@article{SillerElschenbroichGreefrathetal.2023,
  author    = {Siller, Hans-Stefan and Elschenbroich, Hans-J{\"u}rgen and Greefrath, Gilbert and Vorh{\"o}lter, Katrin},
  title     = {Mathematical modelling of exponential growth as a rich learning environment for mathematics classrooms},
  series = {ZDM Mathematics Education},
  volume    = {55},
  journal   = {ZDM Mathematics Education},
  number    = {1},
  issn      = {1863-9690},
  doi       = {10.1007/s11858-022-01433-8},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324393},
  pages     = {17-33},
  year      = {2023},
  abstract  = {Mathematical concepts are regularly used in media reports concerning the Covid-19 pandemic. These include growth models, which attempt to explain or predict the effectiveness of interventions and developments, as well as the reproductive factor. Our contribution has the aim of showing that basic mental models about exponential growth are important for understanding media reports of Covid-19. Furthermore, we highlight how the coronavirus pandemic can be used as a context in mathematics classrooms to help students understand that they can and should question media reports on their own, using their mathematical knowledge. Therefore, we first present the role of mathematical modelling in achieving these goals in general. The same relevance applies to the necessary basic mental models of exponential growth. Following this description, based on three topics, namely, investigating the type of growth, questioning given course models, and determining exponential factors at different times, we show how the presented theoretical aspects manifest themselves in teaching examples when students are given the task of reflecting critically on existing media reports. Finally, the value of the three topics regarding the intended goals is discussed and conclusions concerning the possibilities and limits of their use in schools are drawn.},
  language  = {en}
}
@article{GreefrathOldenburgSilleretal.2023,
  author    = {Greefrath, Gilbert and Oldenburg, Reinhard and Siller, Hans-Stefan and Ulm, Volker and Weigand, Hans-Georg},
  title     = {Mathematics students' characteristics of basic mental models of the derivative},
  series = {Journal f{\"u}r Mathematik-Didaktik},
  volume    = {44},
  journal   = {Journal f{\"u}r Mathematik-Didaktik},
  number    = {1},
  issn      = {0173-5322},
  doi       = {10.1007/s13138-022-00207-9},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324317},
  pages     = {143-169},
  year      = {2023},
  abstract  = {The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students' characteristics of basic mental models is presented and analysed regarding quality criteria. Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students' previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students' basic mental models of the derivative are presented and discussed in detail.},
  language  = {en}
}