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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. T...

## Contexts in source publication

**Context 1**

... accepts that as correct and asks the class to talk about why that would be correct, while showing the diagram on Figure 7. Students offer various ideas, including that the height of the median is the same. After rehearsing various arguments with figure 8, Megan indicates that "Jade is right, that's the first part of the theorem… the second part of the theorem I don't think you're gonna guess. Okay, I originally was gonna have you measure stuff to figure this out, so I'm gonna show you a picture with measurements. ...

**Context 2**

... regards to the angles outside the rhombus, Tarina contributes that since GR is a radius, the triangle PGR is isosceles and so the angles ∠GPR and ∠GRP would be 30 degrees each. Cecilia notes that there are a couple of reasons that make that true: one because they need to add up to 60 to make 180 degrees along with the given ∠PGR which was known to be 120 degrees and two that they needed to be equal since the triangle PGR is isosceles (Figure 18). ...

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## Citations

... The work-specific practice of StoryCircles coupled with the possibility to retrofit MKT-G items to a DCM model suggests the possibility to align StoryCircles content and the attributes being assessed. In a future study, StoryCircles could focus on lessons that make use of instructional situations such as calculating in geometry, doing proofs, or exploring figures (Herbst, 2010) and we could use DCM and the national sample to assess change, providing even more focused evidence than what could be provided in this study. ...

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.

... Some resources, such as diagrams, construction tools and software, particular symbols and language, and mathematical argument play a special role in geometry that they don't play in other courses of study (Chazan & Sandow, 2011). Moreover, as these resources are used in the context of specific instructional situations (such as constructing or proving; see Herbst, 2010), the relevance of norms of usage becomes paramount. While some expectations for improving instruction may be general (e.g., the need to increase students' agency, engagement, or participation in mathematical work), in order for such expectations to become operational in a teacher-learner's practice, they need to be anchored in specific mathematical work, with its specific instructional norms. ...

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.

... As operations, students are expected to produce chains of true statements that they can recognize in the given diagram, justified deductively by retrieving more general reasons from their prior knowledge resources (Herbst et al. 2009). In contrast with proof tasks, conjecturing tasks, sometimes referred to informally as open-ended problems, are subject to a different framing, which Herbst et al. (2010) describes as situations of exploration: Students are usually provided resources in the form of a diagram, and they are asked to explore it, sometimes with some guidance from the teacher in terms of the operations that they could perform on that diagram (e.g., ''draw some angle bisectors,'' ''see if you can relate those angle measures'') . The exploration of a diagram often pursues the formation of generalizations about classes of figures. 2 The notion of instructional situation is a helpful lens for understanding why teachers may view the work of making conjectures as distinct from doing proofs: Within the situation of making conjectures, the statement is unknown and general, within the situation of doing proofs the claim is provided and often particular to a specific geometric object (Otten et al. 2014). ...

Inside the discipline, mathematical work consists of the interplay between stating and refining conjectures and attempting to prove those conjectures. However, the mathematical practices of conjecturing and proving are traditionally separated in high school geometry classrooms, despite some research showing that students can successfully navigate the interplay between the two. In this manuscript, we share perspectives from secondary mathematics teachers regarding what conjecturing and proving look like in geometry classrooms and possible rationales for why they are separated. We document that from the teachers’ perspective, the activities of conjecturing and proving have different goals, draw on different resources, and require different actions from students. In teachers’ eyes, these differences necessitate the separation of conjecturing and proving. By understanding teachers’ perspectives on the activities of conjecturing and proving, we can better consider the constraints of the classroom environment and possibly design activities in which conjecturing and proving could be reunited to allow students a more authentic mathematical experience.

... By task of teaching I mean a mathematical but general (that is, not specific to particular mathematical content) description of what the teacher is doing: creating problems for students to solve, observing students' work on problems, responding to students' work on problems, explaining new ideas, etc. By instructional situation (Herbst, 2006) I refer to the types of mathematical work that students do in particular courses of study: In algebra 1, these include solving, graphing, simplifying, calculating; in geometry, these include constructing, doing proofs, calculating, exploring figures (see Herbst et al., 2010). The two concepts, task of teaching and instructional situation, are described in greater detail in 3.3. ...

... As different instructional situations frame the way students undertake mathematical work differently, teachers' mathematical knowledge called for in the same task of teaching (e.g., evaluating students' work) may differ depending on the instructional situation. For example, in an instance of the instructional situation of calculation in geometry, the teacher task of evaluating student work may include verifying whether students used the known properties of a figure, set up the equations matched to those properties, and found an unknown measure of a single geometric object (Herbst, 2010). In an instance of another instructional situation, such as the situation of doing proofs, evaluating student work may include verifying whether students associated each statement with a correct reason and showed all necessary steps for the proof. ...

... Specifically, students may not measure the diagram in a situation of calculation in geometry, because diagrams are typically given with some extra signs (numbers or hash marks) revealing the properties of a figure. In another situation, students may be expected to measure or mark in a diagram and state properties they found when exploring a figure (Herbst, 2010). Thus, teachers' knowledge used to choose an appropriate diagram that can possibly engage students in intended mathematical work will be different between these two instructional situations. ...

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.

... That is, there are two kinds of problems that students are likely to have been socialized into by the time they are in high school geometry and that can serve as background against which to inscribe this task. One is the situation of exploration of a figure, in which students are given a diagram and various tools (e.g., rulers, protractors) and asked to find information about the figure represented (Chazan, 1995;Herbst, 2010). The other is the situation of geometric calculation in number (Hsu & Silver, 2014), in which students are given some dimensions of a figure and asked to find out other dimensions. ...

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.

... Writing about geometry classrooms, Herbst and colleagues have described the teacher's management of 'making conjectures' as a special case of the instructional situation of exploration which is a student-centered alternative for the introduction of a new idea. Herbst (2010) notes that in a situation of exploration, the work to be done includes the students' free choosing among a range of material operations to apply on concrete (physical or pictorial) embodiments of the concept depending the tools available to them, their reading of the particular results of those operations, and the translation of those results into general statements made in the conceptual register. The reasoning that students could thus have the opportunity to engage in can be described as abductive, proceeding from particular to general (p. ...

... (1) students work privately (individually or in groups) making conjectures and (2) the class publicly discusses those conjectures. Herbst (2010) also notes that in an exploration, There are engagement stakes, according to which it is important to involve students in actively doing something self directed in geometry. There are also content stakes that include new general statements about a generic instance of an abstract concept (p. ...

... Within the 'doing proofs' situation students' work is exchanged for the claim that students have developed skill with specific operations, such as applying theorems and identifying giving information. Herbst (2010) had described and illustrated how situations of exploration can be used to engage students in the introduction of new material. Clearly, conjecturing tasks could usher a class into a new theorem and some curricula have been developed to introduce geometric theorems through conjecturing (Serra, 1997). ...

Background: This paper reports on a study examining teachers’ perceptions of students they observed in an animated episode and who were engaged in the work of making conjectures in a geometry classroom. We examined eight conversations among subsets of 29 experienced geometry teachers with respect to how they described students and the mathematical work they perceived students to be engaged in. Results: Across the study group conversations, participants described students in terms of the tasks’ mathematical resources which students could understand or misunderstand and the tasks’ material and social resources which they could use or misuse, but participants paid little attention to the operations that students might employ in the task or the goals that students were working toward in the task. Conclusions: This study suggests that, when supporting students’ work on conjecturing tasks, teachers focus on the tasks’ resources which students use. This conjecture suggests in turn that in exchanging students’ work on conjecturing tasks for claims that students have learned a bit of the geometry curriculum, teachers might deem that particular work valuable on account of the resources used.

... Below, we describe the instruments we have developed for decisions and for norm recognition in the context of one of the norms in geometry: The norm that the givens and the 'prove' for proof problems are provided by the teacher. In fact, this norm might be one element in characterizing a difference between doing proofs, on the one hand, and another instructional situation, making conjectures (see Herbst, 2010). Clearly, our conjecture that this is the norm in geometry classrooms when doing proofs neither means that we endorse it as desirable or correct, nor that we consider it as determining what will inexorably happen. ...

This paper describes instruments designed to use multimedia to study at scale the instructional decisions that mathematics teachers make as well as teachers’ recognition of elements of the context of their work that might influence those decision. This methodological contribution shows how evidence of constructs like instructional norm and professional obligation can be elicited with multimedia questionnaires by describing the construction of items used to gauge recognition of a norm in “doing proofs” and an obligation to the discipline of mathematics. The paper also shows that the evidence can be used in regression models to account for the decisions teachers make in instructional situations. The research designs described in this article illustrate how the usual attention to individual resources in the research on teacher decision making can be complemented by attention to resources available to teachers from the institutional context of instruction.

... This effort contributes to a long-term agenda that seeks to understand the work of teaching in specific instructional systems, such as high school geometry. Our earlier work had been dedicated to conceptualizing and grounding (through examining records of intact classrooms and of instructional interventions) the didactical contract in geometry (Herbst 2002(Herbst , 2003 and instructional situations in geometry (Herbst 2006;Herbst et al. 2009Herbst et al. , 2010. In the context of that work, we developed a proposal for the description of teachers' actions and decisions as responses to norms of the role teachers play in activity systems (such as didactical contracts and their instructional situations) and obligations of the position of mathematics teacher in an institution (Herbst and Chazan 2012). ...

... While the former might involve the teacher in figuring out what the givens should be to make sure the desired proof could be done, the latter might involve the teacher in posing and solving equations and checking that the solutions of those equations represented well the figures at hand. Thus, while a list of generic tasks of teaching was useful to start the drafting of items, this list appeared to grow more sophisticated with attention to tasks that are specific to different instructional situations in geometry teaching (Herbst et al. 2010). As we note below, this observation led to interest in comparing responses to items that were differently related to instructional situations in geometry, and to making some conjectures about the organization of the MKT domains. ...

... In fact, the numbers obtained after solving the equations of that set of expressions would not work well to represent the sides of a triangle in that the triangle inequality 2 would not hold for those numbers. We conjecture that the experienced geometry teachers' familiarity with the instructional situation of "calculating a measure" (Herbst et al. 2010) mattered in their decision to check that the expressions would yield sides with positive lengths and that they would satisfy the triangle inequality. Our conjecture is not that the non-experienced geometry teachers did not know the triangle inequality. ...

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.

... Sostengo que la ambigüedad del problema propuesto hace posible que aquél pueda caber en dos de las situaciones de instrucción que hemos encontrado en las clases de geometría en secundaria (Herbst et al, 2010): el problema puede ser planteado dentro de una situación de construcción, así como también dentro de una situación de exploración. Describo estas situaciones y sus diferencias a continuación. ...

... Por lo general esta clase se ofrece a estudiantes avanzados en el noveno grado (entre 14 y 15 años) y a otros estudiantes en el décimo grado (15 y 16 años). 4 Véanse:Herbst y Brach (2006); Herbst, Chen, Weiss, y González, Nachlieli, Hamlin, y Brach (2009);Herbst, con González, Hsu, y Chen, Weiss, y Hamlin, (2010) ...

This paper shows how the use of problems in geometry can be can be a research tool to bring to the surface some phenomena in the management of instruction. It describes and exemplifies two classes of phenomena: the adaptation of problems so that students’ initial work on them takes advantage of norms of existent instructional situations, and the transition to a different instructional situation that permits the teacher sanction the work done as valuable. The paper discusses these phenomena in the context of an analysis a priori of the problem of the angle bisectors of a quadrilateral.

This paper describes how the notion of instructional situation can serve as a cornerstone for a subject specific theory of mathematics teaching. The high school geometry course in the U.S. (and some of its instructional situations — constructing a figure, exploring a figure, and doing proofs) is used to identify elements of a subject-specific language of description of the work of teaching. We use these examples to analyze records of a geometry lesson and demonstrate that, if one describes the actions of a teacher using descriptors that are independent of the specific knowledge being transacted, one might miss important elements of the instruction being described. However, if the notion of instructional situations is used to frame how one observes mathematics teaching, then one can not only track how teacher and students transact mathematical meanings but also identify alternative instructional moves that might better support those transactions. [Link to preprint of book: https://deepblue.lib.umich.edu/handle/2027.42/140744]