Creativity in students’ modelling competencies: conceptualisation and measurement

Modelling competencies are currently included in numerous curricula worldwide and are generally accepted as a complex, process-oriented construct. Therefore, effective measurement should include multiple dimensions, like the sub-competencies required throughout the modelling process. Departing from the characteristics of modelling problems as open and often underdetermined real-world problems, we propose to enrich the current conceptualisation of mathematical modelling competencies by including creativity, which plays an important role in numerous phases of the mathematical modelling process but has scarcely been considered in modelling discourse. In the study described in this paper, a new instrument for the evaluation of this enriched construct has been developed and implemented. The modelling competencies incorporating creativity of the students were evaluated based on the adequacy of the models and the modelling processes proposed, and the appropriateness and completeness of the approaches were evaluated in detail. Adapting measurement approaches for creativity that have been developed in the problem-solving discourse, certain criteria of creativity were selected to evaluate the creativity of the students’ approaches in tackling modelling problems—namely, usefulness, fluency, and originality. The empirical study was conducted among 107 Chinese students at the upper secondary school level, who attended a modelling camp and independently solved three complex modelling problems. The results reveal significant correlations between fluency and originality in students’ performances across all tasks; however, the relationships between usefulness and the other two creativity aspects were not consistent. Overall, the results of the study support the importance of the inclusion of creativity in the construct of modelling competencies.

The role of reading comprehension in mathematical modelling: improving the construction of a real-world model and interest in Germany and Taiwan

To solve mathematical modelling problems, students must translate real-world situations, which are typically presented in text form, into mathematical models. To complete the translation process, the problem-solver must first understand the real-world situation. Therefore, reading comprehension can be considered an essential part of solving modelling problems, and fostering reading comprehension might lead to better modelling competence. Further, ease of comprehension and involvement have been found to increase interest in the learning material, and thus, improving reading comprehension might also increase interest in modelling. The aims of this study were to (a) determine whether providing students with reading comprehension prompts would improve the modelling sub-competencies needed to construct a model of the real-world situation and their interest in modelling and (b) analyze the hypothesized effects in two different educational environments (Germany and Taiwan). We conducted an experimental study of 495 ninth graders (201 German and 294 Taiwanese students). The results unexpectedly revealed that providing reading comprehension prompts did not affect the construction of a real-world model. Further, providing reading comprehension prompts improved students’ situational interest. The effects of providing reading comprehension prompts on the construction of a real-world model were similar in Germany and Taiwan. Students’ interest in modelling improved more in Germany. An in-depth quantitative analysis of students’ responses to reading prompts, their solutions, and their interest in the experimental group confirmed the positive relation between reading comprehension and modelling and indicated that the reading comprehension prompts were not sufficient for improving reading comprehension. Implications for future research are discussed.

Model-derived uncertainties in the calculation of geological phase equilibria

<p>Phase equilibrium modelling offers a welcome window onto rock-forming processes. It underpins the principles of geothermobarometry, which today is commonly carried out via pseudosection calculations in software such as THERMOCALC and Perple_X. Increasingly, phase equilibrium modelling is combined with complementary approaches such as diffusion or geodynamical calculations, in order to simulate Earth processes.</p><p>However, as anyone with experience of pseudosection calculations will know, it is not always easy to make sense of a rock through phase equilibrium modelling. Problems may relate to: (1) in what way the assumption of thermodynamic equilibrium may, or may not, be applied; (2) uncertainties in compositional analysis; and (3) uncertainties in the composition-dependent equations of state (<em>x</em>-eos). The <em>x</em>-eos are the building blocks of the modelling – one <em>x</em>-eos is needed to represent each of the mineral and fluid phases in the calculation. </p><p>Of the problems listed above, (3) is the most opaque for the user. In this talk I will discuss the uncertainties associated with the <em>x</em>-eos, and the implications of those uncertainties for thermobarometry and the simulation of Earth processes. I will describe two tools, currently in development, for investigating <em>x</em>-eos-derived uncertainty in thermobarometry.</p>

Create your own problem! When given descriptions of real-world situations, do students pose and solve modelling problems?

As problem posing has been shown to foster students’ problem-solving abilities, problem posing might serve as an innovative teaching approach for improving students’ modelling performance. However, there is little research on problem posing regarding real-world situations. The present paper addresses this research gap by using a modelling perspective to examine (1) what types of problems students pose (e.g., modelling vs. word problems) and (2) how students solve different types of self-generated problems. To answer these questions, we recruited 82 ninth- and tenth-graders from German high schools and middle schools to participate in this study. We presented students with different real-world situations. Then we asked them to pose problems that referred to these situations and to solve the problems they posed. We analyzed students’ self-generated problems and their solutions using criteria from research on modelling. Our analysis revealed that students posed problems that were related to reality and required the application of mathematical methods. Therefore, problem posing with respect to given real-world situations can be a beneficial approach for fostering modelling abilities. However, students showed a strong tendency to generate word problems for which important modelling activities (e.g., making assumptions) are not needed. Of the students who generated modelling problems, a few either neglected to make assumptions or made assumptions but were not able to integrate them adequately into their mathematical models, and therefore failed to solve those problems. We conclude that students should be taught to pose problems, in order to benefit more from this powerful teaching approach in the area of modelling.

Development of the Theory of Macrosystems as a Necessary Condition for Improving Quality of Transport Modelling

The issues of development of the theory of transport macrosystems are considered regarding results obtained in the works of A. G. Wilson, Yu. S. Popkov, A. V. Gasnikov, E. V. Gasnikova and others. The transport macrosystem is considered a complex multicomponent system to which series of thermodynamic analogy can be applied (equilibrium state, information entropy as a function of state parameters, the presence of basic phenomenological schemes for filling of states with elements, etc.). To further develop the theory of development of transport modelling, which is the objective of the article, it is proposed to consider circumstances that reflect modern trends in development of transport systems: diversity of transport systems, dynamic nature of its functioning, many different elements that can obey different models for filling of states. To implement this task, various methods are used: equation of the transport process introduced by the author, which makes it quite easy to go to quasi-dynamic formulations of transport modelling problems, as well as a general formal representation of the system as a set of elements, which is determined based on the analysis of many works of domestic authors. The conclusion is dedicated to discussion of issues of further development of the theory of transport macrosystems in a dynamic setting.

EM modelling of arbitrary shaped dispersive chiral dielectric objects using a 3D leapfrog scheme on unstructured meshes

The standard Yee FDTD algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of this classical scheme to 3D unstructured co-volume meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This circumvents the problem of accuracy losses, which are normally associated with the use of a staircased representation of curved material interfaces in the standard Yee scheme. The procedure has been successfully employed for modelling problems involving both isotropic and anisotropic lossy materials. Here, we consider the novel extension of this approach to allow for the challenging modelling of chiral materials, where the material parameters are frequency dependent. To adequately model the dispersive behaviour, the Z-transform is employed, using second order Padé approximations to maintain the accuracy of the basic scheme. To validate the implementation, the numerical results produced are compared with available analytical solutions. The stability of the chiral algorithm is also studied.

Evaluating Pre-Service Teachers’ Design of Mathematical Modelling Tasks

This study aims to examine pre-service teachers’ competencies in designing authentic mathematical modelling tasks. The participants of this study were 22 pre-service teachers enrolled in a mathematical modelling course during their second year in a mathematics teacher education program. The participants designed 20 problems, which were evaluated based on four criteria for mathematical modelling: reality, openness, complexity, and model eliciting. The results indicate that even though the participants were successful in developing problems that had real-world scenarios, only five of them were classified as modelling problems. The majority of the problems fulfilled the reality criterion (12 out of 17); only five of them fulfilled the criteria of openness and model eliciting, and only six of them fulfilled the criterion of complexity. These findings contribute to the importance of supporting teachers’ competencies in the teaching and learning of mathematical modelling for integration into K-12 classrooms.

Optimization of placement in the tasks of rapid prototyping and manufacturing of volumetric parts based on additive technologies

The problem of optimizing the life cycle of complex three-dimensional objects in small-scale production is considered. Additive technologies and optimization algorithms for the placement of three-dimensional objects are considered as technologies to solve this problem. Using the multilevel synthesis method can significantly reduce the time for prototyping new products. It should be noted that since several independent parts can be manufactured at the same time, this problem belongs to the class of optimization geometric modelling problems, namely, the problem of three-dimensional irregular placement. An algorithmic solution is proposed for the most complex task - placing 3D objects in a container. The results of the analysis of the effectiveness of the proposed algorithms are discussed.