Das Kardinalzahlkonzept

This case-study investigates different aspects of the concept of cardinality of an eighteen-year-old student with mental retardation. At the age of six she could not relate number words, finger and objects in counting. These errors still persist in the classroom situation. This investigation shows that nevertheless her concept of cardinality is fairly highly developed. She knows that in counting she must match number words and objects one to one, the number word sequence she uses is stable, and her insight into the irrelevance of order of enumeration when counting, which she finds by trial, is a sign of the robustness of her cardinal concept. She also understands the relationships of equivalence and order of sets, and she solves arithmetical problems by counting on or down, which means that she understands the number words as cardinal and at the same time as sequence numbers. Errors occur in complex situations, where several components have to be considered. But her concept of cardinality is also incomplete: she has special difficulties concerning counting out objects bundled in tens. The same problems occur when she uses multidigit numbers: she does not see a ten-unit as composed of ten single unit items, that is to say, she replaces the hierarchic structure of the number sequence by a concatenated one. These difficulties must be interpreted as a consequence of her special weakness concerning synthetic thinking and simultaneous performing, as similar patterns can be seen in her spatial perception and in her speech. In the syntactic structure of her utterances, too, the combination of simple entities to complicated units is replaced by a mere concatenation. This means that due to brain dysfunction her behavior is determined by a particular pattern which repeatedly appears intrapersonally, and which is characteristic of some mentally retarded persons though not of all of them. Evidently mathematical thinking is also not a determined system, but a variable one. Mentally retarded students may therefore have great difficulties concerning some areas and at the same time make better progress in others. In particular, difficulties in counting objects are no obstacle to knowledge of cardinality.     

Investigating teachers’ appraisal of unexpected moments and underlying values: an exploratory case in the context of changing mathematics classroom discourse

This article provides an exploratory case study that examines what one teacher indicated as unexpected as she worked to become more purposeful about her classroom discourse practices. We found that she highlighted three areas as being unexpected: (1) aspects of lesson enactment; (2) characteristics of student learning and (3) her own intentionality or purposefulness. We interpret these instances through the lens of the systemic functional linguistics (SFL) appraisal framework in order to understand how she evaluates the instances she highlights and connect these instances to the literature on values in the teaching and learning of mathematics.     

Learning How Amanda,a High School Cerebral Palsy Student,Understands Angles

Little is known about how well cerebral palsy students learn high school geometry. A case study was used to better understand how one student, Amanda, understood angles. Three major accommodations were made to assist her in learning: a) a self-paced curriculum, b) The Geometer's Sketchpad, and c) nontraditional assessment (portfolio, interviews, observations). It was found that Amanda needed a lot of time to process visual information. The orientation of angles, the complexity of the diagram, and the length of the side of an angle all had an impact on her understanding. The software was beneficial for Amanda, because she could hide unnecessary and distracting information, she could make her drawings legible, and she could measure the angles without relying on her own visual perception.     

The powers and pitfalls of algorithmic knowledge: a case study

This study examines one child's use of computational procedures over a period of 3 years in an urban elementary school where teachers were using a standards-based curriculum. From a sociocultural perspective, the use of standard algorithms to solve mathematical problems is viewed as a cultural tool that both enables and constrains particular practices. As this student appropriated and mastered procedures for addition, subtraction, multiplication and division, she could solve problems that involved fairly straightforward computations or where she could easily model the action to determine an appropriate computation. At the same time, her use of these algorithms, along with other readily available tools, such as her fingers or multiplication tables, constrained her ability to reflect on the tens-structure of the number system, an effect that had serious consequences for her overall mathematical achievement. The results of this study suggest that even when not directly introduced, algorithms have such strong currency that they can mediate more reform-oriented instruction.     

A climbing girl's reflections about angles

The main research question in this paper is whether a climbing discourse can be a resource for a school-geometry discourse. The text is based on a 12-year old girl's story from an exciting climbing trip during her summer holiday. The girl uncovers some of her knowledge that had been invisible to her; she is guided to see some relations between her climbing and her understanding of angles. In the beginning, this girl believes her story does not concern angles at all. The tools for uncovering angles in her story are based on different levels of visibility and objects of the climbing discourse combined with different conceptions of space. The girl develops her consciousness about angles as natural elements in her climbing activity and she is guided to see the angle as an object of her climbing discourse.     

Olga Taussky,a torchbearer for mathematics and teacher of mathematicians

Olga Taussky-Todd's mathematical and personal life (1906-1995), her achievements and obstacles, her scientific reasoning and teaching all have served as inspiration to many mathematicians. We describe her role in the mathematics world of the previous century as a torchbearer for mathematics and mathematicians, bearing the “torch of scientific truth” that burns inside of mathematics and its applications. Besides her many deep math contributions – too many to elaborate – she excelled at distilling and presenting mathematical concepts and ideas in her work and gave us many visionary papers and math talks. By sharing her mathematical vision freely she has inspired many of us. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim

This paper critically examines the discrepancies among the pre-requisite fractional concepts assumed by a curricular unit on operations with fractions, the teacher's assumptions about those concepts and a particular student's understanding of fractions. The paper focuses on the case of one student (Tim) in the teacher's 6th grade class who was interviewed by one of the authors once a week during the teaching of the unit. The teaching materials and the teacher's instruction were based on the assumption that students understood the concept of a unit fraction as being one of several equal parts of a given whole. The teacher neither emphasized the need for equal parts nor the part-to-whole relation. The teacher's reasonable assumptions about her students’ understanding of fractions were severely challenged by the cognitive constructs that Tim exhibited during his first two interviews. When she viewed tapes of the class instruction and the interviews with Tim she realized Tim lacked essential constructs to make sense of her instruction. She subsequently made adjustments in her instruction, making effective use of more appropriate representations based on tasks from the unit that we modified and used with Tim in our interviews. These adjustments helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This study illustrates the importance of coming to understand a student's mathematical activity in terms of possible conceptual schemes and modifying instructional strategies to build on those schemes. The coordinated design of the research study facilitated these instructional modifications.     

Learning about teaching for understanding through the study of tutoring

In this study, we conducted a fine-grained analysis of an expert tutor's (Nancy Mack) tutorial actions as she attempted, successfully, to help students learn fractions with understanding. Our analysis revealed that, as Mack tutored students in two different research studies, she took two types of tutorial actions previously unrecorded in the literature. By analyzing her actions using a methodology involving production rules, we suggest how her content knowledge, pedagogical content knowledge, and her knowledge of her students were interrelated and how they impacted on her instructional decisions and teaching actions. We also provide an example of how using production rules can be useful to discern some of the complexities involved in teaching and tutoring.     

Risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks

Achievement of the herd immunity is essential for preventing the periodic spreading of an infectious disease such as the flu. If vaccination is voluntary, as vaccination coverage approaches the critical level required for herd immunity, there is less incentive for individuals to be vaccinated; this results in an increase in the number of so-called “free-riders” who craftily avoid infection via the herd immunity and avoid paying any cost. We use a framework originating in evolutionary game theory to investigate this type of social dilemma with respect to epidemiology and the decision of whether to be vaccinated. For each individual in a population, the decision on vaccination is associated with how one assesses the risk of infection. In this study, we propose a new risk-assessment model in a vaccination game when an individual updates her strategy, she compares her own payoff to a net payoff obtained by averaging a collective payoff over individuals who adopt the same strategy as that of a randomly selected neighbor. In previous studies of vaccination games, when an individual updates her strategy, she typically compares her payoff to the payoff of a randomly selected neighbor, indicating that the risk for changing her strategy is largely based on the behavior of one other individual, i.e., this is an individual-based risk assessment. However, in our proposed model, risk assessment by any individual is based on the collective success of a strategy and not on the behavior of any one other individual. For strategy adaptation, each individual always takes a survey of the degree of success of a certain strategy that one of her neighbors has adopted, i.e., this is a strategy-based risk assessment. Using computer simulations, we determine how these two different risk-assessment methods affect the spread of an infectious disease over a social network. The proposed model is found to benefit the population, depending on the structure of the social network and cost of vaccination. Our results suggest that individuals (or governments) should understand the structure of their social networks at the regional level, and accordingly, they should adopt an appropriate risk-assessment methodology as per the demands of the situation.     

A Preservice Secondary Teacher's Moves to Protect Her View of Herself as a Mathematics Expert

This paper discusses the experience of a preservice secondary mathematics teacher during lesson study. Although the preservice teacher was a strong undergraduate mathematics student, she used compensation “moves” to deflect attention away from her insecurities about her conceptual understanding of secondary mathematics. She feared being labeled as “dumb” and redirected conversations in order to protect her identity as a knower of mathematics. This paper investigates the culture in which preservice teachers develop confidence in their personal mathematics knowledge and how that confidence may influence behavior.     

Breaking social barriers: Florentia Fountoukli (1869–1915)

During the nineteenth century the foundations were laid for the advance of female education. At the same time it was still not easy for women to enter the fields of mathematics and science. In Greece the situation was worse than in some other countries: women were not allowed to matriculate at Athens University until the last decades of the century. Florentia Fountoukli was the first Greek woman to attend lectures at the mathematics department. Her dream of becoming a mathematician was partly fulfilled after a long struggle to persuade university professors to accept her in their lectures as a regular student. Florentia was a talented person who in the end had little choice but to teach philosophy and pedagogy in an Arsakeion school. This paper sheds light on her professional life in the school in Corfu. Finally after four years of service she went to Athens where she set up and managed her own school.

(For another article on the mathematical education of women in the nineteenth century see Marit Hartveit's ‘How Flora got her cap’ in BSHM Bulletin, 24 (2009), 147–158.)     

Characterizing the growth of one student’s mathematical understanding in a multi-representational learning environment

The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10^th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results revealed that Elif was able to progress in her mathematical understanding from informal levels to the formal levels in the Pirie-Kieren model as she performed treatments and conversions, movements involving different registers of representations. The results also showed numerous examples of Elif’s mathematical understanding based on folding back activities, complementary aspects of acting and expressing, and interventions. Using the two theories together provides a powerful and holistic approach to a deeper understanding of mathematical learning by characterizing and articulating the growth of mathematical understanding and the way of mathematical thinking.     

Actions of Noncompact Semisimple Groups on Lorentz Manifolds

The above title is the same, but with “semisimple” instead of “simple,” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her premature death. Here, using a different, geometric approach, we generalize her results to the semisimple case, and give proofs of all her announced results. Received: May 2006, Revision: February 2007, Accepted: March 2007     

The Border Crossings of a Multicultural Science Education Enthusiast

Jill was a preservice science education student who wanted to make science more accessible to all students. This study is an examination of the “borders” she encountered as she completed her student teaching in a cultural setting that was different from her own. Her student teaching experience was documented through interviews, participant observations, field notes, lesson plans, and a journal. An inductive analysis of the documents and a context chart of the coded data revealed that Jill encountered the (a) cultural border of her students, (b) cultural border of science instruction, and (c) cultural border of the school. While some borders were crossed, others were not. This study suggests that during field experiences, preservice teachers may encounter multiple cultural borders, some consistent and some inconsistent with their instructional philosophy. As student teachers work with diverse populations, supervisors and cooperating teachers need to recognize the borders student teachers will encounter and encourage student teachers to examine their beliefs about practice as a means to acknowledge and understand the encountered borders.     

Challenges of maintaining cognitive demand during the limit lessons: understanding one mathematician’s class practices

In this study, we examined five limit lessons using Mathematical Tasks Framework to understand students’ opportunities to learn cognitively challenging tasks and maintain cognitive demand during limit lessons. Our analysis of Dr A’s five lessons shows that students rarely had opportunities to maintain or increase cognitive demand. There are two main factors that shaped her instructional practices, students and time. These two factors greatly influenced how she selects and implements limit tasks in her classes. To serve her students’ needs of knowing more rules, formulas and procedures, she selected and discussed those simple tasks a lot. Although Dr A thinks challenging tasks and asking demanding questions can be potentially good instructional practices, she thinks these instructional practices would not serve her students well. With these factors, we made possible recommendations to have more student-centred teaching.     

Solving a bi-objective nurse rerostering problem by using a utopic Pareto genetic heuristic

Nurse rerostering arises when at least one nurse announces that she will be unable to undertake the tasks previously assigned to her. The problem amounts to building a new roster that satisfies the hard constraints already met by the current one and, as much as possible, fulfils two groups of soft constraints which define the two objectives to be attained. A bi-objective genetic heuristic was designed on the basis of a population of individuals characterised by pairs of chromosomes, whose fitness complies with the Pareto ranking of the respective decoded solution. It includes an elitist policy, as well as a new utopic strategy, introduced for purposes of diversification. The computational experiments produced promising results for the practical application of this approach to real life instances arising from a public hospital in Lisbon.     

Princess Elizabeth of Bohemia and Descartes’ letters (1650–1665)

After Descartes’ death in 1650, Princess Elizabeth generously shared with others several letters she had received from the philosopher, which contained philosophically as well as mathematically exciting material. In this article I place the transmission of these copies in context, revealing that Elizabeth steadily became an intellectually inspiring figure, attracting international attention. In the 1650s she stayed at Heidelberg where she discussed Cartesian philosophy with professors and students alike, including the professor of philosophy and mathematics Johann von Leuneschlos. In the mid-1660s, an initiative was taken from the English side of the Channel (Pell, More) to obtain Descartes’ mathematical letters to Elizabeth that had not yet been published. One letter of Elizabeth herself on this very subject has been preserved. The letter, addressed to Theodore Haak, will be published here for the first time. It is of special interest, because the princess supplies a general outline of her solution to the mathematical problem Descartes gave her to solve in 1643. It substantiates the hypothesis regarding Elizabeth’s solution earlier proposed by Henk Bos.     

Farewell to direct source incompatibilism

Traditional theorists about free will and moral responsibility endorse the principle of alternative possibilities (PAP): an agent is morally responsible for an action that she performs only if she can do or could have done otherwise. According to source theorists, PAP is false and an agent is morally responsible for her action only if she is the source of that action. Source incompatibilists accept the source theory but also endorse INC: if determinism is true, then no one is morally responsible for any action. This paper is a critique of a kind of source incompatibilism, namely, direct source incompatibilism. Direct source incompatibilists reject PAP on the basis of Frankfurt-style examples. Since PAP is one of two premises in the traditional argument for INC, direct source incompatibilists opt for a version of the direct argument, which argues for INC with the aid of some non-responsibility transfer principle. I demonstrate that this option is not available, for there is a tension between the following two claims.     

A value for bi-cooperative games

Bi-cooperative games were introduced by Bilbao et al. as a generalization of TU cooperative games, in which each player can participate positively, negatively, or not at all. In this paper, we propose a definition of a share of the worth obtained by some players after they decided on their participation in the game. It turns out that the cost allocation rule does not look for a given player to her contribution at the opposite participation option to the one she chooses. The relevance of the value is discussed on several examples.