David Ben-Chaim

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.

Proportional Reasoning—A Psychological-Didactical View

Proportional reasoning is the human ability to make use of an effective form of the proportional scheme. This ability has a central role in the development of mathematical thinking, and is frequently described as a concept that, on the one hand, is a cornerstone of higher mathematics, and, on the other hand, is the peak of the basic tenets of mathematics (Lesh, Post, & Behr, 1988).

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.

Authentic Investigative Activities—Introduction

This section presents a wide range of authentic ratio-and-proportion investigative activities that represent actual situations relevant to the real world of students and teachers alike. They span various levels of difficulty appropriate for pre-and inservice elementary and middle school teachers, and can be easily adapted for authentic investigative activities appropriate for pupils in elementary and middle school.

Assessing Research Reports and Building A Student Portfolio

The teaching model presented in this book incorporates into the teaching process the reading and analysis of reports pertaining to both the mathematical and pedagogic-didactic perspectives of ratio and proportion. The guidelines presented above, the results of many years of experience working with pre-service teachers, will guide pre- and in-service teachers in efficiently analyzing such reports.

A Model For Teaching Ratio and Proportion Using Authentic Investigative Activities

The model presented below is the basis used for constructing an instructional unit on the topic of ratio and proportion, whether for pre-service mathematics teachers or for in-service mathematics teachers participating in a professional development course.

Introduction to Assessment Tools

The link between assessment and learning is widely acknowledged and highly significant. Sainsbury and Walker (2007), for example, mention four functions of assessment that directly influence student learning: motivating learning, focusing learning, consolidating and structuring learning, and guiding and correcting learning. Rust (2007) concludes that “any scholarship of assessment must therefore be predicated on the value that good assessment supports and positively influences student learning.”

Research and New Approaches in Pre- and in-Service Mathematics Teacher Education

Recent studies in many countries worldwide have pointed out the difficulties that pre-service teachers aspiring to teach mathematics in elementary school have. These difficulties are expressed especially as an inability to understand mathematical content and concepts, and feelings of incompetence in dealing with and teaching the subject (Lamon, 2007; Empson & Junk, 2004), combined, among other things, with a certain negative attitude to the mathematics-teaching profession as a whole (Tirosh & Graeber, 1990).

Diagnostic Questionnaire in Ratio and Proportion

Max, Alice, Alex, and Sima planned a class bicycle trip to the zoo. The pupils gathered in the parking lot of the school and rode their bikes along the bike path leading to the zoo. They watched the animals for a few hours, and then met by the lake for some snacks and cold drinks before riding back to the school.

A Mathematical Perspective of Ratio and Proportion

In mathematics, the concept of ratio is fundamental to many topics. Children encounter the concept in the earliest years of elementary school, even if they are not introduced to the actual word. They first learn the specific word “ratio,” in Grade 6. In fact, many sections of the elementary and middle school curriculum refer to, directly or indirectly, the concept of ratio. Price per item, fractions, percentages, probability, problems in motion, measurement, enlargement and reduction of shapes and figures, and π as a ratio between the circumference of a circle and its diameter, are just a few examples of ratios in the mathematics curriculum of elementary and middle school.