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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.

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... Research that links in various ways to Radford's work are discussed herein: a) research utilizing theories that overlap in part at least with Radford's theoretical framing (e.g., Chen and Herbst 2013;David and Tomaz 2012); b) research employing some of the same analysis tools (Alibali et al. 2013); and c) research findings that may stimulate reflection on how they fit with Radford's (2013) findings. Radford (2013) and Chen and Herbst (2013) focused on interactions between the teacher and the students, and between students involved in multimodal interactions that included gesturing using materials, and experienced emotions (e.g., surprise about findings). ...

... Research that links in various ways to Radford's work are discussed herein: a) research utilizing theories that overlap in part at least with Radford's theoretical framing (e.g., Chen and Herbst 2013;David and Tomaz 2012); b) research employing some of the same analysis tools (Alibali et al. 2013); and c) research findings that may stimulate reflection on how they fit with Radford's (2013) findings. Radford (2013) and Chen and Herbst (2013) focused on interactions between the teacher and the students, and between students involved in multimodal interactions that included gesturing using materials, and experienced emotions (e.g., surprise about findings). Chen and Herbst (2013), studying student interactions during the development of geometrical proofs, found that diagrams that lacked part of the required information were productive tools for generative interactions: "if a diagram did not include signs to represent all the objects that could be talked about, students' allusions to those objects … might be more conjectural than factual" (p. ...

... Radford (2013) and Chen and Herbst (2013) focused on interactions between the teacher and the students, and between students involved in multimodal interactions that included gesturing using materials, and experienced emotions (e.g., surprise about findings). Chen and Herbst (2013), studying student interactions during the development of geometrical proofs, found that diagrams that lacked part of the required information were productive tools for generative interactions: "if a diagram did not include signs to represent all the objects that could be talked about, students' allusions to those objects … might be more conjectural than factual" (p. 304). ...

Learning and cognition is a classical and very vital area in research on mathematics education. Researchers have published many valuable research findings that have contributed to significant development in this area. The continued efforts of researchers now and in the future will, we hope, lead to extensive ‘pay-offs’. Different to many other special and related TSGs, such as teaching and learning of algebra, geometry, measurement, statistics, calculus, reasoning, proving and problem solving, to mention a few, TSG22’s participants will contribute a more general focus on learning and cognitive activity, and insights into students’ characteristics; their strengths and weaknesses in the process of mathematics learning.

... This choice is rooted in a multimodal perspective on conceptual knowledge (Gallese & Lakoff, 2005). In particular, several studies in Mathematics Education have pointed out the added value of also analyzing gestures to gain insight into cognitive processes (Arzarello, 2006;Arzarello et al., 2009;Arzarello et al., 2015;Chen & Herbst, 2013, just to name a few). Indeed, "by virtue of idiosyncrasy, co-expressive, speech-synchronized gestures open a "window" onto thinking that is otherwise curtained." ...

... Gesture and speech are considered jointly, since each of them expand the meaning possibilities of the other. In particular, the study of gesture coupled with speech during the solvers' interaction with drawings was found to be effective in revealing solvers' reasoning in making hypothetical claims about geometrical objects (Chen & Herbst, 2013). The reason is that, besides their communicative function, gestures play a role in shaping thinking, as is pointed out by McNeill (1992): "gestures, together with language, help constitute thought." ...

... When the solver has to do with objects only imagined it is particularly important to look at gestures, because they also help solvers in carrying out reasoning by providing alternative information not conveyed by words (Novack & Goldin-Meadow, 2015). More explicitly, Chen and Herbst (2013) highlight that, during the interaction with drawings, students utilize gestures to: ...

This paper focuses on the theoretical construct of geometric prediction (GP): a cognitive process through which a geometrical figure is manipulated, and its variations imagined, while certain properties are maintained invariant. The aim is to gain insight into this process as it is accomplished by solvers during the resolution of geometrical tasks within the domain of Euclidean Geometry. Interpreting geometrical reasoning in terms of a dialectic between the figural and the conceptual component of a figural concept, I introduce a framework of geometric prediction that I elaborated cyclically by observing, analyzing through a microgenetic approach, and re-analyzing solvers’ resolution of prediction tasks in a paper-and-pencil environment. The framework provides deep insight into geometric prediction, a specific and fundamental process in geometrical reasoning.

... Η σημασία των διαφορετικών τρόπων επικοινωνίας των μαθηματικών τονίζεται από ερευνητές οι οποίοι εστιάζουν στον τρόπο που τα παιδιά, της πρωτοβάθμιας και της δευτεροβάθμιας εκπαίδευσης, περιγράφουν τον τρόπο σκέψης τους (Chen & Herbst, 2013). Διερευνούν πώς ο λόγος και οι χειρονομίες συνεργάζονται και αλληλοεπηρεάζονται (Arzarello, Paola, Robutti & Sabena, 2009. ...

... Τα βοηθάει να διαμορφώσουν τις ιδέες τους και είναι ένας τρόπος για να γίνει κατανοητό τι υπάρχει στο μυαλό τους. Οι λεκτικές δεξιότητες και η ικανότητα να παράγουν γλωσσικούς κώδικες σε αλληλεπιδράσεις, παίζουν κρίσιμο ρόλο τόσο στη διδασκαλία όσο και στη μάθηση των μαθηματικών (Rudd, Lambert, Satterwhite & Zaier, 2008), επειδή εκφράζουν και παράγουν συλλογισμό αντιπροσωπεύοντας τη μαθηματική κατανόηση (Chen & Herbst, 2013). Στην περίπτωση των μαθηματικών, ειδικά των πρώτων εκπαιδευτικών βαθμίδων, κατά τη διάρκεια της διδασκαλίας και της μάθησης χρησιμοποιούνται δύο είδη γλώσσας: η τυπική μαθηματική γλώσσα και η άτυπη (Carruthers & Worthington, 2006). ...

... They emphasize the need for the students to have diagrams, possibilities to create representations of geometrical objects, and the opportunity to anticipate different relationships within and between these objects. When students make these "reasoned conjectures" [42] (p. 134), they construct mathematical knowledge. ...

... 134), they construct mathematical knowledge. Chen and Herbst also emphasize the social aspects of geometrical reasoning, in which the students discuss, make claims, and give arguments for these claims [42]. ...

Mathematical reasoning is gaining increasing significance in mathematics education. It has become part of school curricula and the numbers of empirical studies are growing. Researchers investigate mathematical reasoning, yet, what is being under investigation is diverse—which goes along with a diversity of understandings of the term reasoning. The aim of this article is to provide an overview on kinds of mathematical reasoning that are addressed in mathematics education research. We conducted a systematic review focusing on the question: What kinds of reasoning are addressed in empirical research in mathematics education? We pursued this question by searching for articles in the database Web of Science with the term reason* in the title. Based on this search, we used a systematic approach to inductively find kinds of reasoning addressed in empirical research in mathematics education. We found three domain-general kinds of reasoning (e.g., creative reasoning) as well as six domain-specific kinds of reasoning (e.g., algebraic reasoning). The article gives an overview on these different kinds of reasoning both in a domain-general and domain-specific perspective, which may be of value for both research and practice (e.g., school teaching).

... Consider the following task used by Chen and Herbst (2013). This task was designed to introduce students to the relationships among angles formed by intersecting and parallel lines, including the equivalence between the triangle sum theorem (which we knew they knew from middle school) and the parallel postulate. ...

... The intersecting lines task (reproduced according toHerbst & Chen, 2013, http://link.springer.com.proxy.lib.umich.edu/article/10.1007/s10649-012-9454-2/fulltext.html) ...

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.

... Various representations such as verbal interactions, gestures and written records including drawings, annotations, words, statements and symbolic notations afford W. Widjaja et al. students to reason and communicate justification effectively (Chen and Herbst 2013;Komatsu 2010;Lannin 2005;Lin and Tsai 2016;Yankelewitz et al. 2010). Stylianides (2007) emphasised the importance of using appropriate modes of argumentation and choice of representations such as concrete manipulatives in primary school particularly to reason about numbers. ...

... Lannin (2005) found the use of visual patterns enabled students to formulate appropriate generalisations and offer valid justifications in geometry. Similarly, Chen and Herbst (2013) found that the use of gestures helped students to communicate their reasoning and thinking to their fellow students and teachers particularly when dealing with geometric objects when students had not developed a formal conception of those objects. ...

Engaging students in comparing and contrasting, forming conjectures, generalising and justifying is critical to develop their mathematical reasoning, but there are untapped opportunities for primary school students to improve these reasoning processes in mathematics lessons. Through a case study of one task, this paper reports on levels of justifying and the connections to other reasoning processes of comparing and contrasting, forming conjectures and generalising observed among Year 3–4 Australian and Canadian students (9–10-year olds) using the mathematical reasoning actions and levels (MRAL) framework (Authors 2017). The findings revealed that examining commonalities and differences was critical to allow Year 3–4 students to form conjectures and generalise for themselves. Justifying by examples and seeking counterexamples to evaluate the conjecture were prevalent with some students attempting to develop logical argument. The findings have implications for frameworks that assess students’ levels of justifying and for teacher actions that encourage students to communicate their reasoning through oral and non-verbal as well as written communications.

... Author's personal copy the relationships between these two research domains (e.g., Chen and Herbst 2013;Duval 1999;Haj-Yahya et al. 2016). ...

... The way drawings are used can cause difficulties for students faced with proof assignments in which the private attributes of a single drawing (which are specific to one example of a concept, but are not critical attributes of the concept) are used in the proving processes instead of general attributes. Chen and Herbst (2013) claimed that students used diagrams to create representations of new geometric objects and anticipate geometrical relationships. In addition, Steenpass and Steinbring (2014) showed that children's discussions about visual representations are important for the establishment of conceptual understandings. ...

This study investigated the effects of students’ constructions of geometrical concepts related to the circle on their construction and validation of proofs. The participants were 110 high school students. A questionnaire was administered; both qualitative and quantitative methods were used to analyze the results. Afterwards, in-depth interviews were conducted with some of the participants. The findings from the interviews enriched and strengthened the findings from the questionnaire. Together, the findings highlight the impact of two factors on the ability to construct or evaluate proofs: (1) the use of the self-attributes of a single presented drawing instead of the critical attributes of the concept; and (2) the use of prototypical or non-prototypical examples. In this study, the position of the drawing attached to the assignment affected students’ construction of proofs.

... According to Arzarello (2006), a synchronic analysis enables the study of relationship among different semiotic sets activated simultaneously, while a diachronic analysis studies the same phenomenon in successive moments. Following Arzarello (2006), I used a synchronic lens to examine the interrelationships between linguistic and nonlinguistic modes of communication and a diachronic lens to investigate how this communication changed over time (see also Chen & Herbst, 2012). By performing these analyses, my goal is to highlight the use of touchscreen, dynamic technology for providing bilingual learners with access to calculus, opportunities to engage in mathematical communication, and possibilities to participate as members of the classroom community. ...

... In particular, the underlining of the transcript to keep track of the timing of one's dragging and speaking helped reveal one's routine, namely conjecturing and verifying. Given that recent studies have shown the importance of studying the interplay among linguistic and non-linguistic communication (Chen & Herbst, 2012;Ng, 2016), this study contributes to presenting directions for advancing discursive research, both theoretically and methodologically, in today's increasingly technological and multimodal learning environments. ...

In this paper, I introduce Sfard's discursive framework to examine secondary-school calculus students’ communication during exploratory activities mediated by the use of touchscreen dynamic geometry environments (DGEs). Six pairs of secondary-school students participated in an open-ended task to explore calculus relationships using touchscreen-DGEs. Qualitative data capturing the students’ linguistic communication (speech) and hand movements (gestures and dragging) were analysed when the students interacted with the touchscreen-DGEs used during the task. Findings suggest that new forms of communication were mobilised through the act of dragging on a touchscreen-DGE. In particular, routines for comparing, reasoning, conjecturing and verifying emerged within the use of the haptic DGE interface. In this paper, potentials of the touchscreen-DGEs in facilitating new forms of gestural thinking, as well as theoretical and methodological considerations that recognise the changing ways in which the hand and media interact, are discussed.

... Through the co-verbal responses, the listeners may signal their interest, atention, and understanding [15]. As a result, the role of co-verbal (and nonverbal) behavior in human communication and in human-machine interaction has been increasingly scrutinized over the last few decades, within a wide range of contexts [16][17][18][19][20][21]. Embodied conversational agents (ECAs) are nowadays the most natural selection for the generation of afective and personalized agents. ...

Conversation is becoming one of the key interaction modes in HMI. As a result, the conversational agents (CAs) have become an important tool in various everyday scenarios. From Apple and Microsoft to Amazon, Google, and Facebook, all have adapted their own variations of CAs. The CAs range from chatbots and 2D, carton-like implementations of talking heads to fully articulated embodied conversational agents performing interaction in various concepts. Recent studies in the field of face-to-face conversation show that the most natural way to implement interaction is through synchronized verbal and co-verbal signals (gestures and expressions). Namely, co-verbal behavior represents a major source of discourse cohesion. It regulates communicative relationships and may support or even replace verbal counterparts. It effectively retains semantics of the information and gives a certain degree of clarity in the discourse. In this chapter, we will represent a model of generation and realization of more natural machine-generated output.
https://www.intechopen.com/books/artificial-intelligence-emerging-trends-and-applications/advanced-content-and-interface-personalization-through-conversational-behavior-and-affective-embodie

... All this takes us to an important research connection. When mathematics education researchers started using video recording in their studies of cognition and classrooms, it became possible to conduct studies of the microgenesis of inscriptions such as diagrams or equations (e.g., Chen & Herbst, 2013). Earlier research technologies, such as audio recording or collecting students' written work, might not have allowed researchers to account fully for how students were interacting with figures or in what way a figure had been constructed. ...

This chapter concludes the collection of chapters, 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 articulating a vision for where the field of secondary school geometry could go in the near future, the chapter revisits issues of methodologies for data collection and data analysis. The chapter proposes how new technologies could be integrated into research and practice in secondary school geometry and outlines some of the questions that the field might expect to address with the aid of such technologies.

... Further, it regulates communicative relationships and may support or even replace the verbal communication in order to clarify or re-enforce the information provided by the verbal counterparts [4]. Thus, the co-verbal behavior effectively retains semantics of the information [5], provides suggestive influences [6], and gives a certain degree of clarity in the discourse [7,8]. Researchers such as Allwood [9], McNeill [10], Duncan [11], Bozkurt [12] and Poggi [13], among others, have made a significant effort in order to redefine the theory of communication and to push it well beyond the realm of pure linguistics. ...

Multimodality and multimodal communication is a rapidly evolving research field addressed by scientists working in various perspectives, from psycho-sociological fields, anthropology and linguistics, to communication and multimodal interfaces, companions, smart homes and ambient assisted living etc. Multimodality in human-machine interaction is not just an add-on or a style of information representation. It goes well beyond semantics and semiotic artefacts. It can significantly contribute to representation of the information as well as in interpersonal and textual function of communication. The study in this paper is a part of an ongoing effort in order to empirically investigate in detail relations between verbal and co-verbal behavior expressed during multi-speaker highly spontaneous and live conversations. It utilizes a highly multimodal approach for investigating into relations between the traditional linguistic (such as: paragraphs, sentences, sentence types, words, POS tags etc.) and prosodic features (such as: phrase breaks, prominence, durations, and pitch), and paralinguistic features traditionally interpreted as non-verbal communication or co-verbal behavior (such as: dialog role, semiotic classification of behavior, emotions, facial expressions, head movement, gaze, and hand gestures). The main motivation for this study is to be able to understand especially the informal nature of human-human communication, and to create co-verbal resources for automatic synthesis of highly natural co-verbal behavior from un-annotated text and expressed through embodied conversational agents. The EVA corpus designed by a novel EVA annotation scheme represents a rich empirical resource for performing such studies in conversational phenomena that manifest themselves in highly spontaneous face-to-face conversations. A preliminary analysis regarding emotions within conversations has been also conducted and presented in the paper.

... Bu hususta literatürde birtakım çalışmalar yer almaktadır. Bu çalışmaların bir kısmı öğrencilerin geometrik muhakeme örneklerini incelerken (Balacheff, 1988;Deliyianni ve ark., 2011;Duval, 1988;Karpuz, Koparan ve Güven, 2014;Michael-Chrysanthou & Gagatsis, 2013;Or, 2013;, diğer kısmı geometrik muhakemenin öğrencilere nasıl kazandırılacağı (Chen & Herbst, 2013;Duval, 1994;Gallagher, 2015;Köse, Uygan ve Özen, 2012;Magdaş, 2015;Samson, 2010) konusunda yürütülmüştür. Dinamik geometri ortamında birtakım etkinlikler tasarlayan Or (2013), bu etkinliklerin öğrencilerin işlevsel kavrama türünü kullanarak muhakeme yürütmelerine ve böylelikle geometri öğretiminin geliştirilmesine katkı sağladığını belirtmiştir. ...

... And gesture can play a variety of roles in the social construction of mathematical knowledge (Krause, 2016). For example, collaborators can use gesture to articulate aspects of a concept that has not been fully elaborated (Chen & Herbst, 2013), and to establish joint attention when mutual understanding is not yet accessible in language (Reynolds & Reeve, 2001). Thus, making progress toward conceptualizing and empirically studying relations between tacit knowledge and embodied interactions during collaborative modeling is a critical area for research. ...

... According to some gesture research, they can inform inscriptions. Châtelet (2000) frames this as diagrams capturing or transfixing gestures, while Chen and Herbst (2013) believe diagrams and gestures work together to bring the former to life. For example, a vector could represent the way another vector rotates and dilates when two complex numbers are multiplied together and a flick of the wrist could convey such a motion (Soto-Johnson and Troup 2014). ...

In this article, we describe the results of a case study examining the development of two undergraduate students’ geometric reasoning about the derivative of a complex-valued function with the aid of The Geometer’s Sketchpad (GSP). Initially, our participants saw it in terms of the slope of the tangent line. Without the aid of GSP, they could describe the rotation and dilation aspect of the derivative for linear complex-valued functions, but were unable to generalize this perception to non-linear ones. Participants’ use of GSP assisted with exploring function behavior, generalizing how for non-linear complex-valued functions the derivative describes the rotation and dilation of an image with respect to its pre-image, and recognizing that the derivative is a local property.

... Despite having learned the topics with the same dynamic sketches used in the study, the participants exemplified different discourses when prompted by two different types of environment. This offers important potential implications for classroom teaching, since it shows that mathematical thinking is not located in the heads but in the task and in the kinds of visual representations used (Chen and Herbst 2013). In order to develop and assess certain aspects of students' discourse in the study of functions and calculus, I have argued that providing situations for students to communicate these ideas in both static and dynamic environments, as well as adopting a multimodal view of communication, can be beneficial. ...

In this article, a thinking-as-communicating approach is used to analyse calculus students’ thinking in two environments. The first is a ‘static’ environment in the sense of static visual representations, such as those found in textbook diagrams, while the second is a dynamic environment as exploited by the use of dynamic geometry environments (DGEs). The purpose of the article is to compare calculus students’ communication as it is facilitated by each of these two environments, and to explore the role of paper- and digital-mediated representations for positioning certain ways of thinking about calculus. The analysis provides evidence that the participants employed different modes of communication – utterances, gestures and touchscreen-dragging – and they communicated about fundamental calculus ideas differently when prompted by different types of representations. The study presents implications for teaching dynamic aspects of functions and calculus, and argues for a multimodal view of communication to capture the use of gestures and dragging for communicating dynamic and temporal mathematical relationships.

... The text produced by students when communicating the resolution of a problem may reflect their cognitive styles and the development of these types of processes, as some people reason better with words and others reason better with figures [10]. The interaction between configuration representations and discourse when students are solving geometric exercises of proving provide information on the geometric reasoning of students, as both written discourse and verbal expressions or gestures can be considered semiotic resources used by students when they are engaged in problem solving and in communicating those resolutions [11]. ...

This article presents a study of configural reasoning and written discourse developed by students
of the National Polytechnic School of Ecuador when performing geometrical exercises of proving.

... The target competence for such a task would be for students to coordinate conceptual language and pointing (and potentially other) gestures while formulating a mathematical argument. Chen and Herbst (2013) found that student gesture production is affected by the features of diagrams and that students used metaphoric and iconic gestures when interacting with diagrams that did not have labels. That teachers of geometry could engineer tasks that might specifically help students develop their fluency in nonwritten modes of mathematical communication is an opportunity for teachers in geometry classrooms. ...

This dissertation investigates how teachers expect students to represent mathematical work. The goal of the study is to identify routine ways that students communicate in mathematics classrooms and to determine whether mathematics teachers recognize these routines. The instructional setting of the study is US high school geometry. The study looks specifically at how students are expected to communicate when doing proofs.
The study consists of two parts. The first part of the study examined video episodes of geometry classrooms to identify how students use different modes of communication when presenting and checking proofs in geometry classrooms. From the analysis of video episodes, I ground hypotheses of routine ways in which students use communication modalities; I call them semiotic norms.
The second part of the study is an experiment that uses representations of geometry instruction to investigate the extent to which secondary teachers recognize specific semiotic norms that I call details and sequence. The details norm describes what students are expected to include in the written statements of a proof. The sequence norm describes the expected order of events contributing to the writing and reading of proofs that students present proofs to the class.
The second part of the study used storyboards that represent episodes of geometry classrooms as probes for a multimedia questionnaire. Participants in the experiment viewed storyboards that represented teachers breaching or complying with the hypothesized norms. Seventy-three high school mathematics teachers from schools within a 60 mile radius of Midwestern University completed the questionnaires. The results of the experiment indicate that secondary mathematics teachers recognize that the details and sequence norms describe routine communication practices of the activity of doing proofs in geometry.
The work reported here identifies communication practices that students use in geometry classrooms when doing proofs. By describing these practices, the research reported in this dissertation contributes subject-specific knowledge of what routinely happens in mathematics classrooms. Knowledge of the routine ways that students communicate is valuable because it provides a foundation for developing discipline-specific communication skills in mathematics classrooms. In turn these inform our understanding of literacy practices in the mathematics classroom.

... Progress toward a solution is both indicated and facilitated by modifications to the recorded diagram that successfully capture the inherent structure of the problem (Nunokawa, 1994). Diagrams and other records can be an important multimodal communication resource, combining with gestures and language to form a semiotic bundle (Arzarello, Paola, Robutti & Sabena, 2009) used by students working together to solve a mathematical problem (Chen & Herbst, 2013). ...

This study explores how the records that students make during problem solving assist their cognition
and communication. Grounded in the problem-solving literature and cognitive load theory, we
examine the records that 14 middle grades students make as they solve geometry problems in one-on-one
task based interviews. We identify features of record keeping that assist the students with
cognition and communication and discuss the implications of this work.

... Progress toward a solution is both indicated and facilitated by modifications to the recorded diagram that successfully capture the inherent structure of the problem (Nunokawa, 1994). Diagrams and other records can be an important multimodal communication resource, combining with gestures and language to form a semiotic bundle (Arzarello, Paola, Robutti & Sabena, 2009) used by students working together to solve a mathematical problem (Chen & Herbst, 2013). ...

In this paper, we explore the benefits of record keeping strategies during problem solving, specifically for managing cognitive load and for aiding in communication about mathematical thinking. Using samples of middle grades students’ work, we describe varied strategies that students employed to keep records of their work and how they used these records later in their problem solving and when explaining their solution strategies to others. We argue that students’ record keeping, including records that do not have conceptual mathematical meaning, supports both cognition and communication.

... In tune with previous studies on DGEs-mediated student thinking (Falcade, Laborde and Mariotti, 2007), the students may have communicated about derivatives geometrically and conceptually as they exploited the functionalities offered in the sketch. As Chen and Herbst (2012) contend, "the constraints of diagrams may enable students to use particular gestures and verbal expressions that, rather than using known facts, permit students to make hypothetical claims about diagrams" (p.304). ...

This paper discusses the importance of considering bilingual learners’ non-linguistic forms of communication for understanding their mathematical thinking. In particular, I provide a detailed analysis of communication involving a pair of high school bilingual learners during an exploratory activity where a touchscreen-based dynamic geometry environment (DGE) was used. The paper focuses on the word-use, gestures and dragging actions in student-pair communication about calculus concepts as they interacted with a touchscreen-based DGE. Findings suggest that the students relied on gestures and dragging as non-linguistic features of the mathematical discourse to communicate dynamic aspects of calculus. Moreover, by examining the interplay between language, gestures, dragging and diagrams, it was possible to identify bilingual learners’ competence in mathematical communications. This paper raises questions about new forms of communication mobilised in dynamic, touchscreen environments, particularly for bilingual learners.

... Our primary goal in developing HandWaver was to create an environment where learners could use their hands to act directly on mathematical objects, without the need to mediate their intuitions through equations, symbol systems, keyboards, or mouse clicks (Sinclair, 2014). We designed HandWaver around natural movements of a user's hands-i.e., pinching, stretching, and spinning gestures-to foreground the connection between diagrams and gestures (de Freitas & Sinclair, 2012;Chen & Herbst, 2013). Gestural interfaces (Zuckerman & Gal-Oz, 2013)-where objects can be manipulated in natural, intuitive ways by movements of one's hands-allow a degree of direct access to virtual objects that have been shown to facilitate learning (Abrahamson & Sánchez-García, 2016) while minimizing cognitive barriers (Barrett, Stull, Hsu, & Hegarty, 2015;Sinclair & Bruce, 2015). ...

We report on the design and development of HandWaver, a mathematical making environment for use with immersive, room-scale virtual reality. A beta version of HandWaver was developed at the IMRE Lab at the University of Maine and released in the spring of 2017. Our goal in developing HandWaver was to harness the modes of representation and interaction available in virtual environments and use them to create experiences where learners use their hands to make and modify mathematical objects. In what follows, we describe the sandbox construction environment, an experience within HandWaver where learners construct geometric figures using a series of gesture-based operators, such as stretching figures to bring them up into higher dimensions, or revolving figures around axes. We describe plans for research and future development.

... Indeed, using the work of the historian of mathematics Gilles Châtelet, who studied the pivotal role of diagramming in mathematical inventions, de Freitas and Sinclair (2012) examined the interplay of gesturing and diagramming in undergraduate students' drawings, highlighting the way in which these drawings can be seen as gestures in "mid-flight" and thus capturing on the page the mobile actions of the hand. Also with an attention to the interplay between gestures, diagrams, and speech, Chen and Herbst (2013) compared the interactions of two groups of high school students: one working with a diagram that contained relevant labels (for vertices and angles) and another working with a diagram that contained no labels. While the students in the first group only used pointing gestures, those in the other group made gestures that extended the existing diagram (by extending a segment, for example) and thus created new geometric elements. ...

This chapter focuses on the relations between spatial reasoning, drawing and mathematics learning. Based on the strong link that has been found in educational psychology between children’s finished drawings and their mathematical achievement, and the central importance of diagramming in mathematics thinking and learning, we wanted to study children’s actual drawing process in order to gain insight into how the movements of their hands and eyes can play a role in perceiving, creating, and interpreting geometric shapes and patterns. We pay particular attention to the interplay between children’s drawings and their gestures, to the role of language in modulating children’s perceptions, and to the back and forth that drawing seems to invite between two-dimensional and three-dimensional perceptions of geometric figures. We seek to forge new ways of including drawing as part of the teaching and learning of geometry and offer new ways of thinking about and analyzing the types of spatial/geometric reasoning young children are capable of.

... This simple, highly accessible feature of the artifact helped him to quantify the squares in the grid and to compute the slope of the line. The important educational function of the virtual figures created by his gestures has been highlighted in other studies, which showed how this feature contributed to exploration and argumentation processes both in geometry (Chen and Herbst 2013) and in calculus (Yoon et al. 2011). The implication of the case reported here is that the combination of the virtual figures with those visible on the screen forms a new landscape, in which students can dynamically explore the mathematical concepts embedded in the function graphs. ...

This article describes construction processes of mathematical meaning of the function–derivative relationship, as it is studied graphically with a dynamic digital artifact. The discussion centres on a case study involving one student during his interaction with the artifact. He was asked to explain the connection between two linked dynamic graphs: the graph of a function and the graph of its derivative function. The study was guided by the semiotic mediation approach, which treats artifacts as fundamental to cognition and views learning as the evolution from meanings connected to the use of a certain artifact to those recognizable as mathematical, that is, connected directly to the mathematical object. In the course of three rounds of data analysis, the student was shown to progress from a point-specific view to an interval one, and to move toward a construction of the meaning of the derivative as a function. The actions of the student and his interactions with the artifact that enabled him to construct the mathematical meanings of the function–derivative relationship are identified and described.

... Our primary goal in developing HandWaver was to provide a space where learners at all levels could use their hands to act directly on mathematical objects, without the need to mediate intuitions through equations, symbol systems, keyboards, or mouse clicks (Sinclair, 2014). We designed the environment around natural movements of user's hands to foreground the connection between diagrams and gestures (de Freitas & Sinclair, 2012;Chen & Herbst, 2013). As one example of how the environment realizes this connection, the stretch operator multiplies (Davis, 2015) single points into many to form a segment, or multiples single segments into many to form a plane figure, or multiplies a single plane figure into many to form a solid. ...

This study aims at understanding the effect of collaborative and iterative GeoGebra intervention on
in-service mathematics teachers GeoGebra adoption in their teaching and the factors that mediate
that adoption. This article is one out of four parts of the study. The type of the study is a multiple
case studying in depth the effect of a GeoGebra (a free mathematics software) intervention on the
Technological Pedagogical Content Knowledge (TPACK) of in-service mathematics teachers in
secondary schools who follow the Lebanese curriculum. The methodology used is Design-Based
Research that focuses on working closely with practitioners in collaborative and iterative manner
in the real context to add principles to theory and practice. Results showed an increase in the level
of TPACK domains of teachers especially in their student-centered teaching approach.
Keywords: In-service secondary teachers PD, GeoGebra, TPACK.

... in the maze layout (Figure 3), as a substantial improvement for young students, to help the visualisation of the step between one square and another, so that the maze becomes a state diagram which is a main programming element and it is proven to improve understanding of problems (Chen & Herbst, 2013;Durak & Saritepeci, 2018;Watanabe, 2015) Expert judgement procedure ...

Computational thinking (CT) is a cognitive ability that is considered one of the core skills to be developed in order to successfully adapt to the future. Therefore, it is being included in school curricula all over the world and, gradually, at an earlier age. However, as the incorporation of CT learning in schools is recent, there is still no consensus on its exact definition or on how it should be assessed. Recent research suggests that systems of assessments should be used for this purpose, using various instruments, and thus covering the different CT dimensions. However, there is a lack of validated instruments
for the assessment of CT, particularly for early ages. Taking as a reference a three-dimensional CT framework, based on a validated CT test, and aimed at early ages (five- to ten-year-old students), the Beginners Computational Thinking Test has been developed as a tool to be used within a system of assessments. This instrument has been designed, submitted to a content validation process through an expert judgement procedure, and administered to primary school students, obtaining very favourable results in terms of its reliability.

... In their study, David and Tomaz (2012, p. 413) presented an illustrative episode that shows how drawing geometrical figures can play a major part in structuring and modifying mathematical activity in the classroom. Chen and Herbst (2013) agree that diagrams can play an important role in students' geometrical reasoning and help them to make reasoned conjectures (p. 304). ...

This study sets out to analyse the co-emergence of visualisation and reasoning processes when selected learners engaged in solving word problems. The study argues that visualisation processes and mathematical reasoning processes are closely interlinked in the process of engaging in any mathematical activity.
This qualitative research project adopted a case study methodology embedded within a broader interpretative orientation. The research participants were a cohort of 17 mixed-gender and mixed-ability Grade 11 learners from a private school in southern Namibia. Data was collected in three phases and comprised of one-on-one task-based interviews in the first phase, focus group task-based interviews in the second, and semi-structured reflective interviews in the third. The analytical framework was informed by elements of enactivism and consisted of a hybrid of observable visualisation and mathematical reasoning indicators.
The study was framed by an enactivist perspective that served as a linking mediator to bring visualisation and reasoning processes together, and as a lens through which the co-emergence of these processes was observed and analysed. The key enactivist concepts of structural coupling and co-emergence were the two mediating ideas that enabled me to discuss the links between visualisation and reasoning that emerged whilst my participants solved the set word problems. The study argues that the visualisation processes enacted by the participants when solving these problems are inseparable from the reasoning processes that the participants brought to bear; that is, they co-emerged.

... A particular focus should lie on the interaction with the (virtual or physical) material, which is realized by touching and moving it and, in the case at hand, performing touch gestures on a touchscreen. Spontaneous gestures in mathematics teaching and learning have been studied intensively in the past years (Krause 2016;Chen and Herbst 2013;Goldin-Meadow 2010;Edwards 2009;Arzarello et al. 2009). Touch gestures, however, are different as they are defined in the design of digital environments. ...

Physical models for equation solving typically lack feedback regarding their appropriate use. Such feedback is possible in virtual environments and could be implemented in hybrid models. Based on an epistemological analysis, this article presents a touch gesture as a way for users to signal they want to divide both sides of an equation and a design for feedback on the use of this so-called ‘division gesture’. The design is investigated by contrasting a case study, in which students used an app with the division gesture, with a preparatory study where students had to perform corresponding actions on physical manipulatives. This investigation revealed insight into feedback functions, steps of understanding dividing with this touch gesture and, furthermore, showed problems that students have with the boundary case where the dividend is 0. The study informs possible improvements of the design of the division gesture and of the overall learning environment. The results are reflected on, in order to illuminate known problems of learning how to solve linear equations, and theorized to contribute to the wider discussion around the design of digital and physical manipulatives, in particular the design of modes of interaction enabled by new technologies.

... This is intended to be a substantial improvement in maze layouts as our hypothesis is that difficulties with this type of layouts at early ages are related with disorientation and hesitations about whether the current and target square, at each step, should be part of the path sequence or ignored. Besides, adding transitions turn the maze in a state diagram, a main item in algorithms and coding which has proved to improve the capability to understand problems [29][30][31]. BCTt v.1 response alternatives are laid out as sequences of thick arrows, numbers and colors, depending on the computational concept involved in each question. ...

Computational Thinking (CT) is a fundamental skill that is not only confined to computer scientists' activities but can be widely applied in daily life and is required in order to adapt to the future and, therefore, should be taught at early ages. Within this framework, assessing CT is an indispensable part to consider in order to introduce CT in the school curricula. Nevertheless, efforts involving the formal assessment of computational thinking has primarily focused on middle school grades and above; and are mostly based on the analysis of projects in specific programming environments. A Beginners Computational Thinking Test (BCTt), aimed at early ages, and based on the Computational Thinking Test [1], has been designed including several improvements; submitted to a content validation process through expert´s judgement procedure; and administered to Primary School students. The BCTt design is considered adequate by experts and results show a high reliability for the assessment of CT in Primary School, particularly in first educational stages.

... The possible answers are sequences of movements represented by arrows, symbols, and numbers. Visual transitions were added in the maze layout (Figure 3), as a substantial improvement for young students, to help the visualisation of the step between one square and another, so that the maze becomes a state diagram which is a main programming element and it is proven to improve understanding of problems (Chen & Herbst, 2013;Durak & Saritepeci, 2018;Watanabe, 2015) Expert judgement procedure ...

The intended curriculum — the curriculum that is
intended to be taught through policy, curriculum
documents, or other required mandates — and
the enacted curriculum — the curriculum that
is actually taught in classrooms by teachers
— are ideally aligned. However, often there is
a chasm between the two. With computing
education being relatively new to schools and
teachers across many countries, we wanted
to learn if a chasm existed and, if it did, how
wide it is across different countries. Working
as part of an international team, we created
a set of templates for measuring intended
curricula and a survey instrument, MEasuring
TeacheR Enacted Curriculum (METRECC), to
measure enacted curricula. The original pilot
investigated the enacted curriculum in seven
countries (with 244 teacher participants). Our
research found that both visual and text-based
programming languages are being used across
K-12, warranting further research into potential
impact on student learning and motivations.
Unplugged activities are commonly used across
K-12, extending into later years despite not
being explicitly defined in intended curricula.
Further, teachers’ motivations for programming
language choice are consistent across countries
and our study revealed that student-driven
factors motivate selection. This initial study was
followed by additional analysis with respect to
teacher self-esteem that was found to differ across multiple factors such as experience in
teaching CS in years and gender. We punctuate
our work with the adaptation of the instrument
for use in South Asia and a call to the community
to consider middle- and low-income nations in
future research.

... Gestures and oral discursive representations are also languages that may express arguments and the structure of proof. In the classroom, different representations are used such as gestures, oral and/or written discourse, diagrams, and so forth (Chen & Herbst, 2013). Since the formulation of proof is concerned with ordinary language, this aspect shows strong cultural effects at both the grammatical and semantic levels. ...

This theoretical paper proposes a new perspective on identifying and characterizing the cultural specificities of proof and proving in the classrooms of a given country. To this end, based on the related literature, researchers propose "structure", "language", and "function" as a triplet of aspects that constitute proving activities. Researchers then exemplify each aspect in an example case of proving activities in a Japanese classroom and discuss how it allows us to characterize the cultural specificities of proof and proving.

... In the case of Amir and Muner, the most evident case of this second process is the "staircase shape," which is made present with the pointer and plays a role in the disclosure of the function graph as the derivative graph. The educational function of virtual figures created by the students' gestures, such as Muner's "staircase shape," has been highlighted in other studies showing how this feature contributed to exploration and argumentation processes both in geometry (Chen and Herbst 2013) and in calculus (Yoon et al. 2011). In our case, the combination of the virtual figures with those actually visible on the screen forms a new landscape in which the students can dynamically explore the mathematical concepts embedded in the function graphs. ...

This paper examines mathematical meaning-making from a phenomenological perspective and considers how a specific dynamic digital tool can prompt students to disclose the relationships between a function and its antiderivatives. Drawing on case study methodology, we focus on a pair of grade 11 students and analyze how the tool’s affordances and the students’ engagement in the interrogative processes of sequential questioning and answering allow them to make sense of the mathematical objects and their relationships and, lastly, of the mathematical activity in which they are engaged. A three-layer model of meaning of the students’ disclosure process emerges, namely, (a) disclosing objects, (b) disclosing relationships, and (c) disclosing functional relationships. The model sheds light on how the students’ interrogative processes help them make sense of mathematical concepts as they work on tasks with a digital tool, an issue that has rarely been explored. The study’s implications and limitations are discussed.

... Given Ng's interests in studying the interplay between language, gestures, dragging, and diagrams, she transcribed the videos by attending to turn, utterances, actions, the speaker, the dragger, the gesturer, and which events happened synchronously. She analyzed her transcripts by performing two types of analyses: a synchronic and a diachronic analysis (Arzarello, 2006;Chen & Herbst, 2013). The synchronic analysis allowed her to explore the relations between language, gestures, and diagrams, whereas the diachronic analysis provided insight into "whether certain utterances, gestures and dragging actions remained prevalent or changed over time" (Ng, 2016, p. 314). ...

Since 2008, Sfard’s theory of commognition has gained considerable traction in the study of discourse and learning in mathematics education and beyond. But researchers who have wanted to conduct commognitive studies have been faced with the challenge of figuring out how to operationalize the commognitive framework—work about which Sfard (2008) was not explicit. Thus, this article reviews different authors’ operationalizations of the commognitive framework to show those new to commognition how they might conduct a commognitive study while providing commognitivists with an opportunity to reflect on core meanings. First, I provide a brief introduction to the commognitive framework, highlighting the four critical components of mathematical discourses: word use, visual mediators, narratives, and routines. I then examine how commognition has been operationalized by: (a) reviewing who has been studied in which mathematical contexts, (b) describing commognitive transcribing practices, (c) examining how each critical component has been interpreted in commognitive research, and (d) identifying issues that commognitivists have left un(der)explored.

... Our primary goal in developing HandWaver was to provide a space where learners at all levels could use their hands to act directly on mathematical objects, without the need to mediate intuitions through equations, symbol systems, keyboards, or mouse clicks (Sinclair, 2014). We designed the environment around natural movements of user's hands to foreground the connection between diagrams and gestures (de Freitas & Sinclair, 2012;Chen & Herbst, 2013). As one example of how the environment realizes this connection, the stretch operator multiplies (Davis, 2015) single points into many to form a segment, or multiples single segments into many to form a plane figure, or multiplies a single plane figure into many to form a solid. ...

We report on the design and development of HandWaver, a gesture-based mathematical making environment for use with immersive, room-scale virtual reality. A beta version of HandWaver was developed at the IMRE Lab at the University of Maine and released in the spring of 2017. Our goal in developing HandWaver was to harness the modes of representation and interaction available in virtual environments and use them to create experiences where learners use their hands to make and modify mathematical objects. In what follows, we describe the sandbox construction environment, an experience within HandWaver where learners construct geometric figures using a series of gesture-based operators, such as stretching figures to bring them up into higher dimensions, or revolving figures around axes that learners can position by dragging and locking. We describe plans for research and future development. OVERVIEW OF HANDWAVER HandWaver is a gesture-based virtual mathematical making environment, currently optimized for in-room (as opposed to seated) immersive virtual reality platforms (such as the HTC Vive) that support gesture recognition. From points in space, users can construct uni-, two-, and three-dimensional mathematical objects through iterations of gesture-based operators. Figure 1 shows iterations of the stretch operator: a point is stretched into a line segment; the line segment is stretched into a plane figure; the plane figure is stretched into a prism. The hands that are shown in the images are virtual renderings of a user's actual hands, tracked via a Leap Motion sensor that is mounted to the virtual reality headset (see Figure 2). Figure 1. Different cases of the stretch operator: a point is stretched into a line segment, the segment is stretched into a plane figure, and the plane figure is stretched into a prism. Figure 2. A user (red sweatshirt) in the virtual space. The large monitor displays a 2D view of the user's first-person view of the virtual world. The device that tracks the user's hand movements is mounted to the front of the headset he is wearing. ICTMT 13 323 Lyon 3-6 July 2017 A second gesture-based operator is revolve. Users can position an axis in space, select objects to rotate around the axis, and then spin a wheel to revolve the selected objects around the axis. Revolving objects in this way creates surfaces of revolution. Figures 3 and 4 show different cases of the revolve operator. In Figure 3, a point is revolved to create a circle; the circle is then revolved around itself to create a sphere; and the circle is revolved around an axis to create a torus. Figure 3. Different cases of the revolve operator. The ship's wheel is a spindle that users turn to revolve figures. The line through the ship's wheel is the axis of rotation. In Figure 4, a segment is revolved parallel to an axis of rotation to create a cylinder; a segment is revolved perpendicular to an axis of rotation to create an annulus; the annulus is revolved around itself to create a sphere with a hole in its center. Figure 4. Different cases of revolving a segment. When the segment is parallel to the axis of rotation, the result is a cylinder. When the segment is perpendicular, the result is an annulus. The last image shows an annulus revolved around itself to create a sphere with a hole in its center (note: the hole is visible in the image by slicing the sphere). We organized the sandbox environment around the stretch and revolve operators to help learners train their dimensional deconstruction skills (Duval, 2014). Dimensional deconstruction is the process of resolving geometric figures into lower-dimensional components, rather than seeing them as whole, fixed shapes. In the HandWaver sandbox, learners can fluidly move from lower-dimensional shapes (e.g., circles) to their higher dimensional analogs (e.g., spheres) and vice versa. The environment brings plane and solid geometry together-subjects that have been separated from each other in the usual presentation of geometry in K-12 schools. The solid analogs of plane figures, in particular sphere-and-plane constructions, are "seldom developed" or "slighted...owing to their theoretic nature" (Franklin, 1919, p. 147). Three-dimensional dynamic geometry software (e.g., GeoGebra or Cabri 3D) has made it possible to engage in such constructions, however the limitations of two-dimensional screens has constrained their practicability. But for users immersed in a three-dimensional space-where the user has natural control over the angle at which an object is viewed, is able to move and manipulate the object in space, and can readily select the components of a figure to be incorporated into a new construction-three-dimensional constructive geometry becomes more feasible. Thus, a final feature of the sandbox environment is three-dimensional analogs of classic construction tools. The arctus tool (Figure 5) allows users to make a sphere centered at a point, through any other point. The size of the arc shown in the figure is variable, and the midpoint of the arctus tool can be locked to any point in the display. Arctus is a spatial compass that creates spheres. The user sets the arc to have the desired radius and then generates a sphere by spinning the arc through space. Figure 5. The arctus tool being used to inscribe a sphere. Users position the tool on a center point and on a point on the surface of the sphere. To generate the sphere, one turns the circle through space by spinning the blue wheel. The flatface tool (Figure 6) allows users to define a plane through any three points. A user sets one of the lines of the flatface to coincide with two of the three points. Once in place, the user sets the second line so that it is collinear with the third point. To generate the plane, one acts with the stretch gesture on one of the lines of the flatface. We implemented plane-and-sphere constructions via gesture-(and motion-) based virtual tools to mimic the physical actions of spinning a compass or drawing a line with a straightedge. Our goal in doing so was to highlight the manual history of making geometric figures. Figure 6. Series of images showing the flatface tool being used to spawn a plane. With arctus and flatface, learners can complete solid geometry construction tasks that are inherently virtual, such as constructing a tetrahedron from three spheres (see Figure 7). Figure 7. Constructing a tetrahedron from three-spheres in the HandWaver sandbox. These tools provide an occasion for learners to explore how plane geometry construction protocols can be extended to higher dimensions. Other experiences within the HandWaver environment include a volume lab, an operator lab, and LatticeLand, which is a spatial analog of the geoboard (Kennedy & McDowell, 1998). Users can define the edges or faces of polyhedra by selecting a circuit of lattice points with a virtual pin (see Figure 8). Figure 8. Connecting the dots in LatticeLand to define a the edges of a cube (second frame), a parallelepiped (third frame), a pyramid (fourth frame), and a trapezoid (fifth frame); the sixth frame shows the trapezoid cut into components (the orange triangle, the blue trapezoid). MOTIVATION AND DESIGN CONSIDERATIONS Our primary goal in developing HandWaver was to provide a space where learners at all levels could use their hands to act directly on mathematical objects, without the need to mediate intuitions through equations, symbol systems, keyboards, or mouse clicks (Sinclair, 2014). We designed the environment around natural movements of user's hands to foreground the connection between diagrams and gestures (de Freitas & Sinclair, 2012; Chen & Herbst, 2013). As one example of how the environment realizes this connection, the stretch operator multiplies (Davis, 2015) single points into many to form a segment, or multiples single segments into many to form a plane figure, or multiplies a single plane figure into many to form a solid. The notion that n-dimensional figures consist of adjoined (n-1)-dimensional figures is foregrounded by the generative use of the stretching gesture.

... Students' learning and thinking occur when they interact with and think "in and through" (Radford, 2009, p. 113) these resources, illustrating the importance of examining these relationships. Diagrams are key resources in students' geometric thinking (Chen & Herbst, 2013). Learning with diagrams and text promote better comprehension compared with text only (e.g., Mayer & Anderson, 1992). ...

College is a critical time when changes in students' attitudes, knowledge, personality characteristics, and self-concepts are affected by their face-to-face and online interactions with educators, peers, and the campus climate (Astin, 1997). The growing use of big data and analytics in higher education has fostered research that supports human judgement in the analysis of information about learning and the application of interventions that can aid students in their development and improve retention rates (Siemens & Baker, 2012). This information is often displayed in the form of learning analytics dashboards (LADs), which are individual displays with multiple visualizations of indicators about learners, their learning activities, and/or features of the learning context both at the individual and group levels (Schwendimann et al., 2017). The information presented in LADs is intended to support students' learning competencies that include metacognitive, cognitive, behavioral, and emotional self-regulation (Jivet et al., 2018). We investigated the impact of a student-facing LAD on students' self-concepts and viewing preferences to address the following questions: What are students' viewing preferences (i.e., for individual vs. comparative performance feedback)? How does viewing performance information affect the development of students' metacognitive skills and self-concepts? And, what are students' perceptions about the usability of LADs? In an end-of-term survey, 111 students at a large research university responded to 10 Likert scale and three open-ended questions. Overall, the students reported understanding the information that was presented to them through the LAD and that it was useful, although some students expressed concerns about its accuracy and wanted more detailed information. Students also reported that they preferred to view comparisons to other students over just viewing their own performance information, and that LAD use increased positive affect about performance. Students also reported that dashboard use affected how much they believed they understood the course material, the time and effort they were willing to put into the course, and that it lessened their anxiety. We concluded that course-specific or program-specific related outcomes may require different LAD design and evaluation approaches, and the nonuse of the LAD may be linked to self-confidence, forgetfulness, and a lack of innovative dashboard features. Our study was limited by the analysis of survey data (without trace data), and the sample size. This research contributes to the literature on student-facing learning analytics dashboards (LADs) by investigating students' reasons for interacting with dashboards, their viewing preferences, and how their interactions affect their performance and tying these insights to educational concepts that were a part of the LAD design. Further research is needed to determine whether presenting students with the option to turn on the dashboard for any or all of their courses over the course of the semester is important,

... Anxiety over performing in Geometry is characterized by feelings of tension, nervousness, and fear. Students become less competent in Geometry as a result of their fear, and they develop a negative attitude toward the subject [40]. ...

This study aims to reveal how diagramming in the everyday mathematics classroom supports students’ mathematical learning, based on the Châteletean perspective. To this end, we analyzed the diagramming of two 9th grade students who participated in the task of proving the Pythagorean theorem through diagrams in a geometry lesson. In particular, we focused on the epistemic distance as well as the material directness between the students and the diagrams. As a result, the students discovered mathematical ideas while they engaged in direct or indirect diagramming; they actualized virtual mathematical objects and relationships that were not visible in the given diagrams. During diagramming, the epistemic distance between the students and the diagrams was also dynamically changing. The findings suggest that diagramming in mathematics classrooms is not just a static representational activity, but an indeterminate and mobile engagement, which materially interacts with diagrams at varying degrees of epistemic distance.

This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.

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 chapter develops around two fundamental ideas, namely, that (1) the perception of the affordances of a certain digital tool is essential to solving mathematical problems with that particular technology and that (2) the activity thus undertaken stimulates different mathematising processes which, in turn, result in different conceptual models. Looking thoroughly, from an interpretative perspective, at four solutions to a particular geometry problem from participants who decided to use dynamic geometry software at some point of their solving activity, our main purpose is to illustrate the ways in which the same tool affords different approaches to the problem in terms of the conceptual models developed for studying and justifying the invariance of the area of a triangle. Their different ways of dealing with the tool and with mathematical knowledge are interpreted as instances of students-with-media engaged in a “solving-with-dynamic-geometry-software” activity, enclosing a range of procedures brought forth by the symbioses between the affordances of the dynamic geometry software and the youngsters’ aptitudes. The analysis shows that different people solving the same problem with the same digital media and recognising a relatively similar set of affordances of the tool produce different digital solutions, but they also generate qualitatively different conceptual models, in this case, for the invariance of the area.

Resumen Esta investigación estudia la relación entre la identificación de figuras prototípicas y el conocimiento de geometría durante la resolución de problemas de probar como una manifestación del razonamiento configural. Se han analizado las respuestas de 182 estudiantes para maestro a dos problemas de probar que proporcionaban una configuración geométrica y pedían probar un hecho geométrico. Los resultados indican que la identificación de una figura prototípica en la configuración inicial tiene un efecto heurístico que activa determinados conocimientos de geometría, que favorecen el cambio del anclaje visual al anclaje discursivo en la resolución del problema. Estos resultados subrayan la importancia de la relación entre la visualización y el conocimiento geométrico en el desarrollo del razonamiento configural durante la resolución de problemas de probar. Finalmente, desde estos resultados generamos algunas implicaciones para la formación de los maestros.

In a German tradition, the concept of ‘Grundvorstellungen’ (GV) concerns mental models that carry the meaning of mathematical concepts or procedures. While research on these GVs is mostly based on the analysis of verbal utterances or written products, we turn towards an embodied approach and investigate the question of how gestures can take part in revealing the activation of students’ GVs, focusing on the specific case of linear functions. In the presented empirical study, we found that gestures are used not only to specify deictic terms, but that they can also reveal a link between a mathematical idea and its non-mathematical grounding as it becomes explicit in speech. How gestures might contribute not only to the descriptive analysis of GVs, but also to the mathematical learning process, is discussed against the background of re-analyzing selected results within two different semiotic approaches.

Despite decades of research revealing the importance of and need for developing students’ spatial reasoning skills, geometry receives the least attention in North American K-12 mathematics classrooms. This chapter focuses on three grade one children as they worked on a spatial-geometric task. The study as part of a larger research project inquired into the actual forms, activities and processes that constituted the children’s reasonings and geometry during the three episodes. The findings contribute to current early years research by further explicating the body’s role in the children’s spatial-geometric reasonings, the impact of these on their conceptions, and how geometry emerged as an ongoing creative process of (re)(con)figuring space. Key implications are considered regarding young children’s spatial-geometric reasoning in the mathematics classroom.

Cognitive functions of gestures when working mathematically ---------------
Although gestures and their role in teaching and learning mathematicsgain increasingly interest in mathematics education research, most research focuseson the social aspects of gesture use, like their role in knowledge construction insocial interaction, the representational side of students’ gestures, or teachers’ in-structional gestures. Little is known about the cognitive functions gestures mightfulfil in mathematical learning. To understand better how gestures can contribute tocognitive processes when learning math, we adapt a framework from psychologywhich distinguishes the four cognitive functions of activating, manipulating, packag-ing and exploring. To explore the potential of this model for understanding the roleof gestures in cognitive processes in mathematical contexts better, we analyze twoexisting data sets in which university students think aloud while working by them-selves in the content area of trigonometry, once with worked examples and once witha task on a special class of trigonometric functions. In our analyses, certain cogni-tive functions appear as related to certain typical mathematical situations which willbe illustrated by means of specific cases concerning the validation of hypotheses,situations of functional thinking and when using specific representations.

This study aims to investigate students’ mathematical communication skills through investigating their defining, using mathematical concepts and mathematical language skills. Additionally, this study aims to investigate the relationship between students’ mathematical communication skills with their academic achievements. A seventh grade classroom at a public middle school in the province of Erzurum and 34 students and their mathematics teacher participated in the study. Students were asked to keep journals during seven weeks for sixteen Geometry standards after each standard. Student journals were analyzed in four hierarchical levels—Level 0 (Avaoidance), Level 1 (Incorrect Use), Level 2 (Incomplete Use) and Level 3(Correct and Complete Use) — by two researchers individually. Later, the researchers compared their categorizations and discussed till solving the disagreements. According to the findings of the study, participating students did not comprehend the concept definitions. It was seen that students attempted to use mathematical language in their journals as a way of communication. However, it should be noted that some incorrect use of mathematical language was apparent in student journals. This study also showed that it was hard to conclude that student mathematical communication skills differ based on their academic achievement levels.

This chapter provides the rationale for the book. It explains why we are looking inside the mathematics classroom, and why we are doing this through a sociological lens. In order to situate the volume at the intersection of two strands of research in mathematics education, this introductory chapter embraces research from sociological perspectives on mathematics education on the one hand, and research on pedagogic practice in the mathematics classroom on the other hand. In the final part, the chapter presents the structure of the volume and introduces its sections.

Word problems about motion contain various conjugated verb forms. As students and teachers grapple with such word problems, they jointly operationalize diagrams, gestures, and language. Drawing on findings from a 3-year research project examining the social semiotics of classroom interaction, we show how teachers and students use gesture and diagram to make sense of complex verb forms in such word problems. We focus on the grammatical category of “aspect” for how it broadens the concept of verb tense. Aspect conveys duration and completion or frequency of an event. The aspect of a verb defines its temporal flow (or lack thereof) and the location of a vantage point for making sense of this durational process.

This article is a sequel to the conversation on learning initiated by the editors of Educational Researcher in volume 25, number 4. The author’s first aim is to elicit the metaphors for learning that guide our work as learners, teachers, and researchers. Two such metaphors are identified: the acquisition metaphor and the participation metaphor. Subsequently, their entailments are discussed and evaluated. Although some of the implications are deemed desirable and others are regarded as harmful, the article neither speaks against a particular metaphor nor tries to make a case for the other. Rather, these interpretations and applications of the metaphors undergo critical evaluation. In the end, the question of theoretical unification of the research on learning is addressed, wherein the purpose is to show how too great a devotion to one particular metaphor can lead to theoretical distortions and to undesirable practices.

This book contributes to the current debate about how to think and talk about human thinking so as to resolve or bypass such time-honored quandaries as the controversy of nature vs. nurture, the body and mind problem, the question of learning transfer, and the conundrum of human consciousness. The author responds to the challenge by introducing her own “commognitive” conceptualization of human thinking. She argues for this special approach with the help of examples of mathematical thinking. Except for its contribution to theorizing on human development, the book is relevant to researchers looking for methodological innovations, and to mathematics educators seeking pedagogical insights and improvements.

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Adding gesture to spoken instructions makes those instructions more effective. The question we ask here is why. A group of 49 third and fourth grade children were given instruction in mathematical equivalence with gesture or without it. Children given in- struction that included a correct problem-solving strategy in gesture were signifi- cantly more likely to produce that strategy in their own gestures during the same in- struction period than children not exposed to the strategy in gesture. Those children were then significantly more likely to succeed on a posttest than children who did not produce the strategy in gesture. Gesture during instruction encourages children to produce gestures of their own, which, in turn, leads to learning. Children may be able to use their hands to change their minds.

In this article we examine students' perspectives on the customary, public work of proving in American high school geometry classes. We analyze transcripts from 29 interviews in which 16 students commented on various problems and the likelihood that their teachers would use those problems to engage students in proving. We use their responses to map the boundaries between activities that (from the students' perspective) constitute normal (vs. marginal) occasions for them to engage in proving. We propose a model of how the public work of proving is shared by teacher and students. This division of labor both creates conditions for students to take responsibility for doing proofs and places boundaries on what sorts of tasks can engage students in proving. Furthermore we show how the activity of proving is a site in which complementarity as well as contradiction can be observed between what makes sense for students to do for particular mathematical tasks and what they think they are supposed to do in instructional situations.

The spontaneous hand gestures that accompany children's explanations of concepts have been used by trained experimenters to gain insight into children's knowledge. In this article, 3 experiments tested whether it is possible to teach adults who are not trained investigators to comprehend information conveyed through children's hand gestures. In Experiment 1, we used a questionnaire to explore whether adults benefit from gesture instruction when making assessments of young children's knowledge of conservation problems. In Experiment 2, we used a similar questionnaire, but asked adultstomakeassessmentsofolderchildren'smathematicalknowledge.Experiment3 also concentrated on math assessments, but used a free-recall paradigm to test the ex- tent of the adult's understanding of the child's knowledge. Taken together, the results oftheexperimentssuggestthatinstructingadultstoattendtogestureenhancestheiras- sessment of children's knowledge at multiple ages and across multiple domains.

Gesture, or visible bodily action intimately involved in the activity of speaking, has long fascinated scholars and laymen alike. Written by a leading authority on the subject, this book draws on the analysis of everyday conversations to demonstrate the varied role of gestures in the construction of utterances. Publication of this definitive account of the topic marks a major development in semiotics as well as in the emerging field of gesture studies.

We analyze how 2 students used a computer-based motion detector in the context of individual interviews. Although 1 student's work is exemplified through transcript and commentary, the themes and discussions have evolved from our study of both students' work with the motion detector as they interacted with the interviewer. Our analysis reveals 3 themes: tool perspectives, fusion, and graphical spaces. Both students developed tool perspectives that enabled them to plan how to move so that they could create and interpret graphs by kinesthetic actions. The theme of fusion explores their emergent ways of talking, acting, and gesturing that do not distinguish between symbols and referents. Graphical spaces reflect our account of episodes in which a change in how they used the motion detector prompted them to investigate the tool anew. Our conclusions contribute to a reconceptualization of the nature of symbolizing, the learning of graphing, and the links between children's and scientists' graphing.

The goal of this article is to present a sketch of what, following the German social theorist Arnold Gehlen, may be termed
“sensuous cognition.” The starting point of this alternative approach to classical mental-oriented views of cognition is a
multimodal “material” conception of thinking. The very texture of thinking, it is suggested, cannot be reduced to that of
impalpable ideas; it is instead made up of speech, gestures, and our actual actions with cultural artifacts (signs, objects,
etc.). As illustrated through an example from a Grade 10 mathematics lesson, thinking does not occur solely in the head but also in and through a sophisticated semiotic coordination of speech, body, gestures, symbols and tools.

This paper focuses on children creating representations on paper for situations that change over time. We articulate the distinction between homogeneous and heterogeneous spaces and reflect on children's tendency to create hybrids between them. Through classroom and interview examples we discuss two families of tasks that seem to facilitate children's development of homogeneous spaces: 1) Making selected features directly visible, instead of requiring intermediate steps and calculations; for example, to be able to directly compare different sets of data combined in a single graph, and 2) Exploring well-defined figural components that can be used in graphing, such as line segments or sequencing from left to right, that are introduced as a resource.

This paper is a case study of how a high school student, whom we call Karen, used a computer-based tool, the Contour Analyzer, to create graphs of height vs. distance and slope vs. distance for a flat board that she positioned with different slants and orientations. With the Contour Analyzer one can generate, on a computer screen, graphs representing functions of height and slope vs. distance corresponding to a line traced along the surface of a real object. Karen was interviewed for three one-hour sessions in an individual teaching experiment. In this paper, our focus is on how Karen came to recognize by visual inspection the mathematical behavior of the slope vs. distance function corresponding to contours traced on a flat board. Karen strove to organize her visual experience by distinguishing which aspects of the board are to be noticed and which ones are to be ignored, as well as by determining the point of view that one should adopt in order to see the variation of slope along an object. We have found it inspiring to use Winnicott's (1971) ideas about transitional objects to examine the role of the graphing instrument for Karen. This theoretical background helped us to articulate a perspective on mathematical visualization that goes beyond the dualism between internal and external representations frequently assumed in the literature, and focuses on the lived-in space that Karen experienced which encompassed at once physical attributes of the tool and human possibilities of action.

Four potential modes of interaction with diagrams in geometry are introduced. These are used to discuss how interaction with
diagrams has supported the customary work of ‘doing proofs’ in American geometry classes and what interaction with diagrams
might support the work of building reasoned conjectures. The extent to which the latter kind of interaction may induce tensions
on the work of a teacher as she manages students’ mathematical work is illustrated.
Vier mögliche Formen der Interaktion mit geometrischen Darstellungen werden aufgezeigt. Diese Formen werden thematisiert um
deutlich zu machen, wie visuelle Darbietungen im am erikanischen Geometrieunterricht das alltägliche Geschäft des Beweisens,
unterstützen. Dadurch soll auch gezeigt werden, welche Art der Interaktion mit geometrischen Darstellungen es erlaubt, das
Herstellen begründeter Vermutungen zu unterstützen. Zugleich wird das Ausmaß illustriert, mit welchem die letztere Art von
Interaktion Spannungen innerhalb der unterrichtlichen Arbeit, der Lehrerin hervorruft, die sich darum bemüht, die mathematischen
Beiträge, d.h. die mathematische Arbeit, der Schülerinnen und Schüler zu organisieren.
ZDM-ClassificationC63-C73-D43-E53-G43

This paper reports a part of a study on the construction of mathematical meanings in terms of development of semiotic systems
(gestures, speech in oral and written form, drawings) in a Vygotskian framework, where artefacts are used as tools of semiotic
mediation. It describes a teaching experiment on perspective drawing at primary school (fourth to fifth grade classes), starting
from a concrete experience with a Dürer’s glass to the interpretation of a new artefact. We analyse the long term process
of appropriation of the mathematical model of perspective drawing (visual pyramid) through the development of gestures, speech
and drawings under the teacher’s guidance.

Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have
brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved.
In this paper, after taking a close look at two episodes that give rise to a number of difficult questions, I propose to base
research on a metaphor of thinking-as-communicating.This conceptualization entails viewing learning mathematics as an initiation to a certain well defined discourse. Mathematical discourse is made special by two main factors: first, by its exceptional reliance on symbolic artifacts as
its communication-mediating tools, and second, by the particular meta-rules that regulate this type of communication. The meta-rules are the observer’s construct and they usually remain tacit for the
participants of the discourse. In this paper I argue that by eliciting these special elements of mathematical communication,
one has a better chance of accounting for at least some of the still puzzling phenomena. To show how it works, I revisit the
episodes presented at the beginning of the paper, reformulate the ensuing questions in the language of thinking-as-communication,
and re-address the old quandaries with the help of special analytic tools that help in combining analysis of mathematical
content of classroom interaction with attention to meta-level concerns of the participants.

This article is a sequel to the conversation on learning initiated by the editors of Educational Researcher in volume 25, number 4. The author’s first aim is to elicit the metaphors for learning that guide our work as learners, teachers, and researchers. Two such metaphors are identified: the acquisition metaphor and the participation metaphor. Subsequently, their entailments are discussed and evaluated. Although some of the implications are deemed desirable and others are regarded as harmful, the article neither speaks against a particular metaphor nor tries to make a case for the other. Rather, these interpretations and applications of the metaphors undergo critical evaluation. In the end, the question of theoretical unification of the research on learning is addressed, wherein the purpose is to show how too great a devotion to one particular metaphor can lead to theoretical distortions and to undesirable practices.

Using data from more than ten years of research, David McNeill shows that gestures do not simply form a part of what is said and meant but have an impact on thought itself. Hand and Mind persuasively argues that because gestures directly transfer mental images to visible forms, conveying ideas that language cannot always express, we must examine language and gesture together to unveil the operations of the mind.

Gestures may provide the long sought-for bridge between science laboratory experiences and scientific discourse about abstract entities. In this article, we present our results of analyzing students' gestures and scientific discourse by supporting three assertions about the relationship between laboratory experiences, gestures, and scientific discourse: (1) gestures arise from the experiences in the phenomenal world, most frequently express scientific content before students master discourse, and allow students to construct complex explanations by lowering the cognitive load; (2) gestures provide a medium on which the development of scientific discourse can piggyback; and (3) gestures provide the material that “glues” layers of perceptually accessible entities and abstract concepts. Our work has important implications for laboratory experiments which students should attempt to explain while still in the lab rather than afterwards and away from the materials. © 2000 John Wiley & Sons, Inc. J Res Sci Teach 38: 103–136, 2001

This article is concerned with understanding situations in which speakers talk in the presence of scientific inscriptions (lectures in science classes, public presentations). Drawing on extensive video materials accumulated in middle and high school science classrooms and university lectures, we develop a framework for the resources speakers make available to their audience for understanding what the talk is about. We distinguish three situations according to the nature of reference to the phenomenon talked about: (i) talk is about phenomenon but mediated by reference to a two-dimensional (2-D) inscription; (ii) talk is about phenomenon but mediated by reference to a three-dimensional (3-D) inscription, and (iii) talk is directly about phenomenon. Associated with these three situations are different body orientations, distances from inscriptions, and types of gestures. When speakers laminate talk characteristic of two different types of situations. the orientation "up" can become "down" and "down" can become "up," potentially leading to confusing statements. (Gesture. orientation, spatial arrangements, body movement.)

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.

It seems obvious to contrast physical representation (a drawing on paper or on a screen) of an object with mental images of this same object. However, there is an important contrast which attracts less attention, between a drawing of a physical object (a house, for example, outlined by a square with a triangle at the top and rectangles for the door…) and a drawing of a geometrical figure (square, circle, triangle…).

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List of figures List of tables Preface 1. Introduction: psychology and anthropology I Part I. Theory in Practice: 2. Missionaries and cannibals (indoors) 3. Life after school 4. Psychology and anthropology II Part II. Practice in Theory: 5. Inside the supermarket (outdoors) and from the veranda 6. Out of trees of knowledge into fields for activity 7. Through the supermarket 8. Outdoors: a social anthropology of cognition in practice Notes References.

Photographs are the most frequent inscriptions in high school biology textbooks. However, little is known about how students make sense of and learn from photographs; even less is known about the different resources available for making sense of photographs when they appear in lectures. In this study, the use of photographs during lectures and lecture-type situations was analyzed with re- spect to the semiotic resources that speakers standing next to the projected photographs provided for understanding and learning from them. Our analysis identified eight types of gesture as semi- otic resources that decreased the ambiguity inherent in photographs, and that have the potential to enhance the understanding of photographs and the scientific concepts embodied in them. We sur- mise that teachers can help their students learn to read and interpret photographs from lectures when they project them in such a way that it allows the use of gestures as additional meaning-making resources.

: Gestures may provide the long sought-for bridge between science laboratory experiences and scientific discourse about abstract entities. In this article, we present our results of analyzing students' gestures and scientific discourse by supporting three assertions about the relationship between laboratory experiences, gestures, and scientific discourse: (1) gestures arise from the experiences in the phenomenal world, most frequently express scientific content before students master discourse, and allow students to construct complex explanations by lowering the cognitive load; (2) gestures provide a medium on which the development of scientific discourse can piggyback; and (3) gestures provide the material that glues layers of perceptually accessible entities and abstract concepts. Our work has important implications for laboratory experiments which students should attempt to explain while still in the lab rather than afterwards and away from the materials.

Describes and categorizes patterns we construct between texts of different kinds (patterns of intertextuality). Whether the principles of intertextuality can be applied to other sorts of semiotic "texts" such as computer graphics and hypermedia is raised as a crucial question for the future of educational research. (Contains 48 references.) (JP)

Cites the lack of authenticity in intertextuality (ITX) research and argues that collaborative learning environments offer a better research setting. One particular research context, literature circles, is examined to identify characteristics of generative research environments for the study of ITX. (Contains 27 references.) (JP)

The purpose of this research was to investigate the presence, role, extent, and constraints of visual thinking in conjunction with affective states and with metaphors in the problem-solving processes of graduate students as they solved nonroutine problems. This article gives details of the solving of 3 word problems by 4 students. Visual imagery was evidenced by 3 descriptors, namely drawing, verbal report, and gesture. Visual imagery was used by each of the 4 students for each of the 3 problems. Visual imagery was reported even in instances where no diagram was drawn and the solution appeared to be purely algebraic. The roles of visualization were investigated in 4 main moments of the solution processes, which we have called preparation, solution, conclusion, and hindsight. The types of imagery and their roles in these moments cause us to differentiate between use of imagery to make sense and to solve, as 2 distinct aims of visualization. Base knowledge, spatial reasoning, metaphors that may enable or constrain, and particularly affective issues all played a role in the graduate students' use of visualization.

The situative perspective shifts the focus of analysis from individual behavior and cognition to larger systems that include behaving cognitive agents interacting with each other and with other subsystems in the environment. The first section presents a version of the situative perspective that draws on studies of social interaction, philosophical situation theory, and ecological psychology. Framing assumptions and concepts are proposed for a synthesis of the situative and cognitive theoretical perspectives, and a further situative synthesis is suggested that would draw on dynamic-systems theory. The second section discusses relations between the situative, cognitive, and behaviorist theoretical perspectives and principles of educational practice. The third section discusses an approach to research and social practice called interactive research and design, which fits with the situative perspective and provides a productive, albeit syncretic, combination of theory-oriented and instrumental functions of research. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

This article is concerned with understanding situations in which speakers talk in the presence of scientific inscriptions (lectures in science classes, public presentations). Drawing on extensive video materials accumulated in middle and high school science classrooms and university lectures, we develop a framework for the resources speakers make available to their audience for understanding what the talk is about. We distinguish three situations according to the nature of reference to the phenomenon talked about: (i) talk is about phenomenon but mediated by reference to a two-dimensional (2-D) inscription; (ii) talk is about phenomenon but mediated by reference to a three-dimensional (3-D) inscription; and (iii) talk is directly about phenomenon. Associated with these three situations are different body orientations, distances from inscriptions, and types of gestures. When speakers laminate talk characteristic of two different types of situations, the orientation can become and can become potentially leading to confusing statements.

The main thesis of the present paper is that geometry deals with mental entities (the so-called geometrical figures) which possess simultaneously conceptual and figural characters. A geometrical sphere, for instance, is an abstract ideal, formally determinable entity, like every genuine concept. At the same time, it possesses figural properties, first of all a certain shape. The ideality, the absolute perfection of a geometrical sphere cannot be found in reality. In this symbiosis between concept and figure, as it is revealed in geometrical entities, it is the image component which stimulates new directions of thought, but there are the logical, conceptual constraints which control the formal rigour of the process. We have called the geometrical figuresfigural concepts because of their double nature. The paper analyzes the internal tensions which may appear in figural concepts because of this double nature, development aspects and didactical implications.

In our ongoing qualitative classroom research, we adopt a sociocultural perspective to investigate discourse, and its role
in how children and teachers make meaning of mathematics in a fifth grade inquiry classroom. Our theoretical perspective draws
primarily on Vygotsky (1978, 1986) and Bakhtin (1981, 1986) each of whom examines how social forms of meaning influence individual
cognition. The episode described in this paper examines the process whereby individual and group developmental trajectories
are constructed, and allows us to explore the relationship between discourse and knowing. We combine a longitudinal design
with a case study approach to focus on the collaborative mathematical problem solving. We use video capture to help us listen
to children’s discussions in classroom activities and small group interactions. Our analysis of the verbal data recorded on
video identifies patterns of interaction, development and change in participants’ use of mathematical language and concepts,
and their evolving understanding, through discussion and argument, of an algebraic expression constructed by one of the children.
The findings lead us to argue for i) a more generative view of the zone of proximal development as a site of learning and
of identity formation, ii) an expanded view of the role of the teacher in inquiry classrooms, and iii) an appreciation for
the valuable roles difference and resistance play in knowledge building.

This chapter focuses on the use of diagrams at the point when students are beginning to be taught geometry as a coherent field of objects and relations of a theoretical nature. It investigates the relations between the domain of diagrams in paper-andpencil or software environments and the domain of theoretical objects of geometry, by means of an analysis of students' solution processes when faced with a geometrical task. It is divided into two parts: the first deals with the rules, (sometimes implicit) that govern the use of diagrams in solving school geometry problems; the second describes the actual processes of students in a problem-solving situation, mainly with dynamic geometry software.

In this paper, we consider gestures as part of the resources activated in the mathematics classroom: speech, inscriptions,
artifacts, etc. As such, gestures are seen as one of the semiotic tools used by students and teacher in mathematics teaching–learning.
To analyze them, we introduce a suitable model, the semiotic bundle. It allows focusing on the relationships of gestures with the other semiotic resources within a multimodal approach. It also enables framing the mediating action of the teacher in the classroom: in this respect, we introduce the
notion of semiotic game where gestures are one of the major ingredients.

Where anthropological and psychological studies have shown that gestures are a central feature of communication and cognition, little is known about the role of gesture in learning and instruction. Drawing from a large database on student learning, we show that when students engage in conversations in the presence of material objects, these objects provide a phenomenal ground against which students can enact metaphorical gestures that embody (give a body to) entities that are conceptual and abstract. In such instances, gestures are often subsequently replaced by an increasing reliance upon the verbal mode of communication. If gestures constitute a bridge between experiences in the physical world and abstract conceptual language, as we conjecture here, our study has significant implications for both learning and instruction.

Robotics, artificial intelligence and, in general, any activity involving computer simulation and engineering relies, in a
fundamental way, on mathematics. These fields constitute excellent examples of how mathematics can be applied to some area
of investigation with enormous success. This, of course, includes embodied oriented approaches in these fields, such as Embodied
Artificial Intelligence and Cognitive Robotics. In this chapter, while fully endorsing an embodied oriented approach to cognition,
I will address the question of the nature of mathematics itself, that is, mathematics not as an application to some area of
investigation, but as a human conceptual system with a precise inferential organization that can be investigated in detail
in cognitive science. The main goal of this piece is to show, using techniques in cognitive science such as cognitive semantics
and gestures studies, that concepts and human abstraction in general (as it is exemplified in a sublime form by mathematics)
is ultimately embodied in nature.

Many teaching practices implicitly assume that conceptual knowledge can be abstracted from the situations in which it is learned and used. This article argues that this assumption inevitably limits the effectiveness of such practices. Drawing on recent research into cognition as it is manifest in everyday activity, the authors argue that knowledge is situated, being in part a product of the activity, context, and culture in which it is developed and used. They discuss how this view of knowledge affects our understanding of learning, and they note that conventional schooling too often ignores the influence of school culture on what is learned in school. As an alternative to conventional practices, they propose cognitive apprenticeship (Collins, Brown, & Newman, in press), which honors the situated nature of knowledge. They examine two examples of mathematics instruction that exhibit certain key features of this approach to teaching.