On popularization of Scientific Education in Italy between 12th and 16th Century

Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathematics education is framed changes according to modifications of the social environment and know\u2013how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of mathematical analysis was profound: there were two complex examinations in which the theory was as important as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth\u2013month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and mathematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of \u201cscientific education\u201d (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by \u201cmaestri d\u2019abaco\u201d to the Renaissance humanists, and with respect to mathematics education nowadays is discussed. Particularly the ways through which mathematical knowledge was spread in Italy between late Middle ages and early Modern age is shown. At that time, the term \u201cscientific education\u201d corresponded to \u201cteaching of mathematics, physics\u201d; hence something different from what nowadays is called science education, NoS, etc. Moreover, the relationships between mathematics education and civilization in Italy between the 12th and the 16th century is also popularized within the Abacus schools and Niccol\uf2 Tartaglia. These are significant cases because the events connected to them are strictly interrelated. The knowledge of such significant relationships between society, mathematics education, advanced mathematics and scientific knowledge can be useful for the scholars who are nowadays engaged in mathematics education research

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

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

Pupils' needs for conviction and explanation within the context of dynamic geometry.

Thesis (M.Ed.)-University of Durban-Westville, 1998.Recent literature on mathematics education, and more especially on the teaching and learning of geometry, indicates a need for further investigations into the possibility of devising new strategies, or even developing present methods, in order to avert what might seem to be a "problem" in mathematics education. Most educators and textbooks, it would seem, do not address the need (function and meaning) of proof at all, or those that do, only address it from the limited perspective that the only function of proof is verification. The theoretical part of this study, therefore, analyzed the various functions of proof, in order to identify possible alternate ways of presenting proof meaningfully to pupils. This work further attempted to build on existing research and tested these ideas in a teaching environment. This was done in order to evaluate the feasibility of introducing "proof" as a means of explanation rather than only verification, within the context of dynamic geometry. Pupils, who had not been exposed to proof as yet, were interviewed and their responses were analyzed. The research focused on a few aspects. It attempted to determine whether pupils were convinced about explored geometric statements and their level of conviction. It also attempted to establish whether pupils exhibited an independent desire for why the result, they obtained, is true and if they did, could they construct an explanation, albeit a guided one, on their own. Several useful implications have evolved from this work and may be able to influence, both the teaching and learning, of geometry in school. Perhaps the suggestions may be useful to pre-service and in-service educators

About a constructivist approach for stimulating students' thinking to produce conjectures and their proving in active learning of geometry

The paper describes processes that might lead secondary school students to produce conjectures in a plane geometry. It highlights relationship between conjecturing and proving. The author attempts to construct a teaching-learning environment proposing activities of observation and exploration of key concepts in geometry favouring the production of conjectures and providing motivation for the successive phase of validation, through refutations and proofs. Supporting didactic materials are built up in a way to introduce production of conjectures as a meaningful activity to students

How to teach the Pythagorean theorem: An analysis of lesson plans

This research was conducted among mathematics graduates who participated in a pedagogical formation certificate program. Participants were asked to prepare a lesson plan intended for use in teaching the Pythagorean theorem as part of a ninth grade mathematics course. Eighteen out of 43 participants included a proof of the Pythagorean theorem as a component of their lesson plan. These proofs were classified in three categories: visual proofs (two participants), algebraic proofs (nine participants), and proofs by using triangular similarities (seven participants). In addition, the solved examples, homework, and evaluation questions included in the lesson plans were classified according to TIMSS cognitive levels. Of the 233 questions prepared by 43 participants, 37% of the questions were at the knowledge level, 60% were at the application level, and the remaining 3% were at the reasoning level

Communities in university mathematics

This paper concerns communities of learners and teachers that are formed, develop and interact in university mathematics environments through the theoretical lens of Communities of Practice. From this perspective, learning is described as a process of participation and reification in a community in which individuals belong and form their identity through engagement, imagination and alignment. In addition, when inquiry is considered as a fundamental mode of participation, through critical alignment, the community becomes a Community of Inquiry. We discuss these theoretical underpinnings with examples of their application in research in university mathematics education and, in more detail, in two Research Cases which focus on mathematics students' and teachers' perspectives on proof and on engineering students' conceptual understanding of mathematics. The paper concludes with a critical reflection on the theorising of the role of communities in university level teaching and learning and a consideration of ways forward for future research

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

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

Using materials from the history of mathematics in discovery-based learning

This paper reports on attempt to integrate history of mathematics in discovery-based learning using technology. Theoretical grounding of the idea is discussed. An exploratory environment on triangle geometry is described. It is designed to support and motivate students' activities in learning through inquiry. Conjectures about properties of Lemoine point and Simson line are produced and proved by students using e-learning textbook

Proof in dynamic geometry contexts

Proof lies at the heart of mathematics yet we know from research in mathematics education that proof is an elusive concept for many mathematics students. The question that this paper raises is whether the introduction of dynamic geometry software will improve the situation – or whether it make the transition from informal to formal proof in mathematics even harder. Through discussion of research into innovative teaching approaches with computers the paper examines whether such approaches can assist pupils in developing a conceptual framework for proof, and in appropriating proof as a means to illuminate geometrical ideas