Teaching teachers mathematics through programming

There are two main arguments underlying the claims for the value of interactive computer programming used by students to model mathematical ideas. One is concerned with mathematical content, i.e. with mathematics as an object of study. The other is concerned with mathematical activity, i.e. doing mathematics, or ‘Mathematicking’ [1]. Both content and activity include processes and these provide the main links with programming. Examples of processes in the content of mathematics are addition, transformation and integration, and these can be described by instructions in a computer program. Examples of process in the activity are problem‐solving, proof generation and pattern finding which can be described by analogy to program building and debugging. We assess the arguments for programming, in relation to the training of teachers, and describe a pilot‐study in which student teachers with mathematical difficulties were taught the programming language LOGO. Observation of the students, learning the language and using it to manipulate computer models of mathematical ideas, which they had not understood previously, highlights both advantages and disadvantages in this approach. The problem of the representation of mathematical ideas within programming projects is discussed.     

The graphics calculator in mathematics education

This article covers a project conducted by the Freudenthal Institute from August 1991 to September 1994 entitled “The graphics calculator in mathematics education.” The theory of realistic mathematics education was taken as the point of departure for formulating the hypotheses. The developmental research design was used. Observation of the students' behavior during the experimental lessons supports the premise that the graphics calculator can stimulate the use of realistic contexts, the exploratory and dynamic approach to mathematics, a more integrated view of mathematics, and a more flexible behavior in problem solving.     

Dog and tricks: some mathematics for students of graphics

This paper describes how parametric cubic splines and cubicBezier curves may be used in designing a two dimensional shape.A simple aerofoil shape is designed using both methods. Themathematics is described and the shape drawn using Excel. Theeffect of varying parameters is shown in both methods.     

Teaching mathematics using Steplets

This article evaluates online mathematical content used for teaching mathematics in engineering classes and in distance education for teacher training students. In the EU projects Xmath and dMath online computer algebra modules (Steplets) for undergraduate students assembled in the Xmath eBook have been designed. Two questionnaires, a compulsory student project and teaching in front of class show that using Steplets turn mathematics teaching from drill to understanding. The Steplets use algorithms developed for the Mathematica programming language.     

Discrete mathematics in the high school curriculum

In this paper we present some topics from the field of discrete mathematics which might be suitable for the high school curriculum. These topics yield both easy to understand challenging problems and important applications of discrete mathematics. We choose elements from number theory and various aspects of coding theory. Many examples and problems are included.     

Teaching and learning mathematics in the collective

In this paper we analyse and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented. We frame our discussion within enactivism—a theory of cognition that has helped us to understand classroom processes, particularly at the level of the group. We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective. Our analysis reveals teaching practices that value, capitalize upon, and promote group cognition, our discussion positions such work as teaching a way of being with mathematics, and we close by offering implications for teaching, educational policy, and further research.     

A research journey through mathematics coaching

Classroom coaching in mathematics is flexible in its definition, complex in its enactment, variable in its outcomes, and dependent on setting and circumstances. Multiple lines of inquiry are required to navigate this subjective terrain: research on coaching encompasses understanding perceptions of coaching held by coaches, teachers, and administrators, measuring the effectiveness of coaching in terms of teachers’ content knowledge and instructional practices, and exploring the nature of coaching within an educational ecosystem. This paper describes a cumulative sequence of research studies that inform current understanding of classroom coaching in mathematics, highlighting methodological decisions made at various crossroads and elaborating on the populations, methods, and instruments used to investigate coaching. A presentation of findings related to the three domains of perception, effectiveness, and nature is followed by reflections on features of coaching that pose particular challenges, questions that remain to be answered, and promising avenues of future inquiry.     

A survey of advanced mathematics topics: a new high school mathematics class

The number of students pursuing undergraduate degrees in mathematics is decreasing. Research reveals students who pursue mathematics majors complained about inadequate high school preparation in terms of disciplinary content or depth, conceptual grasp, or study skills. Unfortunately, the decrease in the number of students studying advanced mathematics occurs at a time when the world's technological drive demands students have improved critical thinking and problem-solving skills. This paper suggests one solution for this alarming problem: a high school class offered to seniors as a means of preparing them for the rigours of college level mathematics while simultaneously increasing their motivation to pursue advanced mathematics. This paper provides the course scope, goals, structure, and analysis of how the curriculum aligns to professional standards. Although this programme has not currently been field tested, the authors are convinced of its impact. Once implemented and properly taught, the proposed Survey of Advanced Mathematics Topics class could increase the quantity and quality of students pursuing studies in mathematics at the university level.     

Learning mathematics through algorithmic and creative reasoning

There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning. Although better teaching methods are proposed, there are very limited research insights as to why some methods work better than others, and the conditions under which these methods are applied. The present paper evaluates a model based on students’ own creation of knowledge, denoted creative mathematically founded reasoning (CMR), and compare this to a procedure-based model of teaching that is similar to what is commonly found in schools, denoted algorithmic reasoning (AR). In the present study, CMR was found to outperform AR. It was also found cognitive proficiency was significantly associated to test task performance. However the analysis also showed that the effect was more pronounced for the AR group.     

Reorganizing freshman business mathematics II: authentic assessment in mathematics through professional memos

The first part of this two-part paper (13) described the developmentof a new freshman business mathematics (FBM) course at our college.In this paper, we discuss our assessment tool, the businessmemo, as a venue for students to apply mathematical skills,via mathematical modelling, to realistic business problems.These memos have proven a crucial step in turning our FBM coursearound from a dreaded course with little connection to students’intended careers into a course where students experience thepower of mathematics for solving problems and informing decisions.Comments from students in the course throughout its 6-year historyclearly point to the course's value and importance.     

Reforming mathematics learning in Indonesian classrooms through RME

This paper reports an experimental study on the development of exemplary curriculum materials for the teaching of fractions in Indonesian primary schools. The study’s context is the current reform movement adopting realistic mathematics education (RME) theory, known as Pendidikan Matematika Realistik Indonesia (PMRI), and it looked at the role of design research in supporting the dissemination of PMRI. The study was carried out in two cycles of teaching experiments in two primary schools. The findings of the design research signified the importance of collaboration between mathematics educators and teachers in developing RME curriculum materials. The availability of RME curriculum materials is an important component in the success of the PMRI movement, particularly in supporting students and teachers in activity-based mathematics learning. Most of the students and teachers in the two schools positively appraised teaching and learning with the developed materials. Since the teachers were actively involved in developing the materials, they felt a sense of ownership and recognised that their students’ classroom experiences of the materials helped them avoid standard difficulties. That appears to be a particular benefit of the bottom-up approach characteristic of the PMRI movement.     

Nonparametric high resolution spectral estimation

Summary The uniform rate of convergence of the integrated relative mean square error over a (with the sample sizeT) increasing classI _T of stationary processes is studied for several estimates of the spectral density. The classI _T is chosen in a way such that estimates with a good uniform rate of convergence overI _T may be termed high resolution spectral estimates. By using this criterion several effects are explained theoretically, for example the leakage effect. The advantages uf using data tapers are proved and the use of local and global bandwiths are studied. Furthermore, the behaviors of segment estimates are studied. Simulations are presented for the illustration of some effects.This work has been supported by the Deutsche Forschungsgemeinschaft     

Stepping beyond high school mathematics: a case study of high school women

The Summer Explorations and Research Collaborations for High School Girls (SEARCH) Program, held annually since 2004 at Mount Holyoke College in the US, was created for talented high school girls to explore mathematics beyond that taught in high school. Our study, which focuses on factors that facilitate or inhibit the pursuit of higher level mathematics by girls, is centered on the 2006 SEARCH Program. We present a combination of qualitative and quantitative data drawn from student journals written during SEARCH, program evaluations written at the end of SEARCH, post-program interviews, and comparisons with two peer group samples. From this data we point to important factors, such as developing a mathematical voice, gaining a broader view of advanced mathematics, being challenged in a supportive atmosphere, and having a positive stance toward risk-taking, that may help to maintain the interest of talented girls in advanced mathematical studies.