Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

The creative process is a holistic view of the DI learning experience and is the educational focus of the Destination Imagination Program. Teams begin by imagining the possibilities and throughout the process work to develop an innovation, the solution to their selected Challenge. In a nutshell, the creative process identifies the elements involved in solving a Challenge as a team. The creative process is not a checklist of steps for teams to go through, but an identification of the skills developed through solving challenges. It is important to note that the creative process is not linear. Teams will find themselves returning to various stages of development while developing a solution. Our goal at Destination Imagination is to give students the chance to learn and experience the creative process from imagination to innovation and have fun doing so. The creative process is about thinking and doing in no prescribed order. Some people “do” first and then think about what they have done, while others “think” first, then initiate action. Below are the components of the creative process that our participants experience while solving our Challenges. From imaginaTion To innovaTion

On the creative process

We continue the study of the reproducibility of Propp’s annotations from Bod et al. (2012). We present four experiments in which test subjects were taught Propp’s annotation system; we conclude that Propp’s system needs a significant amount of training, but that with sufficient time investment, it can be reliably trained for simple tales.

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Annotating with Propp’s Morphology of the Folktale: reproducibility and trainability

We present a technique called event mapping that allows to project text representations into event lists , produce an event table , and derive quantitative conclusions to compare the text representations. The main application of the technique is the case where two classes of text representations have been collected in two di ff erent settings (e.g., as annotations in two di ff erent formal frameworks) and we can compare the two classes with respect to their systematic di ff erences in the event table. We illustrate how the technique works by applying it to data collected in two experiments (one using annotations in Vladimir Propp’s framework, the other using natural language summaries).

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Event mappings for comparing formal frameworks for narratives

Mathematical Practices Can Be Metaphysically Laden

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization.

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Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society

/pdf/reflections-on-the-foundations-of-mathematics-univalent-xmtn4jvbao.pdf

Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts

The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case study on extremal proofs. Based on this case study, we compare our analysis of proof understanding to Avigad’s ability-based analysis of proof understanding.

/pdf/how-to-frame-understanding-in-mathematics-a-case-study-using-4275p9xbdy.pdf

How to Frame Understanding in Mathematics: A Case Study Using Extremal Proofs

Frames are a concept in knowledge representation that explains how the receiver, using background information, completes the information conveyed by the sender. This concept is used in different disciplines, most notably in cognitive linguistics and artificial intelligence. This paper argues that frames can serve as the basis for describing mathematical proofs. The usefulness of the concept is illustrated by giving a partial formalisation of proof frames, specifically focusing on induction proofs, and relevant parts of the mathematical theory within which the proofs are conducted; for the latter, we look at natural numbers and trees specifically.

How to Frame a Mathematician

Interested in formally modelling similarity between narratives, we
investigate judgements of similarity between narratives in a small
corpus of film reviews and book–film comparisons. A main finding is
that judgements tend to concern multiple levels of story
representation at once. As these texts are pragmatically related to
reception contexts, we find many references to reception quality and
optimality. We conclude that current formal models of narrative can
not capture the task of naturalistic narrative comparisons given in
the analysed reviews, but that the development of models containing a
more reception-oriented point of view will be necessary.

https://drops.dagstuhl.de/opus/volltexte/2013/4144/pdf/p094-fisseni.pdf/

Deniz Sarikaya

Papers

Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society

Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts

How to Frame Understanding in Mathematics: A Case Study Using Extremal Proofs

How to Frame a Mathematician

Story Comparisons: Evidence from Film Reviews