Magister Educationis - MEd / This study analysed the modelling competencies of grade 11 learners and also explored the degree to which the learners’ competency in setting up a mathematical model inhibits the development of an acceptable solution for word problems. The research data comprised 30 learners drawn from a secondary school in the Western Cape Province, South Africa. Data was collected via a task-based activity response sheet containing five word problems linked to either one of the following concepts: rectangle, two-digit number, average speed and petrol price. Learners’ responses were graded into four categories viz: correct, partially correct, incorrect and no response. Thereafter, the modelling competency framework was used to diagnose the modelling competencies of the sampled learners.

Magister Educationis - MEd / Mathematical modelling is gaining extensive interest across the schooling sector worldwide, as it is deemed to develop learners with competencies set to deal with the demands of the fourth industrial revolution and being creative problem solvers. As mathematical modelling has only recently gained momentum across the mathematics curricula for schools in South Africa, many teachers may not be aware of the competencies that are needed to be developed in their learners through solving word problems, and even learners may not be aware of these essential modelling competencies. Hence, this mixed-methods approach study adopted a case-study design located within an interpretative paradigm to explore the levels of mathematical modelling competencies a sample of Grade 10 learners attending a Western Cape School demonstrated as they solved a set of word problems associated with the use of simultaneous equations. Additionally, data collected through observations and limited sets of semi-structured interviews were considered in the data analysis processes, which were largely driven by qualitative content analysis methods and supplemented with elementary descriptive statistical methods. The findings of this study showed that most of the learners demonstrated non-competency in modelling mainly because of their inability to understand the problem as evident in their failure to comprehend the context of a problem, inability to recognise important quantities associated with a problem, and muddled relationships if any. The study conjecture that the use of the English language could have been a barrier to the sample of English second language speakers understanding the problem. However, a very limited number of students showed partial modelling competency, as they were only able to understand the problem and build a correct model to solve the problem. Regrettably, these students lacked the knowledge of the heuristics for solving a system of linear equations correctly and completely and did not check or verify their answers. The extremely small number of learners, who demonstrated sufficient modelling competency, demonstrated sufficient understanding of the problem, built and solved the system of simultaneous linear equations successfully without necessarily checking or testing whether their answers satisfied the conditions of the problem. Hence, this study recommends that adequate focus be given to the role of language in understanding a problem, heuristic competencies to solve a system of linear equations should be strengthened, and the habit of checking the reasonableness of the solution should be encouraged and developed continuously across problem-solving tasks. Studying learners’ modelling competencies requires further work to add to the repertoire of this knowledge domain.

Der Umgang mit Modellen gehört zu einem durch Naturwissenschaften und Technik geprägten Alltag. Zur Förderung der Modellkompetenz von Schülerinnen und Schülern beschreibt ein theoriegeleitet formuliertes Kompetenzmodell, in welche Inhalte Modellkompetenz strukturiert ist und wie diese Inhalte graduiert sind. Diese Beschreibung von Modellkompetenz wurde in der vorliegenden Arbeit empirisch geprüft. Hierzu wird ein Itempool benötigt, der das Kompetenzmodell adäquat repräsentiert und dessen Bearbeitung als Indikator von Modellkompetenz interpretierbar ist. Beide Kriterien wurden neben psychometrischen Qualitätskriterien im Prozess der Testkonstruktion geprüft. 40 Multiple-Choice Items, die diesen Kriterien genügten, wurden von Schülerinnen und Schülern der Jahrgangsstufen 7 bis 10 bearbeitet. Eine Analyse dieser Daten mit mit IRT-Modellen diente der empirischen Überprüfung der Strukturierung und Graduierung von Modellkompetenz. Darüber hinaus wurden die Unterschiede in den Personenfähigkeiten zwischen den Jahrgangsstufen und die diskriminante und konvergente Validität von Modellkompetenz betrachtet. Es zeigte sich, dass die Indizien für eine eindimensionale Struktur von Modellkompetenz überwiegen. Die Niveaus sowie die Personenfähigkeiten in den untersuchten Jahrgangsstufen unterscheiden sich empirisch signifikant voneinander. Während sich Modellkompetenz in einer diskriminanten Validierung von allgemeinen kognitiven Fähigkeiten und Lesefähigkeiten sinnvoll abgrenzen lässt, bleibt das Bild für die konvergente Validierung noch unklar. Diese Befunde werden vor dem Hintergrund der theoretischen Grundlagen sowie mit Blick auf Implikationen für den Unterricht und weitere Forschungsprojekte diskutiert. / Science and technology belong to everyday life. It therefore requires the ability of dealing with models. To foster students’ model competence, a theoretical cognitive model structures and grades different aspects of model competence. The present research project aims at testing this description of model competence empirically. The empirical testing of a theoretical cognitive model requires psychometrically proven items that represent the cognitive model adequately. Furthermore, the results of the test should be interpretable as indication of the competence tested. These critera – among psychometric criteria – were included in the test and item construction which led to 40 multiple-choice items. Students in grades 7 to 10 dealt with this item pool. The data were analyzed with IRT models to check the structure and graduation of model competence empirically. Moreover, the differences in skills among grades as well as the discriminant and convergent validity of model competence were considered. Most of the empirical evidence points to a one-dimensional structure model of competence. The levels as well as the skills in the tested grades differed significantly. While model competence was distinguishable from general cognitive ability and reading skills in terms of discriminant validation, the picture remains unclear for the convergent validation. These findings are discussed with regard to the background of the theoretical basis as well as to implications for teaching and further research.