Derek Holton – författare

This book is the final report of the ICMI study on the Teaching and Learning of Mathematics at University Level. As such it is one of a number of such studies that ICMI has commissioned. The other Study Volumes cover assessment in mathematics education, gender equity, research in mathematics education, the teaching of geometry, and history in mathematics education. All ofthese Study Volumes represent a statement of the state of the art in their respective areas. We hope that this is also the case for the current Study Volume. The current study on university level mathematics was commissioned for essentially four reasons. First, universities world-wide are accepting a much larger and more diverse group of students than has been the case. Consequently, universities have begun to adopt a role more like that of the school system and less like the elite institutions of the past. As a result the educational and pedagogical issues facing universities have changed. Second, although university student numbers have increased significantly, there has not been a corresponding increase in the number of mathematics majors. Hence mathematics departments have to be more aware of their students’ needs in order to retain the students they have and to attract future students. As part of this awareness, departments of mathematics have to take the teaching and learning of mathematics more seriously than perhaps they have in the past.

Focusing Your Attention The purpose of this book is Cat least) twofold. First, we want to show you what mathematics is, what it is about, and how it is done-by those who do it successfully. We are, in fact, trying to give effect to what we call, in Section 9.3, our basic principle of mathematical instruction, asserting that "mathematics must be taught so that students comprehend how and why mathematics is qone by those who do it successfully./I However, our second purpose is quite as important. We want to attract you-and, through you, future readers-to mathematics. There is general agreement in the (so-called) civilized world that mathematics is important, but only a very small minority of those who make contact with mathematics in their early education would describe it as delightful. We want to correct the false impression of mathematics as a combination of skill and drudgery, and to re­ inforce for our readers a picture of mathematics as an exciting, stimulating and engrossing activity; as a world of accessible ideas rather than a world of incomprehensible techniques; as an area of continued interest and investigation and not a set of procedures set in stone.

Focusing YourAttention We have called this book Mathematical Vistas because we have already published a companion book MathematicalRefiections in the same series;1 indeed, the two books are dedicated to the same principal purpose - to stimulate the interest ofbrightpeople in mathematics.Itis not our intention in writing this book to make the earlier book aprerequisite, but it is, of course, natural that this book should contain several references to its predecessor. This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math­ ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level.

This book is the final report of the ICMI study on the Teaching and Learning of Mathematics at University Level. As such it is one of a number of such studies that ICMI has commissioned. The other Study Volumes cover assessment in mathematics education, gender equity, research in mathematics education, the teaching of geometry, and history in mathematics education. All ofthese Study Volumes represent a statement of the state of the art in their respective areas. We hope that this is also the case for the current Study Volume. The current study on university level mathematics was commissioned for essentially four reasons. First, universities world-wide are accepting a much larger and more diverse group of students than has been the case. Consequently, universities have begun to adopt a role more like that of the school system and less like the elite institutions of the past. As a result the educational and pedagogical issues facing universities have changed. Second, although university student numbers have increased significantly, there has not been a corresponding increase in the number of mathematics majors. Hence mathematics departments have to be more aware of their students’ needs in order to retain the students they have and to attract future students. As part of this awareness, departments of mathematics have to take the teaching and learning of mathematics more seriously than perhaps they have in the past.

The present volume of Research in Collegiate Mathematics Education, like previous volumes in this series, reflects the importance of research in mathematics education at the collegiate level. The editors in this series encourage communication between mathematicians and mathematics educators, and as pointed out by the International Commission of Mathematics Instruction (ICMI), much more work is needed in concert with these two groups. Indeed, editors of RCME are aware of this need and the articles published in this series are in line with that goal. Nine papers constitute this volume. The first two examine problems students experience when converting a representation from one particular system of representations to another. The next three papers investigate students learning about proofs. In the next two papers, the focus is instructor knowledge for teaching calculus. The final two papers in the volume address the nature of "conception" in mathematics. Whether they are specialists in education or mathematicians interested in finding out about the field, readers will obtain new insights about teaching and learning and will take away ideas that they can use.This series is published in cooperation with the Mathematical Association of America.

This book is the final report of the ICMI study on the Teaching and Learning of Mathematics at University Level. As such it is one of a number of such studies that ICMI has commissioned. The other Study Volumes cover assessment in mathematics education, gender equity, research in mathematics education, the teaching of geometry, and history in mathematics education. All ofthese Study Volumes represent a statement of the state of the art in their respective areas. We hope that this is also the case for the current Study Volume. The current study on university level mathematics was commissioned for essentially four reasons. First, universities world-wide are accepting a much larger and more diverse group of students than has been the case. Consequently, universities have begun to adopt a role more like that of the school system and less like the elite institutions of the past. As a result the educational and pedagogical issues facing universities have changed. Second, although university student numbers have increased significantly, there has not been a corresponding increase in the number of mathematics majors. Hence mathematics departments have to be more aware of their students’ needs in order to retain the students they have and to attract future students. As part of this awareness, departments of mathematics have to take the teaching and learning of mathematics more seriously than perhaps they have in the past.

Focusing YourAttention We have called this book Mathematical Vistas because we have already published a companion book MathematicalRefiections in the same series;1 indeed, the two books are dedicated to the same principal purpose - to stimulate the interest ofbrightpeople in mathematics.Itis not our intention in writing this book to make the earlier book aprerequisite, but it is, of course, natural that this book should contain several references to its predecessor. This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math­ ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level.

Focusing Your Attention The purpose of this book is Cat least) twofold. First, we want to show you what mathematics is, what it is about, and how it is done-by those who do it successfully. We are, in fact, trying to give effect to what we call, in Section 9.3, our basic principle of mathematical instruction, asserting that "mathematics must be taught so that students comprehend how and why mathematics is qone by those who do it successfully./I However, our second purpose is quite as important. We want to attract you-and, through you, future readers-to mathematics. There is general agreement in the (so-called) civilized world that mathematics is important, but only a very small minority of those who make contact with mathematics in their early education would describe it as delightful. We want to correct the false impression of mathematics as a combination of skill and drudgery, and to re­ inforce for our readers a picture of mathematics as an exciting, stimulating and engrossing activity; as a world of accessible ideas rather than a world of incomprehensible techniques; as an area of continued interest and investigation and not a set of procedures set in stone.

Focusing Your Attention The purpose of this book is Cat least) twofold. First, we want to show you what mathematics is, what it is about, and how it is done-by those who do it successfully. We are, in fact, trying to give effect to what we call, in Section 9.3, our basic principle of mathematical instruction, asserting that "mathematics must be taught so that students comprehend how and why mathematics is qone by those who do it successfully./I However, our second purpose is quite as important. We want to attract you-and, through you, future readers-to mathematics. There is general agreement in the (so-called) civilized world that mathematics is important, but only a very small minority of those who make contact with mathematics in their early education would describe it as delightful. We want to correct the false impression of mathematics as a combination of skill and drudgery, and to re­ inforce for our readers a picture of mathematics as an exciting, stimulating and engrossing activity; as a world of accessible ideas rather than a world of incomprehensible techniques; as an area of continued interest and investigation and not a set of procedures set in stone.

Focusing YourAttention We have called this book Mathematical Vistas because we have already published a companion book MathematicalRefiections in the same series;1 indeed, the two books are dedicated to the same principal purpose - to stimulate the interest ofbrightpeople in mathematics.Itis not our intention in writing this book to make the earlier book aprerequisite, but it is, of course, natural that this book should contain several references to its predecessor. This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.'' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math­ ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level.

Utilizing the LPS dataset, Algebra Teaching around the World documents eighth grade algebra teaching across a variety of countries that differ geographically and culturally. Different issues in algebra teaching are reported, and different theories are used to characterize algebra lessons or to compare algebra teaching in different countries. Many commonalities in algebra teaching around the world are identified, but there are also striking and deep-rooted differences. The different ways algebra was taught in different countries point to how algebra teaching may be embedded in the culture and the general traditions of mathematics education of the countries concerned. In particular, a comparison is made between algebra lessons in the Confucian-Heritage Culture (CHC) countries and ‘Western’ countries. It seems that a common emphasis of algebra teaching in CHC countries is the ‘linkage’ or ‘coherence’ of mathematics concepts, both within an algebraic topic and between topics. On the other hand, contemporary algebra teaching in many Western school systems places increasing emphasis on the use of algebra in mathematical modeling in ‘real world’ contexts and in the instructional use of metaphors, where meaning construction is assisted by invoking contexts outside the domain of algebraic manipulation, with the intention to helping students to form connections between algebra and other aspects of their experience. Algebra Teaching around the World should be of value to researchers with a focus on algebra, pedagogy or international comparisons of education. Because of the pedagogical variations noted here, there is a great deal of material that will be of interest to both teachers and teacher educators.

Why were most historically important mathematicians wealthy? Why were they often lawyers and many had pastors for fathers? Why were original results sometimes discovered by two mathematicians independently within a short time of each other? Why did the Italian Fibonacci, speak Arabic?It all began a couple of years ago, when one of the authors started to write short biographies of important historical mathematicians for the teaching journal Australian Primary Mathematics Classroom. It was felt that teachers generally knew very little about the way the subject developed or the people who developed it. And it was felt that historical knowledge would help them see how the subject progressed and enable them to fit in with the historical episodes that would be of interest to students.Clearly, the book that developed contains mathematics up to the 17th century, but we are keen to set the subject in those times, to try to give short biographies of the people involved, as well as provide a perspective of the events that led up to the times and led up to the mathematics. Importantly, it is shown that the maths enterprise was not undertaken by a small few, but worked like a relay race. One or a few might take up an idea and develop it, but it often gets only so far. Later, others would take up the idea, the baton, and the relay race to find results continues.

Why were most historically important mathematicians wealthy? Why were they often lawyers and many had pastors for fathers? Why were original results sometimes discovered by two mathematicians independently within a short time of each other? Why did the Italian Fibonacci, speak Arabic?It all began a couple of years ago, when one of the authors started to write short biographies of important historical mathematicians for the teaching journal Australian Primary Mathematics Classroom. It was felt that teachers generally knew very little about the way the subject developed or the people who developed it. And it was felt that historical knowledge would help them see how the subject progressed and enable them to fit in with the historical episodes that would be of interest to students.Clearly, the book that developed contains mathematics up to the 17th century, but we are keen to set the subject in those times, to try to give short biographies of the people involved, as well as provide a perspective of the events that led up to the times and led up to the mathematics. Importantly, it is shown that the maths enterprise was not undertaken by a small few, but worked like a relay race. One or a few might take up an idea and develop it, but it often gets only so far. Later, others would take up the idea, the baton, and the relay race to find results continues.

The International Mathematical Olympiad (IMO) is an annual international mathematics competition held for pre-collegiate students. It is also the oldest of the international science olympiads, and competition for places is particularly fierce. This book is an amalgamation of the first 8 of 15 booklets originally produced to guide students intending to contend for placement on their country's IMO team.The material contained in this book provides an introduction to the main mathematical topics covered in the IMO, which are: Combinatorics, Geometry and Number Theory. In addition, there is a special emphasis on how to approach unseen questions in Mathematics, and model the writing of proofs. Full answers are given to all questions.Though A First Step to Mathematical Olympiad Problems is written from the perspective of a mathematician, it is written in a way that makes it easily comprehensible to adolescents. This book is also a must-read for coaches and instructors of mathematical competitions.

The International Mathematical Olympiad (IMO) is an annual international mathematics competition held for pre-collegiate students. It is also the oldest of the international science olympiads, and competition for places is particularly fierce. This book is an amalgamation of the booklets originally produced to guide students intending to contend for placement on their country's IMO team. See also A First Step to Mathematical Olympiad Problems which was published in 2009.The material contained in this book provides an introduction to the main mathematical topics covered in the IMO, which are: Combinatorics, Geometry and Number Theory. In addition, there is a special emphasis on how to approach unseen questions in Mathematics, and model the writing of proofs. Full answers are given to all questions.Though A Second Step to Mathematical Olympiad Problems is written from the perspective of a mathematician, it is written in a way that makes it easily comprehensible to adolescents. This book is also a must-read for coaches and instructors of mathematical competitions.