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This paper examines some examples (taken from research conducted over the years) that show students’ linguistic attitudes in geometry tasks. The examples are framed within the Theory of Objectification with reference to the notion of sensuous cognition, semiotic means of objectification and levels of generality. We show the struggle students live, at higher levels of generality, in intertwining natural language, specific language and the spontaneous use of geometrical figures, bound to perception and kinaesthetic activity. Within the networking paradigm, we coordinate the Theory of Objectification and Duval’s semio-cognitive approach to frame the interplay between the ideal and the material that occurs in geometrical argumentations and proofs.

This essay explores the research practice of French geometer Michel Chasles (1793–1880), from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing.

Pre-service mathematics teacher (PST) education often addresses within Geometry Classes how to utilize Dynamic Geometric Software (DGS). Other classes may also incorporate teaching pre-service teachers about the history of mathematics. Although research has documented the use of Dynamic Geometric Software (DGS) in teaching the history of mathematics (HoM) (Zengin, 2018), the focus of this research specifically targets the development of proof for pre-service teachers by utilizing DGS to revisit historical proofs with a modern lens. The findings concur with Fujita et.al. (2010), Zengin (2018), and Conners (2007) work on proof. The novelty of this article was the combination of incorporating the history of mathematics (HoM), dynamic geometry software (DGS), and Toulmin’s model of argumentation. A pedagogical approach appeared to emerge: DGS’s dynamic nature allowed PSTs to see several examples of a method to provide them with an illustration that may be used in proofs.

Decades of research have established that solving geometry proof problems is a challenging endeavor for many students. Consequently, researchers have called for investigations that explore which aspects of proving in geometry are difficult and why this is the case. Here, results from a set of 20 interviews with students who were taught proof in school geometry are reported. Students who earned A or B course-grades in the proof unit(s) were asked to share their thinking aloud while solving two proof tasks using smartpens. Student thinking was analyzed for two subgroups—students who were successful with both proofs (n = 7) and who were unsuccessful with both proofs (n = 13). Large differences were observed in how often students in the two groups exhibited certain competencies and behaviors. The largest gaps occurred in the ways in which students attended to the proof assumptions, attended to warrants in their proofs, and demonstrated logical reasoning.

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

Proof is a key indicator for a student in developing mathematical maturity. However, in the process of learning proof, students have the difficulty of being able to explain the proof that has been compiled using good arguments. So we need a strategy that can put students in the process of clarifying proof better. One strategy that can explore student thought processes in explaining geometric proof is self-explanation strategy. This research aimed to analyze the ability to understand the geometric proof of prospective teacher students by implementing a self-explanation strategy in basic geometry classes. This study used a quasi-experimental research type of nonequivalent control group design. The participants of this research were 75 students of mathematics education study programs at one private university in Semarang. This research used four instrument tests of geometric proof. Before being used for research, the instruments were tested for validity and reliability using product-moment and Cronbach's alpha. Data analysis in this study used a two-way ANOVA test. The results showed that: the increased ability to understand the geometric proof of students who used self-explanation strategy was better than those who obtained direct learning; there was a significant difference between the increase of students’ mathematical proof ability in a group of students with a high and moderate level of initial mathematical ability; the initial ability (high, medium, low) of mathematics did not directly influence the learning process to improve the ability to understand the geometric proof. Hence, it can be concluded that the self-explanation strategy is effective to be used to improve the understanding of the geometric proof.

A symbolic geometry system such as Geometry Expressions can generate symbolic measurements in terms of indeterminate inputs from a geometric figure. It has elements of dynamic geometry system and elements of automated theorem prover. Geometry Expressions is based on the analytical geometry method. We describe the method in the style used by expositions of semi-synthetic theorem provers such as the area method. The analytical geometry method differs in that it considers geometry from a traditional Euclidean/Cartesian perspective. To the extent that theorems are proved, they are only proved for figures sufficiently close to the given figure. This clearly has theoretical disadvantages, however they are balanced by the practical advantage that the geometrical model used is familiar to students and engineers. The method decouples constructions from geometrical measurements, and thus admits a broad variety of measurement types and construction types. An algorithm is presented for automatically deriving simple forms for angle expressions and is shown to be equivalent to a class of traditional proofs. A semi-automated proof system comprises the symbolic geometry system, a CAS and the user. The user’s inclusion in the hybrid system is a key pedagogic advantage. A number of examples are presented to illustrate the breadth of applicability of such a system and the user’s role in proof.

The geometry automated theorem proving area distinguishes itself by a large number of specific methods and implementations, different approaches (synthetic, algebraic, semi-synthetic) and different goals and applications (from research in the area of artificial intelligence to applications in education). Apart from the usual measures of efficiency (e.g. CPU time), the possibility of visual and/or readable proofs is also an expected output against which the geometry automated theorem provers (GATP) should be measured. The implementation of a competition between GATP would allow to create a test bench for GATP developers to improve the existing ones and to propose new ones. It would also allow to establish a ranking for GATP that could be used by "clients" (e.g. developers of educational e-learning systems) to choose the best implementation for a given intended use., Comment: In Proceedings ThEdu'19, arXiv:2002.11895

espanolEn este estudio analizamos el cambio de estatus de las afirmaciones matematicas que componen el proceso discursivo en la resolucion de problemas geometricos de prueba. En particular, nos centramos en como se desarrolla y organiza el discurso escrito (respuesta) que comunica la solucion, con el objetivo de identificar relaciones con los desenlaces del razonamiento configural. Para ello, analizamos las respuestas de estudiantes de cuarto curso de educacion secundaria obligatoria a cuatro problemas geometricos de prueba. Los resultados ponen de manifiesto la necesidad de un cambio en el estatus de las afirmaciones matematicas involucradas en el razonamiento que conduce a la solucion y el desarrollo de una argumentacion que progrese desde el modo de acumulacion hasta el modo de sustitucion. No obstante, la presencia de estas caracteristicas del proceso de prueba, no garantiza que se de el truncamiento del razonamiento configural que genera la prueba formal, debido, entre otros factores, a la influencia que ejerce la subconfiguracion relevante identificada en el proceso de razonamiento EnglishIn this study, we analyze the status change of the different mathematical affirmations that compose the discursive process in solving proof problems into geometry context. In particular, we focus on the way in which it is developed and organized the written discourse (answer) which allows communicating the solution, with the aim of identifying relationships with the configural reasoning outcomes. For this, we analyze the answers of students in the four grade of compulsory secondary education to four proof problems in a geometrical context. The results highlight the necessity of a change into the status of the mathematical affirmations involved in the reasoning leading to the solution, as well as the development of an argumentation that it should be developed from the accumulation mode to the substitution one. However, the presence of these characteristics of the proof process don’t guarantee that the configural reasoning “truncation” that generates the formal proof will occur, due to, among other factors, the influence exerted by the relevant subconfiguration identified in the reasoning process.

We survey the status of decidability of the first order consequences in various axiomatizations of Hilbert-style Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski’s conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler’s theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: It was already known that the universal theory of Hilbert planes and Wu’s orthogonal geometry is decidable. We show here using elementary model theoretic tools that The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving.

Providing feedback on peer solutions to geometry proofs can support preservice mathematics teachers' assessment skills of such complex tasks. However, the quality of peer solutions may influence cognitive processing during peer-feedback provision, learning from providing peer-feedback, and peer-feedback content. To investigate this effect, we recorded the eye-movements of fifty-three preservice mathematics teachers while providing feedback on a near-correct or an erroneous peer solution to a geometry proof, and we measured their proof comprehension and peer-feedback content. Results show that the absence of errors earlier in the peer solution facilitated reliance on a figure-based approach, whereas encountering errors earlier in the peer solution was associated with more focus on the text of the proof. Students who provided peer-feedback on the near correct peer solution had better comprehension of the proof, and they provided more accurate peer-feedback. Errors in peer solutions thus appear to hinder positive peer-feedback outcomes.

A formula for the magnetostatic energy of a finite magnet is proven. In contrast to common approaches, the new energy identity does not rely on evaluation of a nonlocal boundary integral inside the magnet or the solution of an equivalent Dirichlet problem. The formula is therefore computationally efficient, which is also shown numerically. Algorithms for the simulation of magnetic materials could benefit from incorporating the presented representation of the energy. In addition, a natural analogue for the energy via the magnetic induction is given. Proofs are carried out within a setting which is suitable for common discretizations in computational micromagnetics.

This article is part of a wider research project that has the educational goal of supporting students in the production of conjectures, arguments and proofs, as well as promoting a move from the production of arguments expressed in colloquial registers to arguments expressed in literate registers. In this regard, we Giovannina Albano, Umberto Dello Iacono and Maria Alessandra Mariotti designed and implemented a digital educational environment that allows students to formulate and prove conjectures; three different working areas are available where students can work on a geometrical open problem sometimes individually, sometimes in collaboration. In this article, I report on an empirical study aimed at investigating the functioning of one of these areas, the ‘Working with others’ area, where small groups of students are expected to discuss and formulate a shared solution to a problem. The research question concerns if and to what extent the communication tools, specifically designed to foster students’ collaboration, can promote the production of mathematically acceptable arguments. The qualitative data analysis shows that the ‘Working with others’ area seems to foster discussion within the group and can make students aware of their mistakes. Moreover, it can bring out some students’ misconceptions and can provide useful information upon which the teacher can trigger fruitful discussions. However, this working area does not appear to foster a significant improvement of the production of mathematically acceptable arguments, produced by students in a collaborative and sharing mode. The integration of specific components within this working area seems to be necessary to support the student in moving from argumentation to proof.

Constructing a mathematical proof is very challenging for many mathematics students. Many of them having difficulties when doing such construction. This study aimed to investigate and categorize mathematics undergraduate students’ character when construct formal proof on geometry. The data were obtained within 2 phases. Participants were asked to write down the proof of proving-question in phase 1 and they were interviewed to deepen the information in phase 2. The results show that just two of eight students who correctly answer the question and are categorized as students in Quadrant I character. None of participants who is categorized in Quadrant II and two of them are categorized in Quadrant III. Most of the participants (5 out of 8) are categorized in Quadrant IV. Thus, means that most participants have wrong proof-structure and poor connection between concept image and concept definition.

Until recently, it was common for instruction in geometry to be delayed until the secondary school course bearing its name. Continuing with historical precedent, geometry was viewed from a formal axioms and proofs standpoint. Not surprisingly, when geometry was finally addressed at the secondary school level it centered around formal proofs—in a seeming effort as much to teach the student to think “like a mathematician” from a formal logical stance as to teach actual geometric content. This made sense to some degree. Since the prevailing view is that young children cannot operate in the arena of formal logic, pushing geometry back as long as possible seemed reasonable. Geometry, to the extent it was addressed in an K-9 curriculum, was often limited to a memory/recall process of the names of geometric figures. In rare cases a few properties such as parallel, perpendicular, and similar were addressed.

We discuss the verification in the origami geometry. In the preceding chapters, we discussed the origami construction using origami language ORIKOTO. During the construction, the logical formulas that describe the geometric configuration are formed and stored. We use those formulas for verifying the geometric properties of the constructed origami. In this chapter, we detail the process of verification. We give three examples of the application of our verification method. One is a simple geometric shape to explain the principle of verification using algebraic methods. The other two are the proofs of a regular pentagon construction and the generalized Morley’s theorem. Through the three examples, we see the computationally streamlined geometric construction and verification.

This report explores pre-service teachers’ proficiency with concepts of transformational geometry at the end of a semester-long advanced geometry course. In the course, the instructor incorporated transformational geometry content, including congruence proofs, in an attempt to prepare the pre-service teachers to teach high school geometry in alignment with the Common Core State Standards for Mathematics. At the conclusion of the course, students expressed a preference for using traditional triangle congruence criteria (SAS, ASA, SSS, and AAS) over using transformations to complete proofs, but were nevertheless generally successful in completing proofs using transformations. Similarly, while the students often described thinking of transformations in terms of analytic forms, they were successfully able to prove triangle congruences in synthetic contexts. Finally, some evidence indicates that students may have motion or process conceptions of transformations, but not map or object conceptions, but this evidence is not conclusive.

The aims of this study were conducted to investigate the effectiveness of pre-service teachers’ geometric exploration activities assisted by Dynamic Geometry Software (DGS) Cabri II Plus computer application in constructing geometry proofs in a teacher education classroom. To these ends, mix-method design. A total of 72 pre-service teachers taking geometry course participated in the study. Findings of the study show that students who participated in geometric exploration activities assisted by DGS Cabri II Plus computer application had better achievement compared to their counterpart. The use of DGS Cabri II Plus computer application was observed to enable the students to present diagrams of verification problems appropriately, determine the valid conjectures, and make justification regarding the statements in the written proof. More importantly, participating in geometric exploration activities assisted by DGS Cabri II Plus computer application provide students with opportunities to explore alternative proofs related to geometry.

The study aimed to expose how pre-service teachers perceived the value of an inquiry-based geometrical task with dynamic geometry software for increasing and deepening their understanding of the task itself, and for heightening their mathematical, pedagogical, and technological knowledge in the case of reflection and transformation. Twenty-seven pre-service teachers studying to become middle-school (junior high) mathematics teachers were assigned an inquiry-based geometrical task to explore – both by conventional, low-tech means (paper and pencil) and then in a dynamic geometry software (DGS) environment – the properties of objects formed by reflections in polygons (triangles and quadrilaterals) and to offer conjectures and proofs regarding the final polygon’s shape and area concerning the original. The results show that the majority of participants indicated that DGS provided a valuable instrument for revising and expanding their knowledge, that the assignment demonstrated the value that technological tools have in teaching and learning processes, and the importance of adapting learning tasks in geometry to a dynamic geometry environment for expanding and deepening geometrical and pedagogical knowledge and reasoning skills.

The pursuit of an Intelligent and Dynamic Geometry Book should involve the study of how currently developing methodologies and technologies of geometry knowledge representation, management, deduction and discovery can be incorporated effectively into future education. Just as Doron Zeilberger pointed out in the Plane Geometry: An Elementary Textbook By Shalosh B. Ekhad (Circa 2050), a geometry book from the future would be a computer program, in which all the theorems can be automatically discovered (and of course proved) by computer and beautiful illustrations can be automatically generated and dynamically modified. Such a prospect motivates studies on how to represent and manage digitised geometric knowledge on computer. The geometry book of the future (the $$\mathcal {I}$$ nt $$\mathcal {D}$$ yn $$\mathcal {G}$$ eo $$\mathcal {B}$$ ook) should be adaptive, collaborative, visual and intelligent. Adaptive because the contents should adapt itself to the curricula and readers. It will allow collaborative work and its contents would be collaboratively formed using a knowledge base open to contributions. Statements and proofs should be en-lighted by dynamic geometry sketches and diagrams, and the correctness of the proofs should be ensured by computer checking. The book will be intelligent, the reader should be able to ask closed or open questions, and can also for proof hints. The book should also provide interactive exercises with automatic correction. Such a cloud platform, freely available in all standard computational platforms and devices, collaborative, adaptive to each and every user’s profiles, should bring together a whole new generation of mathematical tools with impact in all levels of education. To realise such a book, a network of experts must be built, increasing the connections between several research communities, such as: mathematical knowledge management; computer theorem proving and discovery; education, aggregating expertise in areas such as Proofs in a Learning Context; Interfaces and Searching; Tools Integration; Learning Environments in the Cloud. In this paper the author tries to make the case for such an endeavour.

Pictures and diagrams help high school geometry students develop reasoning and proof-writing skills.

When Michel de Montaigne criticizes different forms of knowledge in his Essays, geometry gets a decidedly bad rap. Geometrical assertions rely on abstract thought, and the essayist would rather we concentrate on concrete measurements of the world around us. While we have until now made much of the philosophical consequences these criticisms engendered, namely that Montaigne was heavily persuaded by skepticism, we have not looked at the extent to which Montaigne was fascinated by the language of geometry and even, in his writing, obsessed with a particular geometrical debate that dates back to the very first commentaries on Euclid: the necessary, infinite separation between parallel lines and between asymptotes and their hyperbola. Montaigne’s writing produces counter proofs to non-incident lines by forming parallels and asymptotes into demonstrable, everyday actions, expressing them in a more colloquial, French equivalent, and proposing the essay as a parallel figure that challenges theorems of infinite separation. The following will read these moments in Montaigne’s writing as conscious and unconscious instances of his geometrical language and, in Lucretian terms, his swerve [clinamen]. Montaigne’s explicit opposition to lines that never cross, specifically his aversion to the asymptote proposed to him by Jacques Peletier and the parallel lines asserted in Epicurus’ atomist discourse, constitutes a geometrical language in which crossing and contact are moral imperatives that promote social togetherness and exchange.

The mathematical literature knows several quite different ways to determine the area of a parabolic segment. In this article they are brought together, compared and varied. Finally two new proofs are added to the collection. Each proof is displayed by an expressive figure. A colorful compilation of those figures accompanies the article as an appendix.

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.

Task design is increasingly recognised as crucial for enhancing student learning of mathematics. Even so, and despite the significance of mathematical activity related to proofs and counterexamples in school mathematics, the task design principles underpinning students’ success in proof-related activity remains under-explored in mathematics education research. What is more, although the affordances of using dynamic geometry environments (DGEs) have been established in the literature, task design in DGEs remains somewhat under-studied. In this paper, we address both these issues by theoretically developing, and empirically testing, task design principles for supporting students’ heuristic refutation (revising conjectures/proofs through addressing counterexamples) in DGEs. We use the existing literature to elaborate three design principles: using tasks whose conditions are purposefully implicit, providing tools that enhance the production of counterexamples, and increasing students’ recognition of contradictions. To test these principles empirically, we analyse two sets of task-based interviews, one with a triad of secondary school students and the other with a pair of undergraduates, where tasks designed according to the principles were implemented. Our analysis shows that using the tasks enabled the students to engage successfully in heuristic refutation. The matter of examining mathematical definitions when handling counterexamples emerged as an issue for further research.

Mathematics education continues to emphasize explorative proving, wherein proving involves producing statements, producing proofs, looking back (examining, improving, and advancing) at these products, and the interactions among these aspects. This study aims to develop an intended explorative proving mathematics curriculum by focusing on students’ ability to plan and construct proofs. We first set Levels 1 and 2 of “planning a proof” and “constructing a proof,” respectively, and Level 0 as the starting point of the learning progression where there is no differentiation between planning and constructing. Next, we combined them, and set nine learning levels in addition to “looking back” as the characteristics of explorative proving. Then, we elucidated two learning progressions of explorative proving as a curriculum framework considering the relationship between planning and constructing a proof. To develop the curriculum based on these learning progressions, we made corresponding tables of units with these learning progressions according to the units of Japan’s national Course of Study, and then showed an example of localizing one of the progressions and its effects by the implemented curriculum. By adopting the method of lesson study and a design experiment, we implemented geometry lessons for 8th graders that aim to shift the progression through the learning levels. The results clarify the advantages and limitations of the developed curriculum, which enabled us to refine a more robust curriculum. Finally, we identify the characteristics of this approach to curriculum development of explorative proving and the necessity of the method of lesson study and design experiment as a realistic dimension of curriculum development and improvement.

Given its formal, logical, and spatial properties, geometry is well suited to teaching environments that include dynamic geometry systems (DGSs) , geometry automated theorem provers (GATPs), and repositories of geometric problems. These tools enable students to explore existing knowledge in addition to creating new constructions and testing new conjectures. In this chapter, we trace the evolution of current automatic proving technologies, how these technologies are beginning to be used by geometry practitioners in general to validate geometric conjectures and generate proofs with natural language and visual rendering, and foresee their evolution and applicability in an educational setting.

This is an expository article detailing results concerning large arcs in finite projective spaces. It is not strictly a survey but attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre’s lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.

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

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.

Two results, proved synthetically in plane Euclidean geometry by Chisini in 1924—stating that: (i) if MAB is an isosceles triangle, with \({MA\equiv MB}\), inscribed in a circle \({\mathcal{C}}\), P1 and P2 are two points on \({{\mathcal C}}\) such that {B, Pi} separates {A, M} for \({i \in \{1,2}\}\), and {B, P2} separates {M, P1}, then \({AP_1+BP_1 < AP_2+BP_2}\), and (ii) of all triangles inscribed in a given circle the equilateral triangle has the greatest perimeter—are proved inside Hilbert’s absolute geometry.

This paper reports the results of an international comparative study on the nature of proof to be taught in geometry. Proofs in French and Japanese lower secondary schools were explored by analyzing curricular documents: mathematics textbooks and national curricula. Analyses on the three aspects of proof—statement, proof, and theory—suggested by the notion of Mathematical Theorem showed differences in these aspects and also differences in the three functions of proof—justification, systematization, and communication—that are seemingly commonly performed in these countries. The results of analyses imply two major elements that form the nature of proof: (a) the nature of the geometrical theory that is chosen to teach and (b) the principal function of proof related to that theory. This paper suggests alternative approaches to teach proof and proving and shows that these approaches are deeply related to the way geometry is taught.

This work was carried out in collaboration between both authors. Author HMA suggested the problem and the outline of the work. Authors HMA and HGAA checked all the proofs of the theorems. Author HMA wrote the protocol and supervised the work. Author HGAA managed the analyses of the study. Author HGAA wrote the first draft of the manuscript. Author HMA managed the literature searches and edited the manuscript. Both authors read and approved the final manuscript.

xi CHAPTER 1 : INTRODUCTION 1 Research Questions 7 Overview of the Study 8 CHAPTER 2 : LITERATURE REVIEW 10 New Teacher Induction 11 The Induction Phase 11 Knowledge for Teaching 13 Curriculum in School Mathematics 18 Interactions between Beliefs, Curriculum, and Practice 20 Curriculum Adaptation 22 Curricular Knowledge, Curriculum Vision, and Curriculum Trust 23 The Role of Tasks in Mathematics Classrooms 25 Proof in the Geometry Curriculum 27 Mathematical Proof 30 Overview of the Categories of Study 30 “Doing Proof” in School Mathematics 32 Classroom Discourse 38 Classroom Talk in Mathematics Classrooms 40 Discourse Moves 42 Summary 52 CHAPTER 3 : METHOD AND PROCEDURES 54 Theoretical Perspectives 54 Ethnography 54 Discourse Analysis 56 Pairing Ethnography with Discourse Analysis 57 Guiding Questions 58 Context of the Study 58 Gaining Access to the Site 58

This study examines the quantity and quality of opportunities for reasoning and proving within the geometry content of three secondary school Mathematics textbooks in Trinidad and Tobago. I use an instrument from Otten et al. (Math Thinking Learn 16:51–79, 2014) to code and analyze the opportunities for students to reflect on or engage in reasoning and proof. My analysis suggests that the three textbooks contain opportunities for students to identify patterns, make conjectures, and construct proofs. At least 30% of the student exercises in two of the textbooks promoted Geometric Calculations with Number and Explanation (GCNE), which provide opportunities for students to develop non-proof arguments or rationales. The findings of this examination can potentially help in guiding curriculum developers, policy makers, and textbook authors with the future design of textbooks, curriculum materials, and other instructional resources that foster the intellectual need of reasoning and proof in students’ mathematical experiences.

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

In the chapter Geometry and trigonometry, the majority of problems can be treated either by classical proof techniques and properties provided in the regular school program or by using some additional proof techniques and theorems that are considered to be outside of the scope of the regular school program. The classical proof techniques, theorems, properties or mathematical laws used in the proofs included in this chapter are the properties of congruence of triangles, properties of similarities of triangles, the Pythagorean theorem, the properties of the right-angled triangle, law of sines and law of cosines, the properties of an area, the properties of inscribed and circumscribed quadrilaterals, trigonometric methods, the method of coordinates and applications of geometric inequalities (for more applications, see Sedrakyan and Sedrakyan, Geometric Inequalities, 2017, [1]). A large number of problems are related with the maxima and minima properties, such problems can be treated either by the different methods of estimations or by transformations and simplifications of the trigonometric expressions. In some cases, when we need to find the sum of the given trigonometric expressions, the main trick is to represent each of the summands as the difference of two expressions and then perform the corresponding simplifications in order to find the required sum. For the readers convenience, we list and provide the formulations of all used theorems and proof techniques that are outside of the scope of the regular school program.

Pursuing abstractness and generality, a number of books and articles in econometrics rely heavily on standard algebraic proofs. However, most of the theorems proved in such a way have a strong geometric appeal. This paper demonstrates how shorter, less technical and even more beautiful the proofs can be when they are based on geometric theorems. We show that this technique can be extended to explain concepts in probability and statistics. Although most of the theorems and ideas are not completely new and there are even a few works on the geometry in econometrics and statistics, this paper introduces alternative explanations and more general results in some cases. Further, there are algebraic proofs provided parallel to geometric ones and illustrations which are our own work.

This study inspected the relationships between self-reported implementation of instructional strategies using a dynamic geometry approach and the students’ engagement in making, testing, and proving conjectures Data collected includes a self-reported questionnaire given to all of the project’s participating high school geometry teachers, collecting both quantitative and qualitative data. The results of the linear model, with proving conjectures as a response variable, indicate that students spent less time proving or disproving their conjectures when working alone regardless of whether they were in a regular or advanced level geometry class. Time spent making conjectures and testing conjectures were positively and significantly correlated with the frequency of teachers’ implementation of class discussions. Furthermore, giving instruction that prompted group work had a significant and positive correlation with students proving conjectures in Regular geometry classes.

As international research confirms, many secondary school students can find it difficult to construct mathematical proofs. In this article, we explain the pedagogical and technological underpinnings of a web-based learning support system for students who are just starting to tackle deductive proving in geometry. We show how the system was designed to enable students to access the study of proofs in geometry by tackling proof problems where they can ‘drag’ sides, angles and triangles from the figural diagram of the problem to on-screen cells within a flow-chart proof format. When doing so, the system automatically converts the figural elements to their symbolic form and identifies any of four kinds of errors in the learners’ proof attempts, providing relevant feedback on-screen. We use empirical examples from our pilot studies to illustrate how this combination of technological features and systematic feedback can support student understanding of the structure of proof and how to plan one. Finally, we point out some limitations to mathematical expression and the usage of the flow-chart format, and indicate the prospect of minimizing such limitations by adopting a learning progression for the introductory lessons concerning deductive proofs.

These notes are based on lectures delivered in the summer school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'', held in Bedlewo, Poland, in the summer of 2011. The course is an exposition of Furstenberg's conjectures on ``transversality'' of the maps $x\rightarrow ax $mod1 and $x\mapsto bx$mod1 for multiplicatively independent integers $a,b$, and of the associated problems on intersections and sums of invariant sets for these maps. The first part of the course is a short introduction to fractal geometry. The second part develops the theory of Furstenberg's CP-chains and local entropy averages, ending in proofs of the sumset problem and of the known case of the intersections conjecture.