Tests are an essential tool to assess students’ ability. In online education these tests are mostly of static nature with the same items for each student. In contrast computerized adaptive testing concepts take into account the information about the test user automatically collected in an online test. The aim is a comparably precise test result with fewer test items (questions). An implementation of such a computerized adaptive test (CAT) is presented here. The adaptation process is based on the precise knowledge of the item parameters, e.g. difficulty, in the item pool. An estimation of the knowledge level of the test user has to be performed in real time after each answer. With this information the next item can be selected accordingly. This leads to a highly individualized test for each test user. For all items the parameters were determined with methods of the item response theory (IRT) in the framework of the probabilistic test theory. For that real test results of former first year students in engineering science had been analyzed. The prototype of such a CAT has been developed. It focusses on a physics test for prospective students in the STEM fields. In fall 2021 the pilot phase was conducted with first year students in engineering science. The CAT shows that the same precision can be achieved with a mean of 9.3 items compared to 12 in the static test. The acceptance among the students is high. The correlation between the static test and the CAT is satisfactory.
No abstract
Given a collection of hypergraphs $\fH=(H_1, \ldots, H_m)$ with the same vertex set, an $m$-edge graph $F\subset \cup_{i\in [m]}H_i$ is a transversal if there is a bijection $\phi:E(F)\to [m]$ such that $e\in E(H_{\phi(e)})$ for each $e\in E(F)$. How large does the minimum degree of each $H_i$ need to be so that $\fH$ necessarily contains a copy of $F$ that is a transversal? Each $H_i$ in the collection could be the same hypergraph, hence the minimum degree of each $H_i$ needs to be large enough to ensure that $F\subseteq H_i$. Since its general introduction by Joos and Kim~[Bull. Lond. Math. Soc., 2020, 52(3): 498–504], a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of $rn$ graphs on an $n$-vertex set, each with minimum degree at least $(r/(r+1)+o(1))n$, contains a transversal copy of the $r$-th power of a Hamilton cycle. This can be viewed as a rainbow version of the P\‘osa-Seymour conjecture.
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