Looking back to the roots of partially correct constructs: The case of the area model in probability

We use the notion Partially Correct Constructs (PaCCs) for students’ constructs that partially match the mathematical principles underlying the learning context. A frequent expression of partial construction of mathematical principles is that a student’s words or actions provide an inaccurate or misleading picture of the student’s knowledge. In this study, we analyze the learning process of a grade 8 student, who learns a topic in elementary probability. The student successfully accomplishes a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require nothing beyond his proven competencies, he encounters difficulties. Using the epistemic actions of the RBC model for abstraction in context as tracers, we analyze his knowledge constructing processes while working on the previous tasks, and identify some of his constructs as PaCCs that are concealed in these processes and explain his later difficulties. In addition, our research points to the complexity of the knowledge structures students are expected to deal with in their attempts to learn an elementary mathematical topic with understanding.     

Phenomenal Conservatism and the Subject’s Perspective Objection

For some years now, Michael Bergmann has urged a dilemma against internalist theories of epistemic justification. For reasons I explain below, some epistemologists have thought that Michael Huemer’s principle of Phenomenal Conservatism (PC) can split the horns of Bergmann’s dilemma. Bergmann has recently argued, however, that PC must inevitably, like all other internalist views, fall prey to his dilemma. In this paper, I explain the nature of Bergmann’s dilemma and his reasons for thinking that PC cannot escape it before arguing that he is mistaken: PC can indeed split its horns.     

‘I’m worried about the correctness’: undergraduate students as producers of screencasts of mathematical explanations for their peers – lecturer and student perceptions

Undergraduate mathematics is traditionally designed and taught by content experts with little contribution from students. Indeed, there are signs that there is resistance from mathematics lecturers to involve students in the creation of material to support their peers – notwithstanding the fact that students have been successfully engaged as co-creators of material in other disciplines. There appears to be little research into what issues may lead to reservations to using student-created content in mathematics learning. This paper takes a case study approach to investigate the reasons for lecturers’ resistance to undergraduate student contributions to learning material, in particular with a view to the production of screencasts of mathematical explanations. It also investigates the views of students producing mathematical screencasts. This study is part of a larger research project investigating undergraduate involvement in mathematics module design. Four second-year students, who were producing mathematics screencasts as part of an internship, and five academics, were interviewed to gain an understanding of their views of the value of student screencasts. The interviews focused on the particular contributions students make to screencasts, outcomes for the students and level of lecturer acceptance of these resources. We argue that students benefit from creating screencasts for their peers by gaining deeper mathematical understanding, improved technological skills and developing other generic skills required of today's graduates. In contrast, we confirm lecturer resistance to using student-generated screencasts in their teaching materials. Lecturer reservations pertain to students’ lack of mathematical maturity and concerns over the mathematical integrity of the content that students produce. We conclude that close collaboration between students and lecturers during the design and production phases of screencasts may help lecturers overcome reservations, whilst preserving the benefits for students. In addition, we provide evidence that the process is a valuable professional development opportunity for the lecturers themselves.     

Student problems with a modelling exercise

In this paper an attempt is made to outline some of the problems facing the student who undertakes a course in mathematical modelling. A case study, considered by several cohorts of students, is used as a medium for introducing the various types of obstacles the students encounter. This approach also results in a demonstration of some of the errors that can be made in operating a modelling course and the paper concludes by offering suggestions on how to overcome certain difficulties that are exposed.     

The finite difference mesh on an air-to-dielectric interface

Cognitive obstacles that arise in the teaching and learning of scalar line integrals, derived from cognitive aids provided to students when first learning about integration of single variable functions are described. A discussion of how and why the obstacles cause students problems is presented and possible strategies to overcome the obstacles are outlined.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.     

The Paul A M Dirac papers at Florida State University: a search for informal mathematical investigations

The research discussed in this article is an archival study of pages of mathematical work produced by the physicist Paul A M Dirac. The pages, referred to as the ‘shoebox papers’, are thought to date back at least to Dirac's time as a student at the University of Cambridge in the 1920s. Florida State University, where the research was conducted and where Dirac worked for the last fourteen years of his life, received the entirety of his papers after his death in 1984. The research so far has identified major themes that recur throughout the collection of papers, including an interest in combinatorics and their relation to algebra problems. Due to Dirac's importance as a physicist and a possible relation to combinatorics work by Leibniz, the collection may have significant implications for the history of mathematics.     

Developing the concept of linear function: one student’s experiences with dynamic physical models

Developing a view of functions as systematic processes involving co-variation among variables has been identified as a goal for mathematics learners at the pre-university level. In this investigation I examined the processes used by an eighth-grade student to interpret linear functions originating in dynamic physical models and the processed he used to link his interpretations to tables, equations, and graphs. The student deepened his understanding of functions by generalizing his view of multiplication beyond that of products resulting from the multiplication of individual factors or from repeated addition. He was enabled to do this by building links between graphic and tabular representations of the functions generated from his exploration with dynamic physical models and by comparing tables of different linear functions. This paper suggests that the development of a student’s reasoning about functions originating in dynamic physical models can be interpreted in terms of generalized multiplicative processes that may occur thorough mapping variations and that a student who interprets such functions as generalized multiplicative processes may use notational variations to generate representing equations.     

A Model of the joint motion of agents with a three-level hierarchy based on a cellular automaton

The collective interaction of agents for jointly overcoming (negotiating) obstacles is simulated. The simulation uses a cellular automaton. The automaton’s cells are filled with agents and obstacles of various complexity. The agents' task is to negotiate the obstacles while moving to a prescribed target point. Each agent is assigned to one of three levels, which specifies a hierarchy of subordination between the agents. The complexity of an obstacle is determined by the amount of time needed to overcome it. The proposed model is based on the probabilities of going from one cell to another.     

Retracing our steps: Fodor’s new old way with concept acquisition

The acquisition of concepts has proven especially difficult for philosophers and psychologists to explain. In this paper, I examine Jerry Fodor’s most recent attempt to explain the acquisition of concepts relative to experiences of their referents. In reevaluating his earlier position, Fodor attempts to co-opt informational semantics into an account of concept acquisition that avoids the radical nativism of his earlier views. I argue that Fodor’s attempts ultimately fail to be persuasive. He must either accept his earlier nativism or adopt a rational causal model of concept acquisition. His animus towards the latter dictates, in my view, a return to the nativism with which he began.     

Physics and Mathematics Without Coordinates

The author presents his views on how physics should be phrased in a coordinate-free manner. Examples are given, which range from classical Continuum Mechanics to Special and General Relativity, in a narrative that hinges on the author’s original contributions to these fields.     

What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views

We present a study in which mathematicians and undergraduate students were asked to explain in writing what mathematicians mean by proof. The 175 responses were evaluated using comparative judgement: mathematicians compared pairs of responses and their judgements were used to construct a scaled rank order. We provide evidence establishing the reliability, divergent validity and content validity of this approach to investigating individuals’ written conceptions of mathematical proof. In doing so, we compare the quality of student and mathematician responses and identify which features the judges collectively valued. Substantively, our findings reveal that despite the variety of views in the literature, mathematicians broadly agree on what people should say when asked what mathematicians mean by proof. Methodologically, we provide evidence that comparative judgement could have an important role to play in investigating conceptions of mathematical ideas, and conjecture that similar methods could be productive in evaluating individuals’ more general (mathematical) beliefs.     

Second-order numerical method for domain optimization problems

This paper is concerned with a second-order numerical method for shape optimization problems. The first variation and the second variation of the objective functional are derived. These variations are discretized by introducing a set of boundary-value problems in order to derive the second-order numerical method. The boundary-value problems are solved by the conventional finite-element method.The authors would like to express their thanks to Mr. T. Masanao, who was an undergraduate student, for his cooperation and comments. They also thank Professor Y. Sakawa of Osaka University for his encouragement.A part of this paper was presented at the IFIP Conference on Control of Boundaries and Stabilization, Clermont-Ferrand, France, 1988.     

In celebration of Prof. Dr. Rolf Klötzler 65th birthday

On January 11, 1996, Professor Klötzler celebrated his 65th birthday. This special issue of the journal “Optimization”, consisting of contributions by his friends, students and colleagues, is dedicated to him. Prof. Klötzler can proudly look back on forty successful years as a university teacher and mathematician. After positions at the Academy of Science, the Hochschule für Bauwesen in Leipzig and the Martin-Luther-University in Halle, where he was head of the Mathematics Institute from 1965 to 1971, he returned to the University of Leipzig in 1972 as a full Professor. He was a student there from 1949 to 1953 and had attended lectures by Ernst Hölder and Herbert Beckert. His scientific treatises are extremely extensive and have made great contributions to many branches of calculus of variations, optimal control and mathematical programming. The common thread running through his work is the solution of Hilbert’s problems concerning the calculus of variations and its development,control theory. Coming from the school of calculus of variations founded by Leon Lichtenstein and Otto Hölder in Leipzig, his scientific interest first focussed on questions concerning extension of field theory of calculus of variations and existence theory of global geodesic fields, from which optimality criteria could be derived. In addition, he was able to establish eigen value criteria for weak optimality of extremals of regular variational problems with multiple integrals. Fundamental results from these works appear in the frequently quoted textbook “MehrdimensionaleVariationsrechnung”, published in 1969.     

Capitalizing on Emerging Technologies: A Case Study of Classroom Blogging

The challenge many teachers face is how to incorporate new technology into their classrooms that strengthens classroom learning by capitalizing on students’ media literacies. Blogs, a new and innovative technological tool, can be used in math and science classrooms to support student learning by capitalizing on students’ interests and familiarity with on‐line communication. This study explores the emerging blogging practices of one high school mathematics teacher and his class to explore issues of intent, use, and perceived value. Data sources for this case included one year's worth of blog content, an interview with the facilitating teacher, and students ‘perceptions of classroom blogging practices. Findings indicate that (1) teachers’ intentions focused on creating additional forms of participation as well as increasing student exposure time with content; (2) blogs were used in a wide variety of ways that likely afforded particular benefits; and (3) both teacher and students perceived the greater investment to be worthwhile. The findings are used to critically consider claims made in the literature about the potential of blogging to effectively support classroom learning.     

General Relativity in the Making, 1916–1918: Rudolf J. Humm in Göttingen and Berlin

Mathematische Semesterberichte - In 1916, the Swiss student Rudolf J. Humm arrived in Göttingen to study relativity theory under Hilbert. He enrolled in his courses, attended Klein’s...     

Examining preservice teachers’ noticing of equity-based teaching practices to empower students engaging in productive struggle

Our study examined ways preservice teachers (PSTs) make connections between teaching practices and use of student resources that support productive struggle and promote equity. Our research questions are: (1) How do PSTs notice and describe the equity-based mathematics teaching practice of leveraging student resources to support student struggles? and (2) In what ways do PSTs make connections to and interpret the role student resources play in the resolution of students’ mathematical struggles? The qualitative study examined 39 PSTs in a mathematics content course for PSTs. Data come from a video analysis assignment where PSTs described their mathematical interpretations of the student struggle(s) and teacher’s use of student resources to support the struggle resolution. Findings show that PSTs noticed teacher moves that leveraged student’s mathematical thinking and linguistic funds of knowledge and based the productive level of the struggle on actions built upon peers, linguistic knowledge and prior mathematical knowledge.     

Systems thinking and mathematical problem solving

Problem solving lies at the core of engineering and remains central in school mathematics. Word problems are a traditional instructional mechanism for learning how to apply mathematics to solving problems. Word problems are formulated so that a student can identify data relevant to the question asked and choose a set of mathematical operations that leads to the answer. However, the complexity and interconnectedness of contemporary problems demands that problem‐solving methods be shaped by systems thinking. This article presents results from three clinical interviews that aimed at understanding the effects that traditional word problems have on a student’s ability to use systems thinking. In particular, the interviews examined how children parse word problems and how they update their answers when contextual information is provided. Results show that traditional word problems create unintended dispositions that limit systems thinking.