Moshe Stupel

A Special Application of Absolute Value Techniques in Authentic Problem Solving.

There are at least five different equivalent definitions of the absolute value concept. In instances where the task is an equation or inequality with only one or two absolute value expressions, it is a worthy educational experience for learners to solve the task using each one of the definitions. On the other hand, if more than two absolute value expressions are involved, the definition that is most helpful is the one involving solving by intervals and evaluating critical points. In point of fact, application of this technique is one reason that the topic of absolute value is important in mathematics in general and in mathematics teaching in particular. We present here an authentic practical problem that is solved using absolute values and the ‘intervals’ method, after which the solution is generalized with surprising results. This authentic problem also lends itself to investigation using educational technological tools such as GeoGebra dynamic geometry software: mathematics teachers can allow their students to initially cope with the problem by working in an inductive environment in which they conduct virtual experiments until a solid conjecture has been reached, after which they should prove the conjecture deductively, using classic theoretical mathematical tools.

Enhancing elementary-school mathematics teachers' efficacy beliefs: a qualitative action research

Individuals and societies that can use mathematics effectively in this period of rapid changes will have a voice on increasing the opportunities and potentials which can shape their future. This has brought affective characteristics, such as self-efficacy, that affect mathematics achievement into focus of the research. Teacher efficacy refers to the extent to which a teacher feels capable to help students learn, influence students’ performance and commitment, and thus plays a crucial role in developing the student in all aspects. In this study, we used two sources of efficacy beliefs, mastery experiences and physiological and emotional states, in an interesting and challenging seven month workshop, as tools to foster teacher efficacy for six elementary-school teachers who were frustrated and wanted to leave their job. Our aim was to study the nature of these teachers’ efficacy in order to change it. In this qualitative action research, we used open interviews, non-participant observations and field notes. Results show that these teachers became efficacious, their students’ achievements and motivation were enhanced, and the school climate was changed. Qualitative inquiry of this construct sheds light on efficacy beliefs of mathematics teachers. Nurturing teacher efficacy has borne much fruit in the field of mathematics in school.

The standard proof, the elegant proof, and the proof without words of tasks in geometry, and their dynamic investigation

ABSTRACT 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.

Absolute value equations – what can we learn from their graphical representation?

Understanding graphical representations of algebraic equations, particularly graphical representations of absolute value equations, significantly improves students’ mathematical comprehension and ignites within them an appreciation of the beauty and aesthetics of mathematics. In this paper, we focus on absolute value equations of linear and quadratic expressions, by examining various cases, presenting different methods of solving them by graphical representation, exhibiting the advantage of using dynamic software such as GeoGebra in solving them, and illustrating some examples of interesting graphical solutions. We recommend that teachers take advantage of the rapid development in technology to help learners tangibly visualize the solutions of absolute value equations before proceeding to the analytical solutions.

Pizza Again? On the Division of Polygons into Sections with a Common Origin.

ABSTRACT The paper explores the division of a polygon into equal-area pieces using line segments originating at a common point. The mathematical background of the proposed method is very simple and belongs to secondary school geometry. Simple examples dividing a square into two, four or eight congruent pieces provide a starting point to discovering how to divide a regular polygon into any number of equal-area pieces using line segments originating from the centre. Moreover, it turns out that there are infinite ways to do the division. Discovering the basic invariant involved allows application of the same procedure to divide any tangential polygon, as after suitable adjustment, it can be used also for rectangles and parallelograms. Further generalization offers many additional solutions of the problem, and some of them are presented for the case of an arbitrary triangle and a square. Links to dynamic demonstrations in GeoGebra serve to illustrate the main results.

The concept of invariance in school mathematics

ABSTRACT In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change leaves some relationships or properties invariant, these properties prove to be inherently interesting to teachers and students.

The Properties of Special Points on the Brocard Circle in a Triangle

In this paper we investigate some properties related to the Brocard circle of a given triangle ∆ABC. Four of them correspond to the special point of a triangle that satisfies some conditions. Another one corresponds to the special case of ∆ABC for which the relation \( ({\text{AC}}^{2} + {\text{AB}}^{2} )/2 = {\text{BC}}^{2} \) holds. We analyzed the case where a symmedian is a tangent to the Brocard circle. We found some interesting properties related to the Brocard circle for right angle triangle and for isosceles triangle.

Proof Without Words: An Elegant Property of a Triangle Having an Angle of 60 Degrees

Summary In a triangle ABC in which angle A measures 60 degrees, the bisectors of angles B and C are used to construct a cyclic quadrilateral with two congruent sides.

Using multiple solutions to mathematical problems to develop pedagogical and mathematical thinking: A case study in a teacher education program

ABSTRACT Mathematics educators agree that linking mathematical ideas by using multiple approaches for solving problems (or proving statements) is essential for the development of mathematical reasoning. In this sense, geometry provides a goldmine of multiple-solution tasks, where a myriad of different methods can be employed: either from the geometry topic under discussion or from other mathematical areas—analytic geometry, trigonometry, vectors, complex number, etc. Employing multiple proofs fosters better comprehension and increased creativity in mathematics for the student/learner, enriching teachers’ pedagogical accomplishments and promoting lively class discussion. Given the important role of multiple-solution problems within and between mathematical topics, the evidence is astonishing that classroom teachers rarely introduce their students to multiple-solution tasks. Hence, one can conjecture that this gap between theory and practice could turn connecting tasks with the employment of technological tools into a powerful environment for the development of pre- and in-service mathematics teachers’ knowledge. For this reason the authors believe that exposing and providing mathematics teachers with an arsenal of specific tasks with a variety of solutions from different mathematical areas is essential. Based on a conducted case study, both teacher trainees and lecturers clearly indicated that solving problems in multiple ways is valuable in developing thinking ability for both students and teachers, encouraging creativity and increasing the quality of teaching—hence this technique should be included in the secondary school curriculum as well as in teacher training programs.

Dynamic investigation of triangles inscribed in a circle, which tend to an equilateral triangle

ABSTRACT We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner. First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.