Using a virtual population to authentically teach epidemiology and biostatistics

Epidemiology is the study of the distribution of disease in human populations. This means that authentically teaching primary data collection in epidemiology is difficult as students cannot easily access suitable human populations. Using an action research methodology, this paper studied the use of a virtual human population (called The Island) to enable students to experience many features of authentic primary data collection in epidemiological research. The Island was used in a course introducing epidemiology and biostatistics for students in non-quantitative disciplines. This paper discusses how The Island was introduced into the course, and then evaluates the change. Students were highly engaged, and students and teaching staff responded favourably to the use of The Island, with 70% of students agreeing or strongly agreeing that The Island was easy to use, and 64% agreeing or strongly agreeing that the use of a virtual population was beneficial to their understanding of epidemiology.     

Two faces of mathematics lessons in Korea: conventional lessons and innovative lessons

This study identifies the characteristics of mathematics classrooms in Korea. First, conventional Korean mathematics lessons are analyzed from the perspective of “theory of variation”. Second, an innovative lesson for gifted children is reported in detail and analyzed from the perspective of “Lakatos’ proofs and refutations”. Third, the classroom characteristics identified in both the conventional lessons and the innovative lesson are interpreted in terms of the underlying cultural values that they share with other East Asian countries. The study concludes that although the two faces of Korean mathematics lessons look different, they may flow from the same “heart”—that of the common Confucian heritage culture culture, and in particular East Asian pragmatism.     

Two lessons from fractals and chaos

We used to think that a good measurement is characterized by its mean and variance and that a good theory is characterized by its ability to predict the values measured in an experiment. The properties of nonlinear systems called fractals and chaos have now taught us that this isn't necessarily true. Data from fractal systems extend over many scales and so cannot be characterized by a single characteristic average number. Data from chaotic systems do not repeat the same time series of values, even if they are started very close to the same initial conditions. This means that a valid mathematical model will not be able to predict the values of the time series.     

Mathematical features of lessons in the TIMSS video study

The video component of the Third International Mathematics and Science Study marked the first time that national samples of teachers were videotaped at work in their classrooms. In this article we review some of the results of the study, with special attention to the nature of the mathematics evident in these eighth-grade lessons from Germany, Japan, and the United States. We conclude by proposing that many lessons within a country follow a cultural pattern of teaching, and that differences among countries on individual indicators of teaching must be understood in reference to these patterns.     

Mathematics and life: lessons from history

Developments in mathematics, technology, science, culture and society are closely interwoven with each other. Beyond that, these interrelations tend to be very complex due to the multiplicity of factors involved. In this contribution, three examples from the human story are analysed in order to sensitize for possible characteristics of the interrelation. For the historical cases of early Mesopotamia, the Inca state and Renaissance Italy, it is demonstrated how specific socio-cultural conditions affect the development of mathematics and technology as well as their use. These findings lead to questions about our use of mathematics and the ways in which mathematics, society, technology, culture and science may be linked, today.     

Exemplary mathematics lessons: a view from the West

Cross-cultural comparative research provides a powerful means of achieving better understanding of one’s own practice. This article contrasts exemplary mathematics teaching, as highlighted in this special issue on its practice and development in East Asia, with the increasing emphasis on effective teaching in the West. It provides an overview of the papers in this issue and examines the similarities and differences between what is seen as exemplary practice and the ways in which its development is supported in the educational systems represented. It attempts to identify some of the cultural factors that influence our understanding of what constitutes exemplary practice and the possibilities for its development in the East and the West.     

Dee and his books: lessons from the history of mathematics for primary and middle school teachers

This article describes a cross-curricular project based on Shakespeare's Tempest, in which the life and work of John Dee was used to inspire lessons in mathematics.     

Problem types used in math lessons: the relationship between student achievement and teacher preferences

The purpose of this study was to determine the relationship between the problems teachers preferred in mathematics lessons and student achievement in different types of problems. In accordance with this purpose, nine mathematics teachers were interviewed, and corresponding problems were prepared and administered to 225 eighth-grade students. The findings indicate that problem types are dependent on teacher preferences. It was found that curriculum-dependent and routine problems were dominant for teacher preferences. Students are more successful at with missing data, problems that are visual and do not require the use of different strategies. They have lower success at long problems, those that contain irrelevant data, problems that require the use of different strategies and difficult problem types. It was found that problem types at which students were successful and which teachers preferred were related. These results relay information about problems used in the learning environment and effect of problem-solving experiences on students' success.     

From lessons to lectures: NCEA mathematics results and first-year mathematics performance

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.     

Competitive dynamics and gaming simulation: lessons from a fishing industry simulator

Experiments with a gaming simulator of the fishing industry show that a wide range of firm and industry performance can arise from players’ differing perceptions of the competitive environment. When transferred into the arena of applied strategy development, the results suggest that modellers should give more attention to modelling alternative conceptualizations of strategic intent in the minds of rival business leaders and the firms they create. This new interpretive emphasis should complement the traditional modelling of cross-functional coordination that has been the hallmark of much published work in strategic modelling and simulation. When leaders and firms in the same industry adopt quite different views of the overall system of resources in the industry, it is important to model the heterogeneity of rival firms in order to understand the dynamic performance of the firm and the industry. We propose a modelling approach that captures heterogeneity among the strategic resources that rival firms seek to build and in the operating goals and coordinating processes they use to build them.     

Towards a competence model for the use of symbolic calculators in mathematics lessons: the case of functions

A long-term project (2003–2012) was started to test the use of symbolic calculators (SCs) in Bavarian grammar schools (Gymnasien) in Germany with students attending grades 10, 11 and 12. The project started with an evaluation in grade 10 (see Weigand in Int J Technol Math Educ 15(1):19–32, 2008) and accompanied the students in grades 11 and 12. The first years of the project illustrated that a classical quantitative pre–post-test design with project and control groups was not an adequate method to prove the increase of student competency in the project group. Therefore, in the last years, we have concentrated increasingly on an evaluation of new competencies which students developed in SC environments: new working styles, new problem-solving strategies and new ways of achieving a better understanding. A competence model has been developed to evaluate students’ abilities while working with the SC within the context of the concept of functions. This article demonstrates the way of constructing the (theoretical) competence model while working on different levels of SC use and different levels of the understanding of functions.     

Japanese and American teachers' evaluations of mathematics lessons: A new technique for exploring beliefs

In our study, we use a novel technique to explore the beliefs of Japanese and American elementary school teachers. Four American and four Japanese teachers watched a mathematics lesson—videotaped in either Nagano, Japan or Chicago, Illinois—and commented on the lesson's strengths and weaknesses. The major pedagogical issues that differentiated the teachers' comments were: what students should do during a lesson, how instructors should use language, how instructors should pace lessons and address ability differences, and how instructional materials should be used. The specific beliefs of the American and Japanese teachers in this study mapped easily onto common instructional practices in elementary school mathematics classes in the United States and Japan. We conclude that, at least for the teachers in this sample, beliefs are linked to practices and they may help to tie teachers to their culturally preferred method of mathematics instruction.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

Studying sample lessons rather than one excellent lesson: A Japanese perspective on the TIMSS videotape classroom study

The findings of the TIMSS Videotape Classroom Study include aspects of mathematics lessons showing a strong resemblance between Germany and the US in difference to Japan. This paper discusses some of the features that appear to make Japanese lessons different from the other two countries. In particular, the paper examines the goals of lessons described by Japanese teachers, how lessons are structured and implemented, and the emphasis on alternative solutions to a problem in the teaching and learning processes. The characteristics of Japanese lessons identified by the TIMSS Videotape Classroom Study can naturally be interpreted as indications of teachers' efforts to foster students' mathematical thinking in the classroom.     

Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y = ax² + bx + c. Students’ generalizations about these connections led to a surprising finding: two-thirds of the students interviewed identified the parameter a as the “slope” of the parabola. Analysis of qualitative data from interviews and classroom observations led to the development of three focusing phenomena in the classroom environment that inadvertently supported a focus on slope-like properties of quadratic functions: (a) the use of linear analogies, (b) the rise over run method, and (c) viewing a as dynamic rather than static.     

The computer as “an exercise and repetition” medium in mathematics lessons: Educational Effectiveness of Tablet PCs

The request of a new educational culture within the classroom goes hand in hand with the introduction of the new Educational Standards. That is they are essentially connected with a paradigm shift. The Project supports this aim via different tasks administered through Tablet PC's within the scope of exercise and repetition phase of learning. The central concern is to find appropriate lesson approaches through computer use in everyday life at school, which are conductive for the math learners and are at the same time easy to effectively implement in other classrooms. In the summer of 2004 the use of Tablet-PCs in school took place in two 9th classes of an Ostalbkreis secondary school in Baden-Württenberg. We report on the effectiveness of this new technology in the classroom.     

A tale of two sets of norms: Comparing opportunities for student agency in mathematics lessons with and without interactive simulations

Previous research has documented the importance of setting up productive norms in mathematics classrooms. Studies have also shown the potential for activities involving interactive simulations (sims) to support student engagement and learning. In this study, we investigated the relationship between norms and sim-based activities. In particular, we examined the social and sociomathematical norms in lessons taught with and without the use of PhET sims in the same teacher’s middle-school mathematics classroom. There were statistically significant differences in indicators of social norms between the two types of lessons. In sim lessons, the teacher more frequently took the role of a facilitator of mathematical ideas, and students exhibited conceptual agency more often than they did in non-sim lessons. On the other hand, there was substantial overlap: the teacher usually acted as an evaluator, and the students usually exhibited disciplinary agency in both types of lessons. However, there was a stark contrast in sociomathematical norms between the two types of lessons. Students’ specifically mathematical obligations in non-sim lessons consistently included practicing procedures in isolation and appealing to rules. Obligations in sim lessons included developing and sharing strategies, making conjectures and providing justifications. In both types of lessons, students were obligated to recall mathematical facts and vocabulary. Thus, the social norms were broadly consistent except for important differences in frequency, whereas we found substantial qualitative contrasts in the sociomathematical norms in the two types of lessons. This case provides evidence that contrasting norms can exist within the same classroom. We argue from our data that these differences may be mediated by curricular choices—in this case, the use of sims.