The Material Reasoning of Folding Paper

This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails

Reliability of mathematical inference

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. This is also a demand that is especially hard to fulfill, given the fragility and complexity of mathematical proof. This essay considers some of ways that mathematics supports reliable assessment, which is necessary to maintain the coherence and stability of the practice

Reliability of mathematical inference

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification

Using Diagrams to Understand Geometry

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73219/1/0824-7935.00062.pd

INSTRUCTIONAL SITUATIONS AND STUDENTS’ OPPORTUNITIES TO REASON IN THE HIGH SCHOOL GEOMETRY CLASS

We outline a theory of instructional exchanges and characterize a handful of instructional situations in high school geometry that frame some of these exchanges. In each of those instructional situations we inspect the possible role of reasoning and proof, drawing from data collected in intact classrooms as well as in instructional interventions.This manuscript is part of the final report of the NSF grant CAREER 0133619 “Reasoning in high school geometry classrooms: Understanding the practical logic underlying the teacher’s work” to the first author.All opinions are those of the authors and do not represent the views of the National Science Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78372/1/Instructional_Situations_in_Geometry.pd

Proofs, intuitions and diagrams : Kant and the mathematical method of proof

Hoe bewijs je idealiter een stelling uit de wiskunde? Volgens vele filosofen is de wiskundige bewijsmethode exclusief een zaak van de logica. Een wiskundige zou uitsluitend gevolgtrekkingen maken volgens algemeen geldende logische redeneerprincipes. Volgens de beroemde Duitse filosoof Immanuel Kant (1724-1804) is bewijsvoering in de wiskunde essenti?een zaak van mentale constructies. De producten van dergelijke constructies noemt Kant intuities. Voor zover het de meetkunde betreft, beargumenteer ik dat intuities het karakter hebben van een beeld, of een diagram. In Kants optiek bewijs je een stelling door bijvoorbeeld de constructie van een driehoek, waar vervolgens, middels verdere constructies, allerlei hulplijnen aan toegevoegd worden. Op een dergelijke wijze wordt, via een keten van voortschrijdende inzichten, de waarheid van een stelling vastgesteld. Kant volgt in zijn opvattingen nauwgezet de bewijzen uit elementaire meetkundeboeken. Dit maakt Kants visie op het eerste gezicht buitengewoon geloofwaardig. Wat Kants visie bijzonder maakt is dat hij wiskundige bewijsvoering vooral in termen van een specifiek wiskundige bewijsmethode ziet, en niet als een zaak van algemeen geldende logische redeneerprincipes. In mijn proefschrift geef ik een reconstructie van deze bewijsmethode. Kants bewijsconceptie is niet alleen om historische redenen van belang, al was het maar omdat zij tot op de dag van vandaag nog steeds niet goed begrepen is. Kants visie is ook van systematisch belang. Zo stelt Kant het vaak nauw geachte verband tussen logica en methodologie ter discussie. Ook stelt Kant fundamentele vragen bij de verhouding tussen redeneren en taal.Jong, W.R. de [Promotor

Signs as a Theme in the Philosophy of Mathematical Practice

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding

The Material Reasoning of Folding Paper

This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails

Instructional strategies in explicating the discovery function of proof for lower secondary school students

In this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr

Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

This chapter discusses some strands of experimental mathematics from both an epistemological and a didactical point of view. We introduce some ancient and recent historical examples in Western and Eastern cultures in order to illustrate how the use of mathematical tools has driven the genesis of many abstract mathematical concepts. We show how the interaction between concrete tools and abstract ideas introduces an "experimental" dimension in mathematics and a dynamic tension between the empirical nature of the activities with the tools and the deductive nature of the discipline. We then discuss how the heavy use of the new technology in mathematics teaching gives new dynamism to this dialectic, specifically through students' proving activities in digital electronic environments. Finally, we introduce some theoretical frameworks to examine and interpret students' thoughts and actions whilst the students work in such environments to explore problematic situations, formulate conjectures and logically prove them. The chapter is followed by a response by Jonathan Borwein and Judy-anne Osborn