In der Statistik und der Mathematik ist die Normalverteilung der am meisten verbreitete, stochastische Term für die Mehrheit der statistischen Modelle. Wir zeigen, dass der entsprechende stochastische Prozess, die Brownsche Bewegung, drei entscheidende empirische Beobachtungen nicht abbildet: schwere Ränder, Langzeitabhängigkeiten und Skalierungsgesetze. Ein selbstähnlicher Prozess, der in der Lage ist Langzeitabhängigkeiten zu modellieren, ist die Gebrochene Brownsche Bewegung, welche durch die Faltung der Inkremente im Limit nicht normalverteilt sein muss. Die Inkremente der Gebrochenen Brownschen Bewegung können durch einen Parameter H, dem Hurst Exponenten, Langzeitabhängigkeiten darstellt werden. Für die Gebrochene Brownsche Bewegung müssten die Skalierungs-(Hurst-) Exponenten über die Momente verschiedener Ordnung konstant sein. Empirisch beobachten wir variierende Hölder-Exponenten, die multifraktales Verhalten implizieren. Wir erklären dieses multifraktale Verhalten durch die Änderung des alpha-stabilen Indizes der alpha-stabilen Verteilung, indem wir Filter für Saisonalitäten und Langzeitabhängigkeiten über verschiedene Zeitfrequenzen anwenden, startend bei 1-minütigen Hochfrequenzdaten. Durch die Anwendung eines Filters für die Langzeitabhängigkeit zeigen wir, dass die Residuen des stochastischen Prozesses geringer Zeitfrequenz (wöchentlich) durch die alpha-stabile Bewegung beschrieben werden können. Dies erlaubt es uns, den empirischen, hochfrequenten Datensatz auf die niederfrequente Zeitfrequenz zu skalieren. Die generierten wöchentlichen Daten aus der Frequenz-Reskalierungs-Methode (FRM) haben schwerere Ränder als der ursprüngliche, wöchentliche Prozess. Wir zeigen, dass eine Teilmenge des Datensatzes genügt, um aus Risikosicht bessere Vorhersagen für den gesamten Datensatz zu erzielen. Im Besonderen wäre die Frequenz-Reskalierungs-Methode (FRM) in der Lage gewesen, die seltenen Events der Finanzkrise 2008 zu modellieren. / Coming from a sphere in statistics and mathematics in which the Normal distribution is the dominating underlying stochastic term for the majority of the models, we indicate that the relevant diffusion, the Brownian Motion, is not accounting for three crucial empirical observations for financial data: Heavy tails, long memory and scaling laws. A self-similar process, which is able to account for long-memory behavior is the Fractional Brownian Motion, which has a possible non-Gaussian limit under convolution of the increments. The increments of the Fractional Brownian Motion can exhibit long memory through a parameter H, the Hurst exponent. For the Fractional Brownian Motion this scaling (Hurst) exponent would be constant over different orders of moments, being unifractal. But empirically, we observe varying Hölder exponents, the continuum of Hurst exponents, which implies multifractal behavior. We explain the multifractal behavior through the changing alpha-stable indices from the alpha-stable distributions over sampling frequencies by applying filters for seasonality and time dependence (long memory) over different sampling frequencies, starting at high-frequencies up to one minute. By utilizing a filter for long memory we show, that the low-sampling frequency process, not containing the time dependence component, can be governed by the alpha-stable motion. Under the alpha-stable motion we propose a semiparametric method coined Frequency Rescaling Methodology (FRM), which allows to rescale the filtered high-frequency data set to the lower sampling frequency. The data sets for e.g. weekly data which we obtain by rescaling high-frequency data with the Frequency Rescaling Method (FRM) are more heavy tailed than we observe empirically. We show that using a subset of the whole data set suffices for the FRM to obtain a better forecast in terms of risk for the whole data set. Specifically, the FRM would have been able to account for tail events of the financial crisis 2008.

Mathematics learning seems to be a demanding and time-consuming task for many learners. Information and communication technology (ICT) is an attractive tool of learning for students at any level and it can provide an effective atmosphere for understanding mathematics. The question is how to combine mathematics teaching contents, approaches, curricula, and syllabus with new media. The key issue in European educational policy (and other countries as well) is exchange and sharing digital learning resources (learning objects) among countries. In order to accumulate the practice of various countries and use the best digital resources created by different countries, it is necessary to localize learning objects (LO). The paper deals with some problems connected with localization of LO, developed for mathematics education, and presents some solution. Software localization is mainly referred to as language translation (e.g., translation of user interface texts and help documents). However, there are many other important elements depending on the country and people who will use the localized software. In this paper, the main attention is paid to localization of learning objects used for teaching and learning mathematics.

The purpose of this study is to determine the impact on student achievement of elementary school teachers who participated in professional development in the content area of mathematics. Teachers participated in professional development courses and have accumulated a range from three to eighteen total credits from the summers of 1998 through 2007. The impact is measured by student achievement data collected on standardized tests.

Our paper presents some preliminary observation from a collaborative exploratory study linking mathematics, science and reading within a technology enhanced problem-based learning scenario conducted at one French Canadian Elementary and Middle School. Presented in a form of dialogue between teacher and researcher, our findings give some meaningful insight in how an innovative mathematics teaching can be developed and implemented using a real-world problem solving. Instead of a traditional presentation of material about lighting up homes, participating mathematics, science and French teachers were working collaboratively with the ICT integration mentor and two university professors helping students investigate a problem from various perspectives using a variety of cognitive and metacognitive strategies, discussing and sharing the finding with peers and presenting them to a larger audience using media tools. Our preliminary results may prompt further investigation of how innovation in teaching and learning can help students become better critical thinkers and scientifically empowered citizens.

In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student\''s access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student\''s previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems.

The major aims of this paper are to: present the background of the mathematics education problem in the Stamford Public School (SPS) district which is common is most U.S cities; explain the need for change in mathematics education; describe the process to systemically transform both the curriculum and instruction of mathematics thereby ensuring that each and every students is prepared for the 21st century, for higher education, and for success in a global society; and provide ways to measure these changes. The K-12 mathematics education reform model presented can be replicated in other cities and for other academic areas.

This paper addresses folding the circle in half and discussing some of over one hundred different mathematical terms and functions generated in that one fold. The simplicity of process in understanding fundamentals of mathematics by folding circles and observing what is generated is unknown because we only draw pictures of circles. Examples are given about observing and exploring relationships in the circle that are appropriate for first, second, third grade level and beyond. The traditional educational ‘parts-towhole’ approach can only be fully realized through the comprehensive frame of Whole-to-parts by folding the circle. Wholemovement of the circle is not only direct; it is the only context inclusive to progressively understanding parts within unity of the Whole.

Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.

Language is the learning device and the device which forms the student''s knowledge in math, his ability to define concepts, express mathematical ideas and solve mathematical problems. Difficulties in the Language are seen more in word problems, clarity and in the way the text is read by the student have a direct effect on the understanding of the problem and therefore, on its solution, could delay the problem solving process. The connection between language and mathematical achievements has a more distinctive significance regarding the Arab student. This is due to the fact that the language which is used in the schools and in textbooks is classical (traditional) Arabic. It is far different than the language used in everyday conversations with family and friends (the spoken Arabic). Our research examine whether rephrasing word problems can affect the achievements of the Arab students in it. The experimental group received mathematics instruction using learning materials of word problems that were rewritten in a “middle language” closer to the students’ everyday language (spoken Arabic), thus keeping the mathematical level of the problems. The research findings showed that students in the experimental group improved their achievements in word and geometric problems significantly more than students from control group.

The author shares some of the findings of the research he conducted in 2007 on grade 11 mathematics learners in two schools, one experimental and the other one control. In his study, the author claims that an open-ended approach towards teaching and learning of mathematics enhances understanding of mathematics by the learners. The outcomes of the study can be summarised as follows: 1. In the experimental school, where the author intervened by introducing an open-ended approach to teaching mathematics (by means of giving the learners an open-ended approach compliant worksheet to work on throughout the intervention period), the performance of the learners in the post-test was better than that of the learners from the control school. Both schools were of similar performance in the pre-test. The two schools wrote the same pre-test and same post-test. Both schools were following common work schedule. 2. Within the experimental school, post-test performance of the learners in the class where the intervention was monitored throughout the intervention period (thus ensuring compliance of the teacher to the open-ended approach) out-performed those in which monitoring was less frequent. 3. There was no significant difference in performance between learners from the unmonitored experimental class and those from the control class.