When interviewing school students for what constitutes a good mathematics teacher, the first characteristic usually listed is the ability to explain well. Besides well-founded content knowledge most important for classroom episodes of teacher explanations is knowledge about how to present mathematical concepts in a comprehensible way to students. This encompasses competencies in the area of verbal communication as well as the conscious use of means for illustrating and visualising mathematical ideas. We report about an analysis of explanatory processes in math lessons and about an analysis of prospective teachers\' explanatory competencies. As a result we identify improvements in teacher education at university.

The importance of family involvement is highlighted by findings that parents’ behaviours, beliefs and attitudes affect children’s behaviour in a major way. The Family Maths programme, which is the focus of this study, provides support for the transformative education practices targeted by the South African Department of Education by offering an intervention which includes teachers, learners and their families in an affirming learning community. In this study participating parents were interviewed to investigate their perceptions of the Family Maths programme mainly in terms of their engagement, enjoyment and confidence levels. The major themes and ideas that were generated in this study include the development of positive attitudes, parents and children working and talking together, and the skills exhibited by Family Maths facilitators. These findings are analysed within the parameters of complexity science and the pre-requisite conditions for developing a complex learning community, viz. internal diversity, redundancy, decentralized control, organised randomness and neighbour interactions.

Young children’s ‘alternative’ notions of science are well documented but their unorthodox ideas about arithmetic are less well known. For example, studies have shown that young children initially treat numbers as position markers rather than size symbols. Also, children often hold a transformational view of operations; that is, they are reluctant to accept the commutativity of addition and multiplication. This ‘alternative’ view of operations is often overlooked by teachers, keen to demonstrate the so called ‘laws’ of arithmetic. However, this paper argues that we should not be in any haste to replace these primitive intuitions; instead, we should show that transformational operations actually reflect how objects behave when acted on in the physical world. The paper draws on earlier research of the writer in which young children used signs for transformational arithmetic in game scenarios. In particular, it examines the feasibility of ‘sums’ in which the operator is distinguished from the operand. In short, this paper presents the theory behind an entirely new way of teaching arithmetic, based on children’s ‘alternative’ intuitions about numbers and operations.

It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using our examples students were encouraged to think independently and came to improve their understanding of the subject.

In a challenging and constantly changing world, students are required to develop advanced thinking skills such as critical systematic thinking, decision making and problem solving. This challenge requires developing critical thinking abilities which are essential in unfamiliar situations. A central component in current reforms in mathematics and science studies worldwide is the transition from the traditional dominant instruction which focuses on algorithmic cognitive skills towards higher order cognitive skills. The transition includes, a component of scientific inquiry, learning science from the student's personal, environmental and social contexts and the integration of critical thinking. The planning and implementation of learning strategies that encourage first order thinking among students is not a simple task. In an attempt to put the importance of this transition in mathematical education to a test, we propose a new method for mathematical instruction based on the infusion approach put forward by Swartz in 1992. In fact, the model is derived from two additional theories., that of Ennis (1989) and of Libermann and Tversky (2001). Union of the two latter is suggested by the infusion theory. The model consists of a learning unit (30h hours) that focuses primarily on statistics every day life situations, and implemented in an interactive and supportive environment. It was applied to mathematically gifted youth of the Kidumatica project at Ben Gurion University. Among the instructed subjects were bidimensional charts, Bayes law and conditional probability; Critical thinking skills such as raising questions, seeking for alternatives and doubting were evaluated. We used Cornell tests (Ennis 1985) to confirm that our students developed critical thinking skills.

This article describes ‘Kidumatica’ – a highly successful project for the promotion of talented students from underprivileged backgrounds. In its 11 year run, Kidumatica has evolved into a way of life for its many students, allowing them opportunities to realize their potential, enter advanced academic studies, and successfully enter a society rich in knowledge and achievement. Kidumatica is based on academic research in the fields of excellence, cognition and mathematics education, and on the social principle of equal opportunity for all and one’s right to self-realization and aspiration, regardless of ethnic background and socio-economic status. Beyond these social/educational purposes, Kidumatica is also a research model and laboratory for testing new programs and teaching methods for gifted students. The following are the basic premises of the Kidumatica model, its goals and how they are achieved, including the recruitment of club members and the mathematical content.

Most of people have fatal attitude to Mathematics: some of them are capable to learn it form nature, but the others are not. So is their fate – to suffer from it for the whole of life… But it is a rude though natural mistake, as it results from means of mathematical education and its content. Most of parents and teachers are directed on these aspects both in kindergarten and at primary school. Of course, parents are different. Nevertheless so many parents can’t possibly but speak about achievements of their children. Some start making their own children learn better by the example of success of the others. They make their children learn long chains of figures with no understanding. It is even more sad to see how a mom asks her 4-year old son: “How much is two plus three?..’ But he replies just because he learned the answer but not calculated. Not only parents but also kindergarten tutors don’t want to understand that drilling for arithmetic has no sense. For a specialist it would take two days only…But teach him how to think logically – is a goal demanding from him, reached by different means.

The use of technology in Kindergarten to grade 12 classrooms provides opportunities for teachers to employ mathematical rigor, to integrate problem solving strategies and to extend mathematical ways of knowing (Drier, Dawson, & Garofalo, 1999). The presentation consists of two parts. One investigation maps secondary mathematics technology lessons and materials to the elementary school mathematics standards and converts the mathematics concepts to manageable elementary school lessons. The other investigation analyzes pre-service teacher lessons written using ASSURE instructional design format. The major aims of this paper are to present two teacher preparation practices, one for secondary mathematics pre-service teachers (converting secondary materials to elementary materials) and the other for elementary mathematics pre-service teachers (writing lessons using the ASSURE model).

Dynamic interactive technology brings new opportunities for helping students learn central statistical concepts. Research and classroom experience can be help identify concepts with which students struggle, and an \"action-consequence\" pre-made technology document can engage students in exploring these concepts. With the right questions, students can begin to make connections among their background in mathematics, foundational ideas that undergrid statistics and the relationship these ideas. The ultimate goal is to have students think deeply about simple and basic statistical ideas so they can see how they lead to reasoning and sense making about data and about making decisions about characteristics of a population from a sample.Technology has a critical role in teaching and learning statistics, enabling students to use real data in investigations, to model complex situations based on data, to visualize relationships using different representations, to move beyond calculations to interpreting statistical processes such as confidence intervals and correlation, and to generate simulations to investigate a variety of problems including laying a foundation for inference. Thus, graphing calculators, spreadsheets, and interactive dynamic software can all be thought of as tools for statistical sense making in the service of developing understanding.

This paper combines data from two different international research studies that used problem posing in analyzing elementary teacher candidates’ understanding of rational numbers. In 2007, a mathematics educator from the United States and a mathematician from Northern Ireland collaborated to investigate their respective elementary teacher candidates’ understanding of addition and division of fractions. A year later, the same US mathematics educator collaborated with a mathematics educator from South Africa on a similar research project that focused solely on the addition of fractions. The results of both studies show that elementary teacher candidates from the three different continents share similar misconceptions regarding the addition of fractions. The misconceptions that emerged were analyzed and used in designing teaching strategies intended to improve elementary teacher candidates’ understanding of rational numbers. The research also suggests that problem posing may improve their understanding of addition of fractions.