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The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra and geometry, like other branches of mathematics, are interrelated. [ABSTRACT FROM AUTHOR]

In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this. [ABSTRACT FROM AUTHOR]

In this paper a class of regular surfaces, generated by rotation and twist of a rigid line segment, is considered. Firstly a complete description of those surfaces by means of local charts is obtained and then its orientability is characterized in terms of the number of half-twists, taking into account their direction (clockwise or counterclockwise) since changes in the sense of twist are allowed. Indeed a procedure to construct nonorientable surfaces is provided. Illustrative examples and computer-generated plots using the free software Maxima are included. [ABSTRACT FROM AUTHOR]

This paper examines the effect of the use of dynamic geometry environments on young children's (ages 5–6, kindergarten/grade 1) thinking about angle. It provides a detailed description of introductory sessions of a geometry unit about angle, during which children worked in a whole classroom setting in which they could interact directly with Sketchpad through an interactive whiteboard. Using Sfard's [2008, Thinking as communicating: Human development, the growth of discourses and mathematizing. Cambridge: Cambridge University Press] communicational approach, an attempt is made to show how children developed a reified discourse on angles and were able to develop an understanding of angle as 'turn', that is, of angle as describing an amount of turn. This discourse focused on the behaviour of a pre-constructed sketch which instantiated both dynamic and static representations of angle. Extending prior research on children's difficulties in unifying static and dynamic conceptions of angles, this study provides one way of establishing a relationship between angle-as-turn and angle-as-shape conceptions. Gestures and motion played an important role in children's developing conceptions of angles. It presents implications of considering young children's embodied forms for communications along with their verbal communication for understanding their mathematical thinking. [ABSTRACT FROM AUTHOR]

Students' difficulty with reasoning and justification in the context of proofs in geometry has been well documented. Studies reveal that many of the ideas that students are asked to prove are known to be true, and therefore, the process of such proof is likely to remain meaningless and purposeless. The implication of this claim is that students need to be asked to prove mathematical assertions that are not known to them to be true. This puts an enormous responsibility on teachers. The purpose of this paper is to provide examples of 'non-traditional conjectures' that can be explored in the dynamic geometry environment of computer construction programs. [ABSTRACT FROM AUTHOR]

This paper presents apps designed to assist students in understanding and developing proofs in geometric theorems. These technologies focus on triangle congruence, triangle similarity and properties of parallelograms. Focus group discussions and initial testing of the apps revealed that the apps offered a more engaging medium for learning proving and were capable of facilitating proof-writing skills in geometry. [ABSTRACT FROM AUTHOR]

In this paper, some new results related to the integral mean value theorem are proven. [ABSTRACT FROM AUTHOR]

The necessary and sufficient condition for a convex quadrilateral to be cyclic is usually given in terms of its angles, namely that the opposite angles are supplementary. In this paper, two different necessary and sufficient symmetrical conditions are proved, which are expressed in terms of the lengths of the sides of a given quadrilateral, which make it cyclic. [ABSTRACT FROM AUTHOR]

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers' understanding of other functions of proof than just the traditional function of 'verification'. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called 'reconstructive' approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry. [ABSTRACT FROM AUTHOR]

Reports on the design and trialing of a Hypercard testing procedure known as the Recognition and Testing System (RATS) that will enable mathematics teachers and researchers to generate data relating to geometry knowledge representation and its access. Three types of information about students' geometric knowledge provided by RATS; Discussion of mathematical knowledge schemas.

A simple proof to some known results on the convergence of linear recursive sequences with nonnegative coefficients is given, using the technique of monotone convergence. [ABSTRACT FROM AUTHOR]

With the emergence of new instructional technologies, pre-service teachers are expected to develop Mathematical Technological Knowledge (MTK), which we define as teacher knowledge of technology developed as a result of exploring mathematics with technology. We examined 16 middle grades pre-service mathematics teachers' (PSMTs) MTK within a geometry course where they used the dynamic geometry software, Geometer's Sketchpad, as a cognitive tool. The data from this exploratory case study revealed five levels of MTK which can be used to classify the ways PSMTs understand and leverage the capabilities and limitations of this dynamic geometry software. In our conclusion, we recommend mathematics teacher educators attend to PSMTs' different levels of utilizing dynamic geometry software in teaching and learning mathematics. These levels of MTK can also be leveraged to inform mathematics teacher educators' decisions in providing technological experiences within mathematical content and instructional technology courses. [ABSTRACT FROM AUTHOR]

These notes discuss several related propositions in geometry that can be explored in a dynamic geometry environment. The propositions involve an unexpected property of quadrilaterals. [ABSTRACT FROM AUTHOR]

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems. [ABSTRACT FROM AUTHOR]

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students’ perceptions of proofs in the geometry. The study explores whether students know when to use a deductive proof and when an example is sufficient for proving or refuting geometrical claims. The findings indicate that in cases of simple claims, the students corroborate them by using a deductive proof. However, when the claim is more complex, the students tend to present both a proof and an example. Moreover, they are unsure whether using an example can constitute a method for proving a mathematical claim, believing that in mathematics everything must be proven. They believe that examples are used merely for illustration purposes rather than as a means of convincing. The research conclusions support the need for deepening and developing the students’ distinction between cases where examples are insufficient and cases where an example is sufficient for proving a claim. [ABSTRACT FROM PUBLISHER]

For any parent tetrahedron ABCD, centroids of selected sub-tetrahedra form the vertices of an irregularly shaped space-filling truncated octahedron. To reflect these properties, such a figure will be called an ISTO. Each edge of the ISTO is parallel to and one-eighth the length of one of the edges of tetrahedron ABCD and the volume of the ISTO is 3/16-th the volume of the tetrahedron. The ISTO is symmetric about the centroid of tetrahedron ABCD and each face is symmetric about a centre and has an opposite face that is parallel and congruent. The area of the faces of the ISTO is not proportional to that of the generating tetrahedron. [ABSTRACT FROM AUTHOR]

In this study, we attempt to put in question students' spontaneous and uncritical application of the simple and neat mathematical formula of linearity. This is impelled with the help of a written test involving geometrical problems, which is based on an original experimental setting. In this setting, grade 9 and 10 students were instructed first to solve all the geometrical problems and then to select only one problem as the appropriate for a given numerical answer. The difficulty of this choice lied in fact that a superficial handling of each problem would resolve to the same numerical answer as the one given. The results show that students' choices are systematic and based on the solutions given to the tasks. However, the experimental setting has managed to help students question in some degree the applicability of the linear model in situations that appear to be linear but are not. [ABSTRACT FROM AUTHOR]