such assessments have aided in exploring relationships observed between teachers' actions and their level of MKT (Ball et al., 2008;Kersting et al., 2010), as well as relationships between teachers' MKT and their decision-making in hypothetical scenarios (Kosko, in review;Kosko & Herbst, 2012). ...

... Such assessments have aided in exploring relationships observed between teachers' actions and their level of MKT (Ball et al., 2008;Kersting et al., 2010), as well as relationships between teachers' MKT and their decision-making in hypothetical scenarios (Kosko, in review;Kosko & Herbst, 2012). Part of what makes these assessments both reliable and valid is their construction of items surrounding particular tasks of teaching (Herbst & Kosko, 2014). In the case of assessments following Ball and colleagues' approach, items are situated in a task specific to mathematics teaching and the participant reading the item is solicited to make some form of decision (is a child's mathematics correct, do they hold a certain misconception, etc.). ...

... Shulman's (1986) contribution to this line of research was his conceptualization of teacher knowledge, particularly PCK. MKT, developed by Ball and colleagues as an extension of Shulman's work (Ball et al., 2008;Ball & Bass, 2000), has since been shown to be a useful factor in explaining a portion of teachers' decision-making (Kosko, in review;Hill, 2010: Kosko & Herbst, 2012.However, items included in assessments of MKT are situated in tasks of teaching (Herbst & Kosko, 2014). These tasks of teaching can serve as simplistic scenarios of classroom practice, often boiled down to a very particular moment in the potential decision-making process. ...

... Assessments of MKT are designed to measure the mathematical knowledge that teachers use in these teaching practices. A number of practice-based assessments of MKT have recently been developed for teachers of K-12 grades (Herbst & Kosko, 2014;Hill, Ball, & Schilling, 2008;Hill, Schilling, & Ball, 2004;Kersting, 2008;Krauss, Baumert, & Blum, 2008;McCrory, Floden, Ferrini-Mundy, Reckase, & Senk, 2012;Tatto et al., 2008). ...

... Assessments of MKT differ in how teaching practice is represented. Some provide written descriptions, while others incorporate video or animations depicting mathematics teaching (see, for example, Herbst & Kosko, 2014;Hill et al., 2004;Kersting, 2008). These features of context support test takers in recognizing the relevant aspects of the content task, understanding the content problem, or providing a response to the assessment question. ...

Assessments of mathematical knowledge for teaching (MKT), which are often designed to measure specialized types of mathematical knowledge, typically include a representation of teaching practice in the assessment task. This analysis makes use of an existing, validated set of 10 assessment tasks to both describe and explore the function of the teaching contexts represented. We found that teaching context serves a variety of functions, some more critical than others. These context features play an important role in both the design of assessments of MKT and the types of mathematical knowledge assessed. © 2016 The Author(s) & Dept. of Mathematical Sciences-The University of Montana.

... Subsequent work on teacher content knowledge built off and refined the conceptual distinctions proposed by Shulman and his colleagues and eventually led to efforts to assess different components of teacher content knowledge (see, e.g., Gitomer et al., 2014;Herbst & Kosko, 2014;Kersting, 2008;Krauss et al., 2008;Mikeska et al., 2018;Phelps & Schilling, 2004;Sadler et al., 2013;Smith & Banilower, 2015;Tatto et al., 2008). CKT, one of the most widely referenced frameworks, was developed by Ball et al. (2008). ...

... They have included elementary or middle school mathematics (see for example, Kersting, 2008;Phelps et al., 2014;Tatto et al., 2008), science (Mikeska et al., 2017;Mikeska et al., 2018;Sadler et al., 2013;Smith & Banilower, 2015) and RLA (Carlisle et al., 2009;Phelps et al., 2014;Phelps & Schilling, 2004). While the work at the secondary level has been more limited, CKT assessments have also been developed for algebra and geometry (Herbst & Kosko, 2014;Howell et al., 2016;Krauss et al., 2008;McCrory et al., 2012;Mohr-Schroeder et al., 2017;Phelps et al., 2014), physics , and English . ...

In this report we provide preliminary evidence on the measurement characteristics for a new type of teaching performance assessment designed to be combined with complementary assessments of teacher content knowledge. The resulting test, which we refer to as the Foundational Assessment of Competencies for Teaching (FACT), is designed for use as part of initial teacher licensure. Twenty elementary FACT performance tasks (10 for mathematics [MATH] and 10 for reading language arts [RLA]) were developed and then administered to 59 teacher candidates. The results from the pilot indicate that the performance tasks function as designed with candidates completing the tasks on average in approximately 3.5 min. Human raters were able to score the tasks quickly and accurately. All score points were well represented for all the scored tasks. Total scores for all tasks combined and subscores for reading language arts RLA and MATH had respective alpha reliabilities of .86, .77, and .79, with the scores well distributed across the scale. A large majority of teacher candidates participating in the study strongly endorsed the FACT tasks as authentic, assessing valuable competencies, and suitable for use as part of teacher licensure. These preliminary results indicate that the FACT performance tasks show great promise for use in large‐scale, high‐stakes testing programs that seek to provide evidence of both the knowledge and skills needed for effective teaching.

... Extending Ball et al.'s (2008) framework, different scholars have examined teachers CK and PCK on various grounds. Herbst and Kosko (2014) developed an instrument to assess high school teachers' mathematical knowledge for teaching geometry (MKT-G). Khakasa & Berger (2016) applied six domains of MKT to categorize secondary school teachers' mathematical knowledge based on their interpretations of open-ended tasks. ...

... Khakasa & Berger (2016) applied six domains of MKT to categorize secondary school teachers' mathematical knowledge based on their interpretations of open-ended tasks. They found both the amount of experience and quality of experiences affect teachers' MKT (Herbst & Kosko, 2014;Khakasa & Berger, 2016). Of particular interest in the present study are those analyses of teachers' MKT for fractions. ...

Teachers' pedagogical content knowledge (PCK) influences their instruction and, by consequence, their children's opportunities to learn better. Within the domain of fractions, items assessing PCK are nested within larger assessments, with little explicit focus on the PCK domain. We report on the development and initial validity argument for a PCK for Fractions assessment that assesses preservice teachers' (PSTs') knowledge of students' fractional reasoning. Results suggest the assessment can differentiate between PSTs of different levels in their teacher education program, and that items appear to assess the intended construct. Implications for future study, and for how PCK may develop among PSTs is discussed.

... his colleagues (1986, 1987) aptly named it "pedagogical content knowledge," which significantly advanced the field. Researchers around the world probed the mathematical knowledge needed for teaching and began to find better answers (e.g., Adler & Davis, 2006;Ball, Thames, & Phelps, 2008;Baumert et al., 2010;Blömeke et al., 2015;Bruckmaier, Krauss, Blum, & Leiss, 2016;Carrillo, Climent, Contreras, & Muñoz-Catalán, 2013;Herbst & Kosko, 2014;Hill, Schilling, & Ball, 2004;Knievel, Lindmeier, & Heinze, 2015;McCrory, Floden, Ferrini-Mundy, Reckase, & Senk, 2012;Rowland, Huckstep, & Thwaites, 2005;Saderholm, Ronau, Brown, & Collins, 2010;Senk et al., 2012;Tatto et al., 2008;Tchoshanov, 2011). Studies have ranged from investigations of what teachers (and preservice teachers) know (or lack) (e.g., Ball, 1990;Baumert et al., 2010;Hill, 2007;Rowland et al., 2005;Thompson, 1984); what teachers learn from interventions, or other opportunities to learn mathematics (e.g., Borko et al., 1992;Hiebert, Morris, & Glass, 2003); to articulating positions about what teachers should know (e.g., Conference Board of Mathematical Sciences, 2001Sciences, , 2012McCrory et al., 2012;Silverman & Thompson, 2008). ...

... Some scholars developed measures of this special kind of knowledge (e.g., Bruckmaier et al., 2016;Herbst & Kosko, 2014;Hill, Ball, & Schilling, 2008;. It is beyond the scope of this paper to represent or discuss the many projects that sought to understand in more nuanced ways the kind of mathematical skill and insight teaching actually requires. ...

... They have also been able to demonstrate that teachers' awareness of many of these norms increases with experience teaching the courses to which they apply (e.g., Herbst, Aaron, Dimmel, & Erickson, 2013). Last, using a measure of teachers' stances towards breaches of a given task norm that consisted of asking participants to rate the relative appropriateness of two problems that differ only in terms of whether the norm of interest is breached or followed, called the diagrammatic register norm (DRN) instrument 27 (Herbst, Dimmel, & Erickson, 2016;Herbst, Kosko, & Dimmel, 2013), Herbst, Chazan, and colleagues were able to determine that a teacher's willingness to assign a non-normative problem may be related to how much of the knowledge needed to teach mathematics (Ball, Thames, & Phelps, 2008;Herbst & Kosko, 2014) they possess (Boileau, Dimmel, & Herbst, 2016). later described how the virtual breaching experiment methodology could be further developed by randomly assigning participants to either a set of storyboards in which the target norm is breached or a set of storyboards in which it is followed. ...

... Between March 2015 and January 2016, the INR GCA instrument was administered to a national sample of U.S. high school mathematics teachers, 36 along with several other questionnaires, including measures of other constructs (beside norms) that Herbst and Chazan (2012) suggest influence mathematics teachers' instructional decisions, such as the mathematical knowledge needed for teaching geometry (Herbst & Kosko, 2014). The initial design of the sample consisted of two stages: a selection of schools then a selection of one teacher from each school. ...

Studies have demonstrated that norms have considerable influence on human behaviour, in particular, that of teachers and students in mathematics classrooms. Studies have also shown that breaches of norms are frequently sanctioned, sometimes positively, but typically negatively. The present study builds on that literature by investigating two other potential consequences of breaching norms of mathematics instruction: that breaches of one norm of a given instructional situation may lead teachers to abandon their expectations that other norms of that situation will be followed and/or alter their attitudes towards breaches of those norms. I focus on the relationship between two hypothesized norms of geometric calculations with algebra (GCA) in U.S. high school geometry. One of them, the GCA-Figure norm, stipulates that the GCA problems that U.S. high school geometry teachers assign are expected to have geometrically-meaningful solutions. The other, the GCA-Theorem norm, stipulates that, when solving GCA problems, students are expected to document their algebraic work, to occasionally verbally state the geometric properties that warrant the equations that they set up, but not to document those properties. To confirm the existence of those norms and investigate whether breaches of the GCA-Figure norm would have either of the aforementioned consequences, I conducted a virtual breaching experiment. This consisted of randomly assigning U.S. high school mathematics teachers to one of three sets of multimedia questionnaires. Each questionnaire confronted the participant with a storyboard representation of a classroom scenario in which each of the two norms is either breached or followed. Their reactions to each storyboard were captured through a set of open- and closed-response items. Scores, based on coded open-responses and closed-responses, were compared across experimental conditions, using statistical models. This was done to predict whether experienced geometry teachers would be more likely to recognize decisions to breach either norm than decisions to follow it (evidence that the hypothesized norm exists), to deem decisions to breach it more acceptable than decisions to follow it, and/or to remark or disapprove of decisions to breach the GCA-Theorem norm when the GCA-Figure norm is followed than when it is breached. Results suggest that experienced geometry teachers’ expectations of GCA problems are well-represented by the above statement of the GCA-Figure norm, but that their expectations of solutions to GCA problems are slightly different than hypothesized. Namely, they suggest that experienced geometry teachers expect students to document their algebraic work, but not to share their geometric reasoning (verbally or in writing). In terms of attitudes towards breaches, results suggest that experienced geometry teachers are generally opposed to problems that breach the GCA-Figure norm, but do not provide much information about their attitudes towards students sharing their geometric reasoning, suggesting the need to develop alternate ways of measuring such attitudes in future research. Lastly, results suggest that experienced geometry teachers’ attitudes towards breaches of the GCA-Theorem norm are not dependent on whether the GCA-Figure norm is followed, but that such teachers may abandon their expectation that the GCA-Theorem norm will be followed when the GCA-Figure norm is breached. While the dissertation’s main contribution is to our understanding of norms of mathematics instruction, it also has implications for instructional improvement. Namely, the latter result suggests that changing even a very specific behaviour may alter whole systems of expectations—something that reformers must consider when anticipating what their recommendations will require.

... ISTs reported an average of 17.15 years of experience (Range = 4-32 years). Participation of ISTs was limited to those with at least 3 years of experience given such experience often distinguishes between novice and experienced teachers (Herbst & Kosko, 2014). ...

... The purpose of this study was to construct an initial validity argument for an assessment of PSTs' PCK. Following recommendations of others (Herbst & Kosko, 2014;Copur-Gencturk et al., 2019;Hill et al., 2008), we found that designing items with a focus on student conceptions to be a beneficial approach. The validity evidence collected from our pilot study provides support for an initial validity argument. ...

The quality of teachers’ mathematical knowledge for teaching (MKT) is critical for effective teaching and mathematical learning of students. However, most efforts on measuring MKT tend to focus on teachers’ content knowledge (CK), with less attention to teachers’ pedagogical content knowledge for teaching (PCK). This study reports on our initial efforts to develop and pilot a measure for assessing teachers’ PCK for fractions. Analysis of cognitive interviews from two expert teachers combined with Rasch modeling of 85 pre-service and in-service teachers was conducted to examine validity evidence for the PCK-Fractions measure. Results provide useful validity evidence for the initial validity argument of the measure. Namely, evidence suggests differences between pre-service (PSTs) and in-service teachers’ (ISTs) scores based on their professional level (junior PSTs, senior PSTs, & ISTs). Implications of this and additional validity evidence suggest a measure useful for assessing the effect of teacher education and professional experience initiatives, as well as indicators for revising this initial measure.

... There are many studies of teachers' mathematical knowledge for teaching mathematics, mostly at pre-high school levels. A growing number of articles address high school teachers' and university instructors' MKT (e.g., Boston 2012; Herbst and Kosko 2014; Lai and Weber 2013; Lewis and Blunk 2012), but the vast majority of studies are at the elementary level. A major criticism of research on teachers' MKT is that knowledge, the central construct of MKT, is rarely defined and is therefore operationalized inconsistently across investigations (Thompson 2013). ...

This book examines the kinds of transitions that have been studied in mathematics education research. It defines transition as a process of change, and describes learning in an educational context as a transition process. The book focuses on research in the area of mathematics education, and starts out with a literature review, describing the epistemological, cognitive, institutional and sociocultural perspectives on transition. It then looks at the research questions posed in the studies and their link with transition, and examines the theoretical approaches and methods used. It explores whether the research conducted has led to the identification of continuous processes, successive steps, or discontinuities. It answers the question of whether there are difficulties attached to the discontinuities identified, and if so, whether the research proposes means to reduce the gap – to create a transition. The book concludes with directions for future research on transitions in mathematics education.

... There are many studies of teachers' mathematical knowledge for teaching mathematics, mostly at pre-high school levels. A growing number of articles address high school teachers' and university instructors' MKT (e.g., Boston 2012;Herbst and Kosko 2014;Lai and Weber 2013;Lewis and Blunk 2012), but the vast majority of studies are at the elementary level. A major criticism of research on teachers' MKT is that knowledge, the central construct of MKT, is rarely defined and is therefore operationalized inconsistently across investigations ( Thompson 2013). ...

... Para identificar los recursos individuales del maestro nos basamos en contribuciones de otros investigadores. Por ejemplo, basados en la teoría del conocimiento matemático de los maestros (MKT) propuesta por Ball, Thames, y Phelps (2008), hemos desarrollado un test que estima el conocimiento de geometría de los maestros (Herbst & Kosko, 2014a). ...

A program of research on the practical rationality of mathematics teaching is introduced in Spanish, starting from problematizing teachers’ decision making in instruction. References to specific articles covering 15 years of research on the subject are provided. The notions of instructional exchange, instructional situation, situational norm, and professional obligation are introduced, then instruments designed to measure teachers’ recognition of norms and obligations are described, and research questions are proposed.

... There are many studies of teachers' mathematical knowledge for teaching mathematics, mostly at pre-high school levels. A growing number of articles address high school teachers' and university instructors' MKT (e.g., Boston 2012; Herbst and Kosko 2014; Lai and Weber 2013; Lewis and Blunk 2012), but the vast majority of studies are at the elementary level. A major criticism of research on teachers' MKT is that knowledge, the central construct of MKT, is rarely defined and is therefore operationalized inconsistently across investigations (Thompson 2013). ...

In this introductory chapter our aim is to set the scene for the following four chapters and situate them within the broad picture of mathematics education research concerning transitions. What kinds of transitions have been considered by mathematics education research? What research questions were studied and which theoretical approaches and associated methods were used? Did the studies lead to the identification of continuous processes, successive steps, or discontinuities? Are difficulties attached to the discontinuities identified, and does the research propose means to reduce those difficulties in order to foster a transition? These are the questions we study in this short literature review.

... As regards the development of ways of assessing teacher knowledge Manizade and Martinovic demonstrate how they use student work to elicit teachers' responses that allow them to assess what they know about specific geometric topics. In contrast, Smith uses the MKT-G test (Herbst & Kosko, 2014) to measure the amount of mathematical knowledge for teaching geometry of practicing and preservice teachers across the domains hypothesized by Ball, Thames, and Phelps (2008). Additionally, Smith uses a questionnaire to access self-reported pedagogical practices of her participants. ...

This chapter introduces the book by providing an orientation to the field of research and practice in the teaching and learning of secondary school geometry. The introduction then outlines the chapters in the book, each of which expands on the papers presented during the Topic Study Group on the teaching and learning of secondary school geometry at ICME-13, in terms of how they contribute to addressing questions asked in the field.

... Moreover, two issues emerge from the work of Ball and colleagues' work: Firstly, while dimensions of MKT have been conceptualized for practice, empirical support for these dimensions have been gathered via test items that refer to tasks involved in teaching. For example, Herbst and colleagues have been actively pursuing CCK, SCK, KCS, and KCT in secondary school geometry (Herbst & Kosko, 2014). Their work has been valuable in generating geometry problems and analyses of teaching scenarios to measure MKT in geometry (see also Smith, this volume). ...

Teacher knowledge that supports effective mathematics teaching has come under scrutiny alongside associated theoretical developments in the education field. Amongst these developments, the Mathematics Knowledge for Teaching (MKT) framework by Ball et al. (J Teacher Educ 59(5):389–407, 2008) has been one of the most influential. While MKT has been useful in helping us identify the knowledge strands teachers need for effective practice, the interplay among MKT’s knowledge strands during the course of teaching has received less attention. In this study, we address this issue by exploring interaction between Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK) in the domain of secondary geometry. We provide results of a preliminary study of SMK and PCK in the context of a teacher teaching students how to construct and bisect an acute angle with the aid of compass and ruler only. Our analysis suggests future research needs to consider (a) the particular characteristics of the discipline of geometry and (b) the developmental knowledge trajectories of teachers of geometry in order to better understand how teachers’ SMK influences and influenced by PCK.

... There are many studies of teachers' mathematical knowledge for teaching mathematics, mostly at pre-high school levels. A growing number of articles address high school teachers' and university instructors' MKT (e.g., Boston 2012;Herbst and Kosko 2014;Lai and Weber 2013;Lewis and Blunk 2012), but the vast majority of studies are at the elementary level. A major criticism of research on teachers' MKT is that knowledge, the central construct of MKT, is rarely defined and is therefore operationalized inconsistently across investigations (Thompson 2013). ...

This book examines the kinds of transitions that have been studied in mathematics education research. It defines transition as a process of change, and describes learning in an educational context as a transition process. The book focuses on research in the area of mathematics education, and starts out with a literature review, describing the epistemological, cognitive, institutional and sociocultural perspectives on transition. It then looks at the research questions posed in the studies and their link with transition, and examines the theoretical approaches and methods used. It explores whether the research conducted has led to the identification of continuous processes, successive steps, or discontinuities. It answers the question of whether there are difficulties attached to the discontinuities identified, and if so, whether the research proposes means to reduce the gap – to create a transition. The book concludes with directions for future research on transitions in mathematics education.

... However, this domain of mathematics education scholarship still has much to contribute to the development of instructional, curricular, and pedagogical innovations that seek "to improve the learning attained by anyone who studies mathematics" (ibid.). The overwhelming majority of research in this area has attended to one, or more, of the following foci: (1) characterizing the nature of mathematical and pedagogical knowledge teachers must possess to support students' conceptual mathematical learning (e.g., Ball, Thames, & Phelps, 2008;Fennema & Franke, 1992;Rowland, Huckstep, & Thwaites, 2005;Shulman, 1986Shulman, , 1887; (2) understanding the experiences by which teachers might construct such knowledge (e.g., Harel, 2008;Harel & Lim, 2004;Silverman & Thompson, 2008); (3) developing assessments to measure teachers' knowledge (e.g., Hill, Ball, & Schilling, 2008;Herbst & Kosko, 2014, Thompson, 2015, and (4) demonstrating the causal link between teacher knowledge and student achievement (e.g., Baumert et al., 2010;Campbell et al., 2014;Hill, Rowan, & Ball, 2005) or instructional quality (e.g., Charalambous & Hill, 2012;Copur-Gencturk, 2015;Even & Tirosh, 1995;. Stated succinctly, research on teacher knowledge in mathematics education has largely focused on what teachers need to know, how they might come to know it, how one might measure it, and the effect of this knowledge on instructional quality and student performance. ...

I present the results of a study designed to determine if there were incongruities between a secondary teacher's mathematical knowledge and the mathematical knowledge he leveraged in the context of teaching, and if so, to ascertain how the teacher's enacted subject matter knowledge was conditioned by his conscious responses to the circumstances he appraised as constraints on his practice. To address this focus, I conducted three semi-structured clinical interviews that elicited the teacher's rationale for instructional occasions in which the mathematical ways of understanding he conveyed in his teaching differed from the ways of understanding he demonstrated during a series of task-based clinical interviews. My analysis revealed that that the occasions in which the teacher conveyed/demonstrated inconsistent ways of understanding were not occasioned by his reacting to instructional constraints, but were instead a consequence of his unawareness of the mental activity involved in constructing particular ways of understanding mathematical ideas.

... This format might better lend itself to capturing more static aspects of teachers' knowledge, thus failing to tap into teachers' knowledge-in-use, which is how MKT has originally been theorized. This argument is supported by both studies reporting on the difficulties in writing multiple-choice items that capture knowledge-in-use (e.g., Herbst and Kosko 2014;Hill et al. 2008) and studies documenting the potential of more dynamic approaches that engage teachers in analysis of actual (e.g., Kersting et al. 2012) or simulated (e.g., Charalambous 2008) teaching practice in capturing this type of knowledge. This latter explanation surfaces another study limitation that pertains to the fact that we measured more static aspects of teacher knowledge instead of exploring more dynamic aspects that relate to engaging teachers in certain mathematical practices. ...

During the last three decades, scholars have proposed several conceptual structures to represent teacher knowledge. A common denominator in this work is the assumption that disciplinary knowledge and the knowledge needed for teaching are distinct. However, empirical findings on the distinguishability of these two knowledge components, and their relationship with student outcomes, are mixed. In this replication and extension study, we explore these issues, drawing on evidence from a multi-year study of over 200 fourth- and fifth-grade US teachers. Exploratory and confirmatory factor analyses of these data suggested a single dimension for teacher knowledge. Value-added models predicting student test outcomes on both state tests and a test with cognitively challenging tasks revealed that teacher knowledge positively predicts student achievement gains. We consider the implications of these findings for teacher selection and education.

... Ball et al. (2008) defined 1 3 SCK as a type of "pure" mathematical knowledge that is both distinct from PCK and only used in teaching. Work in this area has focused on elementary and middle school number concepts and operations, geometry, and algebra (e.g., Hill, Schilling & Ball, 2004;Phelps, Kelcey, Jones & Liu, 2016), secondary algebra Howell, Lai, & Suh, 2017;Lai & Howell, 2016), and secondary geometry (Herbst & Kosko, 2014;Mohr-Schroeder, Ronau, Peters, Lee & Bush, 2017). While each of these assessments differ somewhat in design and how they conceptualize and assess MKT, they all share a focus on measuring types of mathematical knowledge that are primarily or only used in the work of teaching. ...

Teaching mathematics requires a wide range of knowledge, including types of mathematical knowledge specialized to the work of teaching. Because specialized content knowledge is a form of professional knowledge, it is important to emphasize in professional preparation. In this paper, we present results comparing the performance of prospective and practicing teachers on the Learning Mathematics for Teaching (LMT) assessments. While the results show that practicing teachers score higher on the LMT assessments than prospective teachers, only about half of the individual items are significantly more difficult for prospective teachers. To gain a more nuanced understanding of the mathematical knowledge assessed by the LMT items, we also compared the performance of prospective and practicing teachers for items organized by the mathematical work of teaching. We discuss implications for identifying types of professional mathematical knowledge important to emphasize in teacher education.

... That reliance is based on the assumption that the content covered in that course is connected and relevant to the knowledge needed for teaching geometry in schools. However, secondary mathematics teachers and educational researchers alike continue to question whether the undergraduate mathematics course content is sufficiently connected to the work of a secondary teacher (Herbst & Kosko, 2014;Wu, 2011;Zazkis & Leikin, 2010). ...

... For example, Hill et al. (2008) proposed a model of Mathematical Knowledge for Teaching (MKT) that includes several aspects of both subject-matter knowledge and pedagogical content knowledge. Their work focused on elementary and middle-grades mathematics teaching, but other groups have extended the concepts to mathematics instruction at the secondary level, including geometry (e.g., Herbst & Kosko, 2014;Mohr-Schroeder, Ronau, Peters, Lee, &Bush, 2017) andalgebra (e.g., McCrory, Floden, Ferrini-Mundy, Reckase, &. Internationally, there has also been large-scale work characterizing high school teachers' MKT (e.g., Krauss, Baumert, & Blum, 2008) as well as math teachers' general pedagogical knowledge (e.g., Döhrmann, Kaiser, & Blömeke, 2012;Tatto et al., 2012), which includes knowledge like classroom management techniques. ...

... Substantial efforts have been made to improve teachers' mathematical knowledge and teaching practice to support students' learning and achievement in mathematics (Lampert, 2001; National Commission on Mathematics and Science Teaching for the 21st; Century, 2000;Stigler & Hiebert, 1999). However, the efforts developed to improve preservice teachers' geometry knowledge for teaching at the elementary level are limited when compared to the areas of numbers and algebra (Charalmabous & Pitta-Pantazi, 2015;Hill, Schilling, & Ball, 2004) and to middle or secondary grade levels (Brakoniecki, Glassmeyer, & Amador, 2016;Grover & Connor, 2000;Herbst & Kosko, 2014;Hollebrands, 2007;Jones, 2001;Zambak & Tyminski, 2017). ...

Teaching geometry at the elementary level is challenging. This study examines the impact of van Hiele theory-based instructional activities embedded into an elementary mathematics methods course on preservice teachers’ geometry knowledge for teaching. Pre- and post-assessment data from 111 elementary preservice teachers revealed that van Hiele theory-based instruction can be effective in improving three strands of participants’ geometry knowledge for teaching: geometry content knowledge, knowledge of students’ van Hiele levels, and knowledge of geometry instructional activities. As a result, this paper offers implications for teacher educators and policy makers to better prepare elementary preservice teachers with geometry knowledge for teaching.

... These arguments served to illustrate how teachers need to draw on PCK and other forms of professional knowledge to effectively teach a school subject. More recently, assessments of PCK have been developed across multiple subjects and grade levels (see, for example, Carlisle et al., 2011;Gitomer et al., 2014;Herbst & Kosko, 2014;Hill et al., 2004;Kersting, 2008;Krauss, Brunner et al., 2008;Mikeska et al., 2018;Phelps et al., 2020;Phelps & Schilling, 2004;Sadler et al., 2013;Smith & Banilower, 2015;Tatto et al., 2008). In their review of measures of PCK in mathematics, Depaepe et al. (2013) noted that all scholars agree on the following common characteristics of PCK: "it deals with teachers' knowledge, it connects content and pedagogy, it is specific to teaching a particular subject matter, and content knowledge is an important and necessary prerequisite" (p. ...

This study compared undergraduate physics majors and secondary physics teachers on two fundamental aspects of content knowledge for teaching (CKT). Since these groups have similar content knowledge (CK) backgrounds, they should have similar mastery of physics but should differ on knowledge associated with professional experience (i.e., pedagogical content knowledge [PCK]). Cognitive diagnostic modeling (CDM) was used to assign individuals into groups defined by differences in their physics CK and PCK. Both physics-related background and experience teaching physics were strongly associated with differences in PCK, providing empirical support that CKT is a unique construct associated with both content and professional preparation.

... In the past few decades, new types of assessments of content knowledge for teaching (CKT) have been developed with a focus on the content knowledge that is unique to the work of teachinga form of professional content knowledge distinct from the more general content competencies that have traditionally been assessed (see, for example, Ball et al., 2008;Gitomer, Phelps, Weren, Howell, & Croft, 2014;Herbst & Kosko, 2014;Hill, Schilling, & Ball, 2004;Kersting, 2008;Krauss, Baumert, & Blum, 2008;Mikeska, Kurzum, Steinberg, & Xu, 2018;Phelps & Schilling, 2004;Sadler, Sonnert, Coyle, Cook-Smith, & Miller, 2013;Smith & Banilower, 2015;Tatto et al., 2008). ...

Assessments of teacher content knowledge are increasingly designed to provide evidence of the content knowledge needed to carry out the moment-to-moment work of teaching. Often these assessments focus on content knowledge only used in teaching with the goal of testing types of professional content knowledge. In this paper, we argue that while this general approach has produced powerful exemplars of new types of assessment tasks, it has been less successful in developing tests that provide more general evidence of the range of content knowledge associated with particular teaching practices. To illustrate a more systematic approach, we describe the use of evidence-centered design (ECD) to develop an assessment of content knowledge for teaching (CKT) in the area of secondary physics-energy.

... Substantial efforts have been made to improve teachers' mathematical knowledge and teaching practice to support students' learning and achievement in mathematics (Lampert, 2001; National Commission on Mathematics and Science Teaching for the 21st; Century, 2000;Stigler & Hiebert, 1999). However, the efforts developed to improve preservice teachers' geometry knowledge for teaching at the elementary level are limited when compared to the areas of numbers and algebra (Charalmabous & Pitta-Pantazi, 2015;Hill, Schilling, & Ball, 2004) and to middle or secondary grade levels (Brakoniecki, Glassmeyer, & Amador, 2016;Grover & Connor, 2000;Herbst & Kosko, 2014;Hollebrands, 2007;Jones, 2001;Zambak & Tyminski, 2017). ...

Developing preservice teachers’ geometry knowledge for teaching is pivotal. This study examines the impact of van Hiele based interventions on such knowledge. Results revealed significant improvement in geometry content knowledge, knowledge of students’ van Hiele levels, and ability to choose appropriate geometry activities. It also found that geometry content knowledge and knowledge of students’ van Hiele levels significantly predicted ability to choose appropriate geometry activities.

... Examination of the contrast between experts and novices seeks to evince characteristics of expertise, not to discover causes of expertise or to achieve a conclusive way of identifying experts. To sample experts and novices, we operationalized indicators of expertise using experience teaching the geometry course and scores on a knowledge for teaching test specific to the geometry course: We looked for teachers with more than 5 years of experience teaching geometry and with Mathematical Knowledge for Teaching Geometry (MKT-G; Herbst & Kosko, 2014) scores over the national average. And we sampled novices from those teachers with 1 to 5 years of experience teaching geometry and MKT-G scores below the national average. ...

The investigation at scale of the tensions that teachers need to manage when deciding to follow recommendations for practice has been hampered by the problem of occurrence: The conditions in which those decisions could be made need to occur during an observation in order for observers to document how teachers handle them. Simulations have been recommended as a way to immerse teachers in instructional contexts in which they have the opportunity to follow such recommendations and observe what teachers choose to do in response. In this article, we show an example of how a teaching simulation may be used to support such investigation, in the context of policy recommendations to open up classroom discussion and consider multiple solutions and in the instructional situation of doing proofs in geometry. A contrast between the decisions made by expert and novice teachers (n = 59) in the simulation, analyzed using multiple regression, adds empirical evidence to earlier conjectures based on qualitative analysis of classroom teaching experiments that revealed teachers to be particularly attentive to epistemological and temporal constraints. We found that expert and novice teachers differed in how likely they were to prefer practices recommended by policymakers. Specifically, expert teachers were significantly more likely than novice teachers to open up classroom discussions when they had the knowledge resources to correct a student error. Similarly, expert teachers were significantly more likely than novice teachers to explore multiple solutions when there were no time constraints.

... For example, assessments of this type have been developed for elementary or middle school reading language arts (RLA; Carlisle et al., 2009;Phelps et al., 2014;Phelps & Schilling, 2004), mathematics Kersting, 2008;Phelps et al., 2014;Tatto et al., 2008), and science (Mikeska et al., 2017;Mikeska et al., 2018;Sadler et al., 2013;Smith & Banilower, 2015). Other projects have focused on assessing secondarylevel subjects, including English , algebra and geometry (Herbst & Kosko, 2014;Krauss et al., 2008;McCrory et al., 2012;Mohr-Schroeder et al., 2017;Phelps et al., 2014), and physics (Iaconangelo et al., 2017;Phelps et al., 2020). While most of these projects have focused on a single subject-with assessment frameworks that only refer to the topical organization of that particular subject-a number of projects have set out to develop a common or shared framework that can be used across subjects Phelps et al., 2020). ...

The primary purpose of this report is to provide preliminary evidence on the measurement properties for newly designed assessments of content knowledge for teaching (CKT) in elementary reading language arts (RLA) and mathematics. The goal is to offer the CKT tests through the PRAXIS® assessment. Additional analyses were conducted to provide initial evidence on the validity of the CKT assessments. One set of analyses investigated whether the test scores were sensitive to differences in participants' educational backgrounds that might be associated with opportunities to develop CKT. A second set of analyses involved examining score differences by the race/ethnicity of the participating candidates to provide evidence on whether the sample of participants in this study show group score differences that are comparable to what is typically observed on licensure exams. Finally, participant performance on the CKT tests was compared with performance on comparable PRAXIS assessments to examine potential differences in the difficulty of the items.

... 28-29 teachers outperformed US teachers on the MKT assessment. Other groups have extended the concept past elementary and middle school settings such as in Herbst and Kosko's (2014) high school geometry MKT. In sum, MKT has become a foundational tool across settings in mathematics education. ...

In this paper, we share two conceptual replications of Hill et al.’s (2012c) study linking Mathematical Knowledge for Teaching ( MKT ), Mathematical Quality of Instruction ( MQI ), and student assessment scores. In study 1, we share data from 4th and 5th grade teachers in an urban school district. In study 2, we share data from middle school teachers in a school district with a relatively high proportion of emergent bilingual students. By varying contexts, we found that Hill et al.’s (2012c) suggested use of the MKT cutoff points was not warranted in our differing settings. Further, we found some significant relationships between MKT, MQI , and student assessments; however, we were not able to reproduce these consistently with our data. We suggest that the relationship between teaching practice and MKT may be quite sensitive to contextual factors including grade level, demographics, school effects, and assessments. We recommend that policymakers and researchers take caution when using such instruments to evaluate program initiatives and identify teachers for remediation or leadership positions.
The impact sheet to this article can be accessed at 10.6084/m9.figshare.16610080 .

... The successful construction of proofs by a student depends on several prerequisites (Heinze and Kwak, 2002;Lin, 2005;Ufer et al., 2009). Knowing which prerequisites students need in order to handle mathematical proof and being able to assess students' individual availability of these prerequisites in diagnostic classroom situations enables teachers to adapt their teaching to ensure deep understanding as a learning outcome (Beck et al., 2008;Herbst and Kosko, 2014). Research on proving in mathematics education provides indications as to which student sub-skills and knowledge facets influence the ability to prove (van Dormolen, 1977). ...

Formative assessment of student learning is a challenging task in the teaching profession. Both teachers' professional vision and their pedagogical content knowledge of specific subjects such as mathematics play an important role in assessment processes. This study investigated mathematics preservice teachers' diagnostic activities during a formative assessment task in a video-based simulation. It examined which mathematical content was important for the successful assessment of the simulated students' mathematical argumentation skills. Beyond that, the preservice teachers' use of different diagnostic activities was assessed and used as an indicator of their knowledge-based reasoning during the assessment situation. The results showed that during the assessment, the mathematical content focused on varied according to the level of the simulated students' mathematical argumentation skills. In addition, explaining what had been noticed was found to be the most difficult activity for the participants. The results suggest that the examined diagnostic activities are helpful in detecting potential challenges in the assessment process of preservice teachers that need to be further addressed in teacher education. In addition, the findings illustrate that a video-based simulation may have the potential to train specific diagnostic activities by means of additional instructional support.

The results of numerous studies indicate that male students achieve better results on economic knowledge tests than their female classmates. Although this phenomenon has been known for a long time, the gender-specific mechanisms of this gap have not been explored in depth. According to social and educational scientific theories and the current state of research, interest and media use could be related to gender. In this study, we administered a German adaption of the internationally accepted Test of Economic Literacy to explore the gender gap in the economic knowledge of 983 students from 62 classes at 7 vocational secondary schools and commercial vocational schools specializing in business and economics in Germany to determine whether it is significantly influenced by interest and media use. The results indicate that a considerable part of the gender gap can be clarified by effects of interest in economic topics and the use of media to research them. Implications for social and economic education are discussed.

This chapter conceptualizes and illustrates StoryCircles, a form of professional education that builds on the knowledge of practitioners and engages them in collective, iterative scripting, visualization of, and argumentation about mathematics lessons using multimedia. The drive to invent and study new forms of professional education for mathematics teachers, such as StoryCircles, is predicated on the need to improve mathematics instruction. While many such efforts aim to support teachers to make broad sweeping changes, few take into account the actual predicaments of practice that make such changes difficult. StoryCircles aims to support teachers in making incremental improvements to practice by eliciting teachers’ practical wisdom and enabling participants to use each other’s knowledge and experience as resources for professional learning. In this chapter we outline critical characteristics of the StoryCircles interaction and illustrate how they are connected to seminal anchors in the professional development literature. We also illustrate those features with examples from various instantiations of StoryCircles. We close by providing some considerations for the affordances we see for the model both for the profession and for individual groups of teachers.

In this paper, we examine the relationships between teachers’ subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching algebra 1. We administered the same instrument to two different samples of teachers−high school practicing teachers and community college faculty−who teach the same algebra content in different levels of institutions, and we compared the performance of the two different samples and the relationships between the measured knowledge and their educational and teaching background
across the samples. The comparison suggested that the community college faculty possess a higher level of mathematical knowledge for teaching algebra 1 than high school teachers. The subsequent analyses using the Multiple Indicator Multiple Causes (MIMIC) models based on our hypothesis on the factors contributing to the differences
in the knowledge between the two teacher samples suggest that experience teaching advanced algebra courses has positive effects on the mathematical knowledge for teaching algebra 1 in both groups. Highlighting the positive effect of algebra-based teaching experience on test performance, we discuss the implications of the impact of subject specific experience in teaching on teachers’ mathematical content knowledge for teaching.

The present study explored whether primary grades teachers chose probing questions, given two hypothetical mathematics lesson scenarios. After responding to the mathematics lesson scenarios, participating teachers completed the Problems in Schools survey assessing dispositions to support student autonomy, and the Mathematical Knowledge for Teaching (MKT) assessment for primary grades patterns, functions and algebra. Logistic multiple regression was used to examine the influence of teachers’ MKT and dispositions for supporting student autonomy. Results differed by format of scenario. In the scenario where the choice of a probing question would act as an initial prompt for description, results showed this choice was influenced more strongly by MKT score. In the scenario where a choice of probing question followed an already embedded student description, choosing a probing prompt as a follow-up question was more strongly influenced by support for student autonomy. Additionally, a negative, statistically significant interaction effect was found across both scenarios. Implications for these findings are discussed.

How can basic research on mathematics instruction contribute to instructional improvement? In our research on the practical rationality of geometry teaching we describe existing instruction and examine how existing instruction responds to perturbations. In this talk, I consider the proposal that geometry instruction could be improved by infusing it with activities where students use representations of figures to model their experiences with shape and space and I show how our basic research on high school geometry instruction informs the implementing and monitoring of such modeling perspective. I argue that for mathematics education research on instruction to contribute to improvements that teachers can use in their daily work our theories of teaching need to be mathematics-specific.

This study proposes task of teaching as an organizer of dimensionality in teachers’ subject matter knowledge for teaching (SMK) and investigates it in the context of measuring SMK for teaching high school geometry (SMK-G). We hypothesize that teachers use different SMK-G in different aspects of their teaching work and that such differences can be scaled and associated with key elements of instruction. Analyses of 602 high school teachers’ responses to two sets of items designed to measure the SMK-G used in two particular tasks of teaching—understanding students’ work (USW) and choosing givens for a problem (CGP)—suggested the two scales of SMK-G to be distinguishable and differently related to experience in teaching high school geometry.

IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education. The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction. Areas covered include: teaching and learning secondary geometry through history; the representations of geometric figures; students' cognition in geometry; teacher knowledge, practice and, beliefs; teaching strategies, instructional improvement, and classroom interventions; research designs and problems for secondary geometry. Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students' study of geometry in secondary schools. © 2017 Patricio Herbst, Taro Fujita, Stefan Halverscheid, and Michael Weiss. All rights reserved.

This paper reports on an ongoing project aimed at developing an inter-institutional system of professional support for the improvement of the Geometry for Teachers (GeT) courses that mathematics departments teach to preservice secondary teachers. In alignment with the literature on improvement science (see Bryk et al., 2015; Lewis, 2015), it is essential to develop and deploy practical measurement tools to inform improvement. We describe three key forms of measurement our team has been using to drive this work as well as some preliminary findings.

In the commercial sector, which is of crucial importance to the Swiss economy among other countries, a large number of apprentices are trained on a vocational education and training programme every year. Besides other subjects, the subject Economics and Society forms an integral part of the vocational education and training curriculum and serves to prepare apprentices for professional, economic and civic participation. Although content knowledge is widely considered necessary to both teaching quality and student achievement, little is known about the subject-specific content knowledge of Swiss Economics and Society teachers. As previous research has shown a gender gap in the content knowledge of (pre-service) teachers in economics, we focus on the question as to whether Swiss Economics and Society teachers’ economics content knowledge differs, including in relation to gender. As additional influencing factors, our study included teaching experience and teaching load. We measured the economics content knowledge of 153 Economics and Society teachers with a shortened German version of the Test of Understanding College Economics in the German-speaking part of Switzerland. Multivariate analyses indicated a gender effect that manifested itself in higher test scores among male Economics and Society teachers. These findings are relevant to the training of vocational education and training teachers.

We investigate teachers' decision making in contexts where they could choose to provide students more authentic experiences with proving. Specifically, we investigate their preferences to depart from norms about what proof problems to assign to students. Scenario-based instruments consisting of two sets of items reflecting two hypothesized norms in doing geometric proofs, the given and prove norm and the diagrammatic register norm, were used to operationalize teachers' preference to depart from instructional norms in order to increase students' share of labor. By applying a diagnostic classification model (DCM) to classify teachers with respect to their depart from two norms, this study shows that teachers' decisions depend on the norm at issue. To examine individual factors associated with preference profiles, we use scores of teachers' mathematical knowledge for teaching, beliefs on the importance of student autonomy, and confidence in mathematics teaching. This study also illustrates methodological benefits of a DCM model in estimating a binary construct (i.e., teachers' preference), which has more than one sub-construct, with a small number of items.

This study compares the Geometry Teaching Knowledge of pre-service teachers with that of current high school geometry teachers. Data was collected using items from the Mathematical Knowledge for Teaching Geometry (MKT-G) assessment described by Herbst and Kosko (Mathematical knowledge for teaching and its specificity to high school geometry instruction. Research trends in mathematics teacher education. Springer, New York, pp. 23–45, 2014), and a post-assessment survey. The study focuses on the differences found in responses to items belonging to four domains: Common Content Knowledge-Geometry (CCK-G), Specialized Content Knowledge-Geometry (SCK-G), Knowledge of Content and Students-Geometry (KCS-G), and Knowledge of Content and Teaching-Geometry (KCT-G). Data was analyzed using t-tests for independent groups. Practicing high school geometry teachers outperformed the pre-service teachers on the MKT-G assessment in all four domains. Awareness of geometry instructional techniques and methods used in the current high school geometry classrooms was investigated as well. Practicing high school geometry teachers reported using and learning different instructional techniques and methods in their classrooms and professional development when compared to pre-service teachers’ techniques and methods used or learned in their education and mathematics courses.

Recent studies suggest that change is needed in undergraduate mathematics capstone courses for prospective secondary teachers. One promising but infrequently used strategy for improvement is to incorporate tasks that explicitly focus on pedagogical content knowledge (PCK). Expectancy-value theory provides an account for why instruction of these courses does not more regularly employ this strategy. To make this argument, this paper uses an interview-based study of 9 mathematicians that investigated the process of prioritizing tasks and goals for these courses. As the study found, these mathematicians valued developing teachers’ PCK. However, they were unconfident of their ability to teach with tasks and goals focused on developing teachers’ PCK relative to more purely mathematical tasks and goals. The central implication is that interventions in mathematicians’ teaching must take into account the possibility that it may be just as important to improve confidence and resources as it is to change values.

Las concepciones de los profesores sobre la matemática y sus procesos de enseñanza y aprendizaje fueron el punto de partida de nuestras investigaciones acerca del conocimiento del profesor (Carrillo y Contreras, 1995). Este documento pretende mos- trar cómo ha evolucionado nuestra investigación, donde el Conocimiento Especializado del Profesor de Matemáticas (Mtsk, por sus siglas en inglés) es el hilo conductor y resultado más patente, y donde las concepciones forman parte de su núcleo. En la introducción se justifica la elección de la temática y se presenta su estructura, que contendrá elementos de una trayectoria de investigación, con sus antecedentes, funda- mentos teóricos, fundamentos metodológicos, resultados y conclusiones, enmarcados en un momento temporal y contextual diferente.

This chapter addresses the role of technological tools in mathematics teacher
learning within a perspective that conceives of this learning as practice-based and
work-specific. The notion of practice-based and work-specific mathematics teacher
education is envisioned as a just-in-time endeavor emphasizing important continuities
between prospective and practicing teacher education. It does so by proposing a
conceptualization of mathematics teacher learning along the professional lifespan
as recognized with badges enabling holders to exercise their professional expertise
in particular work assignments. In turn, the procurement of these badges follows
participation in a set of technologically-mediated experiences that approximate the
work of teaching using representations of practice. And a set of diverse badges is
envisioned as available for practitioners to procure the skills needed for the work
they desire to do. Building on scholarship that documents the use of technologies
in mathematics teacher education, the chapter sketches how a combination of
those technologies may serve the achievement of teacher learning outcomes. The
chapter proposes a blend of Engeström’s (1999) model of an activity system with
Herbst and Chazan’s (2012) model of an instructional exchange to identify more
precisely objects of activity and the technological tools teacher-learners can use
in pursuing such work-specific learning. These considerations help the authors
illustrate the roles that various technologies, including technologies for media play
and annotation, social interaction and communication, storyboarding, animation,
gaming, and simulation, and technologies for mathematical work and inscription,
could play in different teacher learning activities. Building on the authors’ earlier
and present work studying geometry teaching and supporting teacher learning
in geometry, examples provided demonstrate how technology could support such
teacher learning activities toward a badge for teaching secondary geometry.

This article describes an investigation into mathematics for teaching in current teacher education practice in South Africa. The study focuses on formal evaluative events across mathematics teacher education courses in a range of institutions. Its theoretical orientation is informed by Bernstein's educational code theory and the analytic frame builds on Ball and Bass' notion of "unpacking" in the mathematical work of teaching. The analysis of formal evaluative events reveals that across the range of courses, and particularly mathematics courses designed specifically for teachers, compression or abbreviation (in contrast to unpacking) of mathematical ideas is dominant. The article offers theoretical and practical explanations for why this might be so, as well as avenues for further research.

We describe the development of measures of teachers' recognition of an instructional norm: that proof problems in high school geometry are presented in a diagrammatic register. A first instrument required participants to openly respond to depictions of classroom scenarios in which the norm was breached. A second instrument was a survey that required participants to rate the extent to which they agreed with various explicit statements about instruction. A third instrument capitalized on pros of the other two. We demonstrate how this instrument development process improved our conceptualization of the components included in the diagrammatic register norm.

While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.

We elaborate on the notion of the instructional triangle, to address the question of how the nature of instructional activity can help justify actions in mathematics teaching. We propose a practical rationality of mathematics teaching composed of norms for the relationships between elements of the instructional system and obligations that a person in the position of the mathematics teacher needs to satisfy. We propose such constructs as articulations of a rationality that can help explain the instructional actions a teacher takes in promoting and recognizing learning, supporting work, and making decisions.

This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. In particular, we focus on teachers' mathematical knowledge for teaching (MKT), which includes both the mathematical knowledge that is common to individuals working in diverse professions and the mathematical knowledge that is specialized to teaching. We use a series of five case studies and associated quantitative data to detail how MKT is associated with the mathematical quality of instruction. Although there is a significant, strong, and positive association between levels of MKT and the mathematical quality of instruction, we also find that there are a number of important factors that mediate this relationship, either supporting or hindering teachers' use of knowledge in practice.

See this Research Note at http://www.rasch.org/rmt/rmt74m.htm

This article presents a way of studying the rationality that mathematics teachers utilize in managing the teaching of theorems in high-school geometry. More generally, the study illustrates how to elicit the rationality that guides teachers in handling the demands of teaching practice. In particular, it illustrates how problematic classroom scenarios represented through animations of cartoon characters can facilitate thought experiments among groups of practitioners. Relying on video records from four study group sessions with experienced teachers of geometry, the study shows how these records can be parsed and inspected to identify categories of perception and appreciation with which experienced practitioners relate to an instance of an instructional situation. The study provides initial evidence that supports a theoretically derived hypothesis, namely that teachers of geometry as a group recognize as normative the expectation that a teacher will sanction or endorse those propositions that are to be remembered as theorems for later use. In interacting with a story in which students had proven a proposition that the teacher had not identified as a theorem, the study also shows the kind of tactical resources that teachers of geometry could use to make it feasible for students to reuse such a proposition.

Concerns about the use of computer-aided empirical verification in geometry classes lead to an investigation of students' understandings of the similarities and differences between the measurement of examples and deductive proof. The study reports in-depth interviews with seventeen high school students from geometry classes which employed empirical evidence. The analysis focuses on students' reasons for viewing empirical evidence as proof and mathematical proof simply as evidence.

Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers
to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been
expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics.
One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has
started to gain attention as an important concept in the mathematics teacher education research community, there is limited
understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article,
we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in
both mathematics education and the learning sciences.

We outline a theory of instructional exchanges and characterize a handful of instructional situations in high school geometry that frame some of these exchanges. In each of those instructional situations we inspect the possible role of reasoning and proof, drawing from data collected in intact classrooms as well as in instructional interventions. This manuscript is part of the final report of the NSF grant CAREER 0133619 “Reasoning in high school geometry classrooms: Understanding the practical logic underlying the teacher’s work” to the first author.All opinions are those of the authors and do not represent the views of the National Science Foundation.

Two questions are asked that concern the work of teaching high school geometry with problems and engaging students in building a reasoned conjecture: What kinds of negotiation are needed in order to engage students in such activity? How do those negotiations impact the mathematical activity in which students participate? A teacher's work is analyzed in two classes with an area problem designed to bring about and prove a conjecture about the relationship between the medians and area of a triangle. The article stresses that to understand the conditions of possibility to teach geometry with problems, questions of epistemological and instructional nature need to be asked not only whether and how certain ideas can be conceived by students as they work on a problem but also whether and how the kind of activity that will allow such conception can be summoned by customary ways of transacting work for knowledge.

The overall aim of our research project is to explore the impact of dynamic geometry environments (DGEs) on children's geometrical thinking. The point of departure for the study presented in this paper is the analytically and empirically grounded assumption that as the geometric discourse develops, the direct visual identification of geometric shapes gives way to discursively mediated identification, that is to a process in which one needs to perform a discursive procedure, prescribed by a formal definition of the shape, in order to ascertain the name of the shape. Previous research, conducted in static geometry environments, has already shown that many children, even in the middle school grades, rely on static, visual prototypes when identifying geometric shapes and that formal definitions, even if known, play no role in this process. Our study aimed at testing the conjecture that DGEs, in which the shapes can be continuously transformed, may flex the routine of identification, allowing for greater diversity in the shapes recognized as deserving a given name (e.g. triangle). This, we believed, would be an important step toward the discursively mediated routine of identification. The study, conducted among 4-5 year-old children working with Sketchpad, furnished some supporting evidence. In this paper, the focus is on one 30-min lesson during which the children observed, described, created and transformed triangles of different sizes, proportions, and orientations. During this one meeting the children's thinking evolved, in that the diversity of three-sided polygons they were prepared to call 'triangle' grew substantially. Not surprisingly, however, this rapidly-induced change was local and object-level rather than meta-level: it changed the children's use of a specific word rather than causing a transition to a discourse-mediated routine of identification.

This article deals with 12th-grade students' conceptions of a mathematical definition. Their conceptions of a definition were revealed through individual and group activities in which they were asked to consider a number of possible definitions of four mathematical concepts: two geometric and two analytic. Data consisted of written responses to questionnaires and transcriptions of videotaped group discussions. The findings point to three types of students' arguments: mathematical, communicative, and figurative. In addition, two types of reasoning were identified surrounding the contemplation of alternative definitions: for the geometric concepts, the dominant type of reasoning was a definition-based reasoning; for the analytic concepts, the dominant type was an example-based reasoning. Students' conceptions of a definition are described in terms of the features and roles they attribute to a mathematical definition.

Using a multimedia questionnaire we explore the extent to which secondary mathematics teachers recognize a hypothesized norm of doing proofs in geometry???that the teacher is in charge of providing the 'given' and the 'prove.' We also explore whether teachers who recognize the norm make a negative appraisal of its breach and find that geometry teachers are able to see some departures from the norm in a positive light. This finding suggests that it may be possible to expand geometry students' involvement in proof problems.

There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items. Our results suggest partial success in measuring this domain among practicing teachers but also identify key areas around which the field must achieve conceptual and empirical clarity. Although this is ongoing work, we believe that the lessons learned from our efforts shed light on teachers' knowledge in this domain and can inform future attempts to develop measures.

Widespread agreement exists that U.S. teachers need improved mathematics knowledge for teaching. Over the past decade, policymakers have funded a range of professional development efforts designed to address this need. However, there has been little success in determining whether and when teachers develop mathematical knowledge from professional development, and if so, what features of professional development contribute to such teacher learning. This was due, in part, to a lack of measures of teachers' content knowledge for teaching mathematics. This article attempts to fill these gaps. In it we describe an effort to evaluate California's Mathematics Professional Development Institutes (MPDIs) using novel measures of knowledge for teaching mathematics. Our analyses showed that teachers participating in the MPDIs improved their performance on these measures during the extended summer workshop portion of their experience. This analysis also suggests that program length as measured in days in the summer workshop and workshop focus on mathematical analysis, reasoning, and communication predicted teachers' learning.

Constructing Measures introduces a way to understand the advantages and disadvantages of measurement instruments, how to use such instruments, and how to apply these methods to develop new instruments or adapt old ones. The book is organized around the steps taken while constructing an instrument. It opens with a summary of the constructive steps involved. Each step is then expanded on in the next four chapters. These chapters develop the "building blocks" that make up an instrument--the construct map, the design plan for the items, the outcome space, and the statistical measurement model. The next three chapters focus on quality control. They rely heavily on the calibrated construct map and review how to check if scores are operating consistently and how to evaluate the reliability and validity evidence. The book introduces a variety of item formats, including multiple-choice, open-ended, and performance items; projects; portfolios; Likert and Guttman items; behavioral observations; and interview protocols. Each chapter includes an overview of the key concepts, related resources for further investigation and exercises and activities. Some chapters feature appendices that describe parts of the instrument development process in more detail, numerical manipulations used in the text, and/or data results. A variety of examples from the behavioral and social sciences and education including achievement and performance testing; attitude measures; health measures, and general sociological scales, demonstrate the application of the material. An accompanying CD features control files, output, and a data set to allow readers to compute the text's exercises and create new analyses and case archives based on the book's examples so the reader can work through the entire development of an instrument. Constructing Measures is an ideal text or supplement in courses on item, test, or instrument development, measurement, item response theory, or rasch analysis taught in a variety of departments including education and psychology. The book also appeals to those who develop instruments, including industrial/organizational, educational, and school psychologists, health outcomes researchers, program evaluators, and sociological measurers. Knowledge of basic descriptive statistics and elementary regression is recommended. © 2005 by Lawrence Erlbaum Associates, Inc. All rights reserved.

This article explores elementary school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teacher characteristics. The Learning Mathematics for Teaching project administered a multiple-choice assessment covering topics in number and operation to a nationally representative sample of teachers (n = 625) and at the same time collected information on teacher and student characteristics. Performance did not vary according to mathematical topic (e.g., whole numbers or rational numbers), and items categorized as requiring specialized knowledge of mathematics proved more difficult for this sample of teachers. There were few substantively significant relationships between mathematical knowledge for teaching and teacher characteristics, including leadership activities and self-reported college-level mathematics preparation. Implications for current policies aimed at improving teacher quality are addressed.

Defining what teachers need to know to teach algebra successfully is important for informing teacher preparation and professional development efforts. Based on prior research, an analysis of video, interviews with teachers, and an analysis of textbooks, the authors define categories of knowledge and practices of teaching for understanding and assessing teachers' knowledge for teaching algebra. They argue that the combination of categories and practices must be covered in assessments of teacher knowledge, if the assessments are to be used in research that investigates the presumed links among teachers' content preparation, their knowledge, their practice, and student learning.

While recent national and international assessments have shown mathematical progress being made by US students, little to no gains are evident in the areas of geometry and measurement. These reports also suggest that practicing teachers have traditionally had few opportunities to engage in content learning around topics in geometry and measurement. This article describes a set of assessment tasks designed to measure teachers’ mathematical knowledge for teaching geometry and measurement in a nuanced way. The tasks, focused on relationships between measurable quantities of figures, adhere to three key design principles: Tasks are grounded in the context of teaching, measure common and specialized content knowledge, and capture nuanced performance beyond correct and incorrect answers. Six tasks are presented that reflect these design principles, with teacher data illustrating the ways in which the tasks differentiate performance and reveal important aspects of teacher knowledge.

.. There are five good reasons to introduce your students to pentominoes. Pentominoe activities can challenge students from third grade through graduate school. The information found within this article is based on experiences with real studentsogrades three through graduate school.

Novel (as opposed to familiar) tasks can be contexts for students’ development of new knowledge. But managing such development is a complex activity for a teacher. The actions that a teacher took in managing the development of the mathematical concept of area in the context of a task comparing cardstock triangles are examined. The observation is made that some of the teacher’s actions shaped the mathematics at play in ways that seemed to counter the goals of the task. This article seeks to explain a possible rationality behind those contradictory actions. The hypothesis is presented that in managing task completion and knowledge development, a teacher has to cope with three subject-specific tensions related to direction of activity, representation of mathematical objects, and elicitation of students’ conceptual actions.

This article explores middle school teachers’ mathematical knowledge for teaching and the relationship between such knowledge and teachers’subject matter preparation, certification type, teaching experience, and their students’ poverty status. The author administered multiple-choice measures to a nationally representative sample of teachers and found that those with more mathematical course work, a subject-specific certification, and high school teaching experience tended to possess higher levels of teaching-specific mathematical knowledge. However, teachers with strong mathematical knowledge for teaching are, like those with full credentials and preparation, distributed unequally across the population of U.S. students. Specifically, more affluent students are more likely to encounter more knowledgeable teachers. The author discusses the implications of this for current U.S. policies aimed at improving teacher quality.

This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of didactical contractand double bindto uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching—what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition—are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.

In this article we discuss efforts to design and empirically test measures of teachers' content knowledge for teaching elementary mathematics. We begin by reviewing the literature on teacher knowledge, noting how scholars have organized such knowledge. Next we describe survey items we wrote to represent knowledge for teaching mathematics and results from factor analysis and scaling work with these items. We found that teachers' knowledge for teaching elementary mathematics was multidimensional and included knowledge of various mathematical topics (e.g., number and operations, algebra) and domains (e.g., knowledge of content, knowledge of students and content). The constructs indicated by factor analysis formed psychometrically acceptable scales.

Techniques emerging from the considerable research on cognitive aspects of survey methodology include various forms of probing and cognitive interviewing. These techniques are used to examine whether respondents' interpretations of self-report items are consistent with researchers' assumptions and intended meanings given the constructs the items are designed to measure. However, although informal procedures are common, such developments have not been systematically applied in educational research. We describe how information derived from the systematic application of cognitive pretesting can contribute to determining the validity—designated cognitive validity—of self-report items. Examples are presented from prominent motivation-related instruments that assess real-world instructional practices, mastery classroom goal structure, and student self-efficacy. The implications and pragmatics of adopting this approach are discussed.

Many researchers who study the relations between school resources and student achievement have worked from a causal model, which typically is implicit. In this model, some resource or set of resources is the causal variable and student achievement is the outcome. In a few recent, more nuanced versions, resource effects depend on intervening influences on their use. We argue for a model in which the key causal agents are situated in instruction; achievement is their outcome. Conventional resources can enable or constrain the causal agents in instruction, thus moderating their impact on student achievement. Because these causal agents interact in ways that are unlikely to be sorted out by multivariate analysis of naturalistic data, experimental trials of distinctive instructional systems are more likely to offer solid evidence on instructional effects.

This study explored whether and how teachers' mathematical knowledge for teaching contributes to gains in students' mathematics achievement. The authors used a linear mixed-model methodology in which first and third graders' mathematical achievement gains over a year were nested within teachers, who in turn were nested within schools. They found that teachers' mathematical knowledge was significantly related to student achievement gains in both first and third grades after controlling for key student-and teacher-level covariates. This result, while consonant with findings from the educational production function literature, was obtained via a measure focusing on the specialized mathematical knowledge and skills used in teach-ing mathematics. This finding provides support for policy initiatives designed to improve students' mathematics achievement by improving teachers' math-ematical knowledge.

The Diagnostic Teacher Assessment in Mathematics and Science (DTAMS) was developed to measure the content knowledge and pedagogical content knowledge of middle-school teachers. Its reliability and validity were initially established by reviewing national standards for content and use of expert question writing teams and reviewers. DTAMS was administered to approximately 1,600 middle-school mathematics teachers in 17 states. Subsequent analyses using structural equation modeling and item response theory were performed as part of a multistage validation process. This evaluation contributes to the body of work describing the reliability and validity of these assessments. The results of this study confirm trends in middle-school mathematics teacher preparation and certification and help explain middle-school student mathematics achievement levels.

Interest in teachers' subject matter knowledge has arisen in recent years. But most of the analysis has been general and not topic-specific. This paper shows how one may approach the question of teachers' knowledge about mathematical topics. It demonstrates the building of an analytic framework of subject matter knowledge for teaching a specific topic in mathematics and then uses the concept of function to provide an illustrative case of a paradigm for analyzing subject matter knowledge for teaching. The choice of the aspects, which form the main facets of the framework, was based on integrated knowledge from several bodies of work: the role and importance of the topic in the discipline of mathematics and in the mathematics curriculum; research and theoretical work on learning, knowledge and understanding of mathematical concepts in general and the specific topic in particular; and research and theoretical work on teachers' subject matter knowledge and its role in teaching. An application of the framework in the case of the concept of function is described and illustrated by anecdotes drawn from a study of prospective secondary teachers' knowledge and understanding of functions.

Foundations and methods of didactique

- G Brousseau

Brousseau, G. (1997). Foundations and methods of didactique. In N. Balacheff, M. Cooper,
R. Sutherland, & V. Warfield (Eds. & Trans.), Theory of didactical situations in mathematics:
Didactique des mathématiques, 1970-1990 (pp. 21-75). Dordrecht: Kluwer.

Expanding students’ involvement in proof problems: Are geometry teachers willing to depart from the norm? Paper presented at the 2013 Annual Meeting of the American Educational Research Association

- P Herbst
- W Aaron
- J Dimmel
- A Erickson

Teaching and learning of proof across the grades: A K-16 perspective

- P Herbst
- C Chen
- M Weiss
- G González
- T Nachlieli
- M Hamlin
- C Brach

Herbst, P., Chen, C., Weiss, M., González, G. with Nachlieli, T., Hamlin, M., & Brach, C. (2009).
"Doing proofs" in geometry classrooms. In M. Blanton, D. Stylianou, & E. Knuth (Eds.),
Teaching and learning of proof across the grades: A K-16 perspective (pp. 250-268). New
York: Routledge.

Expanding students' involvement in proof problems: Are geometry teachers willing to depart from the norm?

- P Herbst
- W Aaron
- J Dimmel
- A Erickson

Herbst, P., Aaron, W., Dimmel, J., & Erickson, A. (April 2013a). Expanding students' involvement
in proof problems: Are geometry teachers willing to depart from the norm? Paper presented
at the 2013 Annual Meeting of the American Educational Research Association. Available
from Deep Blue at the University of Michigan. http://hdl.handle.net/2027.42/97425. Accessed
12 Dec 2013.