No abstract

The COVID-19 pandemic has confronted mathematics teachers with the challenge of developing alternative teaching practices—in many cases at a distance through digital technology—because schools were closed. To investigate what distance practices in secondary mathematics education have emerged and how teachers experienced them, we set out online questionnaires in Flanders—the Dutch-speaking part of Belgium—, Germany, and the Netherlands. The questionnaire focused on teaching practices, teacher beliefs, didactics, and assessment. Data consisted of completed questionnaires by 1719 mathematics teachers. Results show that the use of video conferencing tools increased massively, while the use of mathematics-specific tools that teachers used before the lockdown reduced substantially. Further findings are that teachers' confidence in using digital technologies increased remarkably during the lockdown and that their experiences and beliefs only marginally impacted their distance learning practices. Also, we observed some differences between the three countries that might be explained by differences in educational policies and in technological facilities and support. For future research, it would be relevant to investigate long-term changes in teachers’ practices, as well as students’ views and experiences related to the teacher’s practices.

The Stahel-Donoho estimator is defined as a weighted mean and covariance, where the weight of each observation depends on a measure of its outlyingness. In high dimensions, it can easily happen that an amount of outlying measurements is present in such a way that the majority of the observations is contaminated in at least one of its components. In these situations, the Stahel-Donoho estimator has difficulties in identifying the actual outlyingness of the contaminated observations. An adaptation of the Stahel-Donoho estimator is presented where the data are huberized before the outlyingness is computed. It is shown that the huberized outlyingness better reflects the actual outlyingness of each observation towards the non-contaminated observations. Therefore, the resulting adapted Stahel-Donoho estimator can better withstand large amounts of outliers. It is demonstrated that the Stahel-Donoho estimator based on huberized outlyingness works especially well when the data are heavily contaminated.

Secondary school teaching of evolution through natural selection is very important because for most people, it is the only formal introduction to the scientific understanding of this theory. However, there are major concerns over its unsatisfactory teaching. In several European countries, including the Flanders region in Belgium, natural selection is treated as a side-topic that is referred to only after all other biological content has been covered. It has been suggested that improved understanding can be achieved by teaching it in a more integrated manner throughout the biology curriculum, as is largely the case in the Netherlands. We tested this hypothesis by a standardized comparison of the understanding of natural selection between university freshmen who had completed high level biology secondary education in Flanders or the Netherlands. We used the Conceptual Inventory of Natural Selection (CINS), designed to measure the understanding of 10 underlying key concepts (KC), including four core concepts (CC), and the magnitude of alternative conceptions. Regression analysis was used to control for potentially confounding student parameters. Dutch graduates indeed obtained a significantly higher CINS-score than Flemish graduates. They also scored significantly higher on eight key concepts. The 10 KC were employed to varying degrees, with the relative rank being highly comparable between both student populations, and the CC origin of variation and variation inheritable, both linked to genetics, being more challenging than the CC existence of variation and differential survival. The relative frequency of alternative conceptions elicited by the CINS was almost identical in both student populations.

The Stahel-Donoho estimator is defined as a weighted mean and covariance, where each observation receives a weight which depends on a measure of its outlyingness. Therefore, all variables are treated in the same way whether they are responsible for the outlyingness or not. We present an adaptation of the Stahel-Donoho estimator, where we allow separate weights for each variable. By using cellwise weights, we aim to only downweight the contaminated variables such that we avoid losing the information contained in the other variables. The goal is to increase the precision and possibly the robustness, of the estimator. We compare several variants of our proposal and show to what extent they succeed in identifying and downweighting precisely those variables which are contaminated. We further demonstrate that in many situations the mean-squared error of the adapted estimators is lower than that of the original Stahel-Donoho estimator and that this results in better outlier detection capabilities. We also consider some real data examples