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This research article contributes to the growing literature highlighting the potential for innovation in mathematics education through design cycles that involve creative risk-taking and failure-based learning. Specifically, we explore how "failed" cycles of StoryCircles—a practice-based professional development approach that centers on teacher collaboration—have been productive in fostering innovations within the program. Our focus is on the challenges that arose in our efforts to enable feedback mechanisms within the StoryCircles system that support teachers' interrogation of their own instructional practice, as they collaboratively develop lessons and expand their collective knowledge base for teaching mathematics. Through examples of three challenges, we illustrate how various lesson artifacts, including those constructed by teachers in anticipation of implementation and those extracted from actual implementations, failed to serve as the sole source of feedback for supporting teachers' growth. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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How should we expect growing understandings of the nature of mathematical practice to inform classroom mathematical practice? We address this question from a perspective that takes seriously the notion that mathematics education, as a societal enterprise, is accountable to multiple sets of stakeholders, with the discipline of mathematics being only one of them. As they lead instruction, teachers can benefit from the influence of understandings of mathematical practice but they also need to recognize obligations to other stakeholders. We locate how mathematics instruction may actively respond to the influence of the discipline of mathematics and we exemplify how obligations to other stakeholders may participate in the practical rationality of mathematics teaching as those influences are incorporated into instruction. [ABSTRACT FROM AUTHOR]

We investigated how US secondary mathematics teachers expect students to present geometric proofs at the board. We analyzed video records of geometry classrooms and found students to be engaged in a practice that we call proof transcription—i.e., mark-for-mark reproductions of previously completed proofs that were not reasoned reconstructions of arguments. To investigate whether or not US secondary mathematics teachers recognize transcriptions as routine occurrences when students presented proofs, we conducted a survey experiment. Participants (n = 60) viewed episodes of instruction and answered questions that elicited their reactions to those episodes. The analysis of open- and closed-ended responses to the survey indicated that participants recognized transcriptions as routine. Our study contributes a fine-grained description of what teachers expect from students when students are called to present their work at the board. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This study presents an analysis of whether geometry teachers create opportunities for student discussion when engaging students in proving as revealed through the use of a multimedia survey instrument. We presented 42 secondary mathematics teachers with 8 multimedia narratives set in the situation of doing proofs in high school geometry and asked them to choose what they would do next, presenting options that included a normative instructional action which closed off discussion and less typical actions that encouraged student discussion. Our analysis provides insight into the professional obligations that teachers use to justify their departure from the norm in order to encourage student talk as well as the background variables that are associated with such decisions. We found that while secondary mathematics teachers frequently chose to promote discussion in their classrooms, the rationale that they chose for this decision and the reasons they might choose not to immediately encourage discussion differed according to the amount of experience they had teaching geometry. We use these differences to illustrate how the professional obligations can be used to better understand how teacher decisions are rationalized at the level of the instructional situation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we argue that to prepare pre-service teachers for doing complex work of teaching like leading classroom mathematics discussions requires an implementation of different pedagogies of teacher education in deliberate ways. In supporting our argument, we use two frameworks: one curricular and one pedagogical. The curricular framework is based on the work of Hammerness et al. (Preparing teachers for a changing world. What teachers should learn and be able to do. San Francisco, Jossey-Bass Educational Series, pp 358-388, ) outlining four main goals of teacher learning: a vision of practice, knowledge of students and content, dispositions for using this knowledge, and a repertoire of practices and tools. The pedagogical framework is based on the work of Grossman et al. (Teach Teach Theory Pract 15(2):273-289, ; Teach Coll Record 111(9):2055-2100, ) outlining three pedagogies of practice: representations, decompositions, and approximations of practice. We use the curricular framework to examine the opportunities for teacher learning that were afforded by these three different pedagogies of practice in a unit on leading classroom mathematics discussion in a secondary mathematics methods course. We use evidence from our analysis to show how the coordination of those pedagogies of practice is better than any one of them in addressing important goals for teacher learning about classroom discussions. [ABSTRACT FROM AUTHOR]

Although mathematical practice has traditionally valued two distinct kinds of mathematical work-referred to by Gowers () as theory building and problem solving-activity in classrooms appears to be organized largely around the latter, rather than the former. This study takes up the question of whether there is a customary role for theory building in the secondary geometry course and to what extent teachers of geometry hold students accountable for that disposition. We analyze records from study groups composed of experienced geometry teachers in the USA, and ask to what extent the disposition towards theory building is visible in those records, and how teachers react to the suggestion that it could play a larger role in their practice. Among our findings, we report that participants consistently attached a higher value to work organized around problem solving, rejecting the idea that they might hold their students accountable for theory building. [ABSTRACT FROM AUTHOR]

This study compared conversations among groups of teachers of high school geometry that had been elicited by a representation of instruction (either a video or an animation) and facilitated with an open-ended agenda. All artifacts used represented instruction scenarios that departed from what, according to prior work, had been hypothesized as normative. We used as the dependent variable the proportion of modal statements about instructional practice made by a group, which we argue is a good quantitative indicator that the statement appeals to the group's knowledge of the norms of practice. Animations and videos produced similar proportion of modal statements, but the types of modal statements differed-with animations being associated with more statements of probability and obligation and videos being associated with more statements of inclination. Overall, the results suggest that animations are just as useful as videos in eliciting these sorts of orientational meanings. [ABSTRACT FROM AUTHOR]

Mathematical argumentation in classrooms has been linked with students' increased understandings of and achievement in mathematics. A key component of mathematical argumentation is the teacher's use of questioning strategies. In particular, effective questioning asks students to explain and justify their work and probes student understanding. However, literature has consistently shown that instead of asking questions that probe student understanding, teachers generally ask leading or recall-oriented questions. Though reform efforts in the US and elsewhere have called for teachers to adopt more effective questioning strategies, the trend in teacher questioning suggested in the literature continues. One possible explanation for this trend is that teachers' interpretation of facilitating mathematical argumentation is not aligned with what reformers in mathematics education envision. Examining teacher-created representations of practice, the present study explores this possible difference in interpretation and presents preliminary evidence suggesting that the teachers' conceptions of facilitating mathematical argumentation may indeed be substantially different from what reformers intend. [ABSTRACT FROM AUTHOR]

This study explores interactions with diagrams that are involved in geometrical reasoning; more specifically, how students publicly make and justify conjectures through multimodal representations of diagrams. We describe how students interact with diagrams using both gestural and verbal modalities, and examine how such multimodal interactions with diagrams reveal their reasoning. We argue that when limited information is given in a diagram, students make use of gestural and verbal expressions to compensate for those limitations as they engage in making and proving conjectures. The constraints of a diagram, gestures and linguistic systems are semiotic resources that students may use to engage in geometrical reasoning. [ABSTRACT FROM AUTHOR]

This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that (a) help learners see the mathematical work of teaching in lessons they observe, (b) allow candidates to explore tactical decision-making in teaching, and (c) support preservice teachers in rehearsing classroom interactions. [ABSTRACT FROM AUTHOR]

Learning to teach is difficult for prospective teachers because of the complex nature of the work of teaching. Practicing (Lampert in J Teach Educ 61(1-2):21-34, ), interacting with the practice of teaching from a first-person perspective, may give them a unique experience in learning to teach. Computer-based simulators in which the apprentice teacher can interact with virtual students may be used to create that kind of experience. In this paper, we show how to apply techniques in artificial intelligence to design an intelligent learning environment. We show how to model the apprentice's decision making and resources that can help him or her improve the practice of teaching. [ABSTRACT FROM AUTHOR]

We describe the process followed to design representations of mathematics teaching in a community college. The end product sought are animated videos to be used in investigating the practical rationality that community college instructors use to justify norms of the didactical contract or possible departures from those norms. We have chosen to work within the trigonometry course, in the context of an instructional situation, 'finding the values of trigonometric functions,' and specifically on a case of this situation that occurs as instructors and students are working on examples on the board. We describe the design of the material needed to produce the animations: (1) identifying an instructional situation, (2) identifying norms of the contract that are key in that situation, (3) selecting or creating a scenario that illustrates those norms, (4) proposing alternative scenarios that instantiate breaches of those norms, and (5) anticipating justifications or rebuttals for the breaches that could be found in instructors' reactions. We illustrate the interplay of contextual and theoretical elements as we make decisions and state hypothesis about the situation that will be prototyped. [ABSTRACT FROM AUTHOR]

The article discusses several papers published within the issue including the availability to urban mathematics teachers of contexts through video, the role of representations in research and the use of representations of teaching in teacher education.

We investigate experienced high school geometry teachers’ perspectives on “authentic mathematics” and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving. [ABSTRACT FROM AUTHOR]

While every theorem has a proof in mathematics, in US geometry classrooms not every theorem is proved. How can one explain the practitioner's perspective on which theorems deserve proof? Toward providing an account of the practical rationality with which practitioners handle the norm that every theorem has a proof we have designed a methodology that relies on representing classroom instruction using animations. We use those animations to trigger commentary from experienced practitioners. In this article we illustrate how we model instructional situations as systems of norms and how we create animated stories that represent a situation. We show how the study of those stories as prototypes of a basic model can help anticipate the response from practitioners as well as suggest issues to be considered in improving a model. [ABSTRACT FROM AUTHOR]

Provides a historical account for how proving evolved as a task for students in school geometry, starting from the time when geometry became a high school subject and continuing to the time when proof became the centerpiece of the geometry curriculum. Stability brought about by two-column proof format; Reduction of students' participation in the development of new ideas.

The article presents a correction notice to the article "The Role of Theory Building in the Teaching of Secondary Geometry," by Michael Weiss and Patricio Herbst, published in volume 89, issue 2 of the June 2015 issue of the journal "Educational Studies in Mathematics."