1,499 research outputs found

    Cabri's role in the task of proving within the activity of building part of an axiomatic system

    Get PDF
    We want to show how we use the software Cabri, in a Geometry class for preservice mathematics teachers, in the process of building part of an axiomatic system of Euclidean Geometry. We will illustrate the type of tasks that engage students to discover the relationship between the steps of a geometric construction and the steps of a formal justification of the related geometric fact to understand the logical development of a proof; understand dependency relationships between properties; generate ideas that can be useful for a proof; produce conjectures that correspond to theorems of the system; and participate in the deductive organization of a set of statements obtained as solution to open-ended problems

    Hybrid-dynamic objects: DGS environments and conceptual transformations

    Get PDF
    A few theoretical perspectives have been taken under consideration the meaning of an object as the result of a process in mathematical thinking. Building on their work, I shall investigate the meaning of ‘object’ in a dynamic geometry environment. Using the recently introduced notions of dynamic-hybrid objects, diagrams and sections which complement our understanding of geometric processes and concepts as we perform actions in the dynamic software, I shall explain what could be considered to be a ‘procept-in-action’. Finally, a few examples will be analyzed through the lenses of hybrid and dynamic objects in terms of how I designed them. A few snapshots of the research process will be presented to reinforce the theoretical considerations. My aim is to contribute to the field of the Didactics of Mathematics using ICT in relation to students’ cognitive developmen

    A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why

    Get PDF
    In the current study the theoretical notion of instrumental learning path or trajectory is analyzed through examples based on a research synthesis. I point out the role of instrumental decoding in a static or dynamic environment, and how the competence of the participants (students –researcher/ teacher) can influence the holistic result of the learning process by creating interdependencies/intra-dependencies during the construction of instrumental learning trajectories. Instrumental trajectories are not just construction instructions, or a set of information which provides the properties of the figure as the figure is constructed. Instrumental trajectories can show us the interdependencies/intra-dependencies that exist or can be created between different tools. Dynamic Geometry allows for the creation of interdependencies and intra-dependencies between mathematical objects, diagrams and tools. In the sections that follow, I shall present three examples of instrumental learning trajectories, in which the interdependencies among the tools and meanings are analyzed. My aim is to combine and synthesize different primary qualitative research studies and make their results more generalizable. Keywords: instrumental decoding, interdependencies/intra-dependencies, instrumental learning trajectories DOI: 10.7176/IKM/11-3-02 Publication date: April 30th 2021

    Instructional strategies in explicating the discovery function of proof for lower secondary school students

    No full text
    In this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr

    How the activity of proving is constituted in a Cypriot classroom for 12 year old students

    Get PDF
    The aim of this study is to identify how the activity of proving is constituted in a Cypriot primary classroom for 12 year old students. Through Cultural-Historical Activity Theory (CHAT), the influence of research literature, curriculum prescriptions, the students and critically the teacher is documented. The evolution of objects, in particular the aims of the teacher, and other components in the activity systems is traced. Within the qualitative enquiry, this study employs CHAT alongside a collaborative design approach to explore the way the teacher is working with the students to foreground mathematical argumentation. The research is situated in a Cypriot primary school classroom with the researcher having the role of teacher researcher. The usual class teacher and researcher co-developed Dynamic Geometry Environment (DGE)-based tasks to be used with the children. As a result it was possible to track how the nature of the teacher’s objects changed and how contradictions emerged. Evidence from the curriculum documentation and from classroom observations was used to develop the activity systems of exploring and explaining. One important finding lies in how exploring and explaining were key sub-systems within the central activity system of proving as they provided a key pathway, which often included defining. Processes of explaining, defining and exploring appeared to create a fertile ground for the development of proving. I refer to these developments as pre-proving. However, it turns out that there are inherent contradictions within explaining and exploring that hinder the constitution of proving in the classroom. An emerging primary contradiction was apparent in the multifaceted nature of the object of both exploring and explaining to both facilitate mathematical argumentation and address a prescribed curriculum. Due to the tension between these objects, the teacher was often faced with dilemmas such as whether to open up playful activity or close it down to focus on the curriculum specifics. These led to a constant struggle in the teacher’s everyday practice. I report also on how primary contradictions led inevitably to higher-level contradictions between other components of the activity systems

    A Technologically Enriched Geometry Curriculum

    Get PDF
    Mathematics education cannot be viewed as a static science. As times change and students\u27 experiences are altered by their constantly evolving environments, educators of all disciplines must make adjustments that reflect these changes. Careful development of technologically enriched and visually stimulating lessons is essential to the success of mathematics education. In order to gain greater understanding of the power of partnering technology and geometry curriculum, this research project asks two main questions: how the use of technology affects student motivation and engagement in a geometry class and how a technologically enriched geometry curriculum can be created and implemented. The researcher notes that despite all the positive aspects from the use of geometry software applications, a proper balance between direct classroom instruction and student-centered discovery learning through technology is paramount. As part of the project a curriculum was developed, keeping in mind the standards set forth by NYSED and NCTM, which studied congruent triangles. Since the congruent triangles unit is often the first introduction to formal proof techniques in today\u27s secondary geometry classroom, this served the study effectively. The research was conducted in an Applied Mathematics class, comprised of ten students of varying grade levels (10-12) and differing abilities. Pre and post assessments were made as well as student surveys on the research experience. Conclusions from the research project data suggest an increased motivation by students, supporting the theory that a technology enriched curriculum can be more engaging in a geometry classroom than the textbook/lecture model. Curriculum resources and assessment surveys are included in the final sections of this project

    Proof and Proving in Mathematics Education

    Get PDF
    undefine

    Mathematics education reform in Trinidad and Tobago: the case of reasoning and proof in secondary school

    Get PDF
    In Trinidad and Tobago, there have been substantive efforts to reform mathematics education. Through the implementation of new policies, the reformers have promoted changes in mathematics curriculum and instruction. A focus of the reform has been that of increasing opportunities for students to engage in reasoning and proving. However, little is known about how these policies affect the opportunities for reasoning and proof in the written curriculum, the teaching of proof, and students' learning. Furthermore, we are yet to know how the high-stake assessment measures interact with these new policies to impact the teaching and learning of proof. In this dissertation, my overarching question asks: What are the implications of reform on the teaching and learning of secondary school mathematics in Trinidad and Tobago? To answer this question, I conducted three studies, which examined the content, teaching, and students’ conceptions. All the studies are situated in the teaching reasoning and proof when introducing geometry concepts. In the first study, I conduct a curriculum analysis focused on examining the opportunities for reasoning and proof in the three recommended secondary school textbooks. In the second study, I conduct classroom observations of teachers’ geometry instruction focusing on opportunities for engaging students in reasoning and proof. In the third study, I examine geometry students’ conceptions of proof. The three studies are intended to provide an overview of the impact of reform on instructional issues in relation to the dynamics between teachers, student, and content (Cohen, Raudenbush, & Ball, 2003). For the first study, I adapt a framework developed by Otten, Gilbertson, Males, and Clark (2014) to investigate the quality and quantity of the opportunities for students to engage in or reflect on reasoning and proof. The findings highlight some unique characteristics of the recommended textbooks such as, (a) the promotion of the explanatory role of proof through the affordances of what I define as the Geometric Calculation with Number and Explanation (GCNE) exercises, (b) the necessary scaffolding of proof construction through activities and exercises promoting pattern identification, conjecturing, and developing of informal non-proof arguments, and (c) the varying advocacy for Geometry as an area in the curriculum where students can experience the work of real mathematicians and see the intellectual of proof in their mathematical experiences. All these characteristics align with the reformers’ vision for the teaching and learning of reasoning and proof in secondary school mathematics. In the second study, I examine the nature of the teaching of reasoning and proof in secondary school. I use classroom observations along with pre- and post-observations interview data of three teachers to determine (a) the classroom microculture (i.e., classroom mathematical practices and sociomathematical norms), (b) teachers’ pedagogical decisions, and (c) teachers’ use of the Caribbean Secondary Examination Certificate (CSEC) examination materials and textbooks. I also determine whether the teachers’ instruction demonstrate the four characteristics of reform-based mathematics teaching (Hufferd-Ackles, Fuson, & Sherin, 2004). My analysis of classroom observations of the three teachers suggests that their instructional practices exhibit elements of reform-based instruction. These include teachers’ use of open-ended and direct questions to solicit students’ mathematical ideas and teachers’ consideration of students as the source of mathematical ideas. Each teacher established sociomathematical norms that governed how and when a student can ask questions. In this case, questioning helped students articulate their ideas when responding to questions and clarifying their understanding of other's ideas when they posed a question. Teachers also established sociomathematical norms that outlined what counts as a valid proof and what counts as an acceptable answer during instruction. The aforementioned norms supported the expectation that students must always provide explanations for their mathematical thinking, which is another characteristic of reform-based teaching. Teachers used group work and whole class discussions to offer opportunities for collaborative learning, which facilitated their creation of a social constructivist environment for learning reasoning and proof. Teachers used the reform-oriented curriculum materials to provide opportunities for construction of proofs. However, the textbooks and curriculum were limited in their support for proving some geometrical results. Overall, the teachers emphasized the making and testing of conjectures, which afforded students with authentic mathematical experiences that promoted the development of mathematical knowledge. In the third study, I use the six principles of proof understanding (McCrone & Martin, 2009) to examine 21 students’ conceptions of proof. I use semi-structured interviews to gather students’ perspectives of (a) the roles of proof, (b) structure and generality of proof, and (c) the opportunities for proof in the curriculum materials. The findings indicate that the students considered proof as serving the roles of explanation, verification, systemization, and appreciation in mathematics. The latter role helps students see the value and purpose of the mathematical results they learn (a) for applications during problem solving and (b) within the axiomatic system of Geometry results. The aforementioned roles also help students see the intellectual need for reasoning and proof in their mathematical experiences. Students’ talk suggests that, their teachers’ and the external examiners’ expectations of the structure, generality, and validity of proof influence their notions of what constitutes a proof. Students also consider the examination opportunities that require the development of reasoned explanations as possible opportunities to construct proof arguments. The combined findings of these three studies could help researchers understand the implications of the recent reform recommendations on the teaching and learning of proof in Trinidad and Tobago. Firstly, these findings can be useful to policy makers and education stakeholders in their future efforts for developing the national curriculum, revision or development of instructional policies, and recommendations of textbooks and instructional support materials. Secondly, these findings can help curriculum designers, examiners, and teachers in creating future opportunities in the national curriculum and CSEC mathematics syllabus to support students’ learning of proof in Trinidad and Tobago. Thirdly, these findings can help educational stakeholders understand the type of support that is needed for teachers’ future professional development and students’ competency with reasoning and proof on CSEC examinations. This international study is a case of the larger issues surrounding reform implications in a centralized governed educational system, which offers uniform prescriptive guidance for teaching and uniform curriculum support for learning. Furthermore this work potentially adds to the ongoing discussions in mathematics education about the interplay between policy, practice, and student learning

    An analysis of teacher competences in a problem-centred approach to dynamic geometry teaching

    Get PDF
    The subject of teacher competences or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competences necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer’s Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competences were identified: mathematical/geometrical competences, pedagogical competences, computer and software competences, language and assessment competencies.Mathematics EducationM. Ed. (Mathematics Education
    • …
    corecore