Abstracts

The role of visualization in mathematics for all

Abraham Arcavi

Weizmann Institute, Israel

I will discuss several pedagogical approaches to engage students of different mathematical inclinations who study within the same class. In these approaches, I will describe and analyze the role of visualization through several tasks suitable for junior high and high school mathematics.  

Posing Problems Properly: Creating and Maximizing Learning Opportunities for All Students

Jinfa Cai

University of Delaware, USA

This presentation is situated in the context of teaching mathematics through problem posing and problem solving.  It consists of three parts.  I will first present the theoretical bases and empirical evidence for teaching mathematics through problem posing and problem solving.  Then I present samples tasks that can create and maximize learning opportunities for all students.

This presentation will end with a set of principles and recommendations for designing tasks that can create and maximize learning opportunities for all students in the context of teaching mathematics through problem posing and problem solving. 

The personalized mathematics inquiry model for developing Cyprus mathematics textbooks for middle schools

Constantinos Christou, Demetra Pitta-Pantazi

University of Cyprus, Cyprus

The aim of our talk is to present and discuss the theoretical model which supported the design and development of the Cyprus Mathematics Textbooks, with special focus on Middle School Mathematics Textbooks.  Our purpose is to highlight the theoretical model used in selecting, presenting, and organizing the content in the mathematics textbooks for the last 10 years. The application of this theoretical model is exemplified through specific examples taken from the mathematics textbooks.

The framework purports to contribute to Personalized Mathematics Inquiry (PMI). PMI involves a set of practices in which students actively engage, discuss, create, and reflect on mathematical content. These practices integrate principles of inquiry-based learning with elements of cognitive apprenticeship and ideas associated with connected learning. Mathematics instruction, within this framework, seeks to actively involve students in authentic, and personally relevant learning experiences that foster academic achievement, and reflection. 

Inquiry is at the core of PMI framework and consists of explorations and investigations. Both explorations and investigations are based on worthwhile tasks as defined by NCTM (2007), i.e., those problems that capture students’ curiosity and invite students to speculate and to pursue their hunches. 

The goal of explorations is to elicit curiosity and motivation, which will ignite students’ personal inquiry. Textbooks, at the beginning of each unit, which has a specific mathematical content, present a challenging context (or problem) and students are given the time to engage, analyze, experiment and discuss this challenging problem. Inevitably, students bring in these explorations their own background experiences and this is the reason that these explorations supports differentiation and persona learning. Through an exploration students come to a realization of the mathematical concept they need to conquer and make conceptual links either with other mathematical concepts or real life. Each exploration is followed by an investigation. 

 Investigations are again challenging problems which however, are more closed and guided than explorations. To respond to these investigations students often need to analyze, form hypotheses and work in a systematic way by creating and applying various strategies and using different resources such as mathematical tools, manipulatives and applets. Students also need to explain, justify and reflect on their workings. During these investigations, the role of the teachers is to facilitate students to control their own learning. At this stage, new vocabulary and new mathematical concepts are also introduced. 

Meanings and Rhythms behind the Comical Hieroglyphics: Some Thoughts on Mathematical Problem Solving

Alexander Karp

Teachers College Columbia University,  USA

This presentation draws on the experience of writing problem books for different categories of students, and above all for Russian upper-level high school students in so-called classes with in-depth study of the humanities, in other words, students who are given a brief course in mathematics. In what way such a problem book ought to be written, however, is a question that leads to considerations not only about strictly methodological or mathematical issues, but also about the history, role, and uses of problem solving. The presenter will share some of his thoughts on these topics.

On exponents, irrational and rational: a story of one task

Rina Zazkis

Simon Fraser University, Canada 

I will describe implementation of tasks related to rational and irrational exponents with a group of teachers. I will attend to students’ –  in this case teachers’— difficulties in establishing the meaning of irrational number in the exponent, and to ways in which this search for meaning leads to consideration (or reconsideration) of rational exponents. 

I will argue that by attempting to make sense of irrational exponents, learners deepened their comprehension of rational exponents in a meaningful way. 

Design patterns of interactive Example-Eliciting-Tasks

Michal Yerushalmy

University of Haifa, Israel

I will present and discuss several design patterns of interactive Example-Eliciting-Tasks (EETs) and argue that using EETs within  interactive platform that offers immediate feedback to student's submissions is an appropriate way for teachers to  dynamically focus on what is important. Because student responses to EETs can provide teachers with information about a variety of concept definitions and concept images that students hold, teachers can use the online analysis  to make instructional decisions in the heterogeneous classroom.