The transition from school to tertiary mathematics courses, which involve advanced mathematics, is a challenge for many students. Prior research has established the central role of prior mathematical knowledge for successfully dealing with challenges in learning processes during the study entrance phase. However, beyond knowing that more prior knowledge is beneficial for study success, especially passing courses, it is not yet known how a level of prior knowledge can be characterized that is sufficient for a successful start into a mathematics program. The aim of this contribution is to specify the appropriate level of mathematical knowledge that predicts study success in the first semester. Based on theoretical analysis of the demands in tertiary mathematics courses, we develop a mathematical test with 17 items in the domain of Analysis. Thereby, we focus on different levels of conceptual understanding by linking between different (in)formal representation formats and different levels of mathematical argumentations. The empirical results are based on a re-analysis of five studies in which in sum 1553 students of bachelor mathematics and mathematics teacher education programs deal with some of these items in each case. By identifying four levels of knowledge, we indicate that linking multiple representations is an important skill at the study entrance phase. With these levels of knowledge, it might be possible to identify students at risk of failing. So, the findings could contribute to more precise study advice and support before and while studying advanced mathematics at university.

中文翻译：

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在大学学习入学阶段，成功学习需要哪些先验数学知识？基于对现有研究数据的重新分析，得出了一种新的知识水平模型

从学校到高等数学课程的过渡涉及许多高级数学，这对许多学生来说都是一个挑战。先验研究已经确立了先验数学知识对于成功应对入学阶段学习过程中挑战的核心作用。但是，除了知道更多的先验知识对学习成功尤其是通过课程学习有益之外，还不知道如何表征足以成功进入数学课程的先验知识水平。这项贡献的目的是指定适当的数学知识水平，以预测第一学期的学习成功。基于对高等数学课程需求的理论分析，我们在分析领域开发了包含17个项目的数学测试。从而，我们通过将不同的（形式上）形式表示形式与不同水平的数学论证联系起来，专注于不同层次的概念理解。实证结果基于对五项研究的重新分析，在这些研究中，总共有1553名学士数学和数学师范教育课程的学生在每种情况下都处理了其中一些项目。通过确定四个知识水平，我们表明在研究进入阶段，链接多种表示形式是一项重要技能。有了这些知识水平，就有可能确定有失败风险的学生。因此，这些发现可能有助于在大学学习高级数学之前和期间提供更精确的学习建议和支持。