Bridging knowing and proving in mathematics An essay from a didactical perspective

Text of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006International audienceThe learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory

Vignette of Doing Mathematics: A Meta-cognitive Tour of the Production of Some Elementary Mathematics

Mathematics educators, including some mathematicians, have, in various ways, urged that the school curriculum provide opportunities for learners to have some authentic experience of doing mathematics, opportunities to experience and develop the practices, dispositions, sensibilities, habits of mind characteristic of the generation of new mathematical knowledge and understanding – questioning, exploring, representing, conjecturing, consulting the literature, making connections, seeking proofs, proving, making aesthetic judgments, etc. (Polya 1954, Cuoco et al 2005, NCTM 2000 - Standard on Reasoning and Proof). While this inclination in curricular design has a certain appeal and merit, its curricular and instructional expressions are often contrived, or superficial, or no more than caricatures of what they are meant to emulate. One likely source of the difficulty is that most mathematics educators have little or no direct experience of doing a substantial piece of original mathematics, in part because the technical demands are often too far beyond the school curriculum. Studying the history and evolution of important mathematical developments can be helpful, but provides a less immediate and direct experience

Non-Classical Approaches to Logic and Quantification as a Means for Analysis of Classroom Argumentation and Proof in Mathematics Education Research

Background: While it is usually taken for granted that logic taught in the mathematics classroom should consist of elements of classical propositional or first-order predicate logic, the situation may differ when referring to students’ discursive productions. Objectives: The paper aims to highlight how classical logic cannot grasp some epistemic aspects, such as evolution over time, uncertainty, and quantification on blurred domains, because it is specifically tailored to capture the set-theoretic language and to validate, rather than to consider epistemic aspects. The aim is to show that adopting classical and non-classical lenses might lead to different results in analysis. Design: Nyaya pragmatic and empiricist logic, with Peircean non-standard quantification, both linked by the concept of free logic, are used as theoretical lenses in analysing two paradigmatic examples of classroom argumentation. Setting and Participants: excerpts from a set of data collected by prof. Paolo Boero from the University of Genoa during research activities in a secondary school mathematical class. Methodology: The examples are discussed by adopting a hermeneutic approach. Results: The analysis shows that different logical lenses can lead to varying interpretations of students’ behaviour in argumentation and presenting proof in mathematics and that the adopted non-classical lenses expand the range of possible explanations of students’ behaviour. Conclusion: In mathematics education research, the need to consider an epistemic dimension in the analysis of classroom argumentation and proof production leads to the necessity to consider and combine logical tools in a way specific to the discipline, which might differ from those usually required in mathematics

An Introduction to Ricci Flow for Two-Dimensional Manifolds

The study of diferentiable manifolds is a deep an extensive area of mathematics. A technique such as the study of the Ricci flow turns out to be a very useful tool in this regard. This flow is an evolution of a Riemannian metric driven by a parabolic type of partial differential equation. It has attracted great interest recently due to its important achievements in geometry such as Perelman\u27s proof of the geometrization conjecture and Brendle-Schoen\u27s proof of the differentiable sphere theorem. It is the purpose here to give a comprehensive introduction to the Ricci flow on manifolds of dimension two which can be done in a reasonable fashion when the Euler characteristic is negative or zero. A brief introduction will be given to the case in which the Euler characteristic is positive

Keterampilan Berpikir Kritis Siswa SMP dalam Memecahkan Masalah Matematika Kontekstual Ditinjau dari Kemampuan Matematika dan Perbedaan Jenis Kelamin

This research aims to describe the critical thinking skills of junior high school students in solving contextual math problems in terms of mathematical ability and gender differences. The type of research used is descriptive qualitative research. The subjects in this research were 1 male and 1 female student with high mathematics ability, 1 male and 1 female student with moderate mathematics ability, and 1 male and 1 female student with low mathematics ability.Data were collected using test and interview techniques. The instruments used were Mathematics Ability Test (TKM), Problem Solving Test (TPM), and interview guidelines. Based on the results of the research, it can be concluded that the critical thinking skills of (1) male and female students with high mathematical ability met the indicators of interpretation, analysis, evolution (on argument proof, because in argument assessment only male students met the sub-indicator), inference, and explanation. Male students did not fulfill the indicators of self-regulation, while female students did. (2) Male and female students with moderate mathematics ability met the indicators of interpretation, inference, and explanation. Male students did not fulfill the indicators of analysis and self-regulation, while female students did. However, both did not fulfill the evaluation indicator. (3) Male and female students with low mathematics ability have many differences in critical thinking skills. Male students did not fulfill the indicators of interpretation, analysis, explanation, and evaluation. However, the self-regulation indicator is fulfilled. While female students fulfill the indicators of interpretation and analysis. Female students did not fulfill the indicators of evaluation, explanation, and self-regulation

Proving skills in geometry of secondary grammar school leavers specialized in mathematics

We examined the evolution of the van Hiele level of some study groups specialized in mathematics from 2015 to 2018, then selected two of these groups and measured the students’ proof skills by Zalman Usiskin’s proof test. We examined whether students were able to read from the text of the statement the given fact and the fact to be proved, whether they were able to draw a figure and, using the labels, whether they were able to perform a simple proof requiring 2-3 steps