No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
The Nature of Proof
Abstract
I should like to talk about certain changes that are coming in what might be called the ‘philosophy of mathematics’. Part of these changes is due to technology and part simply due to a perception that the classical philosophies do not provide an adequate description of how mathematics is done by those who do it.
Coming from a social perspective, we introduce a classroom organizational frame, where students’ proofs progress from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. This design is informed by a view on proofs as successive social processes in the mathematics community. To illustrate opportunities for mathematics learning of proof progressions, we present a commognitive analysis of a single proof from a small course in topology. The analysis illuminates the processes through which students’ proof was restructured, developed previously unarticulated elements, and became more formal and elaborate. Within this progression, the provers developed their mathematical discourses and the course teacher seized valuable teachable moments. The findings are discussed in relation to key themes within the social perspective on proof.
Bu çalışma ile ortaokul matematik öğretmeni adaylarının ispatın doğası hakkındaki görüşlerini ortaya çıkarmak amaçlanmıştır. Çalışmada nitel araştırma yöntemlerinden durum çalışması kullanılmıştır. Çalışma kapsamında ölçüt örnekleme yöntemi ile seçilen ve bir devlet üniversitesinin ortaokul matematik öğretmenliği programında öğrenim gören üç öğretmen adayının ispatın doğasına ilişkin görüşleri alınmıştır. Matematik öğretmen adaylarına, araştırmacılar tarafından geliştirilen ve açık uçlu sorulardan oluşan “İspatın Doğasına İlişkin Görüşme Formu” yarı yapılandırılmış görüşmeler aracılığıyla yöneltilmiştir. Görüşme verileri içerik analizi yöntemi ile analiz edilmiştir. İçerik analizi sonrasında öğretmen adaylarının ispatın doğasına ilişkin “genelleme”, “yöntem”, “doğruluğa ulaşma”, “problem çözme”, “biçime odaklanma” temaları altında tepkiler verdikleri belirlenmiştir. Çalışmada en sıklıkla ortaya çıkan tema “doğruluğa ulaşma” ; en az sıklıkla ortaya çıkan temalar ise “problem çözme” ve “biçime odaklanma” olarak belirlenmiştir. Bu çalışmada, ortaokul matematik öğretmeni adaylarının ispatın tanımını yapmada, ispatı ispat yapan şeyleri ve başarılı bir ispat için gerekli olan şeyleri belirlemede, kısacası ispatın doğasını anlamada zorluklar yaşadıkları sonucuna ulaşılmıştır.
Principles of task design should have both the fundamental function of a clear relation to the learner’s rules, learning powers or hypothetical learning trajectories and the practical function of easy evaluation of many similar tasks. Drawing on some theories and practical tasks in the literature, we developed a total of 11 principles of task design for learning mathematical conjecturing, transiting between conjecturing and proving, and proving. To further validate the functioning of those principles, more empirical research is encouraged.
The visions of mathematics classrooms called for by current educational reform efforts pose great challenges for kindergarten through Grade 12 schools and teacher education programs. Although a number of colleges and universities throughout the country are making changes in their teacher education programs to reflect these reform recommendations, we have little systematic information on the nature of these programs or their impact on prospective teachers. These issues are of central concern in the study-Learning to Teach Secondary Mathematics in Two Reform-Based Teacher Education Programs-that we draw on in this article. The article focuses on 1 preservice teacher's (Ms. Savant) knowledge, beliefs, and practices related to proof, tasks, and discourse. A situative perspective on cognition and components of teachers' professional knowledge frame our research. We examined data on Ms. Savant's experiences in her teacher education program to understand the influences of teacher education on her development as a mathematics teacher. This research indicates that Ms. Savant's teacher education experiences did make a difference in her development as a teacher. Her mathematics methods course provided a large collection of tasks, engaged her and her preservice colleagues in discourse, and provided her with both formal and informal experiences with proof-all of these experiences reflecting reform-based visions of mathematics classrooms. The situative perspective on cognition directed our attention to issues of compatibility of goals and visions across the various university and kindergarten through Grade 12 classroom settings, and it helped us to understand why some aspects of reform-based pedagogy are more easily learned than others: Why some ideas and practices learned as a student in the university setting are more easily transported to the novice teacher's kindergarten through Grade 12 field setting. We conclude that compatibility of these settings on several key dimensions is essential for the settings to reinforce each other's messages, and thus work in conjunction, rather than in opposition, to prepare reform-minded teachers.
In this paper we explore how the naturalistic perspective in philosophy of mathematics and the situative perspective in mathematics education, while on one level are at odds, might be reconciled by paying attention to actual mathematical practice and activity. We
begin by examining how each approaches mathematical knowledge, and then how mathematical practice manifest itself in these
distinct research areas and gives rise to apparently contrary perspectives. Finally we argue for a deeper agreement and a
reconciliation in the perspectives based on the different projects of justification and explanation in mathematics.
ResearchGate has not been able to resolve any references for this publication.