This paper argues that interactive theorem provers (ITPs) could play an important role in fosteri... more This paper argues that interactive theorem provers (ITPs) could play an important role in fostering students' appreciation and understanding of proof and of mathematics in general. It shows that the ITP Lean has three features that mitigate existing difficulties in teaching and learning mathematical proof. One is that it requires students to identify a proof strategy at the start. The second is that it gives students instant feedback while still allowing them to explore with maximum autonomy. The third is that elementary formal logic finds a natural place in the activity of creating proofs. The challenge in using Lean is that students have to learn its command language, in addition to mathematics course content and elementary logic.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 6, 2019
The mathematics education literature reveals an ongoing interest in fostering students' ability t... more The mathematics education literature reveals an ongoing interest in fostering students' ability to construct and reconstruct proofs. One promising tool is the concept of "key idea". This study examined the relationships between how well students identified and formulated key ideas of the proof the irrationality of √k for non-square k and their subsequent ability to reconstruct the proof. The findings show that students' proof reconstructions were closely aligned with key ideas that they identified. Very few students were able to use precise language and point to an idea that helped them both understand the proof and reconstruct it. The findings suggest that mathematics educators, in their desire to see students enhance their understanding of proof and proving by the use of key ideas, need to create opportunities for key idea identification and proof reconstruction with attention to students' differing approaches to reconstructing a proof.
The fact that proofs can convey new mathematical techniques to students effectively, as shown in ... more The fact that proofs can convey new mathematical techniques to students effectively, as shown in recent literature, is an important advantage of the classroom use of proof, but it is one that mathematics educators seem to have overlooked to a large extent. The paper argues that teachers should make use of the potential of proof for presenting new techniques and demonstrating their value, and that mathematics educators in general should accord this potential its due importance among the many reasons for teaching proof. The teaching of proof in schools has been the topic of extensive investigations over the last two decades in the scholarly literature on mathematics education and in particular in the proceedings of the International Group for the Psychology of Mathematics. In her survey of research on proof in mathematics education, Mariotti (2006) found that most of the investigations on this topic have dealt primarily with the logical aspects of proof and with the cognitive problems...
Our overall concern is with helping students learn to construct and reconstruct proofs. Here we i... more Our overall concern is with helping students learn to construct and reconstruct proofs. Here we investigate an exploratory style which invites learners to think for themselves, with the instructor circulating among them while listening, probing, and suggesting. The objectives of this investigation are, to understand how the actions of teachers can contribute to the development of their students' thinking, and to provide explicit pedagogic strategies that teachers can use to promote their students' appreciation and understanding of mathematical proof.
The purpose of this paper is to investigate gender differences in achievement in statistics, maki... more The purpose of this paper is to investigate gender differences in achievement in statistics, making use of the Population A data of the Second International Mathematics Study (SIMS) conducted by the International Association for the Evaluation of Educational Achievement P A ) . In IEA terms, Population A means all the students in the grade in which most students attain the age of 13.0 to 13.1 1 years by the middle of the school year. The SIMS study used a very large random sample stratified by region and school type in each of the 20 participating "countries", which were in effect 20 different educational systems rather than countries. For example, Canada is represented by two educational systems, British Columbia and Ontario; Belgium by two systems, one Flemish and the other French. A test consisting of 180 multiple-choice items covering five content topics (Arithmetic, Algebra, Geometry, Statistics and Measurement) was administered to the students. Out of these 180 items...
it is commonplace in the educational literature on mathematical practice to argue for a general c... more it is commonplace in the educational literature on mathematical practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians' opinions on mathematical practice. Next, we compare mathematicians' diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of mathematical practice. We examine the career of one much cited and anthologised paper, WP Thurston's 'On Proof and Progress in Mathematics' (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of mathematical practice. The interesting question is not whether mathematicians disagree-they are human so of course they do. The question is how homogenous is their mathematical practice. If there are deep differences in practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for mathematical education.
Efforts to ensure equity for women in scientific and technological disciplines must precede, or a... more Efforts to ensure equity for women in scientific and technological disciplines must precede, or at least accompany, efforts to persuade them to pursue these studies. To achieve gender equity in these disciplines, factors discouraging women from full participation in them should be removed. Many psychological, sociological and institutional factors have been identified as contributors to the under-representation of women in these fields. For the aim of understanding and appraising these factors, this study offers a factual characterization of women's enrolment levels in mathematics, engineering, and computer science in Canadian universities and the change in these levels over the period 1972 to 1995. Findings indicate that patterns of women's enrolment in these three dis- ciplines are vastly different, a fact which suggests that factors specific to each discipline interact with and modify the effects of the more general sociological and psychological obstacles impeding women&...
The contribution of visualisation to mathe-matics and to mathematics education raises a number of... more The contribution of visualisation to mathe-matics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of math-ematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification. 1 Foreword It is a great honour to be asked to contribute to this special issue in memory of Hans Georg Steiner. He was a friend and mentor who had an enormous impact on the field of mathematics education, as acknowledged in the volume dedicated to him, Didactics of mathematics as a scientific discipline, which appeared in 1994 to mark both his 65th birthday and 20 years of work at the Institut für Didaktik der Mathematik (IDM) in Bielefeld. I had the great pleasure and privilege of meeting
This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It ... more This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove.
There has been a recent upsurge in papers on the teaching and learning of proof: Between 1990 and... more There has been a recent upsurge in papers on the teaching and learning of proof: Between 1990 and 1999 the leading journals of mathematics education published over one hundred research papers on this topic. This in itself is an indication that proof remains a prominent issue in mathematics education.
Teaching and Learning Secondary School Mathematics, 2018
To help students improve their grasp of mathematics and make a successful transition from seconda... more To help students improve their grasp of mathematics and make a successful transition from secondary to tertiary mathematics, the chapters in this part share the belief that it is necessary to broaden the students’ mathematical understanding by addressing, at the secondary level, topics such as number theory, probability and statistics, mathematical modelling, and classroom projects making use of innovative technology.
espanolThis paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowe... more espanolThis paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove. EnglishEl objetivo de este articulo es discutir el concepto de amplitud de una demostracion presentado por Timothy Gowers. Se explica el significado de este concepto y se trata de mostrar como se relaciona con otros conceptos discutidos en la literatura existente sobre prueba y demostraciones. Tambien se explora como el concepto de amplitud de una demostracion podria utilizarse productivamente en el curriculo de matematicas y como podria encajar con las diferentes perspectivas sobre el aprendizaje de la demostracion.
Canadian Journal of Science, Mathematics and Technology Education, 2020
En janvier 2001, la Revue canadienne de l'enseignement des sciences, des mathématiques et de la t... more En janvier 2001, la Revue canadienne de l'enseignement des sciences, des mathématiques et de la technologie (CJSMTE) était lancée. Fondée par Derek Hodson, Gila Hanna et Jacques Désautels, il s'agissait de la première revue de recherche en enseignement des sciences, des mathématiques et de la technologie à comité de lecture à être publiée au Canada. Nous souhaitons au journal un joyeux 20 e anniversaire. Les fondateurs étaient convaincus qu'une revue avec une voix distinctive et indépendante, accueillant la diversité et favorisant l'excellence et l'équité dans les sciences, les mathématiques et la technologie serait d'une grande utilité, et cette conviction a été pleinement confirmée. Les rédacteurs et les membres du conseil, présents et passés, ainsi que tous les auteurs publiés, peuvent être à juste titre fiers de cette grande réalisation. Félicitations! En 2001, il a été gratifiant de recevoir de bons voeux et de chaleureuses paroles de bienvenue de la part des rédacteurs de plusieurs revues internationales reconnues d'enseignement des sciences, des mathématiques et de la technologie. Citons, à titre d'exemple, le rédacteur en chef de l'époque de Educational Studies in Mathematics:
This paper argues that interactive theorem provers (ITPs) could play an important role in fosteri... more This paper argues that interactive theorem provers (ITPs) could play an important role in fostering students' appreciation and understanding of proof and of mathematics in general. It shows that the ITP Lean has three features that mitigate existing difficulties in teaching and learning mathematical proof. One is that it requires students to identify a proof strategy at the start. The second is that it gives students instant feedback while still allowing them to explore with maximum autonomy. The third is that elementary formal logic finds a natural place in the activity of creating proofs. The challenge in using Lean is that students have to learn its command language, in addition to mathematics course content and elementary logic.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 6, 2019
The mathematics education literature reveals an ongoing interest in fostering students' ability t... more The mathematics education literature reveals an ongoing interest in fostering students' ability to construct and reconstruct proofs. One promising tool is the concept of "key idea". This study examined the relationships between how well students identified and formulated key ideas of the proof the irrationality of √k for non-square k and their subsequent ability to reconstruct the proof. The findings show that students' proof reconstructions were closely aligned with key ideas that they identified. Very few students were able to use precise language and point to an idea that helped them both understand the proof and reconstruct it. The findings suggest that mathematics educators, in their desire to see students enhance their understanding of proof and proving by the use of key ideas, need to create opportunities for key idea identification and proof reconstruction with attention to students' differing approaches to reconstructing a proof.
The fact that proofs can convey new mathematical techniques to students effectively, as shown in ... more The fact that proofs can convey new mathematical techniques to students effectively, as shown in recent literature, is an important advantage of the classroom use of proof, but it is one that mathematics educators seem to have overlooked to a large extent. The paper argues that teachers should make use of the potential of proof for presenting new techniques and demonstrating their value, and that mathematics educators in general should accord this potential its due importance among the many reasons for teaching proof. The teaching of proof in schools has been the topic of extensive investigations over the last two decades in the scholarly literature on mathematics education and in particular in the proceedings of the International Group for the Psychology of Mathematics. In her survey of research on proof in mathematics education, Mariotti (2006) found that most of the investigations on this topic have dealt primarily with the logical aspects of proof and with the cognitive problems...
Our overall concern is with helping students learn to construct and reconstruct proofs. Here we i... more Our overall concern is with helping students learn to construct and reconstruct proofs. Here we investigate an exploratory style which invites learners to think for themselves, with the instructor circulating among them while listening, probing, and suggesting. The objectives of this investigation are, to understand how the actions of teachers can contribute to the development of their students' thinking, and to provide explicit pedagogic strategies that teachers can use to promote their students' appreciation and understanding of mathematical proof.
The purpose of this paper is to investigate gender differences in achievement in statistics, maki... more The purpose of this paper is to investigate gender differences in achievement in statistics, making use of the Population A data of the Second International Mathematics Study (SIMS) conducted by the International Association for the Evaluation of Educational Achievement P A ) . In IEA terms, Population A means all the students in the grade in which most students attain the age of 13.0 to 13.1 1 years by the middle of the school year. The SIMS study used a very large random sample stratified by region and school type in each of the 20 participating "countries", which were in effect 20 different educational systems rather than countries. For example, Canada is represented by two educational systems, British Columbia and Ontario; Belgium by two systems, one Flemish and the other French. A test consisting of 180 multiple-choice items covering five content topics (Arithmetic, Algebra, Geometry, Statistics and Measurement) was administered to the students. Out of these 180 items...
it is commonplace in the educational literature on mathematical practice to argue for a general c... more it is commonplace in the educational literature on mathematical practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians' opinions on mathematical practice. Next, we compare mathematicians' diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of mathematical practice. We examine the career of one much cited and anthologised paper, WP Thurston's 'On Proof and Progress in Mathematics' (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of mathematical practice. The interesting question is not whether mathematicians disagree-they are human so of course they do. The question is how homogenous is their mathematical practice. If there are deep differences in practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for mathematical education.
Efforts to ensure equity for women in scientific and technological disciplines must precede, or a... more Efforts to ensure equity for women in scientific and technological disciplines must precede, or at least accompany, efforts to persuade them to pursue these studies. To achieve gender equity in these disciplines, factors discouraging women from full participation in them should be removed. Many psychological, sociological and institutional factors have been identified as contributors to the under-representation of women in these fields. For the aim of understanding and appraising these factors, this study offers a factual characterization of women's enrolment levels in mathematics, engineering, and computer science in Canadian universities and the change in these levels over the period 1972 to 1995. Findings indicate that patterns of women's enrolment in these three dis- ciplines are vastly different, a fact which suggests that factors specific to each discipline interact with and modify the effects of the more general sociological and psychological obstacles impeding women&...
The contribution of visualisation to mathe-matics and to mathematics education raises a number of... more The contribution of visualisation to mathe-matics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of math-ematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification. 1 Foreword It is a great honour to be asked to contribute to this special issue in memory of Hans Georg Steiner. He was a friend and mentor who had an enormous impact on the field of mathematics education, as acknowledged in the volume dedicated to him, Didactics of mathematics as a scientific discipline, which appeared in 1994 to mark both his 65th birthday and 20 years of work at the Institut für Didaktik der Mathematik (IDM) in Bielefeld. I had the great pleasure and privilege of meeting
This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It ... more This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove.
There has been a recent upsurge in papers on the teaching and learning of proof: Between 1990 and... more There has been a recent upsurge in papers on the teaching and learning of proof: Between 1990 and 1999 the leading journals of mathematics education published over one hundred research papers on this topic. This in itself is an indication that proof remains a prominent issue in mathematics education.
Teaching and Learning Secondary School Mathematics, 2018
To help students improve their grasp of mathematics and make a successful transition from seconda... more To help students improve their grasp of mathematics and make a successful transition from secondary to tertiary mathematics, the chapters in this part share the belief that it is necessary to broaden the students’ mathematical understanding by addressing, at the secondary level, topics such as number theory, probability and statistics, mathematical modelling, and classroom projects making use of innovative technology.
espanolThis paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowe... more espanolThis paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove. EnglishEl objetivo de este articulo es discutir el concepto de amplitud de una demostracion presentado por Timothy Gowers. Se explica el significado de este concepto y se trata de mostrar como se relaciona con otros conceptos discutidos en la literatura existente sobre prueba y demostraciones. Tambien se explora como el concepto de amplitud de una demostracion podria utilizarse productivamente en el curriculo de matematicas y como podria encajar con las diferentes perspectivas sobre el aprendizaje de la demostracion.
Canadian Journal of Science, Mathematics and Technology Education, 2020
En janvier 2001, la Revue canadienne de l'enseignement des sciences, des mathématiques et de la t... more En janvier 2001, la Revue canadienne de l'enseignement des sciences, des mathématiques et de la technologie (CJSMTE) était lancée. Fondée par Derek Hodson, Gila Hanna et Jacques Désautels, il s'agissait de la première revue de recherche en enseignement des sciences, des mathématiques et de la technologie à comité de lecture à être publiée au Canada. Nous souhaitons au journal un joyeux 20 e anniversaire. Les fondateurs étaient convaincus qu'une revue avec une voix distinctive et indépendante, accueillant la diversité et favorisant l'excellence et l'équité dans les sciences, les mathématiques et la technologie serait d'une grande utilité, et cette conviction a été pleinement confirmée. Les rédacteurs et les membres du conseil, présents et passés, ainsi que tous les auteurs publiés, peuvent être à juste titre fiers de cette grande réalisation. Félicitations! En 2001, il a été gratifiant de recevoir de bons voeux et de chaleureuses paroles de bienvenue de la part des rédacteurs de plusieurs revues internationales reconnues d'enseignement des sciences, des mathématiques et de la technologie. Citons, à titre d'exemple, le rédacteur en chef de l'époque de Educational Studies in Mathematics:
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