Definitions and examples in elementary calculus: the case of monotonicity of functions

This paper is devoted to the investigation of students’ understanding and handling of examples in the framework of an example-based introductory mathematics undergraduate course. The plan of the course included a wide use of graphs in standard lectures, tutoring sessions as well as in examinations. This study deals with the notion of increasing function, which has been introduced by means of both the standard definition and a range of examples and non-examples, most often conveyed through graphs. We have analysed students’ interpretations of the notion of increasing function as they applied them in a set of written examination tests. The data gathered have been completed by a number of interviews of students whose answers were difficult to interpret. The outcomes underline the importance of linguistic and semiotic competence and suggest that the design of innovative teaching paths should take care of the linguistic and semiotic skills needed to handle the representations involved.     

Prevailing educational practices for area measurement and students’ failure in measuring areas

The present research was carried out with the participation of 106 students in their last grade in Elementary School and revealed certain problems that these students faced in understanding the concept of area measurement. The students in the sample persisted on using measurement strategies that often led to failure.Our research plan comprises a comparison between the strategies for area measurement strategies used by two groups: the experimental group (E.G.) and the control group (C.G.). The experimental group attended a special teaching course, which stressed the conceptual characteristics of the area measurement process.The present research aims at revealing the students’ understanding, strategies, and misconceptions regarding area measurement. In addition to that, we examine whether the special teaching course and the use of different measurement tools may lead the two research groups to adopt different measurement strategies.     

Software to support numerical analysis teaching

This paper describes a Fortran90 library designed to support the teaching of numerical analysis and its applications. As well as covering traditional material it introduces recent and important ideas in numerical computation such as interval arithmetic and automatic differentiation. The library rests on a module realpac which provides real arithmetic in a range of precisions with a choice of rounding strategies. This, in turn, supports the implementation of an interval arithmetic module intpac. Derived data types and overloaded operations help inexperienced users to interface with unfamiliar data types such as intervals. The library also includes more conventional modules such as lepac for solving linear systems and minpac for nonlinear optimization. These, however, can be enhanced by being linked to more sophisticated tools for sparse matrix handling and automatic differentiation. As well as showing the main structure and scope of the software, the paper mentions some exercises that have successfully been performed by students.     

Electronic Resources: The Ohio State University EESEE Project

Electronic resources aid in the teaching and learning of statistics by providing data that may be used interactively by teachers and students. By interacting with the data students are encouraged to discover knowledge and thereby gain a deeper understanding of statistical concepts. The Electronic Encyclopedia of Statistical Examples and Exercises (EESEE) is an electronic resource that includes over 80 'real-world' examples about the uses and abuses of statistics. These examples are drawn from published and printed media and the diverse range of subject-matter areas make it suitable for use in any statistics course.     

Teaching Science as Science Is Practiced: Opportunities and Limits for Enhancing Preservice Elementary Teachers' Self‐Efficacy for Science and Science Teaching

Science teaching in elementary schools, or the lack thereof, continues to be an area of concern and criticism. Preservice elementary teachers' lack of confidence in teaching science is a major part of this problem. In this mixed‐methods study, we report the impacts of an inquiry‐based science course on preservice elementary teachers' self‐efficacy for science and science teaching, understanding of science, and willingness to teach it in their future careers. Our findings suggest that for some students, the inquiry‐based science course positively influenced their self‐efficacy for science and science teaching. Gains were made in a majority of students' conceptual understanding of science, understanding of the science process and scientific research, and confidence with science and science teaching. The subjects did not experience the course uniformly, however. Rather, there appeared to be two distinct groups, one on a trajectory of improving their outlook on science teaching and one worsening. The results presented here therefore provoke some interesting questions regarding preservice elementary teachers' preparation for science teaching.     

Statistical literacy and sample survey results

Sample surveys are widely used in the social sciences and business. The news media almost daily quote from them, yet they are widely misused. Using students with prior managerial experience embarking on an MBA course, we show that common sample survey results are misunderstood even by those managers who have previously done a statistics course. In general, they fare no better than managers who have never studied statistics. There are implications for teaching, especially in business schools, as well as for consulting.     

Teaching communication in an MBA operations research/management science course

This paper discusses the inclusion of a writing requirement within a first-year Operations Research/Management Science (OR) course of a Masters of Business Administration (MBA) curriculum. The inclusion of a writing requirement addresses the importance of communication in OR. It is the authors’ supposition that many of the recent criticisms of OR teaching have their roots in poor OR communication. Although OR faculty members can well argue relative values, we have failed to adequately prepare our students to communicate relevant insights from OR studies. The integration of a writing requirement, consisting of short executive summary-style reports and a major term project, within an MBA OR course can do much in promoting further OR study and OR work. This paper describes our experience in instituting a writing requirement within a first-year MBA course in OR.     

Beating the System: Course Structure and Student Strategies in a Traditional Introductory Undergraduate Physics Course for Nonmajors

The purpose of this study was to describe relationships among instructor and student goals, course design, and student strategies for learning physics in a traditional introductory undergraduate physics course for nonmajors. The procedures included 16 hours ofnonparticipant observation, during which detailed field notes and photographs were taken and documents were collected. Audiotaped open-ended interviews were given to instructors and students, and all students participated in a limited survey. The data were analyzed using qualitative methods of the Chicago School. Results indicated that both differing goals of students and instructors and the structure of the course inhibited the understanding of physics and were conducive to students engaging in unexpected behaviors designed to help them successfully pass the course. The consequences of the structure of the course and subsequent student behaviors inhibited their understanding and appreciation of physics and deterred them from enrolling in further science courses. The paper includes implications for teaching introductory college physics.     

Reforming and Assessing Undergraduate Science Instruction Using Collaborative Action‐Based Research Teams

Faculty members at Purdue University in the departments of Earth and Atmospheric Sciences, Biological Sciences, and Chemistry conducted a reform effort for the undergraduate curriculum utilizing action‐based research teams. These action‐based research teams developed, implemented, and assessed constructivist approaches to teaching undergraduate science content in each department. This effort utilized a partnership of scientists, science educators, master teachers, graduate students, and undergraduate students. Results indicated that the project partners were able to (a) implement more inquiry‐based teaching that emphasized conceptual understanding, (b) provide opportunities for cooperative learning experiences, (c) use models as an ongoing theme, (d) link concepts and models to real‐world situations, e.g., field trips, (e) provide a more diverse range of assessment strategies, and (f) have students present their understandings in a variety of different forms. Further, we found that we were able to (a) involve graduate and undergraduate students, classroom teachers, scientists, and science educators together to work on the reform in a collaborative manner, (b) bring multiple perspectives for teaching and for science to support instruction and, (c) provide scientists and graduate science students (who will become university professors) with more effective teaching models. We also found that the collaborative action‐based research process was effective for contributing to the reform of undergraduate teaching.     

Chance and probability: What do they mean to university engineering students?

The great interest aroused by the incorporation of Statistics and Probability into curricular projects has been accompanied by considerable evidence of significant difficulties in the meaningful learning and application of the concepts. These difficulties have been the subject of many studies, mostly concerning secondary school students. This study seeks to investigate the level of understanding of random phenomena, in a group of university students, on the second year of a Technical and Industrial Engineering Degree at the University of the Basque Country, in Spain. The students, who had all undertaken an introductory course in Probability Theory, were tested through both written questionnaires and one-to-one interviews, where they were presented with situations that demanded the ability to apply the knowledge they had learnt, at a high level, to a range of situations and to justify their reasoning. The results show that the vast majority of students had a poor understanding of random phenomena, applying alternative ideas to the ones taught on their course in Probability. The need to improve teaching strategies in Probability and Statistics, so that students can develop skills in constructing probabilistic models, is highlighted.     

Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add-them-all-up-and-divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

PANEL DISCUSSIONS

Representations of mathematical concepts play an important role in understanding: both in helping learners understand the to-be-learned material and in facilitating teachers’ understanding of pedagogical processes which, in turn, are involved in developing learners’ understanding. In this paper, we report on work with a cohort of pre-service primary teachers, with the aim of developing their understanding of mathematics, their confidence in their subject knowledge and their confidence in teaching mathematics. This was attempted through the introduction and use of a ‘representational approach’ to the teaching of the mathematical concepts required of teachers training to teach in primary schools in the UK. We present the results of attitude measures and a follow-up qualitative questionnaire in identifying whether and how the use of this representational approach supported pre-service teachers’ understanding and their confidence in teaching mathematics. The results suggest that the representational approach used had a positively significant impact on the attitudes towards studying and teaching mathematics.     

Strongarm arcade games: a case study in mathematical modelling

Two linked cases are described, that have been used many times over many years with MBA students to teach both basic spreadsheet design and mathematical modelling and to act as a framework to stimulate thought about the wider considerations associated with the application of mathematical models. The broad aspects of the mathematical modelling methodology that are taught are first briefly considered. The simpler case together with its more demanding extension are both presented. The students' approaches are then categorised and the common mistakes/weaknesses and the strength of submissions are listed. This paper closes with a consideration of the value of the cases in student teaching. Solutions to both cases, applying the mathematical modelling methodology in detail, are given in Appendices.     

Varying routes to the top: identifying different strategies in the MBA marketplace

The market for MBA degrees is changing rapidly. As this market tends to maturity,an increase in the number of universities offering these degrees is evident, andalso in the different formats in which these degrees are offered. In thisincreasingly crowded marketplace, there exist two main sources of informationthat buyers (students as well as prospective employers of MBA students) can useto assess the products on offer—whether or not a Business school/MBAprogramme is accredited, and also the position of the school in availablerankings, a number of which are regularly published. Playing ‘the rankingsgame’ is one that occupies the time and effort of many MBA directorsglobally, as they all try to edge their way upwards in order to attract more andbetter students. The problem with an ordinal ranking, though, is that itsuggests too readily that it is based upon a unidimensional measure. We use thedata behind one of these rankings, that of the Financial Times top 100,full-time MBA programmes in 2008, in order to explore to what extent schools areusing different routes to try to improve their rankings. Using a variety ofmultivariate statistical tools, we base our analysis on the three underlyingfactors of alumni career progress, diversity, and idea generation. What emergesis a clearer picture of the extent to which programmes that are ranked veryclosely do, in fact, base their success on very different routes to the top.     

A touch of mathematics: coming to our senses by observing the visually impaired

During the last three decades, the impact of visualization processes on the teaching and learning of mathematics has been extensively researched. However, considerably less work has been devoted to haptic processes. In this paper, we describe and analyze the role of haptic processes from data collected in mathematics lessons taught in a school for blind students. The analysis builds upon studies from perception and also from teaching experiences in order to examine the teacher’s and the students’ hand movements and metaphors when handling solids of revolution and communicating verbally with each other about their insights. We highlight the powerful combination of the visual and the haptic components of their interactions for the conceptualization of mathematical experiences, and we also note the critical role mathematical language plays in supporting the teaching–learning processes in this context. Finally, we consider important educational implications not only for blind people, but for all students and teachers of mathematics.     

Examining prospective mathematics teachers' pedagogical content knowledge on fractions in terms of students' mistakes

The aim of the study is to examine prospective mathematics teachers’ pedagogical content knowledge in terms of knowledge of understanding students and knowledge of instructional strategies which are the subcomponents of pedagogical content knowledge. The participants of this research consist of 98 prospective teachers who are studying in two universities in Turkey. The participants were selected with the purposive sampling method which is one of the non-random sampling methods. Case study method, which is based on the qualitative research approach, was used. The answers given by secondary school students to fraction-related open-ended questions in the study of Soylu and Soylu were used as the data collection tool. The obtained data were analyzed via the content analysis technique. The analyses showed that the prospective mathematics teachers’ pedagogical content knowledge on fractions was not at an adequate level in identifying and correcting students’ errors. However, it was observed that the prospective teachers experienced more difficulty in the knowledge of instructional strategies compared to the knowledge of understanding students.     

Improving Science Teaching for All Students

An inservice program designed to enhance the knowledge and skills of elementary school teachers with respect to science content, effective teaching strategies, and gender equity was implemented as a semester-long course. During the course, teachers explored new science content in chemistry and physics and then collaboratively developed lesson plans from it based on hands-on, discovery-centered learning, enmeshed in strategies that could maximize female student interest and participation in science. Teachers tried out their lessons between course sessions in their own classrooms and then collaboratively reflected on their progress and problems in subsequent sessions. Program results were positive for both teachers and students. Teachers reported significant increases in both their level of knowledge of and their confidence in teaching chemistry and physics concepts, as well as in their knowledge of strategies for addressing gender inequities. Project students' attitudes, particularly those of the girls, improved for some dimensions, remained stable for others, and declined for one; the girls also increased their level of active participation in science activities. Overall, the project seems to have had a positive impact on science teaching content and pedagogy, and on student (especially girls') interest and active participation in science.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.