On the mathematics of emergence

We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting. This article is based on the 1st Takagi Lectures that the second author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006. Steve Smale Partially supported by an NSF grant.     

Using the mathematics literature

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Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course

The purpose of this study was to investigate pre-service teachers’ views about the history of mathematics course in which GeoGebra was used. The qualitative research design was used in this study. The participants of the study consisted of 23 pre-service mathematics teachers studying at a state university in Turkey. An open-ended questionnaire was used as a data collection tool. Qualitative data obtained from the pre-service teachers were analyzed by means of content analysis. As a result, it was determined that GeoGebra software was an effective tool in the learning and teaching of the history of mathematics.     

Euler and the structure of mathematics

When Euler began his mathematical career, analysis was not an autonomous and self-founding field of mathematics but only a method for solving geometrical problems. During Euler's activity, analysis emerged as a new field of mathematics whose methods differed from those of geometry. Moreover, Euler changed the architecture of this science and regarded analysis as lying at the heart of mathematics, while geometry and mechanics were considered as a sort of applied analysis. The new structure of mathematical sciences was successful, but this success soon showed the inadequacy of the Eulerian foundations of analysis.     

Journey to the center of mathematics

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

Maya mathematics

In this article, I discuss how the unique geographic perspective of ancient Mesoamerican experiences fostered the Maya study of astronomy, their world view, and the development of their mathematics.     

The use of concrete learning objects taken from the history of mathematics in mathematics education

This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25. Data were collected by using worksheets, an achievement exam and written opinion forms. The data from the achievement exam were analysed by using the Mann-Whitney U-test while the data from written opinion forms were analysed through content analysis. The Mann–Whitney U-test results showed a significant difference between the mean ranks of the experimental and control groups in favour of the former. Findings from the written opinion forms suggested that the students found the activities to be instructive and fun, enjoyed using concrete models in their classes, and learned from discovering the rules. It was also found that students had previously not engaged in similar activities and had only experienced the history of mathematics through the life stories and works of mathematicians and the representation of ancient numbers at the beginning of each unit.     

An analysis of history of mathematics research literature in Turkey: the mathematics education perspective1

Research in history of mathematics gained momentum in the past two decades in Turkey. The present paper aims to describe the patterns in the history of mathematics research in Turkey and to analyse the research in Turkey using a mathematics education framework. The qualitative paradigm and a case study design are used in the study. The obtained data were analysed by using the document analysis technique with the help of a content analysis. The study group which is comprised of twenty-two postgraduate theses at master's or doctoral level were purposefully selected from the higher education council postgraduate theses database. Findings indicate a dearth of research in the area and that most of the theses are done in the area of mathematics education. Moreover, the focus, in general, was on attitudinal variables, and cognitive aspects seemed to be ignored.     

Experienced engineers becoming mathematics teachers: preliminary perceptions of mathematics teaching

Engineers who choose to change careers and become mathematics teachers are a specific group as far as their mathematics learning in the context of engineering and their previous work experience are concerned. Regarding mathematics, they mainly engaged in applied mathematics associated with engineering, which is a highly practical field. This research explores experienced engineers’ perceptions of mathematics teaching-related topics, before starting their studies in a pre-service mathematics teacher preparation programme. This research explores their perceptions of mathematics as a discipline, mathematics teaching and mathematical understanding. The qualitative research involves three mechanical engineers, two industrial management engineers, and an electrical engineer. Semi-structured interviews were conducted before the beginning of the programme, and analysed qualitatively. The participants view engineering as an applied and changing discipline while perceiving mathematics as closed, rigorous, accurate, systematic, theoretical and as a tool for engineering. They mostly address general features of mathematics teaching while expressing a more multifaceted view of mathematical understanding. Due to the specific characteristics of the participants, this study may contribute to planning mathematics teacher preparation programmes for engineers.