In this article are the results of research of empirical data from two pedagogical experiences using Mathematical Modeling with two groups: one with 28 students from the last period of a course of mathematics teachers, and another with 21 teachers of a course of continuing education. The objectives of the course were: teach Mathematical Modeling, and in sequence, modeling as a method of teaching. The data about the interest for the proposal and the need of the two groups in learning modeling for use in practice was raised from interviews and issues raised and works done by them. Even though the importance of Mathematical Modeling as a method of teaching is not underestimated, some aspects exemplify the difficulties for the participants in changing the concept of teaching and learning: formation of the participants and the need for formation. Key-words: Mathematical Modeling, possibilities and challenges.

“Origami-mathematics lessons” (Boakes, 2006) blend the ancient art of paper folding with the teaching of mathematics. Though a plethora of publications can be easily found advocating the benefits of Origami in the teaching of mathematics, little research exist to quantify the impact Origami has on the learning and building of mathematical skills. The research presented in this paper targets this common claim focusing on how Origamimathematics lessons taught over an extended period of time impact students’ knowledge of geometry and their spatial visualization abilities. The paper begins with a brief overview of Origami as it relates to teaching mathematics followed by a summary of research done with two age groups: middle school children and college students. Gathered data in these two studies suggest that Origami-mathematics lessons are as beneficial as traditional instructional methods in teaching mathematics.

For several decades, there have been a varying number of books on Calculus following the classic line of mathematical thought, where Mathematics is taught for everybody by means of rigorous definitions, theorems, and carefully detailed and extensive demonstrations. For mathematical education into the XXI Century the students need to achieve ability in handling of present mathematical tools and concepts from the beginning of their courses. These needs can be achieved today by means of a paradigmatic change in the focus of mathematics teaching: to learn to develop ideas and to experiment and test those ideas in such way that students can verify their own inferences. In this paper we report an educational research in teaching and learning functions models according to a new paradigm in hands-on experimental mathematics, with applications in the real world, i.e. sciences and engineering by using Computer Algebra Systems. The study of functions is presented, focused into the framing of Exploratory Learning Systems, where students learn by means of the action and their participation in it. It is designed for teachers working together with students in a computer laboratory like hands-on workshops-type activities on other sciences. In this way students have a more “alive”, “realistic” and “accessible” touch in Calculus.

In recent years, the idea of language influencing the cognitive development of an understanding of place value has received increasing attention. This study explored the influence of using explicit number names on prekindergarten and kindergarten students’ ability to rote count, read two-digit numerals, model two-digit numbers, and identify the place value of individual digits in two-digit numerals. Through individual student interviews, preand post-assessments were administered to evaluate rote counting, reading five two-digit numerals, modeling five two-digit numbers, and identifying place value in two two-digit numerals. Chi-square tests for independence showed two significant relations: (1) the relationship between the control and treatment group membership on the postassessment of modeling two-digit numbers and (2) the relationship between place value identifications and group membership. Analysis of the children’s performance and error patterns revealed interesting differences between children taught with explicit number names and children taught with traditional number names. The improvement of the treatment group overall exceeded the improvement of the control group. This study indicates that teaching children to use explicit number names can, indeed, have a positive influence on their understanding of place value.

This paper aims to present a model of teaching and learning mathematics in China. The model is “Two Basic”, basic knowledge and basic ability. Also, the paper will analyze some of the background of the model and why it is efficient in mathematics education. The model is described by a framework of “slab” and based on a model of learning cycle, allow students to work with mathematical thinking. Though the model looks of demonstration and practice looks very traditional, the explanation behind allows us to understand why Chinese students achieved well in many international studies in mathematics. The innovation of the model is the teacher intervention during the learning process. Such interventions include repeated practice, and working on group of selected related questions so that abstraction of the learning process is possible and student can link up mathematical expression and process. Examples used in class are included and the model can be applied in teaching advanced mathematics, which is usually not the case in some many other existing theories or framework.

A study of pre-service primary school teachers in Singapore and the United States revealed superior performance by the Singaporeans on proportional reasoning problems. Analysis of solutions showed the Singapore future teachers were more likely to use unitary and benchmark approaches than were their American counterparts. Conclusions include suggestions for programs intended to improve the performance of prospective elementary school teachers on proportional reasoning problems.

The study of matrices have been of interest to mathematicians for some time. Recently the use of matrices has assumed greater importance also in the fields of management, social science, and natural science because they are very useful in the organization and presentation of data and in the solution of linear equations. The theory of matrices is yet another type of mathematical model which we can use to solve many problems that arise in these fields. The aim of this paper is to show how we can use matrices and their mathematical model to solve some problems in the process of routing. First we will introduce the term of routing and the new approach in the process of selecting paths. We will show some simple examples. We will also pint out how we can learn about matrices in the classroom. At the end we will discuss about advantages and potential disadvantages that may occur in the described technique.

The purpose of this research was to compare American and Albanian students’ achievement in Algebra 1. The study compared algebraic solving abilities of 219 students in a city of Albania and 242 ninth-grade American students, residents of an American region. Albanian sample did not use calculators on the test. Of the American sample, 97 students used calculators on the test, whereas 145 did not use them. The three research questions addressed: (1) students’ mastering of the overall algebraic achievement, (2) students’ mastering of specific domains of algebraic understanding: knowing, applying, and reasoning, and (3) students’ preference of algebraic strategies for solving word-problems. The study found that Albanian students outperformed American students on the overall achievement. However, American students who used calculators on the test significantly outperformed not only the American group who did not use calculators on the test, but also the entire Albanian sample. In addition, Albanian students scored significantly higher than their American peers both on 2 out of 3 cognitive domains and on using algebraic strategies.

In this workshop we present an innovative teaching, learning and research setting that engages beginning teachers in mathematical inquiry as they investigate, represent and connect mathematical ideas through mathematical conversation, reasoning and argument. This workshop connects to the themes of teacher preparation and teaching through problem solving. Drawing on new paradigms to think about teaching and learning, we orient our work within complexity theory (Davis & Sumara, 2006; Holland, 1998; Johnson, 2001; Maturana & Varela, 1987; Varela, Thompson & Rosch, 1991) to understand teaching|learning as a complex iterative process through which opportunities for learning arise out of dynamic interactions. Varela, Thompson and Rosch, (1991) use the term co-emergence to understand how the individual and the environment inform each other and are “bound together in reciprocal specification and selection” (p.174). In particular we are interested in the conditions that enable the co-emergence of teaching|learning collectives that support the generation of new mathematical and pedagogical ideas and understandings. The setting is a one-week summer math program designed for prospective elementary teachers to deepen particular mathematical concepts taught in elementary school. The program is facilitated by recently graduated secondary mathematics teachers to provide them an opportunity to experience mathematics teaching|learning through rich problems. The data collected include questionnaires, interviews, and video recordings. Our analyses show that many a-ha moments of mathematical and pedagogical insight are experienced by both groups as they work together throughout the week. In this workshop we will actively engage the audience in an exploration of the mathematics problems that we pose in this unique teaching|learning environment. We will present our data on the participants’ mathematical and pedagogical responses and open a discussion of the implications of our work.

The article deals with a special kind of modelling tass. These problems are used for learning and researching as well. So the results of an empirical study on mathematical modelling of pupils in secondary schools are presented. Pupils of forms 8-10 were observed working on open, realistic problems. These observations were recorded and evaluated. The goal of the presented part of the study is a detailed look at the control processes of modelling problems. In this context changes between real life control and mathematical control during the control phases are studied and evaluated. We describe in detail the sub phases of controlling and explain their connection with modelling process. The problems used in this project can also be used in math lessons, so this kind of research can put teachers and researchers together. These tasks are suitable to support ongoing in-service development and teacher education.