Applied mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

The problem of the graphic artist

'The problem of the graphic artist' is a small example of applying elementary mathematics (divisibility of natural numbers) to a real problem which we ourselves have actually experienced. It deals with the possibilities for partitioning a sheet of paper into strips. In this contribution we report on a teaching unit in grade 6 as well as on informal tests with students in school and university. Finally we analyse this example methodologically, summarise our observations with pupils and students, and draw some didactical conclusions.

Cleanliness versus cow comfort – an insolvable problem?

Cubicle should provide good resting comfort as well as clean udders. Dairy cows in cubicle houses often face a restrictive environment with regard to resting behaviour, whereas cleanliness may still be impaired. This study aimed to determine reliable behavioural measures regarding resting comfort applicable in on-farm welfare assessments. Furthermore, relationships between cubicle design, cow sizes, management factors and udder cleanliness (namely teats and teat tips) were investigated. Altogether 15 resting measures were examined in terms of feasibility, inter-observer reliability (IOR) and consistency of results per farm over time. They were recorded during three farm visits on farms in Germany and Austria with cubicle, deep litter and tie stall systems. Seven measures occurred to infrequently to allow reliable recording within a limited observation time. IOR was generally acceptable to excellent except for 'collisions during lying down', which only showed good IOR after improvement of the definition. Only three measures were acceptably repeatable over time: 'duration of lying down', 'percentage of collisions during lying down' and 'percentage of cows lying partly or completely outside lying area'. These measures were evaluated as suitable animal based welfare measures regarding resting behaviour in the framework of an on-farm welfare assessment protocol. The second part of the thesis comprises a cross-sectional study on resting comfort and cow cleanliness including 23 Holstein Friesian dairy herds with very low within-farm variation in cubicle measures. Height at withers, shoulder width and diagonal body length were measured in 79-100 % of the cows (herd size 30 to115 cows). Based on the 25 % largest animals, compliance with recommendations for cubicle measures was calculated. Cleanliness of different body parts, the udder, teats and teat tips was assessed for each cow in the herd prior to morning milking. No significant correlation was found between udder soiling and teat or teat tip soiling on herd level. The final model of a stepwise regression regarding the percentage of dirty teats per farm explained 58.5 % the variance and contained four factors. Teat dipping after milking which might be associated with an overall clean and accurate management style, deep bedded cubicles, increasing cubicle maintenance times and decreasing compliance concerning total cubicle length predicted lower teat soiling. The final model concerning teat tip soiling explained 46.0 % of the variance and contained three factors. Increasing litter height in the rear part of the cubicle and increased alley soiling which is difficult to explain, predicted for less soiled teat tips, whereas increasing compliance concerning resting length was associated with higher percentages of dirty teat tips. The dependent variable ‘duration of lying down’ was analysed using again stepwise regression. The final model explained 54.8 % of the total variance. Lying down duration was significantly shorter in deep bedded cubicles. Further explanatory though not significant factors in the model were neck-rail height, deep bedding or comfort mattresses versus concrete floor or rubber mats and clearance height of side partitions. In the attempt to create a more comprehensive lying down measure, another analysis was carried out with percentage of ‘impaired lying down’ (i.e. events exceeding 6.3 seconds, with collisions or being interrupted) as dependent variable. The explanatory value of this final model was 41.3 %. An increase in partition length, in compliance concerning cubicle width and the presence of straw within bedding predicted a lower proportion of impaired lying down. The effect of partition length is difficult to interpret, but partition length and height were positively correlated on the study farms, possibly leading to a bigger zone of clear space for pelvis freedom. No associations could be found between impaired lying down and teat or teat tip soiling. Altogether, in agreement with earlier studies it was found that cubicle dimensions in practice are often inadequate with regard to the body dimensions of the cows, leading to high proportions of impaired lying down behaviour, whereas teat cleanliness is still unsatisfactory. Connections between cleanliness and cow comfort are far from simplistic. Especially the relationship between cubicle characteristics and lying down behaviour apparently is very complex, so that it is difficult to identify single influential factors that are valid for all farm situations. However, based on the results of the present study the use of deep bedded cubicles can be recommended as well as improved management with special regard to cubicle and litter maintenance in order to achieve both better resting comfort and teat cleanliness.

Identification of Problem-based learning components and their effects on graduates' competencies

This study investigated the relationship between higher education and the requirement of the world of work with an emphasis on the effect of problem-based learning (PBL) on graduates' competencies. The implementation of full PBL method is costly (Albanese & Mitchell, 1993; Berkson, 1993; Finucane, Shannon, & McGrath, 2009). However, the implementation of PBL in a less than curriculum-wide mode is more achievable in a broader context (Albanese, 2000). This means higher education institutions implement only a few PBL components in the curriculum. Or a teacher implements a few PBL components at the courses level. For this kind of implementation there is a need to identify PBL components and their effects on particular educational outputs (Hmelo-Silver, 2004; Newman, 2003). So far, however there has been little research about this topic. The main aims of this study were: (1) to identify each of PBL components which were manifested in the development of a valid and reliable PBL implementation questionnaire and (2) to determine the effect of each identified PBL component to specific graduates' competencies. The analysis was based on quantitative data collected in the survey of medicine graduates of Gadjah Mada University, Indonesia. A total of 225 graduates responded to the survey. The result of confirmatory factor analysis (CFA) showed that all individual constructs of PBL and graduates' competencies had acceptable GOFs (Goodness-of-fit). Additionally, the values of the factor loadings (standardize loading estimates), the AVEs (average variance extracted), CRs (construct reliability), and ASVs (average shared squared variance) showed the proof of convergent and discriminant validity. All values indicated valid and reliable measurements. The investigation of the effects of PBL showed that each PBL component had specific effects on graduates' competencies. Interpersonal competencies were affected by Student-centred learning (β = .137; p < .05) and Small group components (β = .078; p < .05). Problem as stimulus affected Leadership (β = .182; p < .01). Real-world problems affected Personal and organisational competencies (β = .140; p < .01) and Interpersonal competencies (β = .114; p < .05). Teacher as facilitator affected Leadership (β = 142; p < .05). Self-directed learning affected Field-related competencies (β = .080; p < .05). These results can help higher education institution and educator to have informed choice about the implementation of PBL components. With this information higher education institutions and educators could fulfil their educational goals and in the same time meet their limited resources. This study seeks to improve prior studies' research method in four major ways: (1) by indentifying PBL components based on theory and empirical data; (2) by using latent variables in the structural equation modelling instead of using a variable as a proxy of a construct; (3) by using CFA to validate the latent structure of the measurement, thus providing better evidence of validity; and (4) by using graduate survey data which is suitable for analysing PBL effects in the frame work of the relationship between higher education and the world of work.

Das Jahr 2000 Problem - Sorgen mit der Zeit?

Vortrag beim Treffen Lions Club Kassel Brüder Grimm am 20. August 1999. In der Zeit vor der Jahrtausendwende gab es Bedenken, dass größere Probleme durch die übliche Darstellung der Jahresangabe mit nur zwei Ziffern entstehen würden, weil Rechner nicht zwischen 1900 und 2000 unterscheiden könnten. Als Beispiel genannt wurden Fahrstühle, die seit 100 Jahren nicht mehr gewartet wurden und daher stehenbleiben. Tatsächlich ist dann sehr wenig passiert, ob wegen der lebhaften Diskussion vorher oder nicht, ist umstritten. Der Vortrag betrachtet sehr gründlich die technischen Probleme, die sich mit der Zeitdarstellung auf Rechnern ergeben.

Problem??tica de la formaci??n profesional espec??fica

Se presenta experiencia educativa que propone la creaci??n de un grupo de trabajo para analizar la problem??tica de la formaci??n profesional y proponer mejoras. Se realiza en el IES Padre P??veda en Guadix, Granada. Los objetivos son: seguimiento del cat??logo nacional de las cualificaciones profesionales y de los nuevos t??tulos de formaci??n profesional que afecta directamente a nuestro alumnado; estudiar la necesidad de implantar nuevos ciclos formativos en nuestra comunidad relacionados con la demanda de alumnado; estudiar el equilibrio o desequilibrio entre los distintos subsistemas de formaci??n profesional, e intentar evitar un estancamiento en la formaci??n profesional espec??fica; impulsar el establecimiento de los centros integrados en Andaluc??a; estudiar y analizar la legislaci??n vigente, difundi??ndola a toda nuestra comunidad escolar; analizar el plan Andaluz de formaci??n profesional; elaborar informes donde se reflejen los resultados de nuestro trabajo y darle la mayor difusi??n posible.

The Logical Problem of Language Change

This paper considers the problem of language change. Linguists must explain not only how languages are learned but also how and why they have evolved along certain trajectories and not others. While the language learning problem has focused on the behavior of individuals and how they acquire a particular grammar from a class of grammars ${cal G}$, here we consider a population of such learners and investigate the emergent, global population characteristics of linguistic communities over several generations. We argue that language change follows logically from specific assumptions about grammatical theories and learning paradigms. In particular, we are able to transform parameterized theories and memoryless acquisition algorithms into grammatical dynamical systems, whose evolution depicts a population's evolving linguistic composition. We investigate the linguistic and computational consequences of this model, showing that the formalization allows one to ask questions about diachronic that one otherwise could not ask, such as the effect of varying initial conditions on the resulting diachronic trajectories. From a more programmatic perspective, we give an example of how the dynamical system model for language change can serve as a way to distinguish among alternative grammatical theories, introducing a formal diachronic adequacy criterion for linguistic theories.

Electrical Design: A Problem for Artificial Intelligence Research

This report outlines the problem of intelligent failure recovery in a problem-solver for electrical design. We want our problem solver to learn as much as it can from its mistakes. Thus we cast the engineering design process on terms of Problem Solving by Debugging Almost-Right Plans, a paradigm for automatic problem solving based on the belief that creation and removal of "bugs" is an unavoidable part of the process of solving a complex problem. The process of localization and removal of bugs called for by the PSBDARP theory requires an approach to engineering analysis in which every result has a justification which describes the exact set of assumptions it depends upon. We have developed a program based on Analysis by Propagation of Constraints which can explain the basis of its deductions. In addition to being useful to a PSBDARP designer, these justifications are used in Dependency-Directed Backtracking to limit the combinatorial search in the analysis routines. Although the research we will describe is explicitly about electrical circuits, we believe that similar principles and methods are employed by other kinds of engineers, including computer programmers.

Alternative ways to estimate change points in multinomial sequences. An application to an authorship attribution problem

The statistical analysis of literary style is the part of stylometry that compares measurable characteristics in a text that are rarely controlled by the author, with those in other texts. When the goal is to settle authorship questions, these characteristics should relate to the author’s style and not to the genre, epoch or editor, and they should be such that their variation between authors is larger than the variation within comparable texts from the same author. For an overview of the literature on stylometry and some of the techniques involved, see for example Mosteller and Wallace (1964, 82), Herdan (1964), Morton (1978), Holmes (1985), Oakes (1998) or Lebart, Salem and Berry (1998). Tirant lo Blanc, a chivalry book, is the main work in catalan literature and it was hailed to be “the best book of its kind in the world” by Cervantes in Don Quixote. Considered by writters like Vargas Llosa or Damaso Alonso to be the first modern novel in Europe, it has been translated several times into Spanish, Italian and French, with modern English translations by Rosenthal (1996) and La Fontaine (1993). The main body of this book was written between 1460 and 1465, but it was not printed until 1490. There is an intense and long lasting debate around its authorship sprouting from its first edition, where its introduction states that the whole book is the work of Martorell (1413?-1468), while at the end it is stated that the last one fourth of the book is by Galba (?-1490), after the death of Martorell. Some of the authors that support the theory of single authorship are Riquer (1990), Chiner (1993) and Badia (1993), while some of those supporting the double authorship are Riquer (1947), Coromines (1956) and Ferrando (1995). For an overview of this debate, see Riquer (1990). Neither of the two candidate authors left any text comparable to the one under study, and therefore discriminant analysis can not be used to help classify chapters by author. By using sample texts encompassing about ten percent of the book, and looking at word length and at the use of 44 conjunctions, prepositions and articles, Ginebra and Cabos (1998) detect heterogeneities that might indicate the existence of two authors. By analyzing the diversity of the vocabulary, Riba and Ginebra (2000) estimates that stylistic boundary to be near chapter 383. Following the lead of the extensive literature, this paper looks into word length, the use of the most frequent words and into the use of vowels in each chapter of the book. Given that the features selected are categorical, that leads to three contingency tables of ordered rows and therefore to three sequences of multinomial observations. Section 2 explores these sequences graphically, observing a clear shift in their distribution. Section 3 describes the problem of the estimation of a suden change-point in those sequences, in the following sections we propose various ways to estimate change-points in multinomial sequences; the method in section 4 involves fitting models for polytomous data, the one in Section 5 fits gamma models onto the sequence of Chi-square distances between each row profiles and the average profile, the one in Section 6 fits models onto the sequence of values taken by the first component of the correspondence analysis as well as onto sequences of other summary measures like the average word length. In Section 7 we fit models onto the marginal binomial sequences to identify the features that distinguish the chapters before and after that boundary. Most methods rely heavily on the use of generalized linear models

Application of discriminant function analysis and change-point problem in dating volcanic ashes

The application of Discriminant function analysis (DFA) is not a new idea in the study of tephrochrology. In this paper, DFA is applied to compositional datasets of two different types of tephras from Mountain Ruapehu in New Zealand and Mountain Rainier in USA. The canonical variables from the analysis are further investigated with a statistical methodology of change-point problems in order to gain a better understanding of the change in compositional pattern over time. Finally, a special case of segmented regression has been proposed to model both the time of change and the change in pattern. This model can be used to estimate the age for the unknown tephras using Bayesian statistical calibration

Educaci??n ambiental para la acci??n ciudadana : concepciones del profesorado en formaci??n sobre la problem??tica de la energ??a

Determinar el alcance de los objetivos y la naturaleza de las concepciones que tienen los-as estudiantes y los materiales did??cticos, sobre la tem??tica de la energ??a, se formula los siguientes problemas: 1. ??Cu??les son las concepciones de los-las estudiantes de Magisterio sobre los modelos de Educaci??n Ambiental? 2. ??Cu??les son las concepciones de los-las estudiantes sobre los problemas socioambientales que consideran m??s importantes y la idea de riesgo asociada a los mismos? 3. ??Cu??les son las concepciones de los-las estudiantes sobre el papel que juega la participaci??n en el proceso de Educaci??n Ambiental? 4. ??Cu??les son las concepciones de los-las estudiantes sobre la energ??a y el papel que juega la energ??a como problema socioambiental? 5. ??Cu??les son las concepciones did??cticas dominantes de los-las estudiantes sobre el tratamiento did??ctico de la energ??a? 6. ??Cu??les son las concepciones did??cticas dominantes de los materiales seleccionados sobre el tratamiento de la energ??a? 7. ??Existe alguna correspondencia entre las concepciones de los-las estudiantes y las concepciones did??cticas de los materiales seleccionados sobre el tratamiento did??ctico de la energ??a? 8. ??Existen diferencias en las concepciones de los-las estudiantes sobre algunos aspectos de la Educaci??n Ambiental y de la energ??a dependiendo del momento de la investigaci??n? 9. ??Existe coherencia en las concepciones de los-las estudiantes y de los materiales?. El paradigma metodol??gico elegido se ubica en la definici??n del paradigma interpretativo, denominado tambi??n naturalista, de enfoque ecol??gico o etnogr??fico, a trav??s de un estudio de caso. Se selecciona un grupo de estudiantes de la Facultad de Ciencias de la Educaci??n de la Universidad de Sevilla, de las modalidades de Educaci??n Primaria y Especial, que cursan la asignatura optativa de Educaci??n Ambiental (EA). La muestra se compone de 12 hombres y 38 mujeres, sumando un total de 50 personas, solo se tiene materiales escritos de forma individual de 45 personas, se agrupan para trabajar de forma conjunta formando 12 grupos. Dentro del planteamiento metodol??gico adoptado, se utiliza diferentes t??cnicas e instrumentos: la b??squeda y an??lisis de materiales de EA desde un punto de vista did??ctico. Observaci??n externa y recogida de informaci??n en el diario de clase. Cuestionarios y documentos de trabajo. Grabaciones de algunas sesiones de clase. Grupo de discusi??n. Para el tratamiento de los datos se dise??a un sistema de categor??as que sistematizara las respuestas. Las respuestas m??s encontradas, sobre el modelo de EA y el car??cter interdisciplinar, se encuentran en los valores m??s simples respecto a los valores considerados en el sistema de categor??as creados, los cuales mayoritariamente vinculan la EA con un modelo conservacionista y con actividades puntuales, lo cual indica que lo que entiende los-as educadores-as ambientales sobre EA, suele estar 'identificado con el amor a la naturaleza, con las salidas fuera del aula, generalmente 'al campo', con la recogida de muestras o la realizaci??n de an??lisis o reciclado de papel' y 'ven la Educaci??n Ambiental como algo ajeno a las dem??s materias que se lleva a cabo en algunas determinadas fechas y que debe tener un curr??culo establecido y diferente del de otras materias, a excepci??n de las ciencias naturales, con las que de alguna manera se liga la Educaci??n Ambiental y aparece una alta relaci??n entre esta visi??n de la EA y la corriente inductivista del aprendizaje'.