Jurie Conradie

Comprehension Tests in Mathematics.

An alternative way for testing a students understanding of theory in a tertiary mathematics course is presented. Two sample questions are provided and advantages and disadvantages of the method are discussed. We argue that the method is an acceptable and flexible means of testing students, and can be adapted to be used in other contexts as well.

A Developmental Approach To Mathematics Testing for University Admissions and Course Placement.

Decisions on admissions to university and placement into university courses are usually based on the results of achievement (as in secondary school exams) and/or aptitude (in intelligence-type tests and SAT). This paper argues that in a situation where educational provision at secondary school level is highly unequal, a third approach to testing offers an alternative which is preferable both on grounds of theory of cognitive psychology and because it yields much better discrimination.The Alternative Admissions Research Project at University of Cape Town has developed a mathematics test according to the dynamic testing approach as advocated by Miller (1990) for admission of African students from grossly under-resourced schools, as well as for placing these and other students into a diversifying first year curriculum. This approach aims to assess the ability of a candidate to learn from authentic academic material within the test. This paper focuses on the reasons for the development of the mathematics test and the process by which the test questions were developed and piloted. The reliability of the test and correlations of this test with subsequent mathematical performance data are discussed.Following the encouraging data for the test as an admission mechanism, the value of the dynamic testing approach for furnishing additional information for placement into an increasingly varied curriculum at first year level was investigated. This enabled the piloting of more topics and more comprehensive validation of this type of testing. The paper concerns itself with the reliability and predictive value of each of the topics in this placement test for a range of core courses in various faculties and the extent to which these tests can identify potentially at risk students who should be placed onto an appropriate curriculum.

The coarsest Hausdorff Lebesgue topology

If a Riesz space E contains an order dense Riesz subspace which admits a Hausdorff Lebesgue (i.e. order continuous) topology, then there is a coarsest Hausdorff Lebesgue topology on E. This topology extends uniquely to a Hausdorff Lebesgue topology on the universal completion of E, and is always minimal amongst Hausdorff locally solid topologies on E. Convergence in a Riesz space with a complete Lebesgue topology can be characterized in terms of this topology and order precompactness.

Duality results for order precompact sets in locally solid Riesz spaces

Abstract A duality result for order precompact sets in locally solid Riesz spaces is derived from a more general duality result, and it is shown that a number of known results on precompactness in Riesz spaces can be deduced from it. The same approach also yields duality results for order bounded linear maps between Riesz spaces.

Convergence and completeness in asymmetrically normed sequence lattices

Abstract If (X, ∣∣ · ∣∣) is a real normed lattice, then p(x) = ∣∣x + ∣∣ defines an asymmetric norm on X . We give suffcient conditions for (X, p) to be left-K-sequentially complete in the case where X is a normed sequence lattice and investigate the Smyth completeness of the positive cone of such lattices.

The Algebra of τ-measurable Operators

Properties of the measure topology on the algebra of τ-measurable operators affiliated to a semi-finite von Neumann algebra with a distinguished faithful semi-finite normal trace τ, in particular its local convexity and order continuity, are investigated.

A general duality result for precompact sets

Abstract In this paper we present an elementary proof of a general duality result for precompact sets which can be considered as a far-reaching generalization of a well-known result of Grothendieck on precompactness in dual systems. It is then shown that a number of known results can be deduced from it, amongst others a general form of the Arzela-Ascoli theorem and Grothendiecks duality theorem itself.