The Problem is the Problem: How Mathematicians Pose and Select Problems

Before you can solve a problem, you must pose it. However, developing, posing, and choosing suitable problems is not an easy task. In fact, mathematicians considered it hard, complex - and of crucial importance for their work. Based on a qualitative interview study with research mathematicians I will in this talk address how mathematicians pose problems and describe the criteria they use to pick out the most suitable once. Among the criteria are personal interest, continuity with previous work, the danger of getting stuck, and the anticipated reactions from fellow mathematicians. Based on these results I will discuss to what extend knowledge about research mathematicians’ practice in problem choice can be used to inform educational practices at various levels.

Tilings as an Open Problem Field for Mathematically Gifted Children.

In this talk we want to give some ideas how to develop an open problem field for gifted students within enrichment programs for high school students. We give an in depth look what students might discover about tilings. The talk is based on Bedenknecht et al. (MS) and Heuer & Sarikaya (2019), which was and is used for enrichment programs for mathematically gifted children. We are hereby in a tradition in Hamburg (cf. K. Kießwetter & H. Rehlich (2005, 2008) or Nolte & Pamperien (2006, 2017a,b)).

In the first part of the talk we reflect on general goals of the work in open problem fields and possible guidelines for students and teachers. Here it is important that the students look for follow-up questions and understand that mathematical research is not the application of given methods but an open endeavor. Key takeaways for the students are that they might need quite some time to find a solution and that it is fine if a problem remains open. On the other hand, the teachers need to understand that they are not a source of knowledge but rather a companion.

In the second part we give an overview of a possible open problem field: constructing tilings of the plane. Starting from Platonic tilings we then continue with Archimedean ones. Especially in architecture, many of these tilings found concrete realisations, for example in Islamic architecture. Later we analyse Symmetriezeichnungen by Escher and motivate the students to generate their own ones. A further topic follows a common path for generalising results and for finding new problems: the variation of dimension. Following this path, we raise the problem of how to find and construct Platonic and Archimedean solids. Finally, for the last and more complex topic, we will show how the introduction of the notion of ‘periodicity’ gives raise to the question whether ‘aperiodic’ or even ‘quasi-periodic’ tilings of the plane exist. The most prominent example of the last ones are probably Penrose tilings, which found concrete realisation within crystallography in the form of quasicrystals. In total we shall see how quickly we get from a high school problem to open mathematical research, which mostly can be tackled by elementary means.

Do Mathematicians Agree about Mathematical Beauty?

Mathematicians often use aesthetic terms to evaluate mathematical objects. But do they share a consensus on mathematical beauty? If so, what predicts their aesthetic agreement? We used comparative judgement to measure aesthetic intuition among British mathematicians, Chinese mathematicians, and British undergraduates, and found aesthetic agreement both within and across these demographic groups. This agreement was found across mathematicians’ judgements of equations and proofs. Our findings thus go against other recent empirical works that found disagreement in mathematicians’ absolute judgements of beauty.

Value Judgments in Mathematics: G. H. Hardy and the (Non-)Seriousness of Mathematical Theorems

In his famous essay A Mathematician’s Apology (1940), the English mathematician G. H. Hardy provides a justification of a serious study of mathematics, arguing in particular that mathematics must be justified as a creative art. In general, the criteria by which the patterns of a mathematician must be judged are beauty and seriousness. In my talk I will concentrate on the latter notion and will discuss one of Hardy’s examples of non-serious mathematics, namely a theorem about reverse multiples that is (allegedly) not very appealing to a mathematician. This is of particular interest, because more than a dozen papers—including one by the renowned mathematician Neil Sloane—have been published on the general phenomenon of reverse multiples since the publication of the Apology. (A reverse multiple is a number which gives a multiple of itself when its digits are reversed, such as 2178, since 4⋅2178=8712.)

The Lure of the Funny Octahedron

In prior work, I’ve focused on the role aesthetics plays for mathematicians (and students) in devising and choosing problems. In this talk, I will consider a specific type of problem-posing practice in mathematics, which is less concerned with solutions than with bringing into existence and understanding the behaviour of mathematical objects—of turning things into concepts. I will draw on Whitehead’s notion of prehension (or feeling), to discuss some implications both for the epistemology and ontology of mathematics.

Quantitative Studies of the Role of Conjectures in Framing, Motivating, and Choosing Mathematical Problems

The mathematical field of combinatorics is often described as a ‘problem solving’ field (Gowers), and many of the problems arise from sharing conjectures. In this talk, the second author will present an open research question, give some of the background, and present preliminary quantitative findings made on the road to characterizing the roles of conjectures in developing mathematical problems and in motivating results in the mathematical literature. The affordances and constraints of the corpora for such studies of mathematical practice will also be addressed.