Personal Excursions: Investigating the Dynamics of Student Engagement

We investigate the dynamics of student engagement as it is manifest in self-directed, self-motivated, relatively long-term, computer-based scientific image processing activities. The raw data for the study are video records of 19 students, grades 7 to 11, who participated in intensive 6-week, extension summer courses. From this raw data we select episodes in which students appear to be highly engaged with the subject matter. We then attend to the fine-grained texture of students’ actions, identifying a core set of phenomena that cut across engagement episodes. Analyzed as a whole, these phenomena suggest that when working in self-directed, self-motivated mode, students pursue proposed activities but sporadically and spontaneously venture into self-initiated activities. Students’ recurring self-initiated activities – which we call personal excursions – are detours from proposed activities, but which align to a greater or lesser extent with the goals of such activities. Because of the deeply personal nature of excursions, they often result in students collecting resources that feed back into both subsequent excursions and framed activities. Having developed an understanding of students’ patterns of self-directed, self-motivated engagement, we then identify four factors that seem to bear most strongly on such patterns: (1) students’ competence (broadly construed); (2) features of the software-based activities, and how such features allowed students to express their competence; (3) the time allotted for students to pursue proposed activities, as well as self-initiated ones; and (4) the flexibility of the computational environment within which the activities were implemented.     

TDmat–mathematics diagnosis evaluation test for engineering sciences students

Since 1989, the Mathematics Education Project (PmatE–Projecto Matemática Ensino) has developed several strategies to improve the success of students in Mathematics. The most important of these are mathematical games for all grades above primary school. The online evaluation of Mathematics subjects is one of PmatE's goals. The implementation of an on-line diagnosis test–TDmat (Teste Diagnóstico de Matemática), mandatory for all students enrolling for the first time in Science and Engineering courses at the University of Aveiro, was the approach chosen to evaluate the mathematical competences of these students. TDmat is a computer test with automatic evaluation in which the analysis of results is done automatically. This paper describes how it is conceived as well as the main results and conclusions drawn from evaluations of the school years 2003/2004 and 2004/2005.     

An APOS study on pre-service teachers’ understanding of injections and surjections

This study was undertaken to explore pre-service teachers’ understanding of injections and surjections. There were 54 pre-service teachers specialising in the teaching of Mathematics in Grades 10–12 curriculum who participated in the project. The concepts were covered as part of a real analysis course at a South African university. Questionnaires based on an initial genetic decomposition of the concepts of surjective and injective functions were administered to the 54 participants. Their written responses, which were used to identify the mental constructions of these concepts, were analysed using an APOS (action-process-object-schema) framework and five interviews were carried out. The findings indicated that most participants constructed only Action conceptions of bijection and none demonstrated the construction of an Object conception of this concept. Difficulties in understanding can be related to students’ lack of construction of the concepts of functions and sets that are a prerequisite to working with bijections.     

The Development of Comprehension of Chance Language: Evaluation and Interpretation

Comprehension of chance language, such as is found in newspapers, is a fundamental aspect of statistical literacy. In this study, students' understandings of chance language were explored through responses to two items in surveys administered to 2,726 students from grades 5 to 11. One item involved evaluating the chance expressed in phrases from newspaper headlines using a number line, and responses were described in four levels of chance language evaluation. The other item involved interpreting, in context, an expression of percent chance, and responses were described in four levels of chance language interpretation. Students in higher grades were more likely to demonstrate higher levels of both evaluation and interpretation. The association between levels of evaluation and interpretation was further explored generally and in relation to one of the headlines involving percent. Implications for mathematics educators in relation to chance language in the curriculum across the years of schooling are discussed.     

Outcomes of a service teaching module on ODEs for physics students

This paper reports on the first part of a multiphase research project that seeks to identify and address the difficulties encountered by physics students when studying differential equations. Differential equations are used extensively by undergraduate physics students, particularly in the advanced modules of their degree. It is, therefore, necessary that students develop conceptual understanding of differential equations in addition to procedural skills. We have investigated the difficulties encountered by third-year students at Dublin City University in an introductory differential equations module. We developed a survey to identify these difficulties and administered it to students who had recently completed the module. We found that students’ mathematical ability in relation to procedural competence is an issue in their study of differential equations, but not as severe an issue as their conceptual understanding. Mathematical competence alone is insufficient if we expect our students to be able to recognize the need for differential equations in a physical context and to be able to set up, solve and interpret the solutions of such equations. We discuss the implications of these results for the next stages of the research project.     

“Last curves not quite correct”: diagnostic competences of future teachers with regard to modelling and graphical representations

The article describes the results of a national enrichment to the six-country study Mathematics Teaching in the 21st century (MT21)—an international comparative study about the efficiency of teacher education. The enrichment focuses on the diagnostic competence of future mathematics teachers as sub-component of teachers’ professional competence for which the evaluation of students’ solutions of a modelling task about the course of a racetrack is demanded. In connection with two sub-facets of the diagnostic competence, namely the competence to recognise students’ misconceptions and the competence of criteria-guided assessment of students’ solutions, typical answer patterns are distinguished as well as the frequency of their occurrence with regard to future teachers’ phase of teacher education and the level of school teaching they are going to teach in.     

The Cumulative and Residual Impact of a Systemic Reform Program on Teacher Change and Student Learning of Science

This longitudinal, five‐year study of teachers and students who had participated in a systemic reform program in science explored if (1) teacher change in practice realized during a three‐year program is sustained one, two, and three years following the program, (2) student performance on state science assessments two years following studying with teachers at this school still demonstrated significant differences from students who attended the control school, and (3) student performance continued to be enhanced for both White and Minority students. Student achievement was assessed using the Discovery Inquiry Test in Science during sixth through eighth grades and the Ohio Graduation Test was used in 10th grade. The same students completed the test in grades 6–8 and 10th grade. Students from the Program school significantly outperformed students who attended the control school on the 10th grade state assessment in science. Findings in this study revealed the ability for sustained, whole‐school, professional development programs to have a cumulative and residual impact on teacher change and student learning of science.     

A possibilistic decision model for new product supply chain design

This paper models supply chain (SC) uncertainties by fuzzy sets and develops a possibilistic SC configuration model for new products with unreliable or unavailable SC statistical data. The supply chain is modeled as a network of stages. Each stage may have one or more options characterized by the cost and lead-time required to fulfill required functions and may hold safety stock to prevent an inventory shortage. The objective is to determine the option and inventory policy for each stage to minimize the total SC cost and maximize the possibility of fulfilling the target service level. A fuzzy SC model is developed to evaluate the performance of the entire SC and a genetic algorithm approach is applied to determine near-optimal solutions. The results obtained show that the proposed approach allows decision makers to perform trade-off analysis among customer service levels, product cost, and inventory investment depending on their risk attitude. It also provides an alternative tool to evaluate and improve SC configuration decisions in an uncertain SC environment.     

Mental Models of Elementary and Middle School Students in Analyzing Simple Battery and Bulb Circuts

Written assessment items were developed to probe students’ understanding of a variety of direct current (DC) resistive electric circuit concepts. The items were used to explore the mental models that grade 3–8 students use in explaining the direction of electric current and how electric current is affected by different configurations of simple battery and bulb circuits. Consistency of applying mental models in different, but equivalent, circuits was also analyzed. Students analyses of current flow direction was categorized into one of two mental models: (1) bidirectional and (2) unidirectional. We found an increase in the consistency of current flow direction mental model use coinciding with grade 4 instruction of batteries and bulbs, however, the proportion of students using a bidirectional flow model was similar in grades 3–8.     

Evolution of the Mathematical Collection of the Polish Virtual Library of Science

The Mathematical Collection of the Polish Virtual Library of Science is an open access repository of Polish mathematical papers. The work on this project started 8 years ago and at present 12,500 articles and 54 books are available in the collection. Statistical analysis of use of the webpage clearly shows the advantages of online publishing of scientific works. Currently, we are preparing the migration of the Mathematical Collection to a new platform for electronic publishing called YADDA. This platform, developed at ICM UW, offers an excellent solution for Open Access paradigm of content publishing, and will ensure improved access to the Mathematical Collection.     

A hybrid model of mathematics support for science students emphasizing basic skills and discipline relevance

The problem of students entering university lacking basic mathematical skills is a critical issue in the Australian higher-education sector and relevant globally. The Maths Skills programme at La Trobe University has been developed to address under preparation in the first-year science cohort in the absence of an institutional mathematics support centre. The programme was delivered through first-year science and statistics subjects with large enrolments and focused on basic mathematical skills relevant to each science discipline. The programme offered a new approach to the traditional mathematical support centre or class. It was designed through close collaboration between science subject coordinators and the project leader, a mathematician, and includes resources relevant to science and mathematics questions written in context. Evaluation of the programme showed it improved the confidence of the participating students who found it helpful and relevant. The programme was delivered through three learning modes to allow students to select activities most suitable for them, which was appreciated by students. Mathematics skills appeared to increase following completion of the programme and student participation in the programme correlated positively and highly with academic grades in their relevant science subjects. This programme offers an alternative model for mathematics support tailored to science disciplines.     

Changes of Working Styles in a Computer Algebra Environment – The Case of Functions

This study is an empirical investigation of 11th graders at a German high school (Gymnasium). Working over a 24-hour period in a computer lab, we investigated students' use of quadratic functions with `Derive', and trigonometric functions with `Mathplus' (or `Theorist' for Macintosh). We were particularly interested in the working styles of students while they solved problems and looked for changes in these styles, as compared to traditional paper and pencil activities. While students worked on the computer, their activities (such as inputs from the keyboard, menu choices or mouse movements) were saved by a special program, which ran in the `background'. We are interested in the possibilities of developing a research method based on these `computer protocols'. The study should be seen as an exploratory study for developing hypotheses for further empirical investigations.This revised version was published online in September 2005 with corrections to the Cover Date.     

Augenbewegungen als Spuren prädikativen oder funktionalen Denkens

We are going to report about methods and results of a study in which pupils of grades 10 and 11 of two grammar schools have been tested in individual examinations using problems of the QuaDiPF-test. The main focus of this investigation was to detect a correspondence between predicative versus functional explanations and patterns of typical eye-movements. While working on the test-problems, the eye-movements of the subjects were recorded. After having worked on a problem, the subjects were asked to explain their procedure and to given reasons for their results. When evaluating these explanations, they were classified according to characteristics of predicative or functional thinking. Moreover, an algorithm was developed that was used to analyse the recorded eye-movements. Both methods of classification of a predicative versus functional cognitive structure showed a very clear correspondence. This study is part of a research project about the importance of individual differences in the preference for predicative versus functional thinking and has been conducted for several years now at the Institute of Cognitive Mathematics. The results of this basic research are especially used to explain the pupils’ understanding of mathematical concepts and their problem solving behaviour in regular mathematics lessons.     

Concept Development of Decimals in Chinese Elementary Students: A Conceptual Change Approach

The aim of this study was to examine the concept development of decimal numbers in 244 Chinese elementary students in grades 4–6. Three grades of students differed in their intuitive sense of decimals and conceptual understanding of decimals, with more strategic approaches used by older students. Misconceptions regarding the density nature of decimals indicated the progress in an ascending spiral trend (i.e., fourth graders performed the worst; fifth graders performed the best; and sixth graders regressed slightly), not in a linear trend. Misconceptions regarding decimal computation (i.e., multiplication makes bigger) generally decreased across grades. However, children's misconceptions regarding the density and infinity features of decimals appeared to be more persistent than misconceptions regarding decimal computation. Some students in higher grades continued to use the discreteness feature of whole numbers to explain the distance between two decimal numbers, indicating an intermediate level of understanding decimals. The findings revealed the effect of symbolic representation of interval end points and students' responses were contingent on the actual representations of interval end points. Students in all three grades demonstrated narrowed application of decimal values (e.g., merchandise), and their application of decimals was largely limited by their learning experiences.     

A discrete approach to the concept of derivative

The concept of derivative is a basic concept of calculus. It is closely related to the concept of function, the idea of rate of change, and the limit concept. In recent decades, teaching the concept of derivative in mathematics classrooms has changed: a quite formal approach—closely linked to the teaching of calculus at university and based on the sequence concept—has been transformed to or substituted by a new one. This means working with rates of change, an intuitive access to the concepts of limit and derivative. It includes working with real functions right from the beginning, a great emphasis on graphs, and the use of digital technologies. The meaning of sequences has decreased to a point where they are sometimes no longer even taught in the calculus course. In recent years this concept has been criticized for not developing adequate perceptions of the basic concepts of calculus and not sufficiently preparing the students for scientific courses at university. In this paper we present an alternative discrete step-by-step approach to the basic concepts of calculus by working with sequences and difference sequences, functions defined on \( \mathbb{Z}\) and discrete domains of \( \mathbb{Q}\) , and by subsequently developing the concept of rate of change in a discrete learning environment. The paper is based on general theoretical considerations and empirical investigations by the author and is meant as a contribution to classroom design-research or “design science” (Wittmann, Educ Stud Math 29(4):355–374, 1995).