In this paper, error analyses are used to reveal the nature of mathematical procedural knowledge. In particular, the aim is to empirically confirm the structure of procedural knowledge described in theory and to further differentiate the corresponding terms. In this context, the coding of the processing of procedural tasks by students at the end of their high school years (n = 455) shows that knowledge of the procedure and arithmetic/algebraic skills can be objectively reconstructed as components of procedural knowledge. The analyses show that procedural knowledge and corresponding difficulties become visible in many ways when working on relevant tasks and that anecdotal experiences about a lack of mathematical procedural knowledge among students must fall short. The didactic implication is that the support of procedural knowledge acquisition needs to be both individualised and task specific. The results also reveal interesting correlations between deficiencies and errors in the processing of procedural tasks and specific item characteristics. For example, deficiencies in knowledge of the procedure tend to occur more frequently in tasks at higher grade levels and in analysis, while specific errors in arithmetic/algebraic skills are more likely to be found in algebra and tasks that require more procedural steps.

The rapid advancements in artificial intelligence (AI) have sparked interest in its application within mathematics education, particularly in automating the coding and grading of student solutions. This study investigates the potential of ChatGPT, specifically the GPT-4 Turbo model, to assess student solutions to procedural mathematics tasks, focusing on its ability to identify correctness and categorize errors into two domains: “knowledge of the procedure” and “arithmetic/algebraic skills.” The research is motivated by the need to reduce the time-intensive nature of coding and grading and to explore AI's reliability in this context. The study employed a two-phase approach using a dataset of handwritten student solutions of a system of linear equations: first, ChatGPT was trained using student solutions that were rewritten by one of the authors to ensure consistency in handwriting style; its performance was then tested with additional solutions, also in the same handwriting. The findings reveal significant challenges, including frequent errors in handwriting recognition, misinterpretation of mathematical symbols, and inconsistencies in the categorization of mistakes. Despite iterative feedback and prompt adjustments, ChatGPT's performance remained inconsistent, with only partial success in accurately coding solutions. The study concludes that while ChatGPT shows promise as a coding aid, its current limitations—particularly in recognizing handwritten inputs and maintaining consistency—highlight the need for improvement. These findings contribute to the growing discourse on AI's role in education, emphasizing the importance of improving AI tools for practical classroom and research applications.

The recent implementation of a standardized school-leaving exam (SSLE) in Austria enables research on the effects of this educational policy decision on mathematics teaching (use of tasks and digital tools) and on mathematics teachers (attitudes, cooperation, satisfaction and self-concept). An interview study with 10 teachers, each of whom prepared students several times for the traditional, individually designed as well as for the new, standardized school-leaving exam, reveals a multi-faceted picture. In particular, it shows that the SSLE as a steering instrument has great influence on the use of tasks in the classroom and has been able to advance the digitization of mathematics teaching within a short period of time. Cooperation among the teachers in the respective schools has clearly increased as a result of the SSLE, even though this is not perceived as unreservedly positive. Interestingly, the interviews did not reveal any evidence of harmful competition between teachers in a school regarding the performance of their students in the SSLE. While the study shows that teachers are basically positive about reforms and are willing to implement them, they also want changes to be well prepared, argued and communicated in time.

The number of complaints university lecturers make about a lack of knowledge, especially first-year students' procedural knowledge, has increased recently. Due to missing adequate empirical evidence, a survey of procedural knowledge among students of Austrian high schools in their final year was conducted. For this purpose, test items for procedural knowledge were created, validated and processed by a total of 455 students without technology and formula booklets. The test items were based on a theoretical model with the dimensions number of procedural steps, curricular grade level, content area and rating of importance. Linear models were used to describe the dependencies between the students' success rate and these dimensions. The results show that the level of procedural knowledge among the students is relatively low overall. A closer look reveals that tasks at lower secondary level have a significantly higher students' success rate than tasks at upper secondary level. Furthermore, there is a positive correlation between the students' success rate of a task and the experts' assessment of whether this task should be able to be solved by students without technological aids. Interestingly, the dimensions number of procedural steps and content area have no significant impact on the students' success rate.

The article at hand deals with students’ procedural knowledge, the frequency of technology use (CAS, graphics calculators) during mathematics education in upper secondary level and their self-assessed technology knowledge. In this study, the participating students (representative sample of Austrian high school students in the final year, n=455) had to solve procedural, curriculum-related tasks without any aids (neither technology nor formula booklets). We examined how the frequency of technology use in the classroom affects the students’ success rate on procedural tasks. On average, GeoGebra or graphic calculators with CAS are used once a week by the teacher and the students in class, respectively, and unexpectedly, there is no significant correlation between the frequency of technology use during mathematics education in upper secondary level and the procedural knowledge acquired. Regardless of the success in solving the procedural tasks, the students rate their technology knowledge for solving the procedural tasks as rather high.

En este artículo queremos compartir principalmente nuestras experiencias y reflexiones sobre un curso especial en la Universidad de Viena (véase el capítulo 2). El tema de este seminario era “modelización y problemas de modelización” y el objetivo era doble: Por un lado, los propios estudiantes de magisterio debían adquirir experiencia en el trabajo con un problema de modelización (en pequeños grupos de 3 a 5 personas), y por otro lado, debían obtener las primeras experiencias para crear problemas de modelización para los alumnos de la escuela (grado 8 - 11) y supervisar sus procesos de modelización durante una jornada de modelización en la escuela. Estas habilidades son cruciales para los futuros profesores si deben implementar la idea de “modelado” más adelante en su vida profesional como profesores en el aula. Para dejar clara nuestra opinión personal sobre la modelización y los problemas de modelización (¿Qué son los problemas de modelización propiamente dichos? ¿Cual es la diferencia entre los problemas de modelización y los problemas de palabras disfrazados? etc.) – que es necesario para entender bien la Sección 2 –, primero esbozamos nuestro correspondiente enfoque en la Sección 1.

Learners’ perspectives on classroom teaching have been explored in various ways. Within the context of past research, the student voice approach has shown the potential of making teaching and learning more effective. This article reports on students’ perceptions with the aim of revealing their predispositions in the sense of Bruner’s perceptual set theory concerning mathematics lessons. By asking students about significant events in specific mathematics lessons they attended and their associated justifications, insights into students’ predispositions (beliefs) about teaching and learning mathematics are gained. We were able to reconstruct a wide range of predispositions, which indicate both a view on mathematics education that is quite positive and as well a willingness to learn comprehensively. As a further result, we obtained information about which kinds of scenes are perceived as significant by students, which was then linked to their decisive predispositions. With our results in mind, teachers could react appropriately to the expectations of students related to these predispositions and thus avoid dissatisfaction and barriers to learning.

Im vorliegenden Artikel wird eine Modellierungsaufgabe präsentiert, die bei einer Modellierungswoche von Schüler/ innen gemeinsam mit Lehrenden bearbeitet wurde. Ziel dieser Aufgabe war es, aus realen Daten einer österreichischen Fluglinie ein so genanntes Overbooking- Management-System zu entwickeln. Ein solches System soll der Fluglinie zu jedem Zeitpunkt angeben, wie viele Flugtickets sie in den einzelnen Buchungsklassen anbieten soll. Wir werden den gut durchdachten und vor allem durchaus erfolgreichen Lösungsweg der Schüler/innen, welcher auf ein lineares Optimierungsproblem führt, verfolgen und Erweiterungsmöglichkeiten für dieses Problem vorstellen. Abschließend geben wir einen Kommentar aus fachdidaktischer Sicht.

Das durch die Deutsche Telekom Stiftung geförderte Projekt „Mathematik besser verstehen“ wurde in den Studienjahren 2009/10 und 2010/11 an der Universität Duisburg- Essen durchgeführt. Es wurden Maßnahmen entwickelt, die den Studierenden des Lehramts für Gymnasien, Gesamtschulen und Berufskollegs den Einstieg in das erste Studienjahr erleichtern sollten. Diese Maßnahmen wurden begleitend zu den Vorlesungen Analysis und Lineare Algebra angeboten, um möglichst wenig Einfluss auf den bestehenden Lehrbetrieb zu nehmen. Das Design des Projektes, einige ausgewählte Materialien sowie Erfahrungen aus der Projektarbeit werden im Artikel vorgestellt.

Ausführliche Lösung 6.1. Wir beweisen mit vollständiger Induktion, da es sich hier- K bei umeine Aussage über die natürlichen Zahlen handelt!.

Anspruch an die Musterlösungen. Im Vergleich zu den üblicherweise im Lehrbetrieb oder in Lehrbüchern eingesetzten komprimierten Musterlösungen möchten die ausführlichen Musterlösungen in diesem Buch zusätzliche Informationen bereitstellen, die für den Lernprozess wichtig sein können. Wir möchten nicht bloß die Lösung der Aufgabe darstellen, sondern auch ein paar Worte darüber verlieren, was eigentlich das Interessante an der Aufgabe ist. Wir möchten klären, inwiefern es gewinnbringend sein kann, sich mit der Aufgabe zu beschäftigen. Außerdem wird auch auf das Entwickeln und Ausformulieren von Lösungsideen Wert gelegt. Wir versuchen zu motivieren, wie man auf eine Lösungsstrategie überhaupt kommt. Manchmal passiert es beim Aufgabenlösen, dass man an irgendeiner Stelle feststeckt und es auf eine andere Art und Weise versuchen muss. Auch solche Situationen haben wir in unseren Musterlösungen abgebildet, sofern der Ablauf dadurch nicht verwirrend wird und zu sehr vom funktionierenden Lösungsweg wegführt. Jedenfalls werden aber die entscheidenden Punkte in der Lösung hervorgehoben und detailliert beschrieben. Nach Möglichkeit sind alle Argumentationsschritte aufgeführt, sodass Sie dem Lösungsstrang in aller Regel gut folgen können sollten.

Ausführliche Lösung 5.1. Wir sollen in dieser Aufgabe feststellen, ob sich die Funk- B tionen in (0, 0) so definieren lassen, dass sie stetig werden. Auf ℝ2 \ {(0, 0)} sind die beiden Funktionen als Kompositionen stetiger Funktionen offensichtlich stetig.