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Doing proofs in geometry classrooms

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... "Doing proofs" is a frame that 3 teachers and students use to exchange the work they do around particulars (a particular statement of the goal, a particular diagram, a particular state of prior knowledge, etc.) for a general claim to the notion of mathematical proof (whether one has done or knows how to do a proof, etc.). Herbst, Chen, Weiss, and González (2009) have modeled "doing proofs" as a system of norms that specify what is exchanged, who has to do what, and how events are to unfold in time. ...
... Research on the role of proof in classroom work often has focused on the activities that students do (Balacheff, 1987;Boero, Garuti, & Mariotti, 1996;Maher & Martino, 1996), but only a portion of this research has begun to uncover how teachers manage justification in classrooms (e.g., Ball & Bass, 2000). Within this focus, Herbst (2002a) has striven to model from an observer's perspective the actions a teacher takes in managing a proof (see also Herbst et al., 2009). ...
... Previous research on the work of engaging students in proving has led to a provisional, descriptive model of the situation of "doing proofs" (Herbst et al., 2009), which describes the exchanges, division of labor, and organization of time in which teacher and students participate when they engage in proving. Such a model describes what Eamonn ordinarily would be expected to do after stating that the angle bisectors of opposite angles in a parallelogram are parallel: He should have justified the assertion, for example, by appealing to two sets of congruent angles formed by a transversal intersecting parallel lines. ...
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This article reports on an investigation of how teachers of geometry perceived an episode of instruction presented to them as a case of engaging students in proving. Confirming what was hypothesized, participants found it remarkable that a teacher would allow a student to make an assumption while proving. But they perceived this episode in various ways, casting the episode as one of as many as 10 different stories. Those different castings of the episode make use of intellectual resources for professional practice that practitioners could use to negotiate the norms of a situation in which they had made a tactical but problematic move. This collection of stories attests to the effectiveness of the technique used for eliciting the rationality of mathematics teaching: By confronting practitioners with episodes of teaching in which some norms have been breached, one can learn about the rationality underlying the norms of customary teaching. ["Seeing a Colleague Encourage a Student to Make an Assumption while Proving: What Teachers Put in Play when Casting an Episode of Instruction" was written with Gloriana Gonzalez.] (Contains 14 footnotes and 3 figures.)
... For example, when doing proofs in U.S. high school geometry, teachers expect that students will produce a mathematical proof and that they should exchange that work for claims on students' understandings of the mathematical properties used to justify their claims as well as their abilities to construct such mathematical arguments, or "do proofs" (Herbst, 2010, p.54). The proof itself should be presented as a sequence of statements and reasons (i.e., justifications of those statements), organized into a two-column table, which begins with the premise(s) of the mathematical proposition to be proved and ends with its conclusion (Herbst, Chen, Weiss, & González, 2009). 8 While there is evidence that students are aware of norms in other literature (e.g., Brousseau & Warfield, 1999;Coob, Wood, Yackel, & McNeal, 1992) Herbst and Chazan have focused their efforts on building a model of teacher decisions making Chazan, Herbst, & Clark, 2016) and have not investigated whether students are aware of the norms of instructional situations that they have described. ...
... For us, normative means not necessarily "correct" or "appropriate" from anybody's perspective but rather "unmarked"; a manner of doing joint activity that is ordinary, in the sense that it goes without saying. (Herbst, Chen, Weiss, & González, 2009) [N]orms [are] expectations of default behavior that, were they to take place in an instance of an instructional situation, they would go without saying but were they to be breached, they would elicit ad hoc 'repairs' (Herbst, Nachlieli, & Chazan, 2011, p.227). ...
... In terms of the types of norms that regulate the actions of teachers and students in these situations, Herbst and Chazan (2012) claim that there are norms of the (standard) didactical contract or contractual norms (e.g., that teachers will assign problems, which students are expected to solve, as Brousseau suggested), norms of instructional situations or situational norms (just defined), and task norms 25 . Herbst, Chen, Weiss, & González (2009) explain that norms of instructional situations fall into three categories: exchange norms, which stipulate "what work is done and what its exchange value is", division of labor norms, which stipulate "how labor is shared between teacher and student", and temporal norms, which stipulate "how that work is organized over time (the sequence and duration of events)" (p.254). ...
Thesis
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.
... A partir de nuestras observaciones de situaciones de demostración en la clase de geometría en Estados Unidos (Herbst, Chen, Weiss, & González, 2009), hemos conjeturado que la norma de enunciación de problemas de demostración es que éstos se enuncien en lo que llamamos el registro diagramático (diagramatic register; Herbst, Kosko, & Dimmel, 2013;ver también Boileau, Dimmel, & Herbst, 2016). Esta norma tiene cinco incisos o subnormas, a saber: ...
... Así pues, el enunciado de la derecha tampoco se adhiere totalmente a la norma del registro diagramático, aunque está más cercano a esa norma. Sin embargo, el hecho de que el enunciado además usa el imperativo "prove" (demuestre) para indicar qué se espera del estudiante (en contraste con el enunciado de la izquierda que sólo pregunta "por qué?"), hace que podamos predecir que quienes han sido socializados en la norma (por ejemplo, maestros que tienen experiencia enseñando geometría en Estados Unidos), reconocerán la opción de la derecha (Figura 2b) como la más apropiada (ver Herbst et al., 2009). ...
... Por ejemplo, en la lección analizada por Herbst & Chazan (2003) tomé yo mismo el papel del maestro y alenté a un estudiante a que prosiguiera escribiendo las afirmaciones de una demostración a pesar de que el estudiante no había podido justificar la última afirmación enunciada. La norma hipotética es que en la escritura de una demostración, cada enunciado debe de ser acompañado de su justificación antes de que se emita el siguiente enunciado (véase Herbst, et al., 2009). Este experimento de transgresión tenía como objetivo, entre otras cosas, ver cómo se adaptaba el resto de la clase, incluyendo el maestro titular que estaba presente, a una situación de demostración que se desviara de la norma, cómo el maestro daba cuenta de que había existido una desviación, y cómo reparaba la presunta transgresión. ...
Article
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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.
... Proof tasks also have common resources; students always have an available diagram, the givens, and the statement to be proved. 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 animated classroom scenario used in this study, The Square, was designed so that it would provoke reactions from experienced geometry teachers based on hypotheses about normal geometry instruction (Herbst et al. 2009). In The Square, a class engages with the Angle Bisectors Problem, which asks, ''What can one say about the angle bisectors of a quadrilateral?'' ...
Article
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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.
... We used cartoon storyboards to represent (part of) a lesson in which a high school geometry class was doing a proof. Herbst et al. (2009) have identi ed a number of norms of the situation of doing proofs in geometry, including how geometric objects are referred to, the order in which different types of contributions to a proof are made, and so on. We started with a scenario that we hypothesized to be normative, 102 that is, an instance of the situation of doing proofs where all norms would be complied with. ...
... The statement is warranted by the series of statements that had been deduced from the givens to that point (and that, hence, were mathematically correct). The situational norm would indicate the next intervention should be a reason for that statement (Herbst et al., 2009). However, mathematically, the statement made is not the most strategic one toward the end of proving that triangle BMN is isosceles: Rather than show that angles ABM and CBN are congruent, then that angles NMB and MNB are congruent, then that triangle BMN is isosceles, one could directly say that BM is congruent to BN then that triangle BMN is isosceles. ...
Chapter
In this chapter, we contribute some insights from project SimTeach (Herbst P, Chieu VM, SIMTEACH: What can practical knowledge modeled in a teaching simulator contribute to support mathematics teacher learning? National Science Foundation Grant, EHR, DRL-1420102, 2014) concerning the role that technologically mediated teaching simulations can play in learning to teach. We examine opportunities to learn about the teaching of mathematics afforded by the use of a digital simulation of a lesson in the context of simulation-based apprenticeship encounters between a novice teacher and four different experienced teachers who, in paired encounters, mentored her by sharing their thinking as they went through the simulation. These encounters gave opportunities to connect abstract concepts of teaching to practical ways in which they might be experienced in classrooms. We present this as a contribution to the perennial problem of connecting theory and practice in learning to teach.
... We used cartoon storyboards to represent (part of) a lesson in which a high school geometry class was doing a proof. Herbst et al. (2009) have identified a number of norms of the situation of doing proofs in geometry, including how geometric objects are referred to, the order in which different types of contributions to a proof are made, and so on. We started with a scenario that we hypothesized to be normative, that is, an instance of the situation of doing proofs where all norms would be complied with. ...
... The statement is warranted by the series of statements that had been deduced from the givens to that point (and that, hence, were mathematically correct). The situational norm would indicate the next intervention should be a reason for that statement (Herbst et al., 2009). However, mathematically, the statement made is not the most strategic one toward the end of proving that triangle BMN is isosceles: Rather than show that angles ABM and CBN are congruent, then that angles NMB and MNB are congruent, then that triangle BMN is isosceles, one could directly say that BM is congruent to BN, then that triangle BMN is isosceles. ...
Chapter
Teacher preparation programs are challenged with simultaneously training teachers who not only utilize effective teaching and classroom management strategies for a wide array of student learners but also boost student achievement and incorporate culturally responsive pedagogy. This chapter highlights how the use of a simulated learning environment (e.g., TeachLivE and Mursion) can provide teacher candidates opportunities for early practice with multiple opportunities to practice, reflect, and increase fluidity in a variety of classroom skills. Additionally, the increased impact of instructional coaching and how it can be used in combination with the use of a simulated learning environment within teacher preparation programs will be discussed.
... In the first iteration of this work, we created means to model those exchanges between mathematical work and knowledge claims: To model them means to represent them as the enactment of norms from subject-specific instructional situations. For example, the instructional situation "doing proofs" includes the norm that every statement in a proof must be justified by a reason (Herbst, Chen, Weiss, & González, 2009). The notion that some actions in the execution of mathematical work are informed by (in the sense of being in the domain of application of) some norms is key in our approach to modeling instruction and in what we can say about decision making at this time. ...
... Our group has been working with three instructional situations, one in geometry (doing proofs ;Herbst, et al., 2009) and two in algebra (solving one variable equations and solving word problems; Chazan & Lueke, 2009;Chazan, Sela, & Herbst, 2012). Our decision instruments concern one norm in each of the instructional situations in algebra and two norms in the instructional situation of doing proofs in geometry. ...
Article
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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.
... In this paper, we show how to design an intelligent teaching simulator, which aims at helping apprentice teachers develop the ability to manage students' engagement in proving in geometry. We have chosen the instructional situation of doing proofs (Herbst and Brach 2006; Herbst, Chen, Weiss, and González 2009) because it provides a simple but sufficiently complex context to illustrate our approach, which we believe could be applied to other instructional situations as well. By instructional situation we mean a system of norms for interaction, usually tacit, that regulate how the teacher and students trade the work they do in and through classroom interaction in exchange for claims on having taught and learned the knowledge at stake. ...
... To build the cK¢ model for the teaching simulator, we used results from studies of geometry classrooms (Herbst, Chen, Weiss, and González 2009) and results of analysis of teacher discussions about an atypical instance of the situation of " doing proofs " in geometry classrooms (Herbst and Chazan 2003; Nachlieli and Herbst 2009; Weiss, Herbst, and Chen 2009). In that instance, the teacher breached a situational norm (that the teacher needs to see that each statement in a proof is justified by a reason before the next statement in the proof is made) to solve the impasse of having a student silent after making a statement; instead the teacher suggested that the student assumed the statement made for the time being and continued on with the proof. ...
Article
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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, 2010), 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.
... Due to its importance on the development of modern society, mathematics teachers have the responsibility to present geometry to students correctly, precisely, and holistically, starting from primary schools (Bayrak et al., 2014;Sinclair & Bruce, 2015;Marchiş, 2008;Rolet, 2003;Boo & Leong, 2016) or even kindergartens (Casey et al., 2008;Zaranis & Synodi, 2017;Huleihil & Huleihil, 2011). The teaching of geometry is important because students learn how to justify their answers (Bayrak et al., 2014), as well as how to prove theorems (Herbst et al., 2010;Harel, 1999). Moreover, the teaching of geometry develops students' abilities of visualizing, critical thinking, intuition, perspective, problem-solving, guessing, and deductive thinking (Jones, 2002;Jones et al., 2012;Juperi, 2018). ...
Article
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While there are several scales that measure students’ mathematical attitudes, few are focused on measuring students’ attitudes towards geometry. Some of the scales present in literature measure different dimensions of students’ attitudes towards geometry, such as their enjoyment of the subject, motivation to learn it, and perception of the usefulness of learning geometry. In the present study, we present an instrument that can be used to measure students’ attitudes towards geometry by focusing on four factors: (1) students’ enjoyment, (2) perception of the usefulness of learning geometry, (3) negative (or positive) factors underlying the learning of geometry, and (4) students’ motivation to learn geometry. The instrument has been tested on a sample of 242 Italian high school students. An exploratory factor analysis has shown that the instrument is valid and the reliability of the instrument is high. Keywords: geometry, attitude, PCA, EFA, high school.
... Un factor que parece incidir en la generación de los procesos deductivos a partir del truncamiento del razonamiento configural, es la posibilidad de considerar un hecho geométrico no solo para identificar una propiedad de una configuración (aprehensión discursiva), sino también como parte de una secuencia de relaciones deductivas. Es decir, la posibilidad de reconocer que un hecho geométrico puede desempeñar papeles diferentes en el proceso de resolución (Herbst, et al. 2009). Esta situación plantea la necesidad de estudiar cómo los resolutores consideran los hechos y proposiciones geométricas identificadas en una configuración, o dadas como datos de un problema, como premisas en secuencias deductivas. ...
Conference Paper
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El objetivo de esta investigación es identificar las relaciones entre el conocimiento de geometría usado durante la resolución de problemas de probar y el truncamiento del razonamiento configural. Los resultados muestran diferentes trayectorias de resolución vinculadas a las sub-configuraciones relevantes. Estos resultados parecen indicar que el truncamiento del razonamiento configural está relacionado con la capacidad de los estudiantes de establecer relaciones significativas entre lo que conocen de la configuración y la tesis que hay que probar a través de algún conocimiento geométrico previamente conocido. The goal of this study is to identify the geometrical knowledge used during the resolution of geometrical proof problems and its relation to the “truncation” of configural reasoning. The findings show different resolution trajectories linked to relevant sub-configurations. These findings seem to indicate that the truncation of configural reasoning is related to how the students are able to related what is known about the configuration (the date of problem, and new generated fact) to thesis of problem (the thesis, what has to be proofed) through of some known geometrical fact.
... Jones (2000) points out that students are unable to distinguish between different forms of mathematical reasoning, such as explanation, argument, verification and proof as the curriculum emphasizes not on the wider reasons for and forms of proof, but on the format of the result. Herbst et al. (2009) shows that it is hard for students to learn to distinguish between what appears to be true and what they can justify as true based on reasons. ...
Chapter
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The teaching of geometry, especially the teaching of proof in geometry, is central to mathematics education in China at the lower secondary school level. This chapter uses a case study of how an expert mathematics teacher in Shanghai taught geometrical theorems to a class of Grade 8 students (aged 13-14 years) to illustrate the ‘shen tou’ (‘permeation’) method of teaching the initial stages of plane geometry. Comprising a set of teaching strategies, the ‘Shen Tou’ method aims gradually to develop the multiple layers of reasoning skills required in geometry, especially the skills to use geometrical language in writing proofs
... The next step in the design was to write a story that we thought would comply with (at least) experienced high school geometry teachers' expectations for how work on that problem might unfold. We did this by having the teacher comply with all of the instructional norms described in ​Herbst, Chen, Weiss, & González (2009);Figure 3shows its beginning. Then, we identified points in that story where we hypothesized an expert teacher might see reason to breach a norm (e.g., in order to comply with one of their professional obligations) and mapped out what alternative decisions would be made at those point; seeFigure 4). ...
Conference Paper
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The primary goal of this paper is to investigate whether a computer-based simulation can detect the difference between novice and expert teachers’ decision-making in mathematics instruction, which is complex in nature. The design of the simulation is grounded in a sociological perspective on practical rationality of mathematics teaching. The simulation consists of classroom scenarios, in the form of cartoon-based storyboards, with a series of decision moments to simulate the instructional situation of doing proofs in geometry. Empirical data helped verify and revise our design hypotheses/principles and showed that the simulation was able to detect some differences between novice-teacher and expert-teacher decision-making. Results of this study could inform the development of more advanced, computational models of mathematics teachers’ decision-making.
... When he was inferring information from the given or postulates, he still stated them as facts. This observation coheres with observations about the " doing proof " situation in high school geometry classes (Herbst, et al., 2009) in which students know the statement to be true before they do the proof. Unlike the dominant use of polar (positive) statements in the intact lesson, in the intervention lessons students used modality expressions when making conjectures and justifying them. ...
Article
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.
... There is a reflexive relationship between students' perceptions of their role, the teacher's role, classroom social norms and what is deemed mathematical activity (Yackel, 1996). "Doing proofs" embodies various actions by teachers and students which are influenced by stated or implicit norms of what work is valued, the structure in which proof ought to be presented, the time allocation for proving, and the responsibility of students and teachers while "doing proofs" (Herbst, 2009). ...
... Among them, content and exercises pertained to RP in mathematics textbooks were the main objects of investigation. Some researchers tend to choose the geometry content, which seems to be much more related to RP (e.g., Hanna & de Bruyn, 1999;Hanna, 2000;Herbst, 2002;Herbst et al., 2009;Sears, 2012). For example, Hanna and de Bruyn (1999) found that only geometry in Canadian textbooks provided opportunities for students to learn proof in an appropriate way. ...
... The fourth teacher, on the other hand, claimed that learning mathematics would not promote the logical ability of the students and that in order to do so there is a need to teach pure logic in the classroom. In line with the study of Herbst et al. (2009), the study presented here aims to acquire information about the views of people in the field regarding the role of learning mathematics in developing deductive reasoning. The study examines the views of people involved in various domains of mathematics education rather than focusing on opinion leaders and policy makers only. ...
Article
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This study examines the views of people involved in mathematics education regarding the commonly stated goal of using mathematics learning to develop deductive reasoning that is usable outside of mathematical contexts. The data source includes 21 individual semi-structured interviews. The findings of the study show that the interviewees ascribed different meanings to the above-stated goal. Moreover, none of them said that it is possible to develop formal logic-based reasoning useful outside of mathematics, but for different reasons. Three distinct views were identified: the intervention–argumentation view, the reservation–deductive view, and the spontaneity–systematic view. Each interviewee’s view was interrelated with the interviewee’s approach to deductive reasoning and its nature in mathematics and outside it. KEY WORDSdeductive reasoning-developing-mathematics educators-mathematics learning-views
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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. 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.
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