In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles.

In this article, we will show that the Pythagorean approximations of Formula coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers converging to different algebraic irrationals. We will see how approximations to some irrational numbers, using known facts from the history of mathematics, may perhaps help to acquire a better comprehension of the real numbers and their properties at further mathematics level.

Teacher-education research lacks a common theoretical basis, which prevents a convincing development of instruments and makes it difficult to connect studies to each other. Our paper models how to measure effective teacher education in the context of the current state of knowledge in the field. First, we conceptualize the central criterion of effective teacher education: “professional competence of future teachers”. Second, individual, institutional, and systemic factors are modeled that may influence the acquisition of this competence during teacher education. In doing this, we turn round the perspective taken by Cochran-Smith and Zeichner (Studying teacher education. The report of the AERA panel on research and teacher education. Lawrence Erlbaum, Mahwah 2005), who mainly take an educational-sociological perspective by focusing on characteristics of teacher education and looking for their effects. In contrast, we take an educational-psychological perspective by focusing on professional competence of teachers and examining influences on this. Challenges connected to an assessment of teacher-education outcomes are discussed as well.

This paper characterizes early mathematics instruction in Hong Kong. The teaching of addition in three pre-primary and three lower primary schools was observed and nine teachers were interviewed about their beliefs about early mathematics teaching. A child-centered, play-based approach was evident but teachers emphasized discipline, diligence and academic success. Observations also revealed practices reflective of both constructivist and instructivist pedagogies. Results from interviews suggest that teachers’ traditional cultural beliefs about instruction were challenged by western ideologies introduced in continuing professional development courses and by notions promulgated by the educational reforms. Both consistencies and inconsistencies between teachers’ beliefs and practices were identified. Implications of the findings are discussed.

Abstract:

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student’s problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.

1. Jeff Babb & James Currie(Canada)
The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context

2. Michael Fried (Israel)
History of Mathematics in Mathematics Education: a Saussurean Perspective

3. Spyros Glenis (Greece)
Comparison of Geometric Figures

4. Giorgio T. Bagni (Italy)
“Obeying a rule”: Ludwig Wittgenstein and the foundations of Set Theory

5. Arnaud Mayrargue (France)
How can science history contribute to the development of new proposals in the teaching of the notion of derivatives?

6. Antti Viholainen (Finland)
Incoherence of a concept image and erroneous conclusions in the case of differentiability

7. Dores Ferreira & Pedro Palhares (Portugal)
Chess and problem solving involving patterns

8. Friðrik Diego & Kristín Halla Jónsdóttir (Iceland)
Associative Operations on a Three-Element Set

9. Jon Warwick (UK)
A Case Study Using Soft Systems Methodology in the Evolution of a Mathematics Module

10. Barbara Garii & Lillian Okumu (New York, USA)
Mathematics and the World: What do Teachers Recognize as Mathematics in Real World Practice?

11. Linda Martin & Kristin Umland (New Mexico, USA)
Mathematics for Middle School Teachers: Choices, Successes, and Challenges

12. Woong Lim (Texas, USA)
Inverses – why we teach and why we need talk more about it more often!

13. Steve Humble (UK)
Magic Math Cards

The issue also contains a couple of articles on logarithms in a historical perspective, a large section of articles with reactions on the report of the National Mathematics Advisory Panel, etc.

Postsecondary remediation is a controversial topic. On one hand, it fills an important and sizeable niche in higher education. On the other hand, critics argue that it wastes tax dollars, diminishes academic standards, and demoralizes faculty. Yet, despite the ongoing debate, few comprehensive, large-scale, multi-institutional evaluations of remedial programs have been published in recent memory. The study presented here constitutes a step forward in rectifying this deficit in the literature, with particular attention to testing the efficacy of remedial math programs. In this study, I use hierarchical multinomial logistic regression to analyze data that address a population of 85,894 freshmen, enrolled in 107 community colleges, for the purpose of comparing the long-term academic outcomes of students who remediate successfully (achieve college-level math skill) with those of students who achieve college-level math skill without remedial assistance. I find that these two groups of students experience comparable outcomes, which indicates that remedial math programs are highly effective at resolving skill deficiencies.

• Integrating supplementary application-based tutorials in the multivariable calculus course by I. M. Verner;
  S. Aroshas; A. Berman
• If not, what yes? by Boris Koichu
• Mathematical e-learning: state of the art and experiences at the Open University of Catalonia by A. Juan; A. Huertas; C. Steegmann; C. Corcoles; C. Serrat
• Unique factorization in cyclotomic integers of degree seven by W. Ethan Duckworth
• A college lesson study in calculus, preliminary report by Joy Becker; Petre Ghenciu; Matt Horak; Helen Schroeder