An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals, and percents

Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N = 581).In addition to using geometric figures and manipulatives, teachers used natural language such as the words nanny and house to characterize mathematical procedures or algorithms. Some teachers used the words or phrases bigger, smaller, doubling, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers and students, specifically equations with missing equals signs and not multiply/dividing by one to find equivalent fractions. No evidence though of a relationship between representational forms and degree of correctness of solutions was found on student work. However, when students exhibited misconceptions, those misconceptions were linked to teachers’ use of idiosyncratic representations.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

Some aspects of causal & neutral equations used in modelling

The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the inter-connection between ordinary differential equations, delay-differential equations, neutral delay-differential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delay-differential equations) rôles for well-defined adjoints and ‘quasi-adjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

Mimicry of proofs with computers: the case of Linear Algebra

In common teaching practice the habit of connecting mathematics classroom activities with reality is still substantially delegated to wor(l)d problems. During recent decades, a growing body of empirical research has documented that the practice of word problem solving in school mathematics does not match this idea of mathematical modelling and mathematization. If we wish to construct ‘real problems arising from real experiences of the child’ following the spirit of these new suggestions, we have to make changes. On the one hand we have to replace the type of activity in which we delegate the process of creating an interplay between reality and mathematics by substituting the word problems with an activity of realistic mathematical modelling, i.e. of both real-world based and quantitatively constrained sense-making; and, on the other hand, to ask for a change in teacher beliefs; furthermore, a directed effort to change the classroom socio-math norms will be needed. This paper discusses some classroom activities that takes these factors into account.     

How young students communicate their mathematical problem solving in writing

This study investigates young students’ writing in connection to mathematical problem solving. Students’ written communication has traditionally been used by mathematics teachers in the assessment of students’ mathematical knowledge. This study rests on the notion that this writing represents a particular activity which requires a complex set of resources. In order to help students develop their writing, teachers need to have a thorough knowledge of mathematical writing and its distinctive features. The study aims to add to the body of knowledge about writing in school mathematics by investigating young students’ mathematical writing from a communicational, rather than mathematical, perspective. A basic inventory of the communicational choices, that are identifiable across a sample of 519 mathematical texts, produced by 9–12 year old students, is created. The texts have been analysed with multimodal discourse analysis, and the findings suggest diversity in students’ use of images, words, numerals, symbols and layout to organize their texts and to represent their problem-solving process along with an answer to the problem. The inventory and the indication that students have different ideas on how, what, for whom and why they should be writing, can be used by teachers to initiate discussions of what may constitute good communication.     

Characterizing Interaction with Visual Mathematical Representations

This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts. Currently, most mathematical tools provide methods by which a learner can interact with these representations. Interaction, in such cases, mediates between the VMR and the thinking, reasoning, and intentions of the learner, and is often intended to support the cognitive tasks that the learner may want to perform on or with the representation. This paper brings together a diverse set of interaction techniques and categorizes and describes them according to their common characteristics, goals, intended benefits, and features. In this way, this paper aims to provide a preliminary framework to help designers of mathematical cognitive tools in their selection and analysis of different interaction techniques as well as to foster the design of more innovative interactive mathematical tools. An effort is made to demonstrate how the different interaction techniques developed in the context of other disciplines (e.g., information visualization) can support a diverse set of mathematical tasks and activities involving VMRs.     

Modelling and simulation of a lay-up process of composite material tapes

This paper deals with various aspects of mathematical modelling, qualitative analysis and simulation related to the technological manufacturing process of composite material panels. The mathematical modelling problem consists in defining all mathematical aspects of the optimum deposition strategy.     

FEM modelling of the time-harmonic dynamical stress field problem for a pre-stressed plate-strip resting on a rigid foundation

In this paper, the FEM modelling of the time-harmonic dynamical stress field problem for the pre-stressed plate-strip with finite length resting on a rigid foundation is developed. The mathematical formulation of the considered problem is made by the use of the equations and relations of the Three-dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies. The proposed modelling is tested on the concrete problems as an example. The numerical results testing the validity of the developed FEM modelling are presented. Moreover, the numerical results attained for the various values of the problem parameters are also presented.     

Bayesian decision making and military command and control

We discuss firstly the problem of military decision, in the context of the more general development of ideas in the representation of decision making. Within this context, we have considered a mathematical model—Bayesian Decision—of decision making and military command. Previous work has been extended, and applied to this problem. A distribution of belief in outcome, given that a decision is made, and a Loss function—a measure of the effect of this outcome relative to a goal—are formed. The Bayes' Decision is the decision which globally minimises the resultant bimodal (or worse) Expected Loss function. The set of all minimising decisions corresponds to the surface of an elementary Catastrophe. This allows smooth parameter changes to lead to a discontinuous change in the Bayes' decision. In future work this approach will be used to help develop a number of hypotheses concerning command processes and military headquarters structure. It will also be used to help capture such command and control processes in simulation modelling of future defence capability and force structure.     

Sequences—Basic elements for discrete mathematics

Sequences are fundamental mathematical objects with a long history in mathematics. Sequences are also tools for the development of other concepts (e. g. the limit concept), as well as tools for the mathematization of real-life situations (e. g. growth processes). But, sequences are also interesting objects in themselves, with lots of surprising properties (e. g. Fibonacci sequence, sequence of prime numbers, sequences of polygonal numbers). Nowadays, new technologies provide the possibility to generate sequences, to create symbolic, numerical and graphical representations, to change between these different representations. Examples of some empirical investigation are given, how students worked with sequences in a computer-supported environment.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Moving boundary problem for the detachment in multispecies biofilms

The work presents the qualitative analysis of the free boundary value problem related to the detachment process in multispecies biofilms. In the framework of continuum approach to one-dimensional mathematical modelling of multispecies biofilm growth, we consider the system of nonlinear hyperbolic partial differential equations governing the microbial species growth, the differential equation for the biomass velocity, the differential equation that governs the free boundary evolution and also accounts for detachment, and the elliptic system for substrate dynamics. The characteristics are used to convert the original moving boundary equation into a suitable differential equation useful to solve the mathematical problem. We also provide another form of the same equation that could be used in numerical applications. Several properties of the solutions to the free boundary problem are shown, such as positiveness of the functions that describe the microbial concentrations and estimates on the characteristic functions. Uniqueness and existence of solutions are proved by introducing a suitable system of Volterra integral equations and using the fixed point theorem.

     

Beyond realistic considerations: modeling conceptions and controls in task examples with simple word problems

Using simple word problems, we analyze possible teacher conceptions on the process of problem solving, its goals and the choices that a problem solver can make in problem mathematization. We identify several possible teacher conceptions that would be responsible for the different didactical contracts that teachers create in the mathematics class. Using especially chosen and designed task examples, we demonstrate the diagnosis of teacher own controls in solving problems and in evaluating problem solutions. We also discuss characteristics of task examples that might promote a shift from a problem solving perspective to a modeling perspective that goes beyond merely accepting alternative solutions due to realistic considerations. This shift in perspective would be exhibited through a new understanding of the process of fitting mathematical models in problem situations.     

On the modelling of heat exchangers and heat exchanger network dynamics using bond graphs

ABSTRACT

Heat exchanger networks are important systems in most thermal engineering systems and are found in applications ranging from power plants and the process industry to domestic heating. Achieving cost-effective design of heat exchanger networks relies heavily on mathematical modelling and simulation-based design. Today, stationary design calculations are carried out for all new designs, but for some special applications, the transient response of complete networks has been researched. However, simulating large heat exchanger networks poses challenges due to computational speed and stiff initial value problems when flow equations are cast in differential algebraic form. In this article, a systems approach to heat exchanger and heat exchanger network modelling is suggested. The modelling approach aims at reducing the cost of system model development by producing modular and interchangeable models. The approach also aims at improving the capability for large and complex network simulation by suggesting an explicit formulation of the network flow problem.     

Arguments constructed within the mathematical modelling cycle

Socio-cultural theories in mathematics education field recently emphasize the importance of the collective argumentation within small-group work. Since mathematical modelling tasks require a process in which students search for a solution for real life problems through small-group work, the arguments in this process become an issue of concern. This study examines the arguments constructed within the mathematical modelling cycle by considering the participants’ modelling processes. In this context, four primary pre-service mathematics teachers worked on a modelling task and their arguments were explained through the components of Toulmin’s argumentation schema. Findings revealed that the data and the claims of most of the arguments corresponded to the starting and ending points of the modelling transition in which the current arguments constructed. The existence of the arguments corresponded through warrant-claim originated from inquiring the assumptions in the modelling cycle. In addition, the participants made assumptions as warrants to support their arguments and as rebuttals to show the degree of certainty of claims in intra-group challenging situations. Both the warrants and the backings depended on modelling context as well as mathematics context.     

An overview of heuristic solution methods

Besides those analysts who are already familiar with a number of heuristic methods, this paper should also be of interest to those analysts and managers who, although not yet aware of specific heuristic approaches, are quite comfortable with the use of mathematical modelling as an aid to decision making. It is concerned with obtaining usable solutions to well-defined mathematical representations of real-world problem situations. Heuristic procedures are defined and reasons for their importance are listed. A wide variety of heuristic methods, including several metaheuristics, are described. In each case, references for further details, including applications, are provided. There is also considerable discussion related to performance evaluation.     

THE INFLUENCE OF TEACHERS’ PRESENTATIONS ON PUPILS’ MENTAL REPRESENTATIONS

Teachers use a variety of external representations to communicate mathematical ideas to their pupils. This paper reports a preliminary study of the internal mental representations that 6- and 7- year-old pupils form as a result of their interactions with the teacher's verbal, written, pictorial and concrete material representations, involving two-digit numbers and operations on them. The results presented here concern the picture-like mental representations that pupils use in performing two-digit calculations mentally. The evidence suggests that pupils seldom spontaneously visualise teachers’ representations or attempt mental manipulation of visual images to help with calculation. Pupils can, however, have mental representations which reproduce some aspects of the teachers’ representations.     

The singing wineglass: an exercise in mathematical modelling

Lecturers in mathematical modelling courses are always on the lookout for new examples to illustrate the modelling process. A physical phenomenon, documented as early as the nineteenth century, was recalled: when a wineglass ‘sings’, waves are visible on the surface of the wine. These surface waves are used as an exercise in mathematical modelling. Based on assumptions about the wine in the glass and observations illustrated with photographs, a mathematical problem is set up. This problem includes a non-homogeneous Neumann boundary condition on the lateral side of the glass. The solution to the mathematical problem is animated using Mathematica?. The predictions of the model are tested by comparing them with the known facts. The predictions of the model agree with the actual observations.