Diagnostic testing in mathematics: paired questions

Diagnostic testing in mathematics has become a favoured toolat many universities. In this paper we report on a novel methodof diagnostic testing which incorporates a paired question approach,with two questions being set on each of 20 different topics.It was introduced at Loughborough University for use with firstyear engineering students in 2002. The structure was chosento allow easy identification, and subsequent follow up, of topicswhere the students needed extra help. However, it was foundthat many students got one question of a pair correct and oneincorrect. This paper analyses the paired questions in detailand suggests why this happened. It also comments on the validityof the test used. Finally, it provides recommendations for practionersin this field who are considering adopting the paired questionapproach and also for those involved in computer-assisted assessment.     

Interviewing in mathematics education research: Choosing the questions

With the development of qualitative methodologies, interviewing has become one of the main tools in mathematics education research. As the first step in analyzing interviewing in mathematics education we focus here on the stage of planning, specifically, on designing the interview questions. We attempt to outline several features of interview questions and understand what guides researchers in choosing the interview questions. Our observations and conclusions are based on examining research in mathematics education that uses interviews as a data-collection tool and on interviews with practicing researchers reflecting on their practice.     

Aporism: Uncertainty about mathematics

Neither absolutism nor aposteriorism have questioned the progressive elements associated with the applications and the social functions of mathematical knowledge. Aporism raises this question by discussing the thesis of the formatting power of mathematics. This thesis unites linguistic relativism applied to mathematics and the idea that technology is a structuring principle in society. We are no longer surrounded by “nature”, instead we live in a techno-nature. Mathematical abstractions can be projected outside the sphere of mathematics, and in this way they modulate and eventually constitute fundamental categories of techno-nature. The Vico paradox expresses the difficulties of specifying the nature and function of technological actions. We are not even able to grasp and to understand what we have ourselves constructed. A critique cannot be guaranteed by scientific (or mathematical) thinking itself. Critique becomes a much more complex activity including reflections on technological actions. A crifique includes ethical considerations, and therefore a critique of mathematics is also ethical.     

How much time do students have to think about teacher questions? An investigation of the quick succession of teacher questions and student responses in the German mathematics classroom

Several studies have shown that the style of the German mathematics classroom at secondary level is mostly based on the so called “fragend-entwickelnde” teaching style which means developing the lesson content by a teacher directed sequence of teacher questions and student responses. In this article we describe a study on the time the students have for thinking about a teacher question in the public classroom interaction. Our investigation is based on a reanalyasis of 22 geometry lessons from grade 8 classes which mainly deal with a challenging proving content. The results show that the average time between a teacher question and a student response is 2.5 seconds. There are no remarkable differences between different phases of the lessons like comparing homework, repetition of content or working on new content. Moreover, for 75% of the teacher questions the first student was called to answer within a three second time interval.     

On two questions about circular choosability

We answer two questions of Zhu on circular choosability of graphs. We show that the circular list chromatic number of an even cycle is equal to 2 and give an example of a graph for which the infimum in the definition of the circular list chromatic number is not attained. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:261‐269, 2008     

On some questions about selective separability

CH implies that selective separability is not preserved by finite powers (solving [3, Problems 3.7 and 3.9]). In ZFC, selective separability does not imply H‐separability (solving [4, Problem 34]) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some open questions about variational inequalities

We discuss some open questions especially concerning the coincidence set and the curve of separation in two-dimensional variational inequalities.     

On two questions about feasibly constructive arithmetic

IPV is the intuitionistic theory axiomatized by Cook's equational theory PV plus PIND on NP‐formulas. Two extensions of IPV were introduced by Buss and by Cook and Urquhart by adding PIND for formulas of the form A(x) ∨ B, respectively ¬¬A(x), where A(x) is NP and x is not free in B. Cook and Urquhart posed the question of whether these extensions are proper. We show that in each of the two cases the extension is proper unless the polynomial hierarchy collapses.     

Solutions to two questions about the Weyl algebras

Affirmative answers are given to the following two questions about the Weyl algebras: a question of J. Alev: Does the first Weyl algebra contain a non-noetherian subalgebra?, and a question of T. Lenagan: Is there a uniserial module of length over the Weyl algebra with a holonomic submodule such that is non-holonomic?

     

Some heuristics about the ecosystem of mathematics research data

Like in other sciences, research data play a growing role in mathematics, but in contrast to classical objects like measurements in physics they are much more heterogeneous. They may take the shape of abstract objects like the collection of integer sequences in OEIS, algorithms and their implementations as mathematical software, libraries of test problems or statistical data. From an infrastructure viewpoint, which aims at sustainable and connected data repositories which facilitate researchers to use existing information efficiently, it is essential to define an appropriate framework that allows not just storage but also connection and retrieval of the various types of data. Recently, there have been promising attempts to define standards for mathematical software, but the general task remains a big challenge, which is also addressed within the recently initiated GDML working group of the IMU. This is especially important in the fields of applied mathematics where research is often connected to research data originating from applications. The goal of this talk is a first attempt to analyse the diverse ecosystem of research data based on reference data from zbMATH. This approach has worked quite well for mathematical software, resulting in the formation of the swMATH database. Though reference data involve always a bias, the collected information of about 16 million reference data in zbMATH may be useful to identify the recent needs of researchers in different fields pertaining mathematical research data, and we discuss several aspects of such an analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

What environmental studies students think about mathematics and what could be done about it

Mathematics is playing an increasing role in environmental studies, which causes problems for many students because of their low mathematical ability. A survey among first year environmental studies students revealed a significant number both of negative experiences in learning mathematics and negative feelings towards the subject. A proposal to design an alternative curriculum which aims to link the maths more closely with environmental studies is outlined as an answer to the problem of negative attitudes to the required mathematics.     

Some questions about the dimension of a group action

This article discusses three families of groups: Z Zⁿ, PL(Iⁿ),and PL(Sⁿ) (the last two being the families of groups of piecewise-linearhomeomorphisms of standard n-dimensional spaces). It is shownthat for positive n N, Z Zⁿ embeds in PL(Iⁿ), which embedsin PL(Sⁿ). It is known that Z Z² fails to embed in PL(I¹),and this article extends that previous result to show that Z Z² also fails to embed in PL(S¹). The nature of the proofsof these embedding and non-embedding results hints that theremay be corresponding non-embedding results in higher dimensions.     

Cultivating inquiry about space in a middle school mathematics classroom

During 46 lessons in Euclidean geometry, sixth-grade students (ages 11, 12) were initiated in the mathematical practice of inquiry. Teachers supported inquiry by soliciting student questions and orienting students to related mathematical habits-of-mind such as generalizing, developing relations, and seeking invariants in light of change, to sustain investigations of their questions. When earlier and later phases of instruction were compared, student questions reflected an increasing disposition to seek generalization and to explore mathematical relations, forms of thinking valued by the discipline. Less prevalent were questions directed toward search for invariants in light of change. But when they were posed, questions about change tended to be oriented toward generalizing and establishing relations among mathematical objects and properties. As instruction proceeded, students developed an aesthetic that emphasized the value of questions oriented toward the collective pursuit of knowledge. Post-instructional interviews revealed that students experienced the forms of inquiry and investigation cultivated in the classroom as self-expressive.